mesh free methods - tu/e · the mesh free methods developed so far are not really ideal. jj j n i...

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/ department of mathematics and computer scienceJJ J N I II 1/31JJ J N I II 1/31

Mesh Free MethodsIntroduction

Nico van der Aa

19th November 2003

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Contents

• What is a Mesh Free Method?

– Mesh

– Free

– Method

• Why do we like to have Mesh Free Methods?

• How can we use Mesh Free Methods?

– Node generation

– Shape functions

• Conclusions

• Questions

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What is a mesh?

A mesh is a net that is formed by connecting nodes in a predefined manner.

Other words are grid, volumes, cells or elements.

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What is mesh free?

Mesh Free Methods use a set of nodes scattered within the problem domainas well as sets of nodes scattered on the boundaries of the domain to repre-sent (not discretize) the problem domain and its boundaries.

No mesh implies no information on the relationship between the nodes isrequired.

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When is a method a Mesh Free Method?

• minimum requirement:A predefined mesh is not necessary (at least in field variable interpola-tion).

• ideal requirement:No mesh is necessary at all throughout the process of solving the prob-lem of a given arbitrary geometry governed by partial differential systemequations subject to all kinds of boundary conditions.

The Mesh Free Methods developed so far are not really ideal.

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Why the need for Mesh Free Methods?

• For an engineer the concern is more the manpower time, and less thecomputer time,

• Large deformations can deteriorate the accuracy because of elementdistortions,

• Specific physical problems, such as breakage of a solid and stress calcu-lations, require more flexibility.

Mesh Free Methods are answers to the problems of the Finite ElementsMethod!

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How do Mesh Free Methods work?

Step 1: Domain representation

• The problem domain is defined.

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How do Mesh Free Methods work?

Step 1: Domain representation

• The problem domain is defined.

• A set of nodes is chosen to represent the problem domain and its bound-ary.

• In general, the density of the nodes is not uniform.

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Support Domain

• DefinitionThe support domain for a point x is a sphere of a certain radius thatrelates to the nodal spacing near the point x.

• ReasonDetermines the number of nodes to be used to approximate the functionvalue at x.

• RestrictionThe nodal density does not vary drastically in the problem domain.

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Step 2: Field interpolation

The field variable u at any pointxwithin the problem domain is interpolatedusing the values of this field at all the nodes within the support domain ofx. Mathematically,

u(x) =

n∑j=1

φi(x)ui

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Step 3: Formulation of system equations

• The equations of a mesh free method can be formulated using the shapefunctions and a strong or weak form system equation.

• Result in global system matrices for the entire problem domain.

• The procedures of forming system equations are slightly different fordifferent MFree methods.

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Step 4: Solving the global MFree equations

Numerical method depends on the kind of equations! Choices:

• Direct solver vs. iterative solver;

• Implicit method vs. explicit solver;

• Required accuracy;

• Speed of method;

• Stability of method;

• ...

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Features of a nodal generation

• The nodes must represent both problem domain and boundary;

• The nodes can be chosen arbitrary within reason;

• The node distribution can be uniform or not.

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Support domain vs Influence domain

An influence domain is the domain that a node exerts an influence upon. Itgoes with a node.

influence domain ↔ support domainnode ↔ pointuniform distribution of nodes ↔ nonregularly distributed nodes

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Properties of Mesh Free Shape Functions

• Partition of unity (compulsory condition)

n∑i=1

φi(x) = 1

• Linear field reproduction (preferable condition)

n∑i=1

φi(x)xi = x

• Kronecker delta function property (preferable condition)

φi(xj) =

1 i = j0 i 6= j

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Requirements of a good method of shape func-tion construction:

1. arbitrary nodal distribution

2. stability

3. consistency

4. compact support

5. efficiency

6. delta function property

7. compatibility

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Methods to construct shape functions:

• Finite integral representation methods:Smoothed particle hydrodynamics (SPH) method, ...

f (x) =

∫ x2

x1

f (ξ)W (x− ξ)dξ

• Finite series representation methods:Moving Least Squares (MLS) methods, Point Interpolation Methods(PIMs), ...

f (x) = a0 + a1p1(x) + a2p2(x) + ...

• Finite differential representation methods:

f (x) = f (x0) + f ′(x0)(x− a) +1

2!f ′′(x0)(x− a)2 + ...

