ss 2009numerische simulationsverfahren prof. dr.-ing. timon rabczuk numerical simulation methods...
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SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Numerical Simulation Methods
Prof. Dr.-Ing. Timon Rabczuk
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
• Gleichungslöser• Zeitintegrationsverfahren• Eigenwertprobleme und Lösungsstrategien• Netzfreie Methoden
Outline
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
• 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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Outline
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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.
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Gauss-Eliminationsverfahren
( , 1, 2,..., )
( 1, 2,..., )
ikij ij kj
ii
iki i k
kk
aa a a i j k k n
aa
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)
nn
nn
n
i ij jj i
iii
bx
a
b a xx i n n
a
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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.
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Matrix-Inversion • Inverse Matrix Methode
1 1
1
A x b
A A x A b
x A b
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
1
detn
iii
A A a
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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.
A LU
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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.
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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.
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Ankündigung
• Am Donnerstag den 12.11.2009 findet von 13:30 bis 15:00 Uhr anstatt der Vorlesung ein Rechnerseminar im Betonpool der Coudraystrasse 13d statt
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Iterative Methoden • Jacobi• Gauss-Seidel• Successive-over-Relaxation• Conjugate Gradient
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Jacobi Iteration • Wähle Startwert x0
( )( 1) ( )
( ) ( )
1
( 1, 2,... )
( 1,2,... )
kk k i
i iii
nk k
i i ij jj
Rx x i n
a
R b a x i n
• Wenn |Δ x| < tol -> beende Iteration
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
1
n
ii
x tol
1/ 22
1
n
ii
x tol
maxi
i
xtol
x
1
ni
i i
xtol
x
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Gauss Seidel • Erfordert diagonale Dominanz zur Sicherung von Konvergenz• Konvergiert schneller als Jacobi-Iteration
( )( 1) ( )
1( ) ( 1) ( )
1 1
( 1,2,... )
( 1,2,... )
kk k i
i iii
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Successive-Over-Relaxation (SOR) • Vorteil: Schnellere Konvergenz
( )( 1) ( )
1( ) ( 1) ( )
1 1
( 1,2,... )
( 1,2,... )
kk k i
i iii
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Conjugate Gradient (CG) • meist benutzter iterativer Löser für grosse Systeme (sparse matrices)• Voraussetzung: A ist positive definit• Quadratische Form
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
CG • Example
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
CG • Start:
mit
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme • Geometrische nicht-linear• physikalisch nicht-linear
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme • Newton-Raphson
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme • Newton-Raphson
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme • Modifiziertes Newton-Raphson
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme • Verzweigungspunkte (bifurcation points)• Limit points• Turning points
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme • Verzweigungspunkte (bifurcation points)• Durchschlagspunkte (Limit points)• Umkehrpunkte (Turning points)
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme •Versagenspunkte (Failure points)
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme • Verzweigungspunkte (bifurcation points)• Durchschlagspunkte (Limit points)• Umkehrpunkte (Turning points)
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme • Verzweigungspunkte (bifurcation points)• Durchschlagspunkte (Limit points)• Umkehrpunkte (Turning points)
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme • Load control
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme • Displacement control
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme • Arc-length control (Bogenlängenverfahren)
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nicht-lineare Probleme • Arc-length control (Bogenlängenverfahren)
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Ankündigung
• Am Donnerstag den 12.11.2009 findet von 13:30 bis 15:00 Uhr anstatt der Vorlesung ein Rechnerseminar im Betonpool der Coudraystrasse 13d statt
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Motivation
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Motivation
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Motivation
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Motivation
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Motivation
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Motivation
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Time Integration
Forward Euler
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Time Integration
Forward Euler
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Time Integration
Backward Euler
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Time Integration
One-step-theta
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Time Integration
Newmark
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Time Integration
Newmark
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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(0)(0)
tx xx v
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Time Integration
0mx cx kx 0
0
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.
