1554:, this does not take advantage of the full power of SPH. By assigning each particle its own smoothing length and allowing it to vary with time, the resolution of a simulation can be made to automatically adapt itself depending on local conditions. For example, in a very dense region where many particles are close together, the smoothing length can be made relatively short, yielding high spatial resolution. Conversely, in low-density regions where individual particles are far apart and the resolution is low, the smoothing length can be increased, optimising the computation for the regions of interest.
189:
solving linear systems of equations. Finally, unlike grid-based techniques, which must track fluid boundaries, SPH creates a free surface for two-phase interacting fluids directly since the particles represent the denser fluid (usually water) and empty space represents the lighter fluid (usually air). For these reasons, it is possible to simulate fluid motion using SPH in real time. However, both grid-based and SPH techniques still require the generation of renderable free surface geometry using a polygonization technique such as
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filtering procedure, the more regular density and pressure fields are obtained. On the other hand, this leads to an increase of the computational costs. In long time simulations, the use of the filtering procedure may lead to the disruption of the hydrostatic pressure component and to an inconsistency between the global volume of fluid and the density field. Further, it does not ensure the enforcement of the dynamic free-surface boundary condition.
4623:
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5178:{\displaystyle \nabla A({\boldsymbol {r}})=\int _{\Omega ({\boldsymbol {r}})}A\left({\boldsymbol {r^{\prime }}}\right)\nabla W({\boldsymbol {r}}-{\boldsymbol {r^{\prime }}},h)d{\boldsymbol {r^{\prime }}}+\int _{B({\boldsymbol {r}})-\Omega ({\boldsymbol {r}})}A\left({\boldsymbol {r^{\prime }}}\right)\nabla W({\boldsymbol {r}}-{\boldsymbol {r^{\prime }}},h)d{\boldsymbol {r^{\prime }}}.}
3348:
6665:
4878:{\displaystyle A({\boldsymbol {r}})=\int _{\Omega ({\boldsymbol {r}})}A\left({\boldsymbol {r^{\prime }}}\right)W(|{\boldsymbol {r}}-{\boldsymbol {r^{\prime }}}|,h)d{\boldsymbol {r^{\prime }}}+\int _{B({\boldsymbol {r}})-\Omega ({\boldsymbol {r}})}A\left({\boldsymbol {r^{\prime }}}\right)W(|{\boldsymbol {r}}-{\boldsymbol {r^{\prime }}}|,h)d{\boldsymbol {r^{\prime }}},}
4511:{\displaystyle {\begin{aligned}{\boldsymbol {v}}_{i}^{n+1/2}&={\boldsymbol {v}}_{i}^{n}+{\boldsymbol {a}}_{i}^{n}{\frac {\Delta t}{2}},\\{\boldsymbol {r}}_{i}^{n+1}&={\boldsymbol {r}}_{i}^{n}+{\boldsymbol {v}}_{i}^{i+1/2}\Delta t,\\{\boldsymbol {v}}_{i}^{n+1}&={\boldsymbol {v}}_{i}^{n+1/2}+{\boldsymbol {a}}_{i}^{i+1}{\frac {\Delta t}{2}},\end{aligned}}}
5347:
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8684:
8252:. This indicates that the inter-particle average velocity and pressure are simply replaced by the solution of the Riemann problem. By comparing both it can be seen that the intermediate velocity and pressure from the inter-particle averages amount to implicit dissipation, i.e. density regularization and numerical viscosity, respectively.
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2100:
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3592:{\displaystyle m_{i}{\frac {\mathrm {d} {\boldsymbol {v}}_{i}}{\mathrm {d} t}}=-\sum _{j}m_{j}{\frac {\partial e_{j}}{\partial {\boldsymbol {r}}_{i}}}+m_{i}{\boldsymbol {g}}=-\sum _{j}m_{j}{\frac {\partial e_{j}}{\partial \rho _{j}}}{\frac {\partial \rho _{j}}{\partial {\boldsymbol {r}}_{i}}}+m_{i}{\boldsymbol {g}}}
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149:
Setting boundary conditions in SPH such as inlets and outlets and walls is more difficult than with grid-based methods. In fact, it has been stated that "the treatment of boundary conditions is certainly one of the most difficult technical points of the SPH method". This challenge is partly because
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In
Antuono et al. a correction to the diffusive term of Molteni was proposed to remove some inconsistencies close to the free-surface. In this case the adopted diffusive term is equivalent to a high-order differential operator on the density field. The scheme is called Ξ΄-SPH and preserves all the
329:
Another important advantage of meshfree methods in general, and of SPH in particular, is that mesh dependence problems are naturally avoided given the meshfree nature of the method. In particular, mesh alignment is related to problems involving cracks and it is avoided in SPH due to the isotropic
5473:
Probably the most popular methodology, or at least the most traditional one, to impose boundary conditions in SPH, is Fluid
Extension technique. Such technique is based on populating the compact support across the boundary with so-called ghost particles, conveniently imposing their field values.
2865:
Notice also that this choice leads to a symmetric divergence operator and antisymmetric gradient. Although there are several ways of discretizing the pressure gradient in the Euler equations, the above antisymmetric form is the most acknowledged one. It supports strict conservation of linear and
7140:
The solution of the
Riemann problem results in three waves emanating from the discontinuity. Two waves, which can be shock or rarefaction wave, traveling with the smallest or largest wave speed. The middle wave is always a contact discontinuity and separates two intermediate states, denoted by
325:
Libersky and
Petschek extended SPH to Solid Mechanics. The main advantage of SPH in this application is the possibility of dealing with larger local distortion than grid-based methods. This feature has been exploited in many applications in Solid Mechanics: metal forming, impact, crack growth,
188:
as well. This is due to several benefits over traditional grid-based techniques. First, SPH guarantees conservation of mass without extra computation since the particles themselves represent mass. Second, SPH computes pressure from weighted contributions of neighboring particles rather than by
153:
The computational cost of SPH simulations per number of particles is significantly larger than the cost of grid-based simulations per number of cells when the metric of interest is not (directly) related to density (e.g., the kinetic-energy spectrum). Therefore, overlooking issues of parallel
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The schemes of the first group apply a filter directly on the density field to remove the spurious numerical noise. The most used filters are the MLS (moving least squares) and the
Shepard filter which can be applied at each time step or every n time steps. The more frequent is the use of the
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the velocity vector. In practice a value of c smaller than the real one is adopted to avoid time steps too small in the time integration scheme. Generally a numerical speed of sound is adopted such that density variation smaller than 1% are allowed. This is the so-called weak-compressibility
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et al. showed that the stress-point technique removes the instability due to spurious singular modes, while tensile instabilities can be avoided by using a
Lagrangian kernel. Many other recent studies can be found in the literature devoted to improve the convergence of the SPH method.
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On the other hand, deploying ghost particles in the truncated domain is not a trivial task, such that modelling complex boundary shapes becomes cumbersome. The 2 most popular approaches to populate the empty domain with ghost particles are
Mirrored-Particles and Fixed-Particles.
3726:
8132:
8908:{\displaystyle \Pi _{ij}={\begin{cases}{\dfrac {-\alpha {\bar {c}}_{ij}\phi _{ij}+\beta \phi _{ij}^{2}}{{\bar {\rho }}_{ij}}}&\quad {\boldsymbol {v}}_{ij}\cdot {\boldsymbol {r}}_{ij}<0\\0&\quad {\boldsymbol {v}}_{ij}\cdot {\boldsymbol {r}}_{ij}\geq 0\end{cases}}}
1716:
8250:
212:
or particles will be used to fill water volumes that are never rendered. However, accuracy can be significantly higher with sophisticated grid-based techniques, especially those coupled with particle methods (such as particle level sets), since it is easier to enforce the
6458:
Usually the weakly-compressible schemes are affected by a high-frequency spurious noise on the pressure and density fields. This phenomenon is caused by the nonlinear interaction of acoustic waves and by fact that the scheme is explicit in time and centered in space .
3959:
10241:
6008:
2857:
11615:{\displaystyle {\begin{cases}p_{a}^{d}=p_{i}+\rho _{i}\max(0,(\mathbf {g} -{\frac {{\text{d}}\mathbf {v} _{a}}{{\text{d}}t}})\cdot \mathbf {n} ^{S})(\mathbf {r} _{ia}\cdot \mathbf {n} ^{S})\\\mathbf {v} _{a}^{d}=2\mathbf {v} _{i}-\mathbf {v} _{a}.\end{cases}}}
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1953:
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support of the kernel functions. However, classical SPH formulations suffer from tensile instabilities and lack of consistency. Over the past years, different corrections have been introduced to improve the accuracy of the SPH solution, leading to the
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The main benefit of this boundary condition is its obvious simplicity. However, several consistency issues shall be considered when this boundary technique is applied. That's in fact a heavy limitation on its potential applications.
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From the work done in the 80's and 90's on numerical integration of point-like particles in large accelerators, appropriate time integrators have been developed with accurate conservation properties on the long term; they are called
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Other symplectic integrators exist (see the reference textbook). It is recommended to use a symplectic (even low-order) scheme instead of a high order non-symplectic scheme, to avoid error accumulation after many iterations.
5342:{\displaystyle \int _{B({\boldsymbol {r}})-\Omega ({\boldsymbol {r}})}A\left({\boldsymbol {r^{\prime }}}\right)\nabla W({\boldsymbol {r}}-{\boldsymbol {r^{\prime }}},h)d{\boldsymbol {r^{\prime }}}\simeq {\boldsymbol {0}},}
9144:
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The first schemes that adopted such an approach were described in
Ferrari and in Molteni where the diffusive term was modeled as a Laplacian of the density field. A similar approach was also used in Fatehi and Manzari .
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is an open source multi-physics, multi-resolution SPH library. It provides C++ APIs for physical accurate simulation and aims to model coupled industrial dynamic systems including fluid, solid, multi-body dynamics and
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Generalized coordinate SPH (GSPH) allocates particles inhomogeneously in the
Cartesian coordinate system and arranges them via mapping in a generalized coordinate system in which the particles are aligned at a uniform
1908:
10494:
563:. This means that the physical quantity of any particle can be obtained by summing the relevant properties of all the particles that lie within the range of the kernel, the latter being used as a weighting function
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Recent improvements in understanding the convergence and stability of SPH have allowed for more widespread applications in Solid
Mechanics. Other examples of applications and developments of the method include:
5488:
The main benefit of this methodology is the simplicity, provided that the boundary contribution is computed as part of the bulk interactions. Also, this methodology has been deeply analyzed in the literature.
3888:{\displaystyle m_{i}{\frac {\mathrm {d} {\boldsymbol {v}}_{i}}{\mathrm {d} t}}=-\sum _{j}m_{j}{\frac {p_{j}}{\rho _{j}^{2}}}{\frac {\partial \rho _{j}}{\partial {\boldsymbol {r}}_{i}}}+m_{i}{\boldsymbol {g}},}
1403:
10743:
In fluid-structure coupling, the surrounding solid structure is behaving as a moving boundary for fluid, and the no-slip boundary condition is imposed at the fluid-structure interface. The interaction forces
7131:{\displaystyle {\begin{cases}(\rho _{L},U_{L},P_{L})=(\rho _{i},\mathbf {v} _{i}\cdot \mathbf {e} _{ij},P_{i})\\(\rho _{R},U_{R},P_{R})=(\rho _{j},\mathbf {v} _{j}\cdot \mathbf {e} _{ij},P_{j}).\end{cases}}}
5786:
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N. Akinci et al., 2012, introduce a versatile boundary handling and two-way SPH-rigid coupling technique that is completely based on hydrodynamic forces; the approach is applicable to different types of SPH
7287:
7213:
8255:
Since the above discretization is very dissipative a straightforward modification is to apply a limiter to decrease the implicit numerical dissipations introduced by limiting the intermediate pressure by
7633:{\displaystyle {\begin{cases}U^{\ast }={\overline {U}}+{\frac {1}{2}}{\frac {(P_{L}-P_{R})}{{\bar {\rho }}c_{0}}}\\P^{\ast }={\overline {P}}+{\frac {1}{2}}{\bar {\rho }}c_{0}{(U_{L}-U_{R})},\end{cases}}}
4928:. Hence, imposing boundary conditions in SPH is completely based on approximating the second integral on the right hand side. The same can be of course applied to the differential operators computation,
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9635:
Often in astrophysics, one wishes to model self-gravity in addition to pure hydrodynamics. The particle-based nature of SPH makes it ideal to combine with a particle-based gravity solver, for instance
1595:
201:, or 'carpet' visualization. For gas dynamics it is more appropriate to use the kernel function itself to produce a rendering of gas column density (e.g., as done in the SPLASH visualisation package).
8139:
247:
N. Akinci et al., 2013, introduce a versatile surface tension and two-way fluid-solid adhesion technique that allows simulating a variety of interesting physical effects that are observed in reality
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4123:{\displaystyle {\frac {\mathrm {d} {\boldsymbol {v}}_{i}}{\mathrm {d} t}}=-\sum _{j}m_{j}\left({\frac {p_{i}}{\rho _{i}^{2}}}+{\frac {p_{j}}{\rho _{j}^{2}}}\right)\nabla W_{ij}+{\boldsymbol {g}},}
131:
The computational cost of SPH simulations per number of particles is significantly less than the cost of grid-based simulations per number of cells when the metric of interest is related to fluid
10432:{\displaystyle m_{a}{\frac {{\text{d}}\mathbf {v} }{{\text{d}}t}}=2\sum _{b}V_{a}V_{b}{\tilde {\mathbb {P} }}_{ab}\nabla _{a}^{0}W_{ab}+\mathbf {g} +\mathbf {f} _{a}^{F:p}+\mathbf {f} _{a}^{F:v}}
6453:
4205:
6673:
conservation properties of the SPH without diffusion (e.g., linear and angular momenta, total energy, see ) along with a smooth and regular representation of the density and pressure fields.
5897:
7774:
7707:
2783:{\displaystyle \sum _{i}V_{i}{\boldsymbol {v}}_{i}\cdot \operatorname {\mathbf {G} } _{i}\left\{p_{j}\right\}=-\sum _{i}V_{i}p_{i}\operatorname {D} _{i}\left\{{\boldsymbol {v}}_{j}\right\},}
2252:
2215:
1945:
12662:
Shadloo, M. S. and Oger, G. and Touze, D. L.. (2016). "Smoothed particle hydrodynamics method for fluid flows, towards industrial applications: Motivations, current state, and challenges".
4155:
3046:
2799:
2095:{\displaystyle \operatorname {D} _{i}\left\{{\boldsymbol {v}}_{j}\right\}=-{\frac {1}{\rho _{i}}}\sum _{j}m_{j}\left({\boldsymbol {v}}_{i}-{\boldsymbol {v}}_{j}\right)\cdot \nabla W_{ij}.}
9594:{\displaystyle _{ij}=2\nu {\frac {m_{j}}{\rho _{j}}}\,{\frac {{\boldsymbol {r}}_{ij}\cdot \nabla w_{h,ij}}{\Vert {\boldsymbol {r}}_{ij}\Vert ^{2}+\eta _{h}^{2}}}\,{\boldsymbol {v}}_{ij}.}
2999:
1774:
5701:{\displaystyle \nabla A_{i}={\frac {1}{\gamma _{i}}}\left(\sum _{j\in \Omega _{i}}V_{j}A_{j}\nabla W_{ij}+\sum _{j\in \partial \Omega _{i}}S_{j}A_{j}{\boldsymbol {n}}_{j}W_{ij}\right),}
3332:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {v}}_{i}}}={\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {r}}_{i}}}.}
9393:
2131:
755:{\displaystyle A({\boldsymbol {r}})=\int A\left({\boldsymbol {r^{\prime }}}\right)W(|{\boldsymbol {r}}-{\boldsymbol {r^{\prime }}}|,h)\,\mathrm {d} V\!\left({\boldsymbol {r'}}\right).}
13370:
L.D. Libersky; A.G. Petschek; A.G. Carney; T.C. Hipp; J.R. Allahdadi; F.A. High (1993). "Strain Lagrangian hydrodynamics: a three-dimensional SPH code for dynamic material response".
8261:
12223:{\displaystyle \mathbf {f} _{a}^{F:v}=2\sum _{i}\eta V_{a}V_{i}{\frac {\mathbf {v} _{a}^{d}-\mathbf {v} _{i}}{r_{ia}}}{\frac {\partial W(\mathbf {r} _{ia},h)}{\partial {{r}_{ai}}}}.}
11285:{\displaystyle \mathbf {f} _{i}^{S:v}=2\sum _{a}\eta V_{i}V_{a}{\frac {\mathbf {v} _{i}-\mathbf {v} _{a}^{d}}{r_{ia}}}{\frac {\partial W(\mathbf {r} _{ia},h)}{\partial {{r}_{ia}}}}.}
10484:
11780:
11740:
10822:
10782:
10713:
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2618:{\displaystyle \operatorname {\mathbf {G} } _{i}\left\{p_{j}\right\}=\rho _{i}\sum _{j}m_{j}\left({\frac {p_{i}}{\rho _{i}^{2}}}+{\frac {p_{j}}{\rho _{j}^{2}}}\right)\nabla W_{ij},}
4571:
1833:
11372:
10206:
6158:
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Colagrossi (2019). "Viscous flow past a cylinder close to a free surface: benchmarks with steady, periodic and metastable responses, solved by meshfree and mesh-based schemes".
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3022:
3004:
Nevertheless, other operators have been proposed, which may perform better numerically or physically. For instance, one drawback of these operators is that while the divergence
1068:
256:
S. Tavakkol et al., 2016, develop curvSPH, which makes the horizontal and vertical size of particles independent and generates uniform mass distribution along curved boundaries
177:
11665:
9687:
463:
434:
8579:
8519:
6642:{\displaystyle {\displaystyle {\frac {d\rho _{i}}{dt}}=\sum _{j}m_{j}\left({\boldsymbol {v}}_{i}-{\boldsymbol {v}}_{j}\right)\cdot \nabla W_{ij}+{\mathcal {D}}_{i}(\rho ),}}
6391:
6186:
5509:
The newest Boundary technique is the Boundary Integral methodology. In this methodology, the empty volume integral is replaced by a surface integral, and a renormalization:
798:
11697:
1523:
kernel. The latter two kernels are compactly supported (unlike the Gaussian, where there is a small contribution at any finite distance away), with support proportional to
1334:
10228:
9138:
The artificial viscosity also has shown to improve the overall stability of general flow simulations. Therefore, it is applied to inviscid problems in the following form
8365:
1463:
204:
One drawback over grid-based techniques is the need for large numbers of particles to produce simulations of equivalent resolution. In the typical implementation of both
8986:
5824:
Another way to determine the density is based on the SPH smoothing operator itself. Therefore, the density is estimated from the particle distribution utilizing the SPH
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7828:
7801:
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6038:
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Symplectic schemes are conservative but explicit, thus their numerical stability requires stability conditions, analogous to the Courant-Friedrichs-Lewy condition (see
3654:
1587:
1288:
1095:
4542:
471:
8936:
8605:
12008:{\displaystyle \mathbf {f} _{a}^{F:p}=-2\sum _{i}V_{a}V_{i}{\frac {p_{a}^{d}\rho _{i}+p_{i}\rho _{a}^{d}}{\rho _{i}+\rho _{a}^{d}}}\nabla _{a}W(\mathbf {r} _{ai},h)}
11070:{\displaystyle \mathbf {f} _{i}^{S:p}=-2\sum _{a}V_{i}V_{a}{\frac {p_{i}\rho _{a}^{d}+p_{a}^{d}\rho _{i}}{\rho _{i}+\rho _{a}^{d}}}\nabla _{i}W(\mathbf {r} _{ia},h)}
8956:
8479:
1804:
14016:
A. Mayrhofer; B.D. Rogers; D. Violeau; M. Ferrand (2013). "Investigation of wall bounded flows using SPH and the unified semi-analytical wall boundary conditions".
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6115:
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790:
405:
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Smoothed-particle hydrodynamics's adaptive resolution, numerical conservation of physically conserved quantities, and ability to simulate phenomena covering many
80:
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10862:
10842:
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needs to be estimated, for e.g. by Torricelli's law or an educated guess. Since only small density variations occur, a linear equation of state can be adopted:
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581:
561:
14187:
Andrea Colagrossi; Matteo Antuono; David Le Touzè (2009). "Theoretical considerations on the free-surface role in the smoothed-particle-hydrodynamics model".
253:
A. Mahdavi and N. Talebbeydokhti, 2015, propose a hybrid algorithm for implementation of solid boundary condition and simulate flow over a sharp crested weir
12829:
9847:{\displaystyle \mathbb {B} _{a}^{0}=\left(\sum _{b}V_{b}^{0}\left(\mathbf {r} _{b}^{0}-\mathbf {r} _{a}^{0}\right)\otimes \nabla _{a}^{0}W_{ab}\right)^{-1}}
15449:
259:
W. Kostorz and A. Esmail-Yakas, 2020, propose a general, efficient and simple method for evaluating normalization factors near piecewise-planar boundaries
13714:
J. Bonet; S. Kulasegaram (2000). "Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations".
8666:
differential operator. Since the direct computation does not provide satisfactory results, several approaches to model the diffusion have been proposed.
53:
10023:{\displaystyle \nabla _{a}^{0}W_{a}={\frac {\partial W\left(|\mathbf {r} _{ab}^{0}|,h\right)}{\partial |\mathbf {r} _{ab}^{0}|}}\mathbf {e} _{ab}^{0}}
9266:{\displaystyle \Pi _{ij}=\alpha hc{\frac {{\boldsymbol {v}}_{ij}\cdot {\boldsymbol {r}}_{ij}}{\Vert {\boldsymbol {r}}_{ij}\Vert ^{2}+\eta _{h}^{2}}}.}
7981:{\displaystyle {\frac {d\rho _{i}}{dt}}=2\rho _{i}\sum _{j}{\frac {m_{j}}{\rho _{j}}}(\mathbf {v} _{i}-\mathbf {v} ^{\ast })\cdot \nabla _{i}W_{ij},}
3182:{\displaystyle {\mathcal {L}}=\sum _{j}m_{j}\left({\tfrac {1}{2}}{\boldsymbol {v}}_{j}^{2}-e_{j}+{\boldsymbol {g}}\cdot {\boldsymbol {r}}_{j}\right)}
105:, which makes it ideally suited to simulate problems dominated by complex boundary dynamics, like free surface flows, or large boundary displacement.
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The anti-symmetric property of the derivative of the kernel function will ensure the momentum conservation for each pair of interacting particles
10612:{\displaystyle {\tilde {\mathbb {P} }}_{ab}={\frac {1}{2}}\left(\mathbb {P} _{a}\mathbb {B} _{a}^{0}+\mathbb {P} _{b}\mathbb {B} _{b}^{0}\right).}
6676:
In the third group there are those SPH schemes which employ numerical fluxes obtained through Riemann solvers to model the particle interactions.
1123:
4617:, then the integral support is truncated. Indeed, when the convolution is affected by a boundary, the convolution shall be split in 2 integrals,
14235:
Bejamin Bouscasse; Andrea Colagrossi; Salvatore Marrone; Matteo Antuono (2013). "Nonlinear water wave interaction with floating bodies in SPH".
13868:
S. Yashiro; T. Okabe (2015). "Smoothed particle hydrodynamics in a generalized coordinate system with a finite-deformation constitutive model".
1853:
9110:{\displaystyle \phi _{ij}={\frac {h{\boldsymbol {v}}_{ij}\cdot {\boldsymbol {r}}_{ij}}{\Vert {\boldsymbol {r}}_{ij}\Vert ^{2}+\eta _{h}^{2}}},}
2437:{\displaystyle {\frac {d{\boldsymbol {v}}_{i}}{dt}}=-{\frac {1}{\rho }}\operatorname {\mathbf {G} } _{i}\left\{p_{j}\right\}+{\boldsymbol {g}}}
14655:
M. Antuono, A. Colagrossi, S. Marrone, D. Molteni (2010). "Free-surface flows solved by means of SPH schemes with numerical diffusive terms".
3659:
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13630:
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Muller, M.; Charypar, D.; and Gross, M.; "Particle-based Fluid Simulation for Interactive Applications", in Breen, D; and Lin, M. (eds.),
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operator. This explains why linear momentum is conserved, and allows conservation of angular momentum and energy to be conserved as well.
14311:
Jose Luis Cercos-Pita; Matteo Antuono; Andrea Colagrossi; Antonio Souto (2017). "SPH energy conservation for fluid--solid interactions".
13471:
13229:
Price, Daniel J; Rockefeller, Gabriel; Warren, Michael S (2006). "SNSPH: A Parallel 3-D Smoothed Particle Radiation Hydrodynamics Code".
12589:
6490:
A different way to smooth out the density and pressure field is to add a diffusive term inside the continuity equation (group 2) :
6462:
Through the years, several techniques have been proposed to get rid of this problem. They can be classified in three different groups:
1339:
234:
K. Bodin et al., 2011, replace the standard equation of state pressure with a density constraint and apply a variational time integrator
83:(where the co-ordinates move with the fluid), and the resolution of the method can easily be adjusted with respect to variables such as
12738:
Kostorz (2020). "A semi-analytical boundary integral method for radial functions with application to Smoothed Particle Hydrodynamics".
5828:. To overcome undesired errors at the free surface through kernel truncation, the density formulation can again be integrated in time.
8127:{\displaystyle {\frac {d\mathbf {v} _{i}}{dt}}=-2\sum _{j}m_{j}\left({\frac {P^{\ast }}{\rho _{i}\rho _{j}}}\right)\nabla _{i}W_{ij}.}
5712:
334:
by Liu et al. Randles and Libersky and Johnson and Beissel tried to solve the consistency problem in their study of impact phenomena.
15958:
15540:
7218:
7144:
15258:
is a mostly open source SPH code based on SPHysics and using GPU computing. The open source components are available under the LGPL.
13922:
1543:. This has the advantage of saving computational effort by not including the relatively minor contributions from distant particles.
15490:
1836:
1711:{\displaystyle {\frac {d\rho _{i}}{dt}}=\sum _{j}m_{j}\left({\boldsymbol {v}}_{i}-{\boldsymbol {v}}_{j}\right)\cdot \nabla W_{ij},}
14872:
Chi Zhang and Xiangyu Hu and Nikolaus Adams (2017). "A weakly compressible SPH method based on a low-dissipation Riemann solver".
8245:{\displaystyle \mathbf {v} ^{\ast }=U^{\ast }\mathbf {e} _{ij}+({\overline {\mathbf {v} }}_{ij}-{\overline {U}}\mathbf {e} _{ij})}
15466:
15440:
14469:
D. Molteni, A. Colagrossi (2009). "A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH".
13665:
12706:
Fraser, K.and Kiss, L. I. and St-George, L. (2016). "A generalized wall boundary condition for smoothed particle hydrodynamics".
6695:
15214:
14598:
Fatehi, R and Manzari, MT (2011). "A remedy for numerical oscillations in weakly compressible smoothed particle hydrodynamics".
13700:
14507:
Colagrossi, Andrea; Landrini, Maurizio (2003). "Numerical simulation of interfacial flows by smoothed particle hydrodynamics".
12613:
Adami, S. and Hu, X. Y. and Adams, N. A.. (2012). "A generalized wall boundary condition for smoothed particle hydrodynamics".
9276:
It is possible to not only stabilize inviscid simulations but also to model the physical viscosity by this approach. To do so
231:
M. Ihmsen et al., 2010, introduce boundary handling and adaptive time-stepping for PCISPH for accurate rigid body interactions
15963:
15764:
15534:
15390:
14168:
13833:
M.I. Herreros; M. Mabssout (2011). "A two-steps time discretization scheme using the SPH method for shock wave propagation".
13354:
14079:
J. Bonet; T.S. Lok (1999). "Variational and momentum preservation aspects of Smoothed Particle Hydrodynamics formulations".
15834:
15691:
15546:
15217:
is the web-site of The Italian Theoretical Virtual Observatory created to query a database of numerical simulation archive.
10103:
9282:
7355:
7292:
13981:
H. Wendland (1995). "Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree".
6003:{\displaystyle p={\frac {\rho _{0}c^{2}}{\gamma }}\left(\left({\frac {\rho }{\rho _{0}}}\right)^{\gamma }-1\right)+p_{0},}
6399:
244:
M. Macklin et al., 2013 simulates incompressible flows inside the Position Based Dynamics framework, for bigger timesteps
10032:
stands for the gradient of the kernel function evaluated at the initial reference configuration. Note that subscripts
331:
150:
in SPH the particles near the boundary change with time. Nonetheless, wall boundary conditions for SPH are available.
14388:"Unified semi-analytical wall boundary conditions for inviscid, laminar or turbulent flows in the meshless SPH method"
12325:
8630:
3048:
is not. Several techniques have been proposed to circumvent this issue, leading to renormalized operators (see e.g.).
3024:
is zero-order consistent (i.e. yields zero when applied to a constant vector field), it can be seen that the gradient
2852:{\displaystyle \int {\boldsymbol {v}}\cdot \operatorname {grad} p=-\int p\operatorname {div} \cdot {\boldsymbol {v}}.}
15127:
Amada, T.; Imura, M.; Yasumuro, Y.; Manabe, Y.; and Chihara, K. (2004); "Particle-based fluid simulation on GPU", in
8654:
In general, the description of hydrodynamic flows require a convenient treatment of diffusive processes to model the
7712:
7645:
2220:
2183:
1916:
15134:
Desbrun, M.; and Cani, M.-P. (1996). "Smoothed Particles: a new paradigm for animating highly deformable bodies" in
6160:. Hence, pressure information travels fast compared to the actual bulk flow, which leads to very small Mach numbers
381:
The Smoothed-Particle Hydrodynamics (SPH) method works by dividing the fluid into a set of discrete moving elements
15742:
14544:
Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems
13790:
T. Rabczuk; J. Eibl; L. Stempniewski (2003). "Simulation of high velocity concrete fragmentation using SPH/MLSPH".
4136:
3027:
2258:
228:
B. Solenthaler, 2009, develops Predictive-Corrective SPH (PCISPH) to allow for better incompressibility constraints
15759:
15226:
15062:
Chi Zhang; Massoud Rezavand; Xiangyu Hu (2020). "A multi-resolution SPH method for fluid-structure interactions".
14814:
2949:
1724:
15893:
15223:
depicts a wide variety of test cases, experimental validations, and commercial applications of the SPH code SPHC.
6692:
For an SPH method based on Riemann solvers, an inter-particle Riemann problem is constructed along a unit vector
13679:
T. Belytschko; Y. Guo; W.K. Liu; S.P. Xiao (2000). "A unified stability analysis of meshless particle methods".
12492:"GPUs, a new tool of acceleration in CFD: efficiency and reliability on smoothed particle hydrodynamics methods"
6688:
SPH simulation: pressure distribution of a dam-break flow using Riemann solver with the low-dissipation limiter.
1840:
221:
is being used increasingly in real-time animation and games where accuracy is not as critical as interactivity.
60:
and Lucy in 1977, initially for astrophysical problems. It has been used in many fields of research, including
15660:
15649:
15626:
9362:
8607:
is generally effective. Also note that the dissipation introduced by the intermediate velocity is not limited.
2108:
1485:
being the space dimension), and on the particle arrangement in space. The latter effect is still poorly known.
262:
Colagrossi et al., 2019, study flow around a cylinder close to a free-surface and compare with other techniques
117:
SPH can be easily extended to a wide variety of fields, and hybridized with some other models, as discussed in
14704:
M. Antuono, A. Colagrossi, S. Marrone (2012). "Numerical diffusive terms in weakly-compressible SPH schemes".
13440:
T. Belytschko; Y. Krongauz; J. Dolbow; C. Gerlach (1998). "On the completeness of meshfree particle methods".
15632:
13080:
International Journal for Numerical Methods in Fluids (2016). "Curvilinear smoothed particle hydrodynamics".
10458:
14753:
Antuono Matteo and Marrone S and Colagrossi A and Bouscasse B (2015). "Energy balance in the Ξ΄-SPH scheme".
14557:
A. Ferrari, M. Dumbser, E. Toro, A. Armanini (2009). "A new 3D parallel SPH scheme for free surface flows".
11745:
11705:
10787:
10747:
10678:
10638:
6120:
Real nearly incompressible fluids such as water are characterized by very high speeds of sound of the order
15749:
15714:
4547:
1809:
136:
15220:
14346:
J. Campbell; R. Vignjevic; L. Libersky (2000). "A contact algorithm for smoothed particle hydrodynamics".
11343:
10174:
6123:
1216:
15754:
15433:
8659:
5832:
3225:
14978:
13582:
C.T. Dyka; R.P. Ingel (1995). "An approach for tension instability in Smoothed Particle Hydrodynamics".
6330:
3007:
15871:
15856:
15732:
15418:
15300:
is a massively parallel, open-source classical molecular dynamics code that can perform SPH simulations
14942:
J. P. Morris; P. J. Fox; Y. Zhu (1997). "Modeling Low Reynolds Number Incompressible Flows Using SPH".
7415:, a linearized Riemann solver for smooth flows or with only moderately strong shocks can be written as
6269:{\displaystyle {\frac {\delta \rho }{\rho _{0}}}\approx {\frac {|{\boldsymbol {v}}|^{2}}{c^{2}}}=M^{2}}
1051:
900:{\displaystyle A({\boldsymbol {r}})=\sum _{j}V_{j}A_{j}W(|{\boldsymbol {r}}-{\boldsymbol {r}}_{j}|,h),}
224:
Recent work in SPH for fluid simulation has increased performance, accuracy, and areas of application:
108:
The lack of a mesh significantly simplifies the model implementation and its parallelization, even for
14816:
13289:
12490:
Alejandro Crespo; Jose M. Dominguez; Anxo Barreiro; Moncho Gomez-Gesteira; Benedict D. Rogers (2011).
11641:
9663:
439:
410:
15518:
15498:
15480:
9636:
8544:
139:
of density fluctuations). This is the case because in SPH the resolution is put where the matter is.
15291:
15264:
is a simple, open source (Zlib), real-time 3D SPH implementation in C++ for liquids for CPU and GPU.
14048:
13520:
P.W. Randles; L.D. Libersky (1997). "Recent improvements in SPH modelling of hypervelocity impact".
13043:
12991:
12893:
12799:
11391:
8709:
8484:
7429:
6897:
6369:
15841:
15727:
15457:
12489:
11670:
8481:
ensures that there is no dissipation when the fluid is under the action of an expansion wave, i.e.
2628:
which has the property of being skew-adjoint with the divergence operator above, in the sense that
1293:
276:
15000:
Vignjevic, Rade; Reveles, Juan R.; Campbell, James (2006). "SPH in a total Lagrangian formalism".
14859:
Modelisation d'Γ©coulements visqueux par methode SPH en vue d'application Γ l'hydrodynamique navale
13462:
13405:
J.W. Swegle; D.A. Hicks; S.W. Attaway (1995). "Smooth particle hydrodynamics stability analysis".
10211:
5806:-th boundary element. The surface term can be also solved considering a semi-analytic expression.
543:
with characteristic radius known as the "smoothing length", typically represented in equations by
15883:
15861:
15846:
15829:
15737:
15722:
15638:
15503:
15361:
15141:
Hegeman, K.; Carr, N. A.; and Miller, G. S. P.; "Particle-based fluid simulation on the GPU", in
14310:
14234:
13547:
G.R. Johnson; S.R. Beissel (1996). "Normalized smoothing functions for SPH impact computations".
12295:
11667:
denotes the surface normal direction of the solid structure, and the imaginary particle density
8541:, is used to modulate dissipation when the fluid is under the action of a compression wave, i.e.
8351:{\displaystyle P^{\ast }={\overline {P}}+{\frac {1}{2}}\beta {\overline {\rho }}{(U_{L}-U_{R})},}
1428:
529:{\displaystyle {\frac {\mathrm {d} {\boldsymbol {r}}_{i}}{\mathrm {d} t}}={\boldsymbol {v}}_{i}.}
15117:
Hoover, W. G. (2006); Smooth Particle Applied Mechanics: The State of the Art, World Scientific.
14385:
13274:
8961:
15803:
15574:
15426:
15279:
15155:
Kolb, A.; and Cuntz, N. (2005); "Dynamic particle coupling for GPU-based fluid simulation", in
14043:
13457:
12986:
11311:
7806:
7779:
6067:
6016:
5888:
5884:
3632:
3057:
1565:
1266:
1073:
317:
is an active area of research in the astronomical community, and has had some limited success.
14387:
13339:
Smooth Particle Hydrodynamics with Strength of Materials, Advances in the Free Lagrange Method
9359:
is the number of spatial dimensions of the model. This approach introduces the bulk viscosity
4524:
2336:
Similarly to continuity, the task is to define a discrete gradient operator in order to write
2326:{\displaystyle {\frac {d{\boldsymbol {v}}}{dt}}=-{\frac {1}{\rho }}\nabla p+{\boldsymbol {g}}}
15851:
15697:
15613:
15402:
15347:
15306:
is an open source abstraction system that supports real time physics engines with SPH support
14752:
14733:
14684:
14635:
14578:
14427:
13030:
12880:
12786:
12715:
12683:
12642:
10095:
Using the initial configuration as the reference, the solid density is directly evaluated as
8921:
8584:
4167:
353:
SPH-based method SPAM (Smoothed Particle Applied Mechanics) for impact fracture in solids by
198:
13609:
C.T. Dyka; P.W. Randles; R.P. Ingel (1997). "Stress points for tension instability in SPH".
8941:
8464:
1779:
302:
impacts are some of the wide variety of astrophysical and cosmological uses of this method.
15922:
15888:
15561:
15028:
14951:
14916:
14907:
Monaghan, Joseph J.; Gingold, Robert A. (1983). "Shock Simulation by the Particle Method".
14881:
14815:
Marongiu Jean-Christophe and Leboeuf Francis and Caro JoΓ«lle and Parkinson Etienne (2010).
14762:
14713:
14664:
14607:
14516:
14478:
14399:
14355:
14320:
14283:
14269:
14244:
14196:
14123:
14088:
14035:
13937:
13877:
13842:
13799:
13762:
13723:
13688:
13653:
13618:
13556:
13494:
13449:
13414:
13379:
13195:
13142:
13089:
13022:
12747:
12622:
12571:
12552:
Price, D. J. (2011). "Smoothed Particle Hydrodynamics: Things I wish my mother taught me".
12503:
12445:
12408:
8524:
6282:
6093:
5866:
4171:
3605:
3196:
3061:
2156:
1499:
984:
933:
768:
314:
15145:(Reading, UK, May 2006), Lecture Notes in Computer Science v. 3994/2006 (Springer-Verlag).
6668:
SPH simulation: pressure distribution of a dam-break flow using standard Ξ΄-SPH formulation
2946:
including the sign change of the effective direction, thanks to the antisymmetry property
337:
Dyka et al. and Randles and Libersky introduced the stress-point integration into SPH and
44:) is a computational method used for simulating the mechanics of continuum media, such as
8:
15655:
15569:
1844:
384:
15926:
15415:
currently used for internal/external flooding/nuclear/chemical engineering applications.
15032:
14955:
14920:
14885:
14766:
14717:
14668:
14611:
14520:
14482:
14403:
14359:
14324:
14287:
14256:
14248:
14200:
14127:
14092:
14039:
13941:
13881:
13846:
13803:
13766:
13727:
13692:
13657:
13622:
13560:
13498:
13453:
13418:
13383:
13199:
13146:
13093:
12751:
12626:
12575:
12507:
12449:
12412:
6660:
SPH simulation: pressure distribution of a dam-break flow using standard SPH formulation
15878:
15819:
15097:
15071:
15044:
14839:
14623:
14415:
14139:
14061:
14025:
13998:
13963:
13903:
13815:
13301:
13256:
13238:
13211:
13185:
13158:
13132:
13105:
13004:
12959:
12924:
12857:
12763:
12561:
12526:
12491:
12377:
12269:
12249:
11785:
10847:
10827:
10718:
10075:
10055:
10035:
9342:
8674:
Introduced by Monaghan and Gingold the artificial viscosity was used to deal with high
6866:
6846:
6826:
6806:
6786:
6766:
6302:
6163:
6043:
5846:
4177:
2929:
2909:
2889:
2869:
2136:
1526:
1468:
1408:
1100:
1031:
1011:
964:
913:
792:. Secondly, the integral is approximated using a Riemann summation over the particles:
606:
586:
566:
546:
310:
272:
57:
15334:
Open Source python visualisation tool for Smoothed Particle Hydrodynamics simulations.
14528:
14367:
14100:
13533:
33:
Flow around cylinder with free surface modelled with SPH. See for similar simulations.
15508:
15328:
Open Source Framework for Smoothed Particle Hydrodynamics in Python (New BSD License)
15101:
15089:
15048:
14928:
14843:
14627:
14448:
14419:
14212:
14164:
14135:
13907:
13819:
13595:
13350:
13319:
13215:
13109:
13079:
13058:
12849:
12767:
12531:
12381:
12299:
8452:{\displaystyle \beta =\min {\big (}\eta \max(U_{L}-U_{R},0),{\overline {c}}{\big )}.}
5840:
5453:
This is a popular approach when free-surface is considered in monophase simulations.
4607:
In case the SPH convolution shall be practiced close to a boundary, i.e. closer than
4573:
is the particle acceleration, given by the right-hand side of the momentum equation.
3656:, and the latter, on the position of the particles. Using the thermodynamic property
1489:
354:
291:
15121:
14570:
14065:
14002:
13967:
13260:
13008:
12861:
12675:
12373:
5831:
The weakly compressible SPH in fluid dynamics is based on the discretization of the
15930:
15824:
15814:
15703:
15671:
15246:
is the free (GPLv3) SPH of the researchers, by the researchers, for the researchers
15081:
15036:
14959:
14924:
14889:
14831:
14797:
14788:
JP. Vila (1999). "On particle weighted methods and smooth particle hydrodynamics".
14770:
14721:
14672:
14615:
14566:
14524:
14486:
14407:
14363:
14328:
14291:
14272:"Theoretical analysis of the no-slip boundary condition enforcement in SPH methods"
14252:
14204:
14143:
14131:
14096:
14053:
13990:
13953:
13945:
13893:
13885:
13850:
13807:
13770:
13731:
13696:
13661:
13626:
13591:
13564:
13529:
13502:
13467:
13422:
13387:
13342:
13311:
13248:
13203:
13150:
13097:
12996:
12951:
12928:
12916:
12841:
12755:
12671:
12630:
12521:
12511:
12453:
12436:
L.B. Lucy (1977). "A numerical approach to the testing of the fission hypothesis".
12416:
12369:
9404:
283:
237:
R. Hoetzlein, 2012, develops efficient GPU-based SPH for large scenes in Fluids v.3
218:
77:
14835:
14817:"Free surface flows simulations in Pelton turbines using an hybrid SPH-ALE method"
13162:
13025:(2013). "Application of Smoothed Particle Hydrodynamics to Full-Film Lubrication".
12963:
12470:
6684:
407:, referred to as particles. Their Lagrangian nature allows setting their position
15866:
15809:
15798:
15276:
15185:
Simulation and Rendering of a Viscous Fluid Using Smoothed Particle Hydrodynamics
14654:
14158:
13059:"A hybrid solid boundary treatment algorithm for smoothed particle hydrodynamics"
12516:
10715:
correspond to the fluid pressure and viscous forces acting on the solid particle
3221:
540:
205:
102:
45:
15372:
13154:
12590:"The Smoothed Particle Hydrodynamics Method vs. Finite Volume Numerical Methods"
12397:"Smoothed particle hydrodynamics: theory and application to non-spherical stars"
3342:
When applied to the above Lagrangian, it gives the following momentum equation:
21:
15644:
15591:
15129:
Proceedings of ACM Workshop on General-purpose Computing on Graphics Processors
15019:
Han, Luhui; Hu, Xiangyu (2018). "SPH modeling of fluid-structure interaction".
14208:
6061:
5501:
5188:
Several techniques has been introduced in the past to model boundaries in SPH.
338:
287:
194:
49:
15408:
15331:
15085:
15040:
14999:
14893:
14801:
14774:
14725:
14676:
14490:
14332:
14057:
13854:
13631:
10.1002/(sici)1097-0207(19970715)40:13<2325::aid-nme161>3.0.co;2-8
13569:
10.1002/(sici)1097-0207(19960830)39:16<2725::aid-nme973>3.0.co;2-9
13369:
13315:
12759:
12634:
12421:
12396:
4599:
309:. Incorporating other astrophysical processes which may be important, such as
15952:
15513:
15411:
is a commercial fluid simulation software based on implicit SPH developed by
15405:
is a general-purpose computational tool for particle-based numerical methods.
15367:
15093:
14186:
13736:
10.1002/(sici)1097-0207(20000228)47:6<1189::aid-nme830>3.0.co;2-i
13346:
13323:
12661:
9660:
To discretize the governing equations of solid dynamics, a correction matrix
9640:
6803:. In this Riemann problem the initial left and right states are on particles
5481:
can be considered as a particular case of fluid extensions, where the field,
5443:{\displaystyle \nabla A_{i}=\sum _{j\in \Omega _{i}}V_{j}A_{j}\nabla W_{ij}.}
4133:
which is the SPH momentum equation already mentioned, where we recognize the
3946:{\displaystyle {\tfrac {\partial \rho _{j}}{\partial {\boldsymbol {r}}_{i}}}}
14556:
13485:
W.K. Liu; S. Jun; Y.F. Zhang (1995). "Reproducing kernel particle methods".
13472:
10.1002/(sici)1097-0207(19981115)43:5<785::aid-nme420>3.0.co;2-9
13176:
Rosswog, Stephan (2015). "SPH Methods in the Modelling of Compact Objects".
13000:
12977:
Akinci (2013). "Versatile Surface Tension and Adhesion for SPH Fluids SPH".
12955:
12920:
7776:
are inter-particle averages. With the solution of the Riemann problem, i.e.
5465:
1405:. The error done in approximating the integral by a discrete sum depends on
1203:{\displaystyle \rho _{i}=\rho ({\boldsymbol {r}}_{i})=\sum _{j}m_{j}W_{ij},}
15685:
15602:
15579:
14963:
14871:
14216:
14015:
13506:
13439:
13426:
13391:
12875:
Hoetzlein (2012). "Fluids v.3, A Large scale, Open Source Fluid Simulator".
12853:
12705:
12535:
185:
159:
73:
61:
15124:, Memorie della Societa Astronomia Italiana, Vol. 65, p. 1117 (1994).
14270:
Fabricio MaciΓ ; Matteo Antuono; Leo M GonzΓ‘lez; Andrea Colagrossi (2011).
12813:
Imhsen (2010). "Boundary handling and adaptive time-stepping for PCISPH".
6472:
the schemes that employ Riemann solvers to model the particle interaction.
1903:{\displaystyle {\frac {d\rho }{dt}}=-\rho \nabla \cdot {\boldsymbol {v}},}
15934:
15596:
15474:
15399:: A freeware SYMbolic ParticLE simulatoR from the University of Freiburg.
15319:
15136:
Proceedings of Eurographics Workshop on Computer Animation and Simulation
14386:
M. Ferrand, D.R. Laurence, B.D. Rogers, D. Violeau, C. Kassiotis (2013).
14296:
14271:
13644:
P.W. Randles; L.D. Libersky (2000). "Normalized SPH with stress points".
13243:
12845:
12612:
8675:
6321:
5196:
4920:
denotes the part of the compact support inside the computational domain,
1562:
For particles of constant mass, differentiating the interpolated density
69:
15448:
13898:
13775:
13666:
10.1002/1097-0207(20000810)48:10<1445::aid-nme831>3.0.co;2-9
13207:
12907:
Akinci (2012). "Versatile Rigid-Fluid Coupling for Incompressible SPH".
12314:
9650:
8619:
15243:
13994:
13701:
10.1002/1097-0207(20000730)48:9<1359::aid-nme829>3.0.co;2-u
13123:
Price, Daniel J (2009). "Astrophysical Smooth Particle Hydrodynamics".
12815:
Workshop on Virtual Reality Interaction and Physical Simulation VRIPHYS
3713:{\displaystyle \mathrm {d} e=\left(p/\rho ^{2}\right)\mathrm {d} \rho }
305:
SPH is used to model hydrodynamic flows, including possible effects of
295:
65:
29:
15270:
is a GPU-based particle simulation API available from OneZero Software
13958:
3898:
Plugging the SPH density interpolation and differentiating explicitly
2866:
angular momentum. This means that a force that is exerted on particle
2257:
The other important equation for a compressible inviscid fluid is the
15204:
14703:
14619:
14411:
13949:
13889:
13101:
8663:
8655:
190:
109:
15354:
for the study and prediction of complex-fluid rheology and stability
15294:
is a program package for particle-based simulation methods, e.g. SPH
15178:
Proceedings of Eurographics/SIGGRAPH Symposium on Computer Animation
13811:
13750:
13678:
5204:
The most straightforward boundary model is neglecting the integral,
184:
Smoothed-particle hydrodynamics is being increasingly used to model
15252:
is a commercial web-based SPH engineering software for CFD purposes
15227:
A derivation of the SPH model starting from Navier-Stokes equations
15076:
15061:
13252:
12457:
9655:
15914:
15309:
15261:
14345:
14030:
13306:
13190:
13137:
12566:
10233:
We can now discretize the momentum equation in the following form
1398:{\displaystyle W(|{\boldsymbol {r}}_{i}-{\boldsymbol {r}}_{j}|,h)}
15782:
15322:
is a freely available visualisation tool for particle simulations
13404:
306:
155:
132:
84:
15325:
15143:
Proceedings of International Conference on Computational Science
13789:
8958:
acts similar to the Neumann Richtmeyr artificial viscosity. The
6469:
the models that add a diffusive term in the continuity equation,
5781:{\displaystyle \gamma _{i}=\sum _{j\in \Omega _{i}}V_{j}W_{ij},}
583:. This can be understood in two steps. First an arbitrary field
250:
J. Kyle and E. Terrell, 2013, apply SPH to Full-Film Lubrication
15621:
15273:
15255:
15169:
Monaghan, Joseph J. (1992). "Smoothed Particle Hydrodynamics",
15150:
Lagrangian Fluid Dynamics Using Smoothed Particle Hydrodynamics
13608:
12781:
Solenthaler (2009). "Predictive-Corrective Incompressible SPH".
12471:
Takahiro Harada; Seiichi Koshizuka; Yoichiro Kawaguchi (2007).
10092:
is identical to that in the discretization of fluid equations.
9135:) to prevent possible numerical infinities at close distances.
