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# 物理代写|空气动力学代写Aerodynamics代考|ME615 Multi-dimensional Finite Element Schemes with Linear Elements

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## 物理代写|空气动力学代写Aerodynamics代考|Multi-dimensional Finite Element Schemes with Linear Elements

Finite element methods are very easy to define for triangular and tetrahedral meshes using piecewise linear trial solutions. Then we introduce basis functions $\phi_j$ that have the value unity at the $j$ th node and zero at all other nodes, as illustrated in Figure $3.8$ for a triangular mesh. We can visualize the basis function of each node as a tent surrounding the node. Consider now Laplace’s equation in a triangulated domain $\mathcal{D}$ with Dirichlet boundary conditions on the boundary $\mathcal{B}$,
\begin{aligned} &u_{x x}+u_{y y}=0 \text { in } \mathcal{D} \ &u \text { specified on } \mathcal{B} . \end{aligned}

The corresponding weak form is obtained by multiplying $(3.45)$ by a test function $\psi(x, y)$ and integrating by parts to obtain
$$\int_{\mathcal{B}} \psi \nabla u_h \cdot \mathbf{n} d l-\int_{\mathcal{D}} \nabla u_h \cdot \nabla \psi d \mathcal{S}=0,$$
where $\mathbf{n}$ is the unit normal to the boundary. The trial solution is
$$u_h=\sum_{j=1}^n u_j \phi_j(x, y)$$

## 物理代写|空气动力学代写Aerodynamics代考|Further Analysis of the Discrete Laplacian

If we consider the stencil illustrated in Figure $3.8$, evaluation of formulas (3.47), (3.48), and (3.49) reduces the equation for node 0 to the form
$$r_0=\sum s_{k 0}\left(u_k-u_0\right),$$
where $s_{k 0}$ are the entries of the stiffness matrix between node $k$ and 0 . If we consider edge 20 , say, the total contribution of $u_2$, applying the formulas (3.47) and (3.48) to the triangles 012 and 023 with areas $\mathcal{S}1$ and $\mathcal{S}_2$ respectively, is \begin{aligned} s{20}=& \frac{1}{4 \mathcal{S}1}\left(\left(y_0-y_1\right)\left(y_2-y_1\right)+\left(x_0-x_1\right)\left(x_2-x_1\right)\right) \ &+\frac{1}{4 \mathcal{S}_2}\left(\left(y_0-y_3\right)\left(y_2-y_3\right)+\left(x_0-x_3\right)\left(x_2-x_3\right)\right) . \end{aligned} Let $l{k 0}$ be the length of the edge $k 0$. Then, we find that
\begin{aligned} s_{20} &=\frac{l_{01} l_{21} \cos \theta_{012}}{2 l_{01} l_{21} \sin \theta_{012}}+\frac{l_{03} l_{23} \cos \theta_{032}}{2 l_{03} l_{23} \sin \theta_{032}} \ &=\frac{1}{2}\left(\cot \theta_{012}+\cot \theta_{032}\right), \end{aligned}
where $\theta_{012}$ and $\theta_{032}$ are the angles between edges $l_{01}$ and $l_{21}$ and $l_{03}$ and $l_{23}$, as illustrated in Figure 3.9.
In the case of the Poisson equation
$$u_{x x}+u_{y y}=f,$$

we must also evaluate
$$\int f \phi_0 d \mathcal{S}$$
assuming $f$ is represented by a piecewise linear approximation. Carrying out the integrations, we find that the equation for node 0 is
$$\sum_{k=1}^n s_{k 0}\left(u_k-u_0\right)=M_{00} f_0+M_{0 k} f_k,$$
where $n$ is the number of neighbors, and the coefficients
$$M_{00}=\frac{1}{6} \sum_{k=1}^n s_k, \quad M_{0 k}=\frac{s_k+s_{k+1}}{12}$$
are entries of the stiffness matrix.

## 物理代写|空气动力学代写空气动力学代考|具有线性元素的多维有限元格式

\begin{aligned} &u_{x x}+u_{y y}=0 \text { in } \mathcal{D} \ &u \text { specified on } \mathcal{B} . \end{aligned}

$$\int_{\mathcal{B}} \psi \nabla u_h \cdot \mathbf{n} d l-\int_{\mathcal{D}} \nabla u_h \cdot \nabla \psi d \mathcal{S}=0,$$
，其中$\mathbf{n}$是边界的法线单位。试解是
$$u_h=\sum_{j=1}^n u_j \phi_j(x, y)$$

## 物理代写|空气动力学代写空气动力学代考|离散拉普拉斯算子的进一步分析

$$r_0=\sum s_{k 0}\left(u_k-u_0\right),$$
，其中$s_{k 0}$是节点$k$和0之间的刚度矩阵项。如果我们考虑边20，假设$u_2$的总贡献，分别对面积为$\mathcal{S}1$和$\mathcal{S}2$的三角形012和023应用公式(3.47)和(3.48)，为\begin{aligned} s{20}=& \frac{1}{4 \mathcal{S}1}\left(\left(y_0-y_1\right)\left(y_2-y_1\right)+\left(x_0-x_1\right)\left(x_2-x_1\right)\right) \ &+\frac{1}{4 \mathcal{S}_2}\left(\left(y_0-y_3\right)\left(y_2-y_3\right)+\left(x_0-x_3\right)\left(x_2-x_3\right)\right) . \end{aligned}设$l{k 0}$为边$k 0$的长度。然后，我们发现
\begin{aligned} s{20} &=\frac{l_{01} l_{21} \cos \theta_{012}}{2 l_{01} l_{21} \sin \theta_{012}}+\frac{l_{03} l_{23} \cos \theta_{032}}{2 l_{03} l_{23} \sin \theta_{032}} \ &=\frac{1}{2}\left(\cot \theta_{012}+\cot \theta_{032}\right), \end{aligned}
，其中$\theta_{012}$和$\theta_{032}$是边$l_{01}$和$l_{21}$以及$l_{03}$和$l_{23}$之间的角度，如图3.9所示。在泊松方程
$$u_{x x}+u_{y y}=f,$$ 的情况下

$$\int f \phi_0 d \mathcal{S}$$
，假设$f$由分段线性逼近表示。进行积分，我们发现节点0的方程是
$$\sum_{k=1}^n s_{k 0}\left(u_k-u_0\right)=M_{00} f_0+M_{0 k} f_k,$$
，其中$n$是邻居数，系数
$$M_{00}=\frac{1}{6} \sum_{k=1}^n s_k, \quad M_{0 k}=\frac{s_k+s_{k+1}}{12}$$

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