Optimal-Transportation Meshfree Approximation Schemes for Fluid and Plastic Flows
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1 Optimal-Transportation Meshfree Approximation Schemes for Fluid and Plastic Flows M. Ortiz California Institute of Technology In collaboration with: Bo Li, Feras Habbal (Caltech), B. Schmidt (TUM), A. Pandolfi (Milano), F. Fraternali (Salerno) Institut für Baustatik und Baudynamik Universität Stuttgart, September 23, 2009
2 Objective: Hypervelocity impact Hypervelocity impact is of interest to a broad scientific community: Micrometeorite shields, geological impact cratering Hypervelocity impact test of multi-layer micrometeorite shield (Ernst-Mach Institut, Germany) The International Space Station uses 200 different types of shield to protect it from impacts
3 Simulation requirements Hypervelocity impact: Grand challenge in scientific computing Main simulation requirements: 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 a a structure u 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 exact discrete Lagrangians for flow problems Inertial part of discrete Lagrangian measures distance between consecutive mass densities (in sense of Wasserstein) Discrete Hamilton principle i of stationary ti action: Variational time integration scheme: Symplectic, time reversible Exact conservation properties (linear and angular momenta, energy) Strong variational i convergence (in sense of Γ- convergence, non-linear phase error analysis)
10 nodal points: material points OTM Spatial discretization Question: How can we reconstruct from nodal coordinates?
11 OTM Max-ent interpolation Problem: Reconstruct function from nodal sample so that: Reconstruction is least biased Reconstruction is most local Optimal shape functions (Arroyo & MO, IJNME, 2006): shape function width information entropy
12 OTM Max-ent interpolation
13 OTM Max-ent interpolation Max-ent interpolation is smooth, meshfree Finite-element interpolation is recovered in the limit of β Rapid decay, short range Monotonicity, maximum principle Good mass lumping properties Kronecker-delta property at the boundary: Displacement boundary conditions Compatibility with finite elements
14 nodal points: material points OTM Spatial discretization
15 nodal points: material points OTM Spatial discretization
16 OTM Spatial discretization Np = local neighborhood of material point p
17 nodal points: material points OTM Spatial discretization Max-ent interpolation at material point p determined by nodes in its local environment Np 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!
18 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
19 OTM Tensile stability tip velocity (m/s) velocity (m/s) time (s) tip time (s) Finite elements OTM OTM is free from tensile instabilities!
20 OTM Riemann problem (Kg/m 3 ) density 1 error no orm density L convergence rate ~ 1 position (m) computed vs. exact wave structure mesh size (h) density convergence (L 1 norm)
21 OTM Shock tube problem Shock tube problem velocity snapshots
22 OTM Shock tube problem vel ocity L 2 error nor m convergence rate ~ 1 den nsity L 1 error norm convergence rate ~ 1 mesh size (h) mesh size (h) velocity convergence (L 2 norm) density convergence (L 1 norm) Shock tube problem convergence plots
23 OTM Taylor anvil test t=0 t=7.5 µs copper 750 m/s t=15 µs t=28 µs
24 OTM Taylor anvil test t=0 t=7.5 µs copper 750 m/s t=15 µs t=28 µs
25 OTM Bouncing balloons FE membrane (rubber, Kapton) OTM fluid (water, air)
26 OTM Bouncing balloons FE membrane (rubber, Kapton) OTM fluid (water, air)
27 OTM Bouncing balloons FE membrane (rubber, Kapton) OTM fluid (water, air)
28 OTM Bouncing balloons FE membrane (rubber, Kapton) OTM fluid (water, air)
29 OTM Terminal ballistics steel projectile 1500 m/s aluminum plate
30 OTM Seizing contact body 1 body 2 linear momentum cancellation! nodes material points Seizing contact (infinite friction) is obtained for free in OTM! (as in other material point methods)
31 Variational Fracture & fragmentation M. Ortiz and A.E. Giannakopoulos, Int. J. Fracture, 44 (1990)
32 Variational Fracture & fragmentation rate (G) -release r Energycrack extension (Δa) M. Ortiz and A.E. Giannakopoulos, Int. J. Fracture, 44 (1990)
33 OTM Fracture & fragmentation Crack growth in mixed mode M. Ortiz and A.E. Giannakopoulos, Int. J. Fracture, 44 (1990) Fracture energy over-estimated as h 0! Non-convergence for general paths, meshes!
34 OTM Fracture & fragmentation ε-neighborhood g construction
35 Variational Fracture & fragmentation Proof of convergence of variational element erosion to Griffith fracture: Schmidt, B., Fraternali, F. and Ortiz, M. Eigenfracture: An eigendeformation approach to variational fracture, SIAM J. Multiscale Model. Simul., 7(3) (2009)
36 Variational Fracture & fragmentation Displacement convergence in L1-norm Mode I propagation through random mesh (courtesy Anna Pandolfi)
37 OTM Fracture & fragmentation OTM implentation: Variational erosion of material points! (by ε-neighborhood construction) crack
38 OTM Back to terminal ballistics steel projectile 1500 m/s aluminum plate
39 OTM Back to terminal ballistics steel projectile 1500 m/s aluminum plate
40 OTM Summary and outlook Optimum-Transportation-Meshfree method: OT is a useful tool for generating geometrically- exact 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 Extensions include: Contact (seizing contact for free!) Fracture and fragmentation (provably convergent) Outlook: Parallel implementation, UQ
41 OTM Summary and outlook Thank you!
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