Optimal-Transportation Meshfree Approximation Schemes Approximation Schemes for Fluid and ...

48 Pages·2010·3.08 MB·English

Optimal-Transportation Meshfree
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for Fluid and Plastic Flows
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California Institute of Technology
IInn ccoollllaabboorraattiioonn wwiitthh::
BBoo LLii,, FFeerraass HHaabbbbaall ((CCaalltteecchh)),,
B. Schmidt (TUM), A. Pandolfi (Milano),
F. Fraternali (Salerno)
UUCCLLAA,, FFeebbrruuaarryy 2233,, 22001100
Michael Ortiz
UCLA 02/22/10
ASC/PSAAP Centers
Michael Ortiz
UCLA 02/22/10
Caltech PSAAP Center
Objective: Predict hypervelocity
impact phenomena with quantified
margins and uncertainties
NASA Ames Research Center
EEnergy ffllashh ffrom hhypervellociitty
test at 7.9 Km/s
Hypervelocity impact test bumper shield
(Ernst-Mach Institut, Freiburg Germany)
Michael Ortiz
UCLA 02/22/10
QMU – Center’s assets
Experimental Science Simulation codes
SPHIR HSRT VTF OTM
PPhhyyssiiccss mmooddeellss UUQQ ttoooollss
Plasma/EoS Strength/Fracture Probability/CoM UQ pipeline
Michael Ortiz
UCLA 02/22/10
Simulation requirements
• Hypervelocity impact: Grand challenge in
scientific compputingg
• Main simulation requirements:
– Hypersonic dynamics, high-energy density (HED)
– Multiphase flows (solid, fluid, gas, plasma)
– Free boundaries + contact
– FFrraaccttuurree, ffrraaggmmeennttaattiioonn, ppeerrffoorraattiioonn
– Complex material phenomena:
• HED/extreme conditions
• Ionization, excited states, plasma
• Multiphase equation of state, transport
• Viscopplasticityy,, thermomechanical coupplingg
• Brittle/ductile fracture, fragmentation...
Michael Ortiz
UCLA 02/22/10
Optimal-Transportation Meshfree (OTM)
• Time integration (OT):
– OOppttiimmaall ttrraannssppoorrttaattiioonn mmeetthhooddss::
• Geometrically exact, discrete Lagrangians
– Discrete mechanics, variational time integrators:
• Symplecticity, exact conservation properties
– Variational material updates, inelasticity:
• Incrementaal vaariaatioonaal struuctuure
• Spatial discretization (M):
– Max-ent meshfree nodal interpolation:
• Kronecker-delta property at boundary
– Material-point sampling:
•• NNuummeerriiccaall qquuaaddrraattuurree, mmaatteerriiaall hhiissttoorryy
– Dynamic reconnection, ‘on-the-fly’ adaptivity
Michael Ortiz
UCLA 02/22/10
Optimal transportation theory
Gaspard Monge Leonid V. Kantorovich
Beaune (1746), Paris (1818) Saint Petersbourg (1912)
"SSuurr llaa tthhééoorriiee ddeess ddéébbllaaiiss eett ddeess MMoossccooww ((11998866))
remblais" (Mém. de l’acad. Nobel Prize in
de Paris, 1781) Economics (1975)
Michael Ortiz
UCLA 02/22/10
Mass flows ─ Optimal transportation
• Flow of non-interacting particles in
• Initial and final conditions:
Michael Ortiz
UCLA 02/22/10
Mass flows ─ Optimal transportation
• Benamou & Brenier minimum principle:
• Reformulation as optimal transportation problem:
• McCann’s interpolation:
Michael Ortiz
UCLA 02/22/10
Euler flows ─ Optimal transportation
• Semidiscrete action:
inertia internal energy
• DDiissccrreettee EEuulleerr-LLaaggrraannggee eeqquuaattiioonnss::
geometrically exact mass conservation!
Michael Ortiz
UCLA 02/22/10