7282:{\displaystyle (\rho _{R}^{\ast },U_{R}^{\ast },P_{R}^{\ast })}
7208:{\displaystyle (\rho _{L}^{\ast },U_{L}^{\ast },P_{L}^{\ast })}
4544:
is the time step, superscripts stand for time iterations while
1947:; hence one defines a discrete divergence operator as follows:
1913:
it is apparent that its right-hand side is an approximation of
1546:
Although the size of the smoothing length can be fixed in both
958:
299:
209:
180:
Fig. SPH simulation of ocean waves using FLUIDS v.1 (Hoetzlein)
15412:
15393:
is an open source (GPL) visualisation tool for SPH simulations
15303:
15210:
SPHERIC (SPH rEsearch and engineeRing International Community)
13920:
6656:
14976:
13341:. Lecture Notes in Physics. Vol. 395. pp. 248β257.
5839:
for compressible fluids. To close the system, an appropriate
5352:
such that just the bulk interactions are taken into account,
4580:
Integration of density has not been studied extensively (see
1547:
176:
15249:
15239:
Algodoo is a 2D simulation framework for education using SPH
14182:
14180:
15776:
15770:
15585:
15384:
15377:
15357:
15164:
Smoothed Particle Hydrodynamics: a meshfree particle method
14941:
14156:
11608:
10455:
where inter-particle averaged first Piola-Kirchhoff stress
8901:
7626:
7124:
6756:{\displaystyle \mathbf {e} _{ij}=-\mathbf {r} _{ij}/r_{ij}}
6664:
1551:
158:, the simulation of constant-density flows (e.g., external
15297:
15238:
15209:
15157:
Proceedings of the 18th Symposium on Simulation Techniques
13930:
International Journal for Numerical Methods in Engineering
13290:"Smoothed particle hydrodynamics and magnetohydrodynamics"
13056:
9689:
is first introduced to reproducing rigid-body rotation as
162:) is more efficient with grid-based methods than with SPH.
15313:
15267:
14597:
14468:
14230:
14228:
14226:
14177:
14114:
J.J. Monaghan (2005). "Smoothed particle hydrodynamics".
13643:
13519:
10072:
are used to denote solid particles, and smoothing length
9644:
8662:. It needs special consideration because it involves the
6180:. The momentum equation leads to the following relation:
15337:
15285:
13832:
13748:
13546:
13484:
12834:
IEEE Transactions on Visualization and Computer Graphics
10844:, due to the presence of the neighboring solid particle
3056:
The above SPH governing equations can be derived from a
2793:
this being a discrete version of the continuum identity
1557:
360:
Modified SPH (SPH/MLSPH) for fracture and fragmentation.
15343:
363:
Taylor-SPH (TSPH) for shock wave propagation in solids.
14223:
13713:
13228:
3906:
3110:
15396:
15018:
14755:
Computer Methods in Applied Mechanics and Engineering
14600:
International Journal for Numerical Methods in Fluids
14392:
International Journal for Numerical Methods in Fluids
14348:
Computer Methods in Applied Mechanics and Engineering
14313:
Computer Methods in Applied Mechanics and Engineering
13082:
International Journal for Numerical Methods in Fluids
12272:
12252:
12042:
11813:
11788:
11748:
11708:
11673:
11644:
11385:
11346:
11314:
11104:
10875:
10850:
10830:
10790:
10750:
10721:
10681:
10641:
10497:
10461:
10244:
10214:
10177:
10148:{\displaystyle \rho _{a}=\rho _{a}^{0}{\frac {1}{J}}}
10106:
10078:
10058:
10038:
9878:
9700:
9666:
9651:
Solid mechanics and fluid-structure interaction (FSI)
9416:
9365:
9345:
9329:{\displaystyle \alpha hc=2(n+2){\frac {\mu }{\rho }}}
9285:
9147:
8997:
8964:
8944:
8924:
8713:
8687:
8587:
8547:
8527:
8487:
8467:
8368:
8264:
8142:
7995:
7838:
7809:
7782:
7715:
7648:
7423:
7408:{\displaystyle P_{L}^{\ast }=P_{R}^{\ast }=P^{\ast }}
7358:
7345:{\displaystyle U_{L}^{\ast }=U_{R}^{\ast }=U^{\ast }}
7295:
7221:
7147:
6891:
6869:
6849:
6829:
6809:
6789:
6769:
6698:
6501:
6499:
6402:
6372:
6333:
6305:
6285:
6189:
6166:
6126:
6096:
6070:
6046:
6019:
5900:
5869:
5849:
5715:
5518:
5361:
5213:
4937:
4626:
4550:
4527:
4203:
4180:
4139:
3962:
3904:
3729:
3662:
3635:
3608:
3351:
3237:
3199:
3073:
3030:
3010:
2952:
2932:
2912:
2892:
2872:
2862:
This property leads to nice conservation properties.
2802:
2637:
2456:
2345:
2270:
2223:
2186:
2159:
2139:
2111:
1956:
1919:
1856:
1812:
1782:
1727:
1598:
1568:
1529:
1502:
1471:
1431:
1411:
1342:
1296:
1269:
1219:
1126:
1103:
1076:
1054:
1034:
1014:
987:
967:
936:
916:
801:
771:
765:
The error in making the above approximation is order
632:
609:
589:
569:
549:
474:
442:
413:
387:
15450:
Numerical methods for partial differential equations
15351:
14464:
14462:
14460:
14453:. Princeton, New Jersey: Princeton University Press.
14263:
13021:
12355:
12353:
7289:. By assuming that the intermediate state satisfies
14790:
Mathematical Models and Methods in Applied Sciences
14339:
14081:
Computers Methods in Applied Mechanical Engineering
13581:
6448:{\displaystyle p=c^{2}\left(\rho -\rho _{0}\right)}
5200:
SPH free-surface model by means of integral neglect
3228:of variational mechanics reads, for each particle:
15312:is a commercial engineering software developed by
15205:First large simulation of star formation using SPH
14856:
14541:
14506:
13867:
13063:Scientia Iranica, Transaction A, Civil Engineering
12278:
12258:
12222:
12007:
11794:
11774:
11734:
11691:
11659:
11614:
11366:
11332:
11284:
11069:
10856:
10836:
10816:
10776:
10727:
10707:
10667:
10611:
10478:
10431:
10222:
10208:is the Jacobian determinant of deformation tensor
10200:
10147:
10084:
10064:
10044:
10022:
9846:
9681:
9593:
9387:
9351:
9328:
9265:
9109:
8980:
8950:
8930:
8907:
8599:
8573:
8533:
8513:
8473:
8451:
8350:
8244:
8126:
7980:
7822:
7795:
7768:
7701:
7632:
7407:
7344:
7281:
7207:
7130:
6875:
6855:
6835:
6815:
6795:
6775:
6755:
6641:
6447:
6385:
6355:
6311:
6291:
6268:
6172:
6152:
6109:
6082:
6052:
6032:
6002:
5875:
5855:
5780:
5700:
5442:
5341:
5177:
4877:
4565:
4536:
4510:
4186:
4149:
4122:
3945:
3887:
3712:
3648:
3621:
3591:
3331:
3212:
3181:
3040:
3016:
2993:
2938:
2918:
2898:
2878:
2851:
2782:
2617:
2436:
2325:
2246:
2209:
2172:
2145:
2125:
2094:
1939:
1902:
1827:
1798:
1768:
1710:
1581:
1535:
1515:
1477:
1457:
1417:
1397:
1328:
1282:
1255:
1202:
1109:
1089:
1062:
1040:
1020:
1000:
973:
949:
922:
899:
784:
754:
615:
595:
575:
555:
528:
457:
428:
399:
14535:
14502:
14500:
14457:
13540:
12464:
12401:Monthly Notices of the Royal Astronomical Society
12350:
4603:SPH Convolution support split close to a boundary
4588:
4581:
730:
125:
15950:
14993:
14078:
13923:"Truncation error in mesh-free particle methods"
13513:
13336:
12395:Gingold, Robert A.; Monaghan, Joseph J. (1977).
11435:
10184:
9656:Total Lagrangian formulation for solid mechanics
8388:
8375:
4170:. The most popular in the SPH literature is the
15912:
15387:is an open source SPH implementation in Fortran
15360:is a commercial simulation package for SPH and
14906:
13296:. Special Issue: Computational Plasma Physics.
12554:Advances in Computational Astrophysics: Methods
12394:
7769:{\displaystyle {\overline {P}}=(P_{L}+P_{R})/2}
7702:{\displaystyle {\overline {U}}=(U_{L}+U_{R})/2}
5819:
2247:{\displaystyle \left\{\mathbf {v} _{j}\right\}}
2210:{\displaystyle \left\{\mathbf {r} _{j}\right\}}
2153:for a given set of particles with given masses
1940:{\displaystyle -\rho \nabla \cdot \mathbf {v} }
15340:Commercial SPH solver for the cinema industry.
15282:) code for cosmological N-body/SPH simulations
14497:
14381:
14379:
14377:
13751:"SPAM-based recipes for continuum simulations"
12733:
12731:
15434:
15002:Computer Modeling in Engineering and Sciences
14304:
14113:
12483:
11699:is calculated through the equation of state.
9407:the viscosity model by Morris was proposed.
8441:
8380:
4150:{\displaystyle \operatorname {\mathbf {G} } }
3041:{\displaystyle \operatorname {\mathbf {G} } }
128:, the method has great conservation features.
14738:: CS1 maint: multiple names: authors list (
14689:: CS1 maint: multiple names: authors list (
14640:: CS1 maint: multiple names: authors list (
14583:: CS1 maint: multiple names: authors list (
14432:: CS1 maint: multiple names: authors list (
12720:: CS1 maint: multiple names: authors list (
12701:
12699:
12688:: CS1 maint: multiple names: authors list (
12647:: CS1 maint: multiple names: authors list (
12551:
9542:
9523:
9339:is substituted in the equation above, where
9230:
9211:
9074:
9055:
6117:is added to avoid negative pressure values.
2994:{\displaystyle \nabla W_{ij}=-\nabla W_{ji}}
2105:This operator gives an SPH approximation of
1769:{\displaystyle \nabla W_{ij}=-\nabla W_{ji}}
15171:Annual Review of Astronomy and Astrophysics
14697:
14398:(4). Int. J. Numer. Meth. Fluids: 446β472.
14374:
13980:
13921:N.J. Quinlan; M. Basa; M. Lastiwka (2006).
13330:
12780:
12728:
10738:
6679:
6485:
6476:
5485:, vanish outside the computational domain.
2906:equals the one that is exerted on particle
1488:Kernel functions commonly used include the
1070:denotes position. For example, the density
15441:
15427:
15012:
14980:PKDGRAV The Parallel k-D Tree Gravity Code
14446:
12608:
12606:
12547:
12545:
12359:
8610:
7830:, the discretization of the SPH method is
6090:is commonly used. The background pressure
5887:(sometimes mistakenly referred to as the "
3602:where the chain rule has been used, since
930:includes all particles in the simulation.
15913:Benitez-Llambay, Alejandro (2015-07-28),
15075:
14295:
14047:
14029:
13957:
13897:
13774:
13461:
13305:
13242:
13189:
13136:
12990:
12942:Macklin (2013). "Position Based Fluids".
12874:
12696:
12655:
12565:
12525:
12515:
12435:
12420:
12305:
10586:
10574:
10554:
10542:
10503:
10466:
10322:
10216:
10191:
9703:
9669:
9572:
9477:
9388:{\displaystyle \zeta ={\frac {5}{3}}\mu }
2126:{\displaystyle \nabla \cdot \mathbf {v} }
721:
15131:(August, 2004, Los Angeles, California).
14787:
14157:E. Hairer; C. Lubich; G. Wanner (2006).
13275:"Star Formation with Radiative Transfer"
9625:
8938:is controlling a volume viscosity while
6683:
6663:
6655:
5836:
5500:
5464:
5195:
4896:is the compact support ball centered at
4598:
3051:
175:
28:
20:
14861:. PhD Thesis, Ecole centrale de Nantes.
13337:Libersky, L.D.; Petschek, A.G. (1990).
13175:
12941:
12737:
12603:
12542:
12473:Smoothed particle hydrodynamics on GPUs
10479:{\displaystyle {\tilde {\mathbb {P} }}}
9575:
9528:
9483:
9427:
9216:
9196:
9178:
9060:
9040:
9022:
8879:
8861:
8830:
8812:
6574:
6559:
6466:the schemes that adopt density filters,
6224:
5667:
5318:
5294:
5285:
5261:
5243:
5226:
5162:
5138:
5129:
5105:
5087:
5070:
5045:
5021:
5012:
4988:
4970:
4948:
4862:
4833:
4824:
4798:
4780:
4763:
4738:
4709:
4700:
4674:
4656:
4634:
4553:
4465:
4431:
4401:
4357:
4337:
4307:
4268:
4248:
4210:
4113:
3973:
3929:
3878:
3851:
3750:
3585:
3558:
3471:
3444:
3372:
3313:
3277:
3164:
3155:
3123:
2842:
2807:
2763:
2660:
2430:
2354:
2319:
2278:
2055:
2040:
1976:
1893:
1815:
1671:
1656:
1371:
1356:
1148:
1056:
870:
861:
809:
737:
697:
688:
662:
640:
513:
485:
15951:
15187:, MSc Thesis, Umea University, Sweden.
14606:(9). Wiley Online Library: 1100β1114.
12976:
12906:
12812:
11775:{\displaystyle \mathbf {f} _{a}^{F:v}}
11735:{\displaystyle \mathbf {f} _{a}^{F:p}}
10817:{\displaystyle \mathbf {f} _{i}^{S:v}}
10777:{\displaystyle \mathbf {f} _{i}^{S:p}}
10708:{\displaystyle \mathbf {f} _{a}^{F:v}}
10668:{\displaystyle \mathbf {f} _{a}^{F:p}}
8649:
5469:SPH Fluid Extension Boundary technique
4594:
4174:scheme, which reads for each particle
118:
15422:
15364:(DEM) simulations from Fraunhofer IWM
15166:, Singapore: World Scientific (2003).
15120:Stellingwerf, R. F.; Wingate, C. A.;
14977:Marios D. Dikaiakos; Joachim Stadel,
13983:Advances in Computational Mathematics
13287:
13122:
12827:
12708:14th International LS-DYNA Conference
12582:
9609:
5478:
4566:{\displaystyle {\boldsymbol {a}}_{i}}
1828:{\displaystyle {\boldsymbol {r}}_{i}}
1558:Discretization of governing equations
15692:Moving particle semi-implicit method
15603:Weighted essentially non-oscillatory
15288:SPH simulator with viscosity (GPLv3)
13755:Computing in Science and Engineering
12309:
12033:
11804:
11702:Accordingly, the interaction forces
11376:
11367:{\displaystyle \mathbf {v} _{a}^{d}}
11095:
10866:
10488:
10235:
10201:{\displaystyle J=\det(\mathbb {F} )}
10097:
9691:
8614:
6153:{\displaystyle 10^{3}\mathrm {m/s} }
5809:
5496:
1256:{\displaystyle m_{j}=\rho _{j}V_{j}}
25:Schematic view of an SPH convolution
15915:"Py-Sphviewer: Py-Sphviewer V1.0.0"
15316:implementing an implicit SPH method
14830:(S1). Taylor & Francis: 40β49.
14257:10.1016/j.jfluidstructs.2013.05.010
13749:W. G. Hoover; C. G. Hoover (2001).
13057:Mahdavi and Talebbeydokhti (2015).
8581:. Numerical experiments found the
5825:
5191:
4160:
1835:. Comparing this equation with the
539:These particles interact through a
13:
15541:Finite-difference frequency-domain
15192:Fluid Mechanics and the SPH method
15111:
12194:
12159:
11966:
11256:
11221:
11028:
10342:
9964:
9910:
9880:
9803:
9499:
9423:
9149:
8689:
8099:
7953:
6614:
6592:
6356:{\displaystyle c=10v_{\text{max}}}
6320:assumption. This corresponds to a
6146:
6138:
5741:
5634:
5630:
5600:
5569:
5519:
5460:
5421:
5390:
5362:
5275:
5236:
5119:
5080:
5002:
4963:
4938:
4773:
4649:
4528:
4489:
4386:
4286:
4093:
3986:
3967:
3924:
3909:
3846:
3831:
3763:
3744:
3703:
3664:
3553:
3538:
3519:
3504:
3439:
3424:
3385:
3366:
3308:
3301:
3296:
3272:
3265:
3260:
3247:
3241:
3076:
3017:{\displaystyle \operatorname {D} }
3011:
2975:
2953:
2745:
2596:
2309:
2112:
2073:
1958:
1926:
1886:
1750:
1728:
1689:
723:
498:
479:
371:
320:
275:make it ideal for computations in
208:and SPH particle techniques, many
16:Method of hydrodynamics simulation
14:
15975:
15198:
15194:, Oxford University Press (2012).
13835:Comput. Methods Appl. Mech. Engrg
9630:
6324:smaller than 0.1, which implies:
1063:{\displaystyle {\boldsymbol {r}}}
603:is written as a convolution with
436:by integration of their velocity
376:
171:
15959:Numerical differential equations
15138:(August 1996, Poitiers, France).
15064:Journal of Computational Physics
14944:Journal of Computational Physics
14909:Journal of Computational Physics
14874:Journal of Computational Physics
14796:(2). World Scientific: 161β209.
14509:Journal of Computational Physics
14237:Journal of Fluids and Structures
13294:Journal of Computational Physics
12740:Journal of Computational Physics
12615:Journal of Computational Physics
12313:
12170:
12130:
12110:
12045:
11983:
11816:
11751:
11711:
11660:{\displaystyle \mathbf {n} ^{S}}
11647:
11592:
11577:
11554:
11535:
11517:
11499:
11468:
11451:
11349:
11232:
11187:
11172:
11107:
11045:
10878:
10793:
10753:
10684:
10644:
10408:
10382:
10373:
10264:
10002:
9974:
9928:
9779:
9759:
9682:{\displaystyle \mathbb {B} ^{0}}
8618:
8360:where the limiter is defined as
8226:
8193:
8170:
8145:
8004:
7936:
7921:
7089:
7074:
6978:
6963:
6722:
6701:
5814:
5332:
5322:
5298:
5265:
5166:
5142:
5109:
5049:
5025:
4992:
4866:
4837:
4802:
4742:
4713:
4678:
4142:
3033:
2676:
2460:
2394:
2230:
2193:
2119:
1933:
740:
701:
666:
458:{\displaystyle \mathbf {v} _{i}}
445:
429:{\displaystyle \mathbf {r} _{i}}
416:
15906:
15894:Method of fundamental solutions
15680:Smoothed-particle hydrodynamics
15152:, MSc Thesis, Univ. Copenhagen.
15055:
14970:
14935:
14900:
14865:
14850:
14808:
14781:
14746:
14706:Computer Physics Communications
14657:Computer Physics Communications
14648:
14591:
14571:10.1016/j.compfluid.2008.11.012
14550:
14471:Computer Physics Communications
14440:
14276:Progress of Theoretical Physics
14160:Geometric Numerical Integration
14150:
14107:
14072:
14018:Computer Physics Communications
14009:
13974:
13914:
13861:
13826:
13783:
13742:
13707:
13672:
13637:
13602:
13575:
13478:
13433:
13398:
13363:
13288:Price, Daniel J. (2012-02-01).
13281:
13267:
13222:
13169:
13116:
13073:
13050:
13015:
12970:
12935:
12900:
12868:
12821:
12806:
12774:
12676:10.1016/j.compfluid.2016.05.029
12477:Computer Graphics International
12374:10.1016/j.compfluid.2019.01.007
9645:particle-particle particle-mesh
8858:
8809:
8574:{\displaystyle U_{L}\geq U_{R}}
266:
38:Smoothed-particle hydrodynamics
15535:Alternating direction-implicit
14116:Reports on Progress in Physics
12429:
12388:
12189:
12165:
12002:
11978:
11545:
11512:
11509:
11491:
11447:
11438:
11251:
11227:
11064:
11040:
10507:
10470:
10326:
10195:
10187:
9993:
9968:
9947:
9922:
9432:
9417:
9313:
9301:
8789:
8729:
8514:{\displaystyle U_{L}<U_{R}}
8423:
8391:
8341:
8315:
8239:
8186:
7946:
7916:
7755:
7729:
7688:
7662:
7616:
7590:
7573:
7508:
7497:
7471:
7276:
7222:
7202:
7148:
7115:
7056:
7050:
7011:
7004:
6945:
6939:
6900:
6631:
6625:
6386:{\displaystyle v_{\text{max}}}
6230:
6219:
5310:
5281:
5247:
5239:
5230:
5222:
5154:
5125:
5091:
5083:
5074:
5066:
5037:
5008:
4974:
4966:
4952:
4944:
4854:
4844:
4819:
4815:
4784:
4776:
4767:
4759:
4730:
4720:
4695:
4691:
4660:
4652:
4638:
4630:
1392:
1382:
1350:
1346:
1263:denotes the particle mass and
1158:
1143:
891:
881:
856:
852:
813:
805:
718:
708:
683:
679:
644:
636:
326:fracture, fragmentation, etc.
143:
1:
15547:Finite-difference time-domain
14836:10.1080/00221686.2010.9641244
14824:Journal of Hydraulic Research
14529:10.1016/S0021-9991(03)00324-3
14368:10.1016/S0045-7825(99)00442-9
14101:10.1016/S0045-7825(99)00051-1
13534:10.1016/s0734-743x(97)87441-6
12344:
11692:{\displaystyle \rho _{a}^{d}}
11308:Here, the imaginary pressure
6040:is the reference density and
5843:is utilized to link pressure
1425:, on the particle size (i.e.
1329:{\displaystyle W_{ij}=W_{ji}}
1008:is the value of the quantity
95:
15964:Computational fluid dynamics
15586:Advection upstream-splitting
15162:Liu, G. R.; and Liu, M. B.;
14929:10.1016/0021-9991(83)90036-0
13596:10.1016/0045-7949(95)00059-p
12979:ACM Transactions on Graphics
12944:ACM Transactions on Graphics
12909:ACM Transactions on Graphics
12517:10.1371/journal.pone.0020685
10223:{\displaystyle \mathbb {F} }
8434:
8309:
8283:
8219:
8197:
7721:
7654:
7549:
7450:
5891:") is used in SPH. It reads
5820:Weakly compressible approach
5479:integral neglect methodology
1589:with respect to time yields
1290:the particle density, while
137:probability density function
7:
15597:Essentially non-oscillatory
15580:Monotonic upstream-centered
15232:
15122:"Impact Modelling with SPH"
14857:De Leffe, Matthieu (2011).
14712:(12). Elsevier: 2570β2580.
14542:Randall J. LeVeque (2007).
13178:Living Rev Comput Astrophys
13155:10.1016/j.newar.2009.08.007
12236:
12021:
11782:acting on a solid particle
11628:
11298:
11083:
10824:acting on a fluid particle
10625:
10445:
10161:
9860:
6366:where the maximum velocity
5883:. Generally, the so-called
5505:SPH Boundary Integral model
1458:{\displaystyle V_{j}^{1/d}}
215:incompressibility condition
166:
124:As discussed in section on
52:flows. It was developed by
10:
15980:
15857:Infinite difference method
15475:Forward-time central-space
14565:(6). Elsevier: 1203β1217.
14209:10.1103/PhysRevE.79.056701
14136:10.1088/0034-4885/68/8/R01
13870:Int. J. Numer. Methods Eng
13792:Int. J. Numer. Methods Eng
13716:Int. J. Numer. Methods Eng
13681:Int. J. Numer. Methods Eng
13646:Int. J. Numer. Methods Eng
13611:Int. J. Numer. Methods Eng
13549:Int. J. Numer. Methods Eng
13487:Int. J. Numer. Methods Eng
13442:Int. J. Numer. Methods Eng
8981:{\displaystyle \phi _{ij}}
6299:is the density change and
5802:the normal of the generic
350:Metal forming simulations.
217:in these systems. SPH for
101:By construction, SPH is a
15791:
15760:PoincarΓ©βSteklov operator
15713:
15670:
15612:
15560:
15527:
15519:Method of characteristics
15489:
15465:
15456:
15304:Physics Abstraction Layer
15173:(1992). 30 : 543β74.
15086:10.1016/j.jcp.2020.110028
15041:10.1007/s42241-018-0006-9
14894:10.1016/j.jcp.2017.01.027
14802:10.1142/S0218202599000117
14775:10.1016/j.cma.2015.02.004
14726:10.1016/j.cpc.2012.07.006
14677:10.1016/j.cpc.2009.11.002
14491:10.1016/j.cpc.2008.12.004
14333:10.1016/j.cma.2016.12.037
14058:10.1016/j.cpc.2013.07.004
13855:10.1016/j.cma.2011.02.006
13316:10.1016/j.jcp.2010.12.011
12760:10.1016/j.jcp.2020.109565
12635:10.1016/j.jcp.2012.05.005
12289:
11333:{\displaystyle p_{a}^{d}}
8521:, and that the parameter
7823:{\displaystyle P^{\ast }}
7796:{\displaystyle U^{\ast }}
6083:{\displaystyle \gamma =7}
6033:{\displaystyle \rho _{0}}
3649:{\displaystyle \rho _{j}}
3220:is the particle specific
1582:{\displaystyle \rho _{i}}
1283:{\displaystyle \rho _{j}}
1090:{\displaystyle \rho _{i}}
910:where the summation over
90:
15777:Tearing and interconnect
15771:Balancing by constraints
15159:(2005) pp. 722β727.
15021:Journal of Hydrodynamics
14663:(3). Elsevier: 532β549.
13347:10.1007/3-540-54960-9_58
10739:Fluid-structure coupling
6680:Riemann solver technique
6486:Diffusive term technique
6477:Density filter technique
4537:{\displaystyle \Delta t}
1336:is a short notation for
277:theoretical astrophysics
15884:Computer-assisted proof
15862:Infinite element method
15650:Gradient discretisation
15362:Discrete element method
15278:is a freely available (
13001:10.1145/2508363.2508395
12956:10.1145/2461912.2461984
12921:10.1145/2185520.2185558
12422:10.1093/mnras/181.3.375
12296:discrete element method
9127:is a small fraction of
8931:{\displaystyle \alpha }
8660:NavierβStokes equations
8611:Incompressible approach
8600:{\displaystyle \eta =3}
6763:pointing form particle
5833:NavierβStokes equations
3226:EulerβLagrange equation
126:weakly compressible SPH
15872:PetrovβGalerkin method
15633:Discontinuous Galerkin
14964:10.1006/jcph.1997.5776
14559:Computers & Fluids
13507:10.1002/fld.1650200824
13427:10.1006/jcph.1995.1010
13392:10.1006/jcph.1993.1199
13038:Cite journal requires
12888:Cite journal requires
12794:Cite journal requires
12322:This section is empty.
12306:Variants of the method
12298:, used for simulating
12280:
12260:
12224:
12009:
11796:
11776:
11736:
11693:
11661:
11616:
11368:
11334:
11286:
11071:
10864:, can be obtained as
10858:
10838:
10818:
10778:
10729:
10709:
10669:
10613:
10480:
10433:
10224:
10202:
10149:
10086:
10066:
10046:
10024:
9848:
9683:
9595:
9389:
9353:
9330:
9267:
9111:
8982:
8952:
8951:{\displaystyle \beta }
8932:
8909:
8678:fluid flows. It reads
8627:This section is empty.
8601:
8575:
8535:
8515:
8475:
8474:{\displaystyle \beta }
8453:
8352:
8246:
8128:
7982:
7824:
7797:
7770:
7703:
7634:
7409:
7346:
7283:
7209:
7132:
6877:
6857:
6837:
6817:
6797:
6777:
6757:
6689:
6669:
6661:
6643:
6449:
6387:
6357:
6313:
6293:
6270:
6174:
6154:
6111:
6084:
6054:
6034:
6004:
5877:
5857:
5782:
5702:
5506:
5470:
5444:
5343:
5201:
5179:
4879:
4604:
4567:
4538:
4512:
4188:
4168:symplectic integrators
4151:
4124:
3947:
3889:
3714:
3650:
3623:
3593:
3333:
3214:
3183:
3064:of a particle system:
3058:least action principle
3042:
3018:
2995:
2940:
2920:
2900:
2880:
2853:
2784:
2619:
2438:
2327:
2261:for momentum balance:
2248:
2211:
2174:
2147:
2127:
2096:
1941:
1904:
1829:
1800:
1799:{\displaystyle W_{ij}}
1770:
1712:
1583:
1537:
1517:
1479:
1459:
1419:
1399:
1330:
1284:
1257:
1204:
1111:
1091:
1064:
1042:
1022:
1002:
975:
951:
924:
901:
786:
756:
617:
597:
577:
557:
530:
459:
430:
401:
181:
34:
26:
15852:Isogeometric analysis
15698:Material point method
14761:. Elsevier: 209β226.
14450:Underwater Explosions
12302:, is related to SPH.
12281:
12261:
12225:
12010:
11797:
11777:
11737:
11694:
11662:
11617:
11369:
11335:
11287:
11072:
10859:
10839:
10819:
10779:
10730:
10710:
10670:
10614:
10481:
10434:
10225:
10203:
10150:
10087:
10067:
10047:
10025:
9849:
9684:
9626:Multiphase extensions
9596:
9390:
9354:
9331:
9268:
9112:
8983:
8953:
8933:
8910:
8602:
8576:
8536:
8534:{\displaystyle \eta }
8516:
8476:
8454:
8353:
8247:
8129:
7983:
7825:
7798:
7771:
7704:
7635:
7410:
7347:
7284:
7210:
7133:
6878:
6858:
6838:
6818:
6798:
6778:
6758:
6687:
6667:
6659:
6644:
6450:
6388:
6358:
6314:
6294:
6292:{\displaystyle \rho }
6271:
6175:
6155:
6112:
6110:{\displaystyle p_{0}}
6085:
6055:
6035:
6005:
5878:
5876:{\displaystyle \rho }
5858:
5783:
5703:
5504:
5477:Along this line, the
5468:
5445:
5344:
5199:
5180:
4880:
4602:
4568:
4539:
4513:
4189:
4152:
4125:
3948:
3890:
3715:
3651:
3624:
3622:{\displaystyle e_{j}}
3594:
3334:
3215:
3213:{\displaystyle e_{j}}
3184:
3052:Variational principle
3043:
3019:
2996:
2941:
2921:
2901:
2881:
2854:
2785:
2620:
2439:
2328:
2249:
2212:
2175:
2173:{\displaystyle m_{j}}
2148:
2128:
2097:
1942:
1905:
1830:
1801:
1771:
1713:
1584:
1538:
1518:
1516:{\displaystyle C^{2}}
1480:
1460:
1420:
1400:
1331:
1285:
1258:
1205:
1117:can be expressed as:
1112:
1092:
1065:
1043:
1023:
1003:
1001:{\displaystyle A_{j}}
976:
952:
950:{\displaystyle V_{j}}
925:
902:
787:
785:{\displaystyle h^{2}}
757:
618:
598:
578:
558:
531:
460:
431:
402:
179:
32:
24:
15935:10.5281/zenodo.21703
15889:Integrable algorithm
15715:Domain decomposition
14297:10.1143/PTP.125.1091
13841:(21β22): 1833β1845.
13023:Journal of Tribology
12846:10.1109/TVCG.2011.29
12664:Computers and Fluids
12362:Computers and Fluids
12270:
12250:
12040:
11811:
11786:
11746:
11706:
11671:
11642:
11383:
11344:
11312:
11102:
10873:
10848:
10828:
10788:
10748:
10719:
10679:
10639:
10495:
10459:
10242:
10212:
10175:
10104:
10076:
10056:
10036:
9876:
9698:
9664:
9414:
9363:
9343:
9283:
9145:
8995:
8962:
8942:
8922:
8685:
8670:Artificial viscosity
8585:
8545:
8525:
8485:
8465:
8366:
8262:
8140:
7993:
7836:
7807:
7780:
7713:
7646:
7421:
7356:
7293:
7219:
7145:
6889:
6867:
6847:
6843:, respectively. The
6827:
6807:
6787:
6767:
6696:
6497:
6400:
6370:
6331:
6303:
6283:
6187:
6164:
6124:
6094:
6068:
6044:
6017:
5898:
5867:
5847:
5713:
5516:
5359:
5211:
4935:
4624:
4548:
4525:
4201:
4178:
4137:
3960:
3902:
3727:
3660:
3633:
3606:
3349:
3235:
3197:
3071:
3060:, starting from the
3028:
3008:
2950:
2930:
2910:
2890:
2870:
2800:
2635:
2454:
2343:
2268:
2221:
2184:
2157:
2137:
2109:
1954:
1917:
1854:
1845:material derivatives
1810:
1780:
1725:
1596:
1566:
1527:
1500:
1469:
1429:
1409:
1340:
1294:
1267:
1217:
1124:
1101:
1074:
1052:
1032:
1012:
985:
965:
934:
914:
799:
769:
630:
607:
587:
567:
547:
472:
440:
411:
385:
15927:2015zndo.....21703B
15733:Schwarz alternating
15656:Loubignac iteration
15148:Kelager, M. (2006)
15033:2018JHyDy..30...62H
14956:1997JCoPh.136..214M
14921:1983JCoPh..52..374M
14886:2017JCoPh.335..605Z
14767:2015CMAME.289..209A
14718:2012CoPhC.183.2570A
14669:2010CoPhC.181..532A
14612:2011IJNMF..67.1100F
14521:2003JCoPh.191..448C
14483:2009CoPhC.180..861M
14447:H. R. Cole (1948).
14404:2013IJNMF..71..446F
14360:2000CMAME.184...49C
14325:2017CMAME.317..771C
14288:2011PThPh.125.1091M
14249:2013JFS....42..112B
14201:2009PhRvE..79e6701C
14128:2005RPPh...68.1703M
14093:1999CMAME.180...97B
14040:2013CoPhC.184.2515M
13942:2006IJNME..66.2064Q
13882:2015IJNME.103..781Y
13847:2011CMAME.200.1833H
13804:2003IJNME..56.1421R
13776:10.1109/5992.909007
13767:2001CSE.....3b..78H
13728:2000IJNME..47.1189B
13693:2000IJNME..48.1359B
13658:2000IJNME..48.1445R
13623:1997IJNME..40.2325D
13561:1996IJNME..39.2725J
13499:1995IJNMF..20.1081L
13454:1998IJNME..43..785B
13419:1995JCoPh.116..123S
13384:1993JCoPh.109...67L
13208:10.1007/lrca-2015-1
13200:2015LRCA....1....1R
13147:2009NewAR..53...78R
13094:2017IJNMF..83..115T
12830:"Constraint Fluids"
12752:2020JCoPh.41709565K
12627:2012JCoPh.231.7057A
12576:2012ASPC..453..249P
12508:2011PLoSO...620685C
12450:1977AJ.....82.1013L
12413:1977MNRAS.181..375G
12124:
12065:
11961:
11931:
11893:
11836:
11771:
11731:
11688:
11568:
11408:
11363:
11329:
11201:
11127:
11023:
10983:
10965:
10898:
10813:
10773:
10704:
10664:
10600:
10568:
10428:
10402:
10355:
10134:
10019:
9991:
9945:
9893:
9816:
9793:
9773:
9751:
9717:
9568:
9256:
9100:
8780:
8650:Viscosity modelling
7391:
7373:
7328:
7310:
7275:
7257:
7239:
7201:
7183:
7165:
4595:Boundary techniques
4584:for more details).
4485:
4459:
4421:
4385:
4351:
4327:
4282:
4262:
4238:
4085:
4053:
3825:
3137:
2588:
2556:
1843:description (using
1837:continuity equation
1776:is the gradient of
1454:
400:{\displaystyle i,j}
273:orders of magnitude
15879:Validated numerics
15221:SPHC Image Gallery
13995:10.1007/BF02123482
13522:Int. J. Impact Eng
13493:(8β9): 1081β1106.
12300:granular materials
12276:
12256:
12220:
12108:
12081:
12043:
12005:
11947:
11917:
11879:
11855:
11814:
11792:
11772:
11749:
11732:
11709:
11689:
11674:
11657:
11612:
11607:
11552:
11394:
11364:
11347:
11330:
11315:
11282:
11185:
11143:
11105:
11067:
11009:
10969:
10951:
10917:
10876:
10854:
10834:
10814:
10791:
10774:
10751:
10725:
10705:
10682:
10665:
10642:
10609:
10584:
10552:
10476:
10429:
10406:
10380:
10341:
10296:
10220:
10198:
10145:
10120:
10082:
10062:
10042:
10020:
10000:
9972:
9926:
9879:
9844:
9802:
9777:
9757:
9737:
9736:
9701:
9679:
9610:Additional physics
9591:
9554:
9385:
9349:
9326:
9263:
9242:
9107:
9086:
8978:
8948:
8928:
8905:
8900:
8805:
8763:
8597:
8571:
8531:
8511:
8471:
8449:
8348:
8242:
8124:
8043:
7978:
7891:
7820:
7793:
7766:
7699:
7630:
7625:
7405:
7377:
7359:
7342:
7314:
7296:
7279:
7261:
7243:
7225:
7205:
7187:
7169:
7151:
7128:
7123:
6873:
6853:
6833:
6813:
6793:
6773:
6753:
6690:
6670:
6662:
6639:
6637:
6541:
6445:
6383:
6353:
6309:
6289:
6266:
6170:
6150:
6107:
6080:
6050:
6030:
6000:
5873:
5853:
5778:
5751:
5698:
5644:
5579:
5507:
5471:
5440:
5400:
5339:
5202:
5175:
4875:
4605:
4563:
4534:
4508:
4506:
4463:
4429:
4399:
4355:
4335:
4305:
4266:
4246:
4208:
4184:
4147:
4120:
4071:
4039:
4011:
3943:
3941:
3885:
3811:
3788:
3710:
3646:
3619:
3589:
3490:
3410:
3329:
3210:
3179:
3121:
3119:
3093:
3038:
3014:
2991:
2936:
2916:
2896:
2876:
2849:
2780:
2723:
2647:
2615:
2574:
2542:
2514:
2434:
2323:
2244:
2207:
2170:
2143:
2123:
2092:
2022:
1937:
1900:
1825:
1796:
1766:
1708:
1638:
1579:
1533:
1513:
1475:
1455:
1432:
1415:
1395:
1326:
1280:
1253:
1200:
1173:
1107:
1087:
1060:
1038:
1018:
998:
971:
947:
920:
897:
828:
782:
752:
613:
593:
573:
553:
526:
455:
426:
397:
311:radiative transfer
292:stellar collisions
182:
35:
27:
15902:
15901:
15842:Immersed boundary
15835:Method of moments
15750:NeumannβDirichlet
15743:abstract additive
15728:Fictitious domain
15672:Meshless/Meshfree
15556:
15555:
15458:Finite difference
15314:FIFTY2 Technology
14189:Physical Review E
14170:978-3-540-30666-5
14024:(11): 2515β2527.
13936:(13): 2064β2085.
13798:(10): 1421β1444.
13652:(10): 1445β1462.
13617:(13): 2325β2341.
13555:(16): 2725β2741.
13528:(6β10): 525β532.
13356:978-3-540-54960-4
12621:(21): 7057β7075.
12479:. pp. 63β70.
12342:
12341:
12279:{\displaystyle a}
12259:{\displaystyle i}
12244:
12243:
12215:
12154:
12072:
12029:
12028:
11963:
11846:
11795:{\displaystyle a}
11636:
11635:
11489:
11483:
11464:
11306:
11305:
11277:
11216:
11134:
11091:
11090:
11025:
10908:
10857:{\displaystyle a}
10837:{\displaystyle i}
10728:{\displaystyle a}
10633:
10632:
10533:
10510:
10473:
10453:
10452:
10329:
10287:
10279:
10273:
10261:
10169:
10168:
10143:
10085:{\displaystyle h}
10065:{\displaystyle b}
10045:{\displaystyle a}
9998:
9868:
9867:
9727:
9637:tree gravity code
9570:
9475:
9380:
9352:{\displaystyle n}
9324:
9258:
9102:
8804:
8792:
8732:
8647:
8646:
8437:
8312:
8299:
8286:
8222:
8200:
8092:
8034:
8023:
7914:
7882:
7864:
7724:
7657:
7576:
7565:
7552:
7525:
7511:
7466:
7453:
6876:{\displaystyle R}
6856:{\displaystyle L}
6836:{\displaystyle j}
6816:{\displaystyle i}
6796:{\displaystyle j}
6776:{\displaystyle i}
6532:
6527:
6380:
6350:
6312:{\displaystyle v}
6251:
6210:
6173:{\displaystyle M}
6053:{\displaystyle c}
5961:
5934:
5856:{\displaystyle p}
5841:equation of state
5810:Modelling physics
5729:
5619:
5557:
5550:
5497:Boundary Integral
5378:
4499:
4296:
4187:{\displaystyle i}
4086:
4054:
4002:
3994:
3940:
3862:
3826:
3779:
3771:
3569:
3533:
3481:
3455:
3401:
3393:
3324:
3288:
3255:
3118:
3084:
2939:{\displaystyle i}
2919:{\displaystyle j}
2899:{\displaystyle j}
2879:{\displaystyle i}
2714:
2638:
2589:
2557:
2505:
2389:
2373:
2307:
2291:
2146:{\displaystyle i}
2013:
2011:
1875:
1629:
1624:
1536:{\displaystyle h}
1496:and the Wendland
1490:Gaussian function
1478:{\displaystyle d}
1418:{\displaystyle h}
1164:
1110:{\displaystyle i}
1041:{\displaystyle j}
1021:{\displaystyle A}
974:{\displaystyle j}
923:{\displaystyle j}
819:
616:{\displaystyle W}
596:{\displaystyle A}
576:{\displaystyle W}
556:{\displaystyle h}
506:
355:William G. Hoover
119:Modelling Physics
81:Lagrangian method
15971:
15944:
15943:
15942:
15941:
15910:
15847:Analytic element
15830:Boundary element
15723:Schur complement
15704:Particle-in-cell
15639:Spectral element
15463:
15462:
15443:
15436:
15429:
15420:
15419:
15346:is a commercial
15183:Vesterlund, M.;
15106:
15105:
15079:
15059:
15053:
15052:
15016:
15010:
15009:
14997:
14991:
14990:
14989:
14987:
14974:
14968:
14967:
14939:
14933:
14932:
14904:
14898:
14897:
14869:
14863:
14862:
14854:
14848:
14847:
14821:
14812:
14806:
14805:
14785:
14779:
14778:
14750:
14744:
14743:
14737:
14729:
14701:
14695:
14694:
14688:
14680:
14652:
14646:
14645:
14639:
14631:
14620:10.1002/fld.2406
14595:
14589:
14588:
14582:
14574:
14554:
14548:
14547:
14539:
14533:
14532:
14504:
14495:
14494:
14466:
14455:
14454:
14444:
14438:
14437:
14431:
14423:
14412:10.1002/fld.3666
14383:
14372:
14371:
14343:
14337:
14336:
14308:
14302:
14301:
14299:
14282:(6): 1091β1121.
14267:
14261:
14260:
14232:
14221:
14220:
14184:
14175:
14174:
14154:
14148:
14147:
14122:(8): 1703β1759.
14111:
14105:
14104:
14076:
14070:
14069:
14051:
14033:
14013:
14007:
14006:
13978:
13972:
13971:
13961:
13950:10.1002/nme.1617
13927:
13918:
13912:
13911:
13901:
13890:10.1002/nme.4906
13865:
13859:
13858:
13830:
13824:
13823:
13787:
13781:
13780:
13778:
13746:
13740:
13739:
13722:(6): 1189β1214.
13711:
13705:
13704:
13687:(9): 1359β1400.
13676:
13670:
13669:
13641:
13635:
13634:
13606:
13600:
13599:
13579:
13573:
13572:
13544:
13538:
13537:
13517:
13511:
13510:
13482:
13476:
13475:
13465:
13437:
13431:
13430:
13402:
13396:
13395:
13367:
13361:
13360:
13334:
13328:
13327:
13309:
13285:
13279:
13278:
13271:
13265:
13264:
13246:
13244:astro-ph/0512532
13226:
13220:
13219:
13193:
13173:
13167:
13166:
13140:
13120:
13114:
13113:
13102:10.1002/fld.4261
13077:
13071:
13070:
13054:
13048:
13047:
13041:
13036:
13034:
13026:
13019:
13013:
13012:
12994:
12974:
12968:
12967:
12939:
12933:
12932:
12904:
12898:
12897:
12891:
12886:
12884:
12876:
12872:
12866:
12865:
12825:
12819:
12818:
12810:
12804:
12803:
12797:
12792:
12790:
12782:
12778:
12772:
12771:
12735:
12726:
12725:
12719:
12711:
12703:
12694:
12693:
12687:
12679:
12659:
12653:
12652:
12646:
12638:
12610:
12601:
12600:
12598:
12597:
12586:
12580:
12579:
12569:
12549:
12540:
12539:
12529:
12519:
12487:
12481:
12480:
12468:
12462:
12461:
12433:
12427:
12426:
12424:
12392:
12386:
12385:
12357:
12337:
12334:
12324:You can help by
12317:
12310:
12285:
12283:
12282:
12277:
12265:
12263:
12262:
12257:
12238:
12229:
12227:
12226:
12221:
12216:
12214:
12213:
12212:
12211:
12203:
12192:
12182:
12181:
12173:
12157:
12155:
12153:
12152:
12140:
12139:
12138:
12133:
12123:
12118:
12113:
12106:
12104:
12103:
12094:
12093:
12080:
12064:
12053:
12048:
12034:
12023:
12014:
12012:
12011:
12006:
11995:
11994:
11986:
11974:
11973:
11964:
11962:
11960:
11955:
11943:
11942:
11932:
11930:
11925:
11916:
11915:
11903:
11902:
11892:
11887:
11877:
11875:
11874:
11865:
11864:
11854:
11835:
11824:
11819:
11805:
11801:
11799:
11798:
11793:
11781:
11779:
11778:
11773:
11770:
11759:
11754:
11741:
11739:
11738:
11733:
11730:
11719:
11714:
11698:
11696:
11695:
11690:
11687:
11682:
11666:
11664:
11663:
11658:
11656:
11655:
11650:
11630:
11621:
11619:
11618:
11613:
11611:
11610:
11601:
11600:
11595:
11586:
11585:
11580:
11567:
11562:
11557:
11544:
11543:
11538:
11529:
11528:
11520:
11508:
11507:
11502:
11490:
11488:
11484:
11481:
11478:
11477:
11476:
11471:
11465:
11462:
11459:
11454:
11434:
11433:
11421:
11420:
11407:
11402:
11377:
11373:
11371:
11370:
11365:
11362:
11357:
11352:
11339:
11337:
11336:
11331:
11328:
11323:
11300:
11291:
11289:
11288:
11283:
11278:
11276:
11275:
11274:
11273:
11265:
11254:
11244:
11243:
11235:
11219:
11217:
11215:
11214:
11202:
11200:
11195:
11190:
11181:
11180:
11175:
11168:
11166:
11165:
11156:
11155:
11142:
11126:
11115:
11110:
11096:
11085:
11076:
11074:
11073:
11068:
11057:
11056:
11048:
11036:
11035:
11026:
11024:
11022:
11017:
11005:
11004:
10994:
10993:
10992:
10982:
10977:
10964:
10959:
10950:
10949:
10939:
10937:
10936:
10927:
10926:
10916:
10897:
10886:
10881:
10867:
10863:
10861:
10860:
10855:
10843:
10841:
10840:
10835:
10823:
10821:
10820:
10815:
10812:
10801:
10796:
10783:
10781:
10780:
10775:
10772:
10761:
10756:
10735:, respectively.
10734:
10732:
10731:
10726:
10714:
10712:
10711:
10706:
10703:
10692:
10687:
10674:
10672:
10671:
10666:
10663:
10652:
10647:
10627:
10618:
10616:
10615:
10610:
10605:
10601:
10599:
10594:
10589:
10583:
10582:
10577:
10567:
10562:
10557:
10551:
10550:
10545:
10534:
10526:
10521:
10520:
10512:
10511:
10506:
10501:
10489:
10485:
10483:
10482:
10477:
10475:
10474:
10469:
10464:
10447:
10438:
10436:
10435:
10430:
10427:
10416:
10411:
10401:
10390:
10385:
10376:
10368:
10367:
10354:
10349:
10340:
10339:
10331:
10330:
10325:
10320:
10316:
10315:
10306:
10305:
10295:
10280:
10278:
10274:
10271:
10268:
10267:
10262:
10259:
10256:
10254:
10253:
10236:
10229:
10227:
10226:
10221:
10219:
10207:
10205:
10204:
10199:
10194:
10163:
10154:
10152:
10151:
10146:
10144:
10136:
10133:
10128:
10116:
10115:
10098:
10091:
10089:
10088:
10083:
10071:
10069:
10068:
10063:
10051:
10049:
10048:
10043:
10029:
10027:
10026:
10021:
10018:
10013:
10005:
9999:
9997:
9996:
9990:
9985:
9977:
9971:
9962:
9961:
9957:
9950:
9944:
9939:
9931:
9925:
9908:
9903:
9902:
9892:
9887:
9862:
9853:
9851:
9850:
9845:
9843:
9842:
9834:
9830:
9829:
9828:
9815:
9810:
9798:
9794:
9792:
9787:
9782:
9772:
9767:
9762:
9750:
9745:
9735:
9716:
9711:
9706:
9692:
9688:
9686:
9685:
9680:
9678:
9677:
9672:
9600:
9598:
9597:
9592:
9587:
9586:
9578:
9571:
9569:
9567:
9562:
9550:
9549:
9540:
9539:
9531:
9521:
9520:
9519:
9495:
9494:
9486:
9479:
9476:
9474:
9473:
9464:
9463:
9454:
9443:
9442:
9430:
9405:Reynolds numbers
9394:
9392:
9391:
9386:
9381:
9373:
9358:
9356:
9355:
9350:
9335:
9333:
9332:
9327:
9325:
9317:
9272:
9270:
9269:
9264:
9259:
9257:
9255:
9250:
9238:
9237:
9228:
9227:
9219:
9209:
9208:
9207:
9199:
9190:
9189:
9181:
9174:
9160:
9159:
9116:
9114:
9113:
9108:
9103:
9101:
9099:
9094:
9082:
9081:
9072:
9071:
9063:
9053:
9052:
9051:
9043:
9034:
9033:
9025:
9015:
9010:
9009:
8987:
8985:
8984:
8979:
8977:
8976:
8957:
8955:
8954:
8949:
8937:
8935:
8934:
8929:
8914:
8912:
8911:
8906:
8904:
8903:
8891:
8890:
8882:
8873:
8872:
8864:
8842:
8841:
8833:
8824:
8823:
8815:
8806:
8803:
8802:
8794:
8793:
8785:
8781:
8779:
8774:
8756:
8755:
8743:
8742:
8734:
8733:
8725:
8714:
8700:
8699:
8642:
8639:
8629:You can help by
8622:
8615:
8606:
8604:
8603:
8598:
8580:
8578:
8577:
8572:
8570:
8569:
8557:
8556:
8540:
8538:
8537:
8532:
8520:
8518:
8517:
8512:
8510:
8509:
8497:
8496:
8480:
8478:
8477:
8472:
8458:
8456:
8455:
8450:
8445:
8444:
8438:
8430:
8416:
8415:
8403:
8402:
8384:
8383:
8357:
8355:
8354:
8349:
8344:
8340:
8339:
8327:
8326:
8313:
8305:
8300:
8292:
8287:
8279:
8274:
8273:
8251:
8249:
8248:
8243:
8238:
8237:
8229:
8223:
8215:
8210:
8209:
8201:
8196:
8191:
8182:
8181:
8173:
8167:
8166:
8154:
8153:
8148:
8133:
8131:
8130:
8125:
8120:
8119:
8107:
8106:
8097:
8093:
8091:
8090:
8089:
8080:
8079:
8069:
8068:
8059:
8053:
8052:
8042:
8024:
8022:
8014:
8013:
8012:
8007:
7997:
7987:
7985:
7984:
7979:
7974:
7973:
7961:
7960:
7945:
7944:
7939:
7930:
7929:
7924:
7915:
7913:
7912:
7903:
7902:
7893:
7890:
7881:
7880:
7865:
7863:
7855:
7854:
7853:
7840:
7829:
7827:
7826:
7821:
7819:
7818:
7802:
7800:
7799:
7794:
7792:
7791:
7775:
7773:
7772:
7767:
7762:
7754:
7753:
7741:
7740:
7725:
7717:
7708:
7706:
7705:
7700:
7695:
7687:
7686:
7674:
7673:
7658:
7650:
7639:
7637:
7636:
7631:
7629:
7628:
7619:
7615:
7614:
7602:
7601:
7588:
7587:
7578:
7577:
7569:
7566:
7558:
7553:
7545:
7540:
7539:
7526:
7524:
7523:
7522:
7513:
7512:
7504:
7500:
7496:
7495:
7483:
7482:
7469:
7467:
7459:
7454:
7446:
7441:
7440:
7414:
7412:
7411:
7406:
7404:
7403:
7390:
7385:
7372:
7367:
7351:
7349:
7348:
7343:
7341:
7340:
7327:
7322:
7309:
7304:
7288:
7286:
7285:
7280:
7274:
7269:
7256:
7251:
7238:
7233:
7214:
7212:
7211:
7206:
7200:
7195:
7182:
7177:
7164:
7159:
7137:
7135:
7134:
7129:
7127:
7126:
7114:
7113:
7101:
7100:
7092:
7083:
7082:
7077:
7068:
7067:
7049:
7048:
7036:
7035:
7023:
7022:
7003:
7002:
6990:
6989:
6981:
6972:
6971:
6966:
6957:
6956:
6938:
6937:
6925:
6924:
6912:
6911:
6882:
6880:
6879:
6874:
6862:
6860:
6859:
6854:
6842:
6840:
6839:
6834:
6822:
6820:
6819:
6814:
6802:
6800:
6799:
6794:
6782:
6780:
6779:
6774:
6762:
6760:
6759:
6754:
6752:
6751:
6739:
6734:
6733:
6725:
6713:
6712:
6704:
6648:
6646:
6645:
6640:
6638:
6624:
6623:
6618:
6617:
6607:
6606:
6588:
6584:
6583:
6582:
6577:
6568:
6567:
6562:
6551:
6550:
6540:
6528:
6526:
6518:
6517:
6516:
6503:
6454:
6452:
6451:
6446:
6444:
6440:
6439:
6438:
6418:
6417:
6392:
6390:
6389:
6384:
6382:
6381:
6378:
6362:
6360:
6359:
6354:
6352:
6351:
6348:
6318:
6316:
6315:
6310:
6298:
6296:
6295:
6290:
6275:
6273:
6272:
6267:
6265:
6264:
6252:
6250:
6249:
6240:
6239:
6238:
6233:
6227:
6222:
6216:
6211:
6209:
6208:
6199:
6191:
6179:
6177:
6176:
6171:
6159:
6157:
6156:
6151:
6149:
6145:
6136:
6135:
6116:
6114:
6113:
6108:
6106:
6105:
6089:
6087:
6086:
6081:
6059:
6057:
6056:
6051:
6039:
6037:
6036:
6031:
6029:
6028:
6009:
6007:
6006:
6001:
5996:
5995:
5983:
5979:
5972:
5971:
5966:
5962:
5960:
5959:
5947:
5935:
5930:
5929:
5928:
5919:
5918:
5908:
5882:
5880:
5879:
5874:
5862:
5860:
5859:
5854:
5801:
5787:
5785:
5784:
5779:
5774:
5773:
5761:
5760:
5750:
5749:
5748:
5725:
5724:
5707:
5705:
5704:
5699:
5694:
5690:
5689:
5688:
5676:
5675:
5670:
5664:
5663:
5654:
5653:
5643:
5642:
5641:
5615:
5614:
5599:
5598:
5589:
5588:
5578:
5577:
5576:
5551:
5549:
5548:
5536:
5531:
5530:
5484:
5449:
5447:
5446:
5441:
5436:
5435:
5420:
5419:
5410:
5409:
5399:
5398:
5397:
5374:
5373:
5348:
5346:
5345:
5340:
5335:
5327:
5326:
5325:
5303:
5302:
5301:
5288:
5274:
5270:
5269:
5268:
5251:
5250:
5246:
5229:
5192:Integral neglect
5184:
5182:
5181:
5176:
5171:
5170:
5169:
5147:
5146:
5145:
5132:
5118:
5114:
5113:
5112:
5095:
5094:
5090:
5073:
5054:
5053:
5052:
5030:
5029:
5028:
5015:
5001:
4997:
4996:
4995:
4978:
4977:
4973:
4951:
4927:
4919:
4911:
4901:
4895:
4884:
4882:
4881:
4876:
4871:
4870:
4869:
4847:
4842:
4841:
4840:
4827:
4822:
4811:
4807:
4806:
4805:
4788:
4787:
4783:
4766:
4747:
4746:
4745:
4723:
4718:
4717:
4716:
4703:
4698:
4687:
4683:
4682:
4681:
4664:
4663:
4659:
4637:
4616:
4572:
4570:
4569:
4564:
4562:
4561:
4556:
4543:
4541:
4540:
4535:
4517:
4515:
4514:
4509:
4507:
4500:
4495:
4487:
4484:
4473:
4468:
4458:
4454:
4439:
4434:
4420:
4409:
4404:
4384:
4380:
4365:
4360:
4350:
4345:
4340:
4326:
4315:
4310:
4297:
4292:
4284:
4281:
4276:
4271:
4261:
4256:
4251:
4237:
4233:
4218:
4213:
4193:
4191:
4190:
4185:
4161:Time integration
4156:
4154:
4153:
4148:
4146:
4145:
4129:
4127:
4126:
4121:
4116:
4108:
4107:
4092:
4088:
4087:
4084:
4079:
4070:
4069:
4060:
4055:
4052:
4047:
4038:
4037:
4028:
4021:
4020:
4010:
3995:
3993:
3989:
3983:
3982:
3981:
3976:
3970:
3964:
3952:
3950:
3949:
3944:
3942:
3939:
3938:
3937:
3932:
3922:
3921:
3920:
3907:
3894:
3892:
3891:
3886:
3881:
3876:
3875:
3863:
3861:
3860:
3859:
3854:
3844:
3843:
3842:
3829:
3827:
3824:
3819:
3810:
3809:
3800:
3798:
3797:
3787:
3772:
3770:
3766:
3760:
3759:
3758:
3753:
3747:
3741:
3739:
3738:
3719:
3717:
3716:
3711:
3706:
3701:
3697:
3696:
3695:
3686:
3667:
3655:
3653:
3652:
3647:
3645:
3644:
3628:
3626:
3625:
3620:
3618:
3617:
3598:
3596:
3595:
3590:
3588:
3583:
3582:
3570:
3568:
3567:
3566:
3561:
3551:
3550:
3549:
3536:
3534:
3532:
3531:
3530:
3517:
3516:
3515:
3502:
3500:
3499:
3489:
3474:
3469:
3468:
3456:
3454:
3453:
3452:
3447:
3437:
3436:
3435:
3422:
3420:
3419:
3409:
3394:
3392:
3388:
3382:
3381:
3380:
3375:
3369:
3363:
3361:
3360:
3338:
3336:
3335:
3330:
3325:
3323:
3322:
3321:
3316:
3306:
3305:
3304:
3294:
3289:
3287:
3286:
3285:
3280:
3270:
3269:
3268:
3258:
3256:
3254:
3250:
3244:
3239:
3219:
3217:
3216:
3211:
3209:
3208:
3188:
3186:
3185:
3180:
3178:
3174:
3173:
3172:
3167:
3158:
3150:
3149:
3136:
3131:
3126:
3120:
3111:
3103:
3102:
3092:
3080:
3079:
3047:
3045:
3044:
3039:
3037:
3036:
3023:
3021:
3020:
3015:
3000:
2998:
2997:
2992:
2990:
2989:
2968:
2967:
2945:
2943:
2942:
2937:
2925:
2923:
2922:
2917:
2905:
2903:
2902:
2897:
2885:
2883:
2882:
2877:
2858:
2856:
2855:
2850:
2845:
2810:
2789:
2787:
2786:
2781:
2776:
2772:
2771:
2766:
2753:
2752:
2743:
2742:
2733:
2732:
2722:
2707:
2703:
2702:
2686:
2685:
2680:
2679:
2669:
2668:
2663:
2657:
2656:
2646:
2624:
2622:
2621:
2616:
2611:
2610:
2595:
2591:
2590:
2587:
2582:
2573:
2572:
2563:
2558:
2555:
2550:
2541:
2540:
2531:
2524:
2523:
2513:
2504:
2503:
2491:
2487:
2486:
2470:
2469:
2464:
2463:
2443:
2441:
2440:
2435:
2433:
2425:
2421:
2420:
2404:
2403:
2398:
2397:
2390:
2382:
2374:
2372:
2364:
2363:
2362:
2357:
2347:
2332:
2330:
2329:
2324:
2322:
2308:
2300:
2292:
2290:
2282:
2281:
2272:
2253:
2251:
2250:
2245:
2243:
2239:
2238:
2233:
2216:
2214:
2213:
2208:
2206:
2202:
2201:
2196:
2179:
2177:
2176:
2171:
2169:
2168:
2152:
2150:
2149:
2144:
2133:at the particle
2132:
2130:
2129:
2124:
2122:
2101:
2099:
2098:
2093:
2088:
2087:
2069:
2065:
2064:
2063:
2058:
2049:
2048:
2043:
2032:
2031:
2021:
2012:
2010:
2009:
1997:
1989:
1985:
1984:
1979:
1966:
1965:
1946:
1944:
1943:
1938:
1936:
1909:
1907:
1906:
1901:
1896:
1876:
1874:
1866:
1858:
1834:
1832:
1831:
1826:
1824:
1823:
1818:
1806:with respect to
1805:
1803:
1802:
1797:
1795:
1794:
1775:
1773:
1772:
1767:
1765:
1764:
1743:
1742:
1717:
1715:
1714:
1709:
1704:
1703:
1685:
1681:
1680:
1679:
1674:
1665:
1664:
1659:
1648:
1647:
1637:
1625:
1623:
1615:
1614:
1613:
1600:
1588:
1586:
1585:
1580:
1578:
1577:
1542:
1540:
1539:
1534:
1522:
1520:
1519:
1514:
1512:
1511:
1484:
1482:
1481:
1476:
1464:
1462:
1461:
1456:
1453:
1449:
1440:
1424:
1422:
1421:
1416:
1404:
1402:
1401:
1396:
1385:
1380:
1379:
1374:
1365:
1364:
1359:
1353:
1335:
1333:
1332:
1327:
1325:
1324:
1309:
1308:
1289:
1287:
1286:
1281:
1279:
1278:
1262:
1260:
1259:
1254:
1252:
1251:
1242:
1241:
1229:
1228:
1209:
1207:
1206:
1201:
1196:
1195:
1183:
1182:
1172:
1157:
1156:
1151:
1136:
1135:
1116:
1114:
1113:
1108:
1096:
1094:
1093:
1088:
1086:
1085:
1069:
1067:
1066:
1061:
1059:
1047:
1045:
1044:
1039:
1027:
1025:
1024:
1019:
1007:
1005:
1004:
999:
997:
996:
980:
978:
977:
972:
956:
954:
953:
948:
946:
945:
929:
927:
926:
921:
906:
904:
903:
898:
884:
879:
878:
873:
864:
859:
848:
847:
838:
837:
827:
812:
791:
789:
788:
783:
781:
780:
761:
759:
758:
753:
748:
744:
743:
726:
711:
706:
705:
704:
691:
686:
675:
671:
670:
669:
643:
622:
620:
619:
614:
602:
600:
599:
594:
582:
580:
579:
574:
562:
560:
559:
554:
535:
533:
532:
527:
522:
521:
516:
507:
505:
501:
495:
494:
493:
488:
482:
476:
464:
462:
461:
456:
454:
453:
448:
435:
433:
432:
427:
425:
424:
419:
406:
404:
403:
398:
284:galaxy formation
219:fluid simulation
15979:
15978:
15974:
15973:
15972:
15970:
15969:
15968:
15949:
15948:
15947:
15939:
15937:
15911:
15907:
15903:
15898:
15867:Galerkin method
15810:Method of lines
15787:
15755:NeumannβNeumann
15709:
15666:
15608:
15575:High-resolution
15552:
15523:
15485:
15452:
15447:
15352:Lorenz Research
15235:
15201:
15114:
15112:Further reading
15109:
15060:
15056:
15017:
15013:
14998:
14994:
14985:
14983:
14975:
14971:
14940:
14936:
14905:
14901:
14870:
14866:
14855:
14851:
14819:
14813:
14809:
14786:
14782:
14751:
14747:
14731:
14730:
14702:
14698:
14682:
14681:
14653:
14649:
14633:
14632:
14596:
14592:
14576:
14575:
14555:
14551:
14540:
14536:
14505:
14498:
14467:
14458:
14445:
14441:
14425:
14424:
14384:
14375:
14344:
14340:
14309:
14305:
14268:
14264:
14233:
14224:
14185:
14178:
14171:
14155:
14151:
14112:
14108:
14087:(1β2): 97β115.
14077:
14073:
14049:10.1.1.770.4985
14014:
14010:
13979:
13975:
13925:
13919:
13915:
13876:(11): 781β797.
13866:
13862:
13831:
13827:
13812:10.1002/nme.617
13788:
13784:
13747:
13743:
13712:
13708:
13677:
13673:
13642:
13638:
13607:
13603:
13580:
13576:
13545:
13541:
13518:
13514:
13483:
13479:
13438:
13434:
13407:J. Comput. Phys
13403:
13399:
13372:J. Comput. Phys
13368:
13364:
13357:
13335:
13331:
13286:
13282:
13273:
13272:
13268:
13227:
13223:
13174:
13170:
13131:(4β6): 78β104.
13121:
13117:
13078:
13074:
13069:(4): 1457β1469.
13055:
13051:
13039:
13037:
13028:
13027:
13020:
13016:
12992:10.1.1.462.8293
12975:
12971:
12940:
12936:
12905:
12901:
12889:
12887:
12878:
12877:
12873:
12869:
12826:
12822:
12811:
12807:
12795:
12793:
12784:
12783:
12779:
12775:
12736:
12729:
12713:
12712:
12704:
12697:
12681:
12680:
12660:
12656:
12640:
12639:
12611:
12604:
12595:
12593:
12588:
12587:
12583:
12550:
12543:
12488:
12484:
12469:
12465:
12434:
12430:
12393:
12389:
12358:
12351:
12347:
12338:
12332:
12329:
12308:
12292:
12271:
12268:
12267:
12251:
12248:
12247:
12204:
12199:
12198:
12197:
12193:
12174:
12169:
12168:
12158:
12156:
12145:
12141:
12134:
12129:
12128:
12119:
12114:
12109:
12107:
12105:
12099:
12095:
12089:
12085:
12076:
12054:
12049:
12044:
12041:
12038:
12037:
11987:
11982:
11981:
11969:
11965:
11956:
11951:
11938:
11934:
11933:
11926:
11921:
11911:
11907:
11898:
11894:
11888:
11883:
11878:
11876:
11870:
11866:
11860:
11856:
11850:
11825:
11820:
11815:
11812:
11809:
11808:
11787:
11784:
11783:
11760:
11755:
11750:
11747:
11744:
11743:
11720:
11715:
11710:
11707:
11704:
11703:
11683:
11678:
11672:
11669:
11668:
11651:
11646:
11645:
11643:
11640:
11639:
11606:
11605:
11596:
11591:
11590:
11581:
11576:
11575:
11563:
11558:
11553:
11549:
11548:
11539:
11534:
11533:
11521:
11516:
11515:
11503:
11498:
11497:
11480:
11479:
11472:
11467:
11466:
11461:
11460:
11458:
11450:
11429:
11425:
11416:
11412:
11403:
11398:
11387:
11386:
11384:
11381:
11380:
11374:are defined by
11358:
11353:
11348:
11345:
11342:
11341:
11324:
11319:
11313:
11310:
11309:
11266:
11261:
11260:
11259:
11255:
11236:
11231:
11230:
11220:
11218:
11207:
11203:
11196:
11191:
11186:
11176:
11171:
11170:
11169:
11167:
11161:
11157:
11151:
11147:
11138:
11116:
11111:
11106:
11103:
11100:
11099:
11049:
11044:
11043:
11031:
11027:
11018:
11013:
11000:
10996:
10995:
10988:
10984:
10978:
10973:
10960:
10955:
10945:
10941:
10940:
10938:
10932:
10928:
10922:
10918:
10912:
10887:
10882:
10877:
10874:
10871:
10870:
10849:
10846:
10845:
10829:
10826:
10825:
10802:
10797:
10792:
10789:
10786:
10785:
10762:
10757:
10752:
10749:
10746:
10745:
10741:
10720:
10717:
10716:
10693:
10688:
10683:
10680:
10677:
10676:
10653:
10648:
10643:
10640:
10637:
10636:
10595:
10590:
10585:
10578:
10573:
10572:
10563:
10558:
10553:
10546:
10541:
10540:
10539:
10535:
10525:
10513:
10502:
10500:
10499:
10498:
10496:
10493:
10492:
10465:
10463:
10462:
10460:
10457:
10456:
10417:
10412:
10407:
10391:
10386:
10381:
10372:
10360:
10356:
10350:
10345:
10332:
10321:
10319:
10318:
10317:
10311:
10307:
10301:
10297:
10291:
10270:
10269:
10263:
10258:
10257:
10255:
10249:
10245:
10243:
10240:
10239:
10215:
10213:
10210:
10209:
10190:
10176:
10173:
10172:
10135:
10129:
10124:
10111:
10107:
10105:
10102:
10101:
10077:
10074:
10073:
10057:
10054:
10053:
10037:
10034:
10033:
10014:
10006:
10001:
9992:
9986:
9978:
9973:
9967:
9963:
9946:
9940:
9932:
9927:
9921:
9920:
9916:
9909:
9907:
9898:
9894:
9888:
9883:
9877:
9874:
9873:
9835:
9821:
9817:
9811:
9806:
9788:
9783:
9778:
9768:
9763:
9758:
9756:
9752:
9746:
9741:
9731:
9726:
9722:
9721:
9712:
9707:
9702:
9699:
9696:
9695:
9673:
9668:
9667:
9665:
9662:
9661:
9658:
9653:
9633:
9628:
9615:Surface tension
9612:
9579:
9574:
9573:
9563:
9558:
9545:
9541:
9532:
9527:
9526:
9522:
9506:
9502:
9487:
9482:
9481:
9480:
9478:
9469:
9465:
9459:
9455:
9453:
9435:
9431:
9426:
9415:
9412:
9411:
9372:
9364:
9361:
9360:
9344:
9341:
9340:
9316:
9284:
9281:
9280:
9251:
9246:
9233:
9229:
9220:
9215:
9214:
9210:
9200:
9195:
9194:
9182:
9177:
9176:
9175:
9173:
9152:
9148:
9146:
9143:
9142:
9125:
9095:
9090:
9077:
9073:
9064:
9059:
9058:
9054:
9044:
9039:
9038:
9026:
9021:
9020:
9016:
9014:
9002:
8998:
8996:
8993:
8992:
8988:is defined by
8969:
8965:
8963:
8960:
8959:
8943:
8940:
8939:
8923:
8920:
8919:
8899:
8898:
8883:
8878:
8877:
8865:
8860:
8859:
8856:
8850:
8849:
8834:
8829:
8828:
8816:
8811:
8810:
8807:
8795:
8784:
8783:
8782:
8775:
8767:
8748:
8744:
8735:
8724:
8723:
8722:
8715:
8712:
8705:
8704:
8692:
8688:
8686:
8683:
8682:
8652:
8643:
8637:
8634:
8613:
8586:
8583:
8582:
8565:
8561:
8552:
8548:
8546:
8543:
8542:
8526:
8523:
8522:
8505:
8501:
8492:
8488:
8486:
8483:
8482:
8466:
8463:
8462:
8440:
8439:
8429:
8411:
8407:
8398:
8394:
8379:
8378:
8367:
8364:
8363:
8335:
8331:
8322:
8318:
8314:
8304:
8291:
8278:
8269:
8265:
8263:
8260:
8259:
8230:
8225:
8224:
8214:
8202:
8192:
8190:
8189:
8174:
8169:
8168:
8162:
8158:
8149:
8144:
8143:
8141:
8138:
8137:
8112:
8108:
8102:
8098:
8085:
8081:
8075:
8071:
8070:
8064:
8060:
8058:
8054:
8048:
8044:
8038:
8015:
8008:
8003:
8002:
7998:
7996:
7994:
7991:
7990:
7966:
7962:
7956:
7952:
7940:
7935:
7934:
7925:
7920:
7919:
7908:
7904:
7898:
7894:
7892:
7886:
7876:
7872:
7856:
7849:
7845:
7841:
7839:
7837:
7834:
7833:
7814:
7810:
7808:
7805:
7804:
7787:
7783:
7781:
7778:
7777:
7758:
7749:
7745:
7736:
7732:
7716:
7714:
7711:
7710:
7691:
7682:
7678:
7669:
7665:
7649:
7647:
7644:
7643:
7624:
7623:
7610:
7606:
7597:
7593:
7589:
7583:
7579:
7568:
7567:
7557:
7544:
7535:
7531:
7528:
7527:
7518:
7514:
7503:
7502:
7501:
7491:
7487:
7478:
7474:
7470:
7468:
7458:
7445:
7436:
7432:
7425:
7424:
7422:
7419:
7418:
7399:
7395:
7386:
7381:
7368:
7363:
7357:
7354:
7353:
7336:
7332:
7323:
7318:
7305:
7300:
7294:
7291:
7290:
7270:
7265:
7252:
7247:
7234:
7229:
7220:
7217:
7216:
7196:
7191:
7178:
7173:
7160:
7155:
7146:
7143:
7142:
7122:
7121:
7109:
7105:
7093:
7088:
7087:
7078:
7073:
7072:
7063:
7059:
7044:
7040:
7031:
7027:
7018:
7014:
7008:
7007:
6998:
6994:
6982:
6977:
6976:
6967:
6962:
6961:
6952:
6948:
6933:
6929:
6920:
6916:
6907:
6903:
6893:
6892:
6890:
6887:
6886:
6868:
6865:
6864:
6848:
6845:
6844:
6828:
6825:
6824:
6808:
6805:
6804:
6788:
6785:
6784:
6768:
6765:
6764:
6744:
6740:
6735:
6726:
6721:
6720:
6705:
6700:
6699:
6697:
6694:
6693:
6682:
6619:
6613:
6612:
6611:
6599:
6595:
6578:
6573:
6572:
6563:
6558:
6557:
6556:
6552:
6546:
6542:
6536:
6519:
6512:
6508:
6504:
6502:
6500:
6498:
6495:
6494:
6488:
6479:
6434:
6430:
6423:
6419:
6413:
6409:
6401:
6398:
6397:
6377:
6373:
6371:
6368:
6367:
6347:
6343:
6332:
6329:
6328:
6304:
6301:
6300:
6284:
6281:
6280:
6260:
6256:
6245:
6241:
6234:
6229:
6228:
6223:
6218:
6217:
6215:
6204:
6200:
6192:
6190:
6188:
6185:
6184:
6165:
6162:
6161:
6141:
6137:
6131:
6127:
6125:
6122:
6121:
6101:
6097:
6095:
6092:
6091:
6069:
6066:
6065:
6045:
6042:
6041:
6024:
6020:
6018:
6015:
6014:
5991:
5987:
5967:
5955:
5951:
5946:
5942:
5941:
5940:
5936:
5924:
5920:
5914:
5910:
5909:
5907:
5899:
5896:
5895:
5868:
5865:
5864:
5848:
5845:
5844:
5837:Euler equations
5822:
5817:
5812:
5799:
5792:
5766:
5762:
5756:
5752:
5744:
5740:
5733:
5720:
5716:
5714:
5711:
5710:
5681:
5677:
5671:
5666:
5665:
5659:
5655:
5649:
5645:
5637:
5633:
5623:
5607:
5603:
5594:
5590:
5584:
5580:
5572:
5568:
5561:
5556:
5552:
5544:
5540:
5535:
5526:
5522:
5517:
5514:
5513:
5499:
5482:
5463:
5461:Fluid Extension
5428:
5424:
5415:
5411:
5405:
5401:
5393:
5389:
5382:
5369:
5365:
5360:
5357:
5356:
5331:
5321:
5317:
5316:
5297:
5293:
5292:
5284:
5264:
5260:
5259:
5255:
5242:
5225:
5218:
5214:
5212:
5209:
5208:
5194:
5165:
5161:
5160:
5141:
5137:
5136:
5128:
5108:
5104:
5103:
5099:
5086:
5069:
5062:
5058:
5048:
5044:
5043:
5024:
5020:
5019:
5011:
4991:
4987:
4986:
4982:
4969:
4962:
4958:
4947:
4936:
4933:
4932:
4921:
4913:
4903:
4897:
4889:
4865:
4861:
4860:
4843:
4836:
4832:
4831:
4823:
4818:
4801:
4797:
4796:
4792:
4779:
4762:
4755:
4751:
4741:
4737:
4736:
4719:
4712:
4708:
4707:
4699:
4694:
4677:
4673:
4672:
4668:
4655:
4648:
4644:
4633:
4625:
4622:
4621:
4608:
4597:
4557:
4552:
4551:
4549:
4546:
4545:
4526:
4523:
4522:
4505:
4504:
4488:
4486:
4474:
4469:
4464:
4450:
4440:
4435:
4430:
4422:
4410:
4405:
4400:
4396:
4395:
4376:
4366:
4361:
4356:
4346:
4341:
4336:
4328:
4316:
4311:
4306:
4302:
4301:
4285:
4283:
4277:
4272:
4267:
4257:
4252:
4247:
4239:
4229:
4219:
4214:
4209:
4204:
4202:
4199:
4198:
4179:
4176:
4175:
4163:
4141:
4140:
4138:
4135:
4134:
4112:
4100:
4096:
4080:
4075:
4065:
4061:
4059:
4048:
4043:
4033:
4029:
4027:
4026:
4022:
4016:
4012:
4006:
3985:
3984:
3977:
3972:
3971:
3966:
3965:
3963:
3961:
3958:
3957:
3933:
3928:
3927:
3923:
3916:
3912:
3908:
3905:
3903:
3900:
3899:
3877:
3871:
3867:
3855:
3850:
3849:
3845:
3838:
3834:
3830:
3828:
3820:
3815:
3805:
3801:
3799:
3793:
3789:
3783:
3762:
3761:
3754:
3749:
3748:
3743:
3742:
3740:
3734:
3730:
3728:
3725:
3724:
3702:
3691:
3687:
3682:
3678:
3674:
3663:
3661:
3658:
3657:
3640:
3636:
3634:
3631:
3630:
3613:
3609:
3607:
3604:
3603:
3584:
3578:
3574:
3562:
3557:
3556:
3552:
3545:
3541:
3537:
3535:
3526:
3522:
3518:
3511:
3507:
3503:
3501:
3495:
3491:
3485:
3470:
3464:
3460:
3448:
3443:
3442:
3438:
3431:
3427:
3423:
3421:
3415:
3411:
3405:
3384:
3383:
3376:
3371:
3370:
3365:
3364:
3362:
3356:
3352:
3350:
3347:
3346:
3317:
3312:
3311:
3307:
3300:
3299:
3295:
3293:
3281:
3276:
3275:
3271:
3264:
3263:
3259:
3257:
3246:
3245:
3240:
3238:
3236:
3233:
3232:
3222:internal energy
3204:
3200:
3198:
3195:
3194:
3168:
3163:
3162:
3154:
3145:
3141:
3132:
3127:
3122:
3109:
3108:
3104:
3098:
3094:
3088:
3075:
3074:
3072:
3069:
3068:
3054:
3032:
3031:
3029:
3026:
3025:
3009:
3006:
3005:
2982:
2978:
2960:
2956:
2951:
2948:
2947:
2931:
2928:
2927:
2911:
2908:
2907:
2891:
2888:
2887:
2871:
2868:
2867:
2841:
2806:
2801:
2798:
2797:
2767:
2762:
2761:
2757:
2748:
2744:
2738:
2734:
2728:
2724:
2718:
2698:
2694:
2690:
2681:
2675:
2674:
2673:
2664:
2659:
2658:
2652:
2648:
2642:
2636:
2633:
2632:
2603:
2599:
2583:
2578:
2568:
2564:
2562:
2551:
2546:
2536:
2532:
2530:
2529:
2525:
2519:
2515:
2509:
2499:
2495:
2482:
2478:
2474:
2465:
2459:
2458:
2457:
2455:
2452:
2451:
2429:
2416:
2412:
2408:
2399:
2393:
2392:
2391:
2381:
2365:
2358:
2353:
2352:
2348:
2346:
2344:
2341:
2340:
2318:
2299:
2283:
2277:
2273:
2271:
2269:
2266:
2265:
2234:
2229:
2228:
2224:
2222:
2219:
2218:
2217:and velocities
2197:
2192:
2191:
2187:
2185:
2182:
2181:
2164:
2160:
2158:
2155:
2154:
2138:
2135:
2134:
2118:
2110:
2107:
2106:
2080:
2076:
2059:
2054:
2053:
2044:
2039:
2038:
2037:
2033:
2027:
2023:
2017:
2005:
2001:
1996:
1980:
1975:
1974:
1970:
1961:
1957:
1955:
1952:
1951:
1932:
1918:
1915:
1914:
1892:
1867:
1859:
1857:
1855:
1852:
1851:
1819:
1814:
1813:
1811:
1808:
1807:
1787:
1783:
1781:
1778:
1777:
1757:
1753:
1735:
1731:
1726:
1723:
1722:
1696:
1692:
1675:
1670:
1669:
1660:
1655:
1654:
1653:
1649:
1643:
1639:
1633:
1616:
1609:
1605:
1601:
1599:
1597:
1594:
1593:
1573:
1569:
1567:
1564:
1563:
1560:
1528:
1525:
1524:
1507:
1503:
1501:
1498:
1497:
1470:
1467:
1466:
1445:
1441:
1436:
1430:
1427:
1426:
1410:
1407:
1406:
1381:
1375:
1370:
1369:
1360:
1355:
1354:
1349:
1341:
1338:
1337:
1317:
1313:
1301:
1297:
1295:
1292:
1291:
1274:
1270:
1268:
1265:
1264:
1247:
1243:
1237:
1233:
1224:
1220:
1218:
1215:
1214:
1188:
1184:
1178:
1174:
1168:
1152:
1147:
1146:
1131:
1127:
1125:
1122:
1121:
1102:
1099:
1098:
1081:
1077:
1075:
1072:
1071:
1055:
1053:
1050:
1049:
1033:
1030:
1029:
1013:
1010:
1009:
992:
988:
986:
983:
982:
966:
963:
962:
941:
937:
935:
932:
931:
915:
912:
911:
880:
874:
869:
868:
860:
855:
843:
839:
833:
829:
823:
808:
800:
797:
796:
776:
772:
770:
767:
766:
736:
735:
731:
722:
707:
700:
696:
695:
687:
682:
665:
661:
660:
656:
639:
631:
628:
627:
608:
605:
604:
588:
585:
584:
568:
565:
564:
548:
545:
544:
541:kernel function
517:
512:
511:
497:
496:
489:
484:
483:
478:
477:
475:
473:
470:
469:
449:
444:
443:
441:
438:
437:
420:
415:
414:
412:
409:
408:
386:
383:
382:
379:
374:
372:Numerical tools
323:
321:Solid mechanics
315:magnetic fields
282:Simulations of
269:
199:point splatting
174:
169:
146:
103:meshfree method
98:
93:
46:solid mechanics
17:
12:
11:
5:
15977:
15967:
15966:
15961:
15946:
15945:
15904:
15900:
15899:
15897:
15896:
15891:
15886:
15881:
15876:
15875:
15874:
15864:
15859:
15854:
15849:
15844:
15839:
15838:
15837:
15827:
15822:
15817:
15812:
15807:
15804:Pseudospectral
15801:
15795:
15793:
15789:
15788:
15786:
15785:
15780:
15774:
15768:
15762:
15757:
15752:
15747:
15746:
15745:
15740:
15730:
15725:
15719:
15717:
15711:
15710:
15708:
15707:
15701:
15695:
15689:
15683:
15676:
15674:
15668:
15667:
15665:
15664:
15658:
15653:
15647:
15642:
15636:
15630:
15624:
15618:
15616:
15614:Finite element
15610:
15609:
15607:
15606:
15600:
15594:
15592:Riemann solver
15589:
15583:
15577:
15572:
15566:
15564:
15558:
15557:
15554:
15553:
15551:
15550:
15544:
15538:
15531:
15529:
15525:
15524:
15522:
15521:
15516:
15511:
15506:
15501:
15499:LaxβFriedrichs
15495:
15493:
15487:
15486:
15484:
15483:
15481:CrankβNicolson
15478:
15471:
15469:
15460:
15454:
15453:
15446:
15445:
15438:
15431:
15423:
15417:
15416:
15406:
15400:
15394:
15388:
15382:
15375:
15370:
15365:
15355:
15341:
15335:
15329:
15323:
15317:
15307:
15301:
15295:
15289:
15283:
15271:
15265:
15259:
15253:
15250:dive solutions
15247:
15241:
15234:
15231:
15230:
15229:
15224:
15218:
15212:
15207:
15200:
15199:External links
15197:
15196:
15195:
15188:
15181:
15174:
15167:
15160:
15153:
15146:
15139:
15132:
15125:
15118:
15113:
15110:
15108:
15107:
15054:
15011:
14992:
14969:
14950:(1): 214β226.
14934:
14915:(2): 347β389.
14899:
14864:
14849:
14807:
14780:
14745:
14696:
14647:
14590:
14549:
14534:
14515:(2): 448β475.
14496:
14477:(6): 861β872.
14456:
14439:
14373:
14338:
14303:
14262:
14222:
14176:
14169:
14149:
14106:
14071:
14008:
13989:(4): 389β396.
13973:
13913:
13860:
13825:
13782:
13741:
13706:
13671:
13636:
13601:
13590:(4): 573β580.
13584:Comput. Struct
13574:
13539:
13512:
13477:
13448:(5): 785β819.
13432:
13413:(1): 123β134.
13397:
13362:
13355:
13329:
13300:(3): 759β794.
13280:
13266:
13253:10.1086/501493
13221:
13168:
13125:New Astron.rev
13115:
13088:(2): 115β131.
13072:
13049:
13040:|journal=
13014:
12969:
12934:
12899:
12890:|journal=
12867:
12828:Bodin (2011).
12820:
12805:
12796:|journal=
12773:
12727:
12695:
12654:
12602:
12581:
12541:
12482:
12463:
12458:10.1086/112164
12428:
12387:
12348:
12346:
12343:
12340:
12339:
12320:
12318:
12307:
12304:
12291:
12288:
12275:
12255:
12242:
12241:
12232:
12230:
12219:
12210:
12207:
12202:
12196:
12191:
12188:
12185:
12180:
12177:
12172:
12167:
12164:
12161:
12151:
12148:
12144:
12137:
12132:
12127:
12122:
12117:
12112:
12102:
12098:
12092:
12088:
12084:
12079:
12075:
12071:
12068:
12063:
12060:
12057:
12052:
12047:
12027:
12026:
12017:
12015:
12004:
12001:
11998:
11993:
11990:
11985:
11980:
11977:
11972:
11968:
11959:
11954:
11950:
11946:
11941:
11937:
11929:
11924:
11920:
11914:
11910:
11906:
11901:
11897:
11891:
11886:
11882:
11873:
11869:
11863:
11859:
11853:
11849:
11845:
11842:
11839:
11834:
11831:
11828:
11823:
11818:
11791:
11769:
11766:
11763:
11758:
11753:
11729:
11726:
11723:
11718:
11713:
11686:
11681:
11677:
11654:
11649:
11634:
11633:
11624:
11622:
11609:
11604:
11599:
11594:
11589:
11584:
11579:
11574:
11571:
11566:
11561:
11556:
11551:
11550:
11547:
11542:
11537:
11532:
11527:
11524:
11519:
11514:
11511:
11506:
11501:
11496:
11493:
11487:
11475:
11470:
11457:
11453:
11449:
11446:
11443:
11440:
11437:
11432:
11428:
11424:
11419:
11415:
11411:
11406:
11401:
11397:
11393:
11392:
11390:
11361:
11356:
11351:
11327:
11322:
11318:
11304:
11303:
11294:
11292:
11281:
11272:
11269:
11264:
11258:
11253:
11250:
11247:
11242:
11239:
11234:
11229:
11226:
11223:
11213:
11210:
11206:
11199:
11194:
11189:
11184:
11179:
11174:
11164:
11160:
11154:
11150:
11146:
11141:
11137:
11133:
11130:
11125:
11122:
11119:
11114:
11109:
11089:
11088:
11079:
11077:
11066:
11063:
11060:
11055:
11052:
11047:
11042:
11039:
11034:
11030:
11021:
11016:
11012:
11008:
11003:
10999:
10991:
10987:
10981:
10976:
10972:
10968:
10963:
10958:
10954:
10948:
10944:
10935:
10931:
10925:
10921:
10915:
10911:
10907:
10904:
10901:
10896:
10893:
10890:
10885:
10880:
10853:
10833:
10811:
10808:
10805:
10800:
10795:
10771:
10768:
10765:
10760:
10755:
10740:
10737:
10724:
10702:
10699:
10696:
10691:
10686:
10662:
10659:
10656:
10651:
10646:
10631:
10630:
10621:
10619:
10608:
10604:
10598:
10593:
10588:
10581:
10576:
10571:
10566:
10561:
10556:
10549:
10544:
10538:
10532:
10529:
10524:
10519:
10516:
10509:
10505:
10486:is defined as
10472:
10468:
10451:
10450:
10441:
10439:
10426:
10423:
10420:
10415:
10410:
10405:
10400:
10397:
10394:
10389:
10384:
10379:
10375:
10371:
10366:
10363:
10359:
10353:
10348:
10344:
10338:
10335:
10328:
10324:
10314:
10310:
10304:
10300:
10294:
10290:
10286:
10283:
10277:
10266:
10252:
10248:
10218:
10197:
10193:
10189:
10186:
10183:
10180:
10167:
10166:
10157:
10155:
10142:
10139:
10132:
10127:
10123:
10119:
10114:
10110:
10081:
10061:
10041:
10017:
10012:
10009:
10004:
9995:
9989:
9984:
9981:
9976:
9970:
9966:
9960:
9956:
9953:
9949:
9943:
9938:
9935:
9930:
9924:
9919:
9915:
9912:
9906:
9901:
9897:
9891:
9886:
9882:
9866:
9865:
9856:
9854:
9841:
9838:
9833:
9827:
9824:
9820:
9814:
9809:
9805:
9801:
9797:
9791:
9786:
9781:
9776:
9771:
9766:
9761:
9755:
9749:
9744:
9740:
9734:
9730:
9725:
9720:
9715:
9710:
9705:
9676:
9671:
9657:
9654:
9652:
9649:
9632:
9629:
9627:
9624:
9623:
9622:
9619:
9616:
9611:
9608:
9607:
9606:
9602:
9601:
9590:
9585:
9582:
9577:
9566:
9561:
9557:
9553:
9548:
9544:
9538:
9535:
9530:
9525:
9518:
9515:
9512:
9509:
9505:
9501:
9498:
9493:
9490:
9485:
9472:
9468:
9462:
9458:
9452:
9449:
9446:
9441:
9438:
9434:
9429:
9425:
9422:
9419:
9401:
9400:
9384:
9379:
9376:
9371:
9368:
9348:
9337:
9336:
9323:
9320:
9315:
9312:
9309:
9306:
9303:
9300:
9297:
9294:
9291:
9288:
9274:
9273:
9262:
9254:
9249:
9245:
9241:
9236:
9232:
9226:
9223:
9218:
9213:
9206:
9203:
9198:
9193:
9188:
9185:
9180:
9172:
9169:
9166:
9163:
9158:
9155:
9151:
9123:
9118:
9117:
9106:
9098:
9093:
9089:
9085:
9080:
9076:
9070:
9067:
9062:
9057:
9050:
9047:
9042:
9037:
9032:
9029:
9024:
9019:
9013:
9008:
9005:
9001:
8975:
8972:
8968:
8947:
8927:
8916:
8915:
8902:
8897:
8894:
8889:
8886:
8881:
8876:
8871:
8868:
8863:
8857:
8855:
8852:
8851:
8848:
8845:
8840:
8837:
8832:
8827:
8822:
8819:
8814:
8808:
8801:
8798:
8791:
8788:
8778:
8773:
8770:
8766:
8762:
8759:
8754:
8751:
8747:
8741:
8738:
8731:
8728:
8721:
8718:
8711:
8710:
8708:
8703:
8698:
8695:
8691:
8672:
8671:
8651:
8648:
8645:
8644:
8625:
8623:
8612:
8609:
8596:
8593:
8590:
8568:
8564:
8560:
8555:
8551:
8530:
8508:
8504:
8500:
8495:
8491:
8470:
8448:
8443:
8436:
8433:
8428:
8425:
8422:
8419:
8414:
8410:
8406:
8401:
8397:
8393:
8390:
8387:
8382:
8377:
8374:
8371:
8347:
8343:
8338:
8334:
8330:
8325:
8321:
8317:
8311:
8308:
8303:
8298:
8295:
8290:
8285:
8282:
8277:
8272:
8268:
8241:
8236:
8233:
8228:
8221:
8218:
8213:
8208:
8205:
8199:
8195:
8188:
8185:
8180:
8177:
8172:
8165:
8161:
8157:
8152:
8147:
8123:
8118:
8115:
8111:
8105:
8101:
8096:
8088:
8084:
8078:
8074:
8067:
8063:
8057:
8051:
8047:
8041:
8037:
8033:
8030:
8027:
8021:
8018:
8011:
8006:
8001:
7977:
7972:
7969:
7965:
7959:
7955:
7951:
7948:
7943:
7938:
7933:
7928:
7923:
7918:
7911:
7907:
7901:
7897:
7889:
7885:
7879:
7875:
7871:
7868:
7862:
7859:
7852:
7848:
7844:
7817:
7813:
7790:
7786:
7765:
7761:
7757:
7752:
7748:
7744:
7739:
7735:
7731:
7728:
7723:
7720:
7698:
7694:
7690:
7685:
7681:
7677:
7672:
7668:
7664:
7661:
7656:
7653:
7627:
7622:
7618:
7613:
7609:
7605:
7600:
7596:
7592:
7586:
7582:
7575:
7572:
7564:
7561:
7556:
7551:
7548:
7543:
7538:
7534:
7530:
7529:
7521:
7517:
7510:
7507:
7499:
7494:
7490:
7486:
7481:
7477:
7473:
7465:
7462:
7457:
7452:
7449:
7444:
7439:
7435:
7431:
7430:
7428:
7402:
7398:
7394:
7389:
7384:
7380:
7376:
7371:
7366:
7362:
7339:
7335:
7331:
7326:
7321:
7317:
7313:
7308:
7303:
7299:
7278:
7273:
7268:
7264:
7260:
7255:
7250:
7246:
7242:
7237:
7232:
7228:
7224:
7204:
7199:
7194:
7190:
7186:
7181:
7176:
7172:
7168:
7163:
7158:
7154:
7150:
7125:
7120:
7117:
7112:
7108:
7104:
7099:
7096:
7091:
7086:
7081:
7076:
7071:
7066:
7062:
7058:
7055:
7052:
7047:
7043:
7039:
7034:
7030:
7026:
7021:
7017:
7013:
7010:
7009:
7006:
7001:
6997:
6993:
6988:
6985:
6980:
6975:
6970:
6965:
6960:
6955:
6951:
6947:
6944:
6941:
6936:
6932:
6928:
6923:
6919:
6915:
6910:
6906:
6902:
6899:
6898:
6896:
6872:
6852:
6832:
6812:
6792:
6772:
6750:
6747:
6743:
6738:
6732:
6729:
6724:
6719:
6716:
6711:
6708:
6703:
6681:
6678:
6650:
6649:
6636:
6633:
6630:
6627:
6622:
6616:
6610:
6605:
6602:
6598:
6594:
6591:
6587:
6581:
6576:
6571:
6566:
6561:
6555:
6549:
6545:
6539:
6535:
6531:
6525:
6522:
6515:
6511:
6507:
6487:
6484:
6478:
6475:
6474:
6473:
6470:
6467:
6456:
6455:
6443:
6437:
6433:
6429:
6426:
6422:
6416:
6412:
6408:
6405:
6376:
6364:
6363:
6346:
6342:
6339:
6336:
6308:
6288:
6277:
6276:
6263:
6259:
6255:
6248:
6244:
6237:
6232:
6226:
6221:
6214:
6207:
6203:
6198:
6195:
6169:
6148:
6144:
6140:
6134:
6130:
6104:
6100:
6079:
6076:
6073:
6062:speed of sound
6049:
6027:
6023:
6011:
6010:
5999:
5994:
5990:
5986:
5982:
5978:
5975:
5970:
5965:
5958:
5954:
5950:
5945:
5939:
5933:
5927:
5923:
5917:
5913:
5906:
5903:
5872:
5852:
5821:
5818:
5816:
5813:
5811:
5808:
5797:
5789:
5788:
5777:
5772:
5769:
5765:
5759:
5755:
5747:
5743:
5739:
5736:
5732:
5728:
5723:
5719:
5708:
5697:
5693:
5687:
5684:
5680:
5674:
5669:
5662:
5658:
5652:
5648:
5640:
5636:
5632:
5629:
5626:
5622:
5618:
5613:
5610:
5606:
5602:
5597:
5593:
5587:
5583:
5575:
5571:
5567:
5564:
5560:
5555:
5547:
5543:
5539:
5534:
5529:
5525:
5521:
5498:
5495:
5462:
5459:
5451:
5450:
5439:
5434:
5431:
5427:
5423:
5418:
5414:
5408:
5404:
5396:
5392:
5388:
5385:
5381:
5377:
5372:
5368:
5364:
5350:
5349:
5338:
5334:
5330:
5324:
5320:
5315:
5312:
5309:
5306:
5300:
5296:
5291:
5287:
5283:
5280:
5277:
5273:
5267:
5263:
5258:
5254:
5249:
5245:
5241:
5238:
5235:
5232:
5228:
5224:
5221:
5217:
5193:
5190:
5186:
5185:
5174:
5168:
5164:
5159:
5156:
5153:
5150:
5144:
5140:
5135:
5131:
5127:
5124:
5121:
5117:
5111:
5107:
5102:
5098:
5093:
5089:
5085:
5082:
5079:
5076:
5072:
5068:
5065:
5061:
5057:
5051:
5047:
5042:
5039:
5036:
5033:
5027:
5023:
5018:
5014:
5010:
5007:
5004:
5000:
4994:
4990:
4985:
4981:
4976:
4972:
4968:
4965:
4961:
4957:
4954:
4950:
4946:
4943:
4940:
4902:, with radius
4886:
4885:
4874:
4868:
4864:
4859:
4856:
4853:
4850:
4846:
4839:
4835:
4830:
4826:
4821:
4817:
4814:
4810:
4804:
4800:
4795:
4791:
4786:
4782:
4778:
4775:
4772:
4769:
4765:
4761:
4758:
4754:
4750:
4744:
4740:
4735:
4732:
4729:
4726:
4722:
4715:
4711:
4706:
4702:
4697:
4693:
4690:
4686:
4680:
4676:
4671:
4667:
4662:
4658:
4654:
4651:
4647:
4643:
4640:
4636:
4632:
4629:
4596:
4593:
4560:
4555:
4533:
4530:
4519:
4518:
4503:
4498:
4494:
4491:
4483:
4480:
4477:
4472:
4467:
4462:
4457:
4453:
4449:
4446:
4443:
4438:
4433:
4428:
4425:
4423:
4419:
4416:
4413:
4408:
4403:
4398:
4397:
4394:
4391:
4388:
4383:
4379:
4375:
4372:
4369:
4364:
4359:
4354:
4349:
4344:
4339:
4334:
4331:
4329:
4325:
4322:
4319:
4314:
4309:
4304:
4303:
4300:
4295:
4291:
4288:
4280:
4275:
4270:
4265:
4260:
4255:
4250:
4245:
4242:
4240:
4236:
4232:
4228:
4225:
4222:
4217:
4212:
4207:
4206:
4183:
4162:
4159:
4144:
4131:
4130:
4119:
4115:
4111:
4106:
4103:
4099:
4095:
4091:
4083:
4078:
4074:
4068:
4064:
4058:
4051:
4046:
4042:
4036:
4032:
4025:
4019:
4015:
4009:
4005:
4001:
3998:
3992:
3988:
3980:
3975:
3969:
3936:
3931:
3926:
3919:
3915:
3911:
3896:
3895:
3884:
3880:
3874:
3870:
3866:
3858:
3853:
3848:
3841:
3837:
3833:
3823:
3818:
3814:
3808:
3804:
3796:
3792:
3786:
3782:
3778:
3775:
3769:
3765:
3757:
3752:
3746:
3737:
3733:
3709:
3705:
3700:
3694:
3690:
3685:
3681:
3677:
3673:
3670:
3666:
3643:
3639:
3616:
3612:
3600:
3599:
3587:
3581:
3577:
3573:
3565:
3560:
3555:
3548:
3544:
3540:
3529:
3525:
3521:
3514:
3510:
3506:
3498:
3494:
3488:
3484:
3480:
3477:
3473:
3467:
3463:
3459:
3451:
3446:
3441:
3434:
3430:
3426:
3418:
3414:
3408:
3404:
3400:
3397:
3391:
3387:
3379:
3374:
3368:
3359:
3355:
3340:
3339:
3328:
3320:
3315:
3310:
3303:
3298:
3292:
3284:
3279:
3274:
3267:
3262:
3253:
3249:
3243:
3207:
3203:
3191:
3190:
3177:
3171:
3166:
3161:
3157:
3153:
3148:
3144:
3140:
3135:
3130:
3125:
3117:
3114:
3107:
3101:
3097:
3091:
3087:
3083:
3078:
3053:
3050:
3035:
3013:
2988:
2985:
2981:
2977:
2974:
2971:
2966:
2963:
2959:
2955:
2935:
2915:
2895:
2875:
2860:
2859:
2848:
2844:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2809:
2805:
2791:
2790:
2779:
2775:
2770:
2765:
2760:
2756:
2751:
2747:
2741:
2737:
2731:
2727:
2721:
2717:
2713:
2710:
2706:
2701:
2697:
2693:
2689:
2684:
2678:
2672:
2667:
2662:
2655:
2651:
2645:
2641:
2626:
2625:
2614:
2609:
2606:
2602:
2598:
2594:
2586:
2581:
2577:
2571:
2567:
2561:
2554:
2549:
2545:
2539:
2535:
2528:
2522:
2518:
2512:
2508:
2502:
2498:
2494:
2490:
2485:
2481:
2477:
2473:
2468:
2462:
2447:One choice is
2445:
2444:
2432:
2428:
2424:
2419:
2415:
2411:
2407:
2402:
2396:
2388:
2385:
2380:
2377:
2371:
2368:
2361:
2356:
2351:
2334:
2333:
2321:
2317:
2314:
2311:
2306:
2303:
2298:
2295:
2289:
2286:
2280:
2276:
2259:Euler equation
2242:
2237:
2232:
2227:
2205:
2200:
2195:
2190:
2167:
2163:
2142:
2121:
2117:
2114:
2103:
2102:
2091:
2086:
2083:
2079:
2075:
2072:
2068:
2062:
2057:
2052:
2047:
2042:
2036:
2030:
2026:
2020:
2016:
2008:
2004:
2000:
1995:
1992:
1988:
1983:
1978:
1973:
1969:
1964:
1960:
1935:
1931:
1928:
1925:
1922:
1911:
1910:
1899:
1895:
1891:
1888:
1885:
1882:
1879:
1873:
1870:
1865:
1862:
1822:
1817:
1793:
1790:
1786:
1763:
1760:
1756:
1752:
1749:
1746:
1741:
1738:
1734:
1730:
1719:
1718:
1707:
1702:
1699:
1695:
1691:
1688:
1684:
1678:
1673:
1668:
1663:
1658:
1652:
1646:
1642:
1636:
1632:
1628:
1622:
1619:
1612:
1608:
1604:
1576:
1572:
1559:
1556:
1532:
1510:
1506:
1494:quintic spline
1474:
1452:
1448:
1444:
1439:
1435:
1414:
1394:
1391:
1388:
1384:
1378:
1373:
1368:
1363:
1358:
1352:
1348:
1345:
1323:
1320:
1316:
1312:
1307:
1304:
1300:
1277:
1273:
1250:
1246:
1240:
1236:
1232:
1227:
1223:
1211:
1210:
1199:
1194:
1191:
1187:
1181:
1177:
1171:
1167:
1163:
1160:
1155:
1150:
1145:
1142:
1139:
1134:
1130:
1106:
1084:
1080:
1058:
1037:
1017:
995:
991:
970:
944:
940:
919:
908:
907:
896:
893:
890:
887:
883:
877:
872:
867:
863:
858:
854:
851:
846:
842:
836:
832:
826:
822:
818:
815:
811:
807:
804:
779:
775:
763:
762:
751:
747:
742:
739:
734:
729:
725:
720:
717:
714:
710:
703:
699:
694:
690:
685:
681:
678:
674:
668:
664:
659:
655:
652:
649:
646:
642:
638:
635:
612:
592:
572:
552:
537:
536:
525:
520:
515:
510:
504:
500:
492:
487:
481:
452:
447:
423:
418:
396:
393:
390:
378:
377:Interpolations
375:
373:
370:
369:
368:
364:
361:
358:
351:
339:Ted Belytschko
322:
319:
288:star formation
268:
265:
264:
263:
260:
257:
254:
251:
248:
245:
242:
238:
235:
232:
229:
195:marching cubes
173:
172:Fluid dynamics
170:
168:
165:
164:
163:
151:
145:
142:
141:
140:
129:
122:
114:
113:
112:architectures.
106:
97:
94:
92:
89:
15:
9:
6:
4:
3:
2:
15976:
15965:
15962:
15960:
15957:
15956:
15954:
15936:
15932:
15928:
15924:
15920:
15916:
15909:
15905:
15895:
15892:
15890:
15887:
15885:
15882:
15880:
15877:
15873:
15870:
15869:
15868:
15865:
15863:
15860:
15858:
15855:
15853:
15850:
15848:
15845:
15843:
15840:
15836:
15833:
15832:
15831:
15828:
15826:
15823:
15821:
15818:
15816:
15813:
15811:
15808:
15805:
15802:
15800:
15797:
15796:
15794:
15790:
15784:
15781:
15778:
15775:
15772:
15769:
15766:
15763:
15761:
15758:
15756:
15753:
15751:
15748:
15744:
15741:
15739:
15736:
15735:
15734:
15731:
15729:
15726:
15724:
15721:
15720:
15718:
15716:
15712:
15705:
15702:
15699:
15696:
15693:
15690:
15687:
15684:
15681:
15678:
15677:
15675:
15673:
15669:
15662:
15659:
15657:
15654:
15651:
15648:
15646:
15643:
15640:
15637:
15634:
15631:
15628:
15625:
15623:
15620:
15619:
15617:
15615:
15611:
15604:
15601:
15598:
15595:
15593:
15590:
15587:
15584:
15581:
15578:
15576:
15573:
15571:
15568:
15567:
15565:
15563:
15562:Finite volume
15559:
15548:
15545:
15542:
15539:
15536:
15533:
15532:
15530:
15526:
15520:
15517:
15515:
15512:
15510:
15507:
15505:
15502:
15500:
15497:
15496:
15494:
15492:
15488:
15482:
15479:
15476:
15473:
15472:
15470:
15468:
15464:
15461:
15459:
15455:
15451:
15444:
15439:
15437:
15432:
15430:
15425:
15424:
15421:
15414:
15410:
15407:
15404:
15401:
15398:
15395:
15392:
15389:
15386:
15383:
15379:
15376:
15374:
15371:
15369:
15366:
15363:
15359:
15356:
15353:
15349:
15345:
15342:
15339:
15336:
15333:
15330:
15327:
15324:
15321:
15318:
15315:
15311:
15308:
15305:
15302:
15299:
15296:
15293:
15290:
15287:
15284:
15281:
15277:
15275:
15272:
15269:
15266:
15263:
15260:
15257:
15254:
15251:
15248:
15245:
15242:
15240:
15237:
15236:
15228:
15225:
15222:
15219:
15216:
15213:
15211:
15208:
15206:
15203:
15202:
15193:
15190:Violeau, D.;
15189:
15186:
15182:
15179:
15175:
15172:
15168:
15165:
15161:
15158:
15154:
15151:
15147:
15144:
15140:
15137:
15133:
15130:
15126:
15123:
15119:
15116:
15115:
15103:
15099:
15095:
15091:
15087:
15083:
15078:
15073:
15069:
15065:
15058:
15050:
15046:
15042:
15038:
15034:
15030:
15026:
15022:
15015:
15007:
15003:
14996:
14982:
14981:
14973:
14965:
14961:
14957:
14953:
14949:
14945:
14938:
14930:
14926:
14922:
14918:
14914:
14910:
14903:
14895:
14891:
14887:
14883:
14879:
14875:
14868:
14860:
14853:
14845:
14841:
14837:
14833:
14829:
14825:
14818:
14811:
14803:
14799:
14795:
14791:
14784:
14776:
14772:
14768:
14764:
14760:
14756:
14749:
14741:
14735:
14727:
14723:
14719:
14715:
14711:
14707:
14700:
14692:
14686:
14678:
14674:
14670:
14666:
14662:
14658:
14651:
14643:
14637:
14629:
14625:
14621:
14617:
14613:
14609:
14605:
14601:
14594:
14586:
14580:
14572:
14568:
14564:
14560:
14553:
14545:
14538:
14530:
14526:
14522:
14518:
14514:
14510:
14503:
14501:
14492:
14488:
14484:
14480:
14476:
14472:
14465:
14463:
14461:
14452:
14451:
14443:
14435:
14429:
14421:
14417:
14413:
14409:
14405:
14401:
14397:
14393:
14389:
14382:
14380:
14378:
14369:
14365:
14361:
14357:
14353:
14349:
14342:
14334:
14330:
14326:
14322:
14318:
14314:
14307:
14298:
14293:
14289:
14285:
14281:
14277:
14273:
14266:
14258:
14254:
14250:
14246:
14242:
14238:
14231:
14229:
14227:
14218:
14214:
14210:
14206:
14202:
14198:
14195:(5): 056701.
14194:
14190:
14183:
14181:
14172:
14166:
14162:
14161:
14153:
14145:
14141:
14137:
14133:
14129:
14125:
14121:
14117:
14110:
14102:
14098:
14094:
14090:
14086:
14082:
14075:
14067:
14063:
14059:
14055:
14050:
14045:
14041:
14037:
14032:
14027:
14023:
14019:
14012:
14004:
14000:
13996:
13992:
13988:
13984:
13977:
13969:
13965:
13960:
13955:
13951:
13947:
13943:
13939:
13935:
13931:
13924:
13917:
13909:
13905:
13900:
13895:
13891:
13887:
13883:
13879:
13875:
13871:
13864:
13856:
13852:
13848:
13844:
13840:
13836:
13829:
13821:
13817:
13813:
13809:
13805:
13801:
13797:
13793:
13786:
13777:
13772:
13768:
13764:
13760:
13756:
13752:
13745:
13737:
13733:
13729:
13725:
13721:
13717:
13710:
13702:
13698:
13694:
13690:
13686:
13682:
13675:
13667:
13663:
13659:
13655:
13651:
13647:
13640:
13632:
13628:
13624:
13620:
13616:
13612:
13605:
13597:
13593:
13589:
13585:
13578:
13570:
13566:
13562:
13558:
13554:
13550:
13543:
13535:
13531:
13527:
13523:
13516:
13508:
13504:
13500:
13496:
13492:
13488:
13481:
13473:
13469:
13464:
13463:10.1.1.28.491
13459:
13455:
13451:
13447:
13443:
13436:
13428:
13424:
13420:
13416:
13412:
13408:
13401:
13393:
13389:
13385:
13381:
13377:
13373:
13366:
13358:
13352:
13348:
13344:
13340:
13333:
13325:
13321:
13317:
13313:
13308:
13303:
13299:
13295:
13291:
13284:
13276:
13270:
13262:
13258:
13254:
13250:
13245:
13240:
13236:
13232:
13225:
13217:
13213:
13209:
13205:
13201:
13197:
13192:
13187:
13183:
13179:
13172:
13164:
13160:
13156:
13152:
13148:
13144:
13139:
13134:
13130:
13126:
13119:
13111:
13107:
13103:
13099:
13095:
13091:
13087:
13083:
13076:
13068:
13064:
13060:
13053:
13045:
13032:
13024:
13018:
13010:
13006:
13002:
12998:
12993:
12988:
12984:
12980:
12973:
12965:
12961:
12957:
12953:
12949:
12945:
12938:
12930:
12926:
12922:
12918:
12914:
12910:
12903:
12895:
12882:
12871:
12863:
12859:
12855:
12851:
12847:
12843:
12840:(3): 516β26.
12839:
12835:
12831:
12824:
12816:
12809:
12801:
12788:
12777:
12769:
12765:
12761:
12757:
12753:
12749:
12745:
12741:
12734:
12732:
12723:
12717:
12709:
12702:
12700:
12691:
12685:
12677:
12673:
12669:
12665:
12658:
12650:
12644:
12636:
12632:
12628:
12624:
12620:
12616:
12609:
12607:
12591:
12585:
12577:
12573:
12568:
12563:
12559:
12555:
12548:
12546:
12537:
12533:
12528:
12523:
12518:
12513:
12509:
12505:
12502:(6): e20685.
12501:
12497:
12493:
12486:
12478:
12474:
12467:
12459:
12455:
12451:
12447:
12444:: 1013β1024.
12443:
12439:
12432:
12423:
12418:
12414:
12410:
12407:(3): 375β89.
12406:
12402:
12398:
12391:
12383:
12379:
12375:
12371:
12367:
12363:
12356:
12354:
12349:
12336:
12327:
12323:
12319:
12316:
12312:
12311:
12303:
12301:
12297:
12287:
12273:
12253:
12240:
12233:
12231:
12217:
12208:
12205:
12200:
12186:
12183:
12178:
12175:
12162:
12149:
12146:
12142:
12135:
12125:
12120:
12115:
12100:
12096:
12090:
12086:
12082:
12077:
12073:
12069:
12066:
12061:
12058:
12055:
12050:
12036:
12035:
12032:
12025:
12018:
12016:
11999:
11996:
11991:
11988:
11975:
11970:
11957:
11952:
11948:
11944:
11939:
11935:
11927:
11922:
11918:
11912:
11908:
11904:
11899:
11895:
11889:
11884:
11880:
11871:
11867:
11861:
11857:
11851:
11847:
11843:
11840:
11837:
11832:
11829:
11826:
11821:
11807:
11806:
11803:
11802:are given by
11789:
11767:
11764:
11761:
11756:
11727:
11724:
11721:
11716:
11700:
11684:
11679:
11675:
11652:
11632:
11625:
11623:
11602:
11597:
11587:
11582:
11572:
11569:
11564:
11559:
11540:
11530:
11525:
11522:
11504:
11494:
11485:
11473:
11455:
11444:
11441:
11430:
11426:
11422:
11417:
11413:
11409:
11404:
11399:
11395:
11388:
11379:
11378:
11375:
11359:
11354:
11340:and velocity
11325:
11320:
11316:
11302:
11295:
11293:
11279:
11270:
11267:
11262:
11248:
11245:
11240:
11237:
11224:
11211:
11208:
11204:
11197:
11192:
11182:
11177:
11162:
11158:
11152:
11148:
11144:
11139:
11135:
11131:
11128:
11123:
11120:
11117:
11112:
11098:
11097:
11094:
11087:
11080:
11078:
11061:
11058:
11053:
11050:
11037:
11032:
11019:
11014:
11010:
11006:
11001:
10997:
10989:
10985:
10979:
10974:
10970:
10966:
10961:
10956:
10952:
10946:
10942:
10933:
10929:
10923:
10919:
10913:
10909:
10905:
10902:
10899:
10894:
10891:
10888:
10883:
10869:
10868:
10865:
10851:
10831:
10809:
10806:
10803:
10798:
10769:
10766:
10763:
10758:
10736:
10722:
10700:
10697:
10694:
10689:
10660:
10657:
10654:
10649:
10629:
10622:
10620:
10606:
10602:
10596:
10591:
10579:
10569:
10564:
10559:
10547:
10536:
10530:
10527:
10522:
10517:
10514:
10491:
10490:
10487:
10449:
10442:
10440:
10424:
10421:
10418:
10413:
10403:
10398:
10395:
10392:
10387:
10377:
10369:
10364:
10361:
10357:
10351:
10346:
10336:
10333:
10312:
10308:
10302:
10298:
10292:
10288:
10284:
10281:
10275:
10250:
10246:
10238:
10237:
10234:
10231:
10181:
10178:
10165:
10158:
10156:
10140:
10137:
10130:
10125:
10121:
10117:
10112:
10108:
10100:
10099:
10096:
10093:
10079:
10059:
10039:
10030:
10015:
10010:
10007:
9987:
9982:
9979:
9958:
9954:
9951:
9941:
9936:
9933:
9917:
9913:
9904:
9899:
9895:
9889:
9884:
9871:
9864:
9857:
9855:
9839:
9836:
9831:
9825:
9822:
9818:
9812:
9807:
9799:
9795:
9789:
9784:
9774:
9769:
9764:
9753:
9747:
9742:
9738:
9732:
9728:
9723:
9718:
9713:
9708:
9694:
9693:
9690:
9674:
9648:
9646:
9642:
9641:particle mesh
9638:
9620:
9618:Heat transfer
9617:
9614:
9613:
9604:
9603:
9588:
9583:
9580:
9564:
9559:
9555:
9551:
9546:
9536:
9533:
9516:
9513:
9510:
9507:
9503:
9496:
9491:
9488:
9470:
9466:
9460:
9456:
9450:
9447:
9444:
9439:
9436:
9420:
9410:
9409:
9408:
9406:
9398:
9397:
9396:
9382:
9377:
9374:
9369:
9366:
9346:
9321:
9318:
9310:
9307:
9304:
9298:
9295:
9292:
9289:
9286:
9279:
9278:
9277:
9260:
9252:
9247:
9243:
9239:
9234:
9224:
9221:
9204:
9201:
9191:
9186:
9183:
9170:
9167:
9164:
9161:
9156:
9153:
9141:
9140:
9139:
9136:
9134:
9130:
9126:
9104:
9096:
9091:
9087:
9083:
9078:
9068:
9065:
9048:
9045:
9035:
9030:
9027:
9017:
9011:
9006:
9003:
8999:
8991:
8990:
8989:
8973:
8970:
8966:
8945:
8925:
8895:
8892:
8887:
8884:
8874:
8869:
8866:
8853:
8846:
8843:
8838:
8835:
8825:
8820:
8817:
8799:
8796:
8786:
8776:
8771:
8768:
8764:
8760:
8757:
8752:
8749:
8745:
8739:
8736:
8726:
8719:
8716:
8706:
8701:
8696:
8693:
8681:
8680:
8679:
8677:
8669:
8668:
8667:
8665:
8661:
8657:
8641:
8632:
8628:
8624:
8621:
8617:
8616:
8608:
8594:
8591:
8588:
8566:
8562:
8558:
8553:
8549:
8528:
8506:
8502:
8498:
8493:
8489:
8468:
8459:
8446:
8431:
8426:
8420:
8417:
8412:
8408:
8404:
8399:
8395:
8385:
8372:
8369:
8361:
8358:
8345:
8336:
8332:
8328:
8323:
8319:
8306:
8301:
8296:
8293:
8288:
8280:
8275:
8270:
8266:
8257:
8253:
8234:
8231:
8216:
8211:
8206:
8203:
8183:
8178:
8175:
8163:
8159:
8155:
8150:
8134:
8121:
8116:
8113:
8109:
8103:
8094:
8086:
8082:
8076:
8072:
8065:
8061:
8055:
8049:
8045:
8039:
8035:
8031:
8028:
8025:
8019:
8016:
8009:
7999:
7988:
7975:
7970:
7967:
7963:
7957:
7949:
7941:
7931:
7926:
7909:
7905:
7899:
7895:
7887:
7883:
7877:
7873:
7869:
7866:
7860:
7857:
7850:
7846:
7842:
7831:
7815:
7811:
7788:
7784:
7763:
7759:
7750:
7746:
7742:
7737:
7733:
7726:
7718:
7696:
7692:
7683:
7679:
7675:
7670:
7666:
7659:
7651:
7640:
7620:
7611:
7607:
7603:
7598:
7594:
7584:
7580:
7570:
7562:
7559:
7554:
7546:
7541:
7536:
7532:
7519:
7515:
7505:
7492:
7488:
7484:
7479:
7475:
7463:
7460:
7455:
7447:
7442:
7437:
7433:
7426:
7416:
7400:
7396:
7392:
7387:
7382:
7378:
7374:
7369:
7364:
7360:
7337:
7333:
7329:
7324:
7319:
7315:
7311:
7306:
7301:
7297:
7271:
7266:
7262:
7258:
7253:
7248:
7244:
7240:
7235:
7230:
7226:
7197:
7192:
7188:
7184:
7179:
7174:
7170:
7166:
7161:
7156:
7152:
7138:
7118:
7110:
7106:
7102:
7097:
7094:
7084:
7079:
7069:
7064:
7060:
7053:
7045:
7041:
7037:
7032:
7028:
7024:
7019:
7015:
6999:
6995:
6991:
6986:
6983:
6973:
6968:
6958:
6953:
6949:
6942:
6934:
6930:
6926:
6921:
6917:
6913:
6908:
6904:
6894:
6884:
6870:
6850:
6830:
6810:
6790:
6770:
6748:
6745:
6741:
6736:
6730:
6727:
6717:
6714:
6709:
6706:
6686:
6677:
6674:
6666:
6658:
6654:
6634:
6628:
6620:
6608:
6603:
6600:
6596:
6589:
6585:
6579:
6569:
6564:
6553:
6547:
6543:
6537:
6533:
6529:
6523:
6520:
6513:
6509:
6505:
6493:
6492:
6491:
6483:
6471:
6468:
6465:
6464:
6463:
6460:
6441:
6435:
6431:
6427:
6424:
6420:
6414:
6410:
6406:
6403:
6396:
6395:
6394:
6374:
6344:
6340:
6337:
6334:
6327:
6326:
6325:
6323:
6306:
6286:
6261:
6257:
6253:
6246:
6242:
6235:
6212:
6205:
6201:
6196:
6193:
6183:
6182:
6181:
6167:
6142:
6132:
6128:
6118:
6102:
6098:
6077:
6074:
6071:
6064:. For water,
6063:
6047:
6025:
6021:
5997:
5992:
5988:
5984:
5980:
5976:
5973:
5968:
5963:
5956:
5952:
5948:
5943:
5937:
5931:
5925:
5921:
5915:
5911:
5904:
5901:
5894:
5893:
5892:
5890:
5889:Tait equation
5886:
5885:Cole equation
5870:
5850:
5842:
5838:
5834:
5829:
5827:
5826:interpolation
5815:Hydrodynamics
5807:
5805:
5800:
5796:
5775:
5770:
5767:
5763:
5757:
5753:
5745:
5737:
5734:
5730:
5726:
5721:
5717:
5709:
5695:
5691:
5685:
5682:
5678:
5672:
5660:
5656:
5650:
5646:
5638:
5627:
5624:
5620:
5616:
5611:
5608:
5604:
5595:
5591:
5585:
5581:
5573:
5565:
5562:
5558:
5553:
5545:
5541:
5537:
5532:
5527:
5523:
5512:
5511:
5510:
5503:
5494:
5490:
5486:
5480:
5475:
5467:
5458:
5454:
5437:
5432:
5429:
5425:
5416:
5412:
5406:
5402:
5394:
5386:
5383:
5379:
5375:
5370:
5366:
5355:
5354:
5353:
5336:
5328:
5313:
5307:
5304:
5289:
5278:
5271:
5256:
5252:
5233:
5219:
5215:
5207:
5206:
5205:
5198:
5189:
5172:
5157:
5151:
5148:
5133:
5122:
5115:
5100:
5096:
5077:
5063:
5059:
5055:
5040:
5034:
5031:
5016:
5005:
4998:
4983:
4979:
4959:
4955:
4941:
4931:
4930:
4929:
4925:
4917:
4910:
4906:
4900:
4893:
4872:
4857:
4851:
4848:
4828:
4812:
4808:
4793:
4789:
4770:
4756:
4752:
4748:
4733:
4727:
4724:
4704:
4688:
4684:
4669:
4665:
4645:
4641:
4627:
4620:
4619:
4618:
4615:
4611:
4601:
4592:
4590:
4585:
4583:
4578:
4574:
4558:
4531:
4501:
4496:
4492:
4481:
4478:
4475:
4470:
4460:
4455:
4451:
4447:
4444:
4441:
4436:
4426:
4424:
4417:
4414:
4411:
4406:
4392:
4389:
4381:
4377:
4373:
4370:
4367:
4362:
4352:
4347:
4342:
4332:
4330:
4323:
4320:
4317:
4312:
4298:
4293:
4289:
4278:
4273:
4263:
4258:
4253:
4243:
4241:
4234:
4230:
4226:
4223:
4220:
4215:
4197:
4196:
4195:
4181:
4173:
4169:
4158:
4117:
4109:
4104:
4101:
4097:
4089:
4081:
4076:
4072:
4066:
4062:
4056:
4049:
4044:
4040:
4034:
4030:
4023:
4017:
4013:
4007:
4003:
3999:
3996:
3990:
3978:
3956:
3955:
3954:
3934:
3917:
3913:
3882:
3872:
3868:
3864:
3856:
3839:
3835:
3821:
3816:
3812:
3806:
3802:
3794:
3790:
3784:
3780:
3776:
3773:
3767:
3755:
3735:
3731:
3723:
3722:
3721:
3720:we may write
3707:
3698:
3692:
3688:
3683:
3679:
3675:
3671:
3668:
3641:
3637:
3614:
3610:
3579:
3575:
3571:
3563:
3546:
3542:
3527:
3523:
3512:
3508:
3496:
3492:
3486:
3482:
3478:
3475:
3465:
3461:
3457:
3449:
3432:
3428:
3416:
3412:
3406:
3402:
3398:
3395:
3389:
3377:
3357:
3353:
3345:
3344:
3343:
3326:
3318:
3290:
3282:
3251:
3231:
3230:
3229:
3227:
3223:
3205:
3201:
3175:
3169:
3159:
3151:
3146:
3142:
3138:
3133:
3128:
3115:
3112:
3105:
3099:
3095:
3089:
3085:
3081:
3067:
3066:
3065:
3063:
3059:
3049:
3002:
2986:
2983:
2979:
2972:
2969:
2964:
2961:
2957:
2933:
2913:
2893:
2873:
2863:
2846:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2803:
2796:
2795:
2794:
2777:
2773:
2768:
2758:
2754:
2749:
2739:
2735:
2729:
2725:
2719:
2715:
2711:
2708:
2704:
2699:
2695:
2691:
2687:
2682:
2670:
2665:
2653:
2649:
2643:
2639:
2631:
2630:
2629:
2612:
2607:
2604:
2600:
2592:
2584:
2579:
2575:
2569:
2565:
2559:
2552:
2547:
2543:
2537:
2533:
2526:
2520:
2516:
2510:
2506:
2500:
2496:
2492:
2488:
2483:
2479:
2475:
2471:
2466:
2450:
2449:
2448:
2426:
2422:
2417:
2413:
2409:
2405:
2400:
2386:
2383:
2378:
2375:
2369:
2366:
2359:
2349:
2339:
2338:
2337:
2315:
2312:
2304:
2301:
2296:
2293:
2287:
2284:
2274:
2264:
2263:
2262:
2260:
2255:
2240:
2235:
2225:
2203:
2198:
2188:
2165:
2161:
2140:
2115:
2089:
2084:
2081:
2077:
2070:
2066:
2060:
2050:
2045:
2034:
2028:
2024:
2018:
2014:
2006:
2002:
1998:
1993:
1990:
1986:
1981:
1971:
1967:
1962:
1950:
1949:
1948:
1929:
1923:
1920:
1897:
1889:
1883:
1880:
1877:
1871:
1868:
1863:
1860:
1850:
1849:
1848:
1846:
1842:
1838:
1820:
1791:
1788:
1784:
1761:
1758:
1754:
1747:
1744:
1739:
1736:
1732:
1705:
1700:
1697:
1693:
1686:
1682:
1676:
1666:
1661:
1650:
1644:
1640:
1634:
1630:
1626:
1620:
1617:
1610:
1606:
1602:
1592:
1591:
1590:
1574:
1570:
1555:
1553:
1549:
1544:
1530:
1508:
1504:
1495:
1491:
1486:
1472:
1450:
1446:
1442:
1437:
1433:
1412:
1389:
1386:
1376:
1366:
1361:
1343:
1321:
1318:
1314:
1310:
1305:
1302:
1298:
1275:
1271:
1248:
1244:
1238:
1234:
1230:
1225:
1221:
1197:
1192:
1189:
1185:
1179:
1175:
1169:
1165:
1161:
1153:
1140:
1137:
1132:
1128:
1120:
1119:
1118:
1104:
1082:
1078:
1035:
1028:for particle
1015:
993:
989:
968:
960:
942:
938:
917:
894:
888:
885:
875:
865:
849:
844:
840:
834:
830:
824:
820:
816:
802:
795:
794:
793:
777:
773:
749:
745:
732:
727:
715:
712:
692:
676:
672:
657:
653:
650:
647:
633:
626:
625:
624:
610:
590:
570:
550:
542:
523:
518:
508:
502:
490:
468:
467:
466:
450:
421:
394:
391:
388:
365:
362:
359:
356:
352:
349:
348:
347:
343:
340:
335:
333:
327:
318:
316:
312:
308:
303:
301:
297:
293:
289:
285:
280:
278:
274:
261:
258:
255:
252:
249:
246:
243:
239:
236:
233:
230:
227:
226:
225:
222:
220:
216:
211:
207:
206:uniform grids
202:
200:
196:
192:
187:
178:
161:
157:
152:
148:
147:
138:
134:
130:
127:
123:
120:
116:
115:
111:
107:
104:
100:
99:
88:
86:
82:
79:
75:
71:
67:
63:
59:
55:
51:
47:
43:
39:
31:
23:
19:
15938:, retrieved
15918:
15908:
15686:Peridynamics
15679:
15504:LaxβWendroff
15413:CENTROID LAB
15332:Py-SPHViewer
15256:DualSPHysics
15191:
15184:
15177:
15170:
15163:
15156:
15149:
15142:
15135:
15128:
15067:
15063:
15057:
15027:(1): 62β69.
