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# 数学代写|有限元方法代写finite differences method代考|Weak Forms

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## 数学代写|有限元代写Finite Element Method代考|Weak Forms

Let $w_h^e$ and $\phi_x^e$ denote the finite element approximations of $w$ and $\phi_x$, respectively, on a typical finite element $\Omega^e=\left(x_a^e, x_b^e\right)$. Substitution of $w_h^e$ and $\phi_x^e$ for $w$ and $\phi_x$, respectively, into Eqs. (5.3.5a) and (5.3.5b) give two residual functions. The weighted integrals of these two residuals over an element $\Omega^e$ are used to develop the weak forms, as was already discussed in Example 2.4.3. Suppose that $\left{v_{1 i}^e\right}$ and $\left{v_{2 i}^e\right}$ are the independent sets of weight functions used for the two equations. The physical meaning of these functions will be clear after completing the steps of the weak-form development. At the end of the second step of the three-step procedure (i.e., after integration by parts), we obtain
\begin{aligned} 0= & \int_{x_a^e}^{x_b^e}\left[G_e A_e K_s \frac{d v_{1 i}^e}{d x}\left(\phi_x^e+\frac{d w_h^e}{d x}\right)+k_f^e v_{1 i}^e w_h^e-v_{1 i}^e q_e\right] d x \ & -\left[v_{1 i}^e G_e A_e K_{\mathrm{s}}\left(\phi_x^e+\frac{d w_h^e}{d x}\right)\right]{x_a^e}^{x_b^e} \ 0= & \int{x_a^e}^{x_b^e}\left[E_e I_e \frac{d v_{2 i}^e}{d x} \frac{d \phi_x^e}{d x}+G_e A_e K_s v_{2 i}^e\left(\phi_x^e+\frac{d w_h^e}{d x}\right)\right] d x \ & -\left[v_{2 i}^e E_e I_e \frac{d \phi_h^e}{d x}\right]{x_a^e}^{x_b^e} \end{aligned} The coefficients of the weight functions $v{1 i}^e$ and $v_{2 i}^e$ in the boundary expressions are
$$G_e A_e K_s\left(\phi_x^e+\frac{d w_h^e}{d x}\right) \equiv V_h^e \quad \text { and } \quad E_e I_e \frac{d \phi_x^e}{d x} \equiv M_h^e$$

where $V_{h h}^e$ is the shear force and $M_h^e$ is the bending moment. Therefore, $\left(V_{h^{\prime}}^e M_h^e\right)^h$ constitute the secondary variables dual to the primary variables $\left(w_h^e, \phi_x^e\right)$ of the weak forms. Thus, the duality pairs of primary and secondary variables are
$$\left(w_h^e, V_h^e\right) \text { and }\left(\phi_x^e, M_h^e\right)$$

## 数学代写|有限元代写Finite Element Method代考|General Finite Element Model

A close examination of the terms in the weak forms, Eqs. (5.3.9a) and (5.3.9b), shows that only the first derivatives of $w_h^e$ and $\phi_x^e$ appear, requiring at least linear approximation of $w_h^e$ and $\phi_x^e$. Also, since the list of primary variables contain only the functions $w_h^e$ and $\phi_x^e$ and not their derivatives, the
Lagrange interpolation of $w_h^e$ and $\phi_x^e$ is admissible. Therefore, the Lagrange interpolation functions derived in Chapter 3 can be used. In general, $w_h^e$ and $\phi_x^e$ can be interpolated using different degree polynomials. In fact, the definition of shear strain $\gamma_{x z}^e=\phi_x^e+d w_h^e / d x$ suggests that $w_h$ should be represented by a polynomial of one degree higher than that used to represent $\phi_x^e$
Let us consider Lagrange approximation of $w$ and $\phi$ over an element $\Omega^e=\left(x_a^e, x_b^e\right)$ in the form
$$w \approx w_h^e=\sum_{j=1}^m w_j^e \psi_j^{(1)}, \quad \phi_x \approx \phi_x^e=\sum_{j=1}^n S_j^e \psi_j^{(2)}$$
where $\psi_j^{(1)}$ and $\psi_j^{(2)}$ are the Lagrange interpolation functions of degree $m-1$ and $n-1$, respectively. From the discussion above, it is advisable to use $m=$ $n+1$ for consistency of representing the variables $w$ and $\phi_x$.

## 数学代写|有限元代写Finite Element Method代考|Weak Forms

\begin{aligned} 0= & \int_{x_a^e}^{x_b^e}\left[G_e A_e K_s \frac{d v_{1 i}^e}{d x}\left(\phi_x^e+\frac{d w_h^e}{d x}\right)+k_f^e v_{1 i}^e w_h^e-v_{1 i}^e q_e\right] d x \ & -\left[v_{1 i}^e G_e A_e K_{\mathrm{s}}\left(\phi_x^e+\frac{d w_h^e}{d x}\right)\right]{x_a^e}^{x_b^e} \ 0= & \int{x_a^e}^{x_b^e}\left[E_e I_e \frac{d v_{2 i}^e}{d x} \frac{d \phi_x^e}{d x}+G_e A_e K_s v_{2 i}^e\left(\phi_x^e+\frac{d w_h^e}{d x}\right)\right] d x \ & -\left[v_{2 i}^e E_e I_e \frac{d \phi_h^e}{d x}\right]{x_a^e}^{x_b^e} \end{aligned}边界表达式中权函数$v{1 i}^e$和$v_{2 i}^e$的系数分别为
$$G_e A_e K_s\left(\phi_x^e+\frac{d w_h^e}{d x}\right) \equiv V_h^e \quad \text { and } \quad E_e I_e \frac{d \phi_x^e}{d x} \equiv M_h^e$$

$$\left(w_h^e, V_h^e\right) \text { and }\left(\phi_x^e, M_h^e\right)$$

## 数学代写|有限元代写Finite Element Method代考|General Finite Element Model

$$w \approx w_h^e=\sum_{j=1}^m w_j^e \psi_j^{(1)}, \quad \phi_x \approx \phi_x^e=\sum_{j=1}^n S_j^e \psi_j^{(2)}$$

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