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Introduction Choice of Language Example Conclusion

SfePy - Introduction, Examples, and Plans

Robert Cimrman1 & others2

1Department of Mechanics & New Technologies Research CentreUniversity of West Bohemia in Plzen, Czech Republic

2other guys from around the world, currently including:Vladimır Lukes (Department of Mechanics),

Andre Smit (?, USA),Logan Sorenson (Georgia Institute of Technology, USA),

Eduard Rohan (Department of Mechanics)

3rd European meeting on Python in Science, Paris,Ecole Normale Superieure, July 8-11 2010

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Outline

1 IntroductionThe Finite Element Method OutlineIntroductionSfePy Interfaces

2 Choice of LanguageMixing Languages — Best of Both WorldsSoftware Dependencies

3 ExampleSettingProblem Description FileRunning SfePyTime-dependent ProblemHomogenization Engine

4 Conclusion

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The Finite Element Method Outline

problem: solve a partial differential equation (PDE)

on some domain on a computer

discretize ↓approximate the domain → mesh

approximate the functions u(x) ≈∑N

i=1 uiϕi(x)ui = degrees of freedom, ϕi(x) piece-wise polynomial base

the simplest case: values in mesh nodes

approximate the integral (weak) forms and their derivatives→ vectors, sparse matrices forming a linear system to solve→ vector of ui

→ →

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Introduction

SfePy = simple finite elements in Pythongeneral finite element analysis softwaresolving systems of PDEs

BSD open-source license

multi-platform (Linux, Mac OS X, Windows)

available athttp://sfepy.org (developers)

mailing lists, issue (bug) trackingwe encourage and support everyone who joins!

http://sfepy.kme.zcu.cz (project information)http://docs.sfepy.org (documentation)

selected applications:homogenization of porous media (parallel flows in a deformableporous medium)acoustic band gaps (homogenization of a strongly heterogenouselastic structure: phononic materials)acoustic waves in thin perforated layersshape optimization in incompressible flow problemsfinite element formulation of Schrodinger equation

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SfePy Interfaces

SfePy can be viewed as

collection of modules (a library)containing

FE engineproblem description facilitiesinterfaces to various solverspostprocessing utilities

usable to build custom applications

“black box” PDE solver

no real programming involvedjust prepare a problem description file and solve ithighly customizable behaviour (with a bit of coding)

this presentation is about the latter (high-level) interface

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Mixing Languages — Best of Both Worlds

low level code (C): element matrix evaluations, . . .

high level code (Python): logic of the code, particular applications,configuration files, problem description files

www.python.org

SfePy = Python + C

notable features:

small size (complete sources + documentation + examples withmeshes are just about 5.5 MB, March 2010)problem description files in pure Pythonproblem description form similar to mathematical description “onpaper”isfepy . . . interactive shell based in IPython

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Software Dependencies

to install and use SfePy, several other packages or libraries areneeded:

NumPy and SciPy: free (BSD license) collection of numericalcomputing libraries for Python

enables Matlab-like array/matrix manipulations and indexing

other: UMFPACK, Pyparsing, Matplotlib, Pytables (+ HDF5), swigvisualization of results: ParaView, MayaVi2, or any otherVTK-capable viewerisfepy requires IPython

SfePy has no built-in meshing capabilitiesFE meshes can be prepared using several free tools:

gmsh, netgen, tetgen

there are now open-source tools for everything

still, the situation is not perfect . . .

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Setting

problem description file is a regular Python module, i.e. all Pythonsyntax and power is accessibleconsists of entities defining:

fields of various FE approximations, variablesequations in the weak form, quadraturesboundary conditions (Dirichlet, periodic, “rigid body”)FE mesh file name, options, solvers, . . .

simple example: the Laplace equation:

c∆u = 0 in Ω, u = u on Γ, weak form:

∫Ω

c ∇u · ∇v = 0, ∀v ∈ V0

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Problem Description FileSolving Laplace Equation — FE Approximations

mesh → define FE approximation to Ω:

filename mesh = ’simple.mesh’

regions → define domain Ω, regions Γleft, Γright, Γ = Γleft ∪ Γright:

h omitted from now on . . .regions =

’Omega’ : (’all’, ),’Gamma Left’ : (’nodes in (x < 0.0001)’, ),’Gamma Right’ : (’nodes in (x > 0.0999)’, ),

fields → define space Vh:fields =

’temperature’ : ((1,1), ’real’, ’Omega’,

’Omega’ : ’3 4 P1’)

’3 4 P1’ means P1 approximation, in 3D, on 4-node FEs (tetrahedra)variables → define uh, vh:variables =

