Python + FEM - KME + FEM Introduction to SFE ... low-level ⇒ lots of lines to get ... and data files in HDF5: general purpose library and file format for storing scientific

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Introduction Our choice Example Problem Final slide :-)

Python + FEMIntroduction to SFE

Robert Cimrman

Department of Mechanics & New Technologies Research CentreUniversity of West BohemiaPlzen, Czech Republic

April 3, 2007, Plzen

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Outline

1 IntroductionProgramming LanguagesProgramming Techniques

2 Our choiceMixing Languages — Best of Both WorldsPythonSoftware Dependencies

3 Example ProblemContinuous FormulationWeak FormulationProblem Description File

4 Final slide :-)

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Programming Languages1 compiled (fortran, C, C++, Java, . . . )

Pros

speed, speed, speed, . . . , did Isay speed?

large code base (legacy codes),tradition

Cons

(often) complicated build process,recompile after any change

low-level ⇒ lots of lines to getbasic stuff done

code size ⇒ maintenance problemsstatic!

2 interpreted or scripting (sh, tcl, matlab, perl, ruby, python, . . . )

Pros

no compiling

(very) high-level ⇒ a few of linesto get (complex) stuff done

code size ⇒ easy maintenancedynamic!

(often) large code base

Cons

many are relatively new

lack of speed, or even utterly slow!

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Pros



Cons





Pros

no compiling




Cons



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Pros



Cons





Pros

no compiling




Cons



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Pros



Cons





Pros

no compiling




Cons



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Pros



Cons





Pros

no compiling




Cons



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Pros



Cons





Pros

no compiling




Cons



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Example Python Code

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Example C Code

char s [ ] = ” s e l tudy , mel dudy . a s e l opravdu tudy . ” ;

i n t s t r f i n d ( char ∗ s t r , char ∗ t g t )

i n t t l e n = s t r l e n ( t g t ) ;i n t max = s t r l e n ( s t r ) − t l e n ;r e g i s t e r i n t i ;

f o r ( i = 0 ; i < max ; i ++) i f ( strncmp(& s t r [ i ] , tgt , t l e n ) == 0)

re tu rn i ;re tu rn −1;

i n t i 1 ;

i 1 = s t r f i n d ( s , ” tudy ” ) ;p r i n t f ( ”%d\n” , i 1 ) ;

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Programming Techniques

Random remarks:

spaghetti code

modular programming

OOP

design patterns (gang-of-four, GO4)

post-OOP — dynamic languages

. . . all used, but I try to move down the list! :-)

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Programming Techniques

Random remarks:

spaghetti code

modular programming

OOP

design patterns (gang-of-four, GO4)

post-OOP — dynamic languages

. . . all used, but I try to move down the list! :-)

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Mixing Languages — Best of Both Worlds

Low level: C

FE matrix evaluations

costly mesh-related functions (listing faces, edges, surfacegeneration, . . . )

. . .

can use also fortran

+High level: Python

www.python.org“batteries included”

logic of the code

configuration files!

problem input files!

http://www.python.org

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Mixing Languages — Best of Both Worlds

Low level: C

FE matrix evaluations

costly mesh-related functions (listing faces, edges, surfacegeneration, . . . )

. . .

can use also fortran

+High level: Python

www.python.org“batteries included”

logic of the code

configuration files!

problem input files!

http://www.python.org

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Python I

Python r©is a dynamic object-oriented programminglanguage that can be used for many kinds of softwaredevelopment. It offers strong support for integration with otherlanguages and tools, comes with extensive standard libraries,and can be learned in a few days. Many Python programmersreport substantial productivity gains and feel the languageencourages the development of higher quality, moremaintainable code.

. . .

“batteries included”

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Python II

1.1.16 Why is it called Python?At the same time he began implementing Python, Guido

van Rossum was also reading the published scripts from ”MontyPython’s Flying Circus” (a BBC comedy series from theseventies, in the unlikely case you didn’t know). It occurred tohim that he needed a name that was short, unique, and slightlymysterious, so he decided to call the language Python.

