Center for Computational Visualization

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Lecture 12 Multiscale Bio-Modeling and
VisualizationOrgan Models II Heart,
Cardiovascular Circulation and Reactive Fluid
Transport
Chandrajit Bajaj http//www.cs.utexas.edu/bajaj
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Blood Circulation
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Heart Organ System
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Active Transport
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Transport of Reactive Substances through Fluids
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Transport of Reactive Substances through Fluids

• To extend the model of fluid hydrodynamics with
  chemical kinetics to handle flow of reactive
  substances through fluids.
• To establish a particle-mesh simulation technique
  for reactive flow transport.

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Basic Fluid Dynamics Equations in Stam99

• The incompressible Navier-Stokes equations for
  inviscid fluids
• For the velocity u (u, v, w),
• Conservation of mass
• Conservation of momentum

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Fluids (contd)

• Helmholtz-Hodge decomposition
• Any vector field is the sum of a mass conserving
  field and a gradient field.
• Projection operator P
• The combined Navier-Stokes equations
• Using the fact that and ,
  the following equation is obtained

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Updating the Velocity Field

• The general procedure

• The add force step f
• Update the velocity field for the effect of
  external forces.
• Implementation
• Simple.

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Advect Step

• The advect step
• Use the method of characteristics for the
  effect of advection a semi-Lagrangian scheme
• Implementation
• Build a particle tracer and linear (or cubic)
  interpolator.

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Diffuse step

• The diffuse step
• Use an implicit method for the effect of
  viscosity.
• Implementation
• Use the linear solver POIS3D from FISHPAK after
  discretization.

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Project step

• The project step P(w3)
• Apply the projection operator to make the
  velocity field divergent-free.
• Implementation
• Use the linear solver POIS3D form FISHPAK after
  discretization.

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Moving Substances through the Fluid

• A non-reactive substance is advected by the fluid
  while diffusing at the same time.
• The following equation can be used to evolve
  density, temperature, etc.
• Dissipation term

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Introduction to Chemical Kinetics

• What is chemical kinetics?
• A branch of kinetics that studies the rates and
  mechanisms of chemical reactions.
• Stoichiometric equation
• A, B, E, F chemical species (reactants
  products)
• a, b, e, f stoichiometric coefficients

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Reaction

• Reaction rate (a.k.a. rate law)
• Describes the rate r of change of the
  concentrations, denoted by , of reactants and
  products.

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Reaction Rate

• How to decide the reaction rate r
• r a function of the concentrations of species
  present at time t,
• For a large class of chemical reactions, it is
  proportional to the concentration of each
  reactant/product raised to some power.
• When, for example, only a forward reaction
  occurs,
• Once the rate is determined, A, B, C and
  D are updated by integrating the rate law over
  time interval.

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Rate Coefficient Dependence

• Rate coefficient k
• Is a function of both temperature and pressure.
• Usually, the pressure dependence is ignored.
• For many homogeneous reactions,

Arrhenius equation
A const. Ea activation energy R universal
gas constant 8.314x10-3 kJ/(mol. K)
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Extension to Reactive Fluids
Update of velocity field
Evolution of density and temperature
Application of chemical reaction
Update of reaction- related parameters

Step1
Step3
Step2
Step4
The simulation technique by Sta99 and FSJ01
comprises Step1 and Step2 and IKC04 for
Step3 and Step 4
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Grid values used in this method

• Several values are defined at the center of the
  grid cell

grid cell
defined values
discretized grid
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Added control factors
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Computation Flow
Computation of the fluids velocity field
Update of reaction-related parameters
Evolution of the density temperature
Application of chemical reaction
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Step1 Update of velocity field

• Uses a modified mass conservation equation, as in
  FOA03, to control the expansion/contraction of
  reactive gases
• The divergence constraint ? is determined for
  each cell according to the reaction process that
  occurs in the region.
• Determined in Step4 after the application of
  chemical kinetics.
• The pressure is computed through the modified
  Poisson equation

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Step2 Evolution of density and
temperature

• Density field
• Similarly as in Sta99 and FSJ01 except that
  multiple substances in the gas mixture are
  handled

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Reactive Fluids

• Each substance is evolved separately.
• Molar concentrations and densities are related by
  molar masses .

