1771:
W2 models can be made soft enough (in uniform fashion) to produce upper bound solutions (for force-driving problems). Together with stiff models (such as the fully compatible FEM models), one can conveniently bound the solution from both sides. This allows easy error estimation for generally complicated problems, as long as a triangular mesh can be generated. Typical W2 models are the
Smoothed Point Interpolation Methods (or S-PIM). The S-PIM can be node-based (known as NS-PIM or LC-PIM), edge-based (ES-PIM), and cell-based (CS-PIM). The NS-PIM was developed using the so-called SCNI technique. It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free. The ES-PIM is found superior in accuracy, and CS-PIM behaves in between the NS-PIM and ES-PIM. Moreover, W2 formulations allow the use of polynomial and radial basis functions in the creation of shape functions (it accommodates the discontinuous displacement functions, as long as it is in G1 space), which opens further rooms for future developments. The W2 formulation has also led to the development of combination of meshfree techniques with the well-developed FEM techniques, and one can now use triangular mesh with excellent accuracy and desired softness. A typical such a formulation is the so-called smoothed finite element method (or S-FEM). The S-FEM is the linear version of S-PIM, but with most of the properties of the S-PIM and much simpler.
1754:, which necessitates a mesh to generate quadrature points and weights). Nodal integration however, suffers from numerical instability due to underestimation of strain energy associated with short-wavelength modes, and also yields inaccurate and non-convergent results due to under-integration of the weak form. One major advance in numerical integration has been the development of a stabilized conforming nodal integration (SCNI) which provides a nodal integration method which does not suffer from either of these problems. The method is based on strain-smoothing which satisfies the first order
101:), the connectivity of the mesh can be difficult to maintain without introducing error into the simulation. If the mesh becomes tangled or degenerate during simulation, the operators defined on it may no longer give correct values. The mesh may be recreated during simulation (a process called remeshing), but this can also introduce error, since all the existing data points must be mapped onto a new and different set of data points. Meshfree methods are intended to remedy these problems. Meshfree methods are also useful for:
22:
806:, respectively. Generally in finite differences one can allow very simply for steps variable along the mesh, but all the original nodes should be preserved and they can move independently only by deforming the original elements. If even only two of all the nodes change their order, or even only one node is added to or removed from the simulation, that creates a defect in the original mesh and the simple finite difference approximation can no longer hold.
45:, but are rather based on interaction of each node with all its neighbors. As a consequence, original extensive properties such as mass or kinetic energy are no longer assigned to mesh elements but rather to the single nodes. Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort. The absence of a mesh allows
1758:. However, it was later realized that low-energy modes were still present in SCNI, and additional stabilization methods have been developed. This method has been applied to a variety of problems including thin and thick plates, poromechanics, convection-dominated problems, among others. More recently, a framework has been developed to pass arbitrary-order patch tests, based on a
1778:
The S-PIM and S-FEM works well for solid mechanics problems. For CFD problems, the formulation can be simpler, via strong formulation. A Gradient
Smoothing Methods (GSM) has also been developed recently for CFD problems, implementing the gradient smoothing idea in strong form. The GSM is similar to
1770:
theory. The W2 formulation offers possibilities to formulate various (uniformly) "soft" models that work well with triangular meshes. Because a triangular mesh can be generated automatically, it becomes much easier in re-meshing and hence enables automation in modeling and simulation. In addition,
1782:
Nodal integration has been proposed as a technique to use finite elements to emulate a meshfree behaviour. However, the obstacle that must be overcome in using nodally integrated elements is that the quantities at nodal points are not continuous, and the nodes are shared among multiple elements.
1409:
pioneered the
Element Free Galerkin (EFG) method, which employed MLS with Lagrange multipliers to enforce boundary conditions, higher order numerical quadrature in the weak form, and full derivatives of the MLS approximation which gave better accuracy. Around the same time, the
1265:
1423:. RKPM and other meshfree methods were extensively developed by Chen, Liu, and Li in the late 1990s for a variety of applications and various classes of problems. During the 1990s and thereafter several other varieties were developed including those listed below.
1414:(RKPM) emerged, the approximation motivated in part to correct the kernel estimate in SPH: to give accuracy near boundaries, in non-uniform discretizations, and higher-order accuracy in general. Notably, in a parallel development, the
1774:
It is a general perception that meshfree methods are much more expensive than the FEM counterparts. The recent study has found however, some meshfree methods such as the S-PIM and S-FEM can be much faster than the FEM counterparts.
1004:
532:
626:
73:
were originally defined on meshes of data points. In such a mesh, each point has a fixed number of predefined neighbors, and this connectivity between neighbors can be used to define mathematical operators like the
1735:, where they can be imposed directly. Techniques have been developed to overcome this difficulty and impose conditions strongly. Several methods have been developed to impose the essential boundary conditions
1378:), it would be a waste of effort to calculate the summations above over every particle in a large simulation. So typically SPH simulators require some extra code to speed up this nearest neighbor calculation.
1765:
One recent advance in meshfree methods aims at the development of computational tools for automation in modeling and simulations. This is enabled by the so-called weakened weak (W2) formulation based on the
1542:
1418:
were developed around the same time which offer similar capabilities. Material point methods are widely used in the movie industry to simulate large deformation solid mechanics, such as snow in the movie
1351:
and its derivatives do not depend on any adjacency information about the particles; they can use the particles in any order, so it doesn't matter if the particles move around or even exchange places.
1129:
3466:
Gross, B. J.; Trask, N.; Kuberry, P.; Atzberger, P. J. (15 May 2020). "Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach".
3054:
Zhang, Jian; Liu, G.R.; Lam, K.Y.; Li, Hua; Xu, G. (November 2008). "A gradient smoothing method (GSM) based on strong form governing equation for adaptive analysis of solid mechanics problems".
1630:
1642:
1524:
1590:
1597:
427:
365:
2858:
Liu, G. R. (2009). "A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part II applications to solid mechanics problems".
812:(SPH), one of the oldest meshfree methods, solves this problem by treating data points as physical particles with mass and density that can move around over time, and carry some value
2105:
Liu, W. K.; Chen, Y.; Jun, S.; Chen, J. S.; Belytschko, T.; Pan, C.; Uras, R. A.; Chang, C. T. (March 1996). "Overview and applications of the reproducing Kernel
Particle methods".
1723:
The primary areas of advancement in meshfree methods are to address issues with essential boundary enforcement, numerical quadrature, and contact and large deformations. The common
3437:
Sousa, Washington; de
Oliveira, Rodrigo (April 2015). "Coulomb's Law Discretization Method: a New Methodology of Spatial Discretization for the Radial Point Interpolation Method".
1494:
1560:
1394:
were the first to apply SPH in solid mechanics. The main drawbacks of SPH are inaccurate results near boundaries and tension instability that was first investigated by Swegle.
1714:
Meshfree
Interface-Finite Element Method (MIFEM) (2015) - a hybrid finite element-meshfree method for numerical simulation of phase transformation and multiphase flow problems
1636:
2942:
Liu, G. R.; Zhang, G. Y. (14 May 2008). "Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM)".
1692:
1612:
303:
264:
1081:
196:
1624:
1566:
1123:
is a kernel function that operates on nearby data points and is chosen for smoothness and other useful qualities. By linearity, we can write the spatial derivative as
804:
771:
1349:
1303:
872:
664:
164:
1034:
837:
738:
223:
3373:
Netuzhylov, Hennadiy; Zilian, Andreas (15 October 2009). "Space-time meshfree collocation method: Methodology and application to initial-boundary value problems".
1572:
1548:
1453:
46:
2598:
Boroomand, B.; Soghrati, S.; Movahedian, B. (2009). "Exponential basis functions in solution of static and time harmonic elastic problems in a meshless style".
2210:
Chen, Shang-Ying; Hsu, Kuo-Chin; Fan, Chia-Ming (15 March 2021). "Improvement of generalized finite difference method for stochastic subsurface flow modeling".
1372:
1121:
1101:
1054:
711:
691:
1536:
1530:
1310:
3615:
1584:
1888:
Libersky, Larry D.; Petschek, Albert G.; Carney, Theodore C.; Hipp, Jim R.; Allahdadi, Firooz A. (November 1993). "High Strain
Lagrangian Hydrodynamics".
2892:
Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li GY and Han X, A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems,
880:
2915:
Liu, G. R.; Zhang, G. Y. (20 November 2011). "A normed G space and weakened weak (W2) formulation of a cell-based smoothed point interpolation method".
1431:
The following numerical methods are generally considered to fall within the general class of "meshfree" methods. Acronyms are provided in parentheses.
2633:
Ghoneim, A. (March 2015). "A meshfree interface-finite element method for modelling isothermal solutal melting and solidification in binary systems".
666:
and its spatial and temporal derivatives to write the equation being simulated in finite difference form, then simulate the equation with one of many
438:
3193:
Arroyo, M.; Ortiz, M. (26 March 2006). "Localmaximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods".
2820:
Liu, G. R. (2009). "A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory".
1750:, nodal integration is generally preferred which offers simplicity, efficiency, and keeps the meshfree method free of any mesh (as opposed to using
543:
1313:. In physical terms, this means calculating the forces between the particles, then integrating these forces over time to determine their motion.
2893:
2327:
A. Behzadan; H. M. Shodja; M. Khezri (2011). "A unified approach to the mathematical analysis of generalized RKPM, gradient RKPM, and GMLS".
2774:; Hillman, Michael; RĂĽter, Marcus (3 August 2013). "An arbitrary order variationally consistent integration for Galerkin meshfree methods".
432:
We can define the derivatives that occur in the equation being simulated using some finite difference formulae on this domain, for example
2905:
G.R. Liu, G.R. Zhang. Edge-based
Smoothed Point Interpolation Methods. International Journal of Computational Methods, 5(4): 621–646, 2008
2736:; Wu, Cheng-Tang; Yoon, Sangpil; You, Yang (20 January 2001). "A stabilized conforming nodal integration for Galerkin mesh-free methods".
1354:
One disadvantage of SPH is that it requires extra programming to determine the nearest neighbors of a particle. Since the kernel function
1477:
1374:
only returns nonzero results for nearby particles within twice the "smoothing length" (because we typically choose kernel functions with
2693:
Belytschko, Ted; Guo, Yong; Kam Liu, Wing; Ping Xiao, Shao (30 July 2000). "A unified stability analysis of meshless particle methods".
2418:
Li, B.; Habbal, F.; Ortiz, M. (17 September 2010). "Optimal transportation meshfree approximation schemes for fluid and plastic flows".
1405:
approximation in the
Galerkin solution of partial differential equations, with approximate derivatives of the MLS function. Thereafter
4085:
3706:
1802:
1767:
3656:
1950:
Nayroles, B.; Touzot, G.; Villon, P. (1992). "Generalizing the finite element method: Diffuse approximation and diffuse elements".
1500:
3366:
1260:{\displaystyle {\partial u \over \partial x}=\sum _{i}m_{i}{\frac {u_{i}^{n}}{\rho _{i}}}{\partial W(|x-x_{i}|) \over \partial x}}
3632:
3606:
3327:
2714:
3515:"First-passage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)"
1618:
2757:
4090:
3930:
3700:
3362:
2998:
2140:
Atluri, S. N.; Zhu, T. (24 August 1998). "A new
Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics".
4000:
3857:
3712:
1512:
1459:
1411:
3353:
Netuzhylov, H. (2008), "A Space-Time Meshfree Collocation Method for Coupled Problems on Irregularly-Shaped Domains",
3293:
3278:
3264:
3248:
3183:
3164:
2555:
Zhang, Xiong; Liu, Xiao-Hu; Song, Kang-Zu; Lu, Ming-Wan (30 July 2001). "Least-squares collocation meshless method".
1779:, but uses gradient smoothing operations exclusively in nested fashions, and is a general numerical method for PDEs.
1680:
3908:
2183:
Oliveira, T.; Portela, A. (December 2016). "Weak-form collocation – A local meshless method in linear elasticity".
79:
3925:
115:
Simulations where the problem geometry may move out of alignment with a fixed mesh, such as in bending simulations
4059:
3845:
1554:
1435:
1387:
809:
371:
309:
3826:
3815:
3792:
3011:
Liu, G. R.; Xu, George X. (10 December 2008). "A gradient smoothing method (GSM) for fluid dynamics problems".
1915:
Swegle, J.W.; Hicks, D.L.; Attaway, S.W. (January 1995). "Smoothed Particle Hydrodynamics Stability Analysis".
1797:
1686:
1676:
2255:"Data assimilation for real-time subsurface flow modeling with dynamically adaptive meshless node adjustments"
3798:
2520:
Ooi, E.H.; Popov, V. (May 2012). "An efficient implementation of the radial basis integral equation method".
1606:
1506:
1447:
1375:
3915:
3880:
1755:
1731:
property. This make essential boundary condition enforcement non-trivial, at least more difficult than the
1727:
requires strong enforcement of the essential boundary conditions, yet meshfree methods in general lack the
90:
4080:
3920:
3599:
83:
1401:. This first method called the diffuse element method (DEM), pioneered by Nayroles et al., utilized the
4037:
4022:
3898:
3584:
3134:
Liu, M. B.; Liu, G. R.; Zong, Z. (20 November 2011). "An overview on smoothed particle hydrodynamics".
1759:
3684:
3664:
3646:
2463:"The Repeated Replacement Method: A Pure Lagrangian Meshfree Method for Computational Fluid Dynamics"
75:
3113:
Garg, Sahil; Pant, Mohit (24 May 2018). "Meshfree Methods: A Comprehensive Review of Applications".
4007:
3893:
3623:
3215:
2666:; Hillman, Michael; Chi, Sheng-Wei (April 2017). "Meshfree Methods: Progress Made after 20 Years".
1817:
1670:
Continuous blending method (enrichment and coupling of finite elements and meshless methods) – see
667:
128:
62:
4049:
4027:
4012:
3995:
3903:
3888:
3804:
3669:
2028:
Liu, Wing Kam; Jun, Sukky; Zhang, Yi Fei (30 April 1995). "Reproducing kernel particle methods".
1812:
94:
270:
231:
3969:
3740:
3592:
3210:
1482:
1471:
1441:
1059:
169:
3412:"RBF Based Meshless Method for Large Scale Shallow Water Simulations: Experimental Validation"
2362:
Gauger, Christoph; Leinen, Peter; Yserentant, Harry (January 2000). "The Finite Mass Method".
713:) are constant along all the mesh, and the left and right mesh neighbors of the data value at
4017:
3863:
3779:
3300:
1747:
1732:
1488:
1415:
1306:
776:
743:
70:
2064:
1319:
1273:
842:
634:
134:
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3727:
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3382:
3315:
3202:
3020:
2951:
2783:
2745:
2702:
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2564:
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2427:
2336:
2301:
2219:
2149:
2037:
2002:
1959:
1924:
1860:
1740:
1658:
1402:
1012:
815:
716:
201:
98:
66:
8:
3821:
3735:
3254:
Belytschko, T.; Huerta, A.; Fernández-Méndez, S; Rabczuk, T. (2004), "Meshless methods",
1792:
1751:
1465:
3540:
3489:
3450:
3386:
3319:
3206:
3024:
2955:
2787:
2749:
2706:
2568:
2478:
2431:
2340:
2305:
2223:
2153:
2041:
2006:
1963:
1928:
1864:
1667:
methods (PoUM) – provide general approximation formulation used in some meshfree methods
41:
are those that do not require connection between nodes of the simulation domain, i.e. a
4044:
3985:
3560:
3526:
3501:
3475:
3398:
3341:
3228:
3036:
2967:
2875:
2837:
2799:
2615:
2580:
2537:
2497:
2462:
2443:
2235:
2165:
2122:
2093:
1975:
1807:
1664:
1420:
1357:
1106:
1086:
1039:
696:
676:
34:
2533:
2196:
1993:
Belytschko, T.; Lu, Y. Y.; Gu, L. (30 January 1994). "Element-free Galerkin methods".
1683:(XFEM, GFEM) – variants of FEM (finite element method) combining some meshless aspects
3674:
3564:
3552:
3505:
3402:
3358:
3345:
3289:
3274:
3260:
3244:
3179:
3160:
2994:
2879:
2841:
2803:
2584:
2541:
2502:
2274:
2239:
2126:
2080:
1979:
3232:
3040:
2971:
2447:
118:
Simulations containing nonlinear material behavior, discontinuities or singularities
112:
Simulations where nodes may be created or destroyed, such as in cracking simulations
3990:
3980:
3869:
3544:
3493:
3454:
3423:
3390:
3331:
3323:
3220:
3143:
3122:
3090:
3063:
3028:
2959:
2924:
2867:
2829:
2791:
2753:
2710:
2675:
2642:
2619:
2607:
2572:
2529:
2492:
2482:
2435:
2371:
2344:
2309:
2266:
2227:
2192:
2169:
2157:
2114:
2076:
2045:
2010:
1967:
1932:
1897:
1868:
1736:
1724:
1518:
999:{\displaystyle u(x,t_{n})=\sum _{i}m_{i}{\frac {u_{i}^{n}}{\rho _{i}}}W(|x-x_{i}|)}
78:. These operators are then used to construct the equations to simulate—such as the
4032:
3975:
3964:
2771:
2733:
2663:
2487:
1827:
1728:
1398:
106:
26:
3256:
Encyclopedia of Computational Mechanics Vol. 1 Chapter 10, John Wiley & Sons
3067:
2646:
1849:"Smoothed particle hydrodynamics: theory and application to non-spherical stars"
527:{\displaystyle {\partial u \over \partial x}={u_{i+1}^{n}-u_{i-1}^{n} \over 2h}}
3810:
3757:
2270:
1406:
50:
3548:
3497:
3147:
3126:
3094:
2928:
2375:
2348:
2231:
1873:
1848:
131:
simulation, the domain of a one-dimensional simulation would be some function
4074:
3679:
3556:
3458:
2278:
2254:
621:{\displaystyle {\partial u \over \partial t}={u_{i}^{n+1}-u_{i}^{n} \over k}}
89:
But in simulations where the material being simulated can move around (as in
42:
3514:
3428:
3411:
2292:
W.K. Liu; S. Jun; Y.F. Zhang (1995). "Reproducing kernel particle methods".
21:
3851:
3768:
3745:
2506:
2313:
2049:
2014:
1936:
1901:
1822:
16:
Methods in numerical analysis not requiring knowledge of neighboring points
3355:
Dissertation, TU Braunschweig, CSE – Computational Sciences in Engineering
2161:
3762:
3640:
3614:
3328:
10.1002/1097-0207(20000820)48:11<1615::AID-NME883>3.0.CO;2-S
1305:
and its spatial derivatives to write the equation being simulated as an
3579:
3253:
2715:
10.1002/1097-0207(20000730)48:9<1359::AID-NME829>3.0.CO;2-U
2118:
1971:
1661:(MLS) – provide general approximation method for arbitrary set of nodes
2815:
2813:
3394:
3336:
3224:
3032:
2963:
2871:
2833:
2795:
2758:
10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A
2611:
2439:
3301:"Enrichment and coupling of the finite element and meshless methods"
2576:
3531:
3480:
2810:
1397:
In the 1990s a new class of meshfree methods emerged based on the
3948:
2326:
1316:
The advantage of SPH in this situation is that the formulae for
3787:
3157:
Smoothed Particle Hydrodynamics, a meshfree and Particle Method
1581:
Generalized/Gradient Reproducing Kernel Particle Method (2011)
107:
creating a useful mesh from the geometry of a complex 3D object
3176:
The Meshless Method (MLPG) for Domain & BIE Discretization
3513:
Gross, B. J.; Kuberry, P.; Atzberger, P. J. (15 March 2022).
3410:
Alhuri, Y.; Naji, A.; Ouazar, D.; Taik, A. (26 August 2010).
2692:
2253:
Chen, Shang-Ying; Wei, Jian-Yu; Hsu, Kuo-Chin (2023-10-01).
3942:
3936:
3751:
3299:
Huerta, Antonio; Fernández-Méndez, Sonia (20 August 2000).
2597:
3465:
3375:
International Journal for Numerical Methods in Engineering
3308:
International Journal for Numerical Methods in Engineering
3195:
International Journal for Numerical Methods in Engineering
2944:
International Journal for Numerical Methods in Engineering
2860:
International Journal for Numerical Methods in Engineering
2822:
International Journal for Numerical Methods in Engineering
2776:
International Journal for Numerical Methods in Engineering
2738:
International Journal for Numerical Methods in Engineering
2695:
International Journal for Numerical Methods in Engineering
2600:
International Journal for Numerical Methods in Engineering
2557:
International Journal for Numerical Methods in Engineering
2420:
International Journal for Numerical Methods in Engineering
1995:
International Journal for Numerical Methods in Engineering
1887:
49:
simulations, in which the nodes can move according to the
673:
In this simple example, the steps (here the spatial step
2291:
2063:
Sulsky, D.; Chen, Z.; Schreyer, H.L. (September 1994).
1603:
Local radial basis function collocation Method (LRBFCM)
109:
may be especially difficult or require human assistance
3298:
2361:
1671:
3512:
3013:
International Journal for Numerical Methods in Fluids
2069:
Computer Methods in Applied Mechanics and Engineering
2030:
International Journal for Numerical Methods in Fluids
1704:
Space-Time Meshfree Collocation Method (STMCM) – see
1360:
1322:
1276:
1132:
1109:
1089:
1062:
1042:
1015:
883:
845:
818:
779:
746:
719:
699:
679:
637:
546:
441:
374:
312:
273:
234:
204:
172:
137:
3616:
Numerical methods for partial differential equations
3409:
3081:
Liu, G. R. (20 November 2011). "On G space theory".
2094:
https://www.math.ucla.edu/~jteran/papers/SSCTS13.pdf
1949:
2065:"A particle method for history-dependent materials"
1847:Gingold, R. A.; Monaghan, J. J. (1 December 1977).
