Two Approaches to Multiphysics Modeling

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Two Approaches toMultiphysics Modeling

• Sun, Yongqi
• FAU Erlangen-Nürnberg

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Content

• Introduction
• Patch Simulation
• Mesoscale Simulation

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Introduction

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

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Introduction

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

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Introduction

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

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Introduction

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

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Patch Simulation

• 2.1 Idea
• 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

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Patch Simulation

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

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Patch Simulation

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

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Patch Simulation

• One-dimentional problem

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Patch Simulation

• Space-time plot for the patches

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Patch Simulation

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

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Patch Simulation

• 2.5 Accuracy Assumption
• Macroscale is separated from microscale
• i. e. ?x ltlt?X, ?t ltlt?T
• Macroscale variables are sufficient to determine
  the systems dynamics and define the microscopic
  i. c.
• Microscale model is well defined
• Macroscale model is statistically stable to small
  perturbations in the microscale
• The reference grid accurately resolves the
  macroscale solution

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Patch Simulation

• 2.6 Crucial Steps
• 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 temporal gaps (average time
  derivatives and the time derivatives of the
  spatial moments) microscale integration(e. g.
  Runge-Kutta)

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Mesoscale Simulation

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

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Mesoscale Simulation

• 3.2 Example of mesoscale method
• Dissipative Particle Dynamics model
• Area complex liquids and dense suspensions
• 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

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References

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

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• Thanks for your attention!