Finite element method

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In mathematics, the finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for partial differential equations. It uses variational methods (the calculus of variations) to minimize an error function and produce a stable solution. Analogous to the idea that connecting many tiny straight lines can approximate a larger circle, FEM encompasses all the methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain.

History[edit]

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 [1] and R. Courant.[2] 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 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[3] and others at the University of Swansea, Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher[4] 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[5] widely available. A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by Strang and Fix.[6] 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.[7][8]

Technical discussion[edit]

The Structure of Finite Element Methods[edit]

Finite element methods are numerical methods for approximating the solutions of mathematical models. Mathematical models are mathematical problems 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[edit]

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

\mbox{ P1 }:\begin{cases}
u''(x)=f(x) \mbox{ in } (0,1), \\
u(0)=u(1)=0,
\end{cases}

where f is given, u is an unknown function of x, and u'' is the second derivative of u with respect to x.

P2 is a two-dimensional problem (Dirichlet problem)

\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 \Omega is a connected open region in the (x,y) plane whose boundary \partial \Omega is "nice" (e.g., a smooth manifold or a polygon), and u_{xx} and u_{yy} denote the second derivatives with respect to x and y, respectively.

The problem P1 can be solved "directly" by computing antiderivatives. However, this method of solving the boundary value problem works only when there is one spatial dimension and does not generalize to higher-dimensional problems or to problems like 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[edit]

The first step is to convert P1 and P2 into their equivalent weak formulations.

The weak form of P1[edit]

If u solves P1, then for any smooth function v that satisfies the displacement boundary conditions, i.e. v=0 at x=0 and x=1, we have

(1) \int_0^1 f(x)v(x) \, dx = \int_0^1 u''(x)v(x) \, dx.

Conversely, if u with u(0)=u(1)=0 satisfies (1) for every smooth function v(x) then one may show that this u will solve P1. The proof is easier for twice continuously differentiable u (mean value theorem), but may be proved in a distributional sense as well.

By using integration by parts on the right-hand-side of (1), we obtain

(2)
\begin{align}
 \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{align}

where we have used the assumption that v(0)=v(1)=0.

The weak form of P2[edit]

If we integrate by parts using a form of Green's identities, we see that if u solves P2, then for any v,

\int_{\Omega} fv\,ds = -\int_{\Omega} \nabla u \cdot \nabla v \, ds \equiv -\phi(u,v),

where \nabla denotes the gradient and \cdot denotes the dot product in the two-dimensional plane. Once more \,\!\phi can be turned into an inner product on a suitable space H_0^1(\Omega) of "once differentiable" functions of \Omega that are zero on \partial \Omega. We have also assumed that 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[edit]

We can loosely think of H_0^1(0,1) to be the absolutely continuous functions of (0,1) that are 0 at x=0 and x=1 (see Sobolev spaces). Such functions are (weakly) "once differentiable" and it turns out that the symmetric bilinear map \!\,\phi then defines an inner product which turns H_0^1(0,1) into a Hilbert space (a detailed proof is nontrivial). On the other hand, the left-hand-side \int_0^1 f(x)v(x)dx is also an inner product, this time on the Lp space L^2(0,1). An application of the Riesz representation theorem for Hilbert spaces shows that there is a unique u solving (2) and therefore P1. This solution is a-priori only a member of H_0^1(0,1), but using elliptic regularity, will be smooth if f is.

Discretization[edit]

A function in 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 u \in  H_0^1 such that
\forall v \in H_0^1, \; -\phi(u,v)=\int fv

with a finite-dimensional version:

(3) Find u \in V such that
\forall v \in V, \; -\phi(u,v)=\int fv

where V is a finite-dimensional subspace of H_0^1. There are many possible choices for V (one possibility leads to the spectral method). However, for the finite element method we take V to be a space of piecewise polynomial functions.

For problem P1[edit]

We take the interval (0,1), choose n values of x with 0=x_0<x_1<\cdots<x_n<x_{n+1}=1 and we define V by:

V=\{v:[0,1] \rightarrow \Bbb 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 x_0=0 and x_{n+1}=1. Observe that functions in V are not differentiable according to the elementary definition of calculus. Indeed, if v \in V then the derivative is typically not defined at any x=x_k, k=1,\,n. However, the derivative exists at every other value of x and one can use this derivative for the purpose of integration by parts.

A piecewise linear function in two dimensions

For problem P2[edit]

We need V to be a set of functions of \Omega. In the figure on the right, we have illustrated a triangulation of a 15 sided polygonal region \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 V would consist of functions that are linear on each triangle of the chosen triangulation.

One often reads V_h instead of V in the literature. The reason is that one hopes that as the underlying triangular grid 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. The triangulation is then indexed by a real valued parameter 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 V must also change with h, hence the notation V_h. Since we do not perform such an analysis, we will not use this notation.

Choosing a basis[edit]

Basis functions vk (blue) and a linear combination of them, which is piecewise linear (red)

To complete the discretization, we must select a basis of V. In the one-dimensional case, for each control point x_k we will choose the piecewise linear function v_k in V whose value is 1 at x_k and zero at every x_j,\;j \neq k, i.e.,

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 k=1,\dots,n; this basis is a shifted and scaled tent function. For the two-dimensional case, we choose again one basis function v_k per vertex x_k of the triangulation of the planar region \Omega. The function v_k is the unique function of V whose value is 1 at x_k and zero at every 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[edit]

Solving the two-dimensional problem 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, u(x, y)=1-x^2-y^2.

The primary advantage of this choice of basis is that the inner products

\langle v_j,v_k \rangle=\int_0^1 v_j v_k\,dx

and

\phi(v_j,v_k)=\int_0^1 v_j' v_k'\,dx

will be zero for almost all j,k. (The matrix containing \langle v_j,v_k \rangle in the (j,k) location is known as the Gramian matrix.) In the one dimensional case, the support of v_k is the interval [x_{k-1},x_{k+1}]. Hence, the integrands of \langle v_j,v_k \rangle and \phi(v_j,v_k) are identically zero whenever |j-k|>1.

Similarly, in the planar case, if x_j and x_k do not share an edge of the triangulation, then the integrals

\int_{\Omega} v_j v_k\,ds

and

\int_{\Omega} \nabla v_j \cdot \nabla v_k\,ds

are both zero.

Matrix form of the problem[edit]

If we write u(x)=\sum_{k=1}^n u_k v_k(x) and f(x)=\sum_{k=1}^n f_k v_k(x) then problem (3), taking v(x)=v_j(x) for j=1,\dots,n, becomes

-\sum_{k=1}^n u_k \phi (v_k,v_j) = \sum_{k=1}^n f_k \int v_k v_j dx for j=1,\dots,n. (4)

If we denote by \mathbf{u} and \mathbf{f} the column vectors (u_1,\dots,u_n)^t and (f_1,\dots,f_n)^t, and if we let

L=(L_{ij})

and

M=(M_{ij})

be matrices whose entries are

L_{ij}=\phi (v_i,v_j)

and

M_{ij}=\int v_i v_j dx

then we may rephrase (4) as

-L \mathbf{u} = M \mathbf{f}. (5)

It is not necessary to assume f(x)=\sum_{k=1}^n f_k v_k(x). For a general function f(x), problem (3) with v(x)=v_j(x) for j=1,\dots,n becomes actually simpler, since no matrix M is used,

-L \mathbf{u} = \mathbf{b}, (6)

where \mathbf{b}=(b_1,\dots,b_n)^t and b_j=\int f v_j dx for j=1,\dots,n.

As we have discussed before, most of the entries of L and M are zero because the basis functions v_k have small support. So we now have to solve a linear system in the unknown \mathbf{u} where most of the entries of the matrix L, which we need to invert, are zero.

Such matrices are known as sparse matrices, and there are efficient solvers for such problems (much more efficient than actually inverting the matrix.) In addition, L is symmetric and positive definite, so a technique such as the conjugate gradient method is favored. For problems that are not too large, sparse LU decompositions and Cholesky 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 L is usually referred to as the stiffness matrix, while the matrix M is dubbed the mass matrix.

General form of the finite element method[edit]

In general, the finite element method is characterized by the following process.

  • One chooses a grid for \Omega. In the preceding treatment, the grid consisted of triangles, but one can also use squares or curvilinear polygons.
  • Then, one chooses basis functions. In our discussion, we used piecewise linear basis functions, but it is also common to use piecewise polynomial basis functions.

A separate consideration is the smoothness of the basis functions. For second order elliptic boundary value problems, piecewise polynomial basis function that are 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 u_{xxxx}+u_{yyyy}=f, one may use piecewise quadratic basis functions that are C^1.

Another consideration is the relation of the finite-dimensional space V to its infinite-dimensional counterpart, in the examples above H_0^1. A conforming element method is one in which the space V 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 nonconforming 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 in general discontinuous along the edges, this finite-dimensional space is not a subspace of the original H_0^1.

Typically, one has an algorithm for taking a given mesh and subdividing it. If the main method for increasing precision is to subdivide the mesh, one has an h-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 h is bounded above by Ch^p, for some C<\infty and p>0, then one has an order p method. Under certain hypotheses (for instance, if the domain is convex), a piecewise polynomial of order d method will have an error of order p=d+1.

If instead of making h smaller, one increases the degree of the polynomials used in the basis function, one has a p-method. If one combines these two refinement types, one obtains an hp-method (hp-FEM). In the hp-FEM, the polynomial degrees can vary from element to element. High order methods with large uniform p are called spectral finite element methods (SFEM). These are not to be confused with spectral methods.

For vector partial differential equations, the basis functions may take values in \mathbb{R}^n.

Various types of finite element methods[edit]

AEM[edit]

The Applied Element Method, or AEM combines features of both FEM and Discrete element method, or (DEM).