2728:
2726:
2724:
2658:
2656:
2062:
1914:
1366:
1343:
1297:
1259:
1115:
1095:
1075:
1048:
1028:
998:
866:
831:
798:
765:
732:
705:
685:
658:
620:
526:
421:
359:
297:
258:
217:
190:
158:
1853:Monthly Notices of the Royal Astronomical Society
1578:Reproducing Kernel Particle Method (RKPM) (1995)
4072:
3436:
3372:
2770:
2461:Walker, Wade A.; Langowski, Jörg (6 July 2012).
2107:Archives of Computational Methods in Engineering
1709:
1650:Exponential Basis Functions method (EBFs) (2010)
97:of the material can occur (as in simulations of
2721:
2653:
2460:
2182:
1846:
1647:Least-square collocation meshless method (2001)
1501:Generalized-strain mesh-free (GSMF) formulation
1426:
3136:International Journal of Computational Methods
3115:International Journal of Computational Methods
3083:International Journal of Computational Methods
2917:International Journal of Computational Methods
2894:International Journal of Computational Methods
2662:
2554:
2104:
1992:
3600:
3053:
2853:
2851:
2732:
2417:
1309:, and simulate the equation with one of many
1543:Meshfree moving Kriging interpolation method
3416:Mathematical Modelling of Natural Phenomena
3192:
2522:Engineering Analysis with Boundary Elements
2185:Engineering Analysis with Boundary Elements
2027:
1743:, Nitche's method, and the penalty method.
1699:
3607:
3593:
3352:
3133:
2848:
2252:
2209:
1705:
422:{\displaystyle t_{n+1}-t_{n}=k\ \forall n}
360:{\displaystyle x_{i+1}-x_{i}=h\ \forall i}
3530:
3479:
3427:
3335:
3214:
2496:
2486:
2139:
1872:
839:with them. SPH then defines the value of
3112:
2985:
2983:
2981:
2941:
2914:
2519:
1386:One of the earliest meshfree methods is
20:
2632:
166:, represented as a mesh of data values
4073:
3439:IEEE Antennas and Propagation Magazine
3173:
3154:
3056:Finite Elements in Analysis and Design
2635:Finite Elements in Analysis and Design
2391:
2389:
2387:
2385:
1631:Optimal Transportation Meshfree method
1619:Discrete least squares meshless method
3588:
3010:
2978:
1718:
1643:Radial basis integral equation method
1525:Moving particle finite element method
1270:Then we can use these definitions of
631:Then we can use these definitions of
3858:Moving particle semi-implicit method
3769:Weighted essentially non-oscillatory
3239:Belytschko, T., Chen, J.S. (2007).
1672:Huerta & Fernández-Méndez (2000)
1513:Generalized finite difference method
3080:
2857:
2819:
2382:
1591:Smoothed point interpolation method
13:
3707:Finite-difference frequency-domain
3580:The USACM blog on Meshfree Methods
3105:
2680:10.1061/(ASCE)EM.1943-7889.0001176
2364:SIAM Journal on Numerical Analysis
1698:Local maximum-entropy (LME) – see
1460:Reproducing kernel particle method
1412:reproducing kernel particle method
1248:
1208:
1144:
1136:
558:
550:
453:
445:
413:
351:
14:
4102:
3573:
2534:10.1016/j.enganabound.2011.12.001
2197:10.1016/j.enganabound.2016.09.010
1598:radial point interpolation method
4086:Numerical differential equations
3519:Journal of Computational Physics
3468:Journal of Computational Physics
2668:Journal of Engineering Mechanics
2399:, CRC Press. 978-1-4200-8209-9
2329:Comput. Methods. Appl. Mech. Eng
2212:Journal of Computational Physics
1917:Journal of Computational Physics
1890:Journal of Computational Physics
4060:Method of fundamental solutions
3846:Smoothed-particle hydrodynamics
3074:
3047:
3004:
2991:Smoothed Finite Element Methods
2935:
2908:
2899:
2886:
2764:
2686:
2626:
2591:
2548:
2513:
2454:
2411:
2402:
2355:
2320:
2285:
2246:
2203:
2176:
2133:
1555:Method of fundamental solutions
1436:Smoothed particle hydrodynamics
1388:smoothed particle hydrodynamics
810:Smoothed-particle hydrodynamics
3701:Alternating direction-implicit
2098:
2087:
2056:
2021:
1986:
1943:
1908:
1881:
1840:
1798:Smoothed finite element method
1710:Netuzhylov & Zilian (2009)
1687:Smoothed finite element method
1495:Meshless local Petrov Galerkin
1390:, presented in 1977. Libersky
1338:
1326:
1307:ordinary differential equation
1292:
1280:
1243:
1239:
1218:
1214:
993:
989:
968:
964:
906:
887:
861:
849:
653:
641:
153:
141:
61:Numerical methods such as the
1:
3713:Finite-difference time-domain
3241:Meshfree and Particle Methods
3155:Liu, G.R.; Liu, M.B. (2003).
2408:Sarler B, Vertnik R. Meshfree
1833:
1607:Viscous vortex domains method
1561:Method of particular solution
1507:Moving particle semi-implicit
1448:Dissipative particle dynamics
56:
4091:Computational fluid dynamics
3752:Advection upstream-splitting
3288:, Berlin: Springer Verlag.
3243:, John Wiley and Sons Ltd.
2488:10.1371/journal.pone.0039999
2081:10.1016/0045-7825(94)90112-0
1427:List of methods and acronyms
91:computational fluid dynamics
7:
3763:Essentially non-oscillatory
3746:Monotonic upstream-centered
3068:10.1016/j.finel.2008.06.006
2647:10.1016/j.finel.2014.10.002
1786:
1637:Repeated replacement method
1083:is the density of particle
10:
4107:
4023:Infinite difference method
3641:Forward-time central-space
3284:Li, S., Liu, W.K. (2004).
2294:Int. J. Numer. Methods Eng
2271:10.1007/s00366-023-01897-6
2259:Engineering with Computers
1456:method (EFG / EFGM) (1994)
1381:
298:{\displaystyle n=0,1,2...}
259:{\displaystyle i=0,1,2...}
122:
3957:
3926:Poincaré–Steklov operator
3879:
3836:
3778:
3726:
3693:
3685:Method of characteristics
3655:
3631:
3622:
3549:10.1016/j.jcp.2021.110932
3498:10.1016/j.jcp.2020.109340
3286:Meshfree Particle Methods
3269:Liu, G.R. 1st edn, 2002.
3148:10.1142/S021987620800142X
3127:10.1142/S0219876218300015
3095:10.1142/S0219876209001863
2929:10.1142/S0219876209001796
2376:10.1137/S0036142999352564
2349:10.1016/j.cma.2010.07.017
2232:10.1016/J.JCP.2020.110002
1700:Arroyo & Ortiz (2006)
1693:Gradient smoothing method
1613:Cracking Particles Method
1076:{\displaystyle \rho _{i}}
874:between the particles by
668:finite difference methods
191:{\displaystyle u_{i}^{n}}
3943:Tearing and interconnect
3937:Balancing by constraints
3459:10.1109/MAP.2015.2414571
2395:Liu, G.R. 2nd edn: 2009
1818:Immersed boundary method
1625:Immersed Particle Method
1567:Method of finite spheres
1036:is the mass of particle
63:finite difference method
4050:Computer-assisted proof
4028:Infinite element method
3816:Gradient discretisation
2142:Computational Mechanics
1952:Computational Mechanics
1874:10.1093/mnras/181.3.375
1813:Boundary element method
799:{\displaystyle x_{i+1}}
766:{\displaystyle x_{i-1}}
84:Navier–Stokes equations
4038:Petrov–Galerkin method
3799:Discontinuous Galerkin
3178:. Tech Science Press.
