Smoothed Particle Hydrodynamics Smoothed Particle Hydrodynamics Techniques for the Physics Based Simulation of Fluids and Solids Part I Introduction, Foundations, Neighborhood Search Dan Jan Barbara Matthias Koschier Bender Solenthaler Teschner 1
Speakers Speakers Jan Barbara Matthias Dan Bender Solenthaler Teschner Koschier 2 . 1
Dan Koschier Post-doctoral Researcher at UCL Smart Geometry Processing Group lead by Niloy Mitra Research interests include Physics-based simulation (deformable solids, fluids) Modelling of interface phenomena Cutting and fracture in solids Machine learning enhanced simulation One of main contributors to Developer of supporting libraries CompactNSearch (compact hashing-based neighborhood search algorithm) Discregrid (Parallel higher-order discretization on regular/adaptive grids 2 . 2
Jan Bender Professor at RWTH Aachen University Head of computer animation group Research interests include Physics-based simulation (rigid bodies, solids, fluids, ...) Collision handling Cutting and Fracture Real-time visualization Founder and maintainer of Open source project C++ implementation of many! modern SPH-based simulation techniques Supports fluids, deformables, and coupling with rigid bodies 2 . 3
Barbara Solenthaler Senior Research Scientist at ETH Zürich Head of simulation and animation group Research interests include Physics-based simulation Artist-controllable techniques Data-driven simulation Co-founder of Apagom AG Commercial engine Real-time fluid simulation using MachineLearning More than 10 years of research on SPH-related topics 2 . 4
Matthias Teschner Professor at University of Freiburg Head of computer graphics group Research interests include Physics-based simulation (rigid bodies, solids, fluids, ...) Rendering Computational geometry Cutting edge technology for fluid simulations in Engineering, entertainment, art, medicine, and robotics Co-founder of spin-off technologies Concerned with development of commercial SPH solvers More than 10 years of research on SPH-related topics 2 . 5
Main Goals of this Tutorial Explain the basic concept of SPH and its features What is SPH? For what can it be used? What is not mentioned in papers? What are its strengths and weaknesses? What can be done/has been done in simulation? Showcase Generality and "beauty" of the concept Potential Wide range of applications (with examples) Make state-of-the-art methods tractable Methods sound often more complex than they actually are Motivate other researchers to "work on"/"use" SPH 3 . 1
What can/can't be expected CAN CAN'T Practical introduction to SPH Rigorous derivation of SPH concept Brief introduction fluid/solid sim. Complete introduction to continuum mechanics Methods to realize physical phenomena Detailed discussion of results limitations of Modular/"as building blocks" each approach Videos/demos Clarification of "urban myths" Discussion of act. (C++) implementation Lecture on software architecture (Some) state-of-the-art approaches concepts, current challenges, ongoing trends 3 . 2
"Urban myths" An SPH particle represents a physical atom/molecule a droplet a grain (e.g., in sand simulation) SPH is better than Eulerian approaches Eulerian approaches are better than SPH "Proper" engineering CFD methods are grid-based SPH is (only) 0th-order consistent ... 3 . 3
Outline Lunch break (60min) Block 1 (9:00 - 10:30) Block 3 (13:30 - 15:00) Foundations of SPH Multiphase fluids Governing equations Viscosity Time integration Vorticity and turbulence Example: Our first SPH solver Demo: Neighborhood Search Coffee break (30min) Coffee break (30min) Block 2 (11:00 - 12:30) Block 4 (15:30 - 17:00) Enforcing incompressibility Deformable solids Rigid body simulation State equation solvers Implicit pressure solvers Dynamics and coupling Boundary Handling Data-driven/ML techniques Summary and conclusion Particle-based methods Implicit approaches 4
Foundations of SPH Foundations of SPH 5 . 1
What is SPH? " " A mesh-free method for the discretization of functions and partial differential operators W Functions are discretized into Samples equipped with kernel function W A i Approximates/discretizes differential operators f ( x ) Interpretation of SPH sample Math.: Coefficients "controling" approx. Phys.: Particle "carrying" quantities Useful to simulate continuum media ∂ conservational properties ∇ f greatly handles topological changes ∂ x algorithms parallelize well good for advection-type/transp. problems 5 . 2
