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Smoothed Particle Hydrodynamics Techniques for the Physics Based - - PowerPoint PPT Presentation
Smoothed Particle Hydrodynamics Techniques for the Physics Based - - PowerPoint PPT Presentation
Smoothed Particle Hydrodynamics Techniques for the Physics Based Simulation of Fluids and Solids Elastic Solids Dan Jan Barbara Matthias Koschier Bender Solenthaler Teschner Graphics Research - SPH Solver Fluids Low viscosity
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[Peer et al., presented at Eurographics 2018]
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[Peer et al., presented at Eurographics 2018]
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Outline
Elastic force model SPH implementation Implicit formulation
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Linear Elasticity
Continuum mechanics formulation Goal: Computation of forces from deformations, e.g. shear, compression, stretch Input: Current deformed object state, initial undeformed state Linearity: Forces depend linearly on object positions
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Linear Elasticity – Force Computation
Initial position Current position Displacement Deformation map Deformation gradient Strain Stress Force per volume
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Linear Elasticity - Examples
Object translation by Isotropic compression or stretch with Rotation with
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Linear Elasticity - Examples
Stress scales strain
Strain components are scaled differently with
Body force accelerates material from high to low deformation Isotropic compression / stretch
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Linear Elasticity - Discussion
Enables stable, efficient, simple implicit formulations Cannot handle rotating objects Cannot handle “large” deformations
Limited to incompressible elastic solids
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Outline
Elastic force model SPH implementation Implicit formulation
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Kernel Gradient Correction
Deformation gradient SPH kernel gradient correction, e.g. [Bonet and Lok 1999] First-order consistent deformation gradient
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Corotated Formulation
Linear elasticity misinterprets rotations as deformations Rotation is extracted from deformation gradient , e.g. polar decomposition or [Mueller 2016] Reference configuration is rotated with the object Kernel gradient with rotated reference configuration Deformation map with rotated reference configuration
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Corotated Formulation - Illustration
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SPH Force Computation
Strain Stress Stress with alternative parameter formulation
Separation of shear and bulk stress
Force per volume [Ganzenmueller 2015]
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Zero-Energy Modes [Ganzenmueller 2015]
Certain deformations do not result in forces Compute correction forces for inconsistent states For all vectors , the term should be zero Penalty force minimizes
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Outline
Elastic force model SPH implementation Implicit formulation
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Linear System
Explicit: Implicit: Position update: Implicit: Linear force formulation: Linear system
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Solver
Linear, symmetric system Matrix-free solver implementation
See PPE solver
Iterative solvers, e.g. CG Two iterations over particles and neighbors per iteration
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Framework
Start: Gravity, viscosity, … : Elastic forces: Zero-energy modes: Pressure: Velocity update: Position update:
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Pressure Projection
Handles self-collisions, preserves volume
Elastic forces preserve volume, but do not detect collisions
Boundary handling with other phases
Simply solve pressure for all particles of all interacting phases, e.g. elastic solids, rigid solids, fluids
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[Peer et al., presented at Eurographics 2018]
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[Peer et al., presented at Eurographics 2018]
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