2896:, 2(4): 645–665, 2005.
2314:10.1002/fld.1650200824
2050:10.1002/fld.1650200824
2015:10.1002/nme.1620370205
1937:10.1006/jcph.1995.1010
1902:10.1006/jcph.1993.1199
1760:Petrov–Galerkin method
1573:Discrete vortex method
1483:Natural element method
1472:Finite pointset method
1442:Diffuse element method
1416:Material point methods
1368:
1345:
1344:{\displaystyle u(x,t)}
1299:
1298:{\displaystyle u(x,t)}
1261:
1117:
1097:
1077:
1050:
1030:
1000:
868:
867:{\displaystyle u(x,t)}
833:
800:
767:
734:
707:
687:
660:
659:{\displaystyle u(x,t)}
622:
528:
423:
361:
299:
260:
219:
192:
160:
159:{\displaystyle u(x,t)}
30:
4018:Isogeometric analysis
3864:Material point method
3429:10.1051/mmnp/20105701
3174:Atluri, S.N. (2004).
2162:10.1007/s004660050346
1733:Finite element method
1549:Boundary cloud method
1489:Material point method
1454:Element-free Galerkin
1369:
1346:
1300:
1262:
1118:
1098:
1078:
1051:
1031:
1029:{\displaystyle m_{i}}
1001:
869:
834:
832:{\displaystyle u_{i}}
801:
768:
735:
733:{\displaystyle x_{i}}
708:
688:
661:
623:
529:
424:
362:
300:
261:
220:
218:{\displaystyle x_{i}}
193:
161:
71:finite element method
24:
4055:Integrable algorithm
3881:Domain decomposition
3159:. World Scientific.
1741:Lagrange multipliers
1659:Moving least squares
1537:Boundary node method
1358:
1320:
1274:
1130:
1107:
1087:
1060:
1040:
1013:
881:
843:
816:
777:
744:
717:
697:
677:
635:
544:
439:
372:
310:
271:
232:
202:
170:
135:
67:finite-volume method
25:20 points and their
3899:Schwarz alternating
3822:Loubignac iteration
3541:2022JCoPh.45310932G
3490:2020JCoPh.40909340G
3451:2015IAPM...57..277S
3387:2009IJNME..80..355N
3320:2000IJNME..48.1615H
3207:2006IJNME..65.2167A
3025:2008IJNMF..58.1101L
2956:2008IJNME..74.1128L
2788:2013IJNME..95..387C
2750:2001IJNME..50..435C
2707:2000IJNME..48.1359B
2569:2001IJNME..51.1089Z
2479:2012PLoSO...739999W
2432:2010IJNME..83.1541L
2341:2011CMAME.200..540B
2306:1995IJNMF..20.1081L
2224:2021JCoPh.42910002C
2154:1998CompM..22..117A
2042:1995IJNMF..20.1081L
2007:1994IJNME..37..229B
1964:1992CompM..10..307N
1929:1995JCoPh.116..123S
1865:1977MNRAS.181..375G
1793:Continuum mechanics
1531:Finite cloud method
1466:Finite point method
1192:
948:
611:
593:
512:
488:
187:
4081:Numerical analysis
4045:Validated numerics
2300:(8–9): 1081–1106.
2119:10.1007/BF02736130
2036:(8–9): 1081–1106.
1972:10.1007/BF00364252
1808:Weakened weak form
1719:Recent development
1665:Partition of unity
1585:Finite mass method
1364:
1341:
1295:
1257:
1178:
1165:
1113:
1093:
1073:
1046:
1026:
996:
934:
921:
864:
829:
796:
763:
740:are the values at
730:
703:
683:
656:
618:
597:
573:
524:
492:
468:
419:
357:
295:
256:
215:
188:
173:
156:
105:Simulations where
35:numerical analysis
31:
4068:
4067:
4008:Immersed boundary
4001:Method of moments
3916:Neumann–Dirichlet
3909:abstract additive
3894:Fictitious domain
3838:Meshless/Meshfree
3722:
3721:
3624:Finite difference
3363:978-3-00-026744-4
3314:(11): 1615–1636.
3271:Mesh Free Methods
3201:(13): 2167–2202.
3019:(10): 1101–1133.
2999:978-1-4398-2027-8
2426:(12): 1541–1579.
2397:Mesh Free Methods
1706:Netuzhylov (2008)
1654:Related methods:
1367:{\displaystyle W}
1311:numerical methods
1255:
1203:
1156:
1151:
1116:{\displaystyle W}
1096:{\displaystyle i}
1049:{\displaystyle i}
959:
912:
706:{\displaystyle k}
686:{\displaystyle h}
616:
565:
522:
460:
412:
350:
129:finite difference
127:In a traditional
99:plastic materials
93:) or where large
4098:
4013:Analytic element
3996:Boundary element
3889:Schur complement
3870:Particle-in-cell
3805:Spectral element
3629:
3628:
3609:
3602:
3595:
3586:
3585:
3568:
3534:
3509:
3483:
3462:
3433:
3431:
3406:
3395:10.1002/nme.2638
3357:
3349:
3339:
3305:
3258:
3236:
3225:10.1002/nme.1534
3218:
3189:
3170:
3151:
3130:
3099:
3098:
3078:
3072:
3071:
3051:
3045:
3044:
3033:10.1002/fld.1788
3008:
3002:
2989:Liu, G.R., 2010
2987:
2976:
2975:
2964:10.1002/nme.2204
2950:(7): 1128–1161.
2939:
2933:
2932:
2912:
2906:
2903:
2897:
2890:
2884:
2883:
2872:10.1002/nme.2720
2866:(9): 1127–1156.
2855:
2846:
2845:
2834:10.1002/nme.2719
2828:(9): 1093–1126.
2817:
2808:
2807:
2796:10.1002/nme.4512
2772:Chen, Jiun-Shyan
2768:
2762:
2761:
2734:Chen, Jiun-Shyan
2730:
2719:
2718:
2701:(9): 1359–1400.
2690:
2684:
2683:
2664:Chen, Jiun-Shyan
2660:
2651:
2650:
2630:
2624:
2623:
2612:10.1002/nme.2718
2595:
2589:
2588:
2563:(9): 1089–1100.
2552:
2546:
2545:
2517:
2511:
2510:
2500:
2490:
2458:
2452:
2451:
2440:10.1002/nme.2869
2415:
2409:
2406:
2400:
2393:
2380:
2379:
2370:(6): 1768–1799.
2359:
2353:
2352:
2335:(5–8): 540–576.
2324:
2318:
2317:
2289:
2283:
2282:
2265:(3): 1893–1925.
2250:
2244:
2243:
2207:
2201:
2200:
2180:
2174:
2173:
2137:
2131:
2130:
2102:
2096:
2091:
2085:
2084:
2075:(1–2): 179–196.
