Elmer. Introduction into Elmer multiphysics. Thomas Zwinger. CSC Tieteen tietotekniikan keskus Oy CSC IT Center for Science Ltd.

Transcription

1 Elmer Introduction into Elmer multiphysics FEM package Thomas Zwinger CSC Tieteen tietotekniikan keskus Oy CSC IT Center for Science Ltd.

2 Contents Elmer Background History users, community contacts and links Elmer Features Implemented models parallel computations Elmer Modules Package modules Workflow FEM formulation Weak form Test and weight functions Standard Galerkin Stabilization Something on numerics Elements Pre-conditioners and Solvers

3 Elmer - Background Solution of partial differential equations by FEM (Finite Element Method) Elmer development was started in 1995 as part of a national CFD program, also funded by Tekes (Finnish technol. agency) After the initial phase the development has been driven by applications (MEMS, Glaciology, medical application, )

4 Elmer Background ctd. In September 2005 Elmer was published under GNU Public Licence Goals of the open source publication: Expand the Elmer community New resources for code development Improved verification process Free software good adverticiment for CSC Roughly lines of code! IP stays with CSC

5 Elmer Background

6 Elmer Background Use the force! elmerfem.org Wiki, Forum Main Elmer page sourceforge.net Main distribution channel Manuals and binaries

7 Elmer - Features Fluid Mechanics: RANS, VMS, Reynolds, free surfaces Structural Mechanics: non- /linear elasticity, plates / Heat Transfer: phase change Electro-Magnetics Accoustics: Helmholtz equation Quantum Chemistry: DFT

8 Elmer Features parallel Scaling of wall clock time with dofs in the cavity lid case using GMRES+ILU0. Simulation Juha Ruokolainen, CSC, visualization Matti Gröhn, CSC. Parallel Runs: using MPI; interfaces to external libraries: Hypre, MUMPS, Pardiso, SuperLU

9 Elmer Features parallel

10 Elmer - Modules Generating geometry Meshing the geomtry Determining the physical and numerical simulation parameter Solution Launching the simulation Numbercrunching Monitoring of convergence Preprocessing Postprocessing Visualization of results (3D, 4D) Data output

11 Elmer - Modules ElmerGUI/ElmerGrid ElmerSolver ElmerPost

12 Elmer - Modules

13 FEM formulation Stokes equation: Integration over volume using testfunction :

14 FEM formulation Partial integration of l.h.s.: Divergence theorem on first term l.h.s.:

15 FEM formulation Weak formulation: Linearization of deviatoric stress:

16 FEM formulation Discretization by weighting function :

17 FEM formulation Standard Galerkin: S αβ x β = f α

18 FEM formulation

19 FEM formulation Real Geometry Coordinate system Internal

20 FEM formulation Saddle point problem: S is not positive definite no minimization problem Usual way of solution: X= S -1 F BUT: is S -1 well behaving?

21 FEM formulation What s the problem?: First row: Insert into 2nd row:

22 FEM formulation Condition for stability: Depends on the space of the test-functions Bad news: standard Galerkin + Stokes Stabilization. Methods in Elmer: 1. Residual square methods 2. Residual free bubbles = not stable

23 FEM formulation Residual square method = Stabilized Finite Elements Simplified example: Weak formulation (energy minimization): Adding (element-wise) residual of equation:

24 FEM formulation Residual free bubbles: Additional degrees of freedom Locally on elements Adding to the orginal system (condensation) equivalent to RSM

25 Something on numerics General advection-diffusion problem For instance, heat transfer problem: Coupled to (Navier-)Stokes via velocity: u Non-linearities via material parameters, e.g.,

26 Something on numerics Weak formulation: Time integration Steady State:dependence on other variables Non-linear iteration: internal dependence

27 Something on numerics

28 Something on numerics Time integration two different methods: Crank Nicholson: Crank Nicholson Backward Difference Formula: BDF BDF Order (if 1, then backward Euler only choice for adaptive time-stepping) Additional settings: Time Derivative Order (if 2 then Bossak) Timestep Intervals Timestep Sizes

29 Something on numerics Steady State Problem Mutual dependence between two segregated Solvers (e.g., Flow solution and convected temperature) Steady State Convergence Tolerance Steady State Max Iterations Steady State Relaxation Factor

30 Something on numerics Nonlinear Problem: Nonlinear System Convergence Tolerance Nonlinear System Max Iterations Nonlinear System Relaxation Factor

31 Something on Numerics Solving the Linear(ized) Problem Keyword: Linear System Iterative Method 3 ways to do that in Elmer: 1. Direct methods (= inversion of A) 2. Iterative methods (=working with approximations to A) 3. Multi-grid methods (not discussed here)

32 Something on numerics But, before we solve, we usually apply a pre-conditioner Find P~A, but much easier to invert P -1 A ~ I has favourable condition number Linear System Preconditioning None Diagonal ILUn (n=0,1,2,...) and ILUT

33 Something on numerics Direct linear system solver Keyword: Linear System Direct Method Banded (default) LAPack UMFPACK Unsymmetric MultiFrontal method (only serial) MUMPS Unsymmetric MultiFrontal method (only parallel) Sometimes the only way to go (if bad conditioned) Costly: Elimination takes ~N 3 operations and needs to store N 2 unknows in memory

34 Something on numerics Iterative solvers: Krylov subspace: Linear System Iterative Method GMRES Generalized Minimal Residual Method CG, CGS, BiCGStab Conjugate Gradient TFQMR Transpose-free quasi-minimal residual GCR Generalized Conjugate Residual