Finite element method

Called the finite element method (FEM), also called " finite element method ", is a numerical method for the solution of partial differential equations. It is a widespread modern calculation methods in Engineering and is the standard tool for the hard -body simulation. The process yields an approximate function of the exact solution to the differential equation, the accuracy can be improved by increasing the degrees of freedom and therefore the amount of computation.

• 4.1 Procedure of a linear mechanical calculation ( examples) 4.1.1 Input: preprocessor
• 4.1.2 Processing: equation solver
• 4.1.3 Issue: postprocessor

Introduction

With the finite element method problems from different physical disciplines can be calculated since this is basically a numerical method for solving differential equations. First, the calculation domain is divided into an arbitrarily large number of elements. These elements are "finally" ( finite ) and not infinity ( infinitely ) small. Splitting the area into a certain number of elements of finite size, can be described with a finite number of parameters, the method gave the name " finite element method ".

Within these elements shape functions are defined (eg local Ritz approaches each element). Substituting these trial functions into the differential equation to be solved, we get together with the initial, boundary and transition conditions a system of equations, which is usually solved numerically. The size of the equations to be solved system depends critically on the number of finite elements. His solution is ultimately the numerical solution of the considered differential equation dar.

General Procedure

The investigated solution area is first, the finite element divided into sub-areas.

Within the finite element are defined for the desired solution per batch functions that are only a few of subdivisions equal to zero. (This property is the real reason for the term "finite " elements. ) By a linear combination of basis functions within the element the possible solutions of the numerical approximation are established and set up a matrix element for each element. In the application, the solutions are determined, for example by material properties and constitutive laws.

The differential equations and the boundary conditions are multiplied by test functions () and integrated over the solution area. The integral is replaced by a sum of individual integrals of the finite element, the integration is usually performed by an approximate numerical integration, such as Gaussian quadrature. Since the shape functions only on a few of the elements are equal to zero, resulting in a sparse, often very large linear system of equations, in which the factors of the linear combination are unknown.

This large total matrix is one, by adding the element matrices. The total matrix is square, so it has the same number of rows as columns.

The dimension of the matrix is derived from the number of basis functions multiplied by the physical model underlying degrees of freedom ( eg, shifts in the x- direction, y - direction, rotations about various axes). The dimension of the matrix is the total number of degrees of freedom, wherein the model appropriate considerations regarding the clarity of the problem (such as in the case of an elastic body, the rigid body movements ) must be excluded.

Because each element is connected only to a few adjacent elements, most of the values ​​of the entire matrix is zero, so that they are " sparse " is shown. In most cases, the same functions are used as an approach and test functions. In this case, the matrix is ​​also symmetrical with respect to its diagonal. The matrix is ​​the left part ( A) of the linear system of equations derived below

The basic equation of the FE method.

If the number of degrees of freedom not too large (up to 500,000 ), can be solve this system of equations most efficiently by a direct method, for example with the Gaussian elimination method. Here, the sparse structure of the equation system to be used effectively. While the Gaussian algorithm is the computational effort for equations, the effort can be significantly reduced by clever pivot selection (for example, Markowitz algorithm or graph-theoretic approaches) but.

For more than 500,000 unknowns the poor condition of the equation system prepares the direct solvers increasing difficulty, so that one, used for large problems in general iterative solvers who gradually improve a solution. Simple examples of this are the Jacobi and Gauss - Seidel method, but practically rather multigrid or preconditioned Krylov subspace methods such as the conjugate gradient method or GMRES used. Due to the size of the systems of equations of the use of parallel computers is sometimes necessary.

The partial differential equation is nonlinear, and the resulting system of equations is not linear. Such can only be solved by numerical approximation methods in general. An example of such a method is the Newton's method in which step a linear system is solved.

Today there are a variety of commercial computer programs that operate on elements of the method of finite.

Mathematical derivation

Weak formulation

An elliptic partial differential equation can be formulated weak, ie The problem may be expressed in a manner which requires the solution to less smoothness. This is done as follows.

Given a Hilbert space, a functional ( function of its dual space ), and a, ie on continuous and elliptic bilinear form as solution of the variational problem, if

Existence and uniqueness of the solution provided by the Riesz representation theorem (for the case that the bilinear form is symmetric ) and the Lemma of Lax- Milgram (general case ).

