Monifysikaalisten ongelmien simulointi Elmer-ohjelmistolla. Simulation of Multiphysical Problems with Elmer Software

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1 Monifysikaalisten ongelmien simulointi Elmer-ohjelmistolla Simulation of Multiphysical Problems with Elmer Software Peter Råback Tieteen CSC

2 Definitions for this presentation Model Mathematical description of a system and its numerical implementation. In this presentation the models are based on the well established laws of (classical) physics. Simulation The process of conducting numerical experiments with the model for the purpose of understanding the behavior of the system and/or evaluating various strategies for the operation of the system.

3 Outline Multiphysics Elmer Examples Conclusions

4 What is Multiphysics... when physics is already everything? Computational problems historically solved with application specific software (fluids, structures, electromagnetics,... ) With the increase of computational resources also problems with more than one physical phenomenom have become feasible Software capable of solving these problems are known as multiphysics software (Ansys, FemLab, CFD-ACE+, Elmer,... ) Basically multiphysics is nothing new, most common coupled problems have been solved for decades (e.g. fluid flow and heat transport)

5 Why Multiphysics? Because now we can! Many industrial processes involve a number of different phenomena Paper manufacturing Casting processes Crystal growth... There are new needs in some emerging fields where the computational tool is a necessity Microsystem technology Nanotechnology Smart materials

6 Coupling of different phenomena Physical equations may be coupled in a number of different ways: Equations are inherently coupled and the coupling is explicitly shown in the equations (E and B in Maxwell s eq., p and v in Navier-Stokes eq.). The different energy domains are coupled by a source or drain (viscous dissipation of kinetic energy to heat). Coupling by material law (v and T by temperature dependent density). Implicit coupling by the shape of the computational domain (large displacements fluid flow)....

7 Equation Field T v E, B c u Energy Temperature T - Navier-Stokes Velocity v 1 - Maxwell s Electric & Magnetic E, B Diffusion, Reaction Concentration c Elasticity Displacement u Thermal flow: natural convection 2. Thermal-electrical: Heating by induction 3. Magnetohydrodynamics, Electrokinetics (microfluidics) 4. Temperature dependent chemical reactions and diffusion 5. Reactive flow: CVD, combustion 6. Electrochemistry: batteries, electrodes, surface treatment 7. Thermoelasticity and -plasticity 8. Fluid-structure interaction: hemodynamics 9. Electro-mechanical: MEMS, piezoelectricity 10. Growth phenomena

8 Solution strategies: simple analogy Let s look at phenomena A and B that both depend on field variables a and b. The solution may be obtained from a coupled system of equations: { A(a, b) = 0 B(b, a) = 0 In order to solve the equations we perform linearization, { Aa da + A b db +... = A(a, b) B a da + B b db +... = B(b, a) We also assume that A is the more independent subproblem to be solved first.

9 strong coupling: A 1 a A b B 1 b B a 1 [ a (m+1) b (m+1) ] = [ a (m) b (m) ] [ Aa A b B a B b ] 1 [ A(a m, b m ) B(b m, a m ) ] (monolithic solution of equations) weak coupling: A 1 a A b B 1 b B a 1 { a (m+1) = a (m) A 1 a A(a m, b m ) b (m+1) = b (m) B 1 b B(b m, a m+1 ) (iterative or segregated solution of equations)

10 hierarchical coupling: A b 0 (equations are solved only once) Note: { a (1) = a (0) A 1 a A(a 0 ) b (1) = b (0) B 1 b B(b 0, a 1 ) If A or B are nonlinear they require iteration within. Usually the computation of the derivatives in respect to a and b is not practicle, or even possible. In practice the convergence of the iteration scheme is usually empirically tested.

11 Iterative vs. monolithic solution method Monolithic models are seldom used as they are difficult to implement and often real memory hogs Iteration method can easily be used if there are separate solvers for all equations: equations are solved one by one until the solution satisfies the convergence criteria of all the equations Applicability of the iteration method may be enhanced by under-relaxation or by modifying the equations Verification of the iteration method is straight-forward: prove consistency of each equation and reach convergence. In pathologically coupled cases one may have to resort to the monolithic method.

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