### 物理代写|理论力学代写theoretical mechanics代考|ENGR 1018

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|理论力学代写theoretical mechanics代考|Homogenization of Piezoelectric Composites

To determine the effective properties of piezoelectric composites, in ACELANCOMPOS package we use classical version of the effective moduli method. For piezoelectric composites this method was applied in a large number of papers $[5,9$, $22,24,30]$, with its mathematical basis given in $[22,24]$. In this section, we describe the formulation of the homogenization problem using the Voigt vector-matrix notation, which is generally accepted in the physical and theoretical literature on piezoelectricity.

The input data for the homogenization problem for two-phase piezoelectric (electroelastic) composite material is its representative volume element $\Omega$ together with the parts $\Omega^{(1)}$ and $\Omega^{(2)}$ filled with materials of different phases. In the domains $\Omega^{(j)}, j=1,2$, the following material moduli are known: the elastic stiffnesses $c_{\alpha \beta}^{E}=c_{\alpha \beta}^{E(j)}$, measured at constant electric field; the piezoelectric moduli $e_{k \beta}=e_{k \beta}^{(j)}$; and the dielectric permittivity constants $\varepsilon_{k m}^{S}=\varepsilon_{k m}^{S(j)}$, measured at constant strain; $\alpha, \beta=1,2, \ldots, 6, k, m=1,2,3 ; \mathbf{x} \in \Omega^{(j)} .$

We also introduce the following notation: $\Gamma=\partial \Omega$ is the outer boundary of the volume; $\mathbf{u}=\mathbf{u}(\mathbf{x})$ is the vector function of displacements; $\varphi=\varphi(\mathbf{x})$ is the electric potential function; $\mathbf{T}=\left{\sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{23}, \sigma_{13}, \sigma_{12}\right}$ is the array of stress components $\sigma_{k m} ; \mathbf{S}=\left{\varepsilon_{11}, \varepsilon_{22}, \varepsilon_{33}, 2 \varepsilon_{23}, 2 \varepsilon_{13}, 2 \varepsilon_{12}\right}$ is the array of the strain components $\varepsilon_{k m} ; \mathbf{D}$ is the vector of electric induction or electric displacement; $\mathbf{E}$ is the vector of electric field; $\mathbf{c}^{E}$ is the $6 \times 6$ matrix of elastic stiffness moduli $c_{\alpha \beta}^{E}$, e is the $3 \times 6$ matrix of piezoelectric modui $e_{k \beta} ; \varepsilon^{s}$ is the $3 \times 3$ matrix of dielectric permittivity moduli $\varepsilon_{k m m^{}}^{S}$. In the homogenization problem, it is necessary to determine the effective moduli $\bar{c}{\alpha \beta}^{E}, \bar{e}{k \beta}, \bar{\varepsilon}_{k m m}^{S}$. In order to do this, we need to solve a set of static boundary piezoelectric problems
$$\begin{gathered} \mathbf{L}^{}(\nabla) \cdot \mathbf{T}=0, \quad \nabla \cdot \mathbf{D}=0, \quad \mathbf{x} \in \Omega \ \mathbf{T}=\mathbf{c}^{E} \cdot \mathbf{S}-\mathbf{e}^{*} \cdot \mathbf{E}=0, \quad \mathbf{D}=\mathbf{e} \cdot \mathbf{S}+\boldsymbol{\varepsilon}^{S} \cdot \mathbf{E}=0 \end{gathered}$$

$$\begin{gathered} \mathbf{S}=\mathbf{L}(\nabla) \cdot \mathbf{u}, \quad \mathbf{E}=-\nabla \varphi \ \mathbf{u}=\mathbf{L}^{}(\mathbf{x}) \cdot \mathbf{S}{0}, \quad \varphi=-\mathbf{x} \cdot \mathbf{E}{0}, \quad \mathbf{x} \in \Gamma \end{gathered}$$
where $\mathbf{S}{0}$ is the six-dimensional array of constant values, $\mathbf{E}{0}$ is the constant vector, $(\ldots)^{}$ is the transpose operation, $L(\nabla)$ is the matrix operator of differentiation, which in transposed form is defined as follows
$$\mathbf{L}^{*}(\nabla)=\left[\begin{array}{cccccc} \partial_{1} & 0 & 0 & 0 & \partial_{3} & \partial_{2} \ 0 & \partial_{2} & 0 & \partial_{3} & 0 & \partial_{1} \ 0 & 0 & \partial_{3} & \partial_{2} & \partial_{1} & 0 \end{array}\right]$$

## 物理代写|理论力学代写theoretical mechanics代考|Some Models of Inhomogeneous Polarization

When analyzing the composites with the skeleton made of elastic piezoceramic material containing inclusions or pores, we can expect high inhomogeneity of the residual polarization vector P of piezoceramics. Indeed, even if the piezoceramics are polarized in one direction, the electric field or electric induction vectors inside the composite will not be parallel to this direction but will go around the inhomogeneities of the composite. Then it is logical to assume that the directions of the vector $\mathbf{P}=\mathbf{P}(\mathbf{x})$ at the first approximation can be obtained from the solution of the model problem of the polarization of composite material in linear setting. We will provide the mathematical setting of this problem in relation to the subsequent finite element homogenization problem.

