laboratori de càlcul numèric (lacàn) departament de matemàtica aplicada iii universitat...
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Laboratori de Càlcul Numèric (LaCàN)
Departament de Matemàtica Aplicada III
Universitat Politècnica de Catalunya (Spain)http://www-lacan.upc.es
Laboratori de Càlcul Numèric (LaCàN)
Departament de Matemàtica Aplicada III
Universitat Politècnica de Catalunya (Spain)http://www-lacan.upc.es
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
Antonio Rodríguez-Ferran&
Pedro Díez
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Computing, 14 de noviembre, 2006 · 2
Research group (LaCàN)Research group (LaCàN)
11 doctors
15 PhD students
2 staff
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Computing, 14 de noviembre, 2006 · 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
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Computing, 14 de noviembre, 2006 · 4
Mixing element-free Galerkin and finite elements
Impose now reproducibility of P(x), accounts
for contribution of uh(x), to compute Nj(x)
FE FE (order=p)(order=p)
EFEFGG
nodesnodes::
particleparticles:s:
Mesh-free methodsMesh-free methods
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Computing, 14 de noviembre, 2006 · 5
Bi-linear FE and numerical approximation
Enriched FE mesh and solution
Mesh-free methodsMesh-free methods
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Computing, 14 de noviembre, 2006 · 6
Corrected Smooth Particle Hydrodynamics (CSPH)
Eulerian
Lagrangian
Mesh-free methodsMesh-free methods
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Computing, 14 de noviembre, 2006 · 7
Mesh-free methodsMesh-free methods
Lagrangian CSPH: Punch test
Stabilized Lagrangian CSPH: Punch test
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Computing, 14 de noviembre, 2006 · 8
Discontinuous Galerkin (DG) methods 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
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Computing, 14 de noviembre, 2006 · 9
Discontinuous Galerkin (DG) methods Discontinuous Galerkin (DG) methods
Numerical examples: linear and nonlinear conservation laws• Scattering of electromagnetic
waves by a Perfect Electric
Conductor (PEC) cylinder
• Subsonic compressible flow past a circle
Scattered field after 4 cycles Mach distribution and isolines
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Computing, 14 de noviembre, 2006 · 10
NEFEM (NURBS-enhanced FEM)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.
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Computing, 14 de noviembre, 2006 · 11
NEFEM (NURBS-enhanced FEM)NEFEM (NURBS-enhanced FEM)
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)
DG NEFEMMesh
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Computing, 14 de noviembre, 2006 · 12
X-FEM (eXtended Finite Element Method)
X-FEM (eXtended Finite Element Method)
Some topics that have to be analyzed • Convergence • Dirichlet boundary conditions• Stability of mixed formulations
voids cracks multiphase
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Computing, 14 de noviembre, 2006 · 13
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
X-FEM (eXtended Finite Element Method)
X-FEM (eXtended Finite Element Method)
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Computing, 14 de noviembre, 2006 · 14
X-FEM enrichmentX-FEM enrichment
Enrichment is needed in elements including the interface to account for gradient discontinuities
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Computing, 14 de noviembre, 2006 · 15
X-FEM (eXtended Finite Element Method)
X-FEM (eXtended Finite Element Method)
Incompressible flow problem
Stability condition:
h is the smallest non-zero eigenvalue of
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Computing, 14 de noviembre, 2006 · 16
Convection-diffusionConvection-diffusion
High-order time-integration: Padé approximation
Stabilisation of convective term
Numerical linear algebra: iterative solvers, preconditioners, approximate inverses, domain decomposition
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Computing, 14 de noviembre, 2006 · 17
Evaporative Evaporative emission systememission systemEvaporative Evaporative emission systememission system
TankActive carbon filter
Atmosphere
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Computing, 14 de noviembre, 2006 · 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
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Computing, 14 de noviembre, 2006 · 19
Adaptive Modeling and Simulation(Verification and Validation)
Adaptive Modeling and Simulation(Verification and Validation)
Validation
Verification
REALITY
Initial Boundary Value Problem(partial differential equations)
Algebraic system of (non)linear equations
Approximation
Modeling
FE discretization
(non)linear equation solver
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Computing, 14 de noviembre, 2006 · 20
Adaptivity schemeAdaptivity scheme
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Computing, 14 de noviembre, 2006 · 21
Error bounds for Quantities of InterestError 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
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Computing, 14 de noviembre, 2006 · 22
Classical energy estimates ensuring bounds
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]
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Computing, 14 de noviembre, 2006 · 23
Hybrid fluxes (unknown Neumann boundary conditions)
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]
Neumann type explicit residual estimates
Neumann type explicit residual estimates
Global problem not affordable; elemental/local decomposition
Solvability:
Better:
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Computing, 14 de noviembre, 2006 · 24
Given the equilibrated fluxes solve in a finite-dimensional space; that is, use a “truth” mesh, i.e.
