Finite Element Analysis

Abstract

Finite Element Analysis (FEA) is based on the Finite Element Method (FEM). The FEM is a mathemati-cal method which transforms an analytically difficult to solve or unsolvable problem described by a var-iational formulation or by a system of differential equations into an algebraic problem. The under con-sideration overall system structure is replaced by a calculation model that divides the structure into a number of small subdivisions (finite elements). If the mechanical problem is described by a differential equation, the equation must be transformed into a variational formulation. The unknown and the variational function are then approximated by a simple interpolation polynomial. By determining that the coefficients of the variational function can take every possible value, an algebraic system of equations is obtained, with which the coefficients of the interpolation function for the unknown can be determined. Constructing the system of equations is facilitated by the feature that the global system of equations can be assembled by matrices, created on the element level (Fig. 1). To calculate node displacements, the global stiff-ness matrix K is determined by a simple sum of all the individual element stiffness matrices. The dimension of the global stiffness matrix is deter-mined by considering the sum of the degrees of freedom of all nodes. The relationship between the acting forces E and the nodal displacements U forms the basis of the FEM for calculating mechan-ical problems: F=K U (1) The stiffness matrix K consists of the coefficients of the equation system. It is calculated from the material and geometry of the structure data (Zienkiewicz et al. 2005). If the external loads acting on an element are known, the nodal displace-ments U can be calculated by prior solution of Eq. 1. U=K-1 F (2) The solution process of the differential equa-tions is carried out in a solver. Commercial FEM software basically consists of three components (Fig. 2) (Doege and Behrens 2010): Preprocessor (software for the creation of the FEM model) Solver (assembling and solving the global sys-tem of equations) Postprocessor (software for visualization of calculation results)