有限元分析中英文对照资料

The finite element analysis

Finite element method, the solving area is regarded as made up of many small in the node connected unit (a domain), the model gives the fundamental equation of sharding (sub-domain) approximation solution, due to the unit (a domain) can be divided into various shapes and sizes of different size, so it can well adapt to the complex geometry, complex material properties and complicated boundary conditions

Finite element model: is it real system idealized mathematical abstractions. Is composed of some simple shapes of unit, unit connection through the node, and under a certain load.

Finite element analysis: is the use of mathematical approximation method for real physical systems (geometry and loading conditions were simulated. And by using simple and interacting elements, namely unit, can use a limited number of unknown variables to approaching infinite unknown quantity of the real system.

Linear elastic finite element method is a ideal elastic body as the research object, considering the deformation based on small deformation assumption of. In this kind of problem, the stress and strain of the material is linear relationship, meet the generalized hooke's law; Stress and strain is linear, linear elastic problem boils down to solving linear equations, so only need less computation time. If the efficient method of solving algebraic equations can also help reduce the duration of finite element analysis.

Linear elastic finite element generally includes linear elastic statics analysis and linear elastic dynamics analysis from two aspects. The difference between the nonlinear problem and linear elastic problems:

1) nonlinear equation is nonlinear, and iteratively solving of general;

2) the nonlinear problem can't use superposition principle;

3) nonlinear problem is not there is always solution, sometimes even no solution. Finite element to solve the nonlinear problem can be divided into the following three categories:

1) material nonlinear problems of stress and strain is nonlinear, but the stress and strain is very small, a linear relationship between strain and displacement at this time, this kind of problem belongs to the material nonlinear problems. Due to theoretically also cannot provide the constitutive relation can be accepted, so, general nonlinear relations between stress and strain of the material based on the test data, sometimes, to simulate the nonlinear material properties available mathematical model though these models always have their limitations. More important material nonlinear problems in engineering practice are: nonlinear elastic (including piecewise linear elastic, elastic-plastic and viscoplastic, creep, etc.

2) geometric nonlinear geometric nonlinear problems are caused due to the nonlinear relationship between displacement. When the object the displacement is larger, the strain and displacement relationship is nonlinear relationship. Research on this kind of problem

Is assumes that the material of stress and strain is linear relationship. It consists

of a large displacement problem of large strain and large displacement little strain. Such as the structure of the elastic buckling problem belongs to the large displacement little strain, rubber parts forming process for large strain.

3) nonlinear boundary problem in the processing, problems such as sealing, the impact of the role of contact and friction can not be ignored, belongs to the highly nonlinear contact boundary. At ordinary times some contact problems, such as gear, stamping forming, rolling, rubber shock absorber, interference fit assembly, etc., when a structure and another structure or external boundary contact usually want to consider nonlinear boundary conditions. The actual nonlinear may appear at the same time these two or three kinds of nonlinear problems.

Finite element theoretical basis

Finite element method is based on variational principle and the weighted residual method, and the basic solving thought is the computational domain is divided into a finite number of non-overlapping unit, within each cell, select some appropriate nodes as solving the interpolation function, the differential equation of the variables in the rewritten by the variable or its derivative selected interpolation node value and the function of linear expression, with the aid of variational principle or weighted residual method, the discrete solution of differential equation. Using different forms of weight function and interpolation function, constitute different finite element methods. 1. The weighted residual method and the weighted residual method of weighted residual method of weighted residual method: refers to the weighted function is zero using make allowance for approximate solution of the differential equation method is called the weighted residual method. Is a kind of directly from the solution of differential equation and boundary conditions, to seek the approximate solution of boundary value problems of mathematical methods. Weighted residual method is to solve the differential equation of the approximate solution of a kind of effective method.