We know that the space is a Hilbert space. Based on this, one can define the Sobolev spaces on the so-called weak derivative.

The problem may be considered as a variant of a partial differential equation in a field.

The Poissonproblem as an example:

Wherein here denotes the Laplace operator. Multiplication by infinitely often continuously differentiable functions gives, after an integration

A partial integration ( Green's first formula ) and the zero boundary conditions then yield

Now an elliptical and continuous bilinear form on, and the right side is a continuous linear form on

Does the considered functional space / Hilbert space has a finite basis, so you can win a linear system of equations from the variational formulation.

For function spaces, the choice of the basis of the efficiency of procedure. Accessible in this context are the use of splines with triangulations, and in certain cases, the discrete Fourier transform ( splitting into sine and cosine).

Because of flexibility considerations regarding the geometry of the area following approach is usually chosen.

We discretize the area by dividing asunder into triangles and one uses splines associated with the vertices p to span the finite dimensional function space on. The splines meet at specified points on the triangles (where δ is the Kronecker delta is ). So you can then represent a discrete function by

With the coefficients relative to the base representation. Due to the finite base must be no longer test against all, but only against all the basic functions, the variational formulation is reduced due to the linearity

So we won a linear system of equations to solve

With

And

This result is obtained with any finite basis of the Hilbert space.

Discretization

The given task is discretized by more generally the base area is in simple sub-basin, the so-called elements, finite ( finite ) number, disassembled. For some tasks, the division into elements of the problem is already largely predetermined, for example in spatial frameworks, in which the individual rods form the elements of the construction. This also applies to frame structures, where each bar or divided bar pieces representing elements of the task.

For two-dimensional problems, the fundamental domain into triangles, parallelograms, curvilinear triangles or quadrangles is divided. Even if only linear elements are used, can be reached with a sufficiently fine discretization of a fairly good approximation (approximation ) of the fundamental domain. Curvilinear elements increase the quality of the approximation. Anyway, this discretization allows a flexible and well adapted to the problem identification of the fundamental domain. However, care must be taken that very pointed or obtuse angles are avoided in the element nodes in order to eliminate numerical difficulties. Then the given area will be replaced by the area of the approximating elements. With the patch test can later verify that this is well done.

Spatial problems with a discretization of the three-dimensional region in tetrahedral elements, brick elements or other adapted to the problem, possibly a curved surface with boundary elements, these are typically Serendipity or Lagrange elements processed.

The fineness of the discretization, that is, the density of the network has a significant influence on the accuracy of the results of the approximate calculation. Since at the same time increases the computational cost when using finer and denser networks, it is necessary to develop intelligent networking solutions possible.

Element approach

In each of the elements of the unknown function, or for the chosen general the problem describing functions, a problem equitable approach. More particularly, this whole rational functions are in the independent space coordinates. For one-dimensional elements (beams, bars) polynomials come first, second, third, and sometimes even higher degree in question. For two-dimensional problems, see linear, quadratic or higher-degree polynomials using. The type of approach depends on the one hand on the shape of the element from, on the other hand can influence the approach to be chosen and the problem being addressed. Because the shape functions must satisfy the transition from one element to the neighboring very specific problem- dependent continuity conditions. The continuity requirements are often physical reasons obvious, and for mathematical reasons also required. For example, the displacement of a continuous body in a direction at the transition from one element to the other must be continuous to ensure the continuity of the material. In the case of beam or plate bending, the continuity requirements are higher, since there even the continuity of the first derivative or the two first partial derivatives must be requested from analog to physical reasons. Elements with shape functions which satisfy the continuity conditions, hot compliant.

To actually meet now the continuity requirements, the function curve must ( the nodal point displacements ) in certain points of the element, the nodal points, expressed in the element by function values ​​and also by values ​​of ( partial) derivatives. The function values ​​used in the nodes and values ​​of derivatives is called the node variables of the element. Using these node variables, there is the trial function as a linear combination of so-called shape functions with the node variables as coefficients dar.