2060:
2054:
2053:
2025:
2019:
2018:
1990:
1984:
1983:
1947:
1941:
1940:
1912:
1906:
1905:
1885:
1879:
1878:
1876:
1844:
1752:Gauss quadrature
1519:Particle-in-cell
1373:
1371:
1370:
1365:
1350:
1348:
1347:
1342:
1304:
1302:
1301:
1296:
1266:
1264:
1263:
1258:
1256:
1254:
1246:
1242:
1237:
1236:
1221:
1206:
1204:
1202:
1201:
1191:
1186:
1177:
1175:
1174:
1164:
1152:
1150:
1142:
1134:
1122:
1120:
1119:
1114:
1102:
1100:
1099:
1094:
1082:
1080:
1079:
1074:
1072:
1071:
1055:
1053:
1052:
1047:
1035:
1033:
1032:
1027:
1025:
1024:
1005:
1003:
1002:
997:
992:
987:
986:
971:
960:
958:
957:
947:
942:
933:
931:
930:
920:
905:
904:
873:
871:
870:
865:
838:
836:
835:
830:
828:
827:
805:
803:
802:
797:
795:
794:
772:
770:
769:
764:
762:
761:
739:
737:
736:
731:
729:
728:
712:
710:
709:
704:
692:
690:
689:
684:
665:
663:
662:
657:
627:
625:
624:
619:
617:
612:
610:
605:
592:
581:
571:
566:
564:
556:
548:
533:
531:
530:
525:
523:
521:
513:
511:
506:
487:
482:
466:
461:
459:
451:
443:
428:
426:
425:
420:
410:
403:
402:
390:
389:
366:
364:
363:
358:
348:
341:
340:
328:
327:
304:
302:
301:
296:
265:
263:
262:
257:
224:
222:
221:
216:
214:
213:
197:
195:
194:
189:
186:
181:
165:
163:
162:
157:
39:meshfree methods
33:In the field of
4106:
4105:
4101:
4100:
4099:
4097:
4096:
4095:
4071:
4070:
4069:
4064:
4033:Galerkin method
3976:Method of lines
3953:
3921:Neumann–Neumann
3875:
3832:
3774:
3741:High-resolution
3718:
3689:
3651:
3618:
3613:
3576:
3571:
3303:
3186:
3167:
3108:
3106:Further reading
3103:
3102:
3079:
3075:
3062:(15): 889–909.
3052:
3048:
3009:
3005:
2988:
2979:
2940:
2936:
2913:
2909:
2904:
2900:
2891:
2887:
2856:
2849:
2818:
2811:
2769:
2765:
2731:
2722:
2691:
2687:
2674:(4): 04017001.
2661:
2654:
2631:
2627:
2606:(8): 971–1018.
2596:
2592:
2577:10.1002/nme.200
2553:
2549:
2518:
2514:
2459:
2455:
2416:
2412:
2407:
2403:
2394:
2383:
2360:
2356:
2325:
2321:
2290:
2286:
2251:
2247:
2208:
2204:
2181:
2177:
2138:
2134:
2103:
2099:
2092:
2088:
2061:
2057:
2026:
2022:
1991:
1987:
1948:
1944:
1913:
1909:
1886:
1882:
1845:
1841:
1836:
1828:Particle method
1789:
1729:Kronecker delta
1721:
1681:Generalized FEM
1596:Meshfree local
1593:(S-PIM) (2005).
1429:
1399:Galerkin method
1384:
1376:compact support
1359:
1356:
1355:
1321:
1318:
1317:
1275:
1272:
1271:
1247:
1238:
1232:
1228:
1217:
1207:
1205:
1197:
1193:
1187:
1182:
1176:
1170:
1166:
1160:
1143:
1135:
1133:
1131:
1128:
1127:
1108:
1105:
1104:
1088:
1085:
1084:
1067:
1063:
1061:
1058:
1057:
1041:
1038:
1037:
1020:
1016:
1014:
1011:
1010:
988:
982:
978:
967:
953:
949:
943:
938:
932:
926:
922:
916:
900:
896:
882:
879:
878:
844:
841:
840:
823:
819:
817:
814:
813:
784:
780:
778:
775:
774:
751:
747:
745:
742:
741:
724:
720:
718:
715:
714:
698:
695:
694:
678:
675:
674:
636:
633:
632:
606:
601:
582:
577:
572:
570:
557:
549:
547:
545:
542:
541:
514:
507:
496:
483:
472:
467:
465:
452:
444:
442:
440:
437:
436:
398:
394:
379:
375:
373:
370:
369:
336:
332:
317:
313:
311:
308:
307:
272:
269:
268:
233:
230:
229:
209:
205:
203:
200:
199:
182:
177:
171:
168:
167:
136:
133:
132:
125:
80:Euler equations
59:
17:
12:
11:
5:
4104:
4094:
4093:
4088:
4083:
4066:
4065:
4063:
4062:
4057:
4052:
4047:
4042:
4041:
4040:
4030:
4025:
4020:
4015:
4010:
4005:
4004:
4003:
3993:
3988:
3983:
3978:
3973:
3970:Pseudospectral
3967:
3961:
3959:
3955:
3954:
3952:
3951:
3946:
3940:
3934:
3928:
3923:
3918:
3913:
3912:
3911:
3906:
3896:
3891:
3885:
3883:
3877:
3876:
3874:
3873:
3867:
3861:
3855:
3849:
3842:
3840:
3834:
3833:
3831:
3830:
3824:
3819:
3813:
3808:
3802:
3796:
3790:
3784:
3782:
3780:Finite element
3776:
3775:
3773:
3772:
3766:
3760:
3758:Riemann solver
3755:
3749:
3743:
3738:
3732:
3730:
3724:
3723:
3720:
3719:
3717:
3716:
3710:
3704:
3697:
3695:
3691:
3690:
3688:
3687:
3682:
3677:
3672:
3667:
3665:Lax–Friedrichs
3661:
3659:
3653:
3652:
3650:
3649:
3647:Crank–Nicolson
3644:
3637:
3635:
3626:
3620:
3619:
3612:
3611:
3604:
3597:
3589:
3583:
3582:
3575:
3574:External links
3572:
3570:
3569:
3510:
3463:
3445:(2): 277–293.
3434:
3407:
3381:(3): 355–380.
3370:
3367:electronic ed.
3350:
3296:
3282:
3273:, CRC Press.
3267:
3251:
3237:
3216:10.1.1.68.2696
3190:
3184:
3171:
3165:
3152:
3142:(1): 135–188.
3131:
3121:(4): 1830001.
3109:
3107:
3104:
3101:
3100:
3089:(2): 257–289.
3073:
3046:
3003:
2977:
2934:
2923:(1): 147–179.
2907:
2898:
2885:
2847:
2809:
2782:(5): 387–418.
2763:
2744:(2): 435–466.
2720:
2685:
2652:
2625:
2590:
2547:
2528:(5): 716–726.
2512:
2453:
2410:
2401:
2381:
2354:
2319:
2284:
2245:
2202:
2175:
2148:(2): 117–127.
2132:
2097:
2086:
2055:
2020:
2001:(2): 229–256.
1985:
1958:(5): 307–318.
1942:
1923:(1): 123–134.
1907:
1880:
1859:(3): 375–389.
