Advanced discretization methods in computational mechanics

1
Advanced discretization methods in computational
mechanics Adaptive Modeling and Simulation

• Antonio Rodríguez-FerranPedro Díez

2
Research group (LaCàN)
11 doctors 15 PhD students 2 staff
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

4
Mesh-free methods

• Mixing element-free Galerkin and finite elements
• Impose now reproducibility of P(x), accounts
• for contribution of uh(x), to compute Njr(x)

FE (orderp)
EFG
5
Mesh-free methods

• Bi-linear FE and numerical approximation
• Enriched FE mesh and solution

6
Mesh-free methods

• Corrected Smooth Particle Hydrodynamics (CSPH)
• Eulerian
• Lagrangian

7
Mesh-free methods

• Lagrangian CSPH Punch test
• Stabilized Lagrangian CSPH Punch test

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

9
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

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.

11
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)

12
X-FEM (eXtended Finite Element Method)
voids cracks
multiphase

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

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

14
X-FEM enrichment

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

15
X-FEM (eXtended Finite Element Method)

• Incompressible flow problem
• Stability condition
• ah is the smallest non-zero
• eigenvalue of

16
Convection-diffusion

• High-order time-integration Padé approximation
• Stabilisation of convective term
• Numerical linear algebra iterative solvers,
  preconditioners, approximate inverses, domain
  decomposition

17
Evaporative emission system
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

19
Adaptive Modeling and Simulation(Verification
and Validation)
20
Adaptivity scheme
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

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
  conditionsStein, Aubry, LaCàN
• Postprocess of Neumann estimatesPrudhomme et
  al. IJNME 2003 Díez, Parés Huerta, IJNME 2003

23
Neumann type explicit residual estimates

• 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
24
Asymptotic / exact upper bound

• Given the equilibrated fluxes solve in a
  finite-dimensional space that is, use a truth
  mesh, i.e.

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
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 gt 1
• Not sharp upper bound
• Repeated use of Cauchy-Schwartz inequality in the
  proof

27
Proposed modificationsParés, Díez Huerta,
CMAME 2006
Neglect effectof local residual outside star
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
29
Transient problems Model problem
eventual advection term
initial condition at t0
boundary conditions on

• 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

30
Assessing the QoI dual problem

• QoI for
• Dual problem find such that
• Strong form

initial condition at tT
homogeneous boundary conditions on
Backward in time!
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

32
Mesh generation algorithms

• Tetrahedral meshes
• are easily adapted
• 
  Generation of hexahedral meshes

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

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
  stressesDíez, Ródenas Zienkiewicz, IJNME in
  press

35
Closure and advertising
Closure

• Error assessment and Adaptivity still a lot to
  do
• A forum
• An advanced school?

http//congress.cimne.upc.es/admos07/
36
(No Transcript)
37
3D analysis of carabiner

• Reality

• Experimentation

• Numerical model

38
Energy estimate
Reference error map
Global effectivity 1,92
Estimated error map
39
Nonlinear output (linearization)

• Locally averaged Von Mises stresses
• Nonlinear Output of Interest to be linearized

? (non linear)

Initial Mesh 8,4 -2,8
Refined Mesh -0,94718 -0,94721
? (linearized)
?
?
Linearization requires the mesh to be
sufficently accurate
40
Nonlinear output error estimates

• Error maps
• 
  
• 
  Estimated error
• Reference error

Global effectivity index ? 2,33