Presentation Outline

1. Mathematical Modelingin FEMLABSeminar atUniversity of WaterlooDecember 3, 2003Jean-François Hiller

2. Presentation Outline • Brief FEMLAB History and Overview • FEMLAB 3.0 Preview • Simulation Examples

4. Company background • COMSOL founded in 1986 • MATLAB distributor in Scandinavia • Released PDE Toolbox 1995 • Released FEMLAB 1.0 in 1999 • 120 employees in 9 offices around the world • Distributors world wide

5. What FEMLAB is used for • Structural Mechanics • Heat Transfer • Acoustics • Electromagnetics • Fluid Flow • Chemical Engineering • General Multiphysics and PDEs

6. Structure of FEMLAB Structure of FEMLAB • FEMLAB core package • Arbitrarily-defined Equations: • Coefficient form • General form • Weak form • Equation-defined (Application) Modes: • Classic PDEs • Physics Modes (Heat transfer, Diffusion, Flow, Mechanics, DC) • Application Specific Modules • Electromagnetics Module • Statics, Eddy Currents, Microwaves, Photonics • Structural Mechanics Module • Solids, Shells, Thermo-stress, Beams, Plates • Chemical Engineering Module • Fluid Dynamics, Chemical Reactions, Electrochemical

7. FEMLAB FEMLAB Features • Single GUI – solid modeling, mesh generation, boundary conditions, material properties, solving, visualization, postprocessing • Equations are always available • Specialized application modes to avoid equations • Ready-to-use models in the Model Library • Open – in the MATLAB environment, scripting language

8. FEMLAB 3.0 Preview!

9. FEMLAB 3.0 • Stand alone implementation in C++ and Java • Great speed and efficiency • Challenges specialized competition • Can also be run from MATLAB • Many improvements to GUI

10. Worked Introductory Example: Heat Transfer in a Cooling Fin

11. Introductory example Purpose of the model • Explain the modeling procedure using FEMLAB

12. Introductory example Problem definition • Heat transfer • Linear stationary • Several subdomains

13. Introductory example - Definition Problem definition Step 1 symmetry Step 2

14. Introductory example - Results Results

15. Deformation of a piston due to temperature changes

16. Piston Will a piston withstand stresses asengine load increases? • Simulate increase from low engine load to full in 5 seconds • Temperature increase induces stress • Young’s modulus is temperature dependent

17. Piston – Model definition Coupled heat transfer – stress-strain analysis • Take advantage of symmetry • Temperature increases with time, T = f(t) • Temperature acts as load in stress-strain analyis • Young’s modulus is temperature dependent, E = f(T)

18. Piston – Results von Mises stress, temperature and displacement

19. Example: Porous Reactor

20. Porous Reactor Introduction • This model shows the flow field in an experimental reactor for studies of heterogeneous catalysis. • The model couples the free fluid and porous media flow through the Navier-Stokes equations and Brinkman’s extension of Darcy’s law. • The mass transport of three species in the reactor is modeled through the diffusion and convection equation.

21. Geometry • The reactor consists of a tubular structure with an injection tube that has its main axis perpendicular to the axis of the reactor. • The incoming species in the main and injection tubes reacts in a fixed porous catalyst bed.

22. The fluid flow in the free flow regions are described by the stationary Navier-Stokes equations. The fluid flow in the porous bed is described with Brinkman equations. In the above equations h denotes the viscosity, r the density, and k the permeability. Porous Reactor – Problem Definition Domain Equations – Fluid Flow

23. The mass transport is given by the diffusion and convection equation: where ci denotes the concentration of species i, u the velocity vector and Ri the reaction rate for species i. Porous Reactor – Problem Definition Domain Equations – Mass Transport

24. The figure shows the flowlines of the velocity field and slices of the norm of the velocity field. The flow is almost homogenized in the porous part of the reactor. Porous Reactor – Results Results – Velocity field

25. The figure shows the pressure in the reactor. The pressure drop is mainly in the porous bed Porous Reactor – Results Results – Pressure

