Basic Idea of Finite Element Method

Basic Idea of Finite Element Method

Copyright: © 2024 |Pages: 38
DOI: 10.4018/979-8-3693-0932-2.ch002
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Abstract

The finite element method (FEM) is covered in this chapter. For the numerical solution of different physics and engineering problems, including heat transport issues, the finite element method is a particularly potent tool. An analytical method-like mathematical function cannot be obtained using FEM. This chapter provides a thorough study of several FEM-related topics, including relationships and governing equations, discretization of equations, two- and three-dimensional isoparametric elements, mesh generation, etc.
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1. Steps Of Finite Element Analysis

Using the finite element method in a problem requires doing a series of specific steps accurately. In general, the main steps of solving by the finite element method are as follows:

Step 1. Discretization of the Desired Problem and Selection of Element Type

The first step in the FEM is to divide the desired solution area into finite elements. In this step, we have to divide the object into smaller components by choosing the appropriate element with the specified number of nodes. The selection of the element should be such that it can show the real physical behavior of the problem. Choosing the type of element, the number of its nodes, the number of degrees of freedom of each node, and the number of elements considered for the whole body are among the first steps of a finite element analysis. In a finite element analysis, the most time is spent on elementing and meshing because the higher the quality of the mesh, the higher the accuracy of the obtained results.

Step 2. Selection of Interpolation Functions

The quantity that we want to be calculated and specified after the FEM is applied to the problem is selected in this section. For example, in problems related to fluids, the variable field is velocity, and in problems related to mechanics of solids, the variable field is displacement. To interpolate field variables on the element, interpolation functions are used. It is worth noting that the polynomial degree depends on the number of nodes assigned to the element. Other quantities that we can calculate in the problem are related to the field variable using relationships, and its values are calculated in the nodes and then in the whole object.

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