### 数学代写|有限元方法代写Finite Element Method代考|The Principle of Minimum Total Potential Energy

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## 数学代写|有限元方法代写Finite Element Method代考|The Principle of Minimum Total Potential Energy

The principle of virtual work discussed in the previous section is applicable to any continuous body with arbitrary constitutive behavior (e.g., linear or nonlinear elastic materials). The principle of minimum total potential energy is obtained as a special case from the principle of virtual displacements when the constitutive relations can be obtained from a potential function. Here we restrict our discussion to materials that admit existence of a strain energy potential such that the stress is derivable from it. Such materials are termed hyperelastic.
For elastic bodies (in the absence of temperature variations), there exists a strain energy potential $U_0$ such that [see Eq. (2.3.5)]
$$\sigma_{i j}=\frac{\partial U_0}{\partial \varepsilon_{i j}}$$
The strain energy density $U_0$ is a function of strains at a point and is assumed to be positive definite. The statement of the principle of virtual displacements, $\delta W=0$, can be expressed in terms of the strain energy density $U_0$ as
\begin{aligned} 0=\delta W & =\int_{\Omega} \sigma_{i j} \delta \varepsilon_{i j} d \Omega-\left[\int_{\Omega} \mathbf{f} \cdot \delta \mathbf{u} d \Omega+\int_{\Gamma_\sigma} \hat{\mathbf{t}} \cdot \delta \mathbf{u} d s\right] \ & =\int_{\Omega} \frac{\partial U_0}{\partial \varepsilon_{i j}} \delta \varepsilon_{i j} d \Omega+\delta V_E \ & =\int_{\Omega} \delta U_0 d \Omega+\delta V_E=\delta\left(U+V_E\right) \equiv \delta \Pi \end{aligned}
where
$$V_E=-\left[\int_{\Omega} \mathbf{f} \cdot \mathbf{u} d \Omega+\int_{\Gamma_\sigma} \hat{\mathbf{t}} \cdot \mathbf{u} d s\right]$$
is the potential energy due to external loads and $U$ is the strain energy potential
$$U=\int_{\Omega} U_0 d \Omega$$

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

Consider the problem of solving the differential equation
$$-\frac{d}{d x}\left[a(x) \frac{d u}{d x}\right]+c u=f(x) \text { for } 0<x<L$$
for $u(x)$, which is subject to the boundary conditions
$$u(0)=u_0, \quad\left[a \frac{d u}{d x}+\beta\left(u-u_{\infty}\right)\right]{x=L}=Q_L$$ Here $a(x), c(x)$, and $f(x)$ are known functions of the coordinate $x ; u_0, u{\infty}, \beta$, and $Q_L$ are known values, and $L$ is the size of the one-dimensional domain. When the specified values are nonzero $\left(u_0 \neq 0\right.$ or $\left.Q_L \neq 0\right)$, the boundary conditions are said to be nonhomogeneous; when the specified values are zero the boundary conditions are said to be homogeneous. The homogeneous form of the boundary condition $u(0)=u_0$ is $u(0)=0$, and the homogeneous form of the boundary condition $\left[a(d u / d x)+\beta\left(u-u_{\infty}\right)\right]{x=L}=Q_L$ is $[a(d u / d x)+$ $\left.\beta\left(u-u{\infty}\right)\right]_{x=L}=0$.

Equations of the type in Eq. (2.4.1) arise, for example, in the study of 1-D heat flow in a rod with surface convection (see Example 1.2.2), as shown in Fig. 2.4.1(a). In this case, $a=k A$, with $k$ being the thermal conductivity and $A$ the cross-sectional area, $c=\beta P$, with $\beta$ being the heat transfer coefficient, $P$ the perimeter of the rod, and $L$ the length of the rod; $f$ denotes the heat generation term, $u_0$ is the specified temperature at $x=0, Q_L$ is the specified heat at $x=L$, and $u_{\infty}$ is the temperature of the surrounding medium. Another example where Eqs. (2.4.1) and (2.4.2) arise is provided by the axial deformation of a bar (see Example 1.2.3), as shown in Fig. 2.4.1(b). In this case, $a=E A$, with $E$ being the Young’s modulus and $A$ the cross-sectional area, $c$ is the spring constant associated with the shear resistance offered by the surrounding medium (as discussed in Example 1.2.3), and $L$ is the length of the bar; $f$ denotes the body force term, $u_0$ is the specified displacement at $x$ $=0\left(u_0=0\right), Q_L$ is the specified point load at $x=L$, and $u_{\infty}=0$. Other physical problems are also described by the same equation, but with different meaning of the variables.

## 数学代写|有限元方法代写Finite Element Method代考|The Principle of Minimum Total Potential Energy

$$\sigma_{i j}=\frac{\partial U_0}{\partial \varepsilon_{i j}}$$

\begin{aligned} 0=\delta W & =\int_{\Omega} \sigma_{i j} \delta \varepsilon_{i j} d \Omega-\left[\int_{\Omega} \mathbf{f} \cdot \delta \mathbf{u} d \Omega+\int_{\Gamma_\sigma} \hat{\mathbf{t}} \cdot \delta \mathbf{u} d s\right] \ & =\int_{\Omega} \frac{\partial U_0}{\partial \varepsilon_{i j}} \delta \varepsilon_{i j} d \Omega+\delta V_E \ & =\int_{\Omega} \delta U_0 d \Omega+\delta V_E=\delta\left(U+V_E\right) \equiv \delta \Pi \end{aligned}

$$V_E=-\left[\int_{\Omega} \mathbf{f} \cdot \mathbf{u} d \Omega+\int_{\Gamma_\sigma} \hat{\mathbf{t}} \cdot \mathbf{u} d s\right]$$

$$U=\int_{\Omega} U_0 d \Omega$$

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

$$-\frac{d}{d x}\left[a(x) \frac{d u}{d x}\right]+c u=f(x) \text { for } 0<x<L$$

$$u(0)=u_0, \quad\left[a \frac{d u}{d x}+\beta\left(u-u_{\infty}\right)\right]{x=L}=Q_L$$其中$a(x), c(x)$、$f(x)$为坐标$x ; u_0, u{\infty}, \beta$的已知函数，$Q_L$为已知值，$L$为一维域的大小。当指定值为非零$\left(u_0 \neq 0\right.$或$\left.Q_L \neq 0\right)$时，边界条件是非齐次的;当指定值为零时，边界条件称为齐次。边界条件$u(0)=u_0$的齐次形式为$u(0)=0$，边界条件$\left[a(d u / d x)+\beta\left(u-u_{\infty}\right)\right]{x=L}=Q_L$的齐次形式为$[a(d u / d x)+$$\left.\beta\left(u-u{\infty}\right)\right]_{x=L}=0$。

## 有限元方法代写

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## MATLAB代写

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