Chandrajit Bajaj cs.utexas/~bajaj

1. Lecture 12: Multiscale Bio-Modeling and VisualizationOrgan Models II: Heart, Cardiovascular Circulation and Reactive Fluid Transport Chandrajit Bajaj http://www.cs.utexas.edu/~bajaj Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

2. Blood Circulation Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

3. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

4. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

5. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

6. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

7. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

8. Heart Organ System Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

9. Active Transport Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

10. Transport of Reactive Substances through Fluids Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

11. 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. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

12. external force advection diffusion pressure 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 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

13. 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: Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

14. Updating the Velocity Field • The general procedure • The add forcestep:f • Update the velocity field for the effect of external forces. • Implementation: • Simple. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

15. Advect Step • The advect step: Use the methodof characteristics for the effect of advection: a semi-Lagrangian scheme • Implementation: • Build a particle tracer and linear (or cubic) interpolator. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

16. Diffuse step • The diffuse step: • Use an implicit methodfor the effect of viscosity. • Implementation: • Use the linear solver POIS3D from FISHPAK after discretization. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

17. 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. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

18. 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 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

19. Introduction to Chemical Kinetics • What is chemical kinetics? • A branch of kinetics that studies the rates and mechanisms of chemical reactions. • Stoichiometricequation • A, B, E, F: chemical species (reactants & products) • a, b, e, f: stoichiometric coefficients Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

20. Reaction • Reaction rate (a.k.a. rate law) • Describes the rate r of change of the concentrations, denoted by [*], of reactants and products. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

21. 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. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

22. 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) Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

23. Extension to Reactive Fluids Update of velocity field Evolution of density and temperature Application of chemical reaction Update of reaction- related parameters [Step1] [Step2] [Step3] [Step4] The simulation technique by [Sta99] and [FSJ01] comprises [Step1] and [Step2]and [IKC04] for [Step3] and [Step 4] Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

24. Grid values used in this method • Several values are defined at the center of the grid cell grid cell defined values discretized grid Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

25. Added control factors Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

26. Computation Flow Update of reaction-related parameters Computation of the fluid’s velocity field Evolution of the density & temperature Application of chemical reaction Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

27. [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 Poissonequation: Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

28. [Step2]Evolution of density and temperature • Density field • Similarly as in [Sta99] and [FSJ01] except that multiple substances in the gas mixture are handled: Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

29. Evolve . Reactive Fluids • Each substance is evolved separately. • Molar concentrations and densities are related by molar masses . Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

30. 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. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

31. [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: Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

32. [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: Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

33. 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. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

34. 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 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

35. Computation Flow Update of reaction-related parameters Computation of the fluid’s velocity field Evolution of the density & temperature Application of chemical reaction Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

36. Animation Results – Reactive substance in a gaseous flow Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

37. 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. O’Brien, 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. O’Brien, 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) Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

38. Heart Organ System I Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

39. Heart Disorders I Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

40. Heart Disorder II Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

41. Heart Disorder III Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

42. Heart Disorder IV Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

43. 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. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

44. 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 smoke’s density  Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

45. 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 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

46. Advection • The advect step: - (u• ) u • Use the methodof characteristics for the effect of advection: a semi-Lagrangian scheme • Implementation: • Build a particle tracer and linear interpolator. • Same as [Stam99] Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

47. 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 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

48. Moving Substances through the Fluid • Use the semi-Lagrangian scheme. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin