8819:(XFEM) is a numerical technique based on the generalized finite element method (GFEM) and the partition of unity method (PUM). It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions. Extended finite element methods enrich the approximation space to naturally reproduce the challenging feature associated with the problem of interest: the discontinuity, singularity, boundary layer, etc. It was shown that for some problems, such an embedding of the problem's feature into the approximation space can significantly improve convergence rates and accuracy. Moreover, treating problems with discontinuities with XFEMs suppresses the need to mesh and re-mesh the discontinuity surfaces, thus alleviating the computational costs and projection errors associated with conventional finite element methods at the cost of restricting the discontinuities to mesh edges.
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methods means that an actual image of the microstructure from a microscope can be input to the solver to get a more accurate stress response. Using a real image with FFT avoids meshing the microstructure, which would be required if using FEM simulation of the microstructure, and might be difficult. Because fourier approximations are inherently periodic, FFT can only be used in cases of periodic microstructure, but this is common in real materials. FFT can also be combined with FEM methods by using fourier components as the variational basis for approximating the fields inside an element, which can take advantage of the speed of FFT based solvers.
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being curvilinear. On the other hand, some authors replace "piecewise linear" with "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". The finite element method is not restricted to triangles (tetrahedra in 3-d or higher-order simplexes in multidimensional spaces). Still, it can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3-d, and so on). Higher-order shapes (curvilinear elements) can be defined with polynomial and even non-polynomial shapes (e.g., ellipse or circle).
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and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured. The mesh is an integral part of the model and must be controlled carefully to give the best results. Generally, the higher the number of elements in a mesh, the more accurate the solution of the discretized problem. However, there is a value at which the results converge, and further mesh refinement does not increase accuracy.
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valuable resource as they remove multiple instances of creating and testing complex prototypes for various high-fidelity situations. For example, in a frontal crash simulation, it is possible to increase prediction accuracy in "important" areas like the front of the car and reduce it in its rear (thus reducing the cost of the simulation). Another example would be in
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have been accelerated primarily through improved initial prototype designs using FEM. In summary, benefits of FEM include increased accuracy, enhanced design and better insight into critical design parameters, virtual prototyping, fewer hardware prototypes, a faster and less expensive design cycle, increased productivity, and increased revenue.
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9030:(FVM). CFD problems usually require discretization of the problem into a large number of cells/gridpoints (millions and more). Therefore the cost of the solution favors simpler, lower-order approximation within each cell. This is especially true for 'external flow' problems, like airflow around the car, airplane, or weather simulation.
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A discretization strategy is understood to mean a clearly defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions), and (c) the mapping of reference elements onto the elements of the mesh.
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This powerful design tool has significantly improved both the standard of engineering designs and the design process methodology in many industrial applications. The introduction of FEM has substantially decreased the time to take products from concept to the production line. Testing and development
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Depending on the author, the word "element" in the "finite element method" refers to the domain's triangles, the piecewise linear basis function, or both. So, for instance, an author interested in curved domains might replace the triangles with curved primitives and so might describe the elements as
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FEA may be used for analyzing problems over complicated domains (like cars and oil pipelines) when the domain changes (as during a solid-state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. FEA simulations provide a
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FEM allows detailed visualization of where structures bend or twist, indicating the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of modeling and system analysis. Similarly, the desired level of accuracy required
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Various specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in the design and development of their products. Several modern FEM packages include specific components such as thermal,
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error estimation in terms of the quantities of interest. When the errors of approximation are larger than what is considered acceptable, then the discretization has to be changed either by an automated adaptive process or by the action of the analyst. Some very efficient postprocessors provide for
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The crystal plasticity finite element method (CPFEM) is an advanced numerical tool developed by Franz Roters. Metals can be regarded as crystal aggregates, which behave anisotropy under deformation, such as abnormal stress and strain localization. CPFEM, based on the slip (shear strain rate), can
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The introduction of the scaled boundary finite element method (SBFEM) came from Song and Wolf (1997). The SBFEM has been one of the most profitable contributions in the area of numerical analysis of fracture mechanics problems. It is a semi-analytical fundamental-solutionless method combining the
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based method (2) to simulate deformation in materials, where the FE method is used for the macroscale stress and deformation, and the FFT method is used on the microscale to deal with the effects of microscale on the mechanical response. Unlike FEM, FFT methodsâ similarities to image processing
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More advanced implementations (adaptive finite element methods) utilize a method to assess the quality of the results (based on error estimation theory) and modify the mesh during the solution aiming to achieve an approximate solution within some bounds from the exact solution of the continuum
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9042:(FFT), where the solution is approximated by a fourier series computed using the FFT. For approximating the mechanical response of materials under stress, FFT is often much faster, but FEM may be more accurate. One example of the respective advantages of the two methods is in simulation of
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One hopes that as the underlying triangular mesh becomes finer and finer, the solution of the discrete problem (3) will, in some sense, converge to the solution of the original boundary value problem P2. To measure this mesh fineness, the triangulation is indexed by a real-valued parameter
803:. The method approximates the unknown function over the domain. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then approximates a solution by minimizing an associated error function via the
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Spectral element methods combine the geometric flexibility of finite elements and the acute accuracy of spectral methods. Spectral methods are the approximate solution of weak-form partial equations based on high-order
Lagrangian interpolants and used only with certain quadrature rules.
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The Cut Finite
Element Approach was developed in 2014. The approach is "to make the discretization as independent as possible of the geometric description and minimize the complexity of mesh generation, while retaining the accuracy and robustness of a standard finite element method."
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The most attractive feature of the FEM is its ability to handle complicated geometries (and boundaries) with relative ease. While FDM in its basic form is restricted to handle rectangular shapes and simple alterations thereof, the handling of geometries in FEM is theoretically
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Various numerical solution algorithms can be classified into two broad categories; direct and iterative solvers. These algorithms are designed to exploit the sparsity of matrices that depend on the variational formulation and discretization strategy choices.
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XFEM has also been implemented in codes like Altair Radios, ASTER, Morfeo, and Abaqus. It is increasingly being adopted by other commercial finite element software, with a few plugins and actual core implementations available (ANSYS, SAMCEF, OOFELIE, etc.).
1321:, etc. Each discretization strategy has certain advantages and disadvantages. A reasonable criterion in selecting a discretization strategy is to realize nearly optimal performance for the broadest set of mathematical models in a particular model class.
8424:, an example of which is the space of piecewise linear functions over the mesh, which are continuous at each edge midpoint. Since these functions are generally discontinuous along the edges, this finite-dimensional space is not a subspace of the original
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8248:, piecewise polynomial basis function that is merely continuous suffice (i.e., the derivatives are discontinuous.) For higher-order partial differential equations, one must use smoother basis functions. For instance, for a fourth-order problem such as
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is used to âbondâ these spaces together to form the approximating subspace. The effectiveness of GFEM has been shown when applied to problems with domains having complicated boundaries, problems with micro-scales, and problems with boundary layers.
9058:(a BCC metal). This simulation did not have a sophisticated shape update algorithm for the FFT method. In both cases, the FFT method was more than 10 times as fast as FEM, but in the wire drawing simulation, where there were large deformations in
901:, as indicated by the scale in the inset legend, red being high amplitude. The area inside the cylinder is low amplitude (dark blue, with widely spaced lines of magnetic flux), which suggests that the shield is performing as it was designed to.
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8975:(GDM). Hence the convergence properties of the GDM, which are established for a series of problems (linear and nonlinear elliptic problems, linear, nonlinear, and degenerate parabolic problems), hold as well for these particular FEMs.
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and set the integral to zero. In simple terms, it is a procedure that minimizes the approximation error by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are
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The generalized finite element method (GFEM) uses local spaces consisting of functions, not necessarily polynomials, that reflect the available information on the unknown solution and thus ensure good local approximation. Then a
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calculate dislocation, crystal orientation, and other texture information to consider crystal anisotropy during the routine. It has been applied in the numerical study of material deformation, surface roughness, fractures, etc.
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In step (2) above, a global system of equations is generated from the element equations by transforming coordinates from the subdomains' local nodes to the domain's global nodes. This spatial transformation includes appropriate
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The S-FEM, Smoothed Finite
Element Methods, is a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining mesh-free methods with the finite element method.
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9062:, the FEM method was much more accurate. In the sheet rolling simulation, the results of the two methods were similar. FFT has a larger speed advantage in cases where the boundary conditions are given in the materials
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After this second step, we have concrete formulae for a large but finite-dimensional linear problem whose solution will approximately solve the original BVP. This finite-dimensional problem is then implemented on a
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2835:{\displaystyle {\begin{aligned}\int _{0}^{1}f(x)v(x)\,dx&=\int _{0}^{1}u''(x)v(x)\,dx\\&=u'(x)v(x)|_{0}^{1}-\int _{0}^{1}u'(x)v'(x)\,dx\\&=-\int _{0}^{1}u'(x)v'(x)\,dx\equiv -\phi (u,v),\end{aligned}}}
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electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and minimizing weight, materials, and costs.
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The mixed finite element method is a type of finite element method in which extra independent variables are introduced as nodal variables during the discretization of a partial differential equation problem.
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which one takes to be very small. This parameter will be related to the largest or average triangle size in the triangulation. As we refine the triangulation, the space of piecewise linear functions
5500:{\displaystyle v_{k}(x)={\begin{cases}{x-x_{k-1} \over x_{k}\,-x_{k-1}}&{\text{ if }}x\in ,\\{x_{k+1}\,-x \over x_{k+1}\,-x_{k}}&{\text{ if }}x\in ,\\0&{\text{ otherwise}},\end{cases}}}
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There are reasons to consider the mathematical foundation of the finite element approximation more sound, for instance, because the quality of the approximation between grid points is poor in FDM.
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Post-processing procedures are designed to extract the data of interest from a finite element solution. To meet the requirements of solution verification, postprocessors need to provide for
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Generally, FEM is the method of choice in all types of analysis in structural mechanics (i.e., solving for deformation and stresses in solid bodies or dynamics of structures). In contrast,
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advantages of finite element formulations and procedures and boundary element discretization. However, unlike the boundary element method, no fundamental differential solution is required.
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of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of
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The quality of a FEM approximation is often higher than in the corresponding FDM approach, but this is highly problem-dependent, and several examples to the contrary can be provided.
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In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often
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861:) in light blue; and air in grey. Although the geometry may seem simple, it would be very challenging to calculate the magnetic field for this setup without FEM software using
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Song, Chongmin; Wolf, John P. (5 August 1997). "The scaled boundary finite-element method â alias consistent infinitesimal finite-element cell method â for elastodynamics".
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In the first step, one rephrases the original BVP in its weak form. Little to no computation is usually required for this step. The transformation is done by hand on paper.
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into small elements, as well as the use of software coded with a FEM algorithm. In applying FEA, the complex problem is usually a physical system with the underlying
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Yang and Lui introduced the
Augmented-Finite Element Method, whose goal was to model the weak and strong discontinuities without needing extra DoFs, as PuM stated.
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8154:'s backslash operator (which uses sparse LU, sparse Cholesky, and other factorization methods) can be sufficient for meshes with a hundred thousand vertices.
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Naghibi
Beidokhti, Hamid; Janssen, Dennis; Khoshgoftar, Mehdi; Sprengers, Andre; Perdahcioglu, Emin Semih; Boogaard, Ton Van den; Verdonschot, Nico (2016).
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Then, one chooses basis functions. We used piecewise linear basis functions in our discussion, but it is common to use piecewise polynomial basis functions.
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approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with
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8955:(MFD) methods, is a generalization of the standard finite element method for arbitrary element geometries. This allows admission of general polygons (or
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dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem
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1770:{\displaystyle {\text{P2 }}:{\begin{cases}u_{xx}(x,y)+u_{yy}(x,y)=f(x,y)&{\text{ in }}\Omega ,\\u=0&{\text{ on }}\partial \Omega ,\end{cases}}}
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P1 and P2 are ready to be discretized, which leads to a common sub-problem (3). The basic idea is to replace the infinite-dimensional linear problem:
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is a subspace of the element space for the continuous problem. The example above is such a method. If this condition is not satisfied, we obtain a
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10146:"A finite element perspective on nonlinear FFT-based micromechanical simulations: A FINITE ELEMENT PERSPECTIVE ON NONLINEAR FFT-BASED SIMULATIONS"
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Typically, one has an algorithm for subdividing a given mesh. If the primary method for increasing precision is to subdivide the mesh, one has an
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in 1969 for use in the analysis of ships. A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by
2013:
Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem (BVP) using the FEM.
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In the 1990s FEM was proposed for use in stochastic modeling for numerically solving probability models and later for reliability assessment.
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BeirĂŁo da Veiga, L.; Brezzi, F.; Cangiani, A.; Manzini, G.; Marini, L. D.; Russo, A. (2013). "Basic principles of
Virtual Element Methods".
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While it is difficult to quote the date of the invention of the finite element method, the method originated from the need to solve complex
4511:{\displaystyle V=\{v:\to \mathbb {R} \;:v{\text{ is continuous, }}v|_{}{\text{ is linear for }}k=0,\dots ,n{\text{, and }}v(0)=v(1)=0\}}
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1233:. Further impetus was provided in these years by available open-source finite element programs. NASA sponsored the original version of
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10105:"A Review of FE-FFT-Based Two-Scale Methods for Computational Modeling of Microstructure Evolution and Macroscopic Material Behavior"
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derives from the fact that knowledge of the local shape function basis is not required and is, in fact, never explicitly calculated.
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if the underlying PDE is linear and vice versa. Algebraic equation sets that arise in the steady-state problems are solved using
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One could consider the FDM a particular case of the FEM approach in several ways. E.g., first-order FEM is identical to FDM for
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Zohdi, T. I. (2018) A finite element primer for beginners-extended version including sample tests and projects. Second
Edition
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1973:(BVP) works only when there is one spatial dimension. It does not generalize to higher-dimensional problems or problems like
273:
9986:"Quantitative comparison between fast fourier transform and finite element method for micromechanical modeling of composite"
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10873:
10730:
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Peng Long; Wang
Jinliang; Zhu Qiding (19 May 1995). "Methods with high accuracy for finite element probability computing".
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Some types of finite element methods (conforming, nonconforming, mixed finite element methods) are particular cases of the
8118:, and there are efficient solvers for such problems (much more efficient than actually inverting the matrix.) In addition,
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947:(PDE). To explain the approximation in this process, the finite element method is commonly introduced as a special case of
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is customarily the diameter of the largest element in the mesh.) In this manner, if one shows that the error with a grid
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problem using FEM software. Colors indicate that the analyst has set material properties for each zone, in this case, a
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systematically recombining all sets of element equations into a global system of equations for the final calculation.
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8237:. In the preceding treatment, the grid consisted of triangles, but one can also use squares or curvilinear polygons.
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8644:). In the hp-FEM, the polynomial degrees can vary from element to element. High-order methods with large uniform
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1129:. In the USSR, the introduction of the practical application of the method is usually connected with the name of
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Hinton, Ernest; Irons, Bruce (July 1968). "Least squares smoothing of experimental data using finite elements".
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sets that occur in the transient problems are solved by numerical integration using standard techniques such as
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16 scaled and shifted triangular basis functions (colors) used to reconstruct a zeroeth order Bessel function
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1170:. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by
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in the later 1950s and early 1960s, based on the computations of dam constructions, where it was called the
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FDM is not usually used for irregular CAD geometries but more often for rectangular or block-shaped models.
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method. Under specific hypotheses (for instance, if the domain is convex), a piecewise polynomial of order
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2010:. For this reason, we will develop the finite element method for P1 and outline its generalization to P2.
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8987:(FDM) is an alternative way of approximating solutions of PDEs. The differences between FEM and FDM are:
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1141:. Although the approaches used by these pioneers are different, they share one essential characteristic:
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10227:"A comparison between dynamic implicit and explicit finite element simulations of the native knee joint"
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10008:"Simulation of micromechanical behavior of polycrystals: finite elements versus fast Fourier transforms"
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P. Solin, K. Segeth, I. Dolezel: Higher-Order Finite
Element Methods, Chapman & Hall/CRC Press, 2003
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Numerical
Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation
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Gierden, Christian; Kochmann, Julian; Waimann, Johanna; Svendsen, Bob; Reese, Stefanie (2022-10-01).
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Kiritsis, D.; Eemmanouilidis, Ch.; Koronios, A.; Mathew, J. (2009). "Engineering Asset Management".
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2020:
The second step is discretization, where the weak form is discretized in a finite-dimensional space.
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1476:{\displaystyle {\text{ P1 }}:{\begin{cases}u''(x)=f(x){\text{ in }}(0,1),\\u(0)=u(1)=0,\end{cases}}}
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4821:(above, in color) of this polygon which is linear on each triangle of the triangulation; the space
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3093:{\displaystyle \int _{\Omega }fv\,ds=-\int _{\Omega }\nabla u\cdot \nabla v\,ds\equiv -\phi (u,v),}
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The finite element method obtained its real impetus in the 1960s and 1970s by the developments of
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The virtual element method (VEM), introduced by BeirĂŁo da Veiga et al. (2013) as an extension of
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analogy, while Courant's approach divides the domain into finite triangular subregions to solve
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Zeman, J.; de Geus, T. W. J.; VondĆejc, J.; Peerlings, R. H. J.; Geers, M. G. D. (2017-09-07).