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Time Integration
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Time Integration
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Linear stability analysis
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Linear stability analysis
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Linear stability analysis
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Lecture notes
www.uni-weimar.de/cms/bauing/forschung/institute/ism/lehre/xfem-mfm.html
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
• 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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
• 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?
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Motivation
Idelsohn et al. 2004
Wang XS 2005http://web.njit.edu/~xwang
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Shuttle crash, 2003 Landslide, Colorado
Taiwan earthquake, 2003 Fragmentation of concrete
Motivation
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Concrete under explosive loading
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Perforation of concrete under explosive loading
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Experimental Results
Ockert 1997
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
MotivationFinite elements have difficulties for problems involving weak and strong discontinuities (material interfaces, cracks)
de Borst et al., 2004
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
• 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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
)(XX JSJ
Juu
Central particleNeighbor particle
Domain of influence (support)
Meshfree approximation
Meshfree approximation
FE Meshfree
)()( XuXX JSJ
JJSJ
iJi uu
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
1)( SJ
J X
Partition of unity
Partition of unity
Linear FEM
IJIJ )(X
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
1)( SJ
J X
Partition of unity
Quadratic FEM
IJIJ )(X
1 23
)1()1()()1(5.0)()1(5.0)(
3
2
1
rrrNrrrNrrrN
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Partition of unity
1)( SJ
J X
Partition of unityThe “Kronecker-delta” property is not fulfilledin meshfree methods. This causes difficulties in imposingDirichlet BCs.
IJIJ )(X
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Partition of unity
IIh uu )(x
0hu
0hu
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Completeness
ZZYYXX JSJ
JJSJ
JJSJ
J
SJJ
SJJ
)()()(
0)(1)( 0
XXX
XX
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.
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Completeness
1)(0)(
0)(1)(
0)(0)(
,,
,,
,,
JSJ
YJJSJ
XJ
JSJ
YJJSJ
XJ
SJYJ
SJXJ
YY
XX
XX
XX
XX
The derivative reproducing conditions are also important for several meshfree methods. In two dimensions, the derivative reproducing conditions for a linear field are
ijJjSJ
iJSJ
iJ X
)(,0)( ,, XX
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
0
SJ
JJSJ
JJDtD vmvm
wIISI
JJJ )()( XσXvm
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Completeness and conservation
0
SJ
JJSJ
JJDtD vmvm
wIISI
JJJ )()( XσXvm
0)()( SJ
IISI
JSJ
JJ wXσXvm
This requires
0)(
ISI
J X 1)(
ISI
J X
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
0
SJJJJJJ
SJJJJDt
D vvxvmxvm
JkJ
ImjISI
JmijkSJ
JJJ xwσDtD
i
)()()( XXexvm
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Completeness and conservation
JkJ
ImjISI
JmijkSJ
JJJ xwσDtD
i
)()()( XXexvm
0)(
)()()(
)()(
)()(
)(
)(
)(
wσ
wσwσx
wσx
xwσ
SIImjijm
SImjmjijkImj
JIkI
SIJmijk
ImjJ
IkISI
Jmijk
JkJ
ImjISI
Jmijk
X
JXX
XX
XX
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
k
III
Chu)u(x | |max
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Continuity
A method is considered to be n-th order continuous (Cn) if their shape functions are n times continuous differentiable.
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Meshfree methods
Linear meshfree
Quadratic meshfree
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Kernel function
Weighting/kernel/window functions:
|| Ixxr hrz /
10
103861),(
432
zzzzz
hrW
Cuartic B-Spline
Ir xx
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Kernel function
h
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Kernel function
Requirements usually imposed on the kernel functions:
)(),(lim0
rhrWh
1),(
dhrW
max0),( rrhrW
),(),( hWhW IJJI XXXX
),(),( 00 hWhW IJJI XXXX
max/with)(ofinstead)( rrzrWzW
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Kernel functionExtension of the kernel function into higher order dimensions:Rectangular support:
312111)( XWXWXWW DDDX
Circular support:
||XX ||)( 1DWW
20
21)2(4
1075.05.11
)( 3
32
z
zzhC
zzzhC
rW D
D
3/12)7/(1013/2
DDD
C
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Kernel function
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
The cubic B-Spline
Kernel function
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
The cubic B-Spline WJ(X) The derivative of the Cubic B-spline
Kernel function
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Kernel function
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Kernel function
21 x
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Kernel function
Lagrangian and Eulerian kernels:
),(),( 0
hWWhWW
xX
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.
