Acknowledgements. Prof. Dan Negrut Prof. Darryl Thelen Prof. Michael Zinn. SBEL Colleagues: Hammad Mazar, Toby Heyn, Manoj Kumar

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2 Acknowledgements Prof. Dan Negrut Prof. Darryl Thelen Prof. Michael Zinn SBEL Colleagues: Hammad Mazar, Toby Heyn, Manoj Kumar 2

3 Outline Motivation Lumped Mass Model Model properties Simulation results Smoothed Particle Hydrodynamics (SPH) SPH formulation for compressible fluids Serial and parallel Implementation Numerical Experiments Conclusion 3

4 Technical Background and Advantages of Meshless Methods 4

5 Technical Background Current simulation tools: (BEM, FDTD, FEM, ) The majority solves the linear wave equation Most methods are mesh-based Mostly Eulerian methods Disadvantages: Limited to small amplitudes & low frequencies No incontinuities as given in shock waves No aero-acoustical effects Moving boundaries are hard to model 5

6 Advantages of Meshless Methods Investigated methods: Based on conservation laws & constitutive relations No linearizations Lagrangian particle methods Advantages for: High amplitudes & frequencies possible Incontinuities (shock waves) Aero-acoustic effects Moving boundaries 6

7 An Intermediate Step to Meshless Acoustics 7

8 Model Properties One-dimensional model consists of N masses which are connected via nonlinear springs Masses represent the inertia of certain gas volume Spring forces replace pressure forces Equation of motion for each mass: 8

9 Model Parameters Spring response from adiabatic process equation: Mass from discretization: mean pressure spring elongation cross section mean density 9

10 Properties & Implementation In the continuum limit, linearization leads to the following differential equation: The model does not draw on any linearization Second order accuracy Implementation & visualization in Matlab Example with 9 Masses and Sinusoidal Excitation 10

11 Simulation results Pressure can be calculated from spring forces at each point Example with 300 mass points and sinusoidal excitation: Linear Springs Nonlinear Springs 11

12 Simulation results Soliton Waves Stable soliton waves can be modeled Propagation speed of soliton waves depends on their amplitude 12

13 Conclusion/Limitations Pro: Speed of sound is modeled accurately Known nonlinear effects can be reproduced Implementation is straightforward because of the simple model Contra: Stability of soliton waves depends on discretization Due to the fixed connectivity, it is not a real meshless method The transfer in two or three dimensional implementation is challenging Move to a more promising method, called Smoothed particle Hydrodynamics (SPH) 13

14 For Acoustic Simulations 14

15 SPH - The Basic Idea Mainly used in hydrodynamics and astronomy Lagrangian particle method Each particle carries field variables (density, internal energy, velocity, ) A kernel-function approach defines the influence area of each particle Field variables and their derivatives can be approximated with the following integrations: 15

16 SPH - The Basic Idea With the product rule of differentiation and the divergence theorem, field function derivatives can also be expressed by: The surface integral is zero if the kernel doesn t intersect domain boundaries The Integration can be approximated by a summation 16

17 SPH - Fluid Dynamics & Acoustics Conservation laws are evaluated for every particle at each time step: Mass: Momentum: Energy: The right hand side is replaced by SPH approximations for field function derivatives The equation of state closes the formulation, relating pressure to density and internal energy Ideal Gas: Water: Time evolution of the system trough time integration of conservation laws 17

18 Boundary Formulations The Achilles heel of SPH (due to the kernel approximation) Requirements on boundary formulations in acoustics: No boundary penetration Accurate sound wave reflection Accurate sound excitation (moving boundaries) No disturbances 18

19 Dynamic Boundary Particles Pro: Easy to implement Contra: Moving boundaries cause disturbances Boundary penetration is possible 19

20 Mirror Particles Pro: Theoretical exact boundary treatment due to symmetry Less disturbances Contra: Boundary penetration is possible: 20

21 Repulsive Forces Pro: No boundary penetration Easy to implement Contra: Large disturbances 21

22 Newly developed Boundary Treatment Recall Standard SPH neglects the surface integral in the formulation for field function derivatives This is the root of boundary problems The surface integral can be used efficiently if two assumptions are made: The field function is constant on the boundary The field function value is equal to or slightly higher than the particle field function value (self interaction) 22

23 Newly developed Boundary Treatment If boundaries are smooth, a generic solution of the surface integral can be used in simulations Pro: No boundary penetration Computationally efficient Contra: disturbances 23

24 SPH - Implementation Structure and functionality of a basic SPH implementations: Leap frog integration 24

25 2D Implementation in Matlab 25

26 2D Hydrodynamic Tests Water flow into a basin SPH liquid particles are poured with a constant initial velocity Boundary treatment through the new developed method 26

27 2D Hydrodynamic Tests The classic SPH test simulation: Dam break experiment A square of water is discretized by 7,225 SPH particles Boundary treatment through dynamic boundary particles 27

28 3D Implementation on the GPU Implementation in Matlab is only reasonable for one and twodimensional problems 10,000 particles ( One time step 30s) Solution: Leveraging the multiprocessor architecture of GPUs The SPH algorithm is highly parallelizable Nearest neighbor search in parallel (radixsort algorithm) Derivatives (interaction parallel) Integration (particle parallel) Three dimensional Implementation using C++ and CUDA CUDA API (programming tools for NVIDIA GPUs) Simulations with up to 3.5 million particles are currently possible Speed up of about 4,000 compared to Matlab 28

29 3D Hydrodynamic Tests Droplet simulation Parameters Number of particles: 250,000 Fluid properties: Water Boundary formulation: Surface integrals Time stepping: t=5.0e -5 s Length: T=1.0s Computation time: 2 hours 29

30 3D Hydrodynamic Tests Different Viscosities Parameters Number of particles: 82,000 Fluid properties: Water/Jelly Boundary formulation: Surface integrals Time stepping: t=1.0e -5 s Length: T=1.5s Computation time: 4 hours 30

Fluid-Structure Interaction Boundary formulations typically require the following information: Boundary position Surface normal Can be describe analytically for simple shapes Using

31 Fluid-Structure Interaction Boundary formulations typically require the following information: Boundary position Surface normal Can be describe analytically for simple shapes Using spherical decomposition, arbitrary shaped boundaries can be discretized by boundary particles. position CAD geometry Surface normal Rigid body motion can be determined using fluid-structure forces 31

32 Fluid-Structure Interaction Water flow on a trough Parameters Number of particles: 250,000 Fluid properties: Water Boundary formulation: Surface integrals Time stepping: t=0.5e -5 s Length: T=4.5s Computation time: 10 hours 32

33 SPH in Acoustic Simulations Recall the advantages of SPH: Whole flow process is solved; (speed of sound automatically adapts to changing fluid properties) Complex boundaries are possible; (spherical decomposition) No linearizations ; (based on conservation laws) Large deformations are unproblematic; (meshless Lagrangian method) Highly parallelizable What needs to be analyzed: Sound propagation Sound excitation and reflection Scaling and accuracy 33

34 2D Sound propagation FDTD SPH SPH models sound propagation accurately Speed of sound differs slightly 34

35 Effects of the Smoothing Length 1D Parameter study with different smoothing lengths Velocity excitation from the left Analytic solution is a traveling step function Level of the step can be related to the speed of sound 35

function Modeled with 270,000 SPH particles Boundaries are first modeled with

36 3D Sound Excitation/Reflection 3D sound excitation in a tube Pressure wave excitation through the moving piston Analytic solution is a traveling step function Modeled with 270,000 SPH particles Boundaries are first modeled with dynamic boundary particles and then with a combination of mirror and dynamic boundary particles 36

37 3D Sound Excitation/Reflection Boundaries: Pressure loss at the edges: 37

38 3D Sound Excitation/Reflection Dynamic Boundar y Particles Mirror & Dynamic Boundar y Particles 38

39 Computational Efficiency 3D experiment with concentric sound propagation six different resolutions are analyzed The pressure at 1,000 positions after a certain simulation time is compared with results from FDTD 39

40 Contributions, Limitations, Future Work 40

41 Summary of contributions Analysis of meshless Lagrangian methods with focus on applicability in acoustical engineering Lumped mass model of one-dimensional nonlinear sound propagation Implementation of SPH on the CPU using Matlab and on the GPU using CUDA Method to model fluid structure interaction through spherical decomposition Analysis of the impact of smoothing length on wave speed Analysis of sound excitation due to moving boundaries New, surface integral based, boundary formulation Work-precision diagram for an acoustic SPH simulation 41

42 Limitations / Future Work Limitations Deficient boundary formulations corrupt simulation results Particle placement is critical and results are sensitive on particle disorder Parameters have to be chosen correctly (experience is necessary) Future Work Improve boundary problematic (CSPH) Nonlinear effects need to be analyzed Fluid-structure interaction for shock waves Hydrodynamic simulations with fluid-structure interaction 42

43 Conclusion It is generally possible to use SPH in acoustic simulations The scaling of the GPU implementation is good Boundary formulations need to be improved Simulating acoustic problems is not straightforward in SPH Exact and noise free boundary enforcement Particle placement Right choice of parameters Potential applications of SPH in Acoustics: aero-acoustic problems complex and changing domain topologies domains with multiple propagation media domains with high temperature or density gradients nonlinear acoustics and shock waves with fluid-structure interaction 43

44 44

Example 13 - Shock Tube

Example 13 - Shock Tube Summary This famous experiment is interesting for observing the shock-wave propagation. Moreover, this case uses the representation of perfect gas and compares the different formulations: