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# 数学代写|有限元方法代写finite differences method代考|MEE356 Gauss Integration

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## 数学代写|有限元代写Finite Element Method代考|Gauss Integration

Consider first a one-dimensional integral. Using the Gauss integration scheme, the integral is evaluated simply by a summation of the integrand evaluated at $m$ Gauss points multiplied by corresponding weight coefficients as follows:
$$I=\int_{-1}^{+1} f(\xi) \mathrm{d} \xi=\sum_{j=1}^m w_j f\left(\xi_j\right)$$
The locations of the Gauss points and the weight coefficients have been found for different $m$, and are given in Table 7.1. In general, the use of more Gauss points will produce more accurate results for the integration. However, excessive use of Gauss points will increase the computational time and use up more computational resources, and it may not necessarily give better results. The appropriate number of Gauss points to be used depends upon the complexity of the integrand. It has been proven that the use of $m$ Gauss points gives the exact results of a polynomial integrand of up to an order of $n=2 m-1$. For example, if the integrand is a linear function (straight line), we have $2 m-1=1$, which gives $m=1$. This means that for a linear integrand, one Gauss point will be sufficient to give the exact result of the integration. If the integrand is of a polynomial of a third order, we have $2 m-1=3$, which gives $m=2$. This means that for an integrand of a third order polynomial, the use of two Gauss points will be sufficient to give the exact result. The use of more than two points will still give the same results, but takes more computation time. For two-dimensional integrations, the Gauss integration is sampled in two directions, as follows:
$$I=\int_{-1}^{+1} \int_{-1}^{+1} f(\xi, \eta) \mathrm{d} \xi \mathrm{d} \eta=\sum_{i=1}^{n_x} \sum_{j=1}^{n_y} w_i w_j f\left(\xi_i, \eta_j\right)$$
Figure 7.9(b) shows the locations of four Gauss points used for integration in a square region.

## 数学代写|有限元代写Finite Element Method代考|Coordinate Mapping

Figure 7.10 shows a 2D domain with the shape of an airplane wing. As you can imagine, dividing such a domain into rectangular elements of parallel edges is impossible. The job can be easily accomplished by the use of quadrilateral elements with four straight but unparallel edges, as shown in Figure 7.10. In developing the quadrilateral elements, we use the same coordinate mapping that was used for the rectangular elements in the previous section. Due to the slightly increased complexity of the element shape, the mapping will become a little more involved, but the procedure is basically the same.

Consider now a quadrilateral element with four nodes numbered 1, 2, 3 and 4 in a counter-clockwise direction, as shown in Figure 7.11. The coordinates for the four nodes are indicated in Figure 7.11(a) in the physical coordinate system. The physical coordinate system can be the same as the global coordinate system for the entire structure. As there are two DOFs at a node, a linear quadrilateral element has a total of eight DOFs, like the rectangular element. A local natural coordinate system $(\xi, \eta)$ with its origin at the centre of the squared element mapped from the global coordinate system is used to construct the shape functions, and the displacement is interpolated using the equation
$$\mathbf{U}^h(\xi, \eta)=\mathbf{N}(\xi, \eta) \mathbf{d}_e$$

## 数学代写|有限元代写Finite Element Method代考|Gauss Integration

$$I=\int_{-1}^{+1} f(\xi) \mathrm{d} \xi=\sum_{j=1}^m w_j f\left(\xi_j\right)$$

，这使 $m=2$. 这意味着对于三阶多项式的被积函数，使用两个高斯点就足以给出准确的结果。使用两个以上 的点仍会给出相同的结果，但需要更多的计算时间。对于二维积分，高斯积分在两个方向上采样，如下:
$$I=\int_{-1}^{+1} \int_{-1}^{+1} f(\xi, \eta) \mathrm{d} \xi \mathrm{d} \eta=\sum_{i=1}^{n_x} \sum_{j=1}^{n_y} w_i w_j f\left(\xi_i, \eta_j\right)$$

## 数学代写|有限元代写Finite Element Method代考|Coordinate Mapping

$$\mathbf{U}^h(\xi, \eta)=\mathbf{N}(\xi, \eta) \mathbf{d}_e$$

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