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
SPH method by Lucy and Monaghan [1977]
YJh du,hWu
)()( YY-XX
Central particleNeighbor particle
Domain of influence
Meshfree Methods
11)(
dY,hWJ Y-X
XdYY,hWJ
)( Y-X
1)(Yu
YYu )(
thatimplies11)(
dY,hWJ Y-X
XdYX,hWJ
)( Y-X
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
gives
0)()(
dYYX,hWJ Y-X
If linear consistency is fulfilled, above is guaranteed by the symmetry of the kernel
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
SPH method by Lucy and Monaghan [1977]
)()(, XX JSJ
Jh tutu
)()(, 00 XX JSJ
Jh tutu
00,)( JJJ VhW XXX
Central particleNeighbor particle
Domain of influence
Meshfree Methods
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Meshfree Methods
),()()()(, 0hWWWVtutu JJJJSJ
Jh X-XXXX
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Meshfree Methods
),()()()(, 0hWWWVtutu JJJJSJ
Jh X-XXXX
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Meshfree Methods
),()()()(, 0hWWWVtutu JJJJSJ
Jh X-XXXX
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Meshfree Methods
Different ways to discretize a body
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
SPH method by Lucy and Monaghan [1977]
)()(, XX JSJ
Jh tutu
)()(, 00 XX JSJ
Jh tutu
00,)( JJJ VhW XXX
Symmetrization
)()(, 00 XX JSJ
JIh uutu
Central particleNeighbor particle
Domain of influence
Meshfree Methods
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Shepard functions
SJJ
JSJ W
WW
)()(
)(X
XX
Meshfree methods
)()(, XX S
SJJ
hJ
Wtutu
)()(, 00 XX S
SJJ
hJ
Wtutu
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Krongauz-Belytschko correction
)()()(, 00 tuWtuSJ
JS
Jh
XXaX
SJJ
JSJ W
WW
)()(
)(X
XX
JS
ZJJS
YJJS
XJ
JS
ZJJS
YJJS
XJ
JS
ZJJS
YJJS
XJ
ZWZWZWYWYWYWXWXWXW
,,,
,,,
,,,
A
IaA T
Meshfree methods
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
• RKPM• EFG (MLS shape functions)• Hp-clouds • PUFEM• GFEM• Intrinsic and Extrinsic Enrichment
Outline
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
)(0 XX JSJ
Juu
Derivatives of the approximation
Elementfree Galerkin (EFG) method
i
T
JJi
T
JJi
T
J
x
Wx
Wx
11
1
10
)()()(
)()()()(
)()()()()(
XBXAXp
X-XXpXAXp
X-XXpXAXpX
)()()()( JT
SJJ W
JX-XXpXpXA
)()()( JSJ
J W X-XXpXB
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Elementfree Galerkin (EFG) method
i
TJJ
i
T
JJi
T
J
xW
x
Wx
11
1
10
)()()()()(
)()()()(
BXAXpX-XXpAXp
X-XXpXApX
)()()()( JT
SJJ W
JX-XXpXpXA
)()()( JSJ
J W X-XXpXB
111
)()(
XAAXAA
ii
-
xx
)()()( JSJ
J W X-XXpXB
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Elementfree Galerkin (EFG) method
)()()()()( 1JJ
T
SJJ
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Second derivatives
Elementfree Galerkin (EFG) method
iJ
jjJ
i
T
jiJJJ
ji
T
JJJJii
T
JJji
T
ji
J
xW
xxW
x
xxWW
xx
WWxx
Wxxxx
)()()(
)()()()()(
)()()()()()(
)()()()()(
11
21
12
11
122
XpAXpAXp
XpXAX-XXpAXp
X-XXpXAX-XXpAXp2
X-XXpXAXpX
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Enrichment in EFG
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
22),( yxyxf
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
MLS SPH
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
Partial derivatives in x-direction
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
0.005% 0.2%
MLS SPH
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
)sin(),( 22 yxyxf
)sin()0,( 2xxf
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
)cos(2),( 22 yxxyxf x
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
MLS SPH
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
SPH-symm
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
Uniform particle distribution
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
SPH (approximation itself)
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
SPH –symm.
