Mathematical Physics - Volume II - Numerical Methods

3.2 Finite element method – two-dimensional problem

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3.2 Finite element method – two-dimensional problem

3.2.1 Introduction

As was shown in the previous chapter, basic steps in finite method application are: 1. Variation formulation of the problem, with identifying of the adequate allowable function space H . 2. Defining of a finite element mesh and polynomials in regions which establish a finite dimen sional space within H . 3. Defining an approximation of the variation problem of boundary values in the subspace of finite elements H h within H , which involves calculating of element matrices and defining of a system of linear algebraic equations in terms of unknown node values of the approximate solution. 4. Solving of the linear algebraic equation system, wherein zero terms and symmetry are of great use. 5. Evaluating the properties of the solution, and if possible, determining its error. These steps essentially apply to problems in two or three dimensions. Considering the significant application to such problems, this chapter will present all of the important finite element equations for two-dimensional boundary problems, i.e. differential equations which need to be satisfied on a two-dimensional domain Ω , whose boundary is generally curvilinear. Instead of a line element, finite elements now have simple two-dimensional forms – triangular or rectangular, and the finite element mesh generally approximates the problem domain. Natural ability of these finite elements to represent a domain of any shape without additional efforts is a key advantage of practical application of finite element method. In this chapter, we will analyse problems in which the unknown scalar function u is a function of position, e.g. heat conduction problems ( u in this case is the temperature), flow through a porous medium ( u is the pressure gradient), or transverse deflection of an elastic membrane ( u is the deflection). 3.2.2 Physical base of the problem Let the problem domain ¯ Ω consist of its interior Ω and boundary ∂ Ω . We assume that this domain is finite and has a sufficiently smooth boundary (unit boundary normal n is a continuous function of position along the boundary, with the exception of corner points). Generally, the boundary can be defined using parametric equations x = x ( s ) and y = y ( s ) , where s is the arc length at ∂ Ω , measured from a referent point. Values of an arbitrary function g on the boundary are denoted as g ( s ) ≡ g ( x ( s ) , y ( s )) , s ∈ ∂ Ω . For state variable u ( x , y ) , the basic requirement is that it is a smooth function in Ω , in accordance with a specific problem and functions x ( s ) and y ( s ) which define ∂ Ω , i.e. it should be as smooth as necessary. The physical base of the problem is based on the rate of change of the scalar filed u in terms of its position within the Ω , which is defined by the vector function ∇ u , which is referred to as a gradient: ∇ u ( x , y ) = ∂ u ( x , y ) ∂ x i + ∂ u ( x , y ) ∂ y j , (3.36) where i , j are the unit vectors for axes x and y , respectively. The gradient defines the total scalar field rate of change at point ( x , y ) in any given direction. If t is the unit vector at angle θ relative to the x axis, and t = cos θ i + sin θ j , then the rate of change for a scalar field at ( x , y ) in direction t can be defined as: