SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Numerical Simulation Methods...

SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk

Numerical Simulation Methods

Prof. Dr.-Ing. Timon Rabczuk

• Gleichungslöser• Zeitintegrationsverfahren• Eigenwertprobleme und Lösungsstrategien• Netzfreie Methoden

Outline

• Eigenschaften von Matrizen• Direkte Gleichungslöser

• Iterative Gleichungslöser

Outline

• Cramer’s Regel• Pivoting• Gauss’sche Eliminationsverfahren• Gauss-Jordan Elimination

• Jacobi Iteration• Gauss-Seidel Iteration• Successive-over relaxation• Die Method der konjugierten Gradienten

Outline

Ankündigung

• Am Donnerstag den 5.11.2009 findet von 13:30 bis 15:00 Uhr anstatt der Vorlesung ein Rechnerseminar im Betonpool der Coudraystrasse 13d statt

Pivoting

Einfache Elimination versagt, wenn aii=0

• Full pivoting: Modifizieren der Reihen (Zeilen) und Spalten, so dass der Maximalwert auf die Diagonale verschoben wird.• Beim partial pivoting werden nur die Reihen vertauscht.• Beim scaled pivoting werden die entsprechenden (zu Beginn die erste Spalte) Spalten mit dem groessten Element der zugehoerigen Reihe skaliert -> Verringerung von Rundungsfehlern.

Gauss-Eliminationsverfahren

( , 1, 2,..., )

( 1, 2,..., )

ikij ij kj

iki i k

aa a a i j k k n

b b b i k k na

• Schritt 1: Pivoting• Schritt 2: Gauss-Elimination

• Schritt 3: Lösung nach x mit Rückwärtssubstitution

1 ( 1, 2,...,1)

i ij jj i

b a xx i n n

Gauss-Jordan Elimination

• Gauss-Jordan Elimination is eine Variation der Gauss-Elimination, bei der die Elemente oberhalb und unterhalb der Hauptdiagonalen von der Hauptdiagonalen eliminiert werden. Normaler Weise werden die Diagonalelement skaliert (A -> I), so dass sich die Lösung sofort aus b ergibt.

Matrix-Inversion • Gauss-Jordan Elimination can zur Berechnung der Inverse verwendet werden (durch Augmentierung von I zu A)

1| |A I I A

80 20 20 120 40 20 120 20 130 1

1 0 0 2 /125 1/100 1/ 2500 1 0 1/100 1/ 30 1/1500 0 1 1/ 250 1/150 7 / 750

Gauss-Jordan

Matrix-Inversion • Inverse Matrix Methode

A A x A b

Matrix-Determinante • Die Determinante kann durch Gauss-Elimination zu einer oberen und unteren Dreiecksmatrix durch

berechnet werden. Es sei darauf aufmerksam gemacht, dass einige Operationen den Wert der Determinante verändern:• Multiplikation einer Reihe mit einer Konstanten multipliziert die Determinante mit dieser Konstanten• Vertauschen zweier Reihen verändert das Vorzeichen der Determinante

LU-Faktorisierung

• Die Faktorisierung von A in L und U ist nicht eindeutig. Wenn allerdings L oder U gegeben ist kann Eindeutigkeit der Faktorisierung sichergestellt werden.• Die Faktorisierung, die auf Einheitsdiagonalelemente von L basiert, wird Doolitte Methode (von U Crout Methode) genannt.• L und U werden durch Gauss-Elimination erhalten.

Frontal Solvers • Frontal solvers are used for solving sparse linear systems• They are based on Gauss elimination avoiding large number of operations involving zero terms• usually build LU or LDU decomposition of a sparse matrix given as assembly of element matrices by assembling the matrix and eliminating the equations only on a subset of elements at a time. This subset is called front.• The entire sparse matrix is never created explicitly. Only the front is in the memory.

Probleme von Eliminationsverf. • Bei Gauss Elimination und Varianten sind Schwierigkeiten durch a) Rundungsfehler und b) schlecht-konditionierte Systeme zu erwarten.• Rundungsfehler treten auf wenn exakte Zahlen (infinite precision) durch ‘finite precision numbers’ approximiert werden.• Bei einem gut-konditionierten Problem treten kleine Aenderungen in der Loesung bei kleinen Änderungen in den Elementen der Systemmatrix auf.• Ein schlecht-konditioniertes Problem ist sensitiv bez. kleiner Änderungen der Elemente

Probleme von Eliminationsverf. • Beim scaled pivoting ist die einzige Abhilfe zur Verbesserung der Genauigkeit eines schlecht-konditionierten Problems die Erhöhung der ‘precision’.• Methoden zur Überprüfung der Konditionierung von A

Konditionszahl: Die Konditionszahl beschreibt die Sensitivitaet des Systems bezüglich kleiner Aenderungen.

Ankündigung

Iterative Methoden • Jacobi• Gauss-Seidel• Successive-over-Relaxation• Conjugate Gradient

Iterative Methoden • Iterative Loeser konvergieren schneller bei diagonal dominanten Matrizen.• Matrizen koennen durch vertauschen von Reihen verbessert werden.• Die Anzahl der Iterationen hängen ab von:

• Diagonalen Dominanz,• Iterationsmethode,• Startwert,• Konvergenzkriterium

Jacobi Iteration • Wähle Startwert x0

( )( 1) ( )

( ) ( )

( 1, 2,... )

( 1,2,... )

kk k i

i i ij jj

Rx x i n

R b a x i n

• Wenn |Δ x| < tol -> beende Iteration

Genauigkeit und Konvergenz • Iterative Methoden sind weniger anfällig fuer Rundungsfehler weil:

• Das System ist diagonal dominant• Das System ist sparse• Jede Iteration ist unabhängig von den Rundungsfehlern der vorherigen Iteration

• Genauigkeit: relative Fehler = absoluter Fehler / exakte Loesung• Konvergenz/Abbruchkriterien

maxix tol

Gauss Seidel • Erfordert diagonale Dominanz zur Sicherung von Konvergenz• Konvergiert schneller als Jacobi-Iteration

( )( 1) ( )

1( ) ( 1) ( )

( 1,2,... )

kk k i

i nk k k

i i ij j ij jj j

Rx x i na

R b a x a x i n

• Anmerkung 1: Es werden nur bereits berechnete Werte von zur Berechnung von benötigt

•Anmerkung 2: Der Speicherplatzbedarf ist niedriger als bei der Jacobi-Iteration

( 1)kjx

( 1)kix

Successive-Over-Relaxation (SOR) • Vorteil: Schnellere Konvergenz

( )( 1) ( )

1( ) ( 1) ( )

( 1,2,... )

kk k i

i nk k k

i i ij j ij jj j

Rx x i na

R b a x a x i n

11 ( 2 )1

Gauss Seidelover relaxed method divergesunder relaxed

• Under-relaxation, wenn Gauss-Seidel ‘overshoots” (nicht-lineare Probleme)• Problem: Wahl von ω

Conjugate Gradient (CG) • meist benutzter iterativer Löser für grosse Systeme (sparse matrices)• Voraussetzung: A ist positive definit• Quadratische Form

CG • Example

CG • Start:

Nicht-lineare Probleme • Geometrische nicht-linear• physikalisch nicht-linear

Nicht-lineare Probleme • Newton-Raphson

Nicht-lineare Probleme • Modifiziertes Newton-Raphson

Nicht-lineare Probleme • Verzweigungspunkte (bifurcation points)• Limit points• Turning points

Nicht-lineare Probleme • Verzweigungspunkte (bifurcation points)• Durchschlagspunkte (Limit points)• Umkehrpunkte (Turning points)

Nicht-lineare Probleme •Versagenspunkte (Failure points)

Nicht-lineare Probleme • Load control

Nicht-lineare Probleme • Displacement control

Nicht-lineare Probleme • Arc-length control (Bogenlängenverfahren)

Ankündigung

Motivation

Ankündigung

• Am Dienstag den 17.11.2009 findet von 15:15 bis 16:45 Uhr anstatt der Vorlesung ein Rechnerseminar im Betonpool der Coudraystrasse 13d statt

Motivation

Time Integration

Forward Euler

Time Integration

Forward Euler

Time Integration

Backward Euler

Time Integration

One-step-theta

Time Integration

Newmark

Time Integration

Newmark

Time Integration

A time integration schemes calculates an orbit of the ODE. The time integration scheme is said to be stable if it evolves like the true solution and converges to an equilibrium. In general, a time integration scheme does not evolve towards the equilibrium for anarbitrary step size. The step size must obey a condition, i.e. it has to be smaller than a certain critical size to tend towards the equilibrium. Such schemes are called conditionally stable.

0mx cx kx 0

0(0)(0)

tx xx v

Time Integration

0mx cx kx 0

0(0)(0)

tx xx v

There are schemes which are linearly stable for any step size. If a time integration scheme tends towards the equilibrium in several steps, but each step arbitrarily large, it is called A-stable. If it even tends towards the equilibrium in a single step for any step size, then it is L-stable.

Time Integration

Linear stability analysis

Lecture notes

www.uni-weimar.de/cms/bauing/forschung/institute/ism/lehre/xfem-mfm.html

• Applications of Meshfree Methods• Partition of Unity• Completeness/consistency, stability, convergence, continuity• Meshfree shape functions and kernel functions and their relation• Specific meshfree methods (SPH, corrected SPH forms, EFG,

RKPM, hp-clouds, PUFEM): methods with intrinsic basis vs. methods with extrinsic basis

• Spatial integration in Meshfree Methods (nodal integration, stress-point integration, Gauss quadrature)

Meshfree Methods

• Meshfree methods are well suited for curve fitting• Meshfree methods are well suited for problems with large

deformations (high velocity impacts, solids under explosive loading, free surface flow)

• Meshfree methods are well suited for problems with localization (fracture, fragmentation, cracks, shear bands eventually with high curvature)

For what applications are meshfree methods useful?

Motivation

Idelsohn et al. 2004

Wang XS 2005http://web.njit.edu/~xwang

Shuttle crash, 2003 Landslide, Colorado

Taiwan earthquake, 2003 Fragmentation of concrete

Motivation

Concrete under explosive loading

Perforation of concrete under explosive loading

Experimental Results

Ockert 1997

MotivationFinite elements have difficulties for problems involving weak and strong discontinuities (material interfaces, cracks)

de Borst et al., 2004

• Advantages:

• No need for mesh generation• Higher order continuity• Often better convergence rate• Can handle easily large

deformations• Incorporation of h-adaptivity is• easy• No mesh alignment sensitivity

Drawbacks:

• Computational expensive• Difficulties in imposing

essential boundary conditions• Instabilities

Idelsohn et al. 2004

)(XX JSJ

Central particleNeighbor particle

Domain of influence (support)

Meshfree approximation

FE Meshfree

)()( XuXX JSJ

iJi uu

1)( SJ

Partition of unity

Linear FEM

IJIJ )(X

1)( SJ

Partition of unity

Quadratic FEM

IJIJ )(X

)1()1()()1(5.0)()1(5.0)(

rrrNrrrNrrrN

Partition of unity

1)( SJ

Partition of unityThe “Kronecker-delta” property is not fulfilledin meshfree methods. This causes difficulties in imposingDirichlet BCs.

IJIJ )(X

Partition of unity

IIh uu )(x

Completeness

ZZYYXX JSJ

)()()(

0)(1)( 0

Completeness is expressed in terms of the order of the polynomial which must be represented exactly. Completeness is often referred to reproducing conditions. An approximation is called complete of order n, if the approximation is able to reproduce a polynomial of order n exactly.

Completeness is important for the convergence of a discretization.

Completeness

1)(0)(

0)(1)(

0)(0)(

The derivative reproducing conditions are also important for several meshfree methods. In two dimensions, the derivative reproducing conditions for a linear field are

ijJjSJ

)(,0)( ,, XX

Completeness and conservation

An approximation that is of zeroth-order completeness guarantees gallilean invariance.

An approximation that is of zeroth-order completeness guarantees linear momentum. Conservation of linear momentum requires that the rate of change of linear momentum due to internal forces is zero. Thus, in the absence of external forces and body forces, conservation of linear momentum requires that

JJDtD vmvm

JJJ )()( XσXvm

JJDtD vmvm

JJJ )()( XσXvm

0)()( SJ

JJ wXσXvm

This requires

J X 1)(

Completeness and conservationAn approximation that is linear complete guarantees angular momentum. Conservation of angular momentum requires that any change is exclusively due to external forces. We will show that the change in angular momentum in the absence of external forces vanishes. The time rate of change in angular momentum can be expressed as

SJJJJJJ

SJJJJDt

D vvxvmxvm

ImjISI

JmijkSJ

JJJ xwσDtD

)()()( XXexvm

ImjISI

JmijkSJ

JJJ xwσDtD

)()()( XXexvm

)()()(

wσwσx

SIImjijm

SImjmjijkImj

SIJmijk

ImjISI

Compl., stability and convergenceA method is convergent if it is consistent and stable, Lax-Richtmeyr. According to Strikwerda (1989), a difference scheme Lu=f (L is the differential operator, Lh the corresponding difference operator) is consistent of order k for any smooth function v if:

kh ChvLLv | |max

In Galerkin methods, completeness takes the role of consistency. Stability ensures that a small defect stays small.

A method is convergent of order k (k>0) if

Chu)u(x | |max

Continuity

A method is considered to be n-th order continuous (Cn) if their shape functions are n times continuous differentiable.

Meshfree methods

Linear meshfree

Quadratic meshfree

Kernel function

Weighting/kernel/window functions:

|| Ixxr hrz /

103861),(

Cuartic B-Spline

Kernel function

Requirements usually imposed on the kernel functions:

)(),(lim0

max0),( rrhrW

),(),( hWhW IJJI XXXX

),(),( 00 hWhW IJJI XXXX

max/with)(ofinstead)( rrzrWzW

Kernel functionExtension of the kernel function into higher order dimensions:Rectangular support:

312111)( XWXWXWW DDDX

Circular support:

||XX ||)( 1DWW

21)2(4

1075.05.11

3/12)7/(1013/2

Kernel function

The cubic B-Spline

Kernel function

The cubic B-Spline WJ(X) The derivative of the Cubic B-spline

Kernel function

Lagrangian and Eulerian kernels:

),(),( 0

hWWhWW

Eulerian kernels are usually applied for large deformations. Eulerian kernels show a so-called tensile instability, meaning methods based on Eulerian kernels become instable when tensile stresses occur. Methods based on Eulerian kernels are generally not well-suited to model crack initiation since such methods are usually not capable of capturing the onset of fracture properly. Therefore, we recommend the use of Lagrangian kernels. When the deformations are too large, then the Lagrangian kernels gets instable when the domain of influence in the current configuration is extremely distorted.