26. Porous Reactor – Results Results – Concentrations Concentration species B Concentration species A Concentration species C

27. Porous Reactor – Results Results – Concentrations Concentration species B Concentration species A Concentration species C

28. The figure shows the iso-concentration surfaces of the species reacting with the one injected through the injection channel. This species is consumed in the porous bed and the reaction distribution at the inlet of the bed is shown in a cross-section plot. The reaction is not uniformly distributed in the catalytic bed and has its maximum close to the radial position of the injection pipe. Porous Reactor – Results Results – Concentrations

29. Diffusion-Reaction in a Zebra Fish Embryo

30. 3D diffusion and use of oxygen • Water - source of oxygen • Body - oxygen enters and leaves by diffusion and also is consumed in metabolism • Yolk – oxygen enters and leaves by diffusion from Sander Kranebarg Wageningen University in the Netherlands

31. Domain Equations Water Main Body Yolk Di is the diffusion coefficient, p is partial pressure of O2 m is reaction due to metabolysis, subscripts denote subdomain Boundary Conditions Outer boundary (water)

32. Fish embryo - Results Oxygen partial pressure w w y mb w w y Continuity of solution across subdomain boundaries

33. Eddy currents and inductive heating

34. Problem definition • AC coil surrounding a metal cylinder • Induces eddy currents in the cylinder • Induced currents in the cylinder produce heat • Electric conductivity of the copper changes with temperature J

35. Coupled AC and Heat Equations z J • Axisymmetric • Nonlinear time-dependent model • DAE system r

36. Results Induced Currents Magnetic Field Lines

37. Results 20min 15min 10min 5min Temperature field

38. Support & Web Resources • Experienced engineering staff • Searchable knowledge base • www.comsol.com • Technical support: • support@femlab.com • Also on the web • Hands-on training courses • Seminar info • Customer stories

39. Academic Institutions • MIT • Harvard University • Stanford University • Ohio State University • Cal-Berkeley • Case Western Reserve • UCLA • University of Texas-Austin • Georgia Tech • Virginia Tech • Purdue University • University of Michigan • University of Washington • University of Notre Dame • Michigan State University • Technical University of Delft • University of Illinois-Urbana Champaign • Washington University-St. Louis • University of Maryland • University of Kentucky • Naval Postgraduate School • University of Tennessee • Vanderbilt University • University of Florida • Tennessee Tech • University of Toronto • Iowa State University • University of Alaska • University of Waterloo • Princeton University • Johns Hopkins University • JPL-Caltech

40. Government & Commercial Users of FEMLAB • Procter & Gamble • Volvo • Honda • National Research Council of Canada • Siemens • Duracell • Northrop Grumman • Chevron • Lockheed Martin • Agilent Corporation • Microsoft Corporation • 3M • Naval Research Labs • Lucent Technologies • Daimler/Chrysler • Delphi Automotive • Sensormatic • Dow Chemical Company • Conoco • Millipore Corporation • Ford Motor Company • NASA (Langley, Kennedy, Glenn) • Wright Patterson AFB • Merck • Los Alamos National Labs • Pacific Northwest National Labs • Oak Ridge National Labs • Lawrence Livermore National Labs • Sandia National Labs • National Institute of Health • Raytheon

41. PDEs

42. PDE Formulations used in FEMLAB • Coefficient form • Used for standard linear or weakly nonlinear problems • Coefficients correspond to common physical parameters (e.g., diffusion, advection, etc.) • General form • Used for nonstandard or highly nonlinear problems • Very flexible • Weak form • PDE form that is the foundation of the FEM • Integral form that gives even more flexibility (e.g., nonstandard boundary conditions)

43. Coefficient Form inside subdomain on subdomain boundary Example: Poisson’s equation inside subdomain on subdomain boundary Implies c=f=h=1 and all other coefficients are 0.

44. General Form inside domain on domain boundary For Poisson’s equation, the corresponding general form implies All other coefficients are 0.

45. Weak Form (subdomain, linear) For Poisson’s equation, the corresponding weak form implies