Introduction to Finite Element...

Vincent Chiaruttini, Georges CailletaudVincent Chiaruttini, Georges [email protected]@onera.fr

Non Linear Computational Mechanics Athens MP06/2012–Non Linear Computational Mechanics Athens MP06/2012–

Introduction to Finite Element computationsIntroduction to Finite Element computations

2 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

OutlineOutline

Continuous to discrete problemStrong to weak formulation

Galerkin method for approximate solution computation

Isoparametric finite elementsFinite element mesh

Geometrical element

Interpolation of displacements

FE method for linear elastic problemsVariational formulation with isoparametric elements

Elemental computations

Global problem

Linear solution process

FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Continuous mechanical problemContinuous mechanical problem

Solving PDE on a given time-space domain

Studied domain

Local equations

Boundary conditions

u = ud on @u

¾ ¢ n = F on @f

@u

@f

Unable to get exact solutionUnable to get exact solutionHow to obtain an approximated solution?How to obtain an approximated solution?

r¾ + f = 0

"(u) =1

2

¡ru+rT u

¢

¾ = K : "



Local equations

Admissible spaces for displacements

Variational formulation of the displacement problemFinding that verifies, for all

r¾ + f = 0¾ = K : "

" =1

2

¡ru+rT u

¢

Z

¾ : "¤ ¡ f:u¤d =Z

@F

F:u¤dS

u = ud on @u

¾ ¢ n = F on @f

Uad = fuju continuous and regular on ; u = ud on @ug

U0ad = fuju continuous and regular on ; u = 0 on @ug

u¤ 2 U0adu 2 Uad

, a(u; u¤) = L(u¤)



Potential energy definition

Minimization problemLax-Milgram theorm insures

Variational formulation = weak formulation on a stationary functional

Galerkin based methodsGalerkin based methodsApproximate minimization problemApproximate minimization problem

P (v) = 1

2

Z

"(v) : K : "(v)d¡Z

f:vd¡Z

@F

F:vdS

P (v) = 1

2a(v; v)¡L(v)

u 2 Uad veri¯es 8u¤ 2 U0ad; a(u; u¤) = L(u¤), u = argminv2UadP(v)


From Galerkin method to FE methodFrom Galerkin method to FE method

Galerkin methodFinding the solution of a variational formulation in an admissible subspace ofUsing polynomial base functions defined on the domain, for instance

The variational formulation produces a NxN linear system

The approximate solution is optimal in the sense that it constitutes the projection of the solution on the given subspace (ie the error is orthogonal to the given subspace)

Advantages/drawbacksEasy to define a regular base (polynomials)Optimal approximationNon-sparse operators, not so easy for complex geometries

Mesh and compact supportMesh and compact supportbase functions => FE methodbase functions => FE method

uD(x) 2 Uad Ák(x) 2 U0adv(x) = uD(x) +NX

k=1

®kÁk(x)

Uad

8h 2 f1; : : : ;Ng;NX

k=1

a(Ák; Áh)®k = L(Ák)


Some popular discrete methods for PDESome popular discrete methods for PDE

Finite differences

Strong formulation of PDE

Differential approximations

Regular grid

Finite volume

Integral approximations

Very popular for conservation laws

Structured or unstructured mesh

Finite elements

Variational formulation

Optimal solution for a given approximation space (Galerkin approach)

Free meshes, efficient a priori and a posteriori error estimation

Ti-1 Ti Ti+1

h

dTi/dx=(Ti+1-Ti-1)/(2h)

Flux


OutlineOutline




Geometrical element




Global problem



Isoparametric finite elementIsoparametric finite element

MotivationIndustrial need for a robust and efficient numerical tool to obtain accurate results in the simulation of mechanical problems with complex behaviour and geometries

Meshing processGiven a geometrical model (usually from CSG built using a CAD software)

235 parts45 parts

How to obtain a suitable discrete representationHow to obtain a suitable discrete representationfor FE computations?for FE computations?


Finite element meshFinite element mesh

A 2D example triangular elements–A continuous domain is approximated using a conform partition based onrectilinear triangular regions=> a triangular Mesh

To insure convergence of the solution process the elements size can be reducedIf the studied domain boundary is polygonal, it is possible to get

Interpolation functionsRepresent a scalar field on the domain using nodal values

= h

Ti

h

Ti

T1

T2

12

x 2 in element T

Can functions TCan functions Tii be less dependant of be less dependant of

the mesh?the mesh?x

T (x)

T3

3

Th(x) = N1(x) T1 +N2(x)T2 +N3(x)T3

Th(x)


Deformed and reference configurationsDeformed and reference configurations

Parametrization of an element positioned anywherein the studied space (linear triangular example)

Physical element Reference element

are base or shape functions defined on the reference domain for the geometrical representation of the element, usually linear:

»1

»2

x »

Ni(»)

X3

X1

X2

x = N1(»)X1 +N2(»)X2 +N3(»)X3

8<:N1(»1; »2) = 1¡ »1 ¡ »2N2(»1; »2) = »1N3(»1; »2) = »2

1(0; 0) 2(1; 0)

