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 compod of some simple shapes of unit, unit connection through the node, and under a certain load.
Finite element analysis: is the u of mathematical approximation method for real physical systems (geometry and loading conditions were simulated. And by using simple and interacting elements, namely unit, can u 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 rearch object, consideri
ng the deformation bad 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 u 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 i
s 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 bad on the test data, sometimes, to simulate the nonlinear material properties available mathematical model though the models always have their limitations. More important material nonlinear problems in engineering practice are: nonlinear elastic (including piecewi linear elastic, elastic-plastic and viscoplastic, creep, etc.
2) geometric nonlinear geometric nonlinear problems are caud due to the nonlinear relationship between displacement. When the object the displacement is larger, the strain and displacement relationship is nonlinear relationship. Rearch 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 aling, 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 asmbly, 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 the two or three kinds of nonlinear problems.
Finite element theoretical basis
Finite element method is bad 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, lect some appropriate nodes as solving the interpolation function, the differential equation of the variables in the rewritten by the variable or its derivative lected interpolation node value and the function of linear expression, with the aid of variational principle or weighted residual method, the dis
crete 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 ek 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.