1838:
1837:
1835:
1832:
1831:
1830:
1825:
1820:
1815:
1810:
1805:
1800:
1795:
1788:
1785:
1720:
1717:
1716:
1715:
1712:
1702:
1696:
1690:
1689:(S-FEM) (2007)
1684:
1674:
1668:
1662:
1652:
1651:
1648:
1645:
1640:
1634:
1628:
1622:
1616:
1610:
1604:
1601:
1594:
1588:
1582:
1579:
1576:
1570:
1564:
1558:
1552:
1546:
1540:
1534:
1528:
1522:
1516:
1510:
1504:
1498:
1492:
1486:
1480:
1475:
1469:
1463:
1457:
1451:
1445:
1439:
1428:
1425:
1383:
1380:
1363:
1340:
1337:
1334:
1331:
1328:
1325:
1294:
1291:
1288:
1285:
1282:
1279:
1268:
1267:
1253:
1250:
1245:
1241:
1235:
1231:
1227:
1224:
1220:
1216:
1213:
1210:
1200:
1196:
1190:
1185:
1181:
1173:
1169:
1163:
1159:
1155:
1149:
1146:
1141:
1138:
1112:
1092:
1070:
1066:
1045:
1023:
1019:
1007:
1006:
995:
991:
985:
981:
977:
974:
970:
966:
963:
956:
952:
946:
941:
937:
929:
925:
919:
915:
911:
908:
903:
899:
895:
892:
889:
886:
863:
860:
857:
854:
851:
848:
826:
822:
793:
790:
787:
783:
760:
757:
754:
750:
727:
723:
702:
682:
655:
652:
649:
646:
643:
640:
629:
628:
615:
609:
604:
600:
596:
591:
588:
585:
580:
576:
569:
563:
560:
555:
552:
535:
534:
520:
517:
510:
505:
502:
499:
495:
491:
486:
481:
478:
475:
471:
464:
458:
455:
450:
447:
430:
429:
418:
415:
409:
406:
401:
397:
393:
388:
385:
382:
378:
367:
356:
353:
347:
344:
339:
335:
331:
326:
323:
320:
316:
305:
294:
291:
288:
285:
282:
279:
276:
266:
255:
252:
249:
246:
243:
240:
237:
212:
208:
185:
180:
176:
155:
152:
149:
146:
143:
140:
124:
121:
120:
119:
116:
113:
110:
58:
55:
51:velocity field
15:
9:
6:
4:
3:
2:
4103:
4092:
4089:
4087:
4084:
4082:
4079:
4078:
4076:
4061:
4058:
4056:
4053:
4051:
4048:
4046:
4043:
4039:
4036:
4035:
4034:
4031:
4029:
4026:
4024:
4021:
4019:
4016:
4014:
4011:
4009:
4006:
4002:
3999:
3998:
3997:
3994:
3992:
3989:
3987:
3984:
3982:
3979:
3977:
3974:
3971:
3968:
3966:
3963:
3962:
3960:
3956:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3929:
3927:
3924:
3922:
3919:
3917:
3914:
3910:
3907:
3905:
3902:
3901:
3900:
3897:
3895:
3892:
3890:
3887:
3886:
3884:
3882:
3878:
3871:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3843:
3841:
3839:
3835:
3828:
3825:
3823:
3820:
3817:
3814:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3789:
3786:
3785:
3783:
3781:
3777:
3770:
3767:
3764:
3761:
3759:
3756:
3753:
3750:
3747:
3744:
3742:
3739:
3737:
3734:
3733:
3731:
3729:
3728:Finite volume
3725:
3714:
3711:
3708:
3705:
3702:
3699:
3698:
3696:
3692:
3686:
3683:
3681:
3678:
3676:
3673:
3671:
3668:
3666:
3663:
3662:
3660:
3658:
3654:
3648:
3645:
3642:
3639:
3638:
3636:
3634:
3630:
3627:
3625:
3621:
3617:
3610:
3605:
3603:
3598:
3596:
3591:
3590:
3587:
3581:
3578:
3577:
3566:
3562:
3558:
3554:
3550:
3546:
3542:
3538:
3533:
3528:
3524:
3520:
3516:
3511:
3507:
3503:
3499:
3495:
3491:
3487:
3482:
3477:
3473:
3469:
3464:
3460:
3456:
3452:
3448:
3444:
3440:
3435:
3430:
3425:
3421:
3417:
3413:
3408:
3404:
3400:
3396:
3392:
3388:
3384:
3380:
3376:
3371:
3368:
3364:
3360:
3356:
3351:
3347:
3343:
3338:
3333:
3329:
3325:
3321:
3317:
3313:
3309:
3302:
3297:
3295:
3294:3-540-22256-1
3291:
3287:
3283:
3280:
3279:0-8493-1238-8
3276:
3272:
3268:
3266:
3265:0-470-84699-2
3262:
3257:
3252:
3250:
3249:0-470-84800-6
3246:
3242:
3238:
3234:
3230:
3226:
3222:
3217:
3212:
3208:
3204:
3200:
3196:
3191:
3187:
3185:0-9657001-8-6
3181:
3177:
3172:
3168:
3166:981-238-456-1
3162:
3158:
3153:
3149:
3145:
3141:
3137:
3132:
3128:
3124:
3120:
3116:
3111:
3110:
3096:
3092:
3088:
3084:
3077:
3069:
3065:
3061:
3057:
3050:
3042:
3038:
3034:
3030:
3026:
3022:
3018:
3014:
3007:
3000:
2996:
2993:, CRC Press,
2992:
2986:
2984:
2982:
2973:
2969:
2965:
2961:
2957:
2953:
2949:
2945:
2938:
2930:
2926:
2922:
2918:
2911:
2902:
2895:
2889:
2881:
2877:
2873:
2869:
2865:
2861:
2854:
2852:
2843:
2839:
2835:
2831:
2827:
2823:
2816:
2814:
2805:
2801:
2797:
2793:
2789:
2785:
2781:
2777:
2773:
2767:
2759:
2755:
2751:
2747:
2743:
2739:
2735:
2729:
2727:
2725:
2716:
2712:
2708:
2704:
2700:
2696:
2689:
2681:
2677:
2673:
2669:
2665:
2659:
2657:
2648:
2644:
2640:
2636:
2629:
2621:
2617:
2613:
2609:
2605:
2601:
2594:
2586:
2582:
2578:
2574:
2570:
2566:
2562:
2558:
2551:
2543:
2539:
2535:
2531:
2527:
2523:
2516:
2508:
2504:
2499:
2494:
2489:
2484:
2480:
2476:
2473:(7): e39999.