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1970:
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1294:, a discretization strategy, one or more solution algorithms, and post-processing procedures.
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The most attractive feature of finite differences is that it is straightforward to implement.
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7062:{\displaystyle -\sum _{k=1}^{n}u_{k}\phi (v_{k},v_{j})=\sum _{k=1}^{n}f_{k}\int v_{k}v_{j}dx}
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Hrennikoff, Alexander (1941). "Solution of problems of elasticity by the framework method".
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in the literature. Since we do not perform such an analysis, we will not use this notation.
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A clear, detailed, and practical presentation of this approach can be found in the textbook
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The global system of equations has known solution techniques and can be calculated from the
787:). To solve a problem, the FEM subdivides a large system into smaller, simpler parts called
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9723:
9710:
Burman, Erik; Claus, Susanne; Hansbo, Peter; Larson, Mats G.; Massing, André (2015-11-16).
9428:
9182:
9027:
8632:
smaller, one increases the degree of the polynomials used in the basis function, one has a
8596:
8316:
8045:
6516:
6489:
6264:
6224:
5695:
5628:
5581:
5554:
5145:
5078:
5051:
4914:
4730:
4158:
3369:
2494:
1802:
1226:
1118:
1102:
1039:
850:
544:
480:
453:
9038:
Another method used for approximating solutions to a partial differential equation is the
7647:
2381:
8:
10694:
10608:
9766:(June 2004). "Generalized Finite Element Methods: Main Ideas, Results, and Perspective".
9147:
8928:
6152:
5002:
4979:
4841:
would consist of functions that are linear on each triangle of the chosen triangulation.
3764:
3447:
3421:
2144:
2118:
2092:
1546:
1218:
1202:
1167:
1163:
1106:
969:
878:
757:
753:
576:
561:
462:
326:
165:
132:
123:
45:
Visualization of how a car deforms in an asymmetrical crash using finite element analysis
10070:
10031:
10023:
9817:
9727:
9432:
5766:
Examples of methods that use higher degree piecewise polynomial basis functions are the
3474:). Such functions are (weakly) once differentiable, and it turns out that the symmetric
10917:
10858:
10157:
10145:
9603:
9530:
9401:
9373:
9262:
9070:
is used to apply the boundary conditions, as more iterations of the method are needed.
9067:
9063:
8738:
8576:
8470:
8403:
8347:
8184:
8160:
8121:
8094:
8025:
8005:
7765:
5675:
5655:
5125:
5105:
5031:
4941:
4894:
4874:
4824:
4752:
4712:
4595:
4320:
4210:
4190:
4130:
4106:
4050:
3739:
3668:
3401:
2978:
2923:
2448:
2430:
2410:
2311:
2072:
2052:
1945:
1925:
1591:
1571:
1526:
1506:
1486:
1258:
1230:
1214:
894:
882:
800:
776:
571:
566:
449:
9825:
6291:
4612:
are not differentiable according to the elementary definition of calculus. Indeed, if
889:
cylindrical part shields the area inside the cylinder by diverting the magnetic field
10547:
10357:
10338:
10249:
10126:
10082:
10035:
9923:
9741:
9665:
9637:
9610:
9574:
9569:
Gard Paulsen; HĂ„kon With Andersen; John Petter Collett; Iver Tangen Stensrud (2014).
9405:
9393:
9340:
9315:
9287:
3733:
1612:
1241:
widely available. In Norway, the ship classification society Det Norske Veritas (now
1110:
1067:
1020:
1004:
628:
407:
142:
10245:
9984:
Ma, X; Parvathaneni, K; Lomov, S; Vasiukov, D; Shakoor, M; Park, C (December 2019).
9544:
9495:
9472:
9455:
9142:
1148:
of a continuous domain into a set of discrete sub-domains, usually called elements.
683:
10863:
10853:
10742:
10710:
10334:
10241:
10167:
10116:
10074:
10027:
9915:
9879:
9821:
9777:
9731:
9526:
9467:
9436:
9383:
9217:
8885:
8178:
8143:
3803:
with zero values at the endpoints (blue) and a piecewise linear approximation (red)
2038:
1334:
1266:
1079:
693:
678:
8659:
For vector partial differential equations, the basis functions may take values in
8244:
Separate consideration is the smoothness of the basis functions. For second-order
3310:
10905:
10848:
10837:
10423:
9631:
8653:
8213:
In general, the finite element method is characterized by the following process.
4970:
4124:
4068:
1855:
1298:
1152:
1126:
1122:
1043:
984:
956:
948:
633:
549:
76:
35:
9456:"Variational methods for the solution of problems of equilibrium and vibrations"
688:
10683:
10630:
10121:
10104:
10078:
9594:
9388:
9361:
9009:
6254:
5869:
5863:
in the disk centered at the origin and radius 1, with zero boundary conditions.
5804:
5778:
problem. Mesh adaptivity may utilize various techniques; the most popular are:
3471:
1966:
1274:
1250:
1179:
1145:
1047:
886:
854:
792:
653:
638:
444:
432:
151:
10055:"A variational fast Fourier transform method for phase-transforming materials"
9903:
9883:
9781:
9759:
4736:
10947:
10552:
10397:
The Finite Element Method: Linear Static and Dynamic Finite Element Analysis,
10187:
Proceedings of the 4th World Congress on Engineering Asset Management (WCEAM)
10130:
10086:
10039:
9927:
9919:
9745:
9397:
9247:
9012:
by a regular rectangular mesh with each rectangle divided into two triangles.
8115:
5874:
5548:
3551:
3500:
3304:
1270:
1206:
1194:
1059:
952:
937:
908:
The subdivision of a whole domain into simpler parts has several advantages:
890:
871:
796:
761:
10413:
10409:
10224:
9110:
9098:
3D pollution transport model - concentration field on perpendicular surface
4147:
as a space of piecewise polynomial functions for the finite element method.
3685:
solving (2) and, therefore, P1. This solution is a-priori only a member of
951:. The process, in mathematical language, is to construct an integral of the
10724:
10641:
10618:
10253:
9232:
9051:
8069:
have small support. So we now have to solve a linear system in the unknown
3475:
1186:
1175:
1142:
973:
842:
673:
623:
509:
137:
9847:
9033:
5884:
2427:
will solve P1. The proof is easier for twice continuously differentiable
10635:
10513:
9629:
9362:"Eighty Years of the Finite Element Method: Birth, Evolution, and Future"
9059:
8202:
5958:
The primary advantage of this choice of basis is that the inner products
3147:
1262:
1246:
1035:
1024:
81:
10487:
3554:(a detailed proof is nontrivial). On the other hand, the left-hand-side
9598:
6042:{\displaystyle \langle v_{j},v_{k}\rangle =\int _{0}^{1}v_{j}v_{k}\,dx}
1254:
961:
795:
in the space dimensions, which is implemented by the construction of a
765:
698:
9440:
8344:
Another consideration is the relation of the finite-dimensional space
2285:{\displaystyle \int _{0}^{1}f(x)v(x)\,dx=\int _{0}^{1}u''(x)v(x)\,dx.}
893:
by the coil (rectangular area on the right). The color represents the
756:. Typical problem areas of interest include the traditional fields of
10171:
9736:
9711:
9047:
8956:
1134:
846:
835:
439:
160:
103:
93:
10184:
9867:
8822:
Several research codes implement this technique to various degrees:
8701:
The Applied Element Method or AEM combines features of both FEM and
8636:-method. If one combines these two refinement types, one obtains an
41:
10162:
9985:
9378:
9055:
8959:
in 3D) that are highly irregular and non-convex in shape. The name
8934:
5551:. For the two-dimensional case, we choose again one basis function
3617:
3307:). The existence and uniqueness of the solution can also be shown.
3123:
2026:
1238:
811:
9712:"CutFEM: Discretizing geometry and partial differential equations"
9687:"CutFEM: Discretizing Partial Differential Equations and Geometry"
9094:
9090:
3D pollution transport model - concentration field on ground level
9086:
1345:
The following two problems demonstrate the finite element method.
10821:
10053:
Cruzado, A; Segurado, J; Hartl, D J; Benzerga, A A (2021-06-01).
9252:
6650:{\displaystyle \int _{\Omega }\nabla v_{j}\cdot \nabla v_{k}\,ds}
4794:
4310:{\displaystyle 0=x_{0}<x_{1}<\cdots <x_{n}<x_{n+1}=1}
1859:
1234:
1051:
114:
109:
98:
10324:
9630:
Olek C Zienkiewicz; Robert L Taylor; J.Z. Zhu (31 August 2013).
8998:
FEM generally allows for more flexible mesh adaptivity than FDM.
8840:
6142:{\displaystyle \phi (v_{j},v_{k})=\int _{0}^{1}v_{j}'v_{k}'\,dx}
1237:. UC Berkeley made the finite element programs SAP IV and later
10660:
10354:
Reliability Assessment Using Stochastic Finite Element Analysis
8966:
8766:
8641:
8151:
8138:
is symmetric and positive definite, so a technique such as the
5792:
5767:
3311:
A proof outline of the existence and uniqueness of the solution
1310:
1242:
992:
10102:
9547:. NISEE e-Library, The Earthquake Engineering Online Archive.
8777:
to achieve exceptionally fast, exponential convergence rates.
8364:
to its infinite-dimensional counterpart in the examples above
6540:
do not share an edge of the triangulation, then the integrals
5008:
The linear combination of basis functions (yellow) reproduces
10143:
10059:
Modelling and Simulation in Materials Science and Engineering
10012:
Modelling and Simulation in Materials Science and Engineering
9074:
8978:
2037:
The first step is to convert P1 and P2 into their equivalent
1306:
31:
27:
Numerical method for solving physical or engineering problems
10178:
10052:
9844:
State Key Laboratory of Scientific and Engineering Computing
8797:, and global differentiability of the local approximations (
8208:
3922:{\displaystyle \forall v\in H_{0}^{1},\;-\phi (u,v)=\int fv}
10815:
10809:
10624:
9983:
8312:, one may use piecewise quadratic basis functions that are
5493:
1763:
1469:
858:
10441:, Springer-Verlag New York, ISBN 978-0-387-75933-3 (2008).
10150:
International Journal for Numerical Methods in Engineering
9716:
International Journal for Numerical Methods in Engineering
8691:
2089:
that satisfies the displacement boundary conditions, i.e.
1285:
10415:
Finite Element Methods for Partial Differential Equations
9573:. Lysaker, Norway: Dinamo Forlag A/S. pp. 121, 436.
4709:. However, the derivative exists at every other value of
1305:
Examples of discretization strategies are the h-version,
1301:, the discontinuous Galerkin method, mixed methods, etc.
1023:. The process is often carried out by FEM software using
10446:
https://link.springer.com/book/10.1007/978-3-319-70428-9
10287:
Accuracy and Economy of Finite Element Magnetic Analysis
8142:
is favored. For problems that are not too large, sparse
3172:
can be turned into an inner product on a suitable space
9709:
9034:
Finite element and fast fourier transform (FFT) methods
30:"Finite element" redirects here. For the elements of a
10201:"Finite Element Analysis: How to create a great model"
9066:, and loses some of its efficiency in cases where the
5048:. In the one-dimensional case, for each control point
940:
of the original problem to obtain a numerical answer.
10424:
The Finite Element Method: Its Basis and Fundamentals
9806:
Computer Methods in Applied Mechanics and Engineering
9758:
9633:
The Finite Element Method: Its Basis and Fundamentals
9623:
8665:
8599:
8579:
8549:
8523:
8493:
8473:
8430:
8406:
8370:
8350:
8319:
8254:
8223:
8187:
8163:
8124:
8097:
8075:
8048:
8028:
8008:
7967:
7915:
7902:{\displaystyle \mathbf {b} =(b_{1},\dots ,b_{n})^{t}}
7848:
7793:
7768:
7730:
7679:
7650:
7568:
7510:
7443:
7378:
7336:
7294:
7235:
7176:
7154:
7132:
7075:
6929:
6886:
6835:
6753:
6671:
6600:
6546:
6519:
6492:
6447:
6398:
6352:
6294:
6267:
6227:
6181:
6155:
6055:
5964:
5894:
5814:
5725:
5698:
5678:
5658:
5631:
5611:
5584:
5557:
5515:
5215:
5175:
5148:
5128:
5108:
5081:
5054:
5034:
4944:
4917:
4897:
4877:
4851:
4827:
4803:
4775:
4755:
4715:
4677:
4644:
4618:
4598:
4559:
4526:
4343:
4323:
4233:
4213:
4193:
4161:
4133:
4109:
4077:
4053:
3971:
3944:
3857:
3817:
3774:
3742:
3691:
3671:
3625:
3560:
3509:
3483:
3450:
3424:
3404:
3372:
3321:
3262:
3239:
3219:
3178:
3156:
3132:
3108:
3001:
2981:
2946:
2926:
2864:
2508:
2464:
2433:
2413:
2384:
2334:
2314:
2178:
2147:
2121:
2095:
2075:
2055:
1979:
1948:
1928:
1898:
1868:
1837:
1805:
1785:
1621:
1594:
1574:
1549:
1529:
1509:
1489:
1354:
1139:
finite difference method based on variation principle
1133:. It was also independently rediscovered in China by
991:
These equation sets are element equations. They are
845:
created by an analyst before finding a solution to a
10489:
Numerical methods for partial differential equations
10289:, 33rd Annual National Relay Conference, April 1985.
9360:
Liu, Wing Kam; Li, Shaofan; Park, Harold S. (2022).
8731:
8002:
As we have discussed before, most of the entries of
4789:. In the figure on the right, we have illustrated a
4638:
then the derivative is typically not defined at any
4024:{\displaystyle \forall v\in V,\;-\phi (u,v)=\int fv}
1151:
Hrennikoff's work discretizes the domain by using a
9871:
Mathematical Models and Methods in Applied Sciences
1965:The problem P1 can be solved directly by computing
10421:O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu :
9956:"What's The Difference Between FEM, FDM, and FVM?"
9602:
8931:is an iterative method in finite element methods.
8680:
8617:
8585:
8561:
8535:
8509:
8479:
8448:
8412:
8388:
8356:
8332:
8304:
8229:
8193:
8169:
8130:
8103:
8083:
8061:
8034:
8014:
7991:
7953:
7901:
7818:
7774:
7754:
7716:
7665:
7636:
7538:
7491:
7429:
7364:
7322:
7280:
7221:
7162:
7140:
7102:
7061:
6910:
6872:
6821:
6739:
6649:
6586:
6532:
6505:
6475:
6433:
6384:
6338:
6280:
6245:
6213:
6167:
6141:
6041:
5947:
5855:
5751:
5711:
5684:
5664:
5644:
5617:
5597:
5570:
5539:
5499:
5201:
5161:
5134:
5114:
5094:
5067:
5040:
4950:
4930:
4903:
4883:
4863:
4833:
4809:
4781:
4761:
4721:
4701:
4663:
4630:
4604:
4584:
4545:
4510:
4329:
4309:
4219:
4199:
4179:
4139:
4115:
4095:
4059:
4023:
3956:
3921:
3841:
3795:
3748:
3724:
3677:
3653:
3608:
3542:
3491:
3462:
3436:
3410:
3390:
3354:
3295:
3248:
3225:
3205:
3164:
3138:
3114:
3092:
2987:
2967:
2932:
2900:
2834:
2485:
2439:
2419:
2399:
2370:
2320:
2284:
2159:
2133:
2107:
2081:
2061:
2002:
1954:
1934:
1914:
1884:
1846:
1823:
1791:
1769:
1600:
1580:
1560:
1535:
1515:
1495:
1475:
1340:
10439:The Mathematical Theory of Finite Element Methods
9073:The FE and FFT methods can also be combined in a
8890:
7637:{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}
6822:{\displaystyle f(x)=\sum _{k=1}^{n}f_{k}v_{k}(x)}
6740:{\displaystyle u(x)=\sum _{k=1}^{n}u_{k}v_{k}(x)}
5024:To complete the discretization, we must select a
3484:
3158:
1086:in the ocean) rather than relatively calm areas.
877:FEM solution to the problem at left, involving a
10945:
10404:Finite Elements Methods for Engineering Sciences
10327:Journal of Computational and Applied Mathematics
10298:
10292:
10109:Archives of Computational Methods in Engineering
10005:
9366:Archives of Computational Methods in Engineering
8935:Crystal plasticity finite element method (CPFEM)
8901:
8793:, polynomial degree of the local approximations
8789:combines adaptively elements with variable size
8769:combines adaptively elements with variable size
5785:refining (and unrefined) elements (h-adaptivity)
3665:for Hilbert spaces shows that there is a unique
1297:Examples of the variational formulation are the
1257:. The method has since been generalized for the
1117:. Its development can be traced back to work by
10351:
10301:"McLaren Mercedes: Feature - Stress to impress"
8943:
5788:changing order of base functions (p-adaptivity)
10352:Haldar, Achintya; Mahadevan, Sankaran (2000).
