Finite element method
Finite element method
Navier–Stokes differential equations used to simulate airflow around an obstruction.
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The finite element method (FEM) is a numerical method for solving problems of engineering and mathematical physics. It is also referred to as finite element analysis (FEA). Typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of these problems generally require the solution to boundary value problems for partial differential equations. The finite element method formulation of the problem results in a system of algebraic equations. The method yields approximate values of the unknowns at discrete number of points over the domain. To solve the problem, it subdivides a large problem into smaller, simpler parts that are called finite elements. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.
Basic concepts:
The subdivision of a whole domain into simpler parts has several advantages:
Accurate representation of complex geometry
Inclusion of dissimilar material properties
Easy representation of the total solution
Capture of local effects.
A typical work out of the method involves (1) 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, followed by (2) systematically recombining all sets of element equations into a global system of equations for the final calculation. The global system of equations has known solution techniques, and can be calculated from the initial 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 often partial differential equations (PDE). To explain the approximation in this process, FEM is commonly introduced as a special case of Galerkin method. The process, in mathematical language, is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with
a set of algebraic equations for steady state problems,
a set of ordinary differential equations for transient problems.
These equation sets are the element equations. They are linear if the underlying PDE is linear, and vice versa. Algebraic equation sets that arise in the steady state problems are solved using numerical linear algebra methods, while ordinary differential equation sets that arise in the transient problems are solved by numerical integration using standard techniques such as Euler’s method or the Runge-Kutta method.
In step (2) above, a global system of equations is generated from the element equations through a transformation of coordinates from the subdomains’ local nodes to the domain’s global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system. The process is often carried out by FEM software using coordinate data generated from the subdomains.
FEM is best understood from its practical application, known as finite element analysis (FEA). FEA as applied in engineering is a computational tool for performing engineering analysis. It includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software program coded with FEM algorithm. In applying FEA, the complex problem is usually a physical system with the underlying physics such as the Euler-Bernoulli beam equation, the heat equation, or the Navier-Stokes equations expressed in either PDE or integral equations, while the divided small elements of the complex problem represent different areas in the physical system.
FEA is a good choice 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. For instance, 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 cost of the simulation). Another example would be in numerical weather prediction, where it is more important to have accurate predictions over developing highly nonlinear phenomena (such as tropical cyclones in the atmosphere, or eddies in the ocean) rather than relatively calm areas.
FEM mesh created by an analyst prior to finding a solution to a magnetic problem using FEM software. Colours indicate that the analyst has set material properties for each zone, in this case a conducting wire coil in orange; a ferromagnetic component (perhaps iron) 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 equations alone.
FEM solution to the problem at left, involving a cylindrically shaped magnetic shield. The ferromagnetic cylindrical part is shielding the area inside the cylinder by diverting the magnetic field created by the coil (rectangular area on the right). The color represents the amplitude of the magnetic flux density, 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.
Before FEM:
While it is difficult to quote a date of the invention of the finite element method, the method originated from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering. Its development can be traced back to the work by A. Hrennikoff and R. Courant in the early 1940s. In China, in the later 1950s and early 1960s, based on the computations of dam constructions, K. Feng proposed a systematic numerical method for solving partial differential equations. The method was called the finite difference method based on variation principle, which was another independent invention of the finite element method. Although the approaches used by these pioneers are different, they share one essential characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains, usually called elements.
Hrennikoff’s work discretizes the domain by using a lattice analogy, while Courant’s approach divides the domain into finite triangular subregions to solve second order elliptic partial differential equations (PDEs) that arise from the problem of torsion of a cylinder. Courant’s contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by Rayleigh, Ritz, and Galerkin.
