Finite Element Method Tutorial (with CD-ROM)

Author: Wang Huanding, Wang Wei / Country: Mainland China
Publisher:
Publish Date: 2005-05-01
Features: The Finite Element Method (FEM) is a computational method that emerged with the widespread application of electronic computers. It is a numerical method for approximating the solutions of general continuum problems. From a physical perspective: it replaces the continuum to be analyzed with a composite of elements connected only at their nodes, essentially dividing the continuum into several interconnected elements. By analyzing the properties of these elements, the properties of the entire continuum are determined. From a mathematical perspective: it transforms a continuous, infinite-degree-of-freedom problem into a discrete, finite-degree-of-freedom problem, significantly simplifying the problem or making it solvable. Once the unknowns of the elements are solved, interpolation functions can be used to determine the field functions on the continuum. Clearly, as the number of elements increases, i.e., as the element size decreases, the accuracy of the approximation will continue to improve. If the elements satisfy the convergence requirements, the approximate solution will converge to the exact solution. The Finite Element Method relies on two important tools: matrix methods are used in theoretical derivations, and computer technology is used in practical calculations. This chapter introduces the necessary prerequisite knowledge for learning the Finite Element Method. The matrix representation of the elastic theory equations and the principle of virtual displacement and the principle of potential energy introduced below are important theoretical foundations for establishing finite element equations.
Once we obtain the node displacements of the element, similar to truss structures, we hope to use the node displacements to determine the displacement at any point within the element. When performing element analysis to determine the relationship between the node forces and node displacements of the element, we also need to express the displacement at any point within the element in terms of the node displacements. Generally, the actual displacement of any point within the element is a very complex function of the coordinates and cannot be accurately represented using only the six node displacements of the element. As mentioned in the previous chapter, the displacement within the element represented by node displacements is generally only a part of the actual displacement or an approximation of it. Practice has shown that when reasonably selecting a displacement form that can be determined from node displacements to approximate the actual displacement of the element, the results will converge to the actual displacement as the element size decreases.
We refer to the displacement form used to approximate the actual displacement of the element in finite element analysis as the displacement function. It is clear that the selection of the displacement function directly affects the convergence of the results and is a critical step. Generally, polynomials are chosen as displacement functions based on the concept of Taylor series expansion. This not only simplifies calculations but also allows the accuracy of the results to be directly controlled by the number of terms. Following this approach, for the current element, since there are only six node displacements, the following equation can be chosen as the displacement function of the element. (p1)