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
SPH
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
SJ
L
KJKJJ
h
SJJ
TJJ
h
auu
auu
1
)()(
)()(
XpXX
XpXX
K
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).
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
SJ
L
KJK
h
SJJ
Th
uu
u
J
J
1
0
0
)()(
)()(
XpXX
uXpXX
K
PUFEM
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.
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
L
KJK
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”.
SJ
L
KJKJJ
h auu1
)()( XpXX K
Hp-clouds
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
otherwise0)58.0,42.0(5.0exp5.042)(
1)1(0)0(
)1,0(0)(
2222
,
xxaxaaxb
uu
xxbu xx
PU-MethodsExample
Analytical solution
25.0exp)( xaxxu
)()()( xbuxxu JSJ
Jh
Approximation
25.0exp)( xax
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
PU-Methods
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
PU-Methods
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Spatial integration
00 )()(
0J
SJJ Vfdf
XX
Nodal integration
The quadrature weights are usually associated with the “volume” of the particle
I-1 I+1I
2/110
III xxV
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Spatial integration
00 )()(
0J
SJJ Vfdf
XX
Nodal integration
Delauny triangulation Voronoi cell
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Spatial integration
Stress point integration
NodeStress point
SPN N
J
SPJ
N
J
NJ VVV
11
)( SPiI
SJJ
NiJ
SPiI Xuu
)( SPiI
SJJ
NiJ
SPiI Xvv
)( SPiI
SJJ
NJ
SPI X
vv
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Spatial integration
Stress point integration
NodeStress point
SPSJ
SPJ
SJ
NJ VfVfdf )()()( XXX
)( SPiI
SJJ
NiJ
SPiI Xuu
)( SPiI
SJJ
NiJ
SPiI Xvv
)( SPiI
SJJ
NJ
SPI X
vv
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Cell integration
Spatial integration
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Cell integration:
),(det),(
),(det),()(1
1
1
100
Jfw
ddJfdf
JJJ
J
X
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.
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Cell integration:
),(det),(
),(det),()(1
1
1
100
Jfw
ddJfdf
JJJ
J
X
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Nodal integration: 0
0 )()(0
JSJ
J Vfdf
XX
Cell integration:
),(det),(
),(det),()(1
1
1
100
Jfw
ddJfdf
JJJ
J
X
Spatial integration
Stress point integration:
SPSJ
SPJJ
SJ
NJJ VfVfdf )()()(
00 XXX
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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).
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
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
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
4)~2(36~6
22 DyxxL
IEqyux
2
22 )3(
4)~54(~3~6
xxLxDxLyIE
qyu y
strainplanefor)1/(~;stressplanefor~ 2 EEEEstrainplanefor)1/(~
stressplanefor~
2
2
2Lanal
Lanalh
Lerr
u
uu
dL
uuu2
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
22
3
3
46/),(
0),(12/
)(),(
yDtD
qyx
yxtD
yxLqyx
xy
yy
xx
energyanal
energyanalh
energyerr
u
uu
2/1
:
d)()(Tenergy
uE:CuEu
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
Examples
SS 2009 Numerische SimulationsverfahrenProf. Dr.-Ing. Timon Rabczuk
ExamplesHole in the plate-problem
Put in formula!!!!!!