### 数学代写|有限元方法代写Finite Element Method代考|ENGR7961

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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|General remarks

From the material presented in Sections $1.2-1.4$ it is clear that one could entertain any of the methods of approximation listed in Section 1.2, spacetime coupled, or space-time decoupled approaches for obtaining numerical solutions of the IVPs.

In this book we only consider finite element method in conjunction with space-time coupled and space-time decoupled approaches for obtaining numerical solutions of the IVPs. The finite element method for both approaches has rigorous mathematical foundation, hence in this approach it is always possible to ascertain feasibility, stability, and accuracy of the resulting computational processes. Error estimation, error computation, convergence, and convergence rates are additional meritorious features of the finite element processes for IVPs compared to all other methods listed in Section 1.2.
In the following sections we present a brief description of space-time coupled and space-time decoupled finite element processes, their merits and shortcomings, time integration techniques for ODEs in time resulting from decoupling space and time, stability of computational processes, error estimation, error computation, and convergence.

Some additional topics related to linear structural and linear solid mechanics such as mode superposition techniques of obtaining time evolution are also discussed.

## 数学代写|有限元方法代写Finite Element Method代考|Space-time coupled finite element method

In the initial value problem (1.1), the operator $A$ is a space-time differential operator. Thus, in order to address STFEM for totality of all IVPs in a problem- and application-independent fashion we must mathematically classify space-time differential operators appearing in all IVPs into groups. For these groups of space-time operators we can consider space-time methods of approximation such as space-time Galerkin method (STGM), space-time Petrov-Galerkin method (STPGM), space-time weighted residual method (STWRM), space-time Galerkin method with weak form (STGM/WF), spacetime least squares method or process (STLSM or STLSP), etc., thereby addressing totality of all IVPs. The space-time integral forms resulting from these space-time methods of approximation are necessary conditions.

By making a correspondence of these integral forms to the space-time calculus of variations we can determine which integral forms lead to unconditionally stable computational processes. The space-time integral forms that satisfy all elements of the space-time calculus of variations are termed space-time variationally consistent (STVC) integral forms. These integral forms result in unconditionally stable computational processes during the entire evolution. The integral forms in which one or more aspects of the space-time calculus of variations is not satisfied are termed space-time variationally inconsistent (STVIC) integral forms. In STVIC integral forms, unconditional stability of the computations is not always ensured.

## 有限元方法代写

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## MATLAB代写

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