SPH Discretization Pipeline PDE: ∇ × f + ∇ g = h ( f , g ) Smoothing/ Kernel Numerical Field quantities/ Continuous Particle convolution approx. functions approximations discretization ∇ ∇× x , v , f , g , … x ∗ W , g ∗ W , … Boundary conditions + solving (Partial) Numerical Differential solution operators 5 . 3
Continuous Approximation { ∞ if r = 0 ∫ δ ( r ) = δ ( x ) dv = , 1 Dirac- function δ 0 otherwise R d Dirac- identity δ ∫ ′ ′ A ( x ) = ( A ∗ δ )( x ) = A ( x ) δ ( x − x ) dv R d Approximation with Gaussian kernel 2 2 N ( x ; μ , σ ), N ( x ; 0, σ ) = δ ( x ) lim σ →0 supp( N ) = R d Good choice because normalized, BUT: Instead use "some other" kernel: W : R × R + R → d such that Controls A ( x ) ≈ ( A ∗ W )( x ) variance 5 . 4
Continuous Approximation - Kernel W : R × R + R What's a good choice for the kernel ? → d Desired properties Essential for valid approximation A ( x ) ≈ ( A ∗ W )( x ) (Optional) Ensures exclusively positive weighting, helps meeting physical constraints, e.g. ρ ≥ 0 (Optional) Allows 1st-order consistent approx. (Optional) Drastically improves efficiency. Kernel construction out of scope of this tutorial. See [LL10] for details. 5 . 5
Continuous Approximation - Kernel W : R × R + R Cubic spline kernel; typical choice for → d ⎧ 6( q − q ) + 1 ⎪ 1 3 2 for 0 ≤ q ≤ ⎨ 2 1 W ( r , h ) = σ 2(1 − q ) 3 for < q ≤ 1 ⎪ ⎩ d 2 0 otherwise 1 ∥ r ∥ with q = h denotes "smoothing length" h Controls support domain radius normalization constant σ d -continuous C 2 Good choice? At least: Kernel fulfills all conditions! 5 . 6
Continuous Approximation - Consistency How accurate is approximation? Polynomial error analysis: 3 ] 1 ∫ [ ′ ′ ′ ′ ′ ( A ∗ W )( x ) = A ( x ) + ∇ A ⋅ ( x − x ) + ( x − x ) ⋅ ∇∇ A ( x − x ) + O (∥ r ∥ ) W ( x − x , h ) dv ∣ x ∣ x 2 ∫ ∫ ′ ′ ′ ′ ′ 2 = A ( x ) W ( x − x ) dv + ( x − x ) W ( x − x ) dv + O (∥ r ∥ ) ∇ A ∣ x ⋅ = 1 = 0 If kernel If kernel normalized symmetric Normalized, symmetric kernels lead to (at least) 1st-order consistency Specialized kernels for higher-order consistency can be constructed 5 . 7
SPH Discretization Pipeline PDE: ∇ × f + ∇ g = h ( f , g ) Smoothing/ Kernel Numerical Field quantities/ Continuous Particle convolution approx. functions approximations discretization ∇ ∇× x , v , f , g , … x ∗ W , g ∗ W , … Boundary conditions + solving (Partial) Numerical Differential solution operators 5 . 8
Field Discretization Continuous approximation (convolution) involves integral Analytic evaluation generally not possible Numerical integration required ′ A ( x ) ∫ ′ ′ ′ ( A ∗ W )( x W ( x x , h ) ρ ( x ) dv Requires discretization ) = − i i ρ ( x ) ′ dm ′ ∑ m j W ( x x ⟨ A ( x : ≈ − , h ) = )⟩ A j ρ i j i j j ∈ F ∫ ∑ Monte-Carlo-like numerical integration Each particle carries A i m i Field sample ρ i Particle mass Density not necessary 5 . 9
Field Discretization - Consistency ∫ ∑ What are we sacrificing? ′ A ( x ) ∫ ⟨ A ⟩ ′ ′ W ( x − x , h ) dm ρ ( x ) ′ Polynomial error analysis: m m ∑ ρ ∑ ρ j j 2 ( x x O (∥ r ∥ ) ⟨ A ⟩ = A + ∇ A ∣ x ⋅ − ) W + W i ij j i ij i j j j j For 1st-order consistency: ∑ ρ m ∑ ρ m j j ( x x 0 = 1 − ) W = W ij j i ij j j j j 5 . 10
Field Discretization - Consistency m m ∑ ρ ∑ ρ j j ( x x 0 = 1 − ) W = W What are we sacrificing? ij j i ij j j j j For 0th-order For 1st-order => Particle ordering important => Almost never satisfied Consequence: Without further treatment we even lose 0th-order consistency Is this a problem? Not really! Approximation accuracy usually still high! Alternatively: order recovery (normalization, matrix-inversion) Common practice: Graphics community: Often ignored. Visual quality still good. Engineering community: Similar. Sometimes order recovery Generally: Depends... Polynomial error is only half of the truth! We'll see that later 5 . 11
Field Discretization - Example Setting: Rectangular domain discretized into particles Test function sampled on particles Discretization quality is tested along red line Results despite concistency order 5 . 12
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