1. Multiphysics Modeling in FEMLAB 3 Tuesday, September 14th Woods hole Remi Magnard

2. FEMLAB-overview Introduction to the FEMLAB modeling : diffusion- reaction in a zebra fish embryo Questions & Demo models on request from audience Contents

3. What is FEMLAB used for? Structural mechanics Heat transfer Acoustics Electromagnetics Fluid Flow Chemical Engineering Geophysics General multiphysics and PDEs

4. Structure of FEMLAB FEMLAB core package –GUI, CAD tools, mesher, solvers, post-processing –Basic physics Modes –Arbitrarily-defined Equations Application Specific Modules –Electromagnetics Module Statics, eddy currents, microwaves, photonics,… –Structural Mechanics Module Solids (including thermal stresses, large deformations, elasto-plasticity, buckling), beams, plates, shells,… –Chemical Engineering Module Incompressible NS flow, compressible Euler flow, chemical reactions,… electrochemistry, porous media flows,… Chemical Engineering Module Electromagnetics Module Structural Mechanics Module Heat Transfer Module MEMS Module Earth Science module

5. Equation implementation within FEMLAB Coefficient form: General form: Weak form: -ux_test*ux-y_test*uy+u_test

6. Diffusion-Reaction in a Zebra Fish Embryo

7. Introduction This model shows the modeling of diffusion-reaction processes in FEMLAB. The model shows the possibility of using different properties and different equations in different subdomains. The example is originally defined by Sander Kranebarg from the Wageningen University in the Netherlands.

8. Model Geometry The geometry consists of three different subdomains: –Main body –Yolk –Surrounding water. Problem symmetric => only ½ of the geometry modeled

9. Problem Definition, diffusion Diffusion coefficient of the fish embryo : 5e -12 Diffusion coefficient of the ambient water : 1e -12 Reaction rate in the main body : -2.5e -4 Boundary condition : fixed concentration on the exterior boundaries : c=16 Symmetry boundary conditions :

10. Problem Definition, equation implementation Coefficient form : da=alpha=gamma=beta=a=0 c = diffusion coefficient f = reaction rate Boundary condition:

11. Result, Oxygen partial pressure

12. Conclusions The model is very simple to define and solve in FEMLAB. The results agree with the experiments and simulations done by Sander Kranenbarg at the Wageningen University. The model can be easily expanded to include effects of convection, which in the model are accounted for by an effective diffusivity. Straightforward equation implementation in the GUI

13. New features in FEMLAB 3.1 (released mid-October) Geometry –NURBS supported by IGES import –Live connection to SolidWorks Meshing –Support for Quad/Brick/Prism –Structured mesh Solvers –64 bit FEMLAB server on several platforms –Multigrid preconditioner –Electromagnetics preconditioner based on multigrid Structural Mechanics Module –Piezo application mode –Incompressible materials –Heat-transfer shell Electromagnetics Module –Periodic boundary condition for vector element –Far field postprocessing –3D hybrid modes for waveguide –S-parameter postprocessing –Maxwell stresses postprocessing –Application mode with 2D quasi- static formulation –Floating potential –Nonlinear magnetic materials (example models) Chemical Engineering Module –3D k-epsilon application mode –Divergence free elements

14. 3 New modules! Heat transfer Module: –Conduction, convection and radiation modeling –Highly conductive layer boundary condition and heat transfer in shell –Bio-heat equation Earth-science module: –Porous media flow in variably saturate substrate –Poroelasticity model –Heat transfer in porous media MEMS module: –Piezoelectric material –Microfluidic and micromechanic application : micro-valve model, sensor.

15. Contact Information: Jeanette Littmarck Regional Sales Manager COMSOL, Inc 781-273-3322 jeanette@comsol.com Visit www.comsol.com jeanette@comsol.com

16. Extra slides

17. PDE Formulations used in FEMLAB Can be used for scalar equations or systems (e.g., vector equations) –Note: coefficients may become operators of higher degree, or tensors 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) –The Lagrange multipliers are solved for explicitly

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

19. General Form inside domain on domain boundary For Poisson’s equation, the corresponding general form implies All other coefficients are 0. (For later, note: )

20. Remember for Poisson’s eq: G =[-ux -uy], F=1, R=u (u constr to 0) Weak Form (stationary) General form Multiply by test function v and integrate Perform integration by parts on left hand side Rearranged Subdomain, weak: -ux_test*ux-uy_test*uy+u_test Boundary, constr: u

21. Weak form, cont. We have but we know from the Neumann condition that Hence we have the system –Problem: How to use the constraint R=u-u 0 =0 in the weak equation? –Say, if G=0, then this corresponds to -n  =0. But, R=0 might contradict this. Can we pick a G such that R=0 is automatically fulfilled? –Solution, Lagrange multiplier  in other words: Adjust the boundary flux (force, surface charge etc) such that the Dirichlet condition is fulfilled. This is done by the Lagrange multiplier in FEMLAB This is the full weak formulation of the original PDE problem omit for now