Hybrid method for the trial function selected is the most convenient, but under the condition of the same precision, the workload is the largest. For internal method and the boundary method basis function must be made in advance to meet certain conditions, the analysis of complex structures tend to have certain difficulty, but the trial function is established, the workload is small. No matter what method is used, when set up trial function should be paid attention to are the following: (1) trial function should be composed of a subset of the complete function set. Have been using the trial function has the power series and trigonometric series, spline functions, beisaier, chebyshev, Legendre polynomial, and so on.

(2) the trial function should have until than to eliminate surplus weighted integral expression of the highest derivative low first order derivative continuity. (3) the trial function should be special solution with analytical solution of the problem or problems associated with it. If computing problems with symmetry, should make full use of it. Obviously, any independent complete set of functions can be used as weight function. According to the weight function of the different options

for different weighted allowance calculation method, mainly include: collocation method, subdomain method, least square method, moment method and galerkin method. The galerkin method has the highest accuracy.

Principle of virtual work: balance equations and geometric equations of the equivalent integral form of "weak" virtual work principles include principle of virtual displacement and virtual stress principle, is the floorboard of the principle of virtual displacement and virtual stress theory. They can be considered with some control equation of equivalent integral "weak" form. Principle of virtual work: get form any balanced force system in any state of deformation coordinate condition on the virtual work is equal to zero, namely the system of virtual work force and internal force of the sum of virtual work is equal to zero. The virtual displacement principle is the equilibrium equation and force boundary conditions of the equivalent integral form of "weak"; Virtual stress principle is geometric equation and displacement boundary condition of the equivalent integral form of "weak". Mechanical meaning of the virtual displacement principle: if the force system is balanced, they on the virtual displacement and virtual strain by the sum of the work is zero. On the other hand, if the force system in the virtual displacement (strain) and virtual and is equal to zero for the work, they must balance equation. Virtual displacement principle formulated the system of force balance, therefore, necessary and sufficient conditions. In general, the virtual displacement principle can not only suitable for linear elastic problems, and can be used in the nonlinear elastic and elastic-plastic nonlinear problem.

Virtual mechanical meaning of stress principle: if the displacement is coordinated, the virtual stress and virtual boundary constraint counterforce in which they are the sum of the work is zero. On the other hand, if the virtual force system in which they are and is zero for the work, they must be meet the coordination. Virtual stress in principle, therefore, necessary and sufficient condition for the expression of displacement coordination. Virtual stress principle can be applied to different linear elastic and nonlinear elastic mechanics problem. But it must be pointed out that both principle of virtual displacement and virtual stress principle, rely on their geometric equation and equilibrium equation is based on the theory of small deformation, they cannot be directly applied to mechanical problems based on large deformation theory. 3,,,,, the minimum total potential energy method of minimum total potential energy method, the minimum strain energy method of minimum total potential energy method, the potential energy function in the object on the external load will cause deformation, the deformation force during the work done in the form of elastic energy stored in the object, is the strain energy.

The convergence of the finite element method, the convergence of the finite element method refers to when the grid gradually encryption, the finite element solution sequence converges to the exact solution; Or when the cell size is fixed, the more freedom degree each unit, the finite element solutions tend to be more precise solution. Convergence condition of the convergence condition of the finite element finite element convergence condition of the convergence condition of the finite element finite element includes the following four aspects: 1) within the unit, the

displacement function must be continuous. Polynomial is single-valued continuous function, so choose polynomial as displacement function, to ensure continuity within the unit. 2) within the unit, the displacement function must include often strain. Total can be broken down into each unit of the state of strain does not depend on different locations within the cell strain and strain is decided by the point location of variables. When the size of the units is enough hours, unit of each point in the strain tend to be equal, unit deformation is uniform, so often strain becomes the main part of the strain. To reflect the state of strain unit, the unit must include the displacement functions often strain. 3) within the unit, the displacement function must include the rigid body displacement. Under normal circumstances, the cell for a bit of deformation displacement and displacement of rigid body displacement including two parts. Deformation displacement is associated with the changes in the object shape and volume, thus producing strain; The rigid body displacement changing the object position, don't change the shape and volume of the object, namely the rigid body displacement is not deformation displacement. Spatial displacement of an object includes three translational and three rotational displacement, a total of six rigid body displacements. Due to a unit involved in the other unit, other units do rigid body displacement deformation occurs will drive unit, thus, to simulate real displacement of a unit, assume that the element displacement function must include the rigid body displacement. 4) the displacement function must be coordinated in public boundary of the adjacent cell. For general unit of coordination is refers to the adjacent cell in public node have the same displacement, but also have the same displacement along the edge of the unit, that is to say, to ensure that the unit does not occur from cracking and invade the overlap each other. To do this requires the function on the common boundary can be determined by the public node function value only. For general unit and coordination to ensure the continuity of the displacement of adjacent cell boundaries. However, between the plate and shell of the adjacent cell, also requires a displacement of the first derivative continuous, only in this way, to guarantee the strain energy of the structure is bounded. On the whole, coordination refers to the public on the border between neighboring units satisfy the continuity conditions. The first three, also called completeness conditions, meet the conditions of complete unit is complete unit; Article 4 is coordination requirements, meet the coordination unit coordination unit; Otherwise known as the coordinating units. Completeness requirement is necessary for convergence, all four meet, constitutes a necessary and sufficient condition for convergence. In practical application, to make the selected displacement functions all meet the requirements of completeness and harmony, it is difficult in some cases can relax the requirement for coordination. It should be pointed out that, sometimes the coordination unit than its corresponding coordination unit, its reason lies in the nature of the approximate solution. Assumed displacement function is equivalent to put the unit under constraint conditions, the unit deformation subject to the constraints, this just some alternative structure compared to the real structure. But the approximate structure due to allow cell separation, overlap, become soft, the stiffness of the unit or formed (such

as round degree between continuous plate unit in the unit, and corner is discontinuous, just to pin point) for the coordination unit, the error of these two effects have the possibility of cancellation, so sometimes use the coordination unit will get very good results. In engineering practice, the coordination of yuan must pass to use "small pieces after test". Average units or nodes average processing method of stress stress average units or nodes average processing method of stress average units or nodes average processing method of stress of the unit average or node average treatment method is the simplest method is to take stress results adjacent cell or surrounding nodes, the average value of stress.

1. Take an average of 2 adjacent unit stress. Take around nodes, the average value of stress

The basic steps of finite element method to solve the problem

The structural discretization structure discretization structure discretization structure discretization to discretization of the whole structure, will be divided into several units, through the node connected to each other between the units; 2. The stiffness matrix of each unit and each element stiffness matrix and the element stiffness matrix and the stiffness matrix of each unit (3) integrated global stiffness matrix integrated total stiffness matrix integrated overall stiffness matrix integrated total stiffness matrix and write out the general balance equations and write out the general balance equations and write out the general balance equations and write a general equation 4. Introduction of supporting conditions, the displacement of each node 5. Calculate the stress and strain in the unit to get the stress and strain of each cell and the cell of the stress and strain and the stress and strain of each cell.

For the finite element method, the basic ideas and steps can be summarized as: (1) to establish integral equation, according to the principle of variational allowance and the weight function or equation principle of orthogonalization, establishment and integral expression of differential equations is equivalent to the initial-boundary value problem, this is the starting point of the finite element method. Unit (2) the area subdivision, according to the solution of the shape of the area and the physical characteristics of practical problems, cut area is divided into a number of mutual connection, overlap of unit. Regional unit is divided into finite element method of the preparation, this part of the workload is bigger, in addition to the cell and node number and determine the relationship between each other, also said the node coordinates, at the same time also need to list the natural boundary and essential boundary node number and the corresponding boundary value.

(3) determine the unit basis function, according to the unit and the approximate solution of node number in precision requirement, choose meet certain interpolation condition basis function interpolation function as a unit. Basis function in the finite element method is selected in the unit, due to the geometry of each unit has a rule in the selection of basis function can follow certain rules. (4) the unit will be analysis: to solve the function of each unit with unit basis functions to

approximate the linear combination of expression; Then approximate function generation into the integral equation, and the unit area integral, can be obtained with undetermined coefficient (i.e., cell parameter value) of each node in the algebraic equations, known as the finite element equation. (5) the overall synthesis: after the finite element equation, the area of all elements in the finite element equation according to certain principles of accumulation, the formation of general finite element equations. (6) boundary condition processing: general boundary conditions there are three kinds of form, divided into the essential boundary conditions (dirichlet boundary condition) and natural boundary conditions (Riemann boundary conditions) and mixed boundary conditions (cauchy boundary conditions). Often in the integral expression for natural boundary conditions, can be automatically satisfied. For essential boundary conditions and mixed boundary conditions, should be in a certain method to modify general finite element equations satisfies. Solving finite element equations (7) : based on the general finite element equations of boundary conditions are fixed, are all closed equations of the unknown quantity, and adopt appropriate numerical calculation method, the function value of each node can be obtained.

有限元分析

有限元法求解区域是由许多小的节点连接单元（域），该模型给出了切分的基本方程（子域名）的近似解，由于单位（域）可以分为不同的形状和大小不同的尺寸，所以它能很好的适应复杂的几何形状、材料特性和边界条件复杂，复杂

有限元模型：它是真实系统的理想化的数学抽象。是由一些简单的单元形状，单元连接通过节点，并在一定的负载。

有限元分析：利用数学逼近法对实际物理系统（几何和加载条件）进行了模拟。通过使用简单的交互元素，即单位，可以使用有限数量的未知变量接近无穷未知量的实际系统。

线弹性有限元法是一种理想的弹性体为研究对象，考虑到基于小变形假设。在这类问题中，材料的应力和应变的线性关系，符合广义虎克定律；应力和应变是线性的，线性弹性问题归

结为求解线性方程组，所以只需要更少的计算时间。如果求解代数方程的有效方法也可以帮助减少有限元分析的持续时间。

线弹性有限元一般包括线性弹性静力学分析和线弹性动力学分析两个方面。非线性问题与线弹性问题的区别：

1）非线性方程是非线性的，一般的迭代求解；

2）非线性问题不能采用叠加原理；

3）非线性问题并非总是有解，有时甚至没有解。有限元求解非线性问题可分为以下三类：1）材料非线性应力应变问题是非线性的，但应力应变很小，此时应变与位移呈线性关系，这种问题属于材料非线性问题。由于理论上也不能提供的本构关系是可以接受的，所以，一般的非线性关系的应力测试数据的基础上，有时材料的应变，来模拟非线性材料特性的数学模型，但这些模型可总是有其局限性。工程实践中较为重要的材料非线性问题有：非线性弹性（包括分段线弹性、弹塑性、粘塑性、蠕变等）。

2）几何非线性几何非线性问题是由位移非线性关系引起的。当位移较大时，应力应变关系为非线性关系。这种问题的研究

假定应力应变材料是线性关系。它是由大应变大位移小应变大位移问题。如结构的弹性屈曲问题属于大位移小应变，橡胶件的大应变成形过程。

3）非线性边界问题的处理，如密封问题，接触和摩擦的作用的影响不容忽视，属于高度非线性接触边界。在一般的一些接触问题，如齿轮，冲压成形，滚动，橡胶减震器，干涉配合装配等，当一个结构和另一个结构或外部边界接触通常要考虑非线性边界条件。实际非线性可能同时出现这两个或三类非线性问题。