It is advantageous to use in addition to a local item- a global coordinate system for the node coordinates. Both are linked by transformation functions. If this transformation, the same shape functions as used for the deformation approach, it is isoparametric elements in functions lower or higher grade sub - or super- parametric elements.

Formal definition of the finite element ( according to Ciarlet )

A finite element is a triple, where:

• T is a non-empty field (eg, triangles, quadrilaterals, tetrahedrons, etc. )
• Is a finite-dimensional space of shape functions ( linear, quadratic or cubic shape functions, ie spline and sine, etc. ) shape functions
• Is a set of linearly independent functionals on node variables

It is true of the functionals that they are associated to functions of the base:

Then for every function

For sine function is as a base then

And the Functional

For splines go against it is sufficient to point evaluation at the fixed points of the triangles.

Boundary conditions

After a given problem is discretized and the element matrices are set up to run a predefined boundary conditions. A typical FE- problem can have two types of boundary conditions: Dirichlet boundary conditions and Neumann boundary conditions. They are always ( work ) on the nodes.

A Dirichlet boundary condition is a function value immediately before and a Neumann boundary condition is a derivative of a function value before. Dirichlet boundary condition is specified, this means that the problem gets a less degree of freedom and the corresponding row and column is deleted in the overall matrix. If the Dirichlet boundary condition equal to zero, the value according to its pre-factor of the linear form (" right " ) is added. Depending on the nature of the physical problem, it may be various physical quantities such as shown in the table by way of example. The Neumann boundary conditions further includes a share of the linear form ( "right side" ).

Another variant is periodic boundary conditions in which the values ​​are taken at an edge as data for another boundary and thus a periodic infinite continued area or a rotationally symmetric problem is simulated.

Minimum of the potential energy

At a static node shifts the problem now be determined from the condition that in this equilibrium state, the potential energy is at a minimum. The principle of minimum potential energy is one of the possible methods for the direct determination of varying stiffness finite element equations. The potential energy of a structure is the sum of the internal strain energy ( elastic deformation energy ) and the potential of the applied load ( the work done by external forces work). For other problems there is not such a natural minimum principle. There, one makes do with the fact that minimizes the residual error. This is called the method of weighted residuals Galerkin, also Galerkin method or Galerkin approach.

Application

Originally, the finite element method for the solution of solid-state problems has been developed, although the term " finite elements " was used a little later. In the course of research, the finite element method was always further generalized and can now in many physical problems, including in different coupled field calculations, forecasts, or any technical issues in the automotive, medical technology, aviation and aerospace engineering, mechanical engineering and consumer goods in engineering are used. A main application of the method is the product development, inter alia, mechanical strength calculations of individual components or, for example, complete chassis and body structures are calculated to reduce elaborate crash tests.

Approach a linear mechanical calculation ( examples)

Programs that use the finite element method, operate on the EVA principle: The user created in a CAD program a (part) geometry. Then he gives the so-called FE preprocessor before further entries. A FEM solver performs the actual bill, and the user receives the calculated results, which he can then look at the so-called finite element post-processor in the form of graphical displays. Often, pre-and post-processor are combined in a program or even part of the CAD program.

Input: preprocessor

In the CAD program, the component is constructed and transferred to the FE preprocessor by means of a direct interface or with a neutral exchange format such as STEP. By selecting network parameters, such as element size and element type (eg, tetrahedron, hexahedron in 3D) in the networking module generates the finite elements using a cross-linking algorithm. Analysis of the mechanical strength, the material behavior is to be entered, which indicates significantly, the reactions experienced by the component (for example, deformation) to external loads. Depending on the material, the relationship between stress and strain is different under deformation. If this relationship is linear, only the Young's modulus and Poisson's ratio are required for the FE calculation, otherwise more characteristic material values ​​and inputs in the preprocessor are necessary. Other boundary conditions are, for example, stresses exerted on the component (forces, pressure, etc.). In order to obtain as realistic as possible, and finally the homogeneous ( restraints ) and the inhomogeneous boundary conditions ( displacements ) as well as with all applicable loads are indicated on the model.

Processing: equation solver

Depending on the program now an explicit ( separate program ) or an integral equation solver is used.

