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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|Mode superposition or normal mode synthesis

For complex structures like an automobile the space-time coupled and the space-time decoupled techniques with time integration are not practical for determining transient dynamic response due to excessively large number of degrees of freedom required for a reasonable discretization of the spatial domain. In such applications if we make the assumption that the time response of the structure is linear, then it is possible to devise a very efficient technique for determining the time dependent response of such structures. Mode superposition technique or normal mode synthesis technique is one such methodology.

When a structure is disturbed with a time dependent load, all natural modes of vibration are excited. Thus, the transient dynamic response of the structure can be thought of as a linear combination of these natural modes of vibrations, i.e. superposition of the modes of vibrations with appropriate multipliers (modal participation factors). A continuous medium naturally has infinitely many natural modes of vibration. A discretized continuous medium has finite number of natural modes of vibration depending upon the degrees of freedom involved in spatial discretization. For a complex structure like an automobile, the spatial discretization results in millions of degrees of freedom, hence has as many natural modes of vibration. It has been observed experimentally and has been verified through numerical simulations that:
(1) The lowest frequency natural mode of vibration has the highest energy content.
(2) With progressively higher frequency natural modes of vibration, the energy content progressively decreases.
(3) Based on (1) and (2), only a few of the lowest modes of natural vibrations contain almost all of the energy involved in the dynamic response of a structure.

## 数学代写|有限元方法代写Finite Element Method代考|Space and time

When constructing the mathematical descriptions of the initial value problems and in obtaining their solutions we are faced with two types of spaces: coordinate spaces and function spaces. In the absence of time, $x$ frame consisting of $o x, o y$, and $o z$ orthogonal axes originating from point $o$ is Cartesian coordinate space. Metric spaces such as Euclidean space are an example. In such spaces there is a concept of distance between two distinct elements of the space. On the other hand the collection of all functions $\phi(x, y, z)$ for all admissible choices of $x, y, x$ (coordinate space) constitute a function space. Banach space, Hilbert space, and Sobolev space are examples of such spaces.

In the solutions of boundary value problems (BVPs) we seek a solution $\phi(x, y, z)$ defined over a subset of coordinate space (the domain of the BVP) that satisfies the BVP and its boundary conditions. $\phi(x, y, z)$ belongs to a function space with desired regularity requirements dictated by the boundary value problem. In the initial value problems, in addition to spatial coordinates $x, y, z$ constituting an orthogonal space (Cartesian coordinates, for example), time $t$ is involved in their mathematical description. Thus, now we need to consider function spaces that contain functions $\phi(x, y, z, t)$, i.e. the functions that depend upon spatial coordinates and time. This poses no particular difficulty as we can treat time $t$ as another independent variable in addition to spatial coordinates $x, y$, and $z$.

The issue of whether time $t$ and $x, y, z$ constitute a four-dimensional space in which time $t$ is orthogonal or non-orthogonal to $x, y, z$ is one that needs to be discussed but is of relatively little importance to the subject matter considered in this book. Space and time need to be together (but not necessarily orthogonal) as ‘without time, space is dead,’ i.e. no change in space can occur without elapsed time. So, from the point of view of the subject matter in this book, we view time as something that allows us to measure changes in space. It is now well-accepted that time is relative, i.e. unit of time is in relation to some motion being considered as standard. Newtonian view of time as an orthogonal axis to space along which time flows indefinitely fails to explain motion of objects near the speed of light. Minkowski spaces in $x$, $y, z$, and $t$ that describe manifolds are needed for description of the motion of such objects. However, within Newtonian mechanics in inertial frame, the Newtonian view of time is accepted as feasible.

# 有限元方法代考

## 数学代写|有限元方法代写Finite Element Method代考|Mode superposition or normal mode synthesis

(1) 最低频率的固有振动模式具有最高的能量含量。
(2) 随着振动频率逐渐升高，能量含量逐渐降低。
(3) 基于 (1) 和 (2)，只有少数最低自然振动模式包含了结构动态响应中涉及的几乎所有能量。

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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