Laboratori de Càlcul Numèric (LaCàN) Departament de Matemàtica Aplicada III Universitat...

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


& Adaptive Modeling and Simulation

Antonio Rodríguez-Ferran&

Pedro Díez

Antonio Rodríguez-Ferran&

Pedro Díez

Computing, 14 de noviembre, 2006 · 2

Research group (LaCàN)Research group (LaCàN)

11 doctors

15 PhD students

2 staff



Mesh-free methods Discontinuous Galerkin (DG) methods NEFEM (NURBS-enhanced finite element method) X-FEM (eXtended finite element method) Convection-diffusion


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


Bi-linear FE and numerical approximation

Enriched FE mesh and solution



Corrected Smooth Particle Hydrodynamics (CSPH)

Eulerian

Lagrangian




Lagrangian CSPH: Punch test

Stabilized Lagrangian CSPH: Punch test


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


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


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.


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


X-FEM (eXtended Finite Element Method)


Some topics that have to be analyzed • Convergence • Dirichlet boundary conditions• Stability of mixed formulations

voids cracks multiphase


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 enrichmentX-FEM enrichment

Enrichment is needed in elements including the interface to account for gradient discontinuities




Incompressible flow problem

Stability condition:

h is the smallest non-zero eigenvalue of


Convection-diffusionConvection-diffusion

High-order time-integration: Padé approximation

Stabilisation of convective term

Numerical linear algebra: iterative solvers, preconditioners, approximate inverses, domain decomposition


Evaporative Evaporative emission systememission systemEvaporative Evaporative emission systememission system

TankActive carbon filter

Atmosphere


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


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


Adaptivity schemeAdaptivity scheme


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


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]


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:


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


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]


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


Proposed modifications[Parés, Díez & Huerta, CMAME 2006]

Proposed modifications[Parés, Díez & Huerta, CMAME 2006]

Neglect effectof local residual outside “star”


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


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


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!


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]


Mesh generation algorithmsMesh generation algorithms

Tetrahedral meshes

are easily adapted

Generation of hexahedral meshes


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


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]


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?


3D analysis of carabiner3D analysis of carabiner

Reality

Experimentation

Numerical model


Energy estimateEnergy estimateReference error map

Estimated error map

Global effectivity : 1,92


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


Nonlinear output: error estimatesNonlinear output: error estimates

Error maps

Estimated error

Reference error

Global effectivity index : = 2,33

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