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Finite element method (FEM) is a popular method for numerically solvingdifferential equations arising in engineering andmathematical modeling. Typical problem areas of interest include the traditional fields ofstructural analysis,heat transfer,fluid flow, mass transport, andelectromagnetic potential. Computers are usually used to perform the calculations required. With high-speedsupercomputers, better solutions can be achieved and are often required to solve the largest and most complex problems.
FEM is a generalnumerical method for solvingpartial differential equations in two- or three-space variables (i.e., someboundary value problems). There are also studies about using FEM to solve high-dimensional problems.[1] To solve a problem, FEM subdivides a large system into smaller, simpler parts calledfinite elements. This is achieved by a particular spacediscretization in the space dimensions, which is implemented by the construction of amesh of the object: the numerical domain for the solution that has a finite number of points. FEM formulation of a boundary value problem finally results in a system ofalgebraic equations. The method approximates the unknown function over the domain.[2] The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. FEM then approximates a solution by minimizing an associated error function via thecalculus of variations.
Studying oranalyzing a phenomenon with FEM is often referred to as finite element analysis (FEA).
The subdivision of a whole domain into simpler parts has several advantages:[3]
A typical approach using the method involves the following steps:
The global system of equations uses known solution techniques and can be calculated from theinitial values of the original problem to obtain a numerical answer.
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 oftenpartial differential equations (PDEs). To explain the approximation of this process, FEM is commonly introduced as a special case of theGalerkin method. The process, in mathematical language, is to construct an integral of theinner product of the residual and theweight functions; then, 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 arepolynomial approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally using the following:
These equation sets are element equations. They arelinear if the underlying PDE is linear and vice versa. Algebraic equation sets that arise in the steady-state problems are solved usingnumerical linear algebraic methods. In contrast,ordinary differential equation sets that occur in the transient problems are solved by numerical integrations using standard techniques such asEuler's method or theRunge–Kutta method.
In the second step 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 appropriateorientation adjustments as applied in relation to the referencecoordinate system. The process is often carried out using FEM software withcoordinate data generated from the subdomains.
The practical application of FEM is known as finite element analysis (FEA). FEA, as applied inengineering, is a computational tool for performingengineering analysis. It includes the use ofmesh generation techniques for dividing acomplex problem into smaller elements, as well as the use of software coded with a FEM algorithm. When applying FEA, the complex problem is usually a physical system with the underlyingphysics, such as theEuler–Bernoulli beam equation, theheat equation, or theNavier–Stokes equations, expressed in either PDEs orintegral equations, while the divided, smaller elements of the complex problem represent different areas in the physical system.
FEA may be used for analyzing problems over complicated domains (e.g., cars and oil pipelines) when the domain changes (e.g., 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 valuable resource, as they remove multiple instances of creating and testing complex prototypes for various high-fidelity situations.[citation needed] 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 the rear of the car, thus reducing the cost of the simulation. Another example would be innumerical weather prediction, where it is more important to have accurate predictions over developing highly nonlinear phenomena, such astropical cyclones in the atmosphere oreddies in the ocean, rather than relatively calm areas.
A clear, detailed, and practical presentation of this approach can be found in the textbookThe Finite Element Method for Engineers.[4]
While it is difficult to quote the date of the invention of FEM, the method originated from the need to solve complexelasticity andstructural analysis problems incivil andaeronautical engineering.[5] Its development can be traced back to work byAlexander Hrennikoff[6] andRichard Courant[7] in the early 1940s. Another pioneer wasIoannis Argyris. In the USSR, the introduction of the practical application of FEM is usually connected withLeonard Oganesyan.[8] It was also independently rediscovered in China byFeng Kang in the late 1950s and early 1960s, based on the computations of dam constructions, where it was called the "finite difference method" based on variation principles. Although the approaches used by these pioneers are different, they share one essential characteristic: themeshdiscretization of a continuous domain into a set of discrete sub-domains, usually called elements.
Hrennikoff's work discretizes the domain by using alattice analogy, while Courant's approach divides the domain into finite triangular sub-regions to solvesecond-orderelliptic partial differential equations that arise from the problem of thetorsion of acylinder. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed byLord Rayleigh,Walther Ritz, andBoris Galerkin.
The application of FEM gained momentum in the 1960s and 1970s due to the developments ofJ. H. Argyris and his co-workers at theUniversity of Stuttgart;R. W. Clough and his co-workers atUniversity of California Berkeley;O. C. Zienkiewicz and his co-workersErnest Hinton,Bruce Irons,[9] and others atSwansea University;Philippe G. Ciarlet at the University ofParis 6; andRichard Gallagher and his co-workers atCornell University. During this period, additional impetus was provided by the available open-source FEM programs. NASA sponsored the original version ofNASTRAN. University of California Berkeley made the finite element programs SAP IV[10] and, later,OpenSees widely available. In Norway, the ship classification society Det Norske Veritas (nowDNV GL) developedSesam in 1969 for use in the analysis of ships.[11] A rigorous mathematical basis for FEM was provided in 1973 with a publication byGilbert Strang andGeorge Fix.[12] The method has since been generalized for thenumerical modeling of physical systems in a wide variety ofengineering disciplines, such aselectromagnetism,heat transfer, andfluid dynamics.[13][14]
A finite element method is characterized by avariational formulation, a discretization strategy, one or more solution algorithms, and post-processing procedures.
