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# 数学代写|有限元方法作业代写finite differences method代考|The finite element method for elliptic problems

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## 数学代写|有限元方法作业代写finite differences method代考|Difficulties associated with the application of weighted residual methods

Although the weighted residual methods introduced in Chapter 2 have been used with success in many areas of physics and engineering, there are certain difficulties which prevent them being more widely used for the solution of practical problems.

One obvious problem involves the choice of trial functions. It is clear that for an irregular-shaped boundary, such as in Fig. 3.1, it would in general be impossible to find one function, let alone a sequence of functions, which satisfies every essential boundary condition. Thus, immediately, the class of problems amenable to solution by this method is restricted to those problems with a ‘simple’ geometry.

Even if the geometry is suitable and a sequence of functions satisying essential boundary conditions is available, these functions are usually polynomials. It is not difficult to appreciate that, in general, very high-order polynomials would be required to approach the exact behaviour of the unknown over the whole region. A worse situation than this, however, concerns the case of discontinuous material properties.

## 数学代写|有限元方法作业代写finite differences method代考|Piecewise application of the Galerkin method

We consider an approach in which the region of interest is subdivided into a finite set of elements, connected together at a set of points called the nodes. In each of these elements, the function behaviour is considered individually and then an overall set of equations is assembled from the individual components. These individual components are found by a piecewise application of the Galerkin method.

The distinction between element numbering and nodal numbering can sometimes lead to confusion, and there is no generally accepted notation. We shall adopt the following: subscripts will refer to nodal numbers and superscripts to element numbers; where it is important to distinguish between them, we shall write $i$ for ‘node $i$ ‘ and $[i]$ for ‘element $i$ ‘.

## 数学代写|有限元方法作业代写FINITE DIFFERENCES METHOD代考|Terminology

The solution of boundary-value problems such as that given by eqns (2.1) and (2.39) frequently represents a quantity associated with a scalar field such as a potential. Consequently, we often refer to such problems as field problems.

Because the finite element method was developed in its computational form by structural engineers (Argyris 1964, Zienkiewicz and Cheung 1965), the structural terminology has remained in the generalization to field problems. In Section $3.6$ we shall develop equations of the form
$$\mathbf{K U}=\mathbf{F},$$
where $\mathrm{U}$ is a vector of nodal variables, i.e. values of $u, \partial u / \partial x, \partial u / \partial y$ etc., evaluated at the nodes. The number of nodal variables associated with a particular node is often called the number of degrees of freedom at that node. $\mathbf{u}^{e}$ is the vector of element nodal variables. $\mathbf{K}$ and $\mathbf{F}$ are called the overall stiffness matrix and the overall force vector, respectively, and are assembled from element matrices $\mathbf{k}^{e}$ and $\mathbf{f}^{e}$.

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