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

2 Zeitintegrationsverfahren Eigenwertprobleme und Lösungsstrategien
Outline Gleichungslöser Zeitintegrationsverfahren Eigenwertprobleme und Lösungsstrategien Netzfreie Methoden SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

3 Eigenschaften von Matrizen Direkte Gleichungslöser
Outline Eigenschaften von Matrizen Direkte Gleichungslöser Iterative Gleichungslöser 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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

4 Outline SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

5 Ankündigung Am Donnerstag den findet von 13:30 bis 15:00 Uhr anstatt der Vorlesung ein Rechnerseminar im Betonpool der Coudraystrasse 13d statt SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

6 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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

7 Gauss-Eliminationsverfahren
Schritt 1: Pivoting Schritt 2: Gauss-Elimination Schritt 3: Lösung nach x mit Rückwärtssubstitution SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

8 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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

9 Matrix-Inversion Gauss-Jordan Elimination can zur Berechnung der Inverse verwendet werden (durch Augmentierung von I zu A) Gauss-Jordan SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

10 Matrix-Inversion Inverse Matrix Methode SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

11 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 SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

12 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. SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

13 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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

14 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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

15 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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

16 Ankündigung Am Donnerstag den findet von 13:30 bis 15:00 Uhr anstatt der Vorlesung ein Rechnerseminar im Betonpool der Coudraystrasse 13d statt SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

17 Iterative Methoden Jacobi Gauss-Seidel Successive-over-Relaxation
Conjugate Gradient SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

18 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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

19 Jacobi Iteration Wähle Startwert x0
Wenn |Δ x| < tol -> beende Iteration SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

20 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 SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

21 Gauss Seidel Erfordert diagonale Dominanz zur Sicherung von Konvergenz
Konvergiert schneller als Jacobi-Iteration Anmerkung 1: Es werden nur bereits berechnete Werte von zur Berechnung von benötigt Anmerkung 2: Der Speicherplatzbedarf ist niedriger als bei der Jacobi-Iteration SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

22 Successive-Over-Relaxation (SOR)
Vorteil: Schnellere Konvergenz Under-relaxation, wenn Gauss-Seidel ‘overshoots” (nicht-lineare Probleme) Problem: Wahl von ω SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

23 Conjugate Gradient (CG)
meist benutzter iterativer Löser für grosse Systeme (sparse matrices) Voraussetzung: A ist positive definit Quadratische Form SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

24 CG Example SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

25 CG Start: mit SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

26 Nicht-lineare Probleme
Geometrische nicht-linear physikalisch nicht-linear SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

27 Nicht-lineare Probleme
Newton-Raphson SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

28 Nicht-lineare Probleme
Newton-Raphson SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

29 Nicht-lineare Probleme
Modifiziertes Newton-Raphson SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

30 Nicht-lineare Probleme
Verzweigungspunkte (bifurcation points) Limit points Turning points SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

31 Nicht-lineare Probleme
Verzweigungspunkte (bifurcation points) Durchschlagspunkte (Limit points) Umkehrpunkte (Turning points) SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

32 Nicht-lineare Probleme
Versagenspunkte (Failure points) SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

33 Nicht-lineare Probleme
Verzweigungspunkte (bifurcation points) Durchschlagspunkte (Limit points) Umkehrpunkte (Turning points) SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

34 Nicht-lineare Probleme
Verzweigungspunkte (bifurcation points) Durchschlagspunkte (Limit points) Umkehrpunkte (Turning points) SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

35 Nicht-lineare Probleme
Load control SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

36 Nicht-lineare Probleme
Displacement control SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

37 Nicht-lineare Probleme
Arc-length control (Bogenlängenverfahren) SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

38 Nicht-lineare Probleme
Arc-length control (Bogenlängenverfahren) SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

39 Motivation SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

40 Ankündigung Am Donnerstag den findet von 13:30 bis 15:00 Uhr anstatt der Vorlesung ein Rechnerseminar im Betonpool der Coudraystrasse 13d statt SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

41 Motivation SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

42 Motivation SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

43 Motivation SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

44 Ankündigung Am Dienstag den findet von 15:15 bis 16:45 Uhr anstatt der Vorlesung ein Rechnerseminar im Betonpool der Coudraystrasse 13d statt SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

45 Motivation SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

46 Motivation SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

47 Motivation SS 2009 Numerische Simulationsverfahren
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48 Time Integration Forward Euler SS 2009 Numerische Simulationsverfahren
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49 Time Integration Forward Euler SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

50 Time Integration Backward Euler SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

51 Time Integration One-step-theta SS 2009
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52 Time Integration Newmark SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

53 Time Integration Newmark SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

54 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 an arbitrary 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. SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

55 Time Integration 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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

56 Time Integration SS 2009 Numerische Simulationsverfahren
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57 Time Integration SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

58 Linear stability analysis
SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

59 Linear stability analysis
SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

60 Linear stability analysis
SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

61 Lecture notes SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

62 Meshfree Methods 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) SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

63 For what applications are meshfree methods useful?
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) SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

64 Motivation Wang XS 2005 Idelsohn et al. 2004
Idelsohn et al. 2004 SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

65 Motivation Shuttle crash, 2003 Landslide, Colorado
Taiwan earthquake, 2003 Fragmentation of concrete SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

66 Concrete under explosive loading
SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

67 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk
SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

68 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk
SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

69 Perforation of concrete under explosive loading
SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

70 Ockert 1997 Experimental Results SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

71 Motivation Finite elements have difficulties for problems involving weak and strong discontinuities (material interfaces, cracks) de Borst et al., 2004 SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

72 No need for mesh generation Higher order continuity
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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

73 Meshfree approximation
FE Meshfree Central particle Neighbor particle Meshfree approximation Domain of influence (support) SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

74 Partition of unity Partition of unity Linear FEM SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

75 Partition of unity Quadratic FEM 1 3 2 SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

76 Partition of unity Partition of unity
The “Kronecker-delta” property is not fulfilled in meshfree methods. This causes difficulties in imposing Dirichlet BCs. SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

77 Partition of unity SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

78 Completeness 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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

79 Completeness The derivative reproducing conditions are also important for several meshfree methods. In two dimensions, the derivative reproducing conditions for a linear field are SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

80 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 SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

81 Meshfree methods Here give the equations for conservation (mass, energy, momentum) SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

82 Completeness and conservation
This requires SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

83 Completeness and conservation
An 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 SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

84 Completeness and conservation
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

85 Compl., stability and convergence
A 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: In Galkerin 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 SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

86 Compl., stability and convergence
A 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: 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 SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

87 Continuity A method is considered to be n-th order continuous (Cn) if their shape functions are n times continuous differentiable. SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

88 Meshfree methods Draw a picture that shows difference between completeness and continuity, example: linear FE+derivatives (discontinuous at element boundaries), show the same for meshfree methods SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

89 Meshfree methods Linear meshfree Quadratic meshfree SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

90 This is nonsense here !!! Meshfree methods
Weighting/kernel/window functions: This is nonsense here !!! SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

91 Kernel function Weighting/kernel/window functions: Cuartic B-Spline
SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

92 Kernel function h SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

93 Kernel function Requirements usually imposed on the kernel functions:
SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

94 Kernel function Extension of the kernel function into higher order dimensions: Rectangular support: Circular support: SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

95 Kernel function SS 2009 Numerische Simulationsverfahren
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96 Kernel function The cubic B-Spline SS 2009
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97 Kernel function The derivative of the The cubic B-Spline WJ(X)
SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

98 Kernel function SS 2009 Numerische Simulationsverfahren
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99 Kernel function SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

100 Kernel function Lagrangian and Eulerian kernels:
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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

101 Lagrangian and Eulerian kernels
Instabilities due to (Belytschko et al. 2003): Rank deficiency Tensile instability (Swegle et al. 1993) Material instability Hyperelastic material law with strain softening SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

102 Meshfree Methods SPH method by Lucy and Monaghan [1977]
Central particle SPH method by Lucy and Monaghan [1977] Neighbor particle Domain of influence SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

103 Meshfree Methods SPH method by Lucy and Monaghan [1977] Subtraction of
Central particle SPH method by Lucy and Monaghan [1977] Neighbor particle Subtraction of gives If linear consistency is fulfilled, above is guaranteed by the symmetry of the kernel Domain of influence SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

104 Meshfree Methods SPH method by Lucy and Monaghan [1977]
Central particle SPH method by Lucy and Monaghan [1977] Neighbor particle Domain of influence SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

105 Meshfree Methods Show the example from TB’s paper to show that linear and zeroth-order completeness is not fulfileed. Here make comment about FE shape functions and Jacobian, large elements vs. small elements, quadrature weights, etc. SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

106 Meshfree Methods SS 2009 Numerische Simulationsverfahren
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107 Meshfree Methods SS 2009 Numerische Simulationsverfahren
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108 Meshfree Methods SS 2009 Numerische Simulationsverfahren
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109 Meshfree Methods Different ways to discretize a body SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

110 Meshfree Methods SPH method by Lucy and Monaghan [1977] Symmetrization
Central particle SPH method by Lucy and Monaghan [1977] Neighbor particle Symmetrization Domain of influence SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

111 Meshfree methods Shepard functions SS 2009
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112 Meshfree methods Krongauz-Belytschko correction SS 2009
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113 Meshfree Methods Derive the equations on the board!!!! SS 2009
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114 Meshfree methods Randles-Libersky correction Non-Symmetrized version
SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

115 Outline RKPM EFG (MLS shape functions) Hp-clouds PUFEM GFEM
Intrinsic and Extrinsic Enrichment SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

116 Outline Here describe RKPM in its discrete and continuous form (Haeusler+Fries) on the board At the beginning of EFG, show Least square fite-> Phu example, then go over to weighted least square -> other Phu example, then final MLS fit SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

