The finite element method FEMis a numerical method for Finite elemente simulation dating problems of engineering and mathematical physics. Typical problem areas of interest include structural analysis Finite elemente simulation dating, heat transferfluid flowmass 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 Finite elemente simulation dating algebraic equations.
The method yields approximate values of the unknowns at discrete number of points over the domain. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. FEM then uses variational methods Finite elemente simulation dating the calculus of variations to approximate a solution by minimizing an associated error function. The subdivision of a whole domain into simpler parts has several advantages: A typical work out of the method involves 1 dividing the
Finite elemente simulation dating of the problem into a collection of subdomains, with
Finite elemente simulation dating 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 Finite elemente simulation dating 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 Finite elemente simulation dating equations PDE.
To explain the approximation in this
Finite elemente simulation dating, 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.
These equation sets are the element equations. They are linear if the
Finite elemente simulation dating 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. FEA as applied in engineering is a computational tool Finite elemente simulation dating 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, complex problem is usually a physical system with the underlying physics such as Finite elemente simulation dating Euler-Bernoulli beam equationthe heat equationor the Navier-Stokes equations expressed in either PDE or integral equationswhile 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 pipelinesFinite elemente simulation dating the domain changes as during a solid state reaction a moving boundarywhen the desired precision varies over the domain, or when the solution lacks smoothness.
FEA simulations provide a valuable resource as they remove multiple instances of creation and testing of hard prototypes for various high fidelity situations. Another example would be in numerical weather predictionwhere 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 Finite elemente simulation dating calm areas.
While it is difficult to quote a date of the invention Finite elemente simulation dating 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
Finite elemente simulation dating be traced back to the work by Hrennikoff  and R. Courant  in the early s. Another pioneer Finite elemente simulation dating Ioannis Argyris.
In the USSR, the Finite elemente simulation dating of the practical application of the method is usually connected with name of Leonard Oganesyan. 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: 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
Finite elemente simulation dating of torsion of a cylinder.
Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by RayleighRitzand Galerkin. The finite element method obtained its real impetus in the s and s by the of J.
Argyris with co-workers at the University of StuttgartR. Clough with co-workers at UC BerkeleyO. Further impetus was in these years by available open source finite element software programs. 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 Finite elemente simulation dating of physical reality.
A finite element method is characterized by a variational formulationa discretization strategy, one or more solution algorithms and post-processing procedures. Examples of variational formulation are the Galerkin methodthe discontinuous Galerkin method, mixed methods, etc.
A discretization strategy is understood to mean a clearly 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-versionhp-versionx-FEMisogeometric analysisetc. 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 Finite elemente simulation dating classified into two broad categories; direct and iterative solvers.
These Finite elemente simulation dating 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 of approximation are larger than what is considered acceptable then the
Finite elemente simulation dating 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. We will demonstrate 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. P2 is a two-dimensional problem Dirichlet problem. The problem P1 can be solved "directly" by computing antiderivatives. For this reason, we will develop the
Finite elemente simulation dating 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 Finite elemente simulation dating a boundary value problem BVP using the FEM. After this second step, we have concrete formulae a large but linear problem whose solution approximately solve the original BVP.
This finite-dimensional problem is then implemented on a computer. The first step is to convert P1 and P2 into their equivalent weak formulations. Existence and uniqueness of the solution can also be shown. P1 and P2 are ready to be discretized which leads a common sub-problem 3.
The basic idea is to replace the infinite-dimensional linear problem:. 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. This parameter will be related to the size of the largest or average triangle in the triangulation. Since we do not perform such an analysis, we will not Finite elemente simulation dating this notation.
Depending on the author, the word "element" in "finite element method" refers either
Finite elemente simulation dating the triangles in the domain, the piecewise linear basis function, or both. So for Finite elemente simulation dating, 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 Finite elemente simulation dating multidimensional spacesbut can be defined on quadrilateral 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. 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:.
Such matrices are known
Finite elemente simulation dating sparse matricesand there are efficient solvers for such problems much more efficient than actually inverting the matrix. 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.
A separate consideration is the smoothness of the basis functions. For second order elliptic boundary value problemspiecewise polynomial basis function that are merely continuous suffice i.
For higher order partial differential equations, one must use smoother basis functions. The example above is such a method.
If this condition is not satisfied, we obtain a nonconforming element methodan example of which is the space of piecewise linear functions over the mesh which are continuous at each edge midpoint.
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. If instead of making h smaller, one increases the degree of the polynomials used in the Finite elemente simulation dating 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 to be confused with spectral methods. 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 Finite elemente simulation dating ensure good local approximation.
The effectiveness of GFEM has been shown
Finite elemente simulation dating applied to problems with domains having complicated boundaries, problems with micro-scales, and problems with boundary layers. 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.
The hp-FEM combines adaptively, elements with variable size h and polynomial degree p in order to achieve exceptionally fast, exponential convergence Finite elemente simulation dating. The hpk-FEM combines adaptively, elements with variable size hpolynomial degree of the local approximations p and global differentiability of the local approximations k-1 in order to achieve best convergence rates.
It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions.