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Advanced discretization methods in computational mechanics & Adaptive Modeling and Simulation

Advanced discretization methods in computational mechanics & Adaptive Modeling and Simulation. Antonio Rodríguez-Ferran & Pedro Díez. Research group (LaCàN). 11 doctors 15 PhD students 2 staff. Advanced discretization methods in computational mechanics Mesh-free methods

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Advanced discretization methods in computational mechanics & Adaptive Modeling and Simulation

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  1. Advanced discretization methods in computational mechanics& Adaptive Modeling and Simulation Antonio Rodríguez-Ferran&Pedro Díez

  2. Research group (LaCàN) 11 doctors 15 PhD students 2 staff Computing, 14 de noviembre, 2006 · 2

  3. Advanced discretization methods in computational mechanics • Mesh-free methods • Discontinuous Galerkin (DG) methods • NEFEM (NURBS-enhanced finite element method) • X-FEM (eXtended finite element method) • Convection-diffusion Computing, 14 de noviembre, 2006 · 3

  4. Mesh-free methods nodes: particles: Mixing element-free Galerkin and finite elements Impose now reproducibility of P(x), accounts for contribution of uh(x), to compute Njr(x) FE (order=p) EFG Computing, 14 de noviembre, 2006 · 4

  5. Mesh-free methods • Bi-linear FEandnumerical approximation • Enriched FE meshand solution Computing, 14 de noviembre, 2006 · 5

  6. Mesh-free methods Corrected Smooth Particle Hydrodynamics (CSPH) Eulerian Lagrangian Computing, 14 de noviembre, 2006 · 6

  7. Mesh-free methods • Lagrangian CSPH: Punch test • Stabilized Lagrangian CSPH: Punch test Computing, 14 de noviembre, 2006 · 7

  8. Discontinuous Galerkin (DG) methods • Key ideas: • Weak formulation element-by-element. • Numerical fluxes For a first-order hyperbolic problem • Difficulties • Choice of the numerical flux (exact or approximate Riemann solvers) • Boundary conditions usually imposed in a weak sense • Main properties • Locally conservative and easy to parallelize • Computations and duplication of nodes on element faces Computing, 14 de noviembre, 2006 · 8

  9. Numerical examples: linear and nonlinear conservation laws Scattering of electromagnetic waves by a Perfect Electric Conductor (PEC) cylinder Subsonic compressible flow past a circle Discontinuous Galerkin (DG) methods Mach distribution and isolines Scattered field after 4 cycles Computing, 14 de noviembre, 2006 · 9

  10. NEFEM (NURBS-enhanced FEM) • Goal: • Work with the CAD geometric model (NURBS functions) • Simplify the refinement process • Advantages • Computational cost and memory requirements (more efficient than corresponding standard DG method) • Main advantages are observed in coarse meshes under p-refinement • Challenges • Numerical integration. NURBS are piecewise rational functions. Computing, 14 de noviembre, 2006 · 10

  11. Numerical examples: NEFEM vs. DG Scattering of electromagnetic waves: NEFEM requires 75% of CPU time and 38% of degrees of freedom required by DG for the same accuracy. Compressible flow problem: NEFEM converges to the steady-state solution using linear interpolation (with DG this is not possible) NEFEM (NURBS-enhanced FEM) • Mesh • DG • NEFEM Computing, 14 de noviembre, 2006 · 11

  12. X-FEM (eXtended Finite Element Method) • Some topics that have to be analyzed • Convergence • Dirichlet boundary conditions • Stability of mixed formulations voids cracks multiphase Computing, 14 de noviembre, 2006 · 12

  13. X-FEM (eXtended Finite Element Method) • Proposed approach: Finite elements + Level sets • Tracking the interface with Level sets allows topology changes (detachment…) • Flexibility in the geometry anddiscretization • Adaptivity may be used if needed Computing, 14 de noviembre, 2006 · 13

  14. X-FEM enrichment • Enrichment is needed in elements including the interface to account for gradient discontinuities Computing, 14 de noviembre, 2006 · 14

  15. X-FEM (eXtended Finite Element Method) • Incompressible flow problem Stability condition: ah is the smallest non-zero eigenvalue of Computing, 14 de noviembre, 2006 · 15

  16. Convection-diffusion • High-order time-integration: Padé approximation • Stabilisation of convective term • Numerical linear algebra: iterative solvers, preconditioners, approximate inverses, domain decomposition Computing, 14 de noviembre, 2006 · 16

  17. Evaporative emission system Active carbon filter Tank Atmosphere Computing, 14 de noviembre, 2006 · 17

  18. Adaptive modeling and simulation • Introduction • Goal-oriented adaptivity, Quantities of Interest • Elliptic problems (space errors) • Energy upper bounds: asymptotic / exact bounds hybrid-flux / flux-free estimates • Parabolic problems (transient thermal, space-time errors) • Remeshing strategies for goal-oriented adaptivity • Mesh generators • Current work Computing, 14 de noviembre, 2006 · 18

  19. Adaptive Modeling and Simulation(Verification and Validation) REALITY Validation Modeling Initial Boundary Value Problem (partial differential equations) FE discretization Algebraic system of (non)linear equations Verification (non)linear equation solver Approximation Computing, 14 de noviembre, 2006 · 19

  20. Adaptivity scheme Computing, 14 de noviembre, 2006 · 20

  21. Error bounds for Quantities of Interest • Introduce adjoint (dual) problem: error representation • Bounds of QoI computed from energy bounds • Both upper and lower energy bounds are required(often zero is used as -not sharp- lower bound) • Implicit residual estimates produce upper and lower energy bounds Computing, 14 de noviembre, 2006 · 21