9769:International Journal of Computational Methods
9281:
8745:
1922:denote the second derivatives with respect to
1290:A finite element method is characterized by a
814:a phenomenon with FEM is often referred to as
748:) is a popular method for numerically solving
10473:
10285:Hastings, J. K., Juds, M. A., Brauer, J. R.,
9940:: CS1 maint: DOI inactive as of April 2024 (
9460:Bulletin of the American Mathematical Society
9178:Finite element method in structural mechanics
8841:Scaled boundary finite element method (SBFEM)
7539:{\displaystyle -L\mathbf {u} =M\mathbf {f} .}
6660:
6587:{\displaystyle \int _{\Omega }v_{j}v_{k}\,ds}
5799:
5075:we will choose the piecewise linear function
4740:A piecewise linear function in two dimensions
1030:The practical application of FEM is known as
721:
10390:Numerical methods in finite element analysis
9571:Building Trust, The history of DNV 1864-2014
9312:An Introduction to the Finite Element Method
9026:(CFD) tend to use FDM or other methods like
8967:Link with the gradient discretization method
8648:are called spectral finite element methods (
7819:{\displaystyle -L\mathbf {u} =\mathbf {b} ,}
6379:
6353:
6208:
6182:
5991:
5965:
4505:
4350:
783:in two or three space variables (i.e., some
10432:Introduction to the Finite Element Method,
10281:
10279:
10277:
9516:
9284:A first course in the finite element method
8801:-1) to achieve the best convergence rates.
6385:{\displaystyle \langle v_{j},v_{k}\rangle }
6214:{\displaystyle \langle v_{j},v_{k}\rangle }
3616:is also an inner product, this time on the
915:Inclusion of dissimilar material properties
912:Accurate representation of complex geometry
10480:
10466:
9418:
9114:Finite Element Model of a human knee joint
8979:Comparison to the finite difference method
8864:
7762:becomes actually simpler, since no matrix
5739:
5605:of the triangulation of the planar region
5189:
4382:
3987:
3885:
791:. This is achieved by a particular space
728:
714:
10161:
10120:
10006:Prakash, A; Lebensohn, R A (2009-09-01).
9735:
9684:
9593:
9471:
9387:
9377:
9359:
8668:
8209:General form of the finite element method
7430:{\displaystyle L_{ij}=\phi (v_{i},v_{j})}
6640:
6577:
6132:
6032:
5405:
5380:
5283:
4378:
3485:
3157:
3056:
3018:
2916:If we integrate by parts using a form of
2794:
2725:
2610:
2552:
2272:
2218:
1261:of physical systems in a wide variety of
1038:, is a computational tool for performing
925:Typical work out of the method involves:
918:Easy representation of the total solution
10274:
9803:
9605:An Analysis of The Finite Element Method
9213:List of finite element software packages
9109:
9093:
9085:
8912:
8760:
8091:where most of the entries of the matrix
7492:{\displaystyle M_{ij}=\int v_{i}v_{j}dx}
7281:{\displaystyle (f_{1},\dots ,f_{n})^{t}}
7222:{\displaystyle (u_{1},\dots ,u_{n})^{t}}
5883:
5868:
5803:
4735:
3763:
3150:in the two-dimensional plane. Once more
2378:satisfies (1) for every smooth function
2069:solves P1, then for any smooth function
1125:in the early 1940s. Another pioneer was
1019:as applied in relation to the reference
40:
9453:
9337:The Finite Element Method for Engineers
9334:
8692:Various types of finite element methods
3609:{\displaystyle \int _{0}^{1}f(x)v(x)dx}
3296:{\displaystyle v\in H_{0}^{1}(\Omega )}
2858:where we have used the assumption that
1286:The structure of finite element methods
1280:
1160:elliptic partial differential equations
1091:The Finite Element Method for Engineers
14:
10946:
10454:, SIAM, ISBN 978-1-61197-772-1 (2024).
10452:Mathematical Theory of Finite Elements
10437:Susanne C. Brenner, L. Ridgway Scott:
9901:
9188:Finite volume method for unsteady flow
8923:
4729:, and one can use this derivative for
4103:. There are many possible choices for
2911:
2044:
1969:. However, this method of solving the
10461:
10098:
10096:
10001:
9999:
9904:"Option pricing with finite elements"
9897:
9895:
9893:
9664:. Cambridge, MA: Klaus-JĂŒrgen Bathe.
9656:
9309:
9228:Multidisciplinary design optimization
8652:). These are not to be confused with
8111:, which we need to invert, are zero.
8042:are zero because the basis functions
5547:; this basis is a shifted and scaled
10731:Moving particle semi-implicit method
10642:Weighted essentially non-oscillatory
9305:
9303:
7784:
7501:
6920:
6257:.) In the one dimensional case, the
5948:{\displaystyle u(x,y)=1-x^{2}-y^{2}}
5808:Solving the two-dimensional problem
3934:
3213:of once differentiable functions of
2499:
2169:
1027:data generated from the subdomains.
176:List of named differential equations
8879:
8593:method will have an error of order
8305:{\displaystyle u_{xxxx}+u_{yyyy}=f}
7954:{\displaystyle b_{j}=\int fv_{j}dx}
4961:
4911:. For this reason, one often reads
3932:with a finite-dimensional version:
2032:
249:Dependent and independent variables
24:
10580:Finite-difference frequency-domain
10373:
10093:
9996:
9890:
9531:10.1111/j.1475-1305.1968.tb01368.x
8530:
8224:
6627:
6611:
6606:
6552:
6486:Similarly, in the planar case, if
6434:{\displaystyle \phi (v_{j},v_{k})}
5612:
4804:
4776:
3972:
3858:
3287:
3243:
3240:
3220:
3206:{\displaystyle H_{0}^{1}(\Omega )}
3197:
3109:
3050:
3041:
3036:
3007:
1841:
1838:
1799:is a connected open region in the
1786:
1754:
1751:
1725:
25:
11000:
10234:Medical Engineering & Physics
9300:
9158:Discontinuity layout optimization
8732:Generalized finite element method
4744:
4150:
3759:
1611:P2 is a two-dimensional problem (
825:
10964:Numerical differential equations
9258:Tessellation (computer graphics)
8681:{\displaystyle \mathbb {R} ^{n}}
8246:elliptic boundary value problems
8077:
7850:
7809:
7801:
7529:
7518:
7156:
7134:
5880:of the discretized linear system
5856:{\displaystyle u_{xx}+u_{yy}=-4}
5017:(black) to any desired accuracy.
5001:
4978:
3249:{\displaystyle \partial \Omega }
2458:We define a new operator or map
1847:{\displaystyle \partial \Omega }
1348:P1 is a one-dimensional problem
870:
834:
384:Carathéodory's existence theorem
10933:Method of fundamental solutions
10719:Smoothed-particle hydrodynamics
10427:, Butterworth-Heinemann (2005).
10345:
10318:
10263:from the original on 2018-07-19
10246:10.1016/j.medengphy.2016.06.001
10218:
10193:
10137:
10046:
9977:
9966:from the original on 2017-07-28
9948:
9902:Topper, JĂŒrgen (January 2005).
9861:
9832:
9797:
9788:
9752:
9703:
9678:
9650:
9587:
9562:
9551:from the original on 2013-03-09
9473:10.1090/s0002-9904-1943-07818-4
9314:(Third ed.). McGraw-Hill.
8150:still work well. For instance,
5752:{\displaystyle x_{j},\;j\neq k}
5202:{\displaystyle x_{j},\;j\neq k}
2497:on the right-hand-side of (1):
1341:Illustrative problems P1 and P2
1162:that arise from the problem of
981:ordinary differential equations
10979:Computational electromagnetics
10969:Partial differential equations
10574:Alternating direction-implicit
9537:
9510:
9480:
9447:
9412:
9353:
9328:
9275:
9081:
8973:gradient discretization method
8891:Discontinuous Galerkin methods
8856:Smoothed finite element method
8817:extended finite element method
8811:Extended finite element method
8177:is usually referred to as the
7890:
7857:
7711:
7705:
7689:
7683:
7660:
7654:
7631:
7625:
7578:
7572:
7562:It is not necessary to assume
7424:
7398:
7372:be matrices whose entries are
7359:
7343:
7317:
7301:
7269:
7236:
7210:
7177:
6993:
6967:
6867:
6861:
6845:
6839:
6816:
6810:
6763:
6757:
6734:
6728:
6681:
6675:
6463:
6449:
6441:are identically zero whenever
6428:
6402:
6333:
6295:
6240:
6228:
6085:
6059:
5910:
5898:
5467:
5435:
5351:
5319:
5232:
5226:
4496:
4490:
4481:
4475:
4436:
4404:
4399:
4374:
4371:
4359:
4174:
4162:
4123:(one possibility leads to the
4006:
3994:
3904:
3892:
3842:{\displaystyle u\in H_{0}^{1}}
3736:regularity, will be smooth if
3725:{\displaystyle H_{0}^{1}(0,1)}
3719:
3707:
3648:
3636:
3597:
3591:
3585:
3579:
3543:{\displaystyle H_{0}^{1}(0,1)}
3537:
3525:
3385:
3373:
3355:{\displaystyle H_{0}^{1}(0,1)}
3349:
3337:
3290:
3284:
3200:
3194:
3084:
3072:
2962:
2950:
2940:solves P2, then we may define
2889:
2883:
2874:
2868:
2822:
2810:
2791:
2785:
2774:
2768:
2722:
2716:
2705:
2699:
2658:
2653:
2647:
2641:
2635:
2607:
2601:
2595:
2589:
2549:
2543:
2537:
2531:
2480:
2468:
2394:
2388:
2359:
2353:
2344:
2338:
2269:
2263:
2257:
2251:
2215:
2209:
2203:
2197:
1818:
1806:
1715:
1703:
1694:
1682:
1663:
1651:
1454:
1448:
1439:
1433:
1420:
1408:
1400:
1394:
1385:
1379:
1001:ordinary differential equation
945:partial differential equations
781:partial differential equations
471: / Integral solutions
13:
1:
10586:Finite-difference time-domain
10032:10.1088/0965-0393/17/6/064010
9826:10.1016/S0045-7825(97)00021-2
9545:"SAP-IV Software and Manuals"
9269:
8908:Finite element limit analysis
8902:Finite element limit analysis
8897:Discontinuous Galerkin method
7717:{\displaystyle v(x)=v_{j}(x)}
7103:{\displaystyle j=1,\dots ,n.}
6873:{\displaystyle v(x)=v_{j}(x)}
4702:{\displaystyle k=1,\ldots ,n}
1056:EulerâBernoulli beam equation
10625:Advection upstream-splitting
10339:10.1016/0377-0427(94)00027-X
9421:Journal of Applied Mechanics
9335:Huebner, Kenneth H. (2001).
9024:computational fluid dynamics
8944:Virtual element method (VEM)
8536:{\displaystyle C<\infty }
8422:nonconforming element method
8084:{\displaystyle \mathbf {u} }
7992:{\displaystyle j=1,\dots ,n}
7755:{\displaystyle j=1,\dots ,n}
7499:then we may rephrase (4) as
7163:{\displaystyle \mathbf {f} }
7141:{\displaystyle \mathbf {u} }
6911:{\displaystyle j=1,\dots ,n}
6149:will be zero for almost all
5540:{\displaystyle k=1,\dots ,n}
4817:in the plane (below), and a
4769:to be a set of functions of
4592:. Observe that functions in
3663:Riesz representation theorem
3256:. We have also assumed that
2407:then one may show that this
1568:is the second derivative of
1076:numerical weather prediction
515:Exponential response formula
261:Coupled / Decoupled
7:
10636:Essentially non-oscillatory
10619:Monotonic upstream-centered
9125:
8752:Mixed finite element method
8746:Mixed finite element method
8114:Such matrices are known as
7832:
7552:
7116:
6346:. Hence, the integrands of
5888:(c) The computed solution,
5791:combinations of the above (
5782:moving nodes (r-adaptivity)
4037:
2901:{\displaystyle v(0)=v(1)=0}
2848:
2371:{\displaystyle u(0)=u(1)=0}
2298:
1066:expressed in either PDE or
752:arising in engineering and
10:
11005:
10896:Infinite difference method
10514:Forward-time central-space
10430:N. Ottosen, H. Petersson:
10406:, Springer Verlag, (2008).
10122:10.1007/s11831-022-09735-6
9840:"Spectral Element Methods"
9389:10.1007/s11831-022-09740-9
9223:Movable cellular automaton
8916:
8905:
8894:
8883:
8868:
8853:
8808:
8780:
8749:
8707:
7365:{\displaystyle M=(M_{ij})}
7323:{\displaystyle L=(L_{ij})}
6661:Matrix form of the problem
6476:{\displaystyle |j-k|>1}
5800:Small support of the basis
5652:is the unique function of
4391: is continuous,
3796:{\displaystyle H_{0}^{1},}
3654:{\displaystyle L^{2}(0,1)}
2968:{\displaystyle \phi (u,v)}
2486:{\displaystyle \phi (u,v)}
1523:is an unknown function of
1096:
1046:techniques for dividing a
29:
10830:
10799:PoincarĂ©âSteklov operator
10752:
10709:
10651:
10599:
10566:
10558:Method of characteristics
10528:
10504:
10495:
10379:G. Allaire and A. Craig:
10356:. John Wiley & Sons.
10299:McLaren-Mercedes (2006).
9884:10.1142/S0218202512500492
9782:10.1142/S0219876204000083
9662:Finite Element Procedures
9636:. Butterworth-Heinemann.
9208:Lattice Boltzmann methods
8722:
8449:{\displaystyle H_{0}^{1}}
8398:conforming element method
8389:{\displaystyle H_{0}^{1}}
8140:conjugate gradient method
7644:. For a general function
6829:then problem (3), taking
6253:location is known as the
6175:. (The matrix containing
4819:piecewise linear function
4585:{\displaystyle x_{n+1}=1}
4443: is linear for
4096:{\displaystyle H_{0}^{1}}
3492:{\displaystyle \!\,\phi }
3165:{\displaystyle \,\!\phi }
2451:) but may be proved in a
1042:. It includes the use of
1034:(FEA). FEA as applied in
921:Capture of local effects.
770:electromagnetic potential
649:JĂłzef Maria Hoene-WroĆski
595:Undetermined coefficients
504:Method of characteristics
389:CauchyâKowalevski theorem
10816:Tearing and interconnect
10810:Balancing by constraints
10079:10.1088/1361-651X/abe4c7
9920:10.1002/wilm.42820050119
9685:celledoni (2023-02-27).
9168:Finite difference method
8985:finite difference method
8849:
8714:
8569:, then one has an order
4067:is a finite-dimensional
3661:. An application of the
3315:We can loosely think of
1115:aeronautical engineering
997:numerical linear algebra
955:of the residual and the
374:PicardâLindelöf theorem
368:Existence and uniqueness
10923:Computer-assisted proof
10901:Infinite element method
10689:Gradient discretisation
10392:, Prentice-Hall (1976).
9922:(inactive 2024-04-07).
9282:Daryl L. Logan (2011).
9198:Interval finite element
9193:Infinite element method
9163:Discrete element method
9153:Direct stiffness method
9138:Boundary element method
8871:Spectral element method
8865:Spectral element method
8804:
8703:Discrete element method
8230:{\displaystyle \Omega }
8217:One chooses a grid for
8148:Cholesky decompositions
5618:{\displaystyle \Omega }
4810:{\displaystyle \Omega }
4782:{\displaystyle \Omega }
4664:{\displaystyle x=x_{k}}
4546:{\displaystyle x_{0}=0}
3226:{\displaystyle \Omega }
3115:{\displaystyle \nabla }
2003:{\displaystyle u+V''=f}
1792:{\displaystyle \Omega }
1292:variational formulation
1191:University of Stuttgart
1189:with co-workers at the
1064:Navier-Stokes equations
1032:finite element analysis
1017:orientation adjustments
853:wire coil in orange; a
816:finite element analysis
785:boundary value problems
600:Variation of parameters
590:Separation of variables
379:Peano existence theorem
10911:PetrovâGalerkin method
10672:Discontinuous Galerkin
9173:Finite element machine
9133:Applied element method
9115:
9099:
9091:
9040:Fast Fourier Transform
8773:and polynomial degree
8710:Applied element method
8696:
8682:
8619:
8587:
8563:
8562:{\displaystyle p>0}
8537:
8511:
8510:{\displaystyle Ch^{p}}
8481:
8450:
8414:
8400:is one in which space
8390:
8358:
8334:
8306:
8231:
8195:
8171:
8132:
8105:
8085:
8063:
8036:
8016:
7993:
7955:
7903:
7820:
7776:
7756:
7718:
7667:
7638:
7604:
7540:
7493:
7431:
7366:
7324:
7282:
7223:
7164:
7142:
7104:
7063:
7019:
6953:
6912:
6874:
6823:
6789:
6741:
6707:
6651:
6588:
6534:
6507:
6477:
6435:
6386:
6340:
6282:
6247:
6215:
6169:
6143:
6043:
5955:
5949:
5881:
5866:
5865:(a) The triangulation.
