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# 数学代写|有限元代写Finite Element Method代考|MEE356 HAMILTON’S PRINCIPLE

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## 数学代写|有限元代写Finite Element Method代考|HAMILTON’S PRINCIPLE

Hamilton’s principle is a simple yet powerful tool that can be used to derive discretized dynamic system equations. It states simply that

“Of all the admissible time histories of displacement the most accurate solution makes the Lagrangian functional a minimum.”
An admissible displacement must satisfy the following conditions:
(a) the compatibility equations,
(b) the essential or the kinematic boundary conditions, and
(c) the conditions at initial $\left(t_1\right)$ and final time $\left(t_2\right)$.
Condition (a) ensures that the displacements are compatible (continuous) in the problem domain. As will be seen in Chapter 11, there are situations when incompatibility can occur at the edges between elements. Condition (b) ensures that the displacement constraints are satisfied; and condition (c) requires the displacement history to satisfy the constraints at the initial and final times.
Mathematically, Hamilton’s principle states:
$$\delta \int_{t_1}^{t_2} L \mathrm{~d} t=0$$
The Langrangian functional, $L$, is obtained using a set of admissible time histories of displacements, and it consists of
$$L=T-\Pi+W_f$$
where $T$ is the kinetic energy, $\Pi$ is the potential energy (for our purposes, it is the elastic strain energy), and $W_f$ is the work done by the external forces. The kinetic energy of the entire problem domain is defined in the integral form
$$T=\frac{1}{2} \int_V \rho \dot{\mathbf{U}}^T \dot{\mathbf{U}} \mathrm{d} V$$
where $V$ represents the whole volume of the solid, and $\mathbf{U}$ is the set of admissible time histories of displacements.

## 数学代写|有限元代写Finite Element Method代考|Domain Discretization

The solid body is divided into $N_e$ elements. The procedure is often called meshing, which is usually performed using so-called pre-processors. This is especially true for complex geometries. Figure $3.1$ shows an example of a mesh for a two-dimensional solid.

The pre-processor generates unique numbers for all the elements and nodes for the solid or structure in a proper manner. An element is formed by connecting its nodes in a pre-defined consistent fashion to create the connectivity of the element. All the elements together form the entire domain of the problem without any gap or overlapping. It is possible for the domain to consist of different types of elements with different numbers of nodes, as long as they are compatible (no gaps and overlapping; the admissible condition (a) required by Hamilton’s principle) on the boundaries between different elements. The density of the mesh depends upon the accuracy requirement of the analysis and the computational resources available. Generally, a finer mesh will yield results that are more accurate, but will increase the computational cost. As such, the mesh is usually not uniform, with a finer mesh being used in the areas where the displacement gradient is larger or where the accuracy is critical to the analysis. The purpose of the domain discretization is to make it easier in assuming the pattern of the displacement field.

The FEM formulation has to be based on a coordinate system. In formulating FEM equations for elements, it is often convenient to use a local coordinate system that is defined for an element in reference to the global coordination system that is usually defined for the entire structure, as shown in Figure 3.4. Based on the local coordinate system defined on an element, the displacement within the element is now assumed simply by polynomial interpolation using the displacements at its nodes (or nodal displacements) as
$$\mathbf{U}^h(x, y, z)=\sum_{i=1}^{n_d} \mathbf{N}i(x, y, z) \mathbf{d}_i=\mathbf{N}(x, y, z) \mathbf{d}_e$$ where the superscript $h$ stands for approximation, $n_d$ is the number of nodes forming the element, and $\mathbf{d}_i$ is the nodal displacement at the $i$ th node, which is the unknown the analyst wants to compute, and can be expressed in a general form of \mathbf{d}_i=\left{\begin{array}{c} d_1 \ d_2 \ \vdots \ d{n_f} \end{array}\right} \rightarrow \begin{aligned} &\rightarrow \text { displacement component } 1 \ &\rightarrow \text { displacement component } 2 \ &\vdots \ &\text { displacement component } n_f \end{aligned}

## 数学代写|有限元代写Finite Element Method代考|HAMILTON’S PRINCIPLE

“在所有允许的位移时间历史中，最准确的解快方安使拉格朗日函数最小。”

(a) 相容方程,
(b) 基本或运动边界条件，以及
(c) 初始条件 $\left(t_1\right)$ 最后一次 $\left(t_2\right)$.

$$\delta \int_{t_1}^{t_2} L \mathrm{~d} t=0$$

$$L=T-\Pi+W_f$$

$$T=\frac{1}{2} \int_V \rho \dot{\mathbf{U}}^T \dot{\mathbf{U}} \mathrm{d} V$$

## 数学代写|有限元代写Finite Element Method代考|Domain Discretization

FEM 公式必须基于坐标系。在为单元制定 FEM 方程时，通常方便地使用为单元定义的局部坐标系，参考通常为整个结构定义的全 局坐标系，如图 $3.4$ 所示。基于单元上定义的局部坐标系，单元内的位移现在通过多项式揷值简单地假设，使用其节点处的位移 (或节点位移) 为
$$\mathbf{U}^h(x, y, z)=\sum_{i=1}^{n_d} \mathbf{N} i(x, y, z) \mathbf{d}_i=\mathbf{N}(x, y, z) \mathbf{d}_e$$

〈left 的分隔符缺失或无法识别

## MATLAB代写

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