  22. Classical energy estimates ensuring bounds • Upper bound estimates: • Neumann (imposed flux) local boundary conditions [Ladevèze, Bank & Weiser, Ainsworth & Oden…] • Flux-free estimates • Lower bound estimates: • Dirichlet (imposed displacement) local boundary conditions[Stein, Aubry, LaCàN…] • Postprocess of Neumann estimates[Prudhomme et al. IJNME 2003; Díez, Parés & Huerta, IJNME 2003] Computing, 14 de noviembre, 2006 · 22

  23. Neumann type explicit residual estimates Hybrid fluxes (unknown Neumann boundary conditions) Solvability: Better: • Global problem not affordable; elemental/local decomposition Hybrid fluxes must be chosen to ensure solvability and to approximate “real” ones. There are well established techniques [For instance Ladevèze & Leguillon SINUM83 or Ainsworth & Oden 93] Computing, 14 de noviembre, 2006 · 23

  24. Asymptotic / exact upper bound Then: Upper bound of the “truth” solution. But… is underestimated and the upper bound property is lost • Given the equilibrated fluxes solve in a finite-dimensional space; that is, use a “truth” mesh, i.e. Computing, 14 de noviembre, 2006 · 24

  25. Flux-free “algorithm” [Machiels, Maday & Patera, CRASP 2000, Carstensen & Funken SIAM JSC 2000, Morin, Nochetto & Siebert, MC 2002] Upper bound estimate Local problem No local boundary conditions imposed Computing, 14 de noviembre, 2006 · 25

  26. Drawbacks of the (former) Flux-free approach • Local weighting of • Blow up of stability constant • Sensitivity to anisotropy (?) • Need of equilibration for some problems • Not for order > 1 • Not sharp upper bound • Repeated use of Cauchy-Schwartz inequality in the proof Computing, 14 de noviembre, 2006 · 26

  27. Proposed modifications[Parés, Díez & Huerta, CMAME 2006] Neglect effectof local residual outside “star” Computing, 14 de noviembre, 2006 · 27

  28. Proposed modifications Different (not weighted) l.h.s. in the local problems • Different approach and proof • Sharper estimates • Easy implementation/ parallelization • Preclude constant blow-up Upper bound computed differently Computing, 14 de noviembre, 2006 · 28

  29. Transient problems: Model problem • Space discretization yields a ODE system • Usually ODE system solved by Finite-Difference time marching scheme • Following [Johnson; Rannacher] use Discontinuous Galerkin (DG) to obtain a variational setup • Variational framework induces a sound error characterization (comprehensive residue) and allows defining error estimates • The error assessment tool based on DG may be used for solutions computed with other methods + eventualadvection term + initial condition at t=0 + boundary conditions on Computing, 14 de noviembre, 2006 · 29

  30. Assessing the QoI: dual problem • QoI: for • Dual problem: find such that • Strong form “initial” condition at t=T +homogeneous boundary conditions on ¡Backward in time! Computing, 14 de noviembre, 2006 · 30

  31. Challenges and difficulties in transient problems • Produce error bounds (asymptotic/exact) • Identify space and time errors • Adapt time step and mesh size • Affordable computational cost (parallelization?) Remeshing strategies for goal-oriented adaptivity • Translate local error into desired element size • Furnish proper information to mesh generator (node-based) • Proof of optimality (already available for energy norm)[Díez, Calderón CMAME 2007] Computing, 14 de noviembre, 2006 · 31

  32. Mesh generation algorithms Tetrahedral meshes are easily adapted Generation of hexahedral meshes Computing, 14 de noviembre, 2006 · 32

  33. Open topics / on-going work (1) • Mixed recovery-residual estimates (simple and sharp) • Elliptic / transient [Díez, Calderón CM in press] • Node-based representation • Space-time remeshing strategies • Balance space-time contributions • Optimize global cost • Adaptive modeling • Introduce proper mapping between different models in the hierarchy Computing, 14 de noviembre, 2006 · 33

  34. Open topics / on-going work (2) • Exploring applications for flux-free estimates • Stokes (with Fredrik Larsson from Chalmers) • Exact bounds (vs. asymptotic, without any truth reference mesh) • Analysis of asymptotic behavior. Anisotropy. • Provide exact error bounds for transient problems (including advection) • Generalize steady case[Paraschiviou, Peraire & Patera, CMAME97] • Use ideas from[Machiels, CMAME01] • Work out recovery type estimates to get upper bounds • Based on the idea of recovering admissible stresses[Díez, Ródenas & Zienkiewicz, IJNME in press] Computing, 14 de noviembre, 2006 · 34

  35. Closure Closure and advertising • Error assessment and Adaptivity: still a lot to do • A forum: An advanced school? http://congress.cimne.upc.es/admos07/ Computing, 14 de noviembre, 2006 · 35

  36. Computing, 14 de noviembre, 2006 · 36

  37. 3D analysis of carabiner • Reality • Experimentation • Numerical model Computing, 14 de noviembre, 2006 · 37

  38. Energy estimate Reference error map Global effectivity : 1,92 Estimated error map Computing, 14 de noviembre, 2006 · 38

  39. Nonlinear output (linearization) • Locally averaged Von Mises stresses • Nonlinear Output of Interest to be linearized  (non linear)  (linearized)   Linearization requires the mesh to be sufficently accurate Computing, 14 de noviembre, 2006 · 39

  40. Nonlinear output: error estimates • Error maps Estimated error Reference error Global effectivity index :  = 2,33 Computing, 14 de noviembre, 2006 · 40

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