Mesh Free Methods - Faculteit Wiskunde en Informatica · Seminar 21-01-2004, Scientific Computing Group Contents • Limitations of traditional methods • General procedure of mesh

Seminar 21-01-2004, Scientific Computing Group

Mesh Free Methodsfor Beams and Plates

Jan Kroot21 January, 2004


Contents

• Limitations of traditional methods

• General procedure of mesh free methods

• Construction of shape functions

• Mechanics of solids and structures

• Examples: Beams and plates


Limitations of traditional methods

• For an engineer the concern is more the manpower time,

and less the computer time

• Large deformations can deteriorate the accuracy

because of element distortions

• Breakage and stress calculations require more flexibility

These problems made people start searching for mesh free methods naturally


FEM and Mesh Free Method procedures

GEOMETRY GENERATIONFEM MFree

ELEMENT MESH GENERATION NODAL MESH GENERATION

SHAPE FUNCTION CREATIONBASED ON ELEMENT PREDEFINED

SHAPE FUNCTION CREATIONBASED ON NODES IN LOCAL DOMAIN

SYSTEM EQUATION FOR ELEMENTS SYSTEM EQUATION FOR NODES

GLOBAL MATRIX ASSEMBLY

SOLVE SYSTEM


Domain representation

• Define the problem domain

• Choose a set of nodes to present the problem domain and the boundary

• In general, the nodal distribution is not uniform

• The density of the nodes depends on the accuracy requirement, and on the

gradient of the displacement


Displacement interpolation

The field variable u at any point x within the problem domain is interpolated using the

displacements at its nodes within the support domain of x.

Mathematically,

i

n

ii uu )()(

1

xx ∑=

= φ


Shape function construction

• Arbitrary nodal distribution

• Stability

• Consistency

• Compact support

• Efficiency

• Delta function property

• Compatibility


Methods in general

• Finite integral representation methods (SPH, RKPM)

• Finite series representation methods (MLS, PIM, FEM)

• Finite difference representation methods (FDM)

∫ −= 2

1

)()()(x

xdxWfxf ξξξ

...)()()( 22110 +++= xpaxpaaxf

...))((''!2

1))((')()( 2000 +−+−+= axxfaxxfxfxf


Moving Least Squares Approximation (MLS)

Use polynomial basis pT(x) = 1, x, y, z, xy, yz, xz, x2, y2, z2,…, xm, ym, zm,

to approximate the field function:

)()()()(),(1

xaxpxxxx iT

j

m

jiji

h apu ==∑=

Construct a functional of weighted residuals:

2

1

)](),([)( iih

n

ii uuWJ xxxxx −−=∑

=


MLS

0=∂∂aJ

Minimizing the weighted residuals requires

which yields:

So,

i

n

iiii

n

ii

Tiii uWW ∑∑

==

−=−11

)()()()()()( xpxxxaxpxpxx

sUxBxaxA )()()( =⇒

( )∑∑= =

−=n

i

m

jijij

h upu1 1

1 )()()()( xBxAxx

∑=

=⇒n

iii

h uu1

)()( xx φ


Remarks MLS

• Kronecker delta function is not satisfied

• Capable of producing approximation with desired order of consistency

• Use of weight functions: Nodes far from x have smaller weights

• Field function is continuous and smooth in entire domain

• Behaviour at end of support domain is smooth: Compatible


Point Interpolation Method (PIM)

Use polynomial basis pT(x) = 1, x, y, z, xy, yz, xz, x2, y2, z2,…, xn, yn, zn,

to approximate the field function:

)()()()(),(1

iiT

ij

n

jiji

h apu xaxpxxxx ==∑=

∑=

=⇒n

iii

h uu1

)()( xx φ

PaU =⇒ s

sThu UPxpx 1)()( −=

So,


Properties of PIM shape functions

The shape functions

• are linearly independent

• possess Kronecker delta function property

• are partitions of unity

• have derivatives of simple polynomial form

• are constructed without weight functions

• are not compatible


Comparison MLS and PIM

NoYesCompatibilty

YesNoDelta function property

ConstantFunctionInterpolation coefficients

m = nm << n# basis functions m

# nodes n

PolynomialPolynomialBasis function

PIMMLS


Mechanics of solids and structures

=

+

∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂

..

..

..

0///00/0/0/0//000/

w

v

u

bbb

xyzxzyyzx

z

y

x

xy

xz

yz

zz

yy

xx

ρ

ρρ

σσσσσσ

..ubσL ρ=+T

∂∂∂∂∂∂∂∂∂∂∂∂∂∂

∂∂∂∂

=

wvu

xyxzyzz

yx

xy

xz

yz

zz

yy

xx

0///0///0/000/000/

εεεεεε

Luε =

−

−

−=

xy

xz

yz

zz

yy

xx

xy

xz

yz

zz

yy

xx

cc

cc

ccccccccccc

εεεεεε

σσσσσσ

200000

02

0000

002

000

000000000

1211

1211

1211

111212

121112

121211

Gcc =−2

1211

)1)(21()1(

11 ννν+−

−= Ec

)1(2 ν+= EG

)1)(21(12 ννν

+−= Ec

cεσ =


Simplifications

1-D: Beams

2-D: Plates

ybxvEI =

∂∂

4

4

zbwhyw

yxw

xwEh =+

∂∂+

∂∂∂+

∂∂

−

..

4

4

22

4

4

4

2

3

2)1(12

ρν


Formation of system equations

Four principles of establishing discretized equations

• Variational methods

• Residual methods

• Taylor series

• Control of conservation laws of each volume element

Solids, structures

Fluid flow, heat transfer


Derivation of weak form

Variational method:

– Compute Lagrangian functional: L = T – Es + Wf

– Use Hamilton’s principle:

Weighted residual method:

– The PDE for solid mechanics in functional form:

– Compute the weighted integral:

02

1

=∫ dtLt

tδ

0=Ω+Γ−Ω−Ω⇒ ∫∫∫∫ΩΓΩΩ

dddd TTTT

t

uutubuσε !!ρδδδδ

( ) 0=Ω−+∫Ω

dTT ubσLW !!ρ

0=−+ ubσL !!ρT


Element Free Galerkin Method (EFG)

• MLS for construction of shape functions

• Galerkin weak form to develop discretized system equations

• Cells of background mesh required to carry out integrations in stiffness matrix

0)( =Γ−Γ−+Γ−Ω−Ω⇒ ∫∫∫∫∫ΓΓΓΩΩ

ddddduut

TTTT λuuuλtubuσε δδδδδ

0=+bσLT in Ω,

uu = on Γu , on Γttnσ =


Meshless Local Petrov-Galerkin Method (MLPG)• MLS for construction of shape functions

• Weak form for a node, based on local weighted residual method

• Choose test and trial functions independently (Petrov-Galerkin)

• Only local background mesh to carry out integrations in stiffness matrix

0=+bσLT in Ω,

uu = on Γu , on Γttnσ =

0)()(,

, =Γ−−Ω+⇒ ∫∫ΓΩ

dWuudWb QiiQijij

QuQ

ασ


Point Interpolation Methods (PIM)

• Formulations in both global Galerkin form and local Petrov-Galerkin form

• Weak forms similar to EFG and MLPG, but different shape functions

• Shape functions of polynomial form: Analytical integration possible (truly MFree)


Solving procedure

Usual ways to solve a matrix equation.

Matrix is sparse if support domains are small and if the nodal density does not vary

too drastically.


Examples: Static analysis of thin elastic beams

1. Simply-Simply supported beams under various loads

Method: PIM with local Petrov-Galerkin form (LPIM)

21 irregularly distributed nodes


Examples: Static analysis of thin elastic beams

1. Beams under uniformly distributed load with different BC

Method: LPIM

21 irregularly distributed nodes


Examples: Free-Vibration analysis of thin elastic beams

1. Eigenvalue equation: natural frequencies and free vibration modes

Method: LPIM, 21 irregularly distributed nodes


Examples: Static analysis of thin square elastic plates

1. SSSS supported plate under uniform pressure

Method: MLPG, 81 nodes


Conclusions

• MFree compared with FEM:

• MFree shows good accuracy in some test problems

• The ideal MFree method has not been found yet

Element mesh(local) background meshIntegration

Element meshSupport domainInterpolation

YesSome methodsDelta function property

ElementsNodesDomain

FEMMFree

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Mesh Free Methods - Faculteit Wiskunde en Informatica · Seminar 21-01-2004, Scientific Computing Group Contents • Limitations of traditional methods • General procedure of mesh