1.1.17 Do I have to like ”Monty Python’s Flying Circus”?

No, but it helps. :)

. . . General Python FAQ

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Python III

NASA uses Python. . .

. . . so does Rackspace, Industrial Light and Magic, AstraZeneca,Honeywell, and many others.

http://wiki.python.org/moin/NumericAndScientific


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Python III

NASA uses Python. . .

. . . so does Rackspace, Industrial Light and Magic, AstraZeneca,Honeywell, and many others.



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Software Dependencies

1 libraries & modules SciPy: free (BSD license) collection of numerical computinglibraries for Python

enables Matlab-like array/matrix manipulations and indexing UMFPACK: very fast direct solver for sparse systems

Python wrappers are part of SciPy :-) output files (results) and data files in HDF5:

general purpose library and file format for storing scientificdata, efficient storage and I/O. . . via PyTables

misc: Pyparsing, Matplotlib2 tools

SWIG: automatic generation of Python-C interface code Medit: simple mesh and data viewer ParaView: VTK-based advanced mesh and data viewer inTcl/Tk

MayaVi/MayaVi2: VTK-based advanced mesh and data viewerin Python

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Shape Optimization in Incompressible Flow ProblemsObjective Function

ΨΓ(u) ≡ν

2

∫Ωc|∇u|2 −→ min (1)

minimize gradients of solution (e.g. losses) in Ωc ⊂ Ωby moving design boundary Γ ⊂ ∂Ω

perturbation of Γ by vector field V

Ω(t) = Ω + tV(x)x∈Ω where V = 0 in Ωc ∪ ∂Ω \ Γ (2)

Γin ΓoutΓD

ΓD

ΓC ΩC

ΩD

Ω

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ΨΓ(u) ≡ν

2

∫Ωc|∇u|2 −→ min (1)




Γin ΓoutΓD

ΓD

ΓC ΩC

ΩD

Ω

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ΨΓ(u) ≡ν

2

∫Ωc|∇u|2 −→ min (1)




Γin ΓoutΓD

ΓD

ΓC ΩC

ΩD

Ω

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Appetizerflow and spline boxes, left: initial, right: final

→

connectivity of spline boxes (6 boxes in different colours)

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Continuous Formulation

laminar steady state incompressible Navier-Stokes equations

−ν∇2u + u · ∇u +∇p = 0 in Ω (3)

∇ · u = 0 in Ω (4)

boundary conditions

u = u0 on Γin

−pn + ν∂u∂n= −pn on Γout

u = 0 on Γwalls

(5)

adjoint equations for computing the adjoint solution w :

−ν∇2w +∇u · w − u · ∇w −∇r = χΩc ν∇2u in Ω∇ · w = 0 in Ω

(6)

w = 0 on Γin ∪ Γwalls and χΩc is the characteristic function of Ωc

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Continuous Formulation

laminar steady state incompressible Navier-Stokes equations

−ν∇2u + u · ∇u +∇p = 0 in Ω (3)

∇ · u = 0 in Ω (4)

boundary conditions

u = u0 on Γin

−pn + ν∂u∂n= −pn on Γout

u = 0 on Γwalls

(5)

adjoint equations for computing the adjoint solution w :

−ν∇2w +∇u · w − u · ∇w −∇r = χΩc ν∇2u in Ω∇ · w = 0 in Ω

(6)

w = 0 on Γin ∪ Γwalls and χΩc is the characteristic function of Ωc

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Weak Formulation

state problem: find u, p (from suitable function spaces) such that∫Ω

ν∇v : ∇u +∫Ω(u · ∇u) · v −

∫Ωp∇ · v = −

∫Γoutpv · n ∀v ∈ V0 ,∫

Ωq∇ · u = 0 ∀q ∈ Q

(7)

adjoint equations: find w, r (from suitable function spaces) such that∫Ω

ν∇v : ∇w +∫Ω(v · ∇u) · w +

∫Ω(u · ∇v) · w +

∫Ωr∇ · v

= −∫Ωc

ν∇v : ∇u ∀v ∈ V0 ,∫Ωq∇ · w = 0 ∀q ∈ Q

(8)

+ proper BC . . .

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Weak Formulation


ν∇v : ∇u +∫Ω(u · ∇u) · v −

∫Ωp∇ · v = −


Ωq∇ · u = 0 ∀q ∈ Q

(7)


ν∇v : ∇w +∫Ω(v · ∇u) · w +

∫Ω(u · ∇v) · w +

∫Ωr∇ · v

= −∫Ωc

ν∇v : ∇u ∀v ∈ V0 ,∫Ωq∇ · w = 0 ∀q ∈ Q

(8)

+ proper BC . . .

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Weak Formulation


ν∇v : ∇u +∫Ω(u · ∇u) · v −

∫Ωp∇ · v = −


Ωq∇ · u = 0 ∀q ∈ Q

(7)


ν∇v : ∇w +∫Ω(v · ∇u) · w +

∫Ω(u · ∇v) · w +

∫Ωr∇ · v

= −∫Ωc

ν∇v : ∇u ∀v ∈ V0 ,∫Ωq∇ · w = 0 ∀q ∈ Q

(8)

+ proper BC . . .

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Problem Description File I

a regular python file ⇒ all python machinery is accessible!requires/recognizes several keywords to define

meshregions → subdomains, boundaries, characteristic functions, . . .boundary conditions → Dirichletfields → field approximation per subdomainvariables → unknown fields, test fields, parametersequationsmaterials→ constitutive relations, e.g. newton fluid, elastic solid, . . .solver parameters → nonlinear solver, flow solver, optimizationsolver, . . .. . .

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Problem Description File I

a regular python file ⇒ all python machinery is accessible!requires/recognizes several keywords to define

meshregions → subdomains, boundaries, characteristic functions, . . .boundary conditions → Dirichletfields → field approximation per subdomainvariables → unknown fields, test fields, parametersequationsmaterials→ constitutive relations, e.g. newton fluid, elastic solid, . . .solver parameters → nonlinear solver, flow solver, optimizationsolver, . . .. . .

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Problem Description File IImesh

f i l eName mesh = ’ s imp l e . mesh ’

regionsr e g i o n 1000 =

’ name ’ : ’ Omega ’ ,’ s e l e c t ’ : ’ e l ement s o f group 6 ’ ,

r e g i o n 0 =

’ name ’ : ’ Wal ls ’ ,’ s e l e c t ’ : ’ nodes o f s u r f a c e −n r . Out le t ’ ,

r e g i o n 1 =

’ name ’ : ’ I n l e t ’ ,’ s e l e c t ’ : ’ nodes by c i n c ( x , y , z , 0 ) ’ ,

r e g i o n 2 =

’ name ’ : ’ Out l e t ’ ,’ s e l e c t ’ : ’ nodes by c i n c ( x , y , z , 1 ) ’ ,

r e g i o n 100 =

’ name ’ : ’ Omega C ’ , # Con t r o l domain .’ s e l e c t ’ : ’ nodes i n ( x > −25.9999e−3) & ( x < −18.990e−3) ’ ,

r e g i o n 101 =

’ name ’ : ’ Omega D ’ , # Des ign domain .’ s e l e c t ’ : ’ a l l −e r . Omega C ’ ,

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Problem Description File III

fieldsf i e l d 1 =

’ name ’ : ’ 3 v e l o c i t y ’ ,’ dim ’ : ( 3 , 1 ) ,’ f l a g s ’ : ( ’ BQP’ , ) ,’ bases ’ : ( ( ’ Omega ’ , ’ 3 4 P1B ’ ) , )

f i e l d 2 = ’ name ’ : ’ p r e s s u r e ’ ,’ dim ’ : ( 1 , 1 ) ,’ f l a g s ’ : ( ) ,’ bases ’ : ( ( ’ Omega ’ , ’ 3 4 P1 ’ ) , )

variablesv a r i a b l e s =

’ u ’ : ( ’ f i e l d ’ , ’ parameter ’ , ’ 3 v e l o c i t y ’ , ( 3 , 4 , 5 ) ) ,’ w 0 ’ : ( ’ f i e l d ’ , ’ parameter ’ , ’ 3 v e l o c i t y ’ , ( 3 , 4 , 5 ) ) ,’w ’ : ( ’ f i e l d ’ , ’ unknown ’ , ’ 3 v e l o c i t y ’ , ( 3 , 4 , 5 ) , 0 ) ,’ v ’ : ( ’ f i e l d ’ , ’ t e s t ’ , ’ 3 v e l o c i t y ’ , ( 3 , 4 , 5 ) , ’ w’ ) ,’ p ’ : ( ’ f i e l d ’ , ’ parameter ’ , ’ p r e s s u r e ’ , ( 9 , ) ) ,’ r ’ : ( ’ f i e l d ’ , ’ unknown ’ , ’ p r e s s u r e ’ , ( 9 , ) , 1 ) ,’ q ’ : ( ’ f i e l d ’ , ’ t e s t ’ , ’ p r e s s u r e ’ , ( 9 , ) , ’ r ’ ) ,’Nu ’ : ( ’ f i e l d ’ , ’ meshVe loc i t y ’ , ’ p r e s s u r e ’ , ( 9 , ) ) ,

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Problem Description File III

fieldsf i e l d 1 =

’ name ’ : ’ 3 v e l o c i t y ’ ,’ dim ’ : ( 3 , 1 ) ,’ f l a g s ’ : ( ’ BQP’ , ) ,’ bases ’ : ( ( ’ Omega ’ , ’ 3 4 P1B ’ ) , )

f i e l d 2 = ’ name ’ : ’ p r e s s u r e ’ ,’ dim ’ : ( 1 , 1 ) ,’ f l a g s ’ : ( ) ,’ bases ’ : ( ( ’ Omega ’ , ’ 3 4 P1 ’ ) , )

variablesv a r i a b l e s =

’ u ’ : ( ’ f i e l d ’ , ’ parameter ’ , ’ 3 v e l o c i t y ’ , ( 3 , 4 , 5 ) ) ,’ w 0 ’ : ( ’ f i e l d ’ , ’ parameter ’ , ’ 3 v e l o c i t y ’ , ( 3 , 4 , 5 ) ) ,’w ’ : ( ’ f i e l d ’ , ’ unknown ’ , ’ 3 v e l o c i t y ’ , ( 3 , 4 , 5 ) , 0 ) ,’ v ’ : ( ’ f i e l d ’ , ’ t e s t ’ , ’ 3 v e l o c i t y ’ , ( 3 , 4 , 5 ) , ’ w’ ) ,’ p ’ : ( ’ f i e l d ’ , ’ parameter ’ , ’ p r e s s u r e ’ , ( 9 , ) ) ,’ r ’ : ( ’ f i e l d ’ , ’ unknown ’ , ’ p r e s s u r e ’ , ( 9 , ) , 1 ) ,’ q ’ : ( ’ f i e l d ’ , ’ t e s t ’ , ’ p r e s s u r e ’ , ( 9 , ) , ’ r ’ ) ,’Nu ’ : ( ’ f i e l d ’ , ’ meshVe loc i t y ’ , ’ p r e s s u r e ’ , ( 9 , ) ) ,

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Problem Description File IV

Dirichlet BCebc =

’ Wal ls ’ : ( ( ’ Wal ls ’ , ( 3 , 4 , 5 ) , 0 . 0 ) , ) ,’ I n l e t ’ : ( ( ’ V e l o c i t y I n l e t x ’ , ( 4 , ) , 1 . 0 ) ,

( ’ V e l o c i t y I n l e t y z ’ , ( 3 , 5 ) , 0 . 0 ) ) ,

state equations (compare with (7))

e qua t i o n s = ’ ba lance ’ :”””+ dw d i v g r ad . Omega ( f l u i d , v , w )+ dw convect . Omega ( v , w )− dw grad .Omega ( v , r )= − dw s u r f a c e l t r . Ou t l e t ( bp re s s , v ) ””” ,

’ i n c om p r e s s i b i l i t y ’ :””” dw div . Omega ( q , w ) = 0””” ,

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Problem Description File IV

Dirichlet BCebc =

’ Wal ls ’ : ( ( ’ Wal ls ’ , ( 3 , 4 , 5 ) , 0 . 0 ) , ) ,’ I n l e t ’ : ( ( ’ V e l o c i t y I n l e t x ’ , ( 4 , ) , 1 . 0 ) ,

( ’ V e l o c i t y I n l e t y z ’ , ( 3 , 5 ) , 0 . 0 ) ) ,

state equations (compare with (7))

e qua t i o n s = ’ ba lance ’ :”””+ dw d i v g r ad . Omega ( f l u i d , v , w )+ dw convect . Omega ( v , w )− dw grad .Omega ( v , r )= − dw s u r f a c e l t r . Ou t l e t ( bp re s s , v ) ””” ,


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Problem Description File V

adjoint equations (compare with (8))e q u a t i o n s a d j o i n t =

’ ba lance ’ :”””+ dw d i v g r ad . Omega ( f l u i d , v , w )+ dw ad j convec t1 . Omega ( v , w , u )+ dw ad j convec t2 . Omega ( v , w , u )+ dw grad .Omega ( v , r )= − dw ad j d i v g r a d 2 . Omega C ( one , f l u i d , v , u ) ””” ,


materialsma t e r i a l 2 =

’ name ’ : ’ f l u i d ’ ,’mode ’ : ’ here ’ ,’ r e g i on ’ : ’ Omega ’ ,’ v i s c o s i t y ’ : 1 . 2 5 e−1,

ma t e r i a l 1 0 0 =

’ name ’ : ’ bp re s s ’ ,’mode ’ : ’ here ’ ,’ r e g i on ’ : ’ Omega ’ ,’ va l ’ : 0 . 0 ,

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Problem Description File V

adjoint equations (compare with (8))e q u a t i o n s a d j o i n t =

’ ba lance ’ :”””+ dw d i v g r ad . Omega ( f l u i d , v , w )+ dw ad j convec t1 . Omega ( v , w , u )+ dw ad j convec t2 . Omega ( v , w , u )+ dw grad .Omega ( v , r )= − dw ad j d i v g r a d 2 . Omega C ( one , f l u i d , v , u ) ””” ,


materialsma t e r i a l 2 =

’ name ’ : ’ f l u i d ’ ,’mode ’ : ’ here ’ ,’ r e g i on ’ : ’ Omega ’ ,’ v i s c o s i t y ’ : 1 . 2 5 e−1,

ma t e r i a l 1 0 0 =

’ name ’ : ’ bp re s s ’ ,’mode ’ : ’ here ’ ,’ r e g i on ’ : ’ Omega ’ ,’ va l ’ : 0 . 0 ,

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Yes, the final slide!

What is done

basic FE element engineapproximations up to P2 onsimplexes (possibly withbubble)Q1 tensor-productapproximation on rectangles

handling of fields and variablesboundary conditions, DOFhandling

FE assembling

equations, terms, regions

materials, material caches

generic solvers (Newton iteration,direct linear system solution)

What is not donegeneral FE engine, with symbolicevaluation (a la SyFi - merge it?)

documentation, unit testsorganization of solvers

fast problem-specific solvers

parallelization of both assemblingand solving

use PETSC?

What will not be done (?)

GUI

real symbolic parsing/evaluation ofequations

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What is done



FE assembling








use PETSC?


GUI


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What is done



FE assembling








use PETSC?


GUI


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This is not a slide!

Documents

Python + FEM - KME + FEM Introduction to SFE ... low-level ⇒ lots of lines to get ... and data files in HDF5: general purpose library and file format for storing scientific