Elmer

CSC – Tieteen tietotekniikan keskus Oy CSC – IT Center for Science Ltd.

Introduction into Elmer multi-physics FEM package

Thomas Zwinger

Contents• Elmer – Background

– History

– users, community

– contacts and links

• FEM formulation

– Weak form

– Test and weight functions

– Standard Galerkin• Elmer – Features

– Implemented models

– parallel computations

• Elmer – Modules

– Package modules

– Workflow

– Standard Galerkin

– Stabilization

• Something on numerics

– Elements

– Pre-conditioners and Solvers

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 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, …)

Elmer – Background ctd.• In September 2005 Elmer was published

under GNU Public Licence

• Goals of the open source publication:

– Expand the Elmer community– Expand the Elmer community

– New resources for code development

– Improved verification process

– Free software good adverticiment for CSC

• Roughly 300000 lines of code!

– IP stays with CSC

Elmer – Background

Elmer – BackgroundUse the force!

– elmerfem.org

• Wiki, Forum

– www.csc.fi/elmer– www.csc.fi/elmer

• Main Elmer page

– sourceforge.net

• Main distribution channel

– www.nic.funet.fi

• Manuals and binaries

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

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

Elmer – Features parallel

Elmer - Modules

Pre-processing

• Generating geometry• Meshing the geomtry• Determining the physical and numerical simulation parameter

• Launching the simulation

Solution• Numbercrunching• Monitoring of convergence

Post-processing

• Visualization of results (3D, 4D)• Data output

ElmerGUI/ElmerGrid ElmerSolver

Elmer - ModulesElmerPost

Elmer - Modules

FEM formulation• Stokes equation:

• Integration over volume using test-function :

FEM formulation• Partial integration of l.h.s.:

• Divergence theorem on first term l.h.s.:

FEM formulation

• Weak formulation:

• Linearization of deviatoric stress:

FEM formulation

• Discretization by weighting function :

FEM formulation

• Standard Galerkin:

Sαβ xβ = fα

FEM formulation

FEM formulation

Real Geometry

Coordinate Coordinate system

Internal

FEM formulation

• Saddle point problem:

• S is not positive definite ↔ no minimization problem

• Usual way of solution: X= S-1F

– BUT: is S-1 ”well behaving”?

FEM formulation

• What’s the problem?:

– First row:

– Insert into 2nd row:

FEM formulation• Condition for stability:

– Depends on the space of the test-functions– Depends on the space of the test-functions

– Bad news: standard Galerkin + Stokes

= not stable

– Stabilization. Methods in Elmer:

1. Residual square methods

2. Residual free bubbles

FEM formulation

• Residual square method

= Stabilized Finite Elements

– Simplified example:

– Weak formulation (energy minimization):– Weak formulation (energy minimization):

– Adding (element-wise) residual of equation:

FEM formulation

• Residual free bubbles:

– Additional degrees of freedom

– Locally on elements

– Adding to the orginal system (condensation)– Adding to the orginal system (condensation)

equivalent to RSM

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.,

Something on numerics• Weak formulation:

– Time integration

– Steady State:dependence on other variables

– Non-linear iteration: internal dependence

Something on numerics

Something on numerics• Time integration – two different methods:

– Crank – Nicholson: Crank – Nicholson

– Backward Difference Formula: BDF

• BDF Order (if 1, then backward Euler – only • 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

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

Something on numerics• Nonlinear Problem:

– Nonlinear System Convergence Tolerance

– Nonlinear System Max Iterations

– Nonlinear System Relaxation Factor

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)

Something on numerics

• But, before we solve, we usually apply a pre-conditioner– Find P~A, but much easier to invert

– P-1A ~ I has favourable condition number– P-1A ~ I has favourable condition number

Linear System Preconditioning

• None

• Diagonal

• ILU n (n=0,1,2,...) and ILUT

Something on numerics

• Direct linear system solver

• Keyword: Linear System Direct Method

• Banded (default) LAPack

• UMFPACKUnsymmetric MultiFrontal method (only serial)

• MUMPSUnsymmetric MultiFrontal method (only parallel)

– Sometimes the only way to go (if bad conditioned)

– Costly: Elimination takes ~N3 operations and needs to store N2 unknows in memory

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