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SPH method (finite integral method)

The integral representation of the field function u(x):

u(x) =

∫ ∞

−∞u(ξ)δ(x− ξ)dξ

In SPH:u(x) =

∫ ∞

−∞u(ξ)W (x− ξ, h)dξ

This integral representation is valid and converges when the weight functionsatisfies certain conditions:

1. W (x− ξ, h) > 0 over Ω;

2. W (x− ξ, h) = 0 outside Ω;

3.∫

Ω W (x− ξ, h)dξ = 1;

4. W is a monotonically decreasing function;

5. W (s, h) → δ(s) as h → 0.

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SPH: Discretised form

uh(x) =

n∑i=1

W (x− xi)ul∆Vl =

n∑i=1

Φi(x)ui

The shape function Φi are defined as:

Φi(x) = W (x− xi)∆Vl

Remarks:

• Kronecker delta function is not always satisfied (ul 6= uh(x));

• Compatible in field function approximation;

• Consistent in the interior of the domain, not on the boundary;

• Reproductivity in the interior of the domain, not on the boundary;

• The shape functions need to be scaled to satisfy the condition of unity.

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SPH: consistency

• C0 consistency is ensured by the condition of unity imposed on a weightfunction.

• C1 consistency is not immediately available on the boundary.

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SPH: most commonly used shape function

• Cubic spline weight function (W1):

• Quartic spline weight function (W2):

• Exponential weight function (W3);

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2Weight functions

d

W

W1W2W3

SCALING ISSTILL

NECESSARY

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MLS (Moving Least Squares)

uh(x) =

m∑j=0

pj(x)aj(x) = pT (x)a(x)

withpT (x) =

1, x, x2, ..., xm

pT (x, y) =

1, x, y, xy, x2, y2, ..., xm, ym

pT (x, y, z) =

1, x, y, z, xy, yz, xz, x2, y2, z2, ..., xm, ym, zm

Features:

• finite series representation;

• ensures the approximated field function is continuous and smooth inthe entire problem domain;

• capable of producing an approximation with the desired order of consis-tency.

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MLS: weighted residual

uh(x, xi) = pT (xi)a(x)

Weighted residual:

J =

n∑i=1

W (x− xi)[uh(x, xi)− u(xi)]

2

=

n∑i=1

W (x− xi)[pT (xi)a(x)− ui]

2

In MLS approximation, at an arbitrary point x, a(x) is chosen to minimizethe weighted residual.

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MLS: minimization condition requirement∂J

∂a= 0 ⇒ A(x)a(x) = B(x)U s

with

A(x) =∑n

i=1 Wi(x)p(xi)pT (xi) Wi(x) = W (x− xi)

B(x) = [B1, B2, ...,Bn] Bi = Wi(x)p(xi)

U s = u1, u2, ..., unT

Solutiona(x) = A−1(x)B(x)U s

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MLS: shape function

uh(x) =

n∑i=1

Φi(x)ui

with the shape functions

Φi(x) =

m∑j=0

pj(x)(A−1(x)B(x))ji = pT (x)A−1(x)Bi

Remarks about the MLS shape function:

• Kronecker delta function is not satisfied;

• possesses mth order consistency;

• is reproductive;

• scaling is not necessary to satisfy the unity requirement;

• compatible.

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PIM (Point Interpolation Method)

uh(x, xQ) =

n∑i=1

Bi(x)ai(x)∗=

n∑i=1

pi(x)ai(xQ) = pT (x)a(xQ)

Determination of a: enforce for every node i

ui = pT (xi)a, i = 1, ..., n

or in matrix natotation:

U s = P Qa ⇒ a = P −1Q U s

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PIM: shape function

uh(x) =

n∑i=1

Φi(x)ui = Φ(x)U s

The matrix of shape functions is defined as

Φ(x) = pT (x)P −1Q

Note that P −1Q does not have to exist!

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PIM: properties of the shape function

Shape functions

• have nth order consistency;

• are linearly independent;

• possesses the Kronecker delta function property;

• are partitions of unity;

• possess reproducing properties;

• use no weight functions in their construction;

• are not compatible.

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3. Conclusions

• Mesh Free Methods are a respons to the limitations of Finite ElementMethods.

• Mesh Free Methods do not use meshes, only nodes.

• The implementation of Mesh Free Methods differs from Finite ElementMethods only in the shape function construction and node generation.

• The ideal Mesh Free Method is not found yet.

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References

Mesh free methods:Moving beyond the finite elementmethodGui-Rong LiuCRC Press LLC 2003

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Questions ?

mesh free methods - tu/e · the mesh free methods developed so far are not really ideal. jj j n i...

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