Examples of the variational formulation are theGalerkin method, the discontinuous Galerkin method, mixed methods, etc.
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. Examples of discretization strategies are the h-version,p-version,hp-version,x-FEM,isogeometric analysis, 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.
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.
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 fora posteriori 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 the realization ofsuperconvergence.
The following two problems demonstrate the finite element method.
P1 is a one-dimensional problemwhere is given, is an unknown function of, and is the second derivative of with respect to.
P2 is a two-dimensional problem (Dirichlet problem)
where is a connected open region in the plane whose boundary is nice (e.g., asmooth manifold or apolygon), and and denote the second derivatives with respect to and, respectively.
The problem P1 can be solved directly by computingantiderivatives. However, this method of solving theboundary value problem (BVP) works only when there is one spatial dimension. It does not generalize to higher-dimensional problems or problems like. For this reason, we will develop the finite element method for P1 and outline its generalization to P2.
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.
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 acomputer.
The first step is to convert P1 and P2 into their equivalentweak formulations.
If solves P1, then for any smooth function that satisfies the displacement boundary conditions, i.e. at and, we have
| 1 |
Conversely, if with satisfies (1) for every smooth function then one may show that this will solve P1. The proof is easier for twice continuously differentiable (mean value theorem) but may be proved in adistributional sense as well.
We define a new operator or map by usingintegration by parts on the right-hand-side of (1):
| 2 |
where we have used the assumption that.
If we integrate by parts using a form ofGreen's identities, we see that if solves P2, then we may define for any by
where denotes thegradient and denotes thedot product in the two-dimensional plane. Once more can be turned into an inner product on a suitable space of once differentiable functions of that are zero on. We have also assumed that (seeSobolev spaces). The existence and uniqueness of the solution can also be shown.
We can loosely think of to be theabsolutely continuous functions of that are at and (seeSobolev spaces). Such functions are (weakly) once differentiable, and it turns out that the symmetricbilinear map then defines aninner product which turns into aHilbert space (a detailed proof is nontrivial). On the other hand, the left-hand-side is also an inner product, this time on theLp space. An application of theRiesz representation theorem for Hilbert spaces shows that there is a unique solving (2) and, therefore, P1. This solution is a-priori only a member of, but usingelliptic regularity, will be smooth if is.
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:
with a finite-dimensional version:
| Find such that | 3 |
where is a finite-dimensionalsubspace of. There are many possible choices for (one possibility leads to thespectral method). However, we take as a space of piecewise polynomial functions for the finite element method.
We take the interval, choose values of with and we define by:
where we define and. Observe that functions in are not differentiable according to the elementary definition of calculus. Indeed, if then the derivative is typically not defined at any,. However, the derivative exists at every other value of, and one can use this derivative forintegration by parts.
We need to be a set of functions of. In the figure on the right, we have illustrated atriangulation of a 15-sidedpolygonal region in the plane (below), and apiecewise linear function (above, in color) of this polygon which is linear on each triangle of the triangulation; the space would consist of functions that are linear on each triangle of the chosen triangulation.
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 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 must also change with. For this reason, one often reads instead of in the literature. Since we do not perform such an analysis, we will not use this notation.
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To complete the discretization, we must select abasis of. In the one-dimensional case, for each control point we will choose the piecewise linear function in whose value is at and zero at every, i.e.,
for; this basis is a shifted and scaledtent function. For the two-dimensional case, we choose again one basis function per vertex of the triangulation of the planar region. The function is the unique function of whose value is at and zero at every.
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 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).
Examples of methods that use higher degree piecewise polynomial basis functions are thehp-FEM andspectral FEM.
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 problem. Mesh adaptivity may utilize various techniques; the most popular are:
The primary advantage of this choice of basis is that the inner productsandwill be zero for almost all.(The matrix containing in the location is known as theGramian matrix.)In the one dimensional case, thesupport of is the interval. Hence, the integrands of and are identically zero whenever.
Similarly, in the planar case, if and do not share an edge of the triangulation, then the integralsandare both zero.
If we write and then problem (3), taking for, becomes
| for | 4 |
If we denote by and the column vectors and, and if we letandbe matrices whose entries areandthen we may rephrase (4) as
| 5 |
It is not necessary to assume. For a general function, problem (3) with for becomes actually simpler, since no matrix is used,
| 6 |
where and for.
As we have discussed before, most of the entries of and are zero because the basis functions have small support. So we now have to solve a linear system in the unknown where most of the entries of the matrix, which we need to invert, are zero.