5857:
5753:
5713:
5686:
5666:
5646:
5619:
5599:
5572:
5541:
5501:
5203:
5163:
5136:
5116:
5096:
5069:
5042:
4952:
4932:
4905:
4891:must also change with
4885:
4865:
4864:{\displaystyle h>0}
4835:
4811:
4783:
4763:
4741:
4723:
4703:
4665:
4632:
4631:{\displaystyle v\in V}
4606:
4586:
4547:
4512:
4331:
4311:
4221:
4201:
4181:
4141:
4117:
4097:
4061:
4025:
3958:
3957:{\displaystyle u\in V}
3923:
3843:
3804:
3797:
3750:
3726:
3679:
3655:
3610:
3544:
3493:
3464:
3438:
3412:
3392:
3356:
3297:
3250:
3227:
3207:
3166:
3140:
3139:{\displaystyle \cdot }
3116:
3094:
2989:
2969:
2934:
2902:
2836:
2487:
2441:
2421:
2401:
2372:
2322:
2286:
2161:
2135:
2109:
2083:
2063:
2004:
1971:boundary value problem
1956:
1936:
1916:
1915:{\displaystyle u_{yy}}
1886:
1885:{\displaystyle u_{xx}}
1848:
1825:
1793:
1771:
1602:
1582:
1562:
1537:
1517:
1497:
1477:
1082:in the atmosphere, or
999:methods. In contrast,
805:calculus of variations
768:, mass transport, and
750:differential equations
669:Carl David Tolmé Runge
212:Differential-algebraic
53:Differential equations
46:
10954:Finite element method
10891:Isogeometric analysis
10737:Material point method
10450:Leszek F. Demkowicz:
10434:Prentice-Hall (1992).
10399:Prentice-Hall (1987).
9310:Reddy, J. N. (2006).
9203:Isogeometric analysis
9113:
9097:
9089:
8919:Stretched grid method
8913:Stretched grid method
8761:Variable â polynomial
8683:
8628:If instead of making
8620:
8618:{\displaystyle p=d+1}
8588:
8564:
8538:
8512:
8482:
8451:
8415:
8391:
8359:
8335:
8333:{\displaystyle C^{1}}
8307:
8232:
8196:
8172:
8133:
8106:
8086:
8064:
8062:{\displaystyle v_{k}}
8037:
8017:
7994:
7956:
7904:
7821:
7777:
7757:
7719:
7668:
7639:
7584:
7541:
7494:
7432:
7367:
7325:
7283:
7224:
7165:
7143:
7105:
7064:
6999:
6933:
6913:
6875:
6824:
6769:
6742:
6687:
6652:
6589:
6535:
6533:{\displaystyle x_{k}}
6508:
6506:{\displaystyle x_{j}}
6478:
6436:
6387:
6341:
6283:
6281:{\displaystyle v_{k}}
6248:
6246:{\displaystyle (j,k)}
6216:
6170:
6144:
6044:
5950:
5887:
5872:
5858:
5807:
5754:
5714:
5712:{\displaystyle x_{k}}
5687:
5667:
5647:
5645:{\displaystyle v_{k}}
5620:
5600:
5598:{\displaystyle x_{k}}
5573:
5571:{\displaystyle v_{k}}
5542:
5502:
5204:
5164:
5162:{\displaystyle x_{k}}
5137:
5117:
5097:
5095:{\displaystyle v_{k}}
5070:
5068:{\displaystyle x_{k}}
5043:
4953:
4933:
4931:{\displaystyle V_{h}}
4906:
4886:
4866:
4836:
4812:
4784:
4764:
4739:
4724:
4704:
4666:
4633:
4607:
4587:
4548:
4513:
4332:
4312:
4222:
4202:
4182:
4180:{\displaystyle (0,1)}
4155:We take the interval
4142:
4118:
4098:
4062:
4026:
3959:
3924:
3844:
3798:
3767:
3751:
3727:
3680:
3656:
3611:
3545:
3494:
3465:
3439:
3413:
3393:
3391:{\displaystyle (0,1)}
3364:absolutely continuous
3357:
3298:
3251:
3228:
3208:
3167:
3141:
3117:
3095:
2990:
2970:
2935:
2903:
2837:
2488:
2442:
2422:
2402:
2373:
2323:
2287:
2162:
2136:
2110:
2084:
2064:
2005:
1957:
1937:
1917:
1887:
1849:
1831:plane whose boundary
1826:
1824:{\displaystyle (x,y)}
1794:
1772:
1603:
1583:
1563:
1538:
1518:
1498:
1478:
1319:isogeometric analysis
1221:at the University of
899:magnetic flux density
775:The FEM is a general
754:mathematical modeling
742:finite element method
659:Augustin-Louis Cauchy
644:Joseph-Louis Lagrange
476:Numerical integration
458:Exponential stability
321:Relation to processes
44:
10928:Integrable algorithm
10754:Domain decomposition
10395:Thomas J.R. Hughes:
10205:Coventive Composites
9498:on 30 September 2015
9454:Courant, R. (1943).
9286:. Cengage Learning.
9243:RayleighâRitz method
9183:Finite volume method
9050:(an FCC metal), and
9028:finite volume method
8663:
8597:
8577:
8547:
8521:
8491:
8487:is bounded above by
8471:
8428:
8404:
8368:
8348:
8317:
8252:
8221:
8185:
8161:
8122:
8095:
8073:
8046:
8026:
8006:
7965:
7913:
7846:
7791:
7766:
7728:
7677:
7673:, problem (3) with
7666:{\displaystyle f(x)}
7648:
7566:
7508:
7441:
7376:
7334:
7292:
7233:
7174:
7152:
7130:
7073:
6927:
6884:
6833:
6751:
6669:
6598:
6544:
6517:
6490:
6445:
6396:
6350:
6292:
6265:
6225:
6179:
6153:
6053:
5962:
5892:
5812:
5723:
5696:
5676:
5656:
5629:
5609:
5582:
5555:
5513:
5213:
5173:
5146:
5126:
5106:
5079:
5052:
5032:
4942:
4915:
4895:
4875:
4849:
4825:
4801:
4773:
4753:
4731:integration by parts
4713:
4675:
4642:
4616:
4596:
4557:
4524:
4341:
4321:
4231:
4211:
4191:
4159:
4131:
4127:). However, we take
4107:
4075:
4051:
3969:
3942:
3855:
3815:
3772:
3740:
3689:
3669:
3623:
3558:
3507:
3481:
3448:
3422:
3402:
3370:
3319:
3260:
3237:
3217:
3176:
3154:
3130:
3106:
2999:
2979:
2944:
2924:
2862:
2506:
2495:integration by parts
2462:
2431:
2411:
2400:{\displaystyle v(x)}
2382:
2332:
2312:
2176:
2145:
2119:
2093:
2073:
2053:
1977:
1946:
1926:
1896:
1866:
1835:
1803:
1783:
1619:
1592:
1572:
1547:
1527:
1507:
1487:
1352:
1281:Technical discussion
1119:Alexander Hrennikoff
1040:engineering analysis
481:Dirac delta function
217:Integro-differential
10984:Canadian inventions
10974:Structural analysis
10959:Continuum mechanics
10772:Schwarz alternating
10695:Loubignac iteration
10071:2021MSMSE..29d5001C
10024:2009MSMSE..17f4010P
9990:FiBreMoD Conference
9818:1997CMAME.147..329S
9728:2015IJNME.104..472B
9433:1941JAM.....8A.169H
9148:Computer experiment
8929:Loubignac iteration
8924:Loubignac iteration
8445:
8385:
8181:, while the matrix
7170:the column vectors
6168:{\displaystyle j,k}
6131:
6118:
6105:
6011:
4969:Interpolation of a
4092:
3881:
3838:
3789:
3706:
3575:
3524:
3463:{\displaystyle x=1}
3437:{\displaystyle x=0}
3336:
3283:
3193:
2912:The weak form of P2
2759:
2690:
2672:
2580:
2527:
2242:
2193:
2160:{\displaystyle x=1}
2134:{\displaystyle x=0}
2108:{\displaystyle v=0}
2045:The weak form of P1
1561:{\displaystyle u''}
1333:the realization of
1265:disciplines, e.g.,
1229:with co-workers at
1219:Philippe G. Ciarlet
1197:with co-workers at
1107:structural analysis
970:algebraic equations
857:component (perhaps
801:algebraic equations
758:structural analysis
577:Perturbation theory
572:Integral transforms
463:Rate of convergence
329:(discrete analogue)
166:Population dynamics
133:Continuum mechanics
124:Applied mathematics
10989:Russian inventions
10918:Validated numerics
9762:; Banerjee, Uday;
9263:Weakened weak form
9116:
9100:
9092:
9008:if the problem is
9006:Poisson's equation
8739:partition of unity
8678:
8615:
8583:
8559:
8533:
8507:
8477:
8446:
8431:
8410:
8386:
8371:
8354:
8330:
8302:
8227:
8191:
8167:
8128:
8101:
8081:
8059:
8032:
8012:
7989:
7951:
7899:
7816:
7772:
7752:
7714:
7663:
7634:
7536:
7489:
7427:
7362:
7320:
7278:
7219:
7160:
7138:
7100:
7059:
6908:
6870:
6819:
6737:
6647:
6584:
6530:
6503:
6473:
6431:
6382:
6336:
6278:
6243:
6211:
6165:
6139:
6119:
6106:
6091:
6039:
5997:
5956:
5945:
5882:
5867:
5853:
5749:
5719:and zero at every
5709:
5682:
5662:
5642:
5615:
5595:
5568:
5537:
5497:
5492:
5199:
5169:and zero at every
5159:
5132:
5112:
5092:
5065:
5038:
4948:
4928:
4901:
4881:
4861:
4831:
4807:
4779:
4759:
4742:
4719:
4699:
4661:
4628:
4602:
4582:
4543:
4508:
4327:
4307:
4217:
4197:
4177:
4137:
4113:
4093:
4078:
4057:
4021:
3954:
3919:
3867:
3839:
3824:
3805:
3793:
3775:
3746:
3722:
3692:
3675:
3651:
3606:
3561:
3540:
3510:
3489:
3460:
3434:
3408:
3388:
3352:
3322:
3293:
3269:
3246:
3223:
3203:
3179:
3162:
3136:
3112:
3090:
2985:
2965:
2930:
2918:Green's identities
2898:
2832:
2830:
2745:
2676:
2656:
2566:
2513:
2483:
2449:mean value theorem
2437:
2417:
2397:
2368:
2318:
2282:
2228:
2179:
2157:
2131:
2105:
2079:
2059:
2000:
1952:
1932:
1912:
1882:
1844:
1821:
1789:
1767:
1762:
1598:
1578:
1558:
1533:
1513:
1493:
1473:
1468:
1259:numerical modeling
1231:Cornell University
1215:Swansea University
1068:integral equations
1009:Runge-Kutta method
567:Integrating factor
408:Initial conditions
343:Stochastic partial
47:
10941:
10940:
10881:Immersed boundary
10874:Method of moments
10789:NeumannâDirichlet
10782:abstract additive
10767:Fictitious domain
10711:Meshless/Meshfree
10595:
10594:
10497:Finite difference
10240:(10): 1123â1130.
9643:978-0-08-095135-5
9616:978-0-13-032946-2
9609:. Prentice Hall.
9580:978-82-8071-256-1
9441:10.1115/1.4009129
9346:978-0-471-37078-9
8953:finite difference
8586:{\displaystyle d}
8480:{\displaystyle h}
8413:{\displaystyle V}
8357:{\displaystyle V}
8194:{\displaystyle M}
8170:{\displaystyle L}
8144:LU decompositions
8131:{\displaystyle L}
8104:{\displaystyle L}
8035:{\displaystyle M}
8015:{\displaystyle L}
7840:
7839:
7775:{\displaystyle M}
7560:
7559:
7124:
7123:
5685:{\displaystyle 1}
5665:{\displaystyle V}
5485:
5427:
5420:
5311:
5304:
5135:{\displaystyle 1}
5115:{\displaystyle V}
5041:{\displaystyle V}
4951:{\displaystyle V}
4904:{\displaystyle h}
4884:{\displaystyle V}
4834:{\displaystyle V}
4762:{\displaystyle V}
4722:{\displaystyle x}
4605:{\displaystyle V}
4470:
4444:
4392:
4330:{\displaystyle V}
4220:{\displaystyle x}
4200:{\displaystyle n}
4140:{\displaystyle V}
4116:{\displaystyle V}
4060:{\displaystyle V}
4045:
4044:
3749:{\displaystyle f}
3678:{\displaystyle u}
3411:{\displaystyle 0}
3233:that are zero on
2988:{\displaystyle v}
2933:{\displaystyle u}
2920:, we see that if
2856:
2855:
2440:{\displaystyle u}
2420:{\displaystyle u}
2321:{\displaystyle u}
2306:
2305:
2082:{\displaystyle v}
2062:{\displaystyle u}
2039:weak formulations
1955:{\displaystyle y}
1935:{\displaystyle x}
1854:is nice (e.g., a
1749:
1723:
1625:
1613:Dirichlet problem
1601:{\displaystyle x}
1581:{\displaystyle u}
1536:{\displaystyle x}
1516:{\displaystyle u}
1496:{\displaystyle f}
1406:
1358:
1227:Richard Gallagher
1203:O. C. Zienkiewicz
1131:Leonard Oganesyan
1080:tropical cyclones
1021:coordinate system
738:
737:
629:Gottfried Leibniz
520:Finite difference
312:
311:
173:
172:
143:Dynamical systems
16:(Redirected from
10996:
10886:Analytic element
10869:Boundary element
10762:Schur complement
10743:Particle-in-cell
10678:Spectral element
10502:
10501:
10482:
10475:
10468:
10459:
10458:
10402:J. Chaskalovic:
10368:
10367:
10349:
10343:
10342:
10322:
10316:
10315:
10313:
10312:
10303:. Archived from
10296:
10290:
10283:
10272:
10271:
10269:
10268:
10262:
10231:
10222:
10216:
10215:
10213:
10212:
10197:
10191:
10190:
10182:
10176:
10175:
10172:10.1002/nme.5481
10165:
10141:
10135:
10134:
10124:
10115:(6): 4115â4135.
10100:
10091:
10090:
10050:
10044:
10043:
10003:
9994:
9993:
9981:
9975:
9974:
9972:
9971:
9952:
9946:
9945:
9939:
9931:
9899:
9888:
9887:
9865:
9859:
9858:
9856:
9855:
9846:. Archived from
9836:
9830:
9829:
9812:(3â4): 329â355.
9801:
9795:
9792:
9786:
9785:
9756:
9750:
9749:
9739:
9737:10.1002/nme.4823
9707:
9701:
9700:
9698:
9697:
9682:
9676:
9675:
9654:
9648:
9647:
9627:
9621:
9620:
9608:
9591:
9585:
9584:
9566:
9560:
9559:
9557:
9556:
9541:
9535:
9534:
9514:
9508:
9507:
9505:
9503:
9494:. Archived from
9484:
9478:
9477:
9475:
9451:
9445:
9444:
9416:
9410:
9409:
9391:
9381:
9372:(6): 4431â4453.
9357:
9351:
9350:
9332:
9326:
9325:
9307:
9298:
9297:
9279:
9218:Meshfree methods
8992:straightforward.
8886:Meshfree methods
8880:Meshfree methods
8687:
8685:
8684:
8679:
8677:
8676:
8671:
8654:spectral methods
8624:
8622:
8621:
8616:
8592:
8590:
8589:
8584:
8568:
8566:
8565:
8560:
8542:
8540:
8539:
8534:
8516:
8514:
8513:
8508:
8506:
8505:
8486:
8484:
8483:
8478:
8455:
8453:
8452:
8447:
8444:
8439:
8419:
8417:
8416:
8411:
8395:
8393:
8392:
8387:
8384:
8379:
8363:
8361:
8360:
8355:
8339:
8337:
8336:
8331:
8329:
8328:
8311:
8309:
8308:
8303:
8295:
8294:
8273:
8272:
8236:
8234:
8233:
8228:
8200:
8198:
8197:
8192:
8179:stiffness matrix
8176:
8174:
8173:
8168:
8137:
8135:
8134:
8129:
8110:
8108:
8107:
8102:
8090:
8088:
8087:
8082:
8080:
8068:
8066:
8065:
8060:
8058:
8057:
8041:
8039:
8038:
8033:
8021:
8019:
8018:
8013:
7998:
7996:
7995:
7990:
7960:
7958:
7957:
7952:
7944:
7943:
7925:
7924:
7908:
7906:
7905:
7900:
7898:
7897:
7888:
7887:
7869:
7868:
7853:
7834:
7825:
7823:
7822:
7817:
7812:
7804:
7785:
7781:
7779:
7778:
7773:
7761:
7759:
7758:
7753:
7723:
7721:
7720:
7715:
7704:
7703:
7672:
7670:
7669:
7664:
7643:
7641:
7640:
7635:
7624:
7623:
7614:
7613:
7603:
7598:
7554:
7545:
7543:
7542:
7537:
7532:
7521:
7502:
7498:
7496:
7495:
7490:
7482:
7481:
7472:
7471:
7456:
7455:
7436:
7434:
7433:
7428:
7423:
7422:
7410:
7409:
7391:
7390:
7371:
7369:
7368:
7363:
7358:
7357:
7329:
7327:
7326:
7321:
7316:
7315:
7288:, and if we let
7287:
7285:
7284:
7279:
7277:
7276:
7267:
7266:
7248:
7247:
7228:
7226:
7225:
7220:
7218:
7217:
7208:
7207:
7189:
7188:
7169:
7167:
7166:
7161:
7159:
7147:
7145:
7144:
7139:
7137:
7126:If we denote by
7118:
7109:
7107:
7106:
7101:
7068:
7066:
7065:
7060:
7052:
7051:
7042:
7041:
7029:
7028:
7018:
7013:
6992:
6991:
6979:
6978:
6963:
6962:
6952:
6947:
6921:
6917:
6915:
6914:
6909:
6879:
6877:
6876:
6871:
6860:
6859:
6828:
6826:
6825:
6820:
6809:
6808:
6799:
6798:
6788:
6783:
6746:
6744:
6743:
6738:
6727:
6726:
6717:
6716:
6706:
6701:
6656:
6654:
6653:
6648:
6639:
6638:
6623:
6622:
6610:
6609:
6593:
6591:
6590:
6585:
6576:
6575:
6566:
6565:
6556:
6555:
6539:
6537:
6536:
6531:
6529:
6528:
6512:
6510:
6509:
6504:
6502:
6501:
6482:
6480:
6479:
6474:
6466:
6452:
6440:
6438:
6437:
6432:
6427:
6426:
6414:
6413:
6391:
6389:
6388:
6383:
6378:
6377:
6365:
6364:
6345:
6343:
6342:
6339:{\displaystyle }
6337:
6332:
6331:
6313:
6312:
6288:is the interval
6287:
6285:
6284:
6279:
6277:
6276:
6252:
6250:
6249:
6244:
6220:
6218:
6217:
6212:
6207:
6206:
6194:
6193:
6174:
6172:
6171:
6166:
6148:
6146:
6145:
6140:
6127:
6114:
6104:
6099:
6084:
6083:
6071:
6070:
6048:
6046:
6045:
6040:
6031:
6030:
6021:
6020:
6010:
6005:
5990:
5989:
5977:
5976:
5954:
5952:
5951:
5946:
5944:
5943:
5931:
5930:
5862:
5860:
5859:
5854:
5843:
5842:
5827:
5826:
5758:
5756:
5755:
5750:
5735:
5734:
5718:
5716:
5715:
5710:
5708:
5707:
5691:
5689:
5688:
5683:
5671:
5669:
5668:
5663:
5651:
5649:
5648:
5643:
5641:
5640:
5624:
5622:
5621:
5616:
5604:
5602:
5601:
5596:
5594:
5593:
5577:
5575:
5574:
5569:
5567:
5566:
5546:
5544:
5543:
5538:
5506:
5504:
5503:
5498:
5496:
5495:
5486:
5483:
5466:
5465:
5447:
5446:
5428:
5425:
5421:
5419:
5418:
5417:
5404:
5403:
5387:
5379:
5378:
5362:
5350:
5349:
5337:
5336:
5312:
5309:
5305:
5303:
5302:
5301:
5282:
5281:
5271:
5270:
5269:
5247:
5225:
5224:
5208:
5206:
5205:
5200:
5185:
5184:
5168:
5166:
5165:
5160:
5158:
5157:
5141:
5139:
5138:
5133:
5121:
5119:
5118:
5113:
5101:
5099:
5098:
5093:
5091:
5090:
5074:
5072:
5071:
5066:
5064:
5063:
5047:
5045:
5044:
5039:
5005:
4982:
4962:Choosing a basis
4957:
4955:
4954:
4949:
4937:
4935:
4934:
4929:
4927:
4926:
4910:
4908:
4907:
4902:
4890:
4888:
4887:
4882:
4870:
4868:
4867:
4862:
4840:
4838:
4837:
4832:
4816:
4814:
4813:
4808:
4788:
4786:
4785:
4780:
4768:
4766:
4765:
4760:
4728:
4726:
4725:
4720:
4708:
4706:
4705:
4700:
4670:
4668:
4667:
4662:
4660:
4659:
4637:
4635:
4634:
4629:
4611:
4609:
4608:
4603:
4591:
4589:
4588:
4583:
4575:
4574:
4552:
4550:
4549:
4544:
4536:
4535:
4520:where we define
4517:
4515:
4514:
4509:
4471:
4468:
4445:
4442:
4440:
4439:
4435:
4434:
4416:
4415:
4402:
4393:
4390:
4381:
4336:
4334:
4333:
4328:
4316:
4314:
4313:
4308:
4300:
4299:
4281:
4280:
4262:
4261:
4249:
4248:
4226:
4224:
4223:
4218:
4206:
4204:
4203:
4198:
4186:
4184:
4183:
4178:
4146:
4144:
4143:
4138:
4122:
4120:
4119:
4114:
4102:
4100:
4099:
4094:
4091:
4086:
4066:
4064:
4063:
4058:
4039:
4030:
4028:
4027:
4022:
3963:
3961:
3960:
3955:
3935:
3928:
3926:
3925:
3920:
3880:
3875:
3848:
3846:
3845:
3840:
3837:
3832:
3802:
3800:
3799:
3794:
3788:
3783:
3755:
3753:
3752:
3747:
3731:
3729:
3728:
3723:
3705:
3700:
3684:
3682:
3681:
3676:
3660:
3658:
3657:
3652:
3635:
3634:
3615:
3613:
3612:
3607:
3574:
3569:
3549:
3547:
3546:
3541:
3523:
3518:
3499:then defines an
3498:
3496:
3495:
3490:
3469:
3467:
3466:
3461:
3443:
3441:
3440:
3435:
3417:
3415:
3414:
3409:
3397:
3395:
3394:
3389:
3361:
3359:
3358:
3353:
3335:
3330:
3302:
3300:
3299:
3294:
3282:
3277:
3255:
3253:
3252:
3247:
3232:
3230:
3229:
3224:
3212:
3210:
3209:
3204:
3192:
3187:
3171:
3169:
3168:
3163:
3145:
3143:
3142:
3137:
3121:
3119:
3118:
3113:
3099:
3097:
3096:
3091:
3040:
3039:
3011:
3010:
2994:
2992:
2991:
2986:
2974:
2972:
2971:
2966:
2939:
2937:
2936:
2931:
2907:
2905:
2904:
2899:
2850:
2841:
2839:
2838:
2833:
2831:
2784:
2767:
2758:
2753:
2735:
2715:
2698:
2689:
2684:
2671:
2666:
2661:
2634:
2620:
2588:
2579:
2574:
2526:
2521:
2500:
2492:
2490:
2489:
2484:
2446:
2444:
2443:
2438:
2426:
2424:
2423:
2418:
2406:
2404:
2403:
2398:
2377:
2375:
2374:
2369:
2327:
2325:
2324:
2319:
2300:
2291:
2289:
2288:
2283:
2250:
2241:
2236:
2192:
2187:
2170:
2166:
2164:
2163:
2158:
2140:
2138:
2137:
2132:
2114:
2112:
2111:
2106:
2088:
2086:
2085:
2080:
2068:
2066:
2065:
2060:
2033:Weak formulation
2009:
2007:
2006:
2001:
1993:
1962:, respectively.
1961:
1959:
1958:
1953:
1941:
1939:
1938:
1933:
1921:
1919:
1918:
1913:
1911:
1910:
1891:
1889:
1888:
1883:
1881:
1880:
1853:
1851:
1850:
1845:
1830:
1828:
1827:
1822:
1798:
1796:
1795:
1790:
1776:
1774:
1773:
1768:
1766:
1765:
1750:
1747:
1724:
1721:
1681:
1680:
1650:
1649:
1626:
1623:
1607:
1605:
1604:
1599:
1588:with respect to
1587:
1585:
1584:
1579:
1567:
1565:
1564:
1559:
1557:
1542:
1540:
1539:
1534:
1522:
1520:
1519:
1514:
1502:
1500:
1499:
1494:
1482:
1480:
1479:
1474:
1472:
1471:
1407:
1404:
1378:
1359:
1356:
1335:superconvergence
1267:electromagnetism
1205:with co-workers
957:weight functions
874:
838:
777:numerical method
730:
723:
716:
694:Phyllis Nicolson
679:Rudolf Lipschitz
562:Green's function
538:Infinite element
529:
494:Solution methods
472:
330:
241:By variable type
195:
194:
77:Natural sciences
70:
69:
49:
48:
21:
11004:
11003:
10999:
10998:
10997:
10995:
10994:
10993:
10944:
10943:
10942:
10937:
10906:Galerkin method
10849:Method of lines
10826:
10794:NeumannâNeumann
10748:
10705:
10647:
10614:High-resolution
10591:
10562:
10524:
10491:
10486:
10376:
10374:Further reading
10371:
10364:
10350:
10346:
10323:
10319:
10310:
10308:
10297:
10293:
10284:
10275:
10266:
10264:
10260:
10229:
10223:
10219:
10210:
10208:
10199:
10198:
10194:
10183:
10179:
10156:(10): 903â926.
10142:
10138:
10101:
10094:
10051:
10047:
10004:
9997:
9982:
9978:
9969:
9967:
9954:
9953:
9949:
9933:
9932:
9900:
9891:
9866:
9862:
9853:
9851:
9838:
9837:
9833:
9802:
9798:
9793:
9789:
9764:Osborn, John E.
9757:
9753:
9708:
9704:
9695:
9693:
9683:
9679:
9672:
9655:
9651:
9644:
9628:
9624:
9617:
9595:Strang, Gilbert
9592:
9588:
9581:
9567:
9563:
9554:
9552:
9543:
9542:
9538:
9515:
9511:
9501:
9499:
9486:
9485:
9481:
9452:
9448:
9417:
9413:
9358:
9354:
9347:
9333:
9329:
9322:
9308:
9301:
9294:
9280:
9276:
9272:
9267:
9128:
9084:
9036:
8981:
8969:
8946:
8937:
8926:
8921:
8915:
8910:
8904:
8899:
8893:
8888:
8882:
8873:
8867:
8858:
8852:
8843:
8813:
8807:
8783:
8763:
8754:
8748:
8734:
8725:
8717:
8712:
8699:
8694:
8672:
8667:
8666:
8664:
8661:
8660:
8598:
8595:
8594:
8578:
8575:
8574:
8548:
8545:
8544:
8522:
8519:
8518:
8501:
8497:
8492:
8489:
8488:
8472:
8469:
8468:
8440:
8435:
8429:
8426:
8425:
8405:
8402:
8401:
8380:
8375:
8369:
8366:
8365:
8349:
8346:
8345:
8324:
8320:
8318:
8315:
8314:
8281:
8277:
8259:
8255:
8253:
8250:
8249:
8222:
8219:
8218:
8211:
8186:
8183:
8182:
8162:
8159:
8158:
8123:
8120:
8119:
8116:sparse matrices
8096:
8093:
8092:
8076:
8074:
8071:
8070:
8053:
8049:
8047:
8044:
8043:
8027:
8024:
8023:
8007:
8004:
8003:
7966:
7963:
7962:
7939:
7935:
7920:
7916:
7914:
7911:
7910:
7893:
7889:
7883:
7879:
7864:
7860:
7849:
7847:
7844:
7843:
7808:
7800:
7792:
7789:
7788:
7767:
7764:
7763:
7729:
7726:
7725:
7699:
7695:
7678:
7675:
7674:
7649:
7646:
7645:
7619:
7615:
7609:
7605:
7599:
7588:
7567:
7564:
7563:
7528:
7517:
7509:
7506:
7505:
7477:
7473:
7467:
7463:
7448:
7444:
7442:
7439:
7438:
7418:
7414:
7405:
7401:
7383:
7379:
7377:
7374:
7373:
7350:
7346:
7335:
7332:
7331:
7308:
7304:
7293:
7290:
7289:
7272:
7268:
7262:
7258:
7243:
7239:
7234:
7231:
7230:
7213:
7209:
7203:
7199:
7184:
7180:
7175:
7172:
7171:
7155:
7153:
7150:
7149:
7133:
7131:
7128:
7127:
7074:
7071:
7070:
7047:
7043:
7037:
7033:
7024:
7020:
7014:
7003:
6987:
6983:
6974:
6970:
6958:
6954:
6948:
6937:
6928:
6925:
6924:
6885:
6882:
6881:
6855:
6851:
6834:
6831:
6830:
6804:
6800:
6794:
6790:
6784:
6773:
6752:
6749:
6748:
6722:
6718:
6712:
6708:
6702:
6691:
6670:
6667:
6666:
6663:
6657:are both zero.
6634:
6630:
6618:
6614:
6605:
6601:
6599:
6596:
6595:
6571:
6567:
6561:
6557:
6551:
6547:
6545:
6542:
6541:
6524:
6520:
6518:
6515:
6514:
6497:
6493:
6491:
6488:
6487:
6462:
6448:
6446:
6443:
6442:
6422:
6418:
6409:
6405:
6397:
6394:
6393:
6373:
6369:
6360:
6356:
6351:
6348:
6347:
6321:
6317:
6302:
6298:
6293:
6290:
6289:
6272:
6268:
6266:
6263:
6262:
6226:
6223:
6222:
6202:
6198:
6189:
6185:
6180:
6177:
6176:
6154:
6151:
6150:
6123:
6110:
6100:
6095:
6079:
6075:
6066:
6062:
6054:
6051:
6050:
6026:
6022:
6016:
6012:
6006:
6001:
5985:
5981:
5972:
5968:
5963:
5960:
5959:
5939:
5935:
5926:
5922:
5893:
5890:
5889:
5864:
5835:
5831:
5819:
5815:
5813:
5810:
5809:
5802:
5730:
5726:
5724:
5721:
5720:
5703:
5699:
5697:
5694:
5693:
5677:
5674:
5673:
5672:whose value is
5657:
5654:
5653:
5636:
5632:
5630:
5627:
5626:
5625:. The function
5610:
5607:
5606:
5589:
5585:
5583:
5580:
5579:
5562:
5558:
5556:
5553:
5552:
5514:
5511:
5510:
5491:
5490:
5484: otherwise
5482:
5480:
5474:
5473:
5455:
5451:
5442:
5438:
5424:
5422:
5413:
5409:
5393:
5389:
5388:
5368:
5364:
5363:
5361:
5358:
5357:
5345:
5341:
5326:
5322:
5308:
5306:
5291:
5287:
5277:
5273:
5272:
5259:
5255:
5248:
5246:
5239:
5238:
5220:
5216:
5214:
5211:
5210:
5180:
5176:
5174:
5171:
5170:
5153:
5149:
5147:
5144:
5143:
5127:
5124:
5123:
5122:whose value is
5107:
5104:
5103:
5086:
5082:
5080:
5077:
5076:
5059:
5055:
5053:
5050:
5049:
5033:
5030:
5029:
5022:
5021:
5020:
5019:
5018:
5016:
5006:
4997:
4996:
4995:
4993:
4983:
4974:
4973:
4971:Bessel function
4964:
4943:
4940:
4939:
4922:
4918:
4916:
4913:
4912:
4896:
4893:
4892:
4876:
4873:
4872:
4850:
4847:
4846:
4826:
4823:
4822:
4802:
4799:
4798:
4774:
4771:
4770:
4754:
4751:
4750:
4747:
4714:
4711:
4710:
4676:
4673:
4672:
4655:
4651:
4643:
4640:
4639:
4617:
4614:
4613:
4597:
4594:
4593:
4564:
4560:
4558:
4555:
4554:
4531:
4527:
4525:
4522:
4521:
4467:
4441:
4424:
4420:
4411:
4407:
4403:
4398:
4397:
4389:
4377:
4342:
4339:
4338:
4322:
4319:
4318:
4289:
4285:
4276:
4272:
4257:
4253:
4244:
4240:
4232:
4229:
4228:
4212:
4209:
4208:
4192:
4189:
4188:
4160:
4157:
4156:
4153:
4132:
4129:
4128:
4125:spectral method
4108:
4105:
4104:
4087:
4082:
4076:
4073:
4072:
4052:
4049:
4048:
3970:
3967:
3966:
3965:
3943:
3940:
3939:
3876:
3871:
3856:
3853:
3852:
3833:
3828:
3816:
3813:
3812:
3784:
3779:
3773:
3770:
3769:
3762:
3741:
3738:
3737:
3701:
3696:
3690:
3687:
3686:
3670:
3667:
3666:
3630:
3626:
3624:
3621:
3620:
3570:
3565:
3559:
3556:
3555:
3519:
3514:
3508:
3505:
3504:
3482:
3479:
3478:
3449:
3446:
3445:
3423:
3420:
3419:
3403:
3400:
3399:
3371:
3368:
3367:
3331:
3326:
3320:
3317:
3316:
3313:
3278:
3273:
3261:
3258:
3257:
3238:
3235:
3234:
3218:
3215:
3214:
3188:
3183:
3177:
3174:
3173:
3155:
3152:
3151:
3131:
3128:
3127:
3107:
3104:
3103:
3035:
3031:
3006:
3002:
3000:
2997:
2996:
2980:
2977:
2976:
2945:
2942:
2941:
2925:
2922:
2921:
2914:
2863:
2860:
2859:
2829:
2828:
2777:
2760:
2754:
2749:
2733:
2732:
2708:
2691:
2685:
2680:
2667:
2662:
2657:
2627:
2618:
2617:
2581:
2575:
2570:
2559:
2522:
2517:
2509:
2507:
2504:
2503:
2463:
2460:
2459:
2455:sense as well.