Generalized finite element method[edit]

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 partition 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.[9]

Mixed finite element method[edit]

hp-FEM[edit]

The hp-FEM combines adaptively, elements with variable size h and polynomial degree p in order to achieve exceptionally fast, exponential convergence rates.[10]

hpk-FEM[edit]

The hpk-FEM combines adaptively, elements with variable size h, polynomial degree of the local approximations p and global differentiability of the local approximations (k-1) in order to achieve best convergence rates.

XFEM[edit]

S-FEM[edit]

Fiber Beam Method[edit]

Main article: Fiber beam element

Spectral methods[edit]

Main article: Spectral method

Meshfree methods[edit]

Main article: Meshfree methods

Discontinuous Galerkin methods[edit]

Finite element limit analysis[edit]

Stretched grid method[edit]

Main article: Stretched grid method

Comparison to the finite difference method[edit]

The finite difference method (FDM) is an alternative way of approximating solutions of PDEs. The differences between FEM and FDM are:

  • 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 straightforward.
  • The most attractive feature of finite differences is that it can be very easy to implement.
  • There are several ways one could consider the FDM a special case of the FEM approach. E.g., first order FEM is identical to FDM for Poisson's equation, if the problem is discretized by a regular rectangular mesh with each rectangle divided into two triangles.
  • 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.
  • The quality of a FEM approximation is often higher than in the corresponding FDM approach, but this is extremely problem-dependent and several examples to the contrary can be provided.

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) while computational fluid dynamics (CFD) tends to use FDM or other methods like finite volume method (FVM). CFD problems usually require discretization of the problem into a large number of cells/gridpoints (millions and more), therefore cost of the solution favors simpler, lower order approximation within each cell. This is especially true for 'external flow' problems, like air flow around the car or airplane, or weather simulation.

Application[edit]

Visualization of how a car deforms in an asymmetrical crash using finite element analysis.[1]

A variety of specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in 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 also in minimizing weight, materials, and costs.

FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. 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.

This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications.[11] The introduction of FEM has substantially decreased the time to take products from concept to the production line.[11] It is primarily through improved initial prototype designs using FEM that testing and development have been accelerated.[12] 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.[11]

FEA has also been proposed to use in stochastic modelling for numerically solving probability models.[13][14]

See also[edit]

References[edit]

  1. ^ Hrennikoff, Alexander (1941). "Solution of problems of elasticity by the framework method". Journal of applied mechanics 8.4: 169–175. 
  2. ^ Courant, R. (1943). "Variational methods for the solution of problems of equilibrium and vibrations". Bulletin of the American Mathematical Society 49: 1–23. 
  3. ^ Hinton, Ernest; Irons, Bruce (July 1968). "Least squares smoothing of experimental data using finite elements". Strain 4: 24–27. 
  4. ^ "Richard H. Gallagher". NNDB. Retrieved 2014-05-18. 
  5. ^ "SAP-IV Software and Manuals". NISEE e-Library, The Earthquake Engineering Online Archive. 
  6. ^ Strang, Gilbert; Fix, George (1973). An Analysis of The Finite Element Method. Prentice Hall. ISBN 0-13-032946-0. 
  7. ^ Zienkiewicz, O.C.; Taylor, R.L.; Zhu, J.Z. (2005). The Finite Element Method: Its Basis and Fundamentals (Sixth ed.). Butterworth-Heinemann. ISBN 0750663200. 
  8. ^ Bathe, K.J. (2006). Finite Element Procedures. Cambridge, MA: Klaus-Jürgen Bathe. ISBN 097900490X. 
  9. ^ Babuška, Ivo; Banerjee, Uday; Osborn, John E. (June 2004). "Generalized Finite Element Methods: Main Ideas, Results, and Perspective". International Journal of Computational Methods 1 (1): 67–103. doi:10.1142/S0219876204000083. 
  10. ^ P. Solin, K. Segeth, I. Dolezel: Higher-Order Finite Element Methods, Chapman & Hall/CRC Press, 2003
  11. ^ a b c Hastings, J. K., Juds, M. A., Brauer, J. R., Accuracy and Economy of Finite Element Magnetic Analysis, 33rd Annual National Relay Conference, April 1985.
  12. ^ McLaren-Mercedes (2006). "McLaren Mercedes: Feature - Stress to impress". Archived from the original on 2006-10-30. Retrieved 2006-10-03. 
  13. ^ Peng Long; Wang Jinliang; Zhu Qiding (19 May 1995). "Methods with high accuracy for finite element probability computing". Journal of Computational and Applied Mathematics 59 (2): 181–189. doi:10.1016/0377-0427(94)00027-X. 
  14. ^ Haldar, Achintya; Mahadevan, Sankaran (2000). Reliability Assessment Using Stochastic Finite Element Analysis. John Wiley & Sons. ISBN 978-0471369615. 

Further reading[edit]

  • G. Allaire and A. Craig: Numerical Analysis and Optimization:An Introduction to Mathematical Modelling and Numerical Simulation
  • K. J. Bathe : Numerical methods in finite element analysis, Prentice-Hall (1976).
  • J. Chaskalovic, Finite Elements Methods for Engineering Sciences, Springer Verlag, (2008).
  • O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu : The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann, (2005).

External links[edit]