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Temperature Fields

• Temperature field
• Similarly as in Sta99 and FSJ01 except that a
  heat source term is added.
• The heat source term is updated for each cell in
  Step4 to reflect the occurring chemical
  reaction in the region.

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Step3 Application of chemical reaction

• The reaction process is
  applied for each cell in the reaction system.
• Determine the reaction rate
  
• Then, the new concentration vector c is updated
  by integrating the differential equations over
  ?t

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Step4 Update of reaction-related parameters

• The updated density d, temperature T, and
  reaction rate r influence the velocity through
  the heat source term external force f and the
  ? value.
• The temperature update is completed by taking
  care of the heat source term defined by
  
• The buoyancy force, as proposed in FSJ01, is
  updated

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Velocity confinement

• The vorticity confinement force, as proposed in
  FSJ01, is updated according to or
  
• The resulting external force
• is applied to the momentum conservation
  equation in each time frame.
• The ? value, determined by or
  ,is applied to the modified mass
  conservation equation in the next time frame.
  

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Vorticity confinement

• fconf vorticity confinement force
• Use a vorticity confinement method by Steinhoff
  and Underhill.
• Inject the energy lost due to numerical
  dissipation back into the fluid using a forcing
  term.
• Reduce the numerical dissipation inherent in
  semi-Lagrangian schemes.
• Implementation straightforward

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Computation Flow
Computation of the fluids velocity field
Update of reaction-related parameters
Evolution of the density temperature
Application of chemical reaction
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Animation Results Reactive substance in a
gaseous flow
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Additional Reading

• J. Stam Stable Fluids, SIGGRAPH 1999, 121-128.
• N. Foster, D. Metaxas, Modeling the motion of a
  hot turbulent gas, SIGGRAPH 1997, 181-188
• G. Yngve, J. OBrien, J. Hodgins. Animating
  explosions. SIGGRAPH 2000. 29-36
• R. Fedkiw, J. Stam, H. Jensen. Visual simulation
  of smoke. SIGGRAPH 2001, 23-30.
• W. Gates Animation of Reactive Fluids, Ph.D.
  Thesis, UBC, 2002
• B. Feldman, J. OBrien, O. Arikan. Animating
  suspended particle explosions. TOG, 22(3)23-40.
  2003.
• I. Ihm, B. Kang, D. Cha Animation of Reactive
  Gaseous Fluids through Chemical Kinetics,
  ACM/Siggraph Symp. on Computer Animation (2004)

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Heart Organ System I
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Heart Disorders I
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Heart Disorder II
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Heart Disorder III
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Heart Disorder IV
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Summary of Stam99

• Based on the full Navier-Stokes equations
• Based on an unconditionally stable
  computational model
• Semi-Lagrangian integration scheme
• Easy to implement
• Appropriate for gas and smoke
• Suffers from numerical dissipation
• The flow tends to dampen rapidly.
• Fedkiw01 attempts to solve this problem.

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Basic Equations in Fedkiw01

• The incompressible Euler equations
• Gases are modeled as inviscid, incompressible,
  constant density fluids.
• The equations for the evolution of the
  temperature T and the smokes density ?

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Updating the Velocity Field

• The add force step f
• Update the velocity field for the effect of
  forces.
• fuser user-defined force (for any purpose)
• fbuoy gravity and buoyancy forces

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Advection

• The advect step - (u ?) u
• Use the method of characteristics for the effect
  of advection a semi-Lagrangian scheme
• Implementation
• Build a particle tracer and linear interpolator.
• Same as Stam99

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Project step

• The project step P(w3)
• Apply the projection operator to make the
  velocity field divergent-free.
• Same as Stam99
• Implementation
• Impose free Neumann boundary conditions at the
  occupied voxels.
• Use the conjugate gradient method with an
  incomplete Choleski pre-conditioner.

Poisson equation
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Moving Substances through the Fluid

• Use the semi-Lagrangian scheme.