2432:
2429:
2428:
2412:
2409:
2408:
2383:
2380:
2379:
2333:
2330:
2329:
2313:
2310:
2309:
2308:Conversely, if
2243:
2237:
2232:
2188:
2183:
2177:
2174:
2173:
2146:
2143:
2142:
2120:
2117:
2116:
2094:
2091:
2090:
2074:
2071:
2070:
2054:
2051:
2050:
2047:
2035:
1986:
1978:
1975:
1974:
1967:antiderivatives
1947:
1944:
1943:
1927:
1924:
1923:
1903:
1899:
1897:
1894:
1893:
1873:
1869:
1867:
1864:
1863:
1856:smooth manifold
1836:
1833:
1832:
1804:
1801:
1800:
1784:
1781:
1780:
1761:
1760:
1746:
1744:
1732:
1731:
1720:
1718:
1673:
1669:
1642:
1638:
1631:
1630:
1622:
1620:
1617:
1616:
1593:
1590:
1589:
1573:
1570:
1569:
1550:
1548:
1545:
1544:
1528:
1525:
1524:
1508:
1505:
1504:
1488:
1485:
1484:
1467:
1466:
1427:
1426:
1403:
1371:
1364:
1363:
1355:
1353:
1350:
1349:
1343:
1299:Galerkin method
1288:
1283:
1127:Ioannis Argyris
1123:Richard Courant
1099:
1048:complex problem
1044:mesh generation
949:Galerkin method
906:
905:
904:
903:
902:
883:magnetic shield
875:
867:
866:
863:equations alone
839:
828:
789:finite elements
734:
705:
704:
703:
634:Jacob Bernoulli
618:
605:
604:
586:
555:PetrovâGalerkin
523:
508:
495:
487:
486:
485:
467:
413:Boundary values
402:
394:
393:
369:
356:
355:
354:
328:
322:
314:
313:
301:
278:
236:
192:
179:
178:
174:
152:Social sciences
108:
86:
67:
39:
36:compact element
28:
23:
22:
15:
12:
11:
5:
11002:
10992:
10991:
10986:
10981:
10976:
10971:
10966:
10961:
10956:
10939:
10938:
10936:
10935:
10930:
10925:
10920:
10915:
10914:
10913:
10903:
10898:
10893:
10888:
10883:
10878:
10877:
10876:
10866:
10861:
10856:
10851:
10846:
10843:Pseudospectral
10840:
10834:
10832:
10828:
10827:
10825:
10824:
10819:
10813:
10807:
10801:
10796:
10791:
10786:
10785:
10784:
10779:
10769:
10764:
10758:
10756:
10750:
10749:
10747:
10746:
10740:
10734:
10728:
10722:
10715:
10713:
10707:
10706:
10704:
10703:
10697:
10692:
10686:
10681:
10675:
10669:
10663:
10657:
10655:
10653:Finite element
10649:
10648:
10646:
10645:
10639:
10633:
10631:Riemann solver
10628:
10622:
10616:
10611:
10605:
10603:
10597:
10596:
10593:
10592:
10590:
10589:
10583:
10577:
10570:
10568:
10564:
10563:
10561:
10560:
10555:
10550:
10545:
10540:
10538:LaxâFriedrichs
10534:
10532:
10526:
10525:
10523:
10522:
10520:CrankâNicolson
10517:
10510:
10508:
10499:
10493:
10492:
10485:
10484:
10477:
10470:
10462:
10456:
10455:
10448:
10442:
10435:
10428:
10419:
10407:
10400:
10393:
10386:
10375:
10372:
10370:
10369:
10363:978-0471369615
10362:
10344:
10333:(2): 181â189.
10317:
10291:
10273:
10217:
10192:
10177:
10136:
10092:
10045:
9995:
9976:
9962:. 2016-04-18.
9960:Machine Design
9947:
9889:
9878:(1): 199â214.
9860:
9831:
9796:
9787:
9751:
9722:(7): 472â501.
9702:
9677:
9671:978-0979004902
9670:
9649:
9642:
9622:
9615:
9586:
9579:
9561:
9536:
9509:
9479:
9446:
9427:(4): 169â175.
9411:
9352:
9345:
9327:
9320:
9299:
9292:
9273:
9271:
9268:
9266:
9265:
9260:
9255:
9250:
9245:
9240:
9235:
9230:
9225:
9220:
9215:
9210:
9205:
9200:
9195:
9190:
9185:
9180:
9175:
9170:
9165:
9160:
9155:
9150:
9145:
9140:
9135:
9129:
9127:
9124:
9083:
9080:
9035:
9032:
9020:
9019:
9016:
9013:
9002:
8999:
8996:
8993:
8980:
8977:
8968:
8965:
8945:
8942:
8936:
8933:
8925:
8922:
8917:Main article:
8914:
8911:
8906:Main article:
8903:
8900:
8895:Main article:
8892:
8889:
8884:Main article:
8881:
8878:
8869:Main article:
8866:
8863:
8854:Main article:
8851:
8848:
8842:
8839:
8834:
8833:
8830:
8827:
8809:Main article:
8806:
8803:
8782:
8779:
8762:
8759:
8750:Main article:
8747:
8744:
8733:
8730:
8724:
8721:
8716:
8713:
8708:Main article:
8698:
8695:
8693:
8690:
8675:
8670:
8614:
8611:
8608:
8605:
8602:
8582:
8558:
8555:
8552:
8532:
8529:
8526:
8504:
8500:
8496:
8476:
8443:
8438:
8434:
8409:
8383:
8378:
8374:
8353:
8327:
8323:
8301:
8298:
8293:
8290:
8287:
8284:
8280:
8276:
8271:
8268:
8265:
8262:
8258:
8242:
8241:
8238:
8226:
8210:
8207:
8201:is dubbed the
8190:
8166:
8127:
8100:
8079:
8056:
8052:
8031:
8011:
7988:
7985:
7982:
7979:
7976:
7973:
7970:
7950:
7947:
7942:
7938:
7934:
7931:
7928:
7923:
7919:
7896:
7892:
7886:
7882:
7878:
7875:
7872:
7867:
7863:
7859:
7856:
7852:
7838:
7837:
7828:
7826:
7815:
7811:
7807:
7803:
7799:
7796:
7771:
7751:
7748:
7745:
7742:
7739:
7736:
7733:
7713:
7710:
7707:
7702:
7698:
7694:
7691:
7688:
7685:
7682:
7662:
7659:
7656:
7653:
7633:
7630:
7627:
7622:
7618:
7612:
7608:
7602:
7597:
7594:
7591:
7587:
7583:
7580:
7577:
7574:
7571:
7558:
7557:
7548:
7546:
7535:
7531:
7527:
7524:
7520:
7516:
7513:
7488:
7485:
7480:
7476:
7470:
7466:
7462:
7459:
7454:
7451:
7447:
7426:
7421:
7417:
7413:
7408:
7404:
7400:
7397:
7394:
7389:
7386:
7382:
7361:
7356:
7353:
7349:
7345:
7342:
7339:
7319:
7314:
7311:
7307:
7303:
7300:
7297:
7275:
7271:
7265:
7261:
7257:
7254:
7251:
7246:
7242:
7238:
7216:
7212:
7206:
7202:
7198:
7195:
7192:
7187:
7183:
7179:
7158:
7136:
7122:
7121:
7112:
7110:
7099:
7096:
7093:
7090:
7087:
7084:
7081:
7078:
7058:
7055:
7050:
7046:
7040:
7036:
7032:
7027:
7023:
7017:
7012:
7009:
7006:
7002:
6998:
6995:
6990:
6986:
6982:
6977:
6973:
6969:
6966:
6961:
6957:
6951:
6946:
6943:
6940:
6936:
6932:
6907:
6904:
6901:
6898:
6895:
6892:
6889:
6869:
6866:
6863:
6858:
6854:
6850:
6847:
6844:
6841:
6838:
6818:
6815:
6812:
6807:
6803:
6797:
6793:
6787:
6782:
6779:
6776:
6772:
6768:
6765:
6762:
6759:
6756:
6736:
6733:
6730:
6725:
6721:
6715:
6711:
6705:
6700:
6697:
6694:
6690:
6686:
6683:
6680:
6677:
6674:
6662:
6659:
6646:
6643:
6637:
6633:
6629:
6626:
6621:
6617:
6613:
6608:
6604:
6583:
6580:
6574:
6570:
6564:
6560:
6554:
6550:
6527:
6523:
6500:
6496:
6472:
6469:
6465:
6461:
6458:
6455:
6451:
6430:
6425:
6421:
6417:
6412:
6408:
6404:
6401:
6381:
6376:
6372:
6368:
6363:
6359:
6355:
6335:
6330:
6327:
6324:
6320:
6316:
6311:
6308:
6305:
6301:
6297:
6275:
6271:
6255:Gramian matrix
6242:
6239:
6236:
6233:
6230:
6210:
6205:
6201:
6197:
6192:
6188:
6184:
6164:
6161:
6158:
6138:
6135:
6130:
6126:
6122:
6117:
6113:
6109:
6103:
6098:
6094:
6090:
6087:
6082:
6078:
6074:
6069:
6065:
6061:
6058:
6038:
6035:
6029:
6025:
6019:
6015:
6009:
6004:
6000:
5996:
5993:
5988:
5984:
5980:
5975:
5971:
5967:
5942:
5938:
5934:
5929:
5925:
5921:
5918:
5915:
5912:
5909:
5906:
5903:
5900:
5897:
5852:
5849:
5846:
5841:
5838:
5834:
5830:
5825:
5822:
5818:
5801:
5798:
5797:
5796:
5789:
5786:
5783:
5748:
5745:
5742:
5738:
5733:
5729:
5706:
5702:
5681:
5661:
5639:
5635:
5614:
5592:
5588:
5565:
5561:
5536:
5533:
5530:
5527:
5524:
5521:
5518:
5494:
5489:
5481:
5479:
5476:
5475:
5472:
5469:
5464:
5461:
5458:
5454:
5450:
5445:
5441:
5437:
5434:
5431:
5426: if
5423:
5416:
5412:
5408:
5402:
5399:
5396:
5392:
5386:
5383:
5377:
5374:
5371:
5367:
5360:
5359:
5356:
5353:
5348:
5344:
5340:
5335:
5332:
5329:
5325:
5321:
5318:
5315:
5310: if
5307:
5300:
5297:
5294:
5290:
5286:
5280:
5276:
5268:
5265:
5262:
5258:
5254:
5251:
5245:
5244:
5242:
5237:
5234:
5231:
5228:
5223:
5219:
5198:
5195:
5192:
5188:
5183:
5179:
5156:
5152:
5131:
5111:
5089:
5085:
5062:
5058:
5037:
5012:
5007:
5000:
4999:
4998:
4989:
4984:
4977:
4976:
4975:
4968:
4967:
4966:
4965:
4963:
4960:
4947:
4925:
4921:
4900:
4880:
4860:
4857:
4854:
4830:
4806:
4793:of a 15-sided
4778:
4758:
4746:
4745:For problem P2
4743:
4718:
4698:
4695:
4692:
4689:
4686:
4683:
4680:
4658:
4654:
4650:
4647:
4627:
4624:
4621:
4601:
4581:
4578:
4573:
4570:
4567:
4563:
4542:
4539:
4534:
4530:
4507:
4504:
4501:
4498:
4495:
4492:
4489:
4486:
4483:
4480:
4477:
4474:
4466:
4463:
4460:
4457:
4454:
4451:
4448:
4438:
4433:
4430:
4427:
4423:
4419:
4414:
4410:
4406:
4401:
4396:
4388:
4385:
4380:
4376:
4373:
4370:
4367:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4326:
4317:and we define
4306:
4303:
4298:
4295:
4292:
4288:
4284:
4279:
4275:
4271:
4268:
4265:
4260:
4256:
4252:
4247:
4243:
4239:
4236:
4216:
4196:
4176:
4173:
4170:
4167:
4164:
4152:
4151:For problem P1
4149:
4136:
4112:
4090:
4085:
4081:
4056:
4043:
4042:
4033:
4031:
4020:
4017:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3986:
3983:
3980:
3977:
3974:
3953:
3950:
3947:
3930:
3929:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3884:
3879:
3874:
3870:
3866:
3863:
3860:
3850:
3836:
3831:
3827:
3823:
3820:
3792:
3787:
3782:
3778:
3768:A function in
3761:
3760:Discretization
3758:
3745:
3721:
3718:
3715:
3712:
3709:
3704:
3699:
3695:
3674:
3650:
3647:
3644:
3641:
3638:
3633:
3629:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3573:
3568:
3564:
3539:
3536:
3533:
3530:
3527:
3522:
3517:
3513:
3488:
3472:Sobolev spaces
3459:
3456:
3453:
3433:
3430:
3427:
3407:
3387:
3384:
3381:
3378:
3375:
3351:
3348:
3345:
3342:
3339:
3334:
3329:
3325:
3312:
3309:
3305:Sobolev spaces
3292:
3289:
3286:
3281:
3276:
3272:
3268:
3265:
3245:
3242:
3222:
3202:
3199:
3196:
3191:
3186:
3182:
3161:
3135:
3111:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3055:
3052:
3049:
3046:
3043:
3038:
3034:
3030:
3027:
3024:
3021:
3017:
3014:
3009:
3005:
2984:
2964:
2961:
2958:
2955:
2952:
2949:
2929:
2913:
2910:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2854:
2853:
2844:
2842:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2793:
2790:
2787:
2783:
2780:
2776:
2773:
2770:
2766:
2763:
2757:
2752:
2748:
2744:
2741:
2738:
2736:
2734:
2731:
2728:
2724:
2721:
2718:
2714:
2711:
2707:
2704:
2701:
2697:
2694:
2688:
2683:
2679:
2675:
2670:
2665:
2660:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2633:
2630:
2626:
2623:
2621:
2619:
2616:
2613:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2587:
2584:
2578:
2573:
2569:
2565:
2562:
2560:
2558:
2555:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2525:
2520:
2516:
2512:
2511:
2482:
2479:
2476:
2473:
2470:
2467:
2453:distributional
2436:
2416:
2396:
2393:
2390:
2387:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2317:
2304:
2303:
2294:
2292:
2281:
2278:
2275:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2249:
2246:
2240:
2235:
2231:
2227:
2224:
2221:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2191:
2186:
2182:
2156:
2153:
2150:
2130:
2127:
2124:
2104:
2101:
2098:
2078:
2058:
2046:
2043:
2034:
2031:
2022:
2021:
2018:
1999:
1996:
1992:
1989:
1985:
1982:
1951:
1931:
1909:
1906:
1902:
1879:
1876:
1872:
1843:
1840:
1820:
1817:
1814:
1811:
1808:
1788:
1764:
1759:
1756:
1753:
1748: on
1745:
1743:
1740:
1737:
1734:
1733:
1730:
1727:
1722: in
1719:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1679:
1676:
1672:
1668:
1665:
1662:
1659:
1656:
1653:
1648:
1645:
1641:
1637:
1636:
1634:
1629:
1597:
1577:
1556:
1553:
1532:
1512:
1492:
1470:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1405: in
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1377:
1374:
1370:
1369:
1367:
1362:
1357: P1
1342:
1339:
1287:
1284:
1282:
1279:
1275:fluid dynamics
1251:Gilbert Strang
1213:and others at
1180:Boris Galerkin
1146:discretization
1098:
1095:
1005:Euler's method
989:
988:
977:
938:initial values
934:
933:
930:
923:
922:
919:
916:
913:
876:
869:
868:
840:
833:
832:
831:
830:
829:
827:
826:Basic concepts
824:
793:discretization
736:
735:
733:
732:
725:
718:
710:
707:
706:
702:
701:
696:
691:
686:
684:Ernst Lindelöf
681:
676:
671:
666:
661:
656:
654:Joseph Fourier
651:
646:
641:
639:Leonhard Euler
636:
631:
626:
620:
619:
616:
615:
612:
611:
607:
606:
603:
602:
597:
592:
585:
584:
579:
574:
569:
564:
559:
558:
557:
547:
542:
541:
540:
533:Finite element
530:
526:CrankâNicolson
517:
512:
506:
501:
497:
496:
493:
492:
489:
488:
484:
483:
478:
473:
465:
460:
447:
445:Phase portrait
442:
437:
436:
435:
433:Cauchy problem
430:
425:
420:
410:
404:
403:
401:General topics
400:
399:
396:
395:
392:
391:
386:
381:
376:
370:
367:
366:
363:
362:
358:
357:
353:
352:
347:
346:
345:
334:
333:
332:
323:
320:
319:
316:
315:
310:
309:
308:
307:
300:
299:
294:
288:
285:
284:
280:
279:
277:
276:
274:Nonhomogeneous
267:
262:
259:
253:
252:
251:
243:
242:
238:
237:
235:
234:
229:
224:
219:
214:
209:
204:
198:
193:
190:
189:
186:
185:
184:Classification
181:
180:
171:
170:
169:
168:
163:
155:
154:
148:
147:
146:
145:
140:
135:
127:
126:
120:
119:
118:
117:
112:
106:
101:
96:
88:
87:
85:
84:
79:
73:
68:
65:
64:
61:
60:
56:
55:
26:
18:Finite element
9:
6:
4:
3:
2:
11001:
10990:
10987:
10985:
10982:
10980:
10977:
10975:
10972:
10970:
10967:
10965:
10962:
10960:
10957:
10955:
10952:
10951:
10949:
10934:
10931:
10929:
10926:
10924:
10921:
10919:
10916:
10912:
10909:
10908:
10907:
10904:
10902:
10899:
10897:
10894:
10892:
10889:
10887:
10884:
10882:
10879:
10875:
10872:
10871:
10870:
10867:
10865:
10862:
10860:
10857:
10855:
10852:
10850:
10847:
10844:
10841:
10839:
10836:
10835:
10833:
10829:
10823:
10820:
10817:
10814:
10811:
10808:
10805:
10802:
10800:
10797:
10795:
10792:
10790:
10787:
10783:
10780:
10778:
10775:
10774:
10773:
10770:
10768:
10765:
10763:
10760:
10759:
10757:
10755:
10751:
10744:
10741:
10738:
10735:
10732:
10729:
10726:
10723:
10720:
10717:
10716:
10714:
10712:
10708:
10701:
10698:
10696:
10693:
10690:
10687:
10685:
10682:
10679:
10676:
10673:
10670:
10667:
10664:
10662:
10659:
10658:
10656:
10654:
10650:
10643:
10640:
10637:
10634:
10632:
10629:
10626:
10623:
10620:
10617:
10615:
10612:
10610:
10607:
10606:
10604:
10602:
10601:Finite volume
10598:
10587:
10584:
10581:
10578:
10575:
10572:
10571:
10569:
10565:
10559:
10556:
10554:
10551:
10549:
10546:
10544:
10541:
10539:
10536:
10535:
10533:
10531:
10527:
10521:
10518:
10515:
10512:
10511:
10509:
10507:
10503:
10500:
10498:
10494:
10490:
10483:
10478:
10476:
10471:
10469:
10464:
10463:
10460:
10453:
10449:
10447:
10443:
10440:
10436:
10433:
10429:
10426:
10425:
10420:
10417:
10416:
10411:
10408:
10405:
10401:
10398:
10394:
10391:
10388:K. J. Bathe:
10387:
10384:
10383:
10378:
10377:
10365:
10359:
10355:
10348:
10340:
10336:
10332:
10328:
10321:
10307:on 2006-10-30
10306:
10302:
10295:
10288:
10282:
10280:
10278:
10259:
10255:
10251:
10247:
10243:
10239:
10235:
10228:
10221:
10206:
10202:
10196:
10188:
10181:
10173:
10169:
10164:
10159:
10155:
10151:
10147:
10140:
10132:
10128:
10123:
10118:
10114:
10110:
10106:
10099:
10097:
10088:
10084:
10080:
10076:
10072:
10068:
10065:(4): 045001.