3(0; 1)


Deformed and reference configurationsDeformed and reference configurations

Parametrization of an element positioned anywherein the studied space (quadratic triangular example)

Physical element Reference element

Reference to physical transformation regularity ( nodes element)

The transformation between the reference and the physical configurations must be a bijection, the Jacobian must not reach the zero value.Misappropriate physical element geometry can produce such failureElement integration , if N is polynomial then J is also

»1

»2

x »

1 2 3X1

X2

X3

X4X5

X64

5

6

x =

nX

k=1

Nk(»)Xk J(») = detJ(»)

dV (x) = J(»)dV (»)

n

J(») =

·@xi@»j

¸=

"nX

k=1

@Nk@»j

(»)Xkj

#


Interpolation of displacementsInterpolation of displacements

Isoparametric elementThe same base functions are used for geometry and displacement interpolation

where are the vectorial nodal displacementsUkuh(x) =

nX

k=1

Nk(»)Uk


Operational computation of displacementsOperational computation of displacements

Local vector of displacement unknowns for element with nodes A vector of elementary degrees of freedoms is usually built, for a 2D problem by

where is the nodal displacement in direction i at node j Thus for any point inside the specified element of coordinate in the reference system, the displacement is obtained by

where is a matrix built using the base function, in 2D:

To insure correct convergence properties (ie representation of constant gradient and solid body displacement field), the following “partition of unity” property must be verified anywhere in the element:

U(j)i

x »

e

Ue =hU(1)1 ; U

(1)2 ; U

(2)1 ; U

(2)2 ; : : : ; U

(j)i ; : : :

iT

Ne(»)

N(») =

·N1(») 0 N2(») 0 : : : Nj(») 0 : : :0 N1(») 0 N2(») : : : 0 Nj(») : : :

¸

nX

k=1

Nk(») = 1

uh(x) = Ne(») ¢ Ue

n


Operational computation of displacement gradientsOperational computation of displacement gradients

Vectorial Voigt writing of second order tensors Symmetric gradients

Linear behaviour relationship as a matrix-vector product

Gradient computationPhysical system of coordinate gradient definitionTo link with the reference coordinate

with

using we getSymmetric gradient computation

That can be expressed as using the symmetric gradient operator

¾v = [A] "v

duh(x) = ruh(x) ¢ dx

dx = J(») ¢ d»H(») = [Hij ] =

"nX

k=0

@Nk@»j

(»)U(k)i

#duh(x) = H(») ¢ dx

ruh(x) = H(») ¢ J¡1(»)

"v =h"11 "22 "33

p2"12

p2"13

p2"23

iT

¾v =h¾11 ¾22 ¾33

p2¾12

p2¾13

p2¾23

iT

²vh(») = B ¢ Ue

²h[u(»)] =

1

2

³H(») ¢ J¡1(») +

¡H(») ¢ J¡1(»)

¢T´


OutlineOutline




Geometrical element




Global problem



Finite element method for linear elasticityFinite element method for linear elasticity

Global algorithm from the variational formulation in displacementContinuous problem

Finite element discrete problem

@u

@f

Uad = fuju continuous and regular on ; u = ud on @ug

8u¤ 2 U0ad;

Sf

Suh

ek

uh(x) = Nke(») ¢ UeOn each element ek

Uks = Ud(xks)

Verifying null value on Su nodes

Virtual displacement field u¤h(x) = Nke(») ¢ U¤e

Prescribed displacements on Su nodes

Uk¤s = 0

8U j¤;, 8W; [W ]T [K][U ] = [W ]T [F ]

Z

h

"vh ¢ [A] : "v¤h dh =Z

h

f:u¤hdh +Z

@Fh

F:u¤hdS

Z

¾ : "¤d =Z

f:u¤d +Z

@F

F:u¤dS


Computing elementary contributionsComputing elementary contributions

Domain formulation based on element by element contributions

Usually the two parts are separated in

Internal force contributions

External force contributions

[W ]T [F ] =X

fekg

Z

ek

f:u¤hdV +Z

¡eF

F:u¤hdS

[Fext] =X

fekg

Z

ek

f:u¤hdV +Z

¡eF

F:u¤hdS

[Fint] + [Fext] = [0]

[W ]T [K][U ] =X

fekg

Z

ek

"vh ¢ [A] ¢ "v¤h dV

[Fint] = ¡X

fekg

Z

ek

"vh ¢ [A] ¢ "v¤h dV



Elementary rigidity matricesUsing the FE gradient operators and the local vectors of unknowns we need to compute:

Even for polynomial base/shape function, the inverse of Jacobian used to compute the [B] symmetric gradient operator, make impossible to evaluate such elementary integral exactly for non-trivial meshes. Thus a Gauss integration process is used:

Such process requires the knowledge of some predefined integration points(usually called Gauss points) and the associated weight for the reference configuration of the element.The parametric representation of both the geometry and the displacement field of the element constitutes one of most important aspect to insure the generic aspect of the FE method: only a few patterns of integration scheme are required corresponding to the associated finite element in reference configuration.

Z

ek

"vh ¢ [A] ¢ "v¤h dV = [We]T

Z

ek

[B]T [A][B]dV [Ue] = [We][Ke][Ue]

»g

wg

Z

ek

f(»)dV (») ¼GX

g=1

wgf(»g)



Gauss integration1D example



Gauss integrationIn 3D



External forces

For the volume force a Gauss integration process is requiredFor the nodal force nodal value must be correctly calculated

[Fext] =X

fekg

Z

ek

[W ]T [N ][f ]dV +

Z

¡eF

[W ][N ]T [F ]dS



External forces



External forces

For the volume force a Gauss integration process is requiredFor the nodal force nodal value must be correctly calculated

[Fext] =X

fekg

Z

ek

f:u¤hdV +Z

¡eF

F :u¤hdS


Assembling the global problemAssembling the global problem

Local to global DOF indices (thermal problem for shake of simplicity)

1

2

322

11

33

66 88

1010

99

55

9944

77

E1E2

E3 E4

E5

E6E7

E8E9

E10

E3

Global DOF numbering Elementary DOF numbering

1 2 3 4 5 6 7 8 9 10

3

4

5

6

7

8

1

2

9

10

Binary trace operators

Assembling global matrix

[K] =X

fekg[¦e]

T [Ke][¦e]

Global matrix

[U ] = [¦e]T [Ue][Ue] = [¦e][U ]

Prescribed displacementsWhen assembling is processed, prescribed displacement unknowns can be replaced and eliminated for the linear system

band

wid

th


Solving efficiently a sparse linear systemSolving efficiently a sparse linear system

[K][U ] = [F ]Large linear system to be solved

sparse with n equations (n from 104 to 109)K is a symmetric, definite positive for a linear-elastic problemReduce bandwidth using an accurate numbering process (reduce computational cost) Optimization in memory requirement: sparse storage only keep non zero terms

Direct solversFactorization process with L a lower-triangular matrix

Solving (2 successive triangular systems solving)Interest: matrix storage, less memory consuming than required by inverse computation, triangular system solving is of the same complexity order than matrix vector productOptimized multifrontal, dissection, multithreated, solvers

Iterative solversKrylov type solvers (conjugate gradient, GMRes) need good preconditionnersDomain decomposition solvers

Split the structure in many subdomains (for supercomputers parallel computation)Efficient iterative strategy to achieve interface equilibrium condition(very good preconditionners)

[K ] = [L][L]T

[L][Y ] = [F ] then [L]T [U ] = [Y ]


Convergence proprietiesConvergence proprieties

Displacement convergencePolynomial approximations are used, on any element Ee, Taylor expansion gives:

Committed error on displacement using p order shape function and h characteristic size

Strain and stress convergenceFE error

Energy normDefined by

For order the FE method is convergent in energy:

and if the problem is regular enough

(x; x0) 2 E2e u(x) = u(x0) +ru(x0):(x¡ x0) + : : :+O(kx¡ x0kp+1)

k"[u](x)¡ "h[u](x)k = O(hp); x 2 Ee

k¾[u](x)¡ ¾h[u](x)k = O(hp); x 2 Ee

kx¡ x0k < h gives ku(x)¡ uh(x)k = O(hp+1); x 2 Ee

kvk2E =Z

"[v] : A : "[v]d

p ¸ 1 ku¡ uhkE ! 0 if h! 0

ku¡ uhkE · C hp kukE


Global FE solution process for a linear problemGlobal FE solution process for a linear problem

Approximate geometry build a mesh using a priori analysis→

Loop on elements

Compute the local variational formulation contribution

Loop on elements Gauss points

Compute Jacobian, gradient matrices, get behaviour, multiply matrices to get local rigidity contribution

Integrate the rigidity matrix and the local vector for external forces

Assemble local rigidity matrix and local external forces

Apply Dirichlet BC, MPC (linear relationship between unknowns)

Compute global external forces

Solve the linear system


ReferencesReferences

Bonnet M., Frangi A. (2006)Analyse des solides déformables par la méthode des éléments finis. Editions Ecole Polytechnique.Belytschko, T., Liu, W., and Moran, B. (2000).Nonlinear Finite Elements for Continua and Structures.Besson, J., Cailletaud, G., Chaboche, J.-L., and Forest, S. (2001).Mecanique non linéaire des matériaux. Hermes.–Ciarlet, P. and Lions, J. (1995).Handbook of Numerical Analysis : Finite Element Methods (P.1), Numerical Methods for Solids (P.2). North Holland.Dhatt, G. and Touzot, G. (1981).Une présentation de la méthode des élements finis. Maloine.Hughes, T. (1987).The finite element method: Linear static and dynamic finite element analysis. Prentice Hall –Inc.Simo, J. and Hughes, T. (1997).Computational Inelasticity. Springer Verlag.Zienkiewicz, O. and Taylor, R. (2000).The finite element method, Vol. I-III (Vol.1: The Basis, Vol.2: Solid Mechanics, Vol. 3: Fluid dynamics). Butterworth Heinemann.–

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