Such matrices are known assparse matrices, and there are efficient solvers for such problems (much more efficient than actually inverting the matrix.) In addition, is symmetric and positive definite, so a technique such as theconjugate gradient method is favored. For problems that are not too large, sparseLU decompositions andCholesky decompositions still work well. For instance,MATLAB's backslash operator (which uses sparse LU, sparse Cholesky, and other factorization methods) can be sufficient for meshes with a hundred thousand vertices.
The matrix is usually referred to as thestiffness matrix, while the matrix is dubbed themass matrix.
In general, the finite element method is characterized by the following process.
Separate consideration is the smoothness of the basis functions. For second-orderelliptic boundary value problems, 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, one may use piecewise quadratic basis functions that are.
Another consideration is the relation of the finite-dimensional space to its infinite-dimensional counterpart in the examples above. Aconforming element method is one in which space 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 anonconforming element method, 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.
Typically, one has an algorithm for subdividing a given mesh. If the primary method for increasing precision is to subdivide the mesh, one has anh-method (h is customarily the diameter of the largest element in the mesh.) In this manner, if one shows that the error with a grid is bounded above by, for some and, then one has an orderp method. Under specific hypotheses (for instance, if the domain is convex), a piecewise polynomial of order method will have an error of order.
If instead of makingh smaller, one increases the degree of the polynomials used in the basis function, one has ap-method. If one combines these two refinement types, one obtains anhp-method (hp-FEM). In the hp-FEM, the polynomial degrees can vary from element to element. High-order methods with large uniformp are called spectral finite element methods (SFEM). These are not to be confused withspectral methods.
For vector partial differential equations, the basis functions may take values in.
The Applied Element Method or AEM combines features of both FEM andDiscrete element method or (DEM).
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.
The Cut Finite Element Approach was developed in 2014.[15] 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."[16]
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 apartition of unity is used to “bond” 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.[17]
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.
Thehp-FEM combines adaptively elements with variable sizeh and polynomial degreep to achieve exceptionally fast, exponential convergence rates.[18]
Thehpk-FEM combines adaptively elements with variable sizeh, polynomial degree of the local approximationsp, and global differentiability of the local approximations (k-1) to achieve the best convergence rates.
Theextended finite element method (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.
Several research codes implement this technique to various degrees:
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.).
The introduction of the scaled boundary finite element method (SBFEM) came from Song and Wolf (1997).[19] 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 advantages of finite element formulations and procedures and boundary element discretization. However, unlike the boundary element method, no fundamental differential solution is required.
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.
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.[20]
Loubignac iteration is an iterative method in finite element methods.
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 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.
The virtual element method (VEM), introduced by Beirão da Veiga et al. (2013)[21] as an extension ofmimeticfinite difference (MFD) methods, is a generalization of the standard finite element method for arbitrary element geometries. This allows admission of general polygons (orpolyhedra in 3D) that are highly irregular and non-convex in shape. The namevirtual derives from the fact that knowledge of the local shape function basis is not required and is, in fact, never explicitly calculated.
Some types of finite element methods (conforming, nonconforming, mixed finite element methods) are particular cases of thegradient discretization method (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.
Thefinite difference method (FDM) is an alternative way of approximating solutions of PDEs. The differences between FEM and FDM are:
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,computational fluid dynamics (CFD) tend to use FDM or other methods likefinite volume method (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.
Another method used for approximating solutions to a partial differential equation is theFast Fourier Transform (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,[24] but FEM may be more accurate.[25] One example of the respective advantages of the two methods is in simulation ofrolling a sheet ofaluminum (an FCC metal), anddrawing a wire oftungsten (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 ingrains, the FEM method was much more accurate. In the sheet rolling simulation, the results of the two methods were similar.[25] FFT has a larger speed advantage in cases where the boundary conditions are given in the materialsstrain, and loses some of its efficiency in cases where thestress is used to apply the boundary conditions, as more iterations of the method are needed.[26]
The FE and FFT methods can also be combined in avoxel 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.[27] Unlike FEM, FFT methods’ similarities to image processing 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.[27] 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.[28]
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, 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.[29]
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 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.[30]
This powerful design tool has significantly improved both the standard of engineering designs and the design process methodology in many industrial applications.[32] The introduction of FEM has substantially decreased the time to take products from concept to the production line.[32] Testing and development have been accelerated primarily through improved initial prototype designs using FEM.[33] 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.[32]
In the 1990s FEM was proposed for use in stochastic modeling for numerically solving probability models[34] and later for reliability assessment.[35]
FEM is widely applied for approximating differential equations that describe physical systems. This method is very popular in the community ofComputational fluid dynamics, and there are many applications for solvingNavier–Stokes equations with FEM.[36][37][38] Recently, the application of FEM has been increasing in the researches of computational plasma. Promising numerical results using FEM forMagnetohydrodynamics,Vlasov equation, andSchrödinger equation have been proposed.[39][40]
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