Motivation for the Finite Element Method
The approach to the finite element method can be derived from different motivations. Essentially there are three ways: a rather descriptive way, which has its roots in the engineering working method, a physical or mathematically motivated approach. Depend
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Motivation for the Finite Element Method
Abstract The approach to the finite element method can be derived from different motivations. Essentially there are three ways: • a rather descriptive way, which has its roots in the engineering working method, • a physical or • mathematically motivated approach. Depending on the perspective, different formulations result, which however all result in a common principal equation of the finite element method. The different formulations will be elaborated in detail based on the following descriptions: • matrix methods, • physically based working and energy methods and • weighted residual method. The finite element method is used to solve different physical problems. Here solely finite element formulations related to structural mechanics are considered [1, 5–7, 9–12].
2.1 From the Engineering Perspective Derived Methods Matrix methods can be regarded in elastostatics as the initial point for the application of the finite element method to analyze complex structures [2, 3]. As example a plane structure can be given (see Fig. 2.1). This example is adapted from [8]. The structure consists of various substructures I, II, III and IV. The substructures are referred to as elements. The elements are coupled at the nodes 2, 3, 4 and 5. The entire structure is supported on nodes 1 and 6, an external load affects node 4. Unknown are • the displacement and reaction forces on every single inner node and A. Öchsner and M. Merkel, One-Dimensional Finite Elements, DOI: 10.1007/978-3-642-31797-2_2, © Springer-Verlag Berlin Heidelberg 2013
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2 Motivation for the Finite Element Method
Fig. 2.1 Plane structure, adapted from [8]
• the support reactions in consequence of the acting load. To solve the problem, matrix methods can be used. In the matrix methods one distinguishes between the force method (static methods), which is based on a direct determination of the statically indeterminate forces, and the displacement method (kinematic method), which considers the displacements as unknown parameters. Both methods allow the determination of the unknown parameters. The decisive advantage of the displacement method is that during the application it is not necessary to distinguish between statically determined and statically undeterminate structures. Due to the generality this method is applied in the following.
2.1.1 The Matrix Stiffness Method It is the primary subgoal to establish the stiffness relation for the entire structure from Fig. 2.1. The following stiffness relation serves as the basis for the matrix displacement method: F = K u. (2.1) F and u are column matrices, K is a square matrix. F summarizes all nodal forces and u summarizes all nodal displacements. The matrix K represents the stiffness matrix of the entire structure. One single element is identified as the basic unit for the problem and is characterized by the fact that it is coupled with other elements via nodes. Displacements and forces are introduced at every single node.
2.1 From the Engineering Perspective Derived Me
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