有限元理论基础

有限元方法是基于变分原理和加权余量法，及其解决的基本思想是将计算域划分为有限个互不重叠的单元，在每个单元，选择一些合适的节点作为解决插值函数，在改写的变量或其导数插值节点选择的价值和功能线性表达变量的微分方程，用变分原理和加权残值法的帮助下，微分方程的离散解。利用不同形式的权函数和插值函数，构成不同的有限元方法。1。加权残值法的加权残数法和加权残数法是指加权函数为零，采用偏微分方程法的近似解称为加权残值法。是一类直接从微分方程和边界条件的解出发，寻求数学方法边值问题的近似解。加权残值法是求解微分方程近似解的一种有效方法。

混合方法对试验函数的选择是最方便的，但在相同精度的条件下，工作量最大。对于内方法和边界法的基函数必须事先作出满足一定的条件，复杂结构的分析往往有一定的难度，但试验功能成立，工作量小。无论采用何种方法，设置试功能时应注意以下几点：

（1）试验函数应由完备函数集的子集组成。已使用的试函数的幂级数和三角级数，样条函数，贝赛尔，切比雪夫，勒让德多项式，等等。

（2）试用函数应具有比消除剩余加权积分表达式的最高导数低阶一阶导数连续性。

（3）试验功能应与问题的分析解或与之相关的问题进行特殊解答。如果计算问题具有对称性，应充分利用它。显然，任何独立完备的函数集都可以作为权重函数。根据不同的加权余量的计算方法，对不同方案的权重函数主要包括：配置法、子域法、最小二乘法、矩量法和伽辽金法。伽辽金法具有最高的精度。

虚功原理：平衡方程和几何方程的等效积分形式的“弱”的虚拟工作原则包括虚拟位移和有效应力原理的总称，是原则的虚拟位移和虚拟应力理论。他们可以考虑一些控制方程的等效积分“弱”形式。虚功原理：获得任何形式的平衡力系统，在任何状态下的变形坐标条件下，虚功等于零，即虚功力和虚功之和的内力等于零。虚位移原理是“弱”等效积分形式的平衡方程和力边界条件；虚应力原理是“弱”等效积分形式的几何方程和位移边界条件。虚位移原理的力学意义：如果力系是平衡的，则虚位移和虚应变由功之和为零。另一方面，如果力

系统在虚位移（应变）与虚和等于零为功时，则必须平衡方程。虚位移原理制定了系统的力平衡，因此，必要条件和充分条件。虚位移原理不仅适用于线弹性问题，而且可用于非线性弹塑性非线性问题。

应力原理虚拟力学意义：如果位移协调，虚拟应力和虚拟边界约束反力在他们的工作和为零。另一方面，如果虚拟力系统中它们是零和为零的工作，则它们必须满足协调。因此，虚应力原理是位移协调表达式的充要条件.。虚拟应力原理适用于不同的线弹性和非线性弹性力学问题。但必须指出的是，虚位移原理和虚应力原理，依赖于它们的几何方程和平衡方程是基于小变形理论，它们不能直接应用于基于大变形理论的力学问题。3 ,,,,,最小总势能法最小总势能法、最小总势能法最小应变能方法，潜在的目标能量函数在外部荷载会引起变形，在工作过程中的变形力在弹性势能的形式储存在对象中，是应变能。