10064:
10060:
10056:
10049:
10041:
10037:
10033:
10029:
10025:
10021:
10018:(6): 064010.
10017:
10013:
10009:
10002:
10000:
9991:
9987:
9980:
9965:
9961:
9957:
9951:
9943:
9937:
9929:
9925:
9921:
9917:
9913:
9909:
9905:
9898:
9896:
9894:
9885:
9881:
9877:
9873:
9872:
9864:
9850:on 2017-08-10
9849:
9845:
9841:
9835:
9827:
9823:
9819:
9815:
9811:
9807:
9800:
9791:
9783:
9779:
9776:(1): 67â103.
9775:
9771:
9770:
9765:
9761:
9755:
9747:
9743:
9738:
9733:
9729:
9725:
9721:
9717:
9713:
9706:
9692:
9688:
9681:
9673:
9667:
9663:
9659:
9653:
9645:
9639:
9635:
9634:
9626:
9618:
9612:
9607:
9606:
9600:
9596:
9590:
9582:
9576:
9572:
9565:
9550:
9546:
9540:
9532:
9528:
9524:
9520:
9513:
9497:
9493:
9489:
9488:"ĐĄĐб ĐĐĐ Đ ĐĐ"
9483:
9474:
9469:
9465:
9461:
9457:
9450:
9442:
9438:
9434:
9430:
9426:
9422:
9415:
9407:
9403:
9399:
9395:
9390:
9385:
9380:
9375:
9371:
9367:
9363:
9356:
9348:
9342:
9338:
9331:
9323:
9321:9780071267618
9317:
9313:
9306:
9304:
9295:
9293:9780495668275
9289:
9285:
9278:
9274:
9264:
9261:
9259:
9256:
9254:
9251:
9249:
9248:Space mapping
9246:
9244:
9241:
9239:
9236:
9234:
9231:
9229:
9226:
9224:
9221:
9219:
9216:
9214:
9211:
9209:
9206:
9204:
9201:
9199:
9196:
9194:
9191:
9189:
9186:
9184:
9181:
9179:
9176:
9174:
9171:
9169:
9166:
9164:
9161:
9159:
9156:
9154:
9151:
9149:
9146:
9144:
9141:
9139:
9136:
9134:
9131:
9130:
9123:
9120:
9112:
9108:
9104:
9096:
9088:
9079:
9076:
9071:
9069:
9065:
9061:
9057:
9053:
9049:
9045:
9041:
9031:
9029:
9025:
9017:
9014:
9011:
9007:
9003:
9000:
8997:
8994:
8990:
8989:
8988:
8986:
8976:
8974:
8964:
8962:
8958:
8954:
8951:
8941:
8932:
8930:
8920:
8909:
8898:
8887:
8877:
8872:
8862:
8857:
8847:
8838:
8831:
8828:
8825:
8824:
8823:
8820:
8818:
8812:
8802:
8800:
8796:
8792:
8788:
8778:
8776:
8772:
8768:
8758:
8753:
8743:
8740:
8729:
8720:
8711:
8706:
8704:
8689:
8673:
8657:
8655:
8651:
8647:
8643:
8639:
8635:
8631:
8626:
8612:
8609:
8606:
8603:
8600:
8580:
8572:
8556:
8553:
8550:
8527:
8524:
8502:
8498:
8494:
8474:
8466:
8462:
8457:
8441:
8436:
8432:
8423:
8407:
8399:
8381:
8376:
8372:
8351:
8342:
8340:
8325:
8321:
8299:
8296:
8291:
8288:
8285:
8282:
8278:
8274:
8269:
8266:
8263:
8260:
8256:
8247:
8239:
8216:
8215:
8214:
8206:
8204:
8188:
8180:
8164:
8155:
8153:
8149:
8145:
8141:
8125:
8117:
8112:
8098:
8054:
8050:
8029:
8009:
8000:
7986:
7983:
7980:
7977:
7974:
7971:
7968:
7948:
7945:
7940:
7936:
7932:
7929:
7926:
7921:
7917:
7894:
7884:
7880:
7876:
7873:
7870:
7865:
7861:
7854:
7836:
7829:
7827:
7813:
7805:
7797:
7794:
7787:
7786:
7783:
7769:
7749:
7746:
7743:
7740:
7737:
7734:
7731:
7708:
7700:
7696:
7692:
7686:
7680:
7657:
7651:
7628:
7620:
7616:
7610:
7606:
7600:
7595:
7592:
7589:
7585:
7581:
7575:
7569:
7556:
7549:
7547:
7533:
7525:
7522:
7514:
7511:
7504:
7503:
7500:
7486:
7483:
7478:
7474:
7468:
7464:
7460:
7457:
7452:
7449:
7445:
7419:
7415:
7411:
7406:
7402:
7395:
7392:
7387:
7384:
7380:
7354:
7351:
7347:
7340:
7337:
7312:
7309:
7305:
7298:
7295:
7273:
7263:
7259:
7255:
7252:
7249:
7244:
7240:
7214:
7204:
7200:
7196:
7193:
7190:
7185:
7181:
7120:
7113:
7111:
7097:
7094:
7091:
7088:
7085:
7082:
7079:
7076:
7056:
7053:
7048:
7044:
7038:
7034:
7030:
7025:
7021:
7015:
7010:
7007:
7004:
7000:
6996:
6988:
6984:
6980:
6975:
6971:
6964:
6959:
6955:
6949:
6944:
6941:
6938:
6934:
6930:
6923:
6922:
6919:
6905:
6902:
6899:
6896:
6893:
6890:
6887:
6864:
6856:
6852:
6848:
6842:
6836:
6813:
6805:
6801:
6795:
6791:
6785:
6780:
6777:
6774:
6770:
6766:
6760:
6754:
6731:
6723:
6719:
6713:
6709:
6703:
6698:
6695:
6692:
6688:
6684:
6678:
6672:
6658:
6644:
6641:
6635:
6631:
6624:
6619:
6615:
6602:
6581:
6578:
6572:
6568:
6562:
6558:
6548:
6525:
6521:
6498:
6494:
6484:
6470:
6467:
6459:
6456:
6453:
6423:
6419:
6415:
6410:
6406:
6399:
6374:
6370:
6366:
6361:
6357:
6328:
6325:
6322:
6318:
6314:
6309:
6306:
6303:
6299:
6273:
6269:
6260:
6256:
6237:
6234:
6231:
6203:
6199:
6195:
6190:
6186:
6162:
6159:
6156:
6136:
6133:
6128:
6124:
6120:
6115:
6111:
6107:
6101:
6096:
6092:
6088:
6080:
6076:
6072:
6067:
6063:
6056:
6036:
6033:
6027:
6023:
6017:
6013:
6007:
6002:
5998:
5994:
5986:
5982:
5978:
5973:
5969:
5940:
5936:
5932:
5927:
5923:
5919:
5916:
5913:
5907:
5904:
5901:
5895:
5886:
5879:
5876:
5875:sparse matrix
5871:
5850:
5847:
5844:
5839:
5836:
5832:
5828:
5823:
5820:
5816:
5806:
5794:
5793:hp-adaptivity
5790:
5787:
5784:
5781:
5780:
5779:
5775:
5773:
5769:
5764:
5760:
5746:
5743:
5740:
5736:
5731:
5727:
5704:
5700:
5679:
5659:
5637:
5633:
5590:
5586:
5563:
5559:
5550:
5549:tent function
5534:
5531:
5528:
5525:
5522:
5519:
5516:
5507:
5487:
5477:
5470:
5462:
5459:
5456:
5452:
5448:
5443:
5439:
5432:
5429:
5414:
5410:
5406:
5400:
5397:
5394:
5390:
5384:
5381:
5375:
5372:
5369:
5365:
5354:
5346:
5342:
5338:
5333:
5330:
5327:
5323:
5316:
5313:
5298:
5295:
5292:
5288:
5284:
5278:
5274:
5266:
5263:
5260:
5256:
5252:
5249:
5240:
5235:
5229:
5221:
5217:
5196:
5193:
5190:
5186:
5181:
5177:
5154:
5150:
5129:
5109:
5087:
5083:
5060:
5056:
5035:
5027:
5015:
5011:
5004:
4992:
4988:
4981:
4972:
4959:
4945:
4923:
4919:
4898:
4878:
4858:
4855:
4852:
4842:
4828:
4820:
4796:
4792:
4791:triangulation
4756:
4738:
4734:
4732:
4716:
4696:
4693:
4690:
4687:
4684:
4681:
4678:
4656:
4652:
4648:
4645:
4625:
4622:
4619:
4599:
4579:
4576:
4571:
4568:
4565:
4561:
4540:
4537:
4532:
4528:
4518:
4502:
4499:
4493:
4487:
4484:
4478:
4472:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4431:
4428:
4425:
4421:
4417:
4412:
4408:
4394:
4386:
4383:
4368:
4365:
4362:
4356:
4353:
4347:
4344:
4324:
4304:
4301:
4296:
4293:
4290:
4286:
4282:
4277:
4273:
4269:
4266:
4263:
4258:
4254:
4250:
4245:
4241:
4237:
4234:
4214:
4194:
4171:
4168:
4165:
4148:
4134:
4126:
4110:
4088:
4083:
4079:
4070:
4054:
4041:
4034:
4032:
4018:
4015:
4012:
4009:
4003:
4000:
3997:
3991:
3988:
3984:
3981:
3978:
3975:
3951:
3948:
3945:
3937:
3936:
3933:
3916:
3913:
3910:
3907:
3901:
3898:
3895:
3889:
3886:
3882:
3877:
3872:
3868:
3864:
3861:
3851:
3834:
3829:
3825:
3821:
3818:
3810:
3809:
3808:
3790:
3785:
3780:
3776:
3766:
3757:
3743:
3735:
3716:
3713:
3710:
3702:
3697:
3693:
3672:
3664:
3645:
3642:
3639:
3631:
3627:
3619:
3603:
3600:
3594:
3588:
3582:
3576:
3571:
3566:
3562:
3553:
3552:Hilbert space
3534:
3531:
3528:
3520:
3515:
3511:
3502:
3501:inner product
3486:
3477:
3473:
3457:
3454:
3451:
3431:
3428:
3425:
3405:
3382:
3379:
3376:
3366:functions of
3365:
3346:
3343:
3340:
3332:
3327:
3323:
3308:
3306:
3279:
3274:
3270:
3266:
3263:
3189:
3184:
3180:
3159:
3149:
3133:
3125:
3100:
3087:
3081:
3078:
3075:
3069:
3066:
3063:
3060:
3057:
3053:
3047:
3044:
3032:
3028:
3025:
3022:
3019:
3015:
3012:
3003:
2982:
2959:
2956:
2953:
2947:
2927:
2919:
2909:
2895:
2892:
2886:
2880:
2877:
2871:
2865:
2852:
2845:
2843:
2825:
2819:
2816:
2813:
2807:
2804:
2801:
2798:
2795:
2788:
2781:
2778:
2771:
2764:
2761:
2755:
2750:
2746:
2742:
2739:
2737:
2729:
2726:
2719:
2712:
2709:
2702:
2695:
2692:
2686:
2681:
2677:
2673:
2668:
2663:
2650:
2644:
2638:
2631:
2628:
2624:
2622:
2614:
2611:
2604:
2598:
2592:
2585:
2582:
2576:
2571:
2567:
2563:
2561:
2556:
2553:
2546:
2540:
2534:
2528:
2523:
2518:
2514:
2502:
2501:
2498:
2496:
2477:
2474:
2471:
2465:
2456:
2454:
2450:
2434:
2414:
2391:
2385:
2365:
2362:
2356:
2350:
2347:
2341:
2335:
2315:
2302:
2295:
2293:
2279:
2276:
2273:
2266:
2260:
2254:
2247:
2244:
2238:
2233:
2229:
2225:
2222:
2219:
2212:
2206:
2200:
2194:
2189:
2184:
2180:
2172:
2171:
2168:
2154:
2151:
2148:
2128:
2125:
2122:
2102:
2099:
2096:
2076:
2056:
2042:
2040:
2030:
2028:
2019:
2016:
2015:
2014:
2011:
1997:
1994:
1990:
1987:
1983:
1980:
1972:
1968:
1963:
1949:
1929:
1907:
1904:
1900:
1877:
1874:
1870:
1861:
1857:
1815:
1812:
1809:
1777:
1757:
1741:
1738:
1735:
1728:
1712:
1709:
1706:
1700:
1697:
1691:
1688:
1685:
1677:
1674:
1670:
1666:
1660:
1657:
1654:
1646:
1643:
1639:
1632:
1627:
1614:
1609:
1595:
1575:
1554:
1551:
1530:
1510:
1490:
1463:
1460:
1457:
1451:
1445:
1442:
1436:
1430:
1423:
1417:
1414:
1411:
1397:
1391:
1388:
1382:
1375:
1372:
1365:
1360:
1346:
1338:
1336:
1331:
1326:
1322:
1320:
1316:
1312:
1308:
1302:
1300:
1295:
1293:
1278:
1276:
1272:
1271:heat transfer
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1216:
1212:
1208:
1207:Ernest Hinton
1204:
1200:
1196:
1192:
1188:
1187:J. H. Argyris
1183:
1181:
1177:
1173:
1172:Lord Rayleigh
1169:
1165:
1161:
1158:
1154:
1149:
1147:
1144:
1140:
1136:
1132:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1094:
1092:
1087:
1085:
1081:
1077:
1071:
1069:
1065:
1061:
1060:heat equation
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1028:
1026:
1022:
1018:
1012:
1010:
1006:
1002:
998:
994:
986:
982:
978:
975:
971:
967:
966:
965:
963:
958:
954:
953:inner product
950:
946:
941:
939:
931:
928:
927:
926:
920:
917:
914:
911:
910:
909:
900:
896:
892:
888:
887:ferromagnetic
884:
880:
879:cylindrically
873:
864:
860:
856:
855:ferromagnetic
852:
848:
844:
837:
823:
821:
817:
813:
808:
806:
802:
798:
794:
790:
786:
782:
778:
773:
771:
767:
763:
762:heat transfer
759:
755:
751:
747:
743:
731:
726:
724:
719:
717:
712:
711:
709:
708:
700:
697:
695:
692:
690:
687:
685:
682:
680:
677:
675:
672:
670:
667:
665:
662:
660:
657:
655:
652:
650:
647:
645:
642:
640:
637:
635:
632:
630:
627:
625:
622:
621:
614:
613:
609:
608:
601:
598:
596:
593:
591:
588:
587:
583:
580:
578:
575:
573:
570:
568:
565:
563:
560:
556:
553:
552:
551:
548:
546:
545:Finite volume
543:
539:
536:
535:
534:
531:
527:
521:
518:
516:
513:
511:
507:
505:
502:
499:
498:
491:
490:
482:
479:
477:
474:
470:
466:
464:
461:
459:
455:
451:
448:
446:
443:
441:
438:
434:
431:
429:
426:
424:
421:
419:
416:
415:
414:
411:
409:
406:
405:
398:
397:
390:
387:
385:
382:
380:
377:
375:
372:
371:
365:
364:
360:
359:
351:
348:
344:
341:
340:
339:
336:
335:
331:
325:
324:
318:
317:
306:
303:
302:
298:
295:
293:
290:
289:
287:
286:
282:
281:
275:
271:
268:
266:
263:
260:
258:
255:
254:
250:
247:
246:
245:
244:
240:
239:
233:
230:
228:
225:
223:
220:
218:
215:
213:
210:
208:
205:
203:
200:
199:
197:
196:
188:
187:
183:
182:
177:
167:
164:
162:
159:
158:
157:
156:
153:
150:
149:
144:
141:
139:
136:
134:
131:
130:
129:
128:
125:
122:
121:
116:
113:
111:
107:
105:
102:
100:
97:
95:
92:
91:
90:
89:
83:
80:
78:
75:
74:
72:
71:
63:
62:
58:
57:
54:
51:
50:
43:
37:
33:
19:
10725:Peridynamics
10652:
10543:LaxâWendroff
10451:
10438:
10431:
10422:
10414:
10403:
10396:
10389:
10380:
10353:
10347:
10330:
10326:
10320:
10309:. Retrieved
10305:the original
10294:
10286:
10265:. Retrieved
10237:
10233:
10220:
10209:. Retrieved
10207:. 2019-03-18
10204:
10195:
10186:
10180:
10153:
10149:
10139:
10112:
10108:
10062:
10058:
10048:
10015:
10011:
9989:
9979:
9968:. Retrieved
9959:
9950:
9936:cite journal
9914:(1): 84â90.
