# 数学代写|有限元方法代写Finite Element Method代考|JEE350

## 数学代写|有限元方法代写Finite Element Method代考|Method of weighted residuals

We now turn our attention to finding approximate solutions to boundary value problems. In particular, the method of weighted residuals describes a group of solution methods to obtain approximate solutions to an integral equation expressed in the form of Eq. (3.50).

We will seek an approximate solution $\tilde{u}$ for the problem. We chose this approximate solution as a superposition of certain basis functions $\phi_j(x)$, as follows:
$$\tilde{u}(x)=\phi_0(x)+c_1 \phi_1(x)+c_2 \phi_2(x)+\ldots+c_N \phi_N(x)=\phi_0(x)+\sum_{j=1}^N c_j \phi_j(x)$$
where $c_j$ are unknown coefficients. Note that Eq. (3.88) represents a finite series with $N+1$ basis functions. In order to find the $N$ unknown coefficients Eq. (3.50) is generalized by using $N$ weight functions $w_i$,
$$\int_{\Omega} w_i R(\tilde{u}) d \Omega=0 \text { for } i=1, \ldots N$$

The choices for the basis and weight functions are not arbitrary. The following conditions should be followed, in choosing the basis and weight functions.
Basis functions: The basis functions $\phi_0, \phi_i$, should be chosen such that:
(i) The approximate solution $\tilde{u}$ satisfies the boundary conditions of the problem.
(ii) The approximate solution $\tilde{u}$ is as many times differentiable as required by the original differential equation.
(iii) The basis functions $\phi_0, \phi_j$ are linearly independent.
Weight functions: The choice of the weight functions, $w_i$, depends on the method used.

These methods are the Rayleigh-Ritz, Galerkin, Petrov-Galerkin, leastsquares, and collocation methods. The finite element method also uses a variational (weighted residual) approach. As we will see later, the choice of basis functions in the FEM are different from that of classical variational methods. In these notes we only introduce the Rayleigh-Ritz and the Galerkin methods. Detailed discussion of the other methods can be found for example in reference [1].

## 数学代写|有限元方法代写Finite Element Method代考|Rayleigh–Ritz method

The Rayleigh-Ritz method seeks an approximate solution of the form given in Eq. (3.88) to the weak form of the boundary value problem. For example, the weak form of Example-3.1 is approximated as follows:
$$w_i\left(x_L\right) Q_L-\int_{x_0}^{x_L}\left[a \frac{d \tilde{u}}{d x} \frac{d w_i}{d x}-c w_i \tilde{u}-w_i q\right] d x=0 \text { for } i=1 . . N$$
where $w_i(x)$ are the weight functions. An approximate solution of the form,
$$\tilde{u}(x)=\phi_0(x)+\sum_{j=1}^N C_j \phi_j(x)$$
is sought, where $C_j$ are the coefficients to be determined, $\phi_0(x)$ and $\phi_j(x)$ are the basis functions.

The weak form of the boundary value problem has certain advantages that are helpful in the selection of the basis functions. First, the natural boundary conditions of the problem are included in the weak form. This means that the natural boundary conditions can be approximately satisfied, if we happen to pick an approximate solution satisfying only the essential boundary conditions of the problem. Therefore, the condition (C1) can be relaxed. Second, the weak form has a reduced order of differentiation. Therefore, the continuity requirement stated in condition (C2) is reduced. With these caveats, the following conditions should be followed to pick the basis functions for the Rayleigh-Ritz method:
Basis functions: The basis functions $\phi_0, \phi_j$ must be chosen such that:
i. The approximate solution $\tilde{u}$ satisfies at least the essential boundary conditions of the problem
(C4)
ii. The approximate solution $\tilde{u}$ is as many times differentiable as required by the weak form integral
(C5)
iii. The basis functions must be $\phi_0, \phi_j$ linearly independent
(C3)
Weight functions: In the Rayleigh-Ritz method, the weight functions are chosen as the same functions as the basis functions (i.e., $w_i=\phi_i$ ).

## 数学代写|有限元方法代写有限元法代考|加权残差法

$$\tilde{u}(x)=\phi_0(x)+c_1 \phi_1(x)+c_2 \phi_2(x)+\ldots+c_N \phi_N(x)=\phi_0(x)+\sum_{j=1}^N c_j \phi_j(x)$$

$$\int_{\Omega} w_i R(\tilde{u}) d \Omega=0 \text { for } i=1, \ldots N$$

(i)近似解$\tilde{u}$满足问题的边界条件
(ii)近似解$\tilde{u}$是原微分方程要求的可微倍数
(iii)基函数$\phi_0, \phi_j$是线性无关的。

## 数学代写|有限元方法代写有限元法代考|瑞利-里兹法

.有限元法

$$w_i\left(x_L\right) Q_L-\int_{x_0}^{x_L}\left[a \frac{d \tilde{u}}{d x} \frac{d w_i}{d x}-c w_i \tilde{u}-w_i q\right] d x=0 \text { for } i=1 . . N$$

$$\tilde{u}(x)=\phi_0(x)+\sum_{j=1}^N C_j \phi_j(x)$$

i。近似解 $\tilde{u}$ 至少满足问题的基本边界条件
(C4)
ii。近似解 $\tilde{u}$ 可微的次数等于弱形式积分
(C5)
iii所要求的次数。基函数必须是 $\phi_0, \phi_j$ 权重函数:在瑞利-里兹方法中，权重函数被选为与基函数相同的函数(即 $w_i=\phi_i$ ).

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