Two Approaches to Multiphysics Modeling Sun, Yongqi FAU Erlangen-Nürnberg

Content Introduction Introduction Patch Simulation Patch Simulation Mesoscale Simulation Mesoscale Simulation

Introduction 1.1 Objects Multiple simultaneous physical phenomena Multiple simultaneous physical phenomena Multiple physical models - PDEs Multiple physical models - PDEs

Introduction 1.2 Application Examples Space weather: plasma kinetics + magneto hydrodynamics Space weather: plasma kinetics + magneto hydrodynamics Fluid-structure interaction: Fluid-structure interaction: fluid dynamics + structural mechanics fluid dynamics + structural mechanics Materials fracture: molecular dynamics + solid mechanics Materials fracture: molecular dynamics + solid mechanics Distributed network performance: Distributed network performance: discrete event dynamics + stochastic fluid models discrete event dynamics + stochastic fluid models Nanodevice electronics: Nanodevice electronics: quantum kinetic theory + quantum hydrodynamics quantum kinetic theory + quantum hydrodynamics

Introduction 1.3 Scales – Time and Space Macro: continuum reaction and transport (PDEs) Macro: continuum reaction and transport (PDEs) e. g. mass-action chemical kinetics, e. g. mass-action chemical kinetics, hydrodynamics (Navier-Stokes Eq.) hydrodynamics (Navier-Stokes Eq.) Micro: physically motivated discrete models Micro: physically motivated discrete models e. g. molecular dynamics e. g. molecular dynamics Boltzmann kinetic theory Boltzmann kinetic theory crack propagation crack propagation

Introduction 1.4 Approaches Patch dynamics Patch dynamics – when only microscopic model is available – when only microscopic model is available – to predict macroscopic space-time scales behaviour – to predict macroscopic space-time scales behaviour Mesoscale model Mesoscale model – bridge of macro- and microscale methods – bridge of macro- and microscale methods – hybrid models for specific kinds of problems – hybrid models for specific kinds of problems

Patch Simulation 2.1 Idea Predict system-level behaviour (macroscale) from locally averaged properties (microscale) Predict system-level behaviour (macroscale) from locally averaged properties (microscale) Macroscopic equations are unavailable (highly non-linear, singular) or equations for the higher moments on microscale are unavailable Macroscopic equations are unavailable (highly non-linear, singular) or equations for the higher moments on microscale are unavailable

Patch Simulation 2.2 Application Examples Molecular dynamics Molecular dynamics Lattice-Boltzmann particles methods Lattice-Boltzmann particles methods Reaction-diffusion equations Reaction-diffusion equations Epidemiology Epidemiology

Patch Simulation 2.3 Algorithms – Finite difference methods Microscopic initial conditions agree with the macroscale averages at the grid points : lifting Microscopic initial conditions agree with the macroscale averages at the grid points : lifting Interpolate the macroscale averages : m acroscopic solution and microscopic boundary conditions Interpolate the macroscale averages : m acroscopic solution and microscopic boundary conditions Solution in each patch by microscopic model Solution in each patch by microscopic model Integration of the microscopic model: changes in macroscale averages and time derivatives : restriction Integration of the microscopic model: changes in macroscale averages and time derivatives : restriction Advance macroscopic variables in time Advance macroscopic variables in time

Patch Simulation One-dimentional problem One-dimentional problem

Patch Simulation Space-time plot for the patches Space-time plot for the patches

Patch Simulation 2.4 Overview Microscale structure: solution varies rapidly Microscale structure: solution varies rapidly Macroscale structure: smooth locally averaged Macroscale structure: smooth locally averaged Patch boundary conditions: communicate between patches, obtained from macroscale reconstructed solution Patch boundary conditions: communicate between patches, obtained from macroscale reconstructed solution Buffer region: microscale solution’s statistical properties Buffer region: microscale solution’s statistical properties Macroscale reconstructed solution: local interpolation of macroscopic field variables Macroscale reconstructed solution: local interpolation of macroscopic field variables

Patch Simulation 2.5 Accuracy Assumption Macroscale is separated from microscale: Macroscale is separated from microscale: i. e. △ x << △ X, △ t << △ T i. e. △ x << △ X, △ t << △ T Macroscale variables are sufficient to determine the system’s dynamics and define the microscopic i. c. Macroscale variables are sufficient to determine the system’s dynamics and define the microscopic i. c. Microscale model is well defined Microscale model is well defined Macroscale model is statistically stable to small perturbations in the microscale Macroscale model is statistically stable to small perturbations in the microscale The reference grid accurately resolves the macroscale solution The reference grid accurately resolves the macroscale solution

Patch Simulation 2.6 Crucial Steps Lifting the microscale i. c. from macroscopic (balance between microscopic forces): defect-correction algorithm (maximum entropy approach) Lifting the microscale i. c. from macroscopic (balance between microscopic forces): defect-correction algorithm (maximum entropy approach) Bridging the spatial gaps (microscale b. c. must agree with macroscale): polynomial interpolation, global conservation Bridging the spatial gaps (microscale b. c. must agree with macroscale): polynomial interpolation, global conservation Bridging the temporal gaps (average time derivatives and the time derivatives of the spatial moments): microscale integration(e. g. Runge-Kutta) Bridging the temporal gaps (average time derivatives and the time derivatives of the spatial moments): microscale integration(e. g. Runge-Kutta)

Mesoscale Simulation 3.1 Problems with Macro- and Microscale Simulation Microscopic methods are computationally expensive Microscopic methods are computationally expensive e. g. Molecular Dynamics: 100 nanometers, several tens of nanoseconds e. g. Molecular Dynamics: 100 nanometers, several tens of nanoseconds Macroscopic models usually fail with microscale Macroscopic models usually fail with microscale e. g. continuum hypothesis break down for approximately 10 molecules e. g. continuum hypothesis break down for approximately 10 molecules The coupling of two methods has problems The coupling of two methods has problems e. g. noise at the interface, conservation of mass or momentum e. g. noise at the interface, conservation of mass or momentum

Mesoscale Simulation 3.2 Example of mesoscale method – Dissipative Particle Dynamics model Dissipative Particle Dynamics model Area: complex liquids and dense suspensions Area: complex liquids and dense suspensions Content: Model of polymer chains in dilute solutions: different forces on polymer- and solvent particles Content: Model of polymer chains in dilute solutions: different forces on polymer- and solvent particles Different time scale: time-staggered integrating scheme with two different time steps – smaller for polymer particles and larger for solvent particles Different time scale: time-staggered integrating scheme with two different time steps – smaller for polymer particles and larger for solvent particles

References J. M. Hyman, “Patch Dynamics For Multiscale Problems,” Computing in Science & Engineering, May/June J. M. Hyman, “Patch Dynamics For Multiscale Problems,” Computing in Science & Engineering, May/June V. Symeonidis et al., “A Seamless Approach To Multiscale Complex Fluid Simulation,” Computing in Science & Engineering, May/June V. Symeonidis et al., “A Seamless Approach To Multiscale Complex Fluid Simulation,” Computing in Science & Engineering, May/June 2005.

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