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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

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

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

principle

is

geometric

equation

and

displacement

boundary

condition

of

the

equivalent

integral

form of

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

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