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

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

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

In this work, modeling refers to mathematical formulation of a physical process. This requires background in the related subjects, certain mathematical tools, and experimental observations. In Chapter 2, we present the formulation of models for deformation of elastic solids and transfer and storage of thermal energy in solids and fluids. Solution of the mathematical model can be a challenging task and forms the general background of this work. Analytical solutions which can be expressed as relatively straight forward relationships between the dependent and independent variables exist only for a relatively small number of situations where the geometry and the physical nature of the problem can be simplified. Numerical methods are used otherwise. Among the numerical solution methods for solving PDEs are the finite difference, variational, and finite element methods.

The finite difference method (FDM) is implemented on the differential form of the BVP. The derivative operators of the PDE are approximated by finite difference operators. The solution domain is discretized in to a grid, and the unknowns are the values of the dependent variable at the nodes. The discretized version of the PDE is evaluated at each grid point. This results in a set of algebraic equations which can be represented in matrix form,
$$[K]{D}={R}$$
where $[K]$ is the stiffness matrix representing the discretized form of the partial derivatives, ${D}$ is the vector of unknown nodal values of the dependent variable, and ${R}$ is the loading vector representing the external effects. The boundary conditions often require specialized treatment of the finite difference operators and modify the $[K]$ matrix. The FDM is effective over relatively simple shapes such as rectangular and cylindrical domains in two-dimensional problems and parallelepiped or spherical domains in three-dimensional problems.

## 数学代写|有限元方法代写Finite Element Method代考|Mathematical modeling of physical systems

The goal of this chapter is to give brief descriptions to modeling of deformation of linear elastic solids and thermal energy transfer and storage in physical systems. More detailed discussion of these topics can be found in the specialized references provided at the end of this chapter. Our goal is to demonstrate how to obtain mathematical models (representations) of physical systems by using the fundamental laws of physics. Thus, we will show that deformation of elastic solids can be describéd by using Newton’s laws of motion. This will reesult in equations of motion represented as partial differential equations. Vibration of a long and slender bar (Section 2.1), deflection of a general deformable body (Section 2.2), and deflection of beams (Section 2.3) constitute examples of such systems. The principle of conservation of energy will be used to describe effects of heat transfer in a continuum (Section 2.4).

When a deformable body is subjected to external effects such as external forces and/or imposed displacements on its boundary, its shape will change and internal forces will develop throughout its volume. The level of deformation for given external effects depends on the material of the deformable body. In this section, the equations of motion for small deflections of linear, elastic materials are presented. In particular, we are interested in small deformations of linear, elastic solids. To this end, following are discussed: $i$ ) concepts of external and internal forces and the concept of stress, ii) elastic deformations and the concept of small strain, iii) linear elastic constitutive relations, iv) balance laws, and $v$ ) total potential energy of a deformable body.

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

$$[K] D=R$$

## 有限元方法代写

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