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数学代写|有限元方法作业代写finite differences method代考|The boundary element method

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数学代写|有限元方法作业代写finite differences method代考|Integral formulation of boundary-value problems

In principle, the boundary element method is just another aspect of the finite element method. However, there is sufficient difference to warrant the new name, which was first coined by Brebbia and Dominguez (1977).

The underlying idea is that a boundary-value problem such as that in Section $3.4$, involving a partial differential equation of the form
$$\mathcal{L} u=f \quad \text { in } D$$
subject to the boundary condition
$$\mathcal{B} u=g \quad \text { on } C,$$
can be transformed to an integral equation
$$\iint_{D} v \mathcal{L} u d A=\iint_{D} v f d A$$
using any weighting function $v$. However, there are circumstance in which the integral $\iint_{v D} \mathcal{L} u d A$ may be reduced to an integral over the boundary $C$ by use of a reciprocal theorem, for example Green’s theorem for potential-type problems (Green 1828) or Somigliana’s 1886 identity (cited by Becker 1992) for elasticity problems.

数学代写|有限元方法作业代写finite differences method代考|Boundary element idealization for Laplace’s equation

We proceed in a manner analogous to that for the finite element method in Chapter 3 . We shall approximate the boundary $C$ by a polygon, $C_{n}$, and it is the polygon edges which are the boundary elements. We also choose a set of nodes, at which we seek approximations $U$ and $Q$ to $u$ and $q$, respectively. We shall consider in the first instance the so-called constant element. In such an element, the geometry is a straight line segment with just one node at the centre; see Fig. 7.2.

N.B. (i) In this element, the approximation of the function is of a lower order than that for the geometry, and the element is known as a superparametric element. Similarly, we can define subparametric elements (cf. the isoparametric elements of Chapter 4).
(ii) There is no requirement for interelement continuity in the boundary element, non-conforming elements are frequently used.

数学代写|有限元方法作业代写FINITE DIFFERENCES METHOD代考|A constant boundary element for Laplace’s equation

The element is shown in Fig. 7.3, and in Fig. $7.4$ we define some of the geometry of element $[j]$, whose length is $l_{j}$.

We calculate the coefficients $H_{i j}$ and $G_{i j}$. Suppose that the base node $i$ is not in the target element $[j]$. Then
\begin{aligned} \frac{\partial}{\partial n}\left(\ln R_{i j}\right) &=(\operatorname{grad} R \cdot \mathbf{n}){i j} \ &=\frac{1}{R{i j}} \mathbf{R}{i j} \cdot \mathbf{n}{j} \ &=\frac{\cos \theta_{i j}}{R_{i j}} \ &=\frac{d_{i j}}{R_{i j}^{2}} \end{aligned}
so that, using Gauss quadrature, we obtain
\begin{aligned} H_{i j} &=\int_{[j]} \frac{d_{i j}}{R_{i j}^{2}} d s \ &=\frac{l_{j} d_{i j}}{2} \int_{-1}^{1} \frac{1}{R_{i j}^{2}(\xi)} d \xi \ & \approx \frac{l_{j} d_{i j}}{2} \sum_{g=1}^{G} \frac{1}{R_{i j}^{2}\left(\xi_{g}\right)} w_{g} \end{aligned}

数学代写|有限元方法作业代写FINITE DIFFERENCES METHOD代考|INTEGRAL FORMULATION OF BOUNDARY-VALUE PROBLEMS

∬D在大号在d一种=∬D在Fd一种

数学代写|有限元方法作业代写FINITE DIFFERENCES METHOD代考|A CONSTANT BOUNDARY ELEMENT FOR LAPLACE’S EQUATION

\begin{aligned} \frac{\partial}{\partial n}\left(\ln R_{i j}\right) &=(\operatorname{grad} R \cdot \mathbf{n}){i j} \ &=\frac{1}{R{i j}} \mathbf{R}{i j} \cdot \mathbf{n}{j} \ &=\frac{\cos \theta_{i j}}{R_{i j}} \ &=\frac{d_{i j}}{R_{i j}^{2}} \end{aligned}
so that, using Gauss quadrature, we obtain
\begin{aligned} H_{i j} &=\int_{[j]} \frac{d_{i j}}{R_{i j}^{2}} d s \ &=\frac{l_{j} d_{i j}}{2} \int_{-1}^{1} \frac{1}{R_{i j}^{2}(\xi)} d \xi \ & \approx \frac{l_{j} d_{i j}}{2} \sum_{g=1}^{G} \frac{1}{R_{i j}^{2}\left(\xi_{g}\right)} w_{g} \end{aligned}