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

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

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

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

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

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

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

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

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

Jacobi Iteration Wähle Startwert x0 Wenn |Δ x| < tol -> beende Iteration SS 2009 Numerische 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 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Lecture notes www.uni-weimar.de/cms/bauing/forschung/institute/ism/lehre/xfem-mfm.html SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

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

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

Motivation Wang XS 2005 Idelsohn et al. 2004 http://web.njit.edu/~xwang Idelsohn et al. 2004 SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Completeness and conservation Numerische 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, 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Examples 0.005% 0.2% MLS SPH SS 2009 Numerische Simulationsverfahren Prof. Dr.-Ing. Timon Rabczuk

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

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

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

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

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

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

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

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

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

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

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

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

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

Examples SS 2009 Numerische 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 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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