Hyperelastic material lawwith strain softening

Instabilities due to (Belytschko et al. 2003):•Rank deficiency•Tensile instability (Swegle et al. 1993)•Material instability

Lagrangian and Eulerian kernels

SPH method by Lucy and Monaghan [1977]

YJh du,hWu

)()( YY-XX

Domain of influence

Meshfree Methods

dY,hWJ Y-X

XdYY,hWJ

)( Y-X

YYu )(

thatimplies11)(

dY,hWJ Y-X

XdYX,hWJ

)( Y-X

SPH method by Lucy and Monaghan [1977]Central particleNeighbor particle

Domain of influence

Meshfree Methods

XdYY,hWJ

)( Y-X

XdYX,hWJ

)( Y-X

Subtraction of

dYYX,hWJ Y-X

If linear consistency is fulfilled, above is guaranteed by the symmetry of the kernel

)()(, XX JSJ

Jh tutu

)()(, 00 XX JSJ

Jh tutu

00,)( JJJ VhW XXX

Domain of influence

Meshfree Methods

),()()()(, 0hWWWVtutu JJJJSJ

Jh X-XXXX

Meshfree Methods

Jh X-XXXX

Meshfree Methods

Jh X-XXXX

Meshfree Methods

Different ways to discretize a body

)()(, XX JSJ

Jh tutu

)()(, 00 XX JSJ

Jh tutu

00,)( JJJ VhW XXX

Symmetrization

)()(, 00 XX JSJ

JIh uutu

Domain of influence

Meshfree Methods

Shepard functions

Meshfree methods

)()(, XX S

)()(, 00 XX S

Krongauz-Belytschko correction

)()()(, 00 tuWtuSJ

ZWZWZWYWYWYWXWXWXW

Meshfree methods

• RKPM• EFG (MLS shape functions)• Hp-clouds • PUFEM• GFEM• Intrinsic and Extrinsic Enrichment

Outline

Elementfree Galerkin method (EFG)Conditioning of the A-matrix:

• The number of nodes n within a domain of influence has to be larger than the number M of basis monomials.• For linear complete basis polynomials, two of the three nodes have to point in different spatial directions.

A singularA regular

)(0 XX JSJ

Derivatives of the approximation

Elementfree Galerkin (EFG) method

)()()(

)()()()(

)()()()()(

XBXAXp

X-XXpXAXp

X-XXpXAXpX

)()()()( JT

JX-XXpXpXA

)()()( JSJ

J W X-XXpXB

)()()()()(

)()()()(

BXAXpX-XXpAXp

X-XXpXApX

)()()()( JT

JX-XXpXpXA

)()()( JSJ

J W X-XXpXB

XAAXAA

)()()( JSJ

J W X-XXpXB

)()()()()( 1JJ

h Wuu X-XXpXAXpX

Fast computation of the derivatives

)()()()()( 1JJ

TJ W X-XXpXAXpX

)()()()( JJJ W X-XXpXgX T)()()( XpXgXA

T)()()()()( XpXgXAXgXA

)()()()()( 1 XgXAXpXAXg T

Second derivatives

JJJJii

)()()(

)()()()()(

)()()()()()(

)()()()()(

XpAXpAXp

XpXAX-XXpAXp

X-XXpXAX-XXpAXp2

X-XXpXAXpX

Enrichment in EFG

Examples

22),( yxyxf

Examples

MLS SPH

Examples

Partial derivatives in x-direction

Examples

0.005% 0.2%

MLS SPH

Examples

)sin(),( 22 yxyxf

)sin()0,( 2xxf

Examples

)cos(2),( 22 yxxyxf x

Examples

MLS SPH

Examples

SPH-symm

Examples

Uniform particle distribution

Examples

SPH (approximation itself)

Examples

SPH –symm.

Examples

Hp-clouds

Method with extrinsic basis

The hp-cloud method is based on a so-called extrinsic enrichment. The second term is called the extrinsic basis and aJ are additional parameters introduced into the variational formulation and are used to increase the order of completeness (as in a p-refinement sense of finite elements).

Partition of Unity Finite Element Method (PUFEM)

The PUFEM method was developed almost simultaneously as the hp-cloud method and uses Shepard functions as shape functions. It was originally applied for the Helmholtz equation in 1D where the analytical solution was introduced in the basis p.