2472:
2468:
2464:
2457:
2449:
2445:
2441:
2437:
2433:
2429:
2425:
2421:
2414:
2405:
2398:
2392:
2390:
2388:
2386:
2377:
2373:
2369:
2365:
2358:
2350:
2346:
2342:
2338:
2334:
2330:
2323:
2315:
2311:
2307:
2303:
2299:
2295:
2288:
2280:
2276:
2272:
2268:
2264:
2260:
2256:
2249:
2241:
2237:
2233:
2229:
2225:
2221:
2217:
2213:
2206:
2198:
2194:
2190:
2186:
2179:
2171:
2167:
2163:
2159:
2155:
2151:
2147:
2143:
2136:
2128:
2124:
2120:
2116:
2112:
2108:
2101:
2095:
2090:
2082:
2078:
2074:
2070:
2066:
2059:
2051:
2047:
2043:
2039:
2035:
2031:
2024:
2016:
2012:
2008:
2004:
2000:
1996:
1989:
1981:
1977:
1973:
1969:
1965:
1961:
1957:
1953:
1946:
1938:
1934:
1930:
1926:
1922:
1918:
1911:
1903:
1899:
1895:
1891:
1884:
1875:
1870:
1866:
1862:
1858:
1854:
1850:
1843:
1839:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1799:
1796:
1794:
1791:
1790:
1784:
1780:
1776:
1772:
1769:
1763:
1761:
1757:
1753:
1749:
1744:
1742:
1738:
1734:
1730:
1726:
1713:
1711:
1707:
1703:
1701:
1697:
1694:
1691:
1688:
1685:
1682:
1678:
1675:
1673:
1669:
1666:
1663:
1660:
1657:
1656:
1655:
1649:
1646:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1621:(DLSM) (2006)
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1497:(MLPG) (1998)
1496:
1493:
1490:
1487:
1484:
1481:
1479:
1476:
1473:
1470:
1467:
1464:
1462:(RKPM) (1995)
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1433:
1432:
1424:
1422:
1417:
1413:
1408:
1404:
1400:
1395:
1393:
1389:
1379:
1377:
1361:
1352:
1335:
1332:
1329:
1323:
1314:
1312:
1308:
1289:
1286:
1283:
1277:
1251:
1233:
1229:
1225:
1222:
1211:
1198:
1194:
1188:
1183:
1179:
1171:
1167:
1161:
1157:
1153:
1147:
1139:
1126:
1125:
1124:
1110:
1090:
1068:
1064:
1043:
1021:
1017:
983:
979:
975:
972:
961:
954:
950:
944:
939:
935:
927:
923:
917:
913:
909:
901:
897:
893:
890:
884:
877:
876:
875:
858:
855:
852:
846:
824:
820:
811:
807:
791:
788:
785:
781:
758:
755:
752:
748:
725:
721:
700:
693:and timestep
680:
671:
669:
650:
647:
644:
638:
613:
607:
602:
598:
594:
589:
586:
583:
578:
574:
567:
561:
553:
540:
539:
538:
518:
515:
508:
503:
500:
497:
493:
489:
484:
479:
476:
473:
469:
462:
456:
448:
435:
434:
433:
416:
407:
404:
399:
395:
391:
386:
383:
380:
376:
368:
354:
345:
342:
337:
333:
329:
324:
321:
318:
314:
306:
292:
289:
286:
283:
280:
277:
274:
267:
253:
250:
247:
244:
241:
238:
235:
228:
227:
226:
210:
206:
183:
178:
174:
150:
147:
144:
138:
130:
117:
114:
111:
108:
104:
103:
102:
100:
96:
92:
87:
85:
81:
77:
72:
68:
64:
54:
52:
48:
44:
40:
36:
28:
23:
19:
3852:Peridynamics
3837:
3670:Lax–Wendroff
3522:
3518:
3471:
3467:
3442:
3438:
3419:
3415:
3378:
3374:
3354:
3311:
3307:
3285:
3270:
3255:
3240:
3198:
3194:
3175:
3156:
3139:
3135:
3118:
3114:
3086:
3082:
3076:
3059:
3055:
3049:
3016:
3012:
3006:
2990:
2947:
2943:
2937:
2920:
2916:
2910:
2901:
2888:
2863:
2859:
2825:
2821:
2779:
2775:
2766:
2741:
2737:
2698:
2694:
2688:
2671:
2667:
2638:
2634:
2628:
2603:
2599:
2593:
2560:
2556:
2550:
2525:
2521:
2515:
2470:
2466:
2456:
2423:
2419:
2413:
2404:
2396:
2367:
2363:
2357:
2332:
2328:
2322:
2297:
2293:
2287:
2262:
2258:
2248:
2215:
2211:
2205:
2188:
2184:
2178:
2145:
2141:
2135:
2110:
2106:
2100:
2089:
2072:
2068:
2058:
2033:
2029:
2023:
1998:
1994:
1988:
1955:
1951:
1945:
1920:
1916:
1910:
1896:(1): 67–75.
1893:
1889:
1883:
1856:
1852:
1842:
1823:Stencil code
1781:
1777:
1773:
1764:
1745:
1739:, including
1722:
1695:(GSM) (2008)
1677:eXtended FEM
1653:
1639:(RRM) (2012)
1633:(OTM) (2010)
1627:(IPM) (2006)
1615:(CPM) (2004)
1587:(FMM) (2000)
1474:(FPM) (1998)
1468:(FPM) (1996)
1450:(DPD) (1992)
1444:(DEM) (1992)
1438:(SPH) (1977)
1430:
1396:
1391:
1385:
1353:
1315:
1269:
1008:
808:
672:
630:
536:
431:
126:
95:deformations
88:
60:
38:
32:
18:
3986:Collocation
3422:(7): 4–10.
2191:: 144–160.
2113:(1): 3–80.
4075:Categories
3675:MacCormack
3657:Hyperbolic
3532:2102.02421
3525:: 110932.
3481:1905.10469
3474:: 109340.
3365:, also as
2218:: 110002.
1834:References
1756:patch test
1748:quadrature
1407:Belytschko
198:at points
76:derivative
57:Motivation
47:Lagrangian
3991:Level-set
3981:Multigrid
3931:Balancing
3633:Parabolic
3565:231802303
3557:0021-9991
3506:166228451
3403:122969330
3346:122813651
3337:2117/8264
3211:CiteSeerX
2880:119378545
2842:123009384
2804:124640562
2641:: 20–41.
2585:119952479
2542:122004658
2279:1435-5663
2240:228828681
2127:122241092
1980:121511161
1725:weak form
1478:hp-clouds
1249:∂
1226:−
1209:∂
1195:ρ
1158:∑
1145:∂
1137:∂
1065:ρ
976:−
951:ρ
914:∑
756:−
595:−
559:∂
551:∂
501:−
490:−
454:∂
446:∂
414:∀
392:−
352:∀
330:−
3965:Spectral
3904:additive
3827:Smoothed
3793:Extended
3233:15974625
3041:53983110
2972:54088894
2507:22866175
2467:PLOS ONE
2448:18225521
1787:See also
225:, where
3949:FETI-DP
3829:(S-FEM)
3748:(MUSCL)
3736:Godunov
3537:Bibcode
3486:Bibcode
3447:Bibcode
3383:Bibcode
3316:Bibcode
3203:Bibcode
3021:Bibcode
2952:Bibcode
2784:Bibcode
2746:Bibcode
2703:Bibcode
2620:4943418
2565:Bibcode
2498:3391243
2475:Bibcode
2428:Bibcode
2337:Bibcode
2302:Bibcode
2220:Bibcode
2170:3688083
2150:Bibcode
2038:Bibcode
2003:Bibcode
1960:Bibcode
1925:Bibcode
1861:Bibcode
1803:G space
1768:G space
1746:As for
1600:(RPIM).
1527:(MPFEM)
1382:History
123:Example
82:or the
27:Voronoi
3958:Others
3945:(FETI)
3939:(BDDC)
3811:Mortar
3795:(XFEM)
3788:hp-FEM
3771:(WENO)
3754:(AUSM)
3715:(FDTD)
3709:(FDFD)
3694:Others
3680:Upwind
3643:(FTCS)
3563:
3555:
3504:
3401:
3361:
3344:
3292:
3277:
3263:
3247:
3231:
3213:
3182:
3163:
3039:
2997:
2970:
2878:
2840:
2802:
2618:
2583:
2540:
2505:
2495:
2446:
2277:
2238:
2168:
2125:
1978:
1737:weakly
1515:(GFDM)
1503:(2016)
1421:Frozen
1392:et al.
1103:, and
1009:where
411:
349:
69:, and
3972:(DVR)
3933:(BDD)
3872:(PIC)
3866:(MPM)
3860:(MPS)
3848:(SPH)
3818:(GDM)
3807:(SEM)
3765:(ENO)
3703:(ADI)
3561:S2CID
3527:arXiv
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