The finite element method obtained its real impetus in the 1960s and 1970s by the developments of J. H. Argyris with co-workers at the University of Stuttgart, R. W. Clough with co-workers at UC Berkeley, O. C. Zienkiewicz with co-workers Ernest Hinton, Bruce Irons[5] and others at the University of Swansea, Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher with co-workers at Cornell University. Further impetus was provided in these years by available open source finite element software programs. NASA sponsored the original version of NASTRAN, and UC Berkeley made the finite element program SAP IV[6] widely available. In Norway the ship classification society Det Norske Veritas (now DNV GL) developed Sesam in 1969 for use in analysis of ships. A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by Strang and Fix. The method has since been generalized for the numerical modeling of physical systems in a wide variety of engineering disciplines, e.g., electromagnetism, heat transfer, and fluid dynamics.
Technical discussion:
The structure of finite element methods:
Finite element methods are numerical methods for approximating the solutions of mathematical problems that are usually formulated so as to precisely state an idea of some aspect of physical reality.
A finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms and post-processing procedures.
Examples of variational formulation are the Galerkin 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.
There are various numerical solution algorithms that 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 choices of variational formulation and discretization strategy.
Postprocessing procedures are designed for the extraction of the data of interest from a finite element solution. In order to meet the requirements of solution verification, postprocessors need to provide for a 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 action of the analyst. There are some very efficient postprocessors that provide for the realization of superconvergence.
Illustrative problems P1 and P2:
We will illustrate the finite element method using two sample problems from which the general method can be extrapolated. It is assumed that the reader is familiar with calculus and linear algebra.
P1 is a one-dimensional problem
{\displaystyle {\mbox{ P1 }}:{\begin{cases}u”(x)=f(x){\mbox{ in }}(0,1),\\u(0)=u(1)=0,\end{cases}}}
where {\displaystyle f} is given, {\displaystyle u} is an unknown function of {\displaystyle x}, and {\displaystyle u”} is the second derivative of {\displaystyle u} with respect to {\displaystyle x}.
P2 is a two-dimensional problem (Dirichlet problem)
{\displaystyle {\mbox{P2 }}:{\begin{cases}u_{xx}(x,y)+u_{yy}(x,y)=f(x,y)&{\mbox{ in }}\Omega ,\\u=0&{\mbox{ on }}\partial \Omega ,\end{cases}}}
where {\displaystyle \Omega } is a connected open region in the {\displaystyle (x,y)} plane whose boundary {\displaystyle \partial \Omega } is “nice” (e.g., a smooth manifold or a polygon), and {\displaystyle u_{xx}} and {\displaystyle u_{yy}} denote the second derivatives with respect to {\displaystyle x} and {\displaystyle y}, respectively.
The problem P1 can be solved “directly” by computing antiderivatives. However, this method of solving the boundary value problem (BVP) works only when there is one spatial dimension and does not generalize to higher-dimensional problems or to problems like {\displaystyle u+u”=f}. 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.
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.
The second step is the discretization, where the weak form is discretized in a finite-dimensional space.
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 computer.
Weak formulation:
The first step is to convert P1 and P2 into their equivalent weak formulations.
The weak form of P1:
If {\displaystyle u} solves P1, then for any smooth function {\displaystyle v} that satisfies the displacement boundary conditions, i.e. {\displaystyle v=0} at {\displaystyle x=0} and {\displaystyle x=1}, we have
(1) {\displaystyle \int _{0}^{1}f(x)v(x)\,dx=\int _{0}^{1}u”(x)v(x)\,dx.}
Conversely, if {\displaystyle u} with {\displaystyle u(0)=u(1)=0} satisfies (1) for every smooth function {\displaystyle v(x)} then one may show that this {\displaystyle u} will solve P1. The proof is easier for twice continuously differentiable {\displaystyle u} (mean value theorem), but may be proved in a distributional sense as well.
We define a new function {\displaystyle \phi (u,v)} by using integration by parts on the right-hand-side of (1):
(2){\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}}}
where we have used the assumption that {\displaystyle v(0)=v(1)=0}.