Let $\Omega$ be a cubic representative volume of the composite of the size $L \times L \times L$ with the mesh consisting of finite elements $\Omega^{e m}, \Omega=\cup_{m} \Omega^{e m}$. It is assumed that each element $\Omega^{e m}$ belongs to the domain of one of the two phases, namely, the unpolarized piezoceramics $\Omega^{(1)}$ or the inclusion $\Omega^{(2)}$. Consequently, each element $\Omega^{e m}$ has dielectric properties of two phases, which we will consider isotropic materials with dielectric permeabilities $\varepsilon_{i}=\varepsilon_{i}^{(j)}, \mathbf{x} \in \Omega^{(j)}, j=1,2$. We assume that the edges $x_{3}=0$ and $x_{3}=L$ of the volume $\Omega$ are electrodized and are subjected to the potential difference $\Delta V=L E_{}$ with the field value $E_{}$, which is enough for the polarization of homogeneous piezoceramic material.

For the representative volume $\Omega$ with the help of FEM we solve the problem of electrostatics
$$\begin{gathered} \nabla \cdot \mathbf{D}=0, \quad \mathbf{D}=\varepsilon_{i} \mathbf{E}, \quad \mathbf{E}=-\nabla \varphi, \quad \mathbf{x} \in \Omega, \ \varphi=L E_{4}, \quad x_{3}=0 ; \quad \varphi=0, \quad x_{3}=L . \end{gathered}$$

## 物理代写|理论力学代写theoretical mechanics代考|Software Package Concept

ACELAN-COMPOS is a client-server GUI application with a modular structure. The user interface is implemented as an application developed using HTML and JavaScript and runs in a web-browser. The client-side application consists of the following moduli:

1. Graphic 3D preprocessor – a component for creating and viewing the source geometry. It is developed using the WebGL Framework. Currently, to start solving the problem the user provides parameters for the new model, including preferred connectivity type. Then the preprocessor allows analyzing generated mesh.
2. Tools for editing physical models – a set of forms for specifying boundary conditions and material properties with the help of the ACELAN command language.
3. Graphic 3D postprocessor – a module for analyzing the solution obtained, which includes the ability to view the solution both in tabular form and in the form of visualizations over the original geometry. Supported viewing modes include heat maps, vector field visualizations, sections and body viewing capabilities, etc. WebGL Framework is also selected as the implementation tool for the graphic postprocessor.

The server-side part of the package is a cross-platform application, developed using the .Net Core Framework and the $\mathrm{C} #$ programming language. It is responsible for performing calculations and processing the results of solving the problem. It allows performing computations for different users simultaneously. The interaction between the server and the client application is implemented by means of the REST API. The main components are:

1. A set of mesh generators for composites of supported types. Various plug-in mesh generators allow users to get models of composites that meet the required criteria. Currently only two-component composites are supported.
2. The ALGLIB Library and custom implementation of the Page-Sanders algorithm for solving systems of linear equations.
3. Finite element method solvers.

## 物理代写|理论力学代写theoretical mechanics代考|Some Models of Inhomogeneous Polarization

∇⋅D=0,D=e一世和,和=−∇披,X∈Ω, 披=大号和4,X3=0;披=0,X3=大号.

## 物理代写|理论力学代写theoretical mechanics代考|Software Package Concept

ACELAN-COMPOS 是一个具有模块化结构的客户端-服务器 GUI 应用程序。用户界面被实现为使用 HTML 和 JavaScript 开发的应用程序，并在网络浏览器中运行。客户端应用程序由以下模块组成：

1. 图形 3D 预处理器——用于创建和查看源几何图形的组件。它是使用 WebGL 框架开发的。目前，要开始解决问题，用户需要为新模型提供参数，包括首选连接类型。然后预处理器允许分析生成的网格。
2. 用于编辑物理模型的工具——一组在 ACELAN 命令语言的帮助下指定边界条件和材料属性的表格。
3. 图形 3D 后处理器 – 用于分析获得的解决方案的模块，其中包括以表格形式和原始几何图形的可视化形式查看解决方案的能力。支持的查看模式包括热图、矢量场可视化、截面和身体查看功能等。WebGL框架也被选为图形后处理器的实现工具。

1. 一组网格生成器，用于支持类型的组合。各种插件网格生成器允许用户获得满足所需标准的复合材料模型。目前仅支持双组分复合材料。
2. 用于求解线性方程组的 ALGLIB 库和 Page-Sanders 算法的自定义实现。
3. 有限元方法求解器。

## 有限元方法代写

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## MATLAB代写

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