有限元法的收敛性，有限元法的收敛性是指当电网逐渐加密，有限元解序列收敛于精确解；或当单元格的大小是固定的，多自由度的每个单元，有限元解趋于精确解。有限元有限元收敛条件有限元收敛条件收敛条件的收敛条件包括以下四个方面：（1）在单元内，位移函数必须连续。多项式是单值连续函数，所以选择多项式作为位移函数，保证单元内的连续性.。2）在单元内，位移函数必须包含常应变。总可分解成每个单元的应变状态不依赖于细胞内不同的位置应变和应变是由变量的点位置决定的。当单位尺寸足够小时时，应变中各点的单位趋于相等，单位变形均匀，因此应变往往成为应变的主要部分。为了反映应变单元的状态，单元必须包括位移函数往往应变。3）在单元内，位移函数必须包括刚体位移。在正常情况下，该单元为一个变形位移和位移的刚体位移，包括两部分。变形位移与物体形状和体积的变化有关，从而产生应变；刚体位移改变物体的位置，不改变物体的形状和体积，即刚体位移不变形位移。对象的空间位移包括三平移和三个旋转位移，共有六个刚体位移。由于某一单元涉及到另一个单元，其他单元做刚体位移变形时会发生驱动单元，从而，模拟一个单元的实际位移，假设单元位移函数必须包含刚体位移。4）相邻单元的公共边界必须协调位移函数。对于一般单位的协调是指相邻节点在公共节点上有相同的位移，但也有相同的位移沿该单元的边缘，也就是说，以确保该单元不发生从开裂和入侵彼此重叠。要做到这一点就要求公共边界函数只能由公共节点函数值确定.。一般单元和协调，以确保相邻小区边界位移的连续性。然而，相邻单元之间的板和壳，也需要一个连续的一阶导数的位移，只有这样，以保证结构的应变能是有界的。总的来说，协调是指公众对相邻单位之间的边界满足连续性条件。第三条，也叫完整条件，满足条件的是完整的单位是完整的单位；第4条是协调要求，满足协调单位协调单位；否则称为协调单位。收敛的完整性要求是必要的，所有四个满足，构成收敛的充分必要条件。在实际应用中，要使选定的位移函数都满足完备性和协调性的要求，在某些情况下很难放松协调的要求。需要指出的是，有时协调单位比其相应的协调单位

有限元法求解问题的基本步骤

将结构离散化结构离散化结构离散化结构离散化到整体结构离散化，将其划分为若干个单元，通过节点之间相互连接的单元；2。每个单元的刚度矩阵和单元刚度矩阵和单元刚度矩阵和单元刚度矩阵（3）集成总刚度矩阵集成总刚度矩阵集成整体刚度矩阵集成总刚度矩阵和写出一般平衡方程并写出总体平衡方程并写出总体平衡方程和编写一个通用公式4。介绍支撑条件，每个节点位移5。计算单元内的应力应变，得到各单元的应力应变，应力应变的单元和各单元的应力应变。

有限元法的基本思想和步骤，可以概括为：（1）建立积分方程，根据变分原理和津贴的权函数正交化方程原理，及微分方程的积分表达式的建立相当于初边值问题，这是有限元法出发点。单元（2）面积细分，根据解面积的形状和物理特性的实际问题，将截面积划分为若干个相互连接、重叠的单元。区域单元划分为编制有限元方法，这部分的工作量比较大，除了细胞和节点数和确定彼此之间的关系，也表示节点坐标，同时还需要列出自然边界和本质边

界节点个数和相应的边界值。（3）确定单元的基函数，根据单元和节点数的近似解求精度，选择满足一定插值条件的基函数插值函数为单位。基函数是在有限元法中选择的单元，由于每个单元的几何有一个规则，在选择基函数时可以遵循一定的规则。（4）单位将分析：针对每个单元与单元基函数的函数近似表达的线性组合；然后近似函数代入积分方程和单位面积积分，可以得到与待定系数（即细胞参数值）的代数方程组中的每个节点，被称为有限元方程。（5）整体综合：在有限元方程中，根据有限元方程中所有元素的面积，根据一定的积累原理，形成一般的有限元方程。（6）边界条件的处理：一般边界条件有三种形式，分为本质边界条件（Dirichlet边界条件）、自然边界条件（黎曼边界条件）和混合边界条件（柯西边界条件）。在自然边界条件下的积分表达式中，可以自动满足。对于本质边界条件和混合边界条件，应采用一定的方法来修正一般有限元方程的满足。求解有限元方程（7）：基于一般有限元方程的边界条件是固定的，都是封闭方程的未知量，并采用适当的数值计算方法，每个节点的功能值可以得到。

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