9911:
9907:
9875:
9869:
9863:
9852:. Retrieved
9848:the original
9843:
9834:
9809:
9805:
9799:
9790:
9773:
9767:
9760:BabuĆĄka, Ivo
9754:
9719:
9715:
9705:
9694:. Retrieved
9690:
9680:
9661:
9652:
9632:
9625:
9604:
9589:
9570:
9564:
9553:. Retrieved
9539:
9525:(3): 24â27.
9522:
9518:
9512:
9500:. Retrieved
9496:the original
9491:
9482:
9463:
9459:
9449:
9424:
9420:
9414:
9369:
9365:
9355:
9336:
9330:
9311:
9283:
9277:
9233:Multiphysics
9121:
9117:
9105:
9101:
9072:
9037:
9021:
8982:
8970:
8960:
8947:
8938:
8927:
8874:
8859:
8844:
8835:
8821:
8814:
8798:
8794:
8790:
8784:
8774:
8770:
8764:
8755:
8735:
8726:
8718:
8700:
8658:
8645:
8637:
8633:
8629:
8627:
8570:
8464:
8460:
8458:
8343:
8243:
8212:
8156:
8113:
8001:
7841:
7830:
7561:
7550:
7125:
7114:
6665:If we write
6664:
6485:
5957:
5877:
5776:
5772:spectral FEM
5765:
5761:
5508:
5023:
5013:
5009:
4990:
4986:
4843:
4748:
4519:
4154:
4046:
4035:
3931:
3806:
3732:, but using
3503:which turns
3476:bilinear map
3314:
3146:denotes the
3122:denotes the
3101:
2915:
2857:
2846:
2457:
2307:
2296:
2048:
2036:
2023:
2012:
1964:
1778:
1610:
1347:
1344:
1330:a posteriori
1329:
1327:
1323:
1303:
1296:
1289:
1245:) developed
1195:R. W. Clough
1184:
1176:Walther Ritz
1157:second order
1150:
1138:
1109:problems in
1100:
1090:
1088:
1072:
1054:such as the
1031:
1029:
1013:
990:
974:steady state
942:
935:
924:
907:
819:
815:
810:Studying or
809:
788:
779:for solving
774:
745:
741:
739:
689:Ămile Picard
674:Martin Kutta
664:George Green
624:Isaac Newton
532:
456: /
452: /
272: /
138:Chaos theory
10859:Collocation
9658:Bathe, K.J.
9599:Fix, George
9143:CĂ©a's lemma
9082:Application
9046:a sheet of
9010:discretized
8517:, for some
8203:mass matrix
8157:The matrix
5578:per vertex
4938:instead of
4469:, and
3148:dot product
1263:engineering
1211:Bruce Irons
1199:UC Berkeley
1036:engineering
582:RungeâKutta
327:Difference
270:Homogeneous
82:Engineering
10948:Categories
10548:MacCormack
10530:Hyperbolic
10410:Endre SĂŒli
10311:2006-10-03
10267:2019-09-19
10211:2019-04-05
10189:: 591â592.
10163:1601.05970
9970:2017-07-28
9854:2017-07-28
9696:2023-10-13
9555:2013-01-24
9379:2107.04960
9270:References
9238:Patch test
9054:a wire of
8832:openxfem++
8705:or (DEM).
6918:, becomes
4207:values of
3362:to be the
2167:, we have
1503:is given,
1311:hp-version
1255:George Fix
1103:elasticity
1025:coordinate
962:polynomial
851:conducting
766:fluid flow
699:John Crank
500:Inspection
454:Asymptotic
338:Stochastic
257:Autonomous
232:Non-linear
222:Fractional
10864:Level-set
10854:Multigrid
10804:Balancing
10506:Parabolic
10131:1886-1784
10087:0965-0393
10040:0965-0393
9928:1540-6962
9746:0029-5981
9492:emi.nw.ru
9406:235794921
9398:1134-3060
9339:. Wiley.
8957:polyhedra
8640:-method (
8531:∞
8463:-method (
8225:Ω
7981:…
7930:∫
7874:…
7795:−
7782:is used,
7744:…
7586:∑
7512:−
7461:∫
7396:ϕ
7253:…
7194:…
7089:…
7031:∫
7001:∑
6965:ϕ
6935:∑
6931:−
6900:…
6771:∑
6689:∑
6628:∇
6625:⋅
6612:∇
6607:Ω
6603:∫
6553:Ω
6549:∫
6457:−
6400:ϕ
6380:⟩
6354:⟨
6307:−
6209:⟩
6183:⟨
6093:∫
6057:ϕ
5999:∫
5992:⟩
5966:⟨
5933:−
5920:−
5848:−
5744:≠
5613:Ω
5529:…
5433:∈
5407:−
5382:−
5331:−
5317:∈
5296:−
5285:−
5264:−
5253:−
5194:≠
4805:Ω
4795:polygonal
4777:Ω
4691:…
4623:∈
4459:…
4375:→
4267:⋯
4187:, choose
4013:∫
3992:ϕ
3989:−
3979:∈
3973:∀
3964:such that
3949:∈
3911:∫
3890:ϕ
3887:−
3865:∈
3859:∀
3849:such that
3822:∈
3563:∫
3487:ϕ
3398:that are
3288:Ω
3267:∈
3244:Ω
3241:∂
3221:Ω
3198:Ω
3160:ϕ
3134:⋅
3110:∇
3070:ϕ
3067:−
3064:≡
3051:∇
3048:⋅
3042:∇
3037:Ω
3033:∫
3029:−
3008:Ω
3004:∫
2948:ϕ
2808:ϕ
2805:−
2802:≡
2747:∫
2743:−
2678:∫
2674:−
2568:∫
2515:∫
2493:by using
2466:ϕ
2230:∫
2181:∫
1842:Ω
1839:∂
1787:Ω
1755:Ω
1752:∂
1726:Ω
1307:p-version
1135:Feng Kang
1062:, or the
987:problems.
985:transient
979:a set of
976:problems,
968:a set of
895:amplitude
812:analyzing
440:Wronskian
418:Dirichlet
161:Economics
104:Chemistry
94:Astronomy
10838:Spectral
10777:additive
10700:Smoothed
10666:Extended
10258:Archived
10254:27349493
9964:Archived
9660:(2006).
9601:(1973).
9549:Archived
9502:17 March
9466:: 1â23.
9126:See also
9056:tungsten
9048:aluminum
8826:GetFEM++
6129:′
6116:′
5873:(b) The
5209:, i.e.,
4749:We need
4069:subspace
3734:elliptic
3618:Lp space
3124:gradient
2975:for any
2782:′
2765:′
2713:′
2696:′
2632:′
2586:″
2248:″
2027:computer
1991:″
1624:P2
1555:″
1376:″
1239:OpenSees
1168:cylinder
847:magnetic
550:Galerkin
450:Lyapunov
361:Solution
305:Notation
297:Operator
283:Features
202:Ordinary
10822:FETI-DP
10702:(S-FEM)
10621:(MUSCL)
10609:Godunov
10067:Bibcode
10020:Bibcode
9908:Wilmott
9814:Bibcode
9724:Bibcode
9429:Bibcode
9253:STRAND7
9052:drawing
9044:rolling
8961:virtual
8950:mimetic
8787:hpk-FEM
8781:hpk-FEM
6259:support
6221:in the
4994:(black)
4797:region
3550:into a
1862:), and
1860:polygon
1235:NASTRAN
1223:Paris 6
1164:torsion
1153:lattice
1097:History
1052:physics
1007:or the
897:of the
891:created
881:shaped
423:Neumann
207:Partial
115:Geology
110:Biology
99:Physics
10831:Others
10818:(FETI)
10812:(BDDC)
10684:Mortar
10668:(XFEM)
10661:hp-FEM
10644:(WENO)
10627:(AUSM)
10588:(FDTD)
10582:(FDFD)
10567:Others
10553:Upwind
10516:(FTCS)
10360:
10252:
10129:
10085:
10038:
9926:
9744:
9668:
9640:
9613:
9577:
9519:Strain
9404:
9396:
9343:
9318:
9290:
9068:stress
9064:strain
9060:grains
8829:xfem++
8767:hp-FEM
8723:CutFEM
8642:hp-FEM
8152:MATLAB
7842:where
5768:hp-FEM
4047:where
3102:where
1779:where
1543:, and
1483:where
1273:, and
1243:DNV GL
1178:, and
1084:eddies
1058:, the
993:linear
885:. The
610:People
522:
469:Series
227:Linear
66:Fields
34:, see
10845:(DVR)
10806:(BDD)
10745:(PIC)
10739:(MPM)
10733:(MPS)
10721:(SPH)
10691:(GDM)
10680:(SEM)
10638:(ENO)
10576:(ADI)
10261:(PDF)
10230:(PDF)
10158:arXiv
9402:S2CID
9374:arXiv
9075:voxel
8850:S-FEM
8715:A-FEM
5026:basis
4227:with
3938:Find
3811:Find
3470:(see
3303:(see
2328:with
1858:or a
1315:x-FEM
1247:Sesam
1166:of a
1111:civil
510:Euler
428:Robin
350:Delay
292:Order
265:Exact
191:Types
59:Scope
32:poset
10727:(PD)
10674:(DG)
10358:ISBN
10250:PMID
10127:ISSN
10083:ISSN
10036:ISSN
9942:link
9924:ISSN
9912:2005
9742:ISSN
9691:ECMI
9666:ISBN
9638:ISBN
9611:ISBN
9575:ISBN
9504:2018
9394:ISSN
9341:ISBN
9316:ISBN
9288:ISBN
8983:The
8815:The
8805:XFEM
8785:The
8765:The
8650:SFEM
8554:>
8543:and
8528:<
8396:. A
8146:and
8022:and
7961:for
7909:and
7724:for
7437:and
7330:and
7229:and
7148:and
7069:for
6880:for
6747:and
6594:and
6513:and
6468:>
6392:and
6049:and
5770:and
5509:for
4856:>
4553:and
4337:by:
4283:<
4270:<
4264:<
4251:<
3756:is.
3444:and
3126:and
2141:and
1942:and
1892:and
1253:and
1225:and
1143:mesh
1121:and
1113:and
1105:and
983:for
972:for
859:iron
843:mesh
841:FEM
797:mesh
740:The
617:List
10335:doi
10242:doi
10168:doi
10154:111
10117:doi
10075:doi
10028:doi
9916:doi
9880:doi
9822:doi
9810:147
9778:doi
9732:doi
9720:104
9527:doi
9468:doi
9437:doi
9384:doi
8697:AEM
6261:of
5692:at
5142:at
5102:in
5028:of
4071:of
3418:at
2995:by
2115:at
2049:If
822:).
820:FEA
746:FEM
10950::
10412::
10331:59
10329:.
10276:^
10256:.
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10232:.
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9368:.
9364:.
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8688:.
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8638:hp
8625:.
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1277:.
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1193:,
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807:.
772:.
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760:,
10481:e
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10467:v
10418:.
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10314:.
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8799:k
8795:p
8791:h
8775:p
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8674:n
8669:R
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8610:+
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8604:=
8601:p
8581:d
8571:p
8557:0
8551:p
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8499:h
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8465:h
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8437:0
8433:H
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8322:C
8300:f
8297:=
8292:y
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8286:y
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8275:+
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8264:x
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7877:,
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7866:1
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7570:f
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7534:.
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7299:=
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7264:n
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7215:t
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7119:)
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7098:.
7095:n
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4682:=
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4626:V
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4580:1
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4572:1
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4566:n
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4506:}
4503:0
4500:=
4497:)
4494:1
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4488:v
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4482:)
4479:0
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3532:,
3529:0
3526:(
3521:1
3516:0
3512:H
3458:1
3455:=
3452:x
3432:0
3429:=
3426:x
3406:0
3386:)
3383:1
3380:,
3377:0
3374:(
3350:)
3347:1
3344:,
3341:0
3338:(
3333:1
3328:0
3324:H
3291:)
3285:(
3280:1
3275:0
3271:H
3264:v
3201:)
3195:(
3190:1
3185:0
3181:H
3088:,
3085:)
3082:v
3079:,
3076:u
3073:(
3061:s
3058:d
3054:v
3045:u
3026:=
3023:s
3020:d
3016:v
3013:f
2983:v
2963:)
2960:v
2957:,
2954:u
2951:(
2928:u
2896:0
2893:=
2890:)
2887:1
2884:(
2881:v
2878:=
2875:)
2872:0
2869:(
2866:v
2851:)
2849:2
2847:(
2826:,
2823:)
2820:v
2817:,
2814:u
2811:(
2799:x
2796:d
2792:)
2789:x
2786:(
2779:v
2775:)
2772:x
2769:(
2762:u
2756:1
2751:0
2740:=
2730:x
2727:d
2723:)
2720:x
2717:(
2710:v
2706:)
2703:x
2700:(
2693:u
2687:1
2682:0
2669:1
2664:0
2659:|
2654:)
2651:x
2648:(
2645:v
2642:)
2639:x
2636:(
2629:u
2625:=
2615:x
2612:d
2608:)
2605:x
2602:(
2599:v
2596:)
2593:x
2590:(
2583:u
2577:1
2572:0
2564:=
2557:x
2554:d
2550:)
2547:x
2544:(
2541:v
2538:)
2535:x
2532:(
2529:f
2524:1
2519:0
2481:)
2478:v
2475:,
2472:u
2469:(
2447:(
2435:u
2415:u
2395:)
2392:x
2389:(
2386:v
2366:0
2363:=
2360:)
2357:1
2354:(
2351:u
2348:=
2345:)
2342:0
2339:(
2336:u
2316:u
2301:)
2299:1
2297:(
2280:.
2277:x
2274:d
2270:)
2267:x
2264:(
2261:v
2258:)
2255:x
2252:(
2245:u
2239:1
2234:0
2226:=
2223:x
2220:d
2216:)
2213:x
2210:(
2207:v
2204:)
2201:x
2198:(
2195:f
2190:1
2185:0
2155:1
2152:=
2149:x
2129:0
2126:=
2123:x
2103:0
2100:=
2097:v
2077:v
2057:u
1998:f
1995:=
1988:V
1984:+
1981:u
1950:y
1930:x
1908:y
1905:y
1901:u
1878:x
1875:x
1871:u
1819:)
1816:y
1813:,
1810:x
1807:(
1758:,
1742:0
1739:=
1736:u
1729:,
1716:)
1713:y
1710:,
1707:x
1704:(
1701:f
1698:=
1695:)
1692:y
1689:,
1686:x
1683:(
1678:y
1675:y
1671:u
1667:+
1664:)
1661:y
1658:,
1655:x
1652:(
1647:x
1644:x
1640:u
1633:{
1628::
1596:x
1576:u
1552:u
1531:x
1511:u
1491:f
1464:,
1461:0
1458:=
1455:)
1452:1
1449:(
1446:u
1443:=
1440:)
1437:0
1434:(
1431:u
1424:,
1421:)
1418:1
1415:,
1412:0
1409:(
1401:)
1398:x
1395:(
1392:f
1389:=
1386:)
1383:x
1380:(
1373:u
1366:{
1361::
865:.
818:(
744:(
729:e
722:t
715:v
528:)
524:(
38:.
20:)
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