’u’ : (’unknown field’, ’temperature’, 0),

’v’ : (’test field’, ’temperature’, ’u’),

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Problem Description FileSolving Laplace Equation — Equations

materials → define c:

materials = ’m’ : (’c’ : 1.0,),

integrals → define numerical quadrature:

integrals = ’i1’ : (’v’, ’gauss o1 d3’)

equations → define what and where should be solved:

equations = ’eq’ : ’dw laplace.i1.Omega( m.c, v, u ) = 0’

Dirichlet BCs → define u on Γleft, Γright:

ebcs = ’t left’ : (’Gamma Left’, ’u.0’ : 2.0),’t right’ : (’Gamma Right’, ’u.all’ : -2.0),

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Running SfePy

$ s i m p l e . py examples / e x a m p l e l a p l a c e . pys f e p y : r e a d i n g mesh ( examples / s i m p l e . mesh ) . . .s f e p y : . . . done i n 0 . 0 4 ss f e p y : s e t t i n g up domain edges . . .s f e p y : . . . done i n 0 . 0 3 ss f e p y : s e t t i n g up domain f a c e s . . .s f e p y : . . . done i n 0 . 0 2 ss f e p y : c r e a t i n g r e g i o n s . . .s f e p y : Gamma Rights f e p y : Omegas f e p y : Gamma Lefts f e p y : . . . done i n 0 . 0 7 ss f e p y : e q u a t i o n ” L a p l a c e e q u a t i o n ” :s f e p y : d w l a p l a c e . i 1 . Omega( m. c , v , u ) = 0s f e p y : s e t t i n g up d o f c o n n e c t i v i t i e s . . .s f e p y : . . . done i n 0 . 0 0 ss f e p y : d e s c r i b i n g g e o m e t r i e s . . .s f e p y : . . . done i n 0 . 0 1 ss f e p y : u s i n g s o l v e r s :

n l s : newtonl s : l s

s f e p y : m a t r i x shape : ( 30 0 , 300)s f e p y : a s s e m b l i n g m a t r i x graph . . .s f e p y : . . . done i n 0 . 0 1 ss f e p y : m a t r i x s t r u c t u r a l n o n z e r o s : 3538 ( 3 . 9 3 e−02% f i l l )s f e p y : u p d a t i n g m a t e r i a l s . . .s f e p y : ms f e p y : . . . done i n 0 . 0 0 ss f e p y : u p d a t i n g v a r i a b l e s . . .s f e p y : . . . dones f e p y : n l s : i t e r : 0 , r e s i d u a l : 1 .176265 e−01 ( r e l : 1 .000000 e +00)s f e p y : r e z i d u a l : 0 . 0 0 [ s ]s f e p y : s o l v e : 0 . 0 0 [ s ]s f e p y : m a t r i x : 0 . 0 0 [ s ]s f e p y : n l s : i t e r : 1 , r e s i d u a l : 9 .709500 e−17 ( r e l : 8 .254515 e−16)

top level of SfePy code is acollection of executable scriptstailored for various applications

simple.py is dumb script ofbrute force, attempting to solveany equations it finds by theNewton method

. . . exactly what we need here(solver options were omitted inprevious slides)

↓

→

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Time-dependent Problem

adding time to the Laplace equation:

∂u

∂t− c∆u = 0 in Ω, u(·, x) = u on Γ, u(0, ·) = 0,

weak form:

∫Ω

v · ∂u

∂t+

∫Ω

c ∇u · ∇v = 0, ∀v ∈ V0

variables

variables = ’u’ : (’unknown field’, ’temperature’, 0, ’previous’),’v’ : (’test field’, ’temperature’, ’u’),

equations

equations = ’eq’ : """dw mass scalar.i1.Omega( v, du/dt )

+ dw laplace.i1.Omega( m.c, v, u ) = 0"""

+ define time-stepping solver options

↓

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Homogenization Engine

theory of homogenization:

derive macroscopic material parameters from microstructure(topology)kind of “data compression” to avoid brute force

3Y

2Y

1Y1Γ

2ΓΩ

allows recovery of microscopic quantities (e.g. stresses)computing homogenized coefficients requires solution of manyauxiliary problems to get corrector functions

→ homogenization engine

to express data dependencies among subproblems and coefficients

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Conclusion

what is done:basic finite element (FE) engine:

variables, boundary conditions, FE assemblingequations, terms, regionsmaterials, material caches

homogenization enginevarious solvers accessed via abstract interfaceunit tests, sphinx-based documentationmany linear and nonlinear example problems (usually multiphysical)

immediate plans (work in progress):decouple high level API from FE engineimprove interactive API (isfepy)even better documentationfast problem-specific solvers (!)

other plans:adaptive mesh refinement (!) - couple with hermes(http://hpfem.org)parallelization (petsc4py)

http://sfepy.org

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This is not a slide!