- Home
- Documents
*FOUNDMENTALS FOR FINITE ELEMENT METHOD CHAPTER1: The Finite Element Method A Practical Course The...*

prev

next

of 34

View

257Download

21

Embed Size (px)

The Finite Element MethodA Practical Course FOUNDMENTALS FOR FINITE ELEMENTMETHOD

CHAPTER1:

The Finite Element Method by G. R. Liu and S. S. Quek

CONTENTSSTRONG AND WEAK FORMS OF GOVERNING EQUATIONSHAMILTONS PRINCIPLEFEM PROCEDUREDomain discretizationDisplacement interpolationFormation of FE equation in local coordinate systemCoordinate transformationAssembly of FE equationsImposition of displacement constraintsSolving the FE equationsSTATIC ANALYSISEIGENVALUE ANALYSISTRANSIENT ANALYSIS (reading materials)REMARKS

The Finite Element Method by G. R. Liu and S. S. Quek

STRONG AND WEAK FORMS OF GOVERNING EQUATIONSSystem equations: strong form (PDE), difficult to solve.Weak (integral) form: requires weaker continuity on the dependent variables (e.g., u, v, w).Weak form is often preferred for obtaining an approximated solution. Formulation based on a weak form leads to a set of algebraic system equations FEM.

The Finite Element Method by G. R. Liu and S. S. Quek

HAMILTONS PRINCIPLEOf all the admissible time histories of displacement the most accurate solution makes the Lagrangian functional a minimum.

An admissible displacement must satisfy:The compatibility conditionsThe essential or the kinematic boundary conditionsThe conditions at initial (t1) and final time (t2)

The Finite Element Method by G. R. Liu and S. S. Quek

HAMILTONS PRINCIPLEMathematicallywhereL=T-P+Wf(Kinetic energy)(Potential energy)(Work done by external forces)Lagrangian functional

The Finite Element Method by G. R. Liu and S. S. Quek

FEM PROCEDUREStep 1: Domain discretizationStep 2: Displacement interpolationStep 3: Formation of FE equation in local coordinatesStep 4: Coordinate transformationStep 5: Assembly of FE equationsStep 6: Imposition of displacement constraintsStep 7: Solving the FE equations

The Finite Element Method by G. R. Liu and S. S. Quek

Step 1: Domain discretizationThe solid body is divided into Ne elements with proper connectivity compatibility.All the elements form the entire domain of the problem without any gap or overlapping compatibility.There can be different types of element with different number of nodes.The density of the mesh depends upon the accuracy requirement of the analysis.The mesh is usually not uniform, and a finer mesh is often used in the area where the displacement gradient is larger.

Triangular elements

Nodes

The Finite Element Method by G. R. Liu and S. S. Quek

Step 2: Displacement interpolationBases on local coordinate system, the displacement within element is interpolated using nodal displacements.nf: Degree of freedoms at a nodend: number of nodes in an element

fsy

fsx

A

3 (x3, y3)

(u3, v3)

2 (x2, y2)

(u2, v2)

1 (x1, y1)

(u1, v1)

y, v

x, u

The Finite Element Method by G. R. Liu and S. S. Quek

Step 2: Displacement interpolationN is a matrix of shape functionswhereShape function for each displacement component at a node

The Finite Element Method by G. R. Liu and S. S. Quek

Displacement interpolationConstructing shape functionsConsider constructing shape function for a single displacement componentApproximate in the formpT(x)={1, x, x2, x3, x4,..., xp} (1D)Basis function

The Finite Element Method by G. R. Liu and S. S. Quek

Pascal triangle of monomials: 2D

xy

x

x2

x3

x4

x5

y

y2

y3

y4

y5

x2y

x3y

x4y

x3y2

xy2

xy3

xy4

x2y3

x2y2

10 terms

Constant terms: 1

1

6 terms

Quadratic terms: 3

Cubic terms: 4

Quartic terms: 5

Quintic terms: 6

Linear terms: 2

21 terms

15 terms

3 terms

The Finite Element Method by G. R. Liu and S. S. Quek

Pascal pyramid of monomials : 3D

35 terms

20 terms

10 terms

y4

y3

y2

y

x4

x3

x2

x

xy

z

xz

yz

xy2

x2y

x2z

zy2

z2

xz2

yz2

xyz

z3

x3y

x3z

x2y2

x2z2

x2yz

xy3

zy3

z2y2

xy2z

xyz2

xz3

z4

z3y

Quartic terms: 15

Cubic terms: 10

Quadratic terms: 6

Linear terms: 3

1

Constant term: 1

4 terms

The Finite Element Method by G. R. Liu and S. S. Quek

Displacement interpolationEnforce approximation to be equal to the nodal displacements at the nodesdi = pT(xi) i = 1, 2, 3, ,nd orde=P where ,Moment matrix

The Finite Element Method by G. R. Liu and S. S. Quek

Displacement interpolationThe coefficients in can be found byTherefore, uh(x) = N( x) de

The Finite Element Method by G. R. Liu and S. S. Quek

Displacement interpolationSufficient requirements for FEM shape functions

1.(Delta function property)2.(Partition of unity property rigid body movement)3.(Linear field reproduction property)

The Finite Element Method by G. R. Liu and S. S. Quek

Step 3: Formation of FE equations in local coordinatesSince U= Nde Therefore,e = LU e = L N de= B deStrain matrixorwhere(Stiffness matrix)

The Finite Element Method by G. R. Liu and S. S. Quek

Step 3: Formation of FE equations in local coordinatesSince U= Nde orwhere(Mass matrix)

The Finite Element Method by G. R. Liu and S. S. Quek

Step 3: Formation of FE equations in local coordinates(Force vector)

The Finite Element Method by G. R. Liu and S. S. Quek

Step 3: Formation of FE equations in local coordinatesFE Equation(Hamiltons principle)

The Finite Element Method by G. R. Liu and S. S. Quek

Step 4: Coordinate transformation,,where(Local)(Global)

The Finite Element Method by G. R. Liu and S. S. Quek

Step 5: Assembly of FE equationsDirect assembly methodAdding up contributions made by elements sharing the node(Static)

The Finite Element Method by G. R. Liu and S. S. Quek

Step 6: Impose displacement constraintsNo constraints rigid body movement (meaningless for static analysis)Remove rows and columns corresponding to the degrees of freedom being constrainedK is semi-positive definite

The Finite Element Method by G. R. Liu and S. S. Quek

Step 7: Solve the FE equationsSolve the FE equation,

for the displacement at the nodes, D

The strain and stress can be retrieved by using e = LU and s = c e with the interpolation, U=Nd

The Finite Element Method by G. R. Liu and S. S. Quek

STATIC ANALYSISSolve KD=F for D

Gauss eliminationLU decompositionEtc.

The Finite Element Method by G. R. Liu and S. S. Quek

EIGENVALUE ANALYSIS(Homogeneous equation, F = 0)AssumeLet[ K - li M ] fi = 0(Eigenvector)(Roots of equation are the eigenvalues)

The Finite Element Method by G. R. Liu and S. S. Quek

EIGENVALUE ANALYSISMethods of solving eigenvalue equationJacobis methodGivens method and Householders methodThe bisection method (Sturm sequences)Inverse iterationQR methodSubspace iterationLanczos method

The Finite Element Method by G. R. Liu and S. S. Quek

TRANSIENT ANALYSISStructure systems are very often subjected to transient excitation. A transient excitation is a highly dynamic time dependent force exerted on the structure, such as earthquake, impact, and shocks. The discrete governing equation system usually requires a different solver from that of eigenvalue analysis. The widely used method is the so-called direct integration method.

The Finite Element Method by G. R. Liu and S. S. Quek

TRANSIENT ANALYSIS(reading material)The direct integration method is basically using the finite difference method for time stepping.There are mainly two types of direct integration method; one is implicit and the other is explicit.Implicit method (e.g. Newmarks method) is more efficient for relatively slow phenomenaExplicit method (e.g. central differencing method) is more efficient for very fast phenomena, such as impact and explosion.

The Finite Element Method by G. R. Liu and S. S. Quek

REMARKSIn FEM, the displacement field U is expressed by displacements at nodes using shape functions N defined over elements. The strain matrix B is the key in developing the stiffness matrix. To develop FE equations for different types of structure components, all that is needed to do is define the shape function and then establish the strain matrix B. The rest of the procedure is very much the same for all types of elements.

The Finite Element Method by G. R. Liu and S. S. Quek

Newmarks method (Implicit)Assume thatSubstitute intoTypicallyg = 0.5b = 0.25

The Finite Element Method by G. R. Liu and S. S. Quek

Newmarks method (Implicit)whereTherefore,

The Finite Element Method by G. R. Liu and S. S. Quek

Newmarks method (Implicit)Start with D0 and Obtain usingObtain usingObtain Dt and usingMarch forward in time

The Finite Element Method by G. R. Liu and S. S. Quek

Central difference method (explicit)(Lumped mass no need to solve matrix equation)

The Finite Element Method by G. R. Liu and S. S. Quek

Central difference method (explicit)