15024:
15020:
15014:
15005:
15001:
14995:
14984:, retrieved
14979:
14972:
14947:
14943:
14937:
14912:
14908:
14902:
14877:
14873:
14867:
14858:
14852:
14827:
14823:
14810:
14793:
14789:
14783:
14758:
14754:
14748:
14734:cite journal
14709:
14705:
14699:
14685:cite journal
14660:
14656:
14650:
14636:cite journal
14603:
14599:
14593:
14579:cite journal
14562:
14558:
14552:
14543:
14537:
14512:
14508:
14474:
14470:
14449:
14442:
14428:cite journal
14395:
14391:
14354:(1): 49β65.
14351:
14347:
14341:
14316:
14312:
14306:
14279:
14275:
14265:
14240:
14236:
14192:
14188:
14163:. Springer.
14159:
14152:
14119:
14115:
14109:
14084:
14080:
14074:
14021:
14017:
14011:
13986:
13982:
13976:
13933:
13929:
13916:
13899:2324/4476056
13873:
13869:
13863:
13838:
13834:
13828:
13795:
13791:
13785:
13761:(2): 78β85.
13758:
13754:
13744:
13719:
13715:
13709:
13684:
13680:
13674:
13649:
13645:
13639:
13614:
13610:
13604:
13587:
13583:
13577:
13552:
13548:
13542:
13525:
13521:
13515:
13490:
13486:
13480:
13445:
13441:
13435:
13410:
13406:
13400:
13378:(1): 67β75.
13375:
13371:
13365:
13338:
13332:
13297:
13293:
13283:
13269:
13234:
13231:Astrophys. J
13230:
13224:
13181:
13177:
13171:
13128:
13124:
13118:
13085:
13081:
13075:
13066:
13062:
13052:
13031:cite journal
13017:
12982:
12978:
12972:
12947:
12943:
12937:
12912:
12908:
12902:
12881:cite journal
12870:
12837:
12833:
12823:
12814:
12808:
12787:cite journal
12776:
12743:
12739:
12716:cite journal
12707:
12684:cite journal
12667:
12663:
12657:
12643:cite journal
12618:
12614:
12594:. Retrieved
12592:. 2018-03-21
12584:
12557:
12553:
12499:
12495:
12485:
12476:
12472:
12466:
12441:
12437:
12431:
12404:
12400:
12390:
12365:
12361:
12330:
12326:adding to it
12321:
12293:
12245:
12234:
12030:
12019:
11701:
11637:
11626:
11307:
11296:
11092:
11081:
10742:
10634:
10623:
10454:
10443:
10232:
10170:
10159:
10094:
10031:
9872:
9869:
9858:
9659:
9634:
9631:Astrophysics
9402:
9338:
9275:
9137:
9132:
9128:
9121:
9119:
8917:
8673:
8653:
8638:October 2022
8635:
8631:adding to it
8626:
8460:
8362:
8359:
8258:
8254:
8135:
7989:
7832:
7641:
7417:
7139:
6885:
6783:to particle
6691:
6675:
6671:
6651:
6489:
6480:
6461:
6457:
6365:
6278:
6119:
6012:
5863:and density
5830:
5823:
5803:
5794:
5793:
5790:
5508:
5491:
5487:
5476:
5472:
5455:
5452:
5351:
5203:
5187:
4923:
4915:
4908:
4904:
4898:
4891:
4887:
4613:
4609:
4606:
4586:
4579:
4575:
4520:
4164:
4132:
3897:
3601:
3341:
3192:
3055:
3003:
2926:by particle
2886:by particle
2864:
2861:
2792:
2627:
2446:
2335:
2256:
2180:, positions
2104:
1912:
1720:
1561:
1545:
1487:
1212:
1097:of particle
961:of particle
909:
764:
538:
380:
344:
336:
328:
324:
304:
281:
270:
267:Astrophysics
223:
203:
186:fluid motion
183:
160:aerodynamics
74:oceanography
62:astrophysics
41:
37:
36:
18:
15820:Collocation
15350:product by
14986:February 1,
14880:: 605β620.
14319:: 771β791.
14243:: 112β129.
13237:: 292β305.
12950:(4): 1β12.
12368:: 345β363.
8676:Mach number
6883:states are
6322:Mach number
3629:depends on
144:Limitations
135:(e.g., the
70:volcanology
15953:Categories
15940:2022-03-30
15509:MacCormack
15491:Hyperbolic
15262:FLUIDS v.1
15244:AQUAgpusph
15077:1911.13255
15070:: 110028.
15008:: 181β198.
13959:10379/1170
12985:(6): 1β8.
12915:(4): 1β8.
12746:: 109565.
12596:2018-08-30
12345:References
9621:Turbulence
9131:(e.g. 0.01
8461:Note that
3062:Lagrangian
1841:Lagrangian
296:supernovae
96:Advantages
76:. It is a
66:ballistics
15825:Level-set
15815:Multigrid
15765:Balancing
15467:Parabolic
15409:NDYNAMICS
15378:SPHinXsys
15358:SimPARTIX
15102:208513116
15094:0021-9991
15049:125369012
14844:121493014
14628:121381641
14420:124465492
14044:CiteSeerX
14031:1304.3692
13908:121935150
13820:119799557
13458:CiteSeerX
13324:0021-9991
13307:1012.1885
13216:119119783
13191:1406.4224
13138:0903.5075
13110:124425780
12987:CiteSeerX
12768:219438340
12670:: 11β34.
12567:1111.1259
12438:Astron. J
12382:128143912
12333:July 2018
12195:∂
12160:∂
12126:−
12083:η
12074:∑
11967:∇
11949:ρ
11936:ρ
11919:ρ
11896:ρ
11848:∑
11841:−
11676:ρ
11588:−
11531:⋅
11495:⋅
11456:−
11427:ρ
11257:∂
11222:∂
11183:−
11145:η
11136:∑
11029:∇
11011:ρ
10998:ρ
10986:ρ
10953:ρ
10910:∑
10903:−
10508:~
10471:~
10343:∇
10327:~
10289:∑
10122:ρ
10109:ρ
9965:∂
9911:∂
9881:∇
9837:−
9804:∇
9800:⊗
9775:−
9729:∑
9556:η
9543:‖
9524:‖
9500:∇
9497:⋅
9467:ρ
9451:ν
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10383:f
10378:+
10374:g
10370:+
10365:b
10362:a
10358:W
10352:0
10347:a
10337:b
10334:a
10323:P
10313:b
10309:V
10303:a
10299:V
10293:b
10285:2
10282:=
10276:t
10272:d
10265:v
10260:d
10251:a
10247:m
10217:F
10196:)
10192:F
10188:(
10182:=
10179:J
10164:)
10162:2
10160:(
10141:J
10138:1
10131:0
10126:a
10118:=
10113:a
10080:h
10060:b
10040:a
10016:0
10011:b
10008:a
10003:e
9994:|
9988:0
9983:b
9980:a
9975:r
9969:|
9959:)
9955:h
9952:,
9948:|
9942:0
9937:b
9934:a
9929:r
9923:|
9918:(
9914:W
9905:=
9900:a
9896:W
9890:0
9885:a
9863:)
9861:1
9859:(
9840:1
9832:)
9826:b
9823:a
9819:W
9813:0
9808:a
9796:)
9790:0
9785:a
9780:r
9770:0
9765:b
9760:r
9754:(
9748:0
9743:b
9739:V
9733:b
9724:(
9719:=
9714:0
9709:a
9704:B
9675:0
9670:B
9589:.
9584:j
9581:i
9576:v
9565:2
9560:h
9552:+
9547:2
9537:j
9534:i
9529:r
9517:j
9514:i
9511:,
9508:h
9504:w
9492:j
9489:i
9484:r
9471:j
9461:j
9457:m
9448:2
9445:=
9440:j
9437:i
9433:]
9428:v
9418:[
9378:3
9375:5
9370:=
9347:n
9314:)
9311:2
9308:+
9305:n
9302:(
9299:2
9296:=
9293:c
9290:h
9261:.
9253:2
9248:h
9240:+
9235:2
9225:j
9222:i
9217:r
9205:j
9202:i
9197:r
9187:j
9184:i
9179:v
9171:c
9168:h
9162:=
9157:j
9154:i
9133:h
9129:h
9124:h
9122:Ξ·
9105:,
9097:2
9092:h
9084:+
9079:2
9069:j
9066:i
9061:r
9049:j
9046:i
9041:r
9031:j
9028:i
9023:v
9018:h
9012:=
9007:j
9004:i
8974:j
8971:i
8896:0
8888:j
8885:i
8880:r
8870:j
8867:i
8862:v
8854:0
8847:0
8839:j
8836:i
8831:r
8821:j
8818:i
8813:v
8800:j
8797:i
8777:2
8772:j
8769:i
8758:+
8753:j
8750:i
8740:j
8737:i
8727:c
8707:{
8702:=
8697:j
8694:i
8640:)
8636:(
8595:3
8592:=
8567:R
8563:U
8554:L
8550:U
8507:R
8503:U
8494:L
8490:U
8447:.
8442:)
8432:c
8427:,
8424:)
8421:0
8418:,
8413:R
8409:U
8400:L
8396:U
8392:(
8381:(
8373:=
8346:,
8342:)
8337:R
8333:U
8324:L
8320:U
8316:(
8297:2
8294:1
8289:+
8281:P
8276:=
8267:P
8240:)
8235:j
8232:i
8227:e
8217:U
8207:j
8204:i
8194:v
8187:(
8184:+
8179:j
8176:i
8171:e
8160:U
8156:=
8146:v
8122:.
8117:j
8114:i
8110:W
8104:i
8095:)
8087:j
8077:i
8062:P
8056:(
8050:j
8046:m
8040:j
8032:2
8026:=
8020:t
8017:d
8010:i
8005:v
8000:d
7976:,
7971:j
7968:i
7964:W
7958:i
7947:)
7937:v
7927:i
7922:v
7917:(
7910:j
7900:j
7896:m
7888:j
7878:i
7870:2
7867:=
7861:t
7858:d
7851:i
7843:d
7812:P
7785:U
7764:2
7760:/
7756:)
7751:R
7747:P
7743:+
7738:L
7734:P
7730:(
7727:=
7719:P
7697:2
7693:/
7689:)
7684:R
7680:U
7676:+
7671:L
7667:U
7663:(
7660:=
7652:U
7621:,
7617:)
7612:R
7608:U
7599:L
7595:U
7591:(
7585:0
7581:c
7563:2
7560:1
7555:+
7547:P
7542:=
7533:P
7520:0
7516:c
7498:)
7493:R
7489:P
7480:L
7476:P
7472:(
7464:2
7461:1
7456:+
7448:U
7443:=
7434:U
7427:{
7397:P
7393:=
7383:R
7379:P
7375:=
7365:L
7361:P
7334:U
7330:=
7320:R
7316:U
7312:=
7302:L
7298:U
7277:)
7267:R
7263:P
7259:,
7249:R
7245:U
7241:,
7231:R
7223:(
7203:)
7193:L
7189:P
7185:,
7175:L
7171:U
7167:,
7157:L
7149:(
7119:.
7116:)
7111:j
7107:P
7103:,
7098:j
7095:i
7090:e
7080:j
7075:v
7070:,
7065:j
7057:(
7054:=
7051:)
7046:R
7042:P
7038:,
7033:R
7029:U
7025:,
7020:R
7012:(
7005:)
7000:i
6996:P
6992:,
6987:j
6984:i
6979:e
6969:i
6964:v
6959:,
6954:i
6946:(
6943:=
6940:)
6935:L
6931:P
6927:,
6922:L
6918:U
6914:,
6909:L
6901:(
6895:{
6871:R
6851:L
6831:j
6811:i
6791:j
6771:i
6749:j
6746:i
6742:r
6737:/
6731:j
6728:i
6723:r
6715:=
6710:j
6707:i
6702:e
6635:,
6632:)
6626:(
6621:i
6615:D
6609:+
6604:j
6601:i
6597:W
6586:)
6580:j
6575:v
6565:i
6560:v
6554:(
6548:j
6544:m
6538:j
6530:=
6524:t
6521:d
6514:i
6506:d
6442:)
6436:0
6421:(
6415:2
6411:c
6407:=
6404:p
6375:v
6345:v
6338:=
6335:c
6307:v
6262:2
6258:M
6254:=
6247:2
6243:c
6236:2
6231:|
6225:v
6220:|
6206:0
6168:M
6147:s
6143:/
6139:m
6133:3
6103:0
6099:p
6078:7
6075:=
6048:c
6026:0
5998:,
5993:0
5989:p
5985:+
5981:)
5977:1
5964:)
5957:0
5944:(
5938:(
5926:2
5922:c
5916:0
5905:=
5902:p
5851:p
5804:j
5798:j
5795:n
5776:,
5771:j
5768:i
5764:W
5758:j
5754:V
5746:i
5735:j
5727:=
5722:i
5696:,
5692:)
5686:j
5683:i
5679:W
5673:j
5668:n
5661:j
5657:A
5651:j
5647:S
5639:i
5625:j
5617:+
5612:j
5609:i
5605:W
5596:j
5592:A
5586:j
5582:V
5574:i
5563:j
5554:(
5546:i
5538:1
5533:=
5528:i
5524:A
5483:A
5438:.
5433:j
5430:i
5426:W
5417:j
5413:A
5407:j
5403:V
5395:i
5384:j
5376:=
5371:i
5367:A
5337:,
5333:0
5319:r
5314:d
5311:)
5308:h
5305:,
5295:r
5286:r
5282:(
5279:W
5272:)
5262:r
5257:(
5253:A
5248:)
5244:r
5240:(
5231:)
5227:r
5223:(
5220:B
5173:.
5163:r
5158:d
5155:)
5152:h
5149:,
5139:r
5130:r
5126:(
5123:W
5116:)
5106:r
5101:(
5097:A
5092:)
5088:r
5084:(
5075:)
5071:r
5067:(
5064:B
5056:+
5046:r
5041:d
5038:)
5035:h
5032:,
5022:r
5013:r
5009:(
5006:W
4999:)
4989:r
4984:(
4980:A
4975:)
4971:r
4967:(
4956:=
4953:)
4949:r
4945:(
4942:A
4926:)
4924:r
4918:)
4916:r
4909:h
4905:s
4899:r
4894:)
4892:r
4873:,
4863:r
4858:d
4855:)
4852:h
4849:,
4845:|
4834:r
4825:r
4820:|
4816:(
4813:W
4809:)
4799:r
4794:(
4790:A
4785:)
4781:r
4777:(
4768:)
4764:r
4760:(
4757:B
4749:+
4739:r
4734:d
4731:)
4728:h
4725:,
4721:|
4710:r
4701:r
4696:|
4692:(
4689:W
4685:)
4675:r
4670:(
4666:A
4661:)
4657:r
4653:(
4642:=
4639:)
4635:r
4631:(
4628:A
4614:h
4610:s
4559:i
4554:a
4532:t
4502:,
4497:2
4493:t
4482:1
4479:+
4476:i
4471:i
4466:a
4461:+
4456:2
4452:/
4448:1
4445:+
4442:n
4437:i
4432:v
4427:=
4418:1
4415:+
4412:n
4407:i
4402:v
4393:,
4390:t
4382:2
4378:/
4374:1
4371:+
4368:i
4363:i
4358:v
4353:+
4348:n
4343:i
4338:r
4333:=
4324:1
4321:+
4318:n
4313:i
4308:r
4299:,
4294:2
4290:t
4279:n
4274:i
4269:a
4264:+
4259:n
4254:i
4249:v
4244:=
4235:2
4231:/
4227:1
4224:+
4221:n
4216:i
4211:v
4182:i
4143:G
4118:,
4114:g
4110:+
4105:j
4102:i
4098:W
4090:)
4082:2
4077:j
4067:j
4063:p
4057:+
4050:2
4045:i
4035:i
4031:p
4024:(
4018:j
4014:m
4008:j
3997:=
3991:t
3987:d
3979:i
3974:v
3968:d
3935:i
3930:r
3918:j
3883:,
3879:g
3873:i
3869:m
3865:+
3857:i
3852:r
3840:j
3822:2
3817:j
3807:j
3803:p
3795:j
3791:m
3785:j
3774:=
3768:t
3764:d
3756:i
3751:v
3745:d
3736:i
3732:m
3704:d
3699:)
3693:2
3684:/
3680:p
3676:(
3672:=
3669:e
3665:d
3642:j
3615:j
3611:e
3586:g
3580:i
3576:m
3572:+
3564:i
3559:r
3547:j
3528:j
3513:j
3509:e
3497:j
3493:m
3487:j
3476:=
3472:g
3466:i
3462:m
3458:+
3450:i
3445:r
3433:j
3429:e
3417:j
3413:m
3407:j
3396:=
3390:t
3386:d
3378:i
3373:v
3367:d
3358:i
3354:m
3327:.
3319:i
3314:r
3302:L
3291:=
3283:i
3278:v
3266:L
3252:t
3248:d
3242:d
3206:j
3202:e
3189:,
3176:)
3170:j
3165:r
3156:g
3152:+
3147:j
3143:e
3134:2
3129:j
3124:v
3116:2
3113:1
3106:(
3100:j
3096:m
3090:j
3082:=
3077:L
3034:G
3012:D
2987:i
2984:j
2980:W
2970:=
2965:j
2962:i
2958:W
2934:i
2914:j
2894:j
2874:i
2847:.
2843:v
2833:p
2824:=
2821:p
2808:v
2778:,
2774:}
2769:j
2764:v
2759:{
2750:i
2746:D
2740:i
2736:p
2730:i
2726:V
2720:i
2709:=
2705:}
2700:j
2696:p
2692:{
2683:i
2677:G
2666:i
2661:v
2654:i
2650:V
2644:i
2613:,
2608:j
2605:i
2601:W
2593:)
2585:2
2580:j
2570:j
2566:p
2560:+
2553:2
2548:i
2538:i
2534:p
2527:(
2521:j
2517:m
2511:j
2501:i
2493:=
2489:}
2484:j
2480:p
2476:{
2467:i
2461:G
2431:g
2427:+
2423:}
2418:j
2414:p
2410:{
2401:i
2395:G
2384:1
2376:=
2370:t
2367:d
2360:i
2355:v
2350:d
2320:g
2316:+
2313:p
2302:1
2294:=
2288:t
2285:d
2279:v
2275:d
2241:}
2236:j
2231:v
2226:{
2204:}
2199:j
2194:r
2189:{
2166:j
2162:m
2141:i
2120:v
2090:.
2085:j
2082:i
2078:W
2067:)
2061:j
2056:v
2046:i
2041:v
2035:(
2029:j
2025:m
2019:j
2007:i
1999:1
1991:=
1987:}
1982:j
1977:v
1972:{
1963:i
1959:D
1934:v
1898:,
1894:v
1878:=
1872:t
1869:d
1861:d
1821:i
1816:r
1792:j
1789:i
1785:W
1762:i
1759:j
1755:W
1745:=
1740:j
1737:i
1733:W
1706:,
1701:j
1698:i
1694:W
1683:)
1677:j
1672:v
1662:i
1657:v
1651:(
1645:j
1641:m
1635:j
1627:=
1621:t
1618:d
1611:i
1603:d
1575:i
1531:h
1509:2
1505:C
1473:d
1451:d
1447:/
1443:1
1438:j
1434:V
1413:h
1393:)
1390:h
1387:,
1383:|
1377:j
1372:r
1362:i
1357:r
1351:|
1347:(
1344:W
1322:i
1319:j
1315:W
1311:=
1306:j
1303:i
1299:W
1276:j
1249:j
1245:V
1239:j
1231:=
1226:j
1222:m
1198:,
1193:j
1190:i
1186:W
1180:j
1176:m
1170:j
1162:=
1159:)
1154:i
1149:r
1144:(
1138:=
1133:i
1105:i
1083:i
1057:r
1036:j
1016:A
994:j
990:A
969:j
943:j
939:V
918:j
895:,
892:)
889:h
886:,
882:|
876:j
871:r
862:r
857:|
853:(
850:W
845:j
841:A
835:j
831:V
825:j
817:=
814:)
810:r
806:(
803:A
778:2
774:h
750:.
746:)
738:r
733:(
728:V
724:d
719:)
716:h
713:,
709:|
698:r
689:r
684:|
680:(
677:W
673:)
663:r
658:(
654:A
648:=
645:)
641:r
637:(
634:A
611:W
591:A
571:W
551:h
524:.
519:i
514:v
509:=
503:t
499:d
491:i
486:r
480:d
451:i
446:v
422:i
417:r
395:j
392:,
389:i
357:.
121:.
40:(
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