Introduction to Finite Element Method

Currently, the finite element method (FEM) is one of the main engineering analysis tools. Software packages that implement this method used for calculations of mechanical engineering products and building structures, studies of various solids, liquid and gaseous media, as well as their interactions. The advantages of FEM, in addition to the practically unlimited applicability are: validity, reliability and usability in computer modeling. Initially, the appearance of FEM was associated with structural mechanics. However, it was soon discovered that with this The method can also solve other classes of problems. Further rapid development FEM was largely determined by the progress of digital computing technology, since with the increase in computer productivity, effective solving increasingly complex engineering problems. Given the current wide prevalence of FEM, it can be expected that in the future the scope of its application will only expand.

History of the method

The prototype of FEM appeared in connection with the need to solve complex problems theories of elasticity and structural mechanics arising during design aviation structures. This appearance can be established, in particular, based on the works of Alexander Khrennikov (1941) and Richard Courant (1942).Despite Although the approaches they used were different, they had important similarities: both were based on discretizing a continuous region into a grid consisting from a set of individual sub-regions called elements. In Khrennikov's work the area was divided using a grid analogy, and Courant's approach was based on partitioning the region into finite triangular subregions in order to solve second order differential equation that described the problem torsion of a rod of cylindrical cross-section. Courant's contribution was evolutionary nature, his work was based on a large number of earlier results, achieved in solving partial differential equations Rayleigh, Ritz and Galerkin. In 1947, Olgerd Zenkevich, a graduate of the Imperial College of London, combined the approaches of Khrennikov and Courant into one method, later called the “finite element method” (finite element method), and gave it its original mathematical formalism.

The development of FEM continued in the second half of the 1950s, again in relation to calculations of aircraft structures and was associated with the name Ioannis Argyris, professor at the University of Stuttgart. From now on key concepts about the stiffness matrix and finite element ensemble exist almost unchanged.

In 1965, NASA issued a request for software development implementing FEM; as a result, NASTRAN was created. In 1973, in the publication Strang "Finite Element Analysis"for FEM was laid rigorous mathematical foundation; Since then, the method has gradually emerged into the branch of applied mathematics and began to be used for numerical modeling physical processes of various natures, including in the field of electromagnetism and gas dynamics.

Brief characteristics of FEM

FEM proved to be a more effective method almost from the very beginning compared to the finite difference method; he turned out and turns out to this day indispensable when it is necessary to take into account the geometric features of the studied areas and differences in physical properties in their different parts.

From a mathematical point of view, FEM is a generalization of the method Rayleigh-Ritz-Galerkin and therefore applicable to a wide class of equations in partial derivatives. In the Ritz method, however, it is not directly solved differential equation; instead the original problem is presented in an equivalent variational formulation, and then seeks an approximate solution in the form of a combination of trial (approximating the solution) functions of a given species identified throughout the study area. At the same time, the weighting coefficients are calculated from the variational principle corresponding to the problem.

In FEM, test functions are, as a rule, piecewise polynomial. Each function is zero over most of the region and is nonzero only within one element. In this small region the function is composed from polynomials of a certain (mostly low) degree, as a result which makes the calculations relatively simple.

Let the problem to be solved be posed in variational form: you need to find a function uthat minimizes a given potential energy functional. The need to minimize leads to differential equation foruwhich is usually cannot be solved exactly and approximate methods must be used. Idea of the method Rayleigh-Ritz-Galerkin consists in selecting a finite number of trial functions and among all their linear combinations one is sought that delivers minimum functionality. This combination is a Ritz approximation. Unknown weights are determined not from differential equations, but from a system of N discrete algebraic equations; to solve this system it is possible to use a computer. Minimization process automatically gives the combination closest to the ufunction. Theoretically the goal is to select trial features that are comfortable enough for calculations and minimization of potential energy and, at the same time, provide a good approximation to the desired solutionu.

Within the framework of FEM, everything starts with dividing the studied area into small subdomains are finite elements. Their geometric shape should be sufficient easy to set test functions and process using computational cars. In the case of a plane problem, these are usually triangles or rectangles. Then a test function is specified inside each element - usually a polynomial third or fourth degree.Boundary conditions are specified along the corresponding sides of triangles or rectangles, which is much easier than determining them immediately along the entire border of the region. The calculation accuracy increases with more finely dividing the area into elements. The computer is running according to the same program, only longer.

One of the tasks associated with the use of FEM is the assessment of accuracy or the error with which piecewise polynomial functions can approximate the desired solution. How accurate are the finite elements that are being built? based on ease of calculation, and are they capable of providing a good approximation? It is intuitively clear that for any smooth function one can construct the approximation using suitable piecewise functions. Mathematically the problem is in obtaining a reliable estimate of the approximation error and determining the speed this error decreases with increasing number of elements (partition density region) or degrees of polynomials defined within an element.

Thus, the main components of FEM (not always obvious to user of finite element software) are:

Variational formulation of the problem.
Construction of piecewise polynomial test functions.
Calculation of the stiffness matrix (or its analogue, corresponding class of the problem being solved) and solving a system of discrete algebraic equations.
Assessment of the accuracy of the Ritz approximation.