The weak form of P2:
If we integrate by parts using a form of Green’s identities, we see that if {\displaystyle u} solves P2, then we may define {\displaystyle \phi (u,v)} for any {\displaystyle v} by
{\displaystyle \int _{\Omega }fv\,ds=-\int _{\Omega }\nabla u\cdot \nabla v\,ds\equiv -\phi (u,v),}
where {\displaystyle \nabla } denotes the gradient and {\displaystyle \cdot } denotes the dot product in the two-dimensional plane. Once more {\displaystyle \,\!\phi } can be turned into an inner product on a suitable space {\displaystyle H_{0}^{1}(\Omega )} of “once differentiable” functions of {\displaystyle \Omega } that are zero on {\displaystyle \partial \Omega }. We have also assumed that {\displaystyle v\in H_{0}^{1}(\Omega )} (see Sobolev spaces). Existence and uniqueness of the solution can also be shown.
A proof outline of existence and uniqueness of the solution:
We can loosely think of {\displaystyle H_{0}^{1}(0,1)} to be the absolutely continuous functions of {\displaystyle (0,1)} that are {\displaystyle 0} at {\displaystyle x=0} and {\displaystyle x=1} (see Sobolev spaces). Such functions are (weakly) “once differentiable” and it turns out that the symmetric bilinear map {\displaystyle \!\,\phi } then defines an inner product which turns {\displaystyle H_{0}^{1}(0,1)} into a Hilbert space (a detailed proof is nontrivial). On the other hand, the left-hand-side {\displaystyle \int _{0}^{1}f(x)v(x)dx} is also an inner product, this time on the Lp space {\displaystyle L^{2}(0,1)}. An application of the Riesz representation theorem for Hilbert spaces shows that there is a unique {\displaystyle u} solving (2) and therefore P1. This solution is a-priori only a member of {\displaystyle H_{0}^{1}(0,1)}, but using elliptic regularity, will be smooth if {\displaystyle f} is.
Discretization:
A function in {\displaystyle H_{0}^{1},} with zero values at the endpoints (blue), and a piecewise linear approximation (red)
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:
Find {\displaystyle u\in H_{0}^{1}} such that
{\displaystyle \forall v\in H_{0}^{1},\;-\phi (u,v)=\int fv}
with a finite-dimensional version:
(3) Find {\displaystyle u\in V} such that
{\displaystyle \forall v\in V,\;-\phi (u,v)=\int fv}
where {\displaystyle V} is a finite-dimensional subspace of {\displaystyle H_{0}^{1}}. There are many possible choices for {\displaystyle V} (one possibility leads to the spectral method). However, for the finite element method we take {\displaystyle V} to be a space of piecewise polynomial functions.
For problem P1:
We take the interval {\displaystyle (0,1)}, choose {\displaystyle n} values of {\displaystyle x} with {\displaystyle 0=x_{0}<x_{1}<\cdots <x_{n}<x_{n+1}=1} and we define {\displaystyle V} by:
{\displaystyle V=\{v:[0,1]\rightarrow {\mathbb {R}}\;:v{\mbox{ is continuous, }}v|_{[x_{k},x_{k+1}]}{\mbox{ is linear for }}k=0,\dots ,n{\mbox{, and }}v(0)=v(1)=0\}}
where we define {\displaystyle x_{0}=0} and {\displaystyle x_{n+1}=1}. Observe that functions in {\displaystyle V} are not differentiable according to the elementary definition of calculus. Indeed, if {\displaystyle v\in V} then the derivative is typically not defined at any {\displaystyle x=x_{k}}, {\displaystyle k=1,\ldots ,n}. However, the derivative exists at every other value of {\displaystyle x} and one can use this derivative for the purpose of integration by parts.
A piecewise linear function in two dimensions
For problem P2:
We need {\displaystyle V} to be a set of functions of {\displaystyle \Omega }. In the figure on the right, we have illustrated a triangulation of a 15 sided polygonal region {\displaystyle \Omega } in the plane (below), and a piecewise linear function (above, in color) of this polygon which is linear on each triangle of the triangulation; the space {\displaystyle V} 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 {\displaystyle h>0} which one takes to be very small. This parameter will be related to the size of the largest or average triangle in the triangulation. As we refine the triangulation, the space of piecewise linear functions {\displaystyle V} must also change with {\displaystyle h}. For this reason, one often reads {\displaystyle V_{h}} instead of {\displaystyle V} in the literature. Since we do not perform such an analysis, we will not use this notation.
Choosing a basis:
Interpolation of a Bessel function
16 scaled and shifted triangular basis functions (colors) used to reconstruct a zeroeth order Bessel function J0 (black).
The linear combination of basis functions (yellow) reproduces J0(blue) to any desired accuracy.
To complete the discretization, we must select a basis of {\displaystyle V}. In the one-dimensional case, for each control point {\displaystyle x_{k}} we will choose the piecewise linear function {\displaystyle v_{k}} in {\displaystyle V} whose value is {\displaystyle 1} at {\displaystyle x_{k}} and zero at every {\displaystyle x_{j},\;j\neq k}, i.e.,
{\displaystyle v_{k}(x)={\begin{cases}{x-x_{k-1} \over x_{k}\,-x_{k-1}}&{\mbox{ if }}x\in [x_{k-1},x_{k}],\\{x_{k+1}\,-x \over x_{k+1}\,-x_{k}}&{\mbox{ if }}x\in [x_{k},x_{k+1}],\\0&{\mbox{ otherwise}},\end{cases}}}
for {\displaystyle k=1,\dots ,n}; this basis is a shifted and scaled tent function. For the two-dimensional case, we choose again one basis function {\displaystyle v_{k}} per vertex {\displaystyle x_{k}} of the triangulation of the planar region {\displaystyle \Omega }. The function {\displaystyle v_{k}} is the unique function of {\displaystyle V} whose value is {\displaystyle 1} at {\displaystyle x_{k}} and zero at every {\displaystyle x_{j},\;j\neq k}.
Depending on the author, the word “element” in “finite element method” refers either to the triangles in the domain, 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” by “piecewise quadratic” or even “piecewise polynomial”. The author might then say “higher order element” instead of “higher degree polynomial”. Finite element method is not restricted to triangles (or tetrahedra in 3-d, or higher order simplexes in multidimensional spaces), but 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 the hp-FEM and spectral 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 approximate solution within some bounds from the ‘exact’ solution of the continuum problem. Mesh adaptivity may utilize various techniques, the most popular are:
moving nodes (r-adaptivity)
refining (and unrefining) elements (h-adaptivity)
changing order of base functions (p-adaptivity)
combinations of the above (hp-adaptivity).
Small support of the basis:
Solving the two-dimensional problem {\displaystyle u_{xx}+u_{yy}=-4} in the disk centered at the origin and radius 1, with zero boundary conditions.
(a) The triangulation.
(b) The sparse matrix L of the discretized linear system
(c) The computed solution, {\displaystyle u(x,y)=1-x^{2}-y^{2}.}
The primary advantage of this choice of basis is that the inner products
{\displaystyle \langle v_{j},v_{k}\rangle =\int _{0}^{1}v_{j}v_{k}\,dx}and {\displaystyle \phi (v_{j},v_{k})=\int _{0}^{1}v_{j}’v_{k}’\,dx}
will be zero for almost all {\displaystyle j,k}. (The matrix containing {\displaystyle \langle v_{j},v_{k}\rangle } in the {\displaystyle (j,k)} location is known as the Gramian matrix.) In the one dimensional case, the support of {\displaystyle v_{k}} is the interval {\displaystyle [x_{k-1},x_{k+1}]}. Hence, the integrands of {\displaystyle \langle v_{j},v_{k}\rangle } and {\displaystyle \phi (v_{j},v_{k})} are identically zero whenever {\displaystyle |j-k|>1}.
Similarly, in the planar case, if {\displaystyle x_{j}} and {\displaystyle x_{k}} do not share an edge of the triangulation, then the integrals
{\displaystyle \int _{\Omega }v_{