SJ SJJJJ

h auu1

)()(ˆ)( XpXXX K

GFEMGeneralized Finite Element Method (GFEM)

In the GFEM approximation, different partitions of unity are used (for the usual part and the extrinsic basis. The extrinsic basis is often called “enrichment”.

h auu1

)()( XpXX K

Hp-clouds

otherwise0)58.0,42.0(5.0exp5.042)(

1)1(0)0(

)1,0(0)(

xxaxaaxb

xxbu xx

PU-MethodsExample

Analytical solution

25.0exp)( xaxxu

)()()( xbuxxu JSJ

Approximation

25.0exp)( xax

PU-Methods

Spatial integration

00 )()(

SJJ Vfdf

Nodal integration

The quadrature weights are usually associated with the “volume” of the particle

I-1 I+1I

III xxV

Spatial integration

00 )()(

SJJ Vfdf

Nodal integration

Delauny triangulation Voronoi cell

Spatial integration

Stress point integration

NodeStress point

NJ VVV

)( SPiI

SPiI Xuu

)( SPiI

SPiI Xvv

)( SPiI

Spatial integration

Stress point integration

NodeStress point

NJ VfVfdf )()()( XXX

)( SPiI

SPiI Xuu

)( SPiI

SPiI Xvv

)( SPiI

Cell integration

Spatial integration

Cell integration:

),(det),(

),(det),()(1

ddJfdf

Spatial integration

In FE Gauss quadrature, 2nq-1 Gauss points are necessary to reproduce a polynomial of n-th order exactlySince meshfree shape functions are often not polynomials- e.g. MLS shape functions- exact integration of the weak form is difficult to impossible. Usually, a higher number of Gauss points are used to decrease integration errors.

Cell integration:

),(det),(

),(det),()(1

ddJfdf

Spatial integration

Since meshfree shape functions are often not polynomials- e.g. MLS shape functions- exact integration of the weak form is difficult to impossible. Usually, a higher number of Gauss points are used to decrease integration errors.Estimate for the number of Gauss points per background cell in 2D:

cellpernodesofnumber,2 mmnq

Nodal integration: 0

0 )()(0

J Vfdf

Cell integration:

),(det),(

),(det),()(1

ddJfdf

Spatial integration

Stress point integration:

NJJ VfVfdf )()()(

00 XXX

Spatial integrationIntegration over supports

Integration over supports is often used for methods that are based on local weak (the Meshless Petrov Galerkin (MLPG) method is probably the most popular local method).

Meshfree methodsShape functions:• SPH shape functions• SPH corrected derivatives shape functions• Shepard functions (=zero-order complete MLS shape functions)• MLS shape functions• RKPM shape functions (that are very similar to the MLS shape functions)

Integration techniques:• Nodal integration• Stress point integration• Gauss quadrature

Methods:• collocation methods• Bubnov Galerkin methods • Petrov Galerkin methods

)()( iIiiI xxx )()( iIiI xx )()( iIiI xx

Meshfree methodsMethods with intrinsic basis• SPH and corrected SPH versions• RKPM• EFG• MLPG

Integrationstrong form, collocation

weak form, nodal/cell integrationweak form, nodal/SP/cell int.local weak form, integration over support

Methods with extrinsic basis

• PUFEM• hp-clouds• GFEM• XFEM

Examples

4)~2(36~6

22 DyxxL

IEqyux

22 )3(

4)~54(~3~6

xxLxDxLyIE

strainplanefor)1/(~;stressplanefor~ 2 EEEEstrainplanefor)1/(~

stressplanefor~

2Lanal

Lanalh

Examples

46/),(

0),(12/

yxLqyx

energyanal

energyanalh

energyerr

d)()(Tenergy

uE:CuEu

Examples

ExamplesHole in the plate-problem

Put in formula!!!!!!

SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Numerical Simulation Methods...

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