## 物理代写|空气动力学代写Aerodynamics代考|Isoparametric Bilinear and Trilinear Elements

The simplest finite element method for a mesh with quadrilateral cells uses trial solutions with unique nodal values and a bilinear form inside each cell,
$$u_h=a_0+a_1 x+a_2 y+a_3 x y,$$
where the local coefficients are chosen so that $u_h$ matches the nodal values. The resulting trial solutions are continuous across the edges of a Cartesian mesh because either $x$ or $y$ is constant along every edge. Consequently, the trial solutions in the cells

on either side of an edge vary linearly along the edge and must coincide, since they have the same value at each end of the edge. Unfortunately, this is not true for nonrectangular elements, as can be seen from the counterexample shown in Figure $3.12$ of two cells separated by a diagonal edge.

Requiring the use of Cartesian meshes would be far too restrictive. In fact, it would obviate the apparent advantage of finite element over finite difference methods in the treatment of complex geometrical shapes. The need to avoid such a restriction leads us to the introduction of isoparametric bilinear and trilinear elements. These use local bilinear and trilinear formulas of the same type as the formulas used to represent the trial solution to map each element to a square or a cube. This is the reason for calling these elements isoparametric. Now the solution is implicitly defined by the equations for the solution and the local position within each element.

The two-dimensional isoparametric bilinear element uses a local bilinear mapping of the physical domain and a square reference element $-\frac{1}{2} \leq \xi \leq \frac{1}{2},-\frac{1}{2} \leq \eta \leq \frac{1}{2}$, as illustrated in Figure 3.13.

## 物理代写|空气动力学代写Aerodynamics代考|Finite Volume Methods

Finite volume methods directly approximate the integral form of the conservation laws – in the case of fluid dynamics, conservation of mass, momentum, and energy. The domain is divided into a large number of small discrete control volumes, and the conservation laws are approximated for each control volume. This is a conceptually simple and intuitively appealing way to discretize any time-dependent system of conservation laws. In the aerospace community it seems to have been first adopted by MacCormack, although he did not immediately publish his ideas (MacCormack \& Paullay 1972). Finite volume methods were also independently developed for the simulation of complex turbulent flows in a varicty of industrial applications. These developments have been described, for example, by Patankar and Spalding (1972).
In order to illustrate the formulation of a finite volume method, consider the twodimensional time-dependent conservation law
$$\frac{\partial \boldsymbol{w}}{\partial t}+\frac{\partial}{\partial x} f(w)+\frac{\partial}{\partial y} g(w)=0 .$$
This could, for example, be the Euler equations for inviscid compressible flow. This equation can be expressed in integral form for a domain $\mathcal{D}$ and boundary $\mathcal{B}$ as
$$\frac{d}{d t} \int_{\mathcal{D}} w d \mathcal{S}+\oint_{\mathcal{B}}(f d y-g d x)=0$$

where S is the area. Suppose now that the domain is subdivided into small quadrilateral control volumes as illustrated in Figure 3.17. Assuming for the moment a regular
pattern, we can label the cells with double subscripts i,j and obtain a semi-discrete
scheme by applying (3.56) to each control volume.

## 物理代写|空气动力学代写空气动力学代考|等参双线性和三线性元素

$$u_h=a_0+a_1 x+a_2 y+a_3 x y,$$
，其中局部系数的选择使$u_h$与节点值匹配。得到的试验解在笛卡尔网格的边缘上是连续的，因为$x$或$y$在每条边缘上都是常数。因此，细胞中的试验溶液

## 物理代写|空气动力学代写空气动力学代考|有限体积方法

. 有限体积法直接近似于守恒定律的积分形式——在流体动力学的情况下，质量、动量和能量守恒。将该区域划分为大量离散的小控制体，并对每个控制体的守恒律进行近似计算。这是一种概念上简单，直观上吸引人的方法，可以离散任何与时间有关的守恒定律系统。在航空航天界，它似乎是由麦考马克首先采用的，尽管他没有立即发表他的想法(麦考马克＆Paullay 1972)。有限体积方法也被独立开发用于模拟各种工业应用中的复杂湍流流动。例如，帕坦卡尔和斯伯丁(1972)对这些发展进行了描述。为了说明有限体积法的公式，考虑二维时相关守恒律$$\frac{\partial \boldsymbol{w}}{\partial t}+\frac{\partial}{\partial x} f(w)+\frac{\partial}{\partial y} g(w)=0 .$$

$$\frac{d}{d t} \int_{\mathcal{D}} w d \mathcal{S}+\oint_{\mathcal{B}}(f d y-g d x)=0$$

，其中S是面积。假设现在域被细分为小的四次立方控制卷，如图3.17所示。假设现在是一个规则的

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