Optimal-Transportation Meshfree Approximation Schemes
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1 Optimal-Transportation Meshfree Approximation Schemes M. Ortiz California Institute of Technology In collaboration with: Bo Li (Caltech), B. Schmidt (Augsburg), Mini-Workshop on Variational Methods for Evolution Mathematisches Forschungsinstitut Oberwolfach Oberwolfach, Germany, December 4-10, 2011
2 ASC/PSAAP Centers
3 Optimal transportation and numerical simulation Many applications involve complex fluid and solid (plastic flows) Example: Hypervelocity impact: Hypersonic dynamics, high-energy density (HED) Multiphase flows (solid, fluid, gas, plasma) Free boundaries + contact Fracture, fragmentation, perforation Complex material phenomena: HED/extreme conditions Ionization, excited states, plasma Multiphase equation of state, transport Viscoplasticity, thermomechanical coupling Brittle/ductile fracture, fragmentation...
4 Optimal-Transportation Meshfree (OTM) Time integration (OT): Optimal transportation methods: Geometrically exact, discrete Lagrangians Discrete mechanics, variational time integrators: Symplecticity, exact conservation properties Variational material updates, inelasticity: Incremental variational structure Spatial discretization (M): Max-ent meshfree nodal interpolation: Kronecker-delta property at boundary Material-point sampling: Numerical quadrature, material history Dynamic reconnection, on-the-fly adaptivity
5 Optimal transportation theory Gaspard Monge Beaune (1746), Paris (1818) "Sur la théorie des déblais et des remblais" (Mém. de l acad. de Paris, 1781) Leonid V. Kantorovich Saint Petersbourg (1912) Moscow (1986) Nobel Prize in Economics (1975)
6 Mass flows Optimal transportation Flow of non-interacting particles in Initial and final conditions:
7 Mass flows Optimal transportation Benamou & Brenier minimum principle: Reformulation as optimal transportation problem: McCann s interpolation:
8 Euler flows Optimal transportation Semidiscrete action: inertia internal energy Discrete Euler-Lagrange equations: geometrically exact mass conservation!
9 Optimal-Transportation Meshfree (OTM) Optimal transportation theory is a useful tool for generating geometrically-exact discrete Lagrangians for flow problems Inertial part of discrete Lagrangian measures distance between consecutive mass densities (in sense of Wasserstein) Discrete Hamilton principle of stationary action: Variational time integration scheme: Symplectic, time reversible Exact conservation properties: linear and angular momenta, energy (with time-adaption) Strong variational convergence in the sense of Γ- convergence (work in progress )
10 material points OTM Spatial discretization nodal points: - Material pts carry mass, material state - Nodal pts carry field information
11 OTM Spatial discretization Steel projectile/aluminum plate: Nodal set
12 OTM Spatial discretization Steel projectile/aluminum plate: Material point set
13 material points OTM Spatial discretization nodal points: Question: How can we reconstruct from nodal coordinates?
14 OTM Max-ent interpolation Problem: Reconstruct incremental deformation mapping from nodal coordinates: Optimal interpolation (Arroyo & MO, 2006): nodal weight costs information entropy
15 OTM Max-ent interpolation
16 OTM Max-ent interpolation Optimal weight functions can be computed exactly in close form Max-ent interpolation is smooth, meshfree, monotonic, rapid decay, short range Simplicial Delaunay interpolation is recovered in the limit of β Kronecker-delta property at the boundary (interpolation on the boundary depends on boundary data only) Density in W 1,p (A. Bompadre, MO, B. Schmidt)
17 OTM Spatial discretization Np = local neighborhood of material point p
18 OTM Spatial discretization material points nodal points: Max-ent interpolation at material point p determined by nodes in its local environment Np only Local environments determined on-the-fly by range searches Local environments evolve continuously during flow (dynamic reconnection) Dynamic reconnection requires no remapping of history variables!
19 OTM Flow chart (i) Explicit nodal coordinate update: (ii) Material point update: position: deformation: volume: density: (iii) Constitutive update at material points (iv) Reconnect nodal and material points (range searches), recompute max-ext shape functions
20 Example: Water sloshing in tank (free-surface, compressible NS) body of water dropped in tank tank tank motion Dirk Hartmann, Siemens AG, Munich Corporate Research and Technologies
21 Example: Hypervelocity impact Impactor OTM simulation, 5.4 Km/s Nylon/Al6061-T6 Caltech s SPHIR facility
22 Example Bouncing balloons FE membrane (rubber, Kapton) OTM fluid (water, air)
23 OTM Convergence analysis
24 OTM Convergence analysis
25 OTM Convergence analysis
26 OTM Convergence analysis
27 OTM Summary and outlook Optimum-Transportation-Meshfree method: OT is a useful tool for generating geometricallyexact discrete Lagrangians for flow problems Max-ent approach supplies an efficient meshfree, continuously adaptive, remapping-free, FEcompatible, interpolation scheme Material-point sampling effectively addresses the issues of numerical quadrature, history variables Outlook: Extend convergence analysis to compressive-euler flows, solid flows Applications, applications, applications
28 OTM Summary and outlook myocardium (FE) myocardial muscle fibers OTM blood electrophysiology ACME: Advanced Cardiac Mechanics Simulator (IAS/TUM) M. Ortiz (Caltech), W. Wall, M. Gee (TUM) W. Klug (UCLA)
29 OTM Summary and outlook Thank you!
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