Then:
Upper bound of the “truth” solution. But…
is underestimated and the upper bound property is lost
Asymptotic / exact upper boundAsymptotic / exact upper bound
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Computing, 14 de noviembre, 2006 · 25
Flux-free “algorithm”Flux-free “algorithm”
Upper bound estimate
No local boundary conditions imposed
Local problem
[Machiels, Maday & Patera, CRASP 2000, Carstensen & Funken SIAM JSC 2000, Morin, Nochetto & Siebert, MC 2002]
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Computing, 14 de noviembre, 2006 · 26
Drawbacks of the (former) Flux-free approachDrawbacks 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
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Computing, 14 de noviembre, 2006 · 27
Proposed modifications[Parés, Díez & Huerta, CMAME 2006]
Proposed modifications[Parés, Díez & Huerta, CMAME 2006]
Neglect effectof local residual outside “star”
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Computing, 14 de noviembre, 2006 · 28
Proposed modificationsProposed modifications
Different (not weighted) l.h.s. in the local problems
Upper bound computed differently
•Different approach and proof
• Sharper estimates
•Easy implementation/ parallelization
•Preclude constant blow-up
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Computing, 14 de noviembre, 2006 · 29
Transient problems: Model problemTransient problems: Model problem
+ initial condition at t=0
+ boundary conditions on
+ eventual advection term
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
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Computing, 14 de noviembre, 2006 · 30
Assessing the QoI: dual problemAssessing 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!
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Computing, 14 de noviembre, 2006 · 31
Challenges and difficulties in transient problems
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
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]
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Computing, 14 de noviembre, 2006 · 32
Mesh generation algorithmsMesh generation algorithms
Tetrahedral meshes
are easily adapted
Generation of hexahedral meshes
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Open topics / on-going work (1)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
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Computing, 14 de noviembre, 2006 · 34
Open topics / on-going work (2)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]
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Computing, 14 de noviembre, 2006 · 35
Closure and advertisingClosure and advertising
http://congress.cimne.upc.es/admos07/
Closure Closure Error assessment and Adaptivity: still a lot to do A forum:
An advanced school?
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Computing, 14 de noviembre, 2006 · 37
3D analysis of carabiner3D analysis of carabiner
Reality
Experimentation
Numerical model
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Computing, 14 de noviembre, 2006 · 38
Energy estimateEnergy estimateReference error map
Estimated error map
Global effectivity : 1,92
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Computing, 14 de noviembre, 2006 · 39
Nonlinear output (linearization)Nonlinear output (linearization)
Locally averaged Von Mises stresses
Nonlinear Output of Interest to be linearized
(non linear)
(linearized)
Initial Mesh 8,4 -2,8
Refined Mesh -0,94718 -0,94721
Linearization requires the mesh to be sufficently accurate
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Computing, 14 de noviembre, 2006 · 40
Nonlinear output: error estimatesNonlinear output: error estimates
Error maps
Estimated error
Reference error
Global effectivity index : = 2,33