Die Präsentation wird geladen. Bitte warten

Die Präsentation wird geladen. Bitte warten

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

Ähnliche Präsentationen


Präsentation zum Thema: "SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Numerical Simulation Methods Prof. Dr.-Ing. Timon Rabczuk."—  Präsentation transkript:

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

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

3 SS 2009Numerische Simulationsverfahren Prof. 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

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

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

6 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Pivoting Einfache Elimination versagt, wenn a ii =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.

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

8 SS 2009Numerische Simulationsverfahren Prof. 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.

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

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

11 SS 2009Numerische Simulationsverfahren Prof. 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

12 SS 2009Numerische Simulationsverfahren Prof. 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.

13 SS 2009Numerische Simulationsverfahren Prof. 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.

14 SS 2009Numerische Simulationsverfahren Prof. 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

15 SS 2009Numerische Simulationsverfahren Prof. 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.

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

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

18 SS 2009Numerische Simulationsverfahren Prof. 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

19 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Jacobi Iteration Wähle Startwert x 0 Wenn |Δ x| beende Iteration

20 SS 2009Numerische Simulationsverfahren Prof. 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 unabh ä ngig von den Rundungsfehlern der vorherigen Iteration Genauigkeit: relative Fehler = absoluter Fehler / exakte Loesung Konvergenz/Abbruchkriterien

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

54 SS 2009Numerische Simulationsverfahren Prof. 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 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.

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

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

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

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

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

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

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

62 SS 2009Numerische Simulationsverfahren Prof. 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

63 SS 2009Numerische Simulationsverfahren Prof. 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?

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

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

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

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

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

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

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

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

72 SS 2009Numerische Simulationsverfahren Prof. 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

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

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

75 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Partition of unity Quadratic FEM 123

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

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

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

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

80 SS 2009Numerische Simulationsverfahren Prof. 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

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

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

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

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

85 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk 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, L h 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

86 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk 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, L h 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

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

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

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

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

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

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

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

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

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

96 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk The cubic B-Spline Kernel function

97 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk The cubic B-Spline W J (X) The derivative of the Cubic B-spline Kernel function

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

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

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

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

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

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

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

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

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

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

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

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

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

111 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Shepard functions Meshfree methods

112 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Krongauz-Belytschko correction Meshfree methods

113 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Meshfree Methods Derive the equations on the board!!!!

114 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Randles-Libersky correction Symmetrized version Non-Symmetrized version Meshfree methods

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

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

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

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

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

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

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

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

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

124 SS 2009Numerische Simulationsverfahren Prof. 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

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

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

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

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

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

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

131 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Enrichment in EFG

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

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

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

135 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Examples Partial derivatives in x-direction

136 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Examples 0.005% 0.2% MLSSPH

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

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

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

140 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Examples SPH-symm

141 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Examples Uniform particle distribution

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

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

144 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Examples SPH (approximation itself)

145 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Examples SPH –symm.

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

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

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

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

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

151 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk 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 a J 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).

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

153 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk GFEM Generalized 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”. Hp-clouds

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

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

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

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

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

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

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

161 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Spatial integration Stabilized nodal integration Here Chen’s stuff…..

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

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

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

165 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Cell integration: Spatial integration In FE Gauss quadrature, 2n q -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.

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

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

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

169 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk 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).

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

171 SS 2009Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk Meshfree methods Methods with intrinsic basis SPH and corrected SPH versions RKPM EFG MLPG Integration 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

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

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

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

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

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

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

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


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

Ähnliche Präsentationen


Google-Anzeigen