The advantage of a direct solver for the Gauss method is for the practical application in the numerical stability and the preservation of an exact result. Disadvantages are the poor conditioning of the usually sparse stiffness matrices and the high memory requirements, as mentioned above. Iterative solvers are less sensitive in poor conditioning and require less memory when the non-zero elements of storage used. However, iterative solvers use a termination criterion for the calculation of the results when this is reached before a nearly exact solution was found, the result, for example, a voltage profile can be easily misinterpreted.

Edition: postprocessor

In the case of mechanical strength calculation, the user receives as a result of the FEM solver in particular stress, deformation and strain values ​​. It can represent the post-processor, for example, a false-color image.

History

The use of the FEM in the practice started in the 1950s in a structure calculation of aircraft wings in the air and space industry ( Turner, Clough 1956) and very soon also in the vehicle. The method is based on the work here at Daimler AG in Stuttgart, which began the self-developed FEM program ESEM ( electrostatics element method ), long before the computer-aided design (CAD) made ​​her entry in the early 1980s. The term finite element method was first proposed in 1960 by RW Clough and is used since the 1970s everywhere. The most common German -language term for industrial users is calculation engineer.

The history of the finite element method laid out in the research and publications of the following authors ( selection):

• Karl Heinrich Schell Bach: calculus of variations; Solution of a minimal surface problem (1851 /52)
• Gustav serious Kirsch: The fundamental equations of the theory of elasticity of solids derived from the consideration of a system of points which are connected by elastic members ( 1868)
• John William Strutt, 3rd Baron Rayleigh ( 1842-1919 ): On the theory of resonance. 1870
• Walter Ritz (1878-1909): new method for the solution of variational problems, Ritz'sches procedure ( 1908/ 09)
• Boris G. Galerkin (1871-1945): method of weighted residuals (1915 )
• Erich Trefftz (1926 ): localized basis functions; Counterpart to Ritz'schen method
• Hans Ebner ( 1929): thrust sheet as a flat element in aircraft
• Alexander Hrennikoff (1896-1984): stick models, replacing disks by trusses, plates by carrier gratings 1940/41,
• Richard Courant (1888-1972): Variational methods for the solution of problems of equilibrium and vibration ( s ). 1943 ( basis functions with local support each element approaches for vibration problems )
• William Prager (1903-1980), John Lighton Synge (1897-1995): Approximation in Elasticity based on the concept of function space. 1947
• John Argyris (1913-2004): force and displacement method for beam structures, Matrizenformulierung (1954 /55)
• MJ Turner, Ray W. Clough, HC Martin, LJ Topp: Stiffness and deflection analysis of complex structures. 1956 ( first structure calculation of aircraft wings at Boeing, first application of the FEM with computer program, first application of surface elements )
• Spierig (1963 ): Development of triangular elements, transfer to bowls
• Olgierd Cecil Zienkiewicz (1921-2009), pioneer of the FEM and the first standard work ( textbook ): The Finite Element Method in Structural and Continuum Mechanics, 1967 ( with YK Cheung).
• Alfred Zimmer ( * 1920 ) and Peter Groth ( born 1938 ), pioneers of the FEM, the first German FEM textbook. Element method elastostatics, 1969 Oldenbourg Verlag, Munich, Vienna
• Olga Alexandrovna Ladyschenskaja (1922-2004), Ivo Babuška (* 1926) and Franco Brezzi (* 1945) - Ladyschenskaja - Babuška - Brezzi condition for the stability of a mixed finite element problem with the saddle -point structure
• Ivo Babuška (* 1926) - adaptive finite element algorithms

Programs

Finite element software and its application has become an industry with billions in annual sales.

• In practice, many different stand -alone programs are large with a similar range of applications in use; selecting which program is used, does not only depend on the use, but also on factors such as availability, certification standard in the company or licensing costs.
• With the built- in commercial CAD systems, finite element packages simpler (usually linear ) can be computed problems and then analyzed directly using the CAD system. The individual steps, such as the cross-linking process ( meshing ) run automatically in the background.
• There are Prä-/Postprozessoren with graphical user interface and separate FE solvers.
• There are program frameworks without a GUI, usually as a pre-processor with integrated equation solver, which are operated by programming language to control, for example, with self-made additional routines the FE solver.