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the mathematical theory of finite element method

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the mathematical theory of finite

element method

The mathematical theory of the finite element method (FEM) is a branch of

numerical analysis that provides a framework for approximating solutions to partial

differential equations (PDEs) using discretization techniques. The finite element

method is widely used in engineering and scientific disciplines to simulate and

analyze physical phenomena.

At the core of the FEM is the concept of dividing a domain into a finite number

of elements, which are connected at nodes. The unknown solution within each element

is approximated using a simple function, referred to as the basis function. These

basis functions are usually polynomials of a certain degree, and their coefficients

are determined by solving a set of linear equations.

The mathematical theory of the FEM involves several key concepts and techniques.

One of the fundamental principles is the variational formulation, which transforms

the PDE into an equivalent variational problem. This variational problem is then

discretized using the finite element approximation, resulting in a system of

algebraic equations.

Another important aspect is the assembly process, where the contributions from

each element are combined to form the global stiffness matrix and right-hand side

vector. This assembly is based on the integration of the basis functions and their

derivatives over the element domains.

Error estimation and convergence analysis are also essential components of the

mathematical theory of the FEM. Various techniques, such as the energy method and

the posteriori error estimators, are used to assess the accuracy of the finite

element solution and to determine the appropriate mesh refinement for achieving

convergence.

Furthermore, the mathematical theory of the FEM includes the treatment of

boundary conditions, imposition of symmetries, and the development of efficient

solvers for the resulting linear systems. It also addresses issues such as numerical

stability,并行 computing, and adaptivity.

In summary, the mathematical theory of the finite element method provides a

comprehensive framework for numerically solving PDEs. It encompasses concepts such

as element discretization, variational formulation, assembly, error estimation, and

convergence analysis, which collectively enable the accurate and efficient

simulation of a wide range of physical problems.

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有限元法介绍

有限元法介绍

通俗地说，有限元法就是一种计算机模拟技术，使人们能够在计算机上用软件模拟一个工程问题

的发生过程而无需把东西真的做出来。这项技术带来的好处就是，在图纸设计阶段就能够让人们

在计算机上观察到设计出的产品将来在使用中可能会出现什么问题，不用把样机做出来在实验中检验会出现什么问题，可以有效降低产品开发的成本，缩短产品设计的周期。

有限元法也叫有限单元法（finite element method, FEM），是随着电子计算机的发展而

迅速发展起来的一种弹性力学问题的数值求解方法。五十年代初，它首先应用于连续体力学领域—飞机结构静、动态特性分析中，用以求得结构的变形、应力、固有频率以及振型。由于这种方

法的有效性，有限单元法的应用已从线性问题扩展到非线性问题，分析的对象从弹性材料扩展到

塑性、粘弹性、粘塑性和复合材料，从连续体扩展到非连续体。

有限元法最初的思想是把一个大的结构划分为有限个称为单元的小区域，在每一个小区域

里，假定结构的变形和应力都是简单的，小区域内的变形和应力都容易通过计算机求解出来，进

而可以获得整个结构的变形和应力。

事实上，当划分的区域足够小，每个区域内的变形和应力总是趋于简单，计算的结果也就越

接近真实情况。理论上可以证明，当单元数目足够多时，有限单元解将收敛于问题的精确解，但

是计算量相应增大。为此，实际工作中总是要在计算量和计算精度之间找到一个平衡点。

有限元法中的相邻的小区域通过边界上的结点联接起来，可以用一个简单的插值函数描述每

个小区域内的变形和应力，求解过程只需要计算出结点处的应力或者变形，非结点处的应力或者

变形是通过函数插值获得的，换句话说，有限元法并不求解区域内任意一点的变形或者应力。

大多数有限元程序都是以结点位移作为基本变量，求出结点位移后再计算单元内的应力，这

种方法称为位移法。

有限元法本质上是一种微分方程的数值求解方法，认识到这一点以后，从70年代开始，有

限元法的应用领域逐渐从固体力学领域扩展到其它需要求解微分方程的领域，如流体力学、传热

有限单元法基本原理与数值方法

有限单元法基本原理与数值方法

有限单元法基本原理与数值方法

英文回答：

Basic Principles of Finite Element Method.

The finite element method (FEM) is a numerical

technique used to find approximate solutions to boundary

value problems for partial differential equations (PDEs).

It is based on the idea of dividing a complex problem into

smaller, more manageable parts, called finite elements.

The first step in the FEM is to discretize the domain

of the problem into a mesh of finite elements. The mesh is

typically generated using a computer program, and the size

and shape of the elements can be varied to suit the

requirements of the problem.

Once the mesh has been created, the next step is to

define the governing equations for the problem. The

governing equations are mathematical equations that describe the physical behavior of the system being modeled.

finite-element analysis

finite-element analysis

finite-element analysis

FEA is used in various fields of engineering, such as

mechanical, civil, and aerospace engineering, to simulate the

behavior of a structure, system, or material. By analyzing the

structure in smaller segments, FEA allows engineers to examine

the stress, strain, deformation, and other important parameters

at different points of the design. This helps engineers predict

how the structure or system will behave under different loads or

conditions, which can help refine the design to optimize its

performance and safety.

FEA involves creating a 3D virtual model of the structure

and applying a set of mathematical equations to each small

element. These equations relate the movement of each element to

the forces acting upon it in its location. With this information,

engineers can create a visual representation of the structure,

ABAQUS Theory Manual

ABAQUS Theory Manual

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1

ABAQUS Theory Manual

Introduction

The ABAQUS software, developed by Dassault Systèmes,

provides powerful capabilities for finite element analysis (FEA)

and allows engineers and researchers to simulate and analyze

the behavior of structures and materials under various

loadings and boundary conditions. This theory manual aims to

provide a comprehensive overview of the underlying

principles and theories that form the foundation of ABAQUS.

1. Finite Element Method

The finite element method (FEM) is the basis of ABAQUS.

This section will introduce the fundamental concepts and

principles of FEM, including discretization, interpolation,

element types, and numerical integration techniques.

Understanding these concepts is essential for users to

accurately model and analyze their structures using ABAQUS.