tailieunhanh - Ebook A first course in finite element analysis: Part 2

(BQ) Part 2 book "A first course in finite element analysis" has contents: Approximations of trial solutions, weight functions and gauss quadrature for multidimensional problems; finite element formulation for multidimensional scalar field problems, finite element formulation for vector field problems – Linear elasticity; finite element formulation for other contents. | 7 Approximations of Trial Solutions, Weight Functions and Gauss Quadrature for Multidimensional Problems In this chapter, we describe the construction of the weight functions and trial solutions for two-dimensional applications; we will sometimes collectively call these approximations or just functions. In finite element methods, these approximations are constructed from shape functions. As in Chapter 4, where weight functions and trial solutions were constructed for one-dimensional problems, the basic idea is to construct C0 interpolants that are complete. Following the nomenclature introduced in Chapter 4, we will denote the approximation by ðx; yÞ. It represents any scalar function such as temperature or material concentration. We have already noted that the situation in multidimensions is altogether different from that in onedimensional problems, as the exact solution of the partial differential equations in multidimensions is feasible for problems only on simple domains with simple boundary conditions. Thus, numerical solution of the partial differential equations is generally the only possibility for practical problems. The approach of finite element methods remains the same: approximate the weight functions and trial solutions by finite element shape functions so that as the number of elements is increased, the quality of the solution is improved. In the limit as h ! 0 (h being the element size) or as the polynomial order is increased, the finite element solution should converge to the exact solution if the approximations are sufficiently smooth and complete. It is in two-dimensional problems that the power of the finite element method becomes clearly apparent. We will see that the finite element method provides a method for easily constructing approximations to solutions for bodies of arbitrary shape. Furthermore, as will become apparent when we examine the MATLAB programs, finite element methods possess a modularity that enables simple programs to treat a large .