117 Reproducing kernel particle method
(Original SPH) SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

118 RKPM SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

119 RKPM SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

120 Elementfree Galerkin method (EFG)
Least square data fitting Taking derivative with respect to a gives SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

121 Elementfree Galerkin method (EFG)
Example SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

122 Element-free Galerkin (EFG) method
Basic approximation Central particle Neighbor particle Minimize quadratic form leads to linear equations for a can be written in shape function form Domain of influence (support) SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

123 Derive MLS equations in more detail on the board!!!!
Outline Derive MLS equations in more detail on the board!!!! SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

124 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 regular A singular SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

125 Here Haeusler, shifting in p(x- xI), Gram-Schmidt orthogonalization
Outline Here Haeusler, shifting in p(x- xI), Gram-Schmidt orthogonalization SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

126 Elementfree Galerkin (EFG) method
Derivatives of the approximation SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

127 Elementfree Galerkin (EFG) method
SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

128 Here show the formula for derivatives of A-1 on the board!!
Outline Here show the formula for derivatives of A-1 on the board!! SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

129 Elementfree Galerkin (EFG) method
Fast computation of the derivatives SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

130 Elementfree Galerkin (EFG) method
Second derivatives SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

131 Enrichment in EFG SS 2009 Numerische Simulationsverfahren
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132 Outline First derive the derivatives of A-1, give example of MLS shape function including numbers and final matrix forms Discuss ill conditioning of matrix A and necessary conditions such that A stays regular ->Haeusler, Korn, solution: Gram-Schmidt orthogonalization Here derive some stuff about EFG such as centering, effective computations of the derivatives, condition matrix- >Haeusler, Gram-Schmidt orthogonalization SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

133 Examples SS 2009 Numerische Simulationsverfahren
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134 Examples MLS SPH SS 2009 Numerische Simulationsverfahren
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135 Examples Partial derivatives in x-direction SS 2009
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136 Examples 0.005% 0.2% MLS SPH SS 2009 Numerische Simulationsverfahren
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137 Examples SS 2009 Numerische Simulationsverfahren
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138 Examples SS 2009 Numerische Simulationsverfahren
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139 Examples MLS SPH SS 2009 Numerische Simulationsverfahren
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140 Examples SPH-symm SS 2009 Numerische Simulationsverfahren
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141 Examples Uniform particle distribution SS 2009
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142 Examples SS 2009 Numerische Simulationsverfahren
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143 Examples SS 2009 Numerische Simulationsverfahren
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144 Examples SPH (approximation itself) SS 2009
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145 Examples SPH –symm. SS 2009 Numerische Simulationsverfahren
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146 Examples SPH SS 2009 Numerische Simulationsverfahren
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147 Examples SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

148 Here die Herleitung aus meiner Diss
Outline Here die Herleitung aus meiner Diss SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

149 Examples SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

150 Examples SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

151 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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

152 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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

153 GFEM Generalized Finite Element Method (GFEM) Hp-clouds
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”. SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

154 PU-Methods Example Analytical solution Approximation SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

155 PU-Methods SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

156 PU-Methods SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

157 Spatial integration Nodal integration
The quadrature weights are usually associated with the “volume” of the particle I-1 I I+1 SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

158 Spatial integration Nodal integration Delauny triangulation
Voronoi cell SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

159 Spatial integration Nodal integration leads to instability due to rank deficiency similar to reduced integrated finite elements. This instability is a weak instability and grows linear in time. SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

160 Spatial integration Example: 4 node quadrilateral: SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

161 Spatial integration Stabilized nodal integration Here Chen’s stuff…..
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

162 Spatial integration Stress point integration Node Stress point SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

163 Spatial integration Stress point integration Node Stress point SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

164 Spatial integration Cell integration SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

165 Spatial integration Cell integration:
In FE Gauss quadrature, 2nq-1 Gauss points are necessary to reproduce a polynomial of n-th order exactly 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. SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

166 Spatial integration Cell 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: SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

167 Spatial integration Nodal integration: Stress point integration:
Cell integration: SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

168 Discrete internal forces
Nodal integration: Stress point integration: Cell integration: SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

169 Spatial integration Integration 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 Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

170 Meshfree methods Shape 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 SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

171 Meshfree methods Methods with intrinsic basis Integration
SPH and corrected SPH versions RKPM EFG MLPG strong form, collocation weak form, nodal/cell integration weak form, nodal/SP/cell int. local weak form, integration over support Methods with extrinsic basis PUFEM hp-clouds GFEM XFEM SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

172 Meshfree methods Show the formula for deformation gradient and internal forces from my paper on the board SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

173 Examples SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

174 Examples SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

175 Meshfree methods E Explain how to compute the error in the energy norm via the relation sigma=E : epsilon…. SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

176 Examples SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

177 Examples SS 2009 Numerische Simulationsverfahren
Prof. Dr.-Ing. Timon Rabczuk

178 Examples Hole in the plate-problem Put in formula!!!!!! SS 2009
Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk


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