Cimrman Python Fem 04-03-2007

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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 Bohemia

Plzen, Czech Republic

April 3, 2007, Plzen
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Outline

1 IntroductionProgramming LanguagesProgramming Techniques

2 Our choice

Mixing Languages Best of Both WorldsPythonSoftware Dependencies

3 Example ProblemContinuous FormulationWeak FormulationProblem Description File

4 Final slide :-)
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Programming Languages

1 compiled (fortran, C, C++, Java, . . . )

Prosspeed, speed, speed, . . . , did Isayspeed?

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 problems

static!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 maintenance

dynamic!

(often) large code base

Cons

many are relatively newlack of speed, or even utterly slow!

O )
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Pros



dynamic!


Cons

many are relatively newlack of speed, or even utterlyslow!

I t d ti O h i E l P bl Fi l lid )
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Pros



dynamic!


Cons


Introduction Our choice Example Problem Final slide : )
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Pros



dynamic!


Cons


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code size maintenance problemsstatic!

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

Pros



dynamic!


Cons


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code size maintenance problemsstatic!

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

Pros



dynamic!


Cons


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Example Python Code

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p )

Example C Code

c h a r s [ ] = s e l tudy , mel dudy . a s e l o pr av du t ud y . ;

i n t s t r f i n d (c h a r s t r , c h a r 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 ( s tr nc mp (& s t r [ i ] , t g t , t l e n ) = = 0 )

r e t u r n i ;}r e t u r n 1;

}

i n t i 1 ;

i 1 = s t r f i n d ( s , tu d y ) ;

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 alsofortran

+

High level: Python

www.python.orgbatteries included

logic of the code

configurationfiles!

probleminputfiles!

http://www.python.org/http://www.python.org/http://find/


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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 alsofortran

+

High level: Python

www.python.orgbatteries included

logic of the code

configurationfiles!

probleminputfiles!

http://www.python.org/http://www.python.org/http://find/


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Python I

Python ris a dynamic object-oriented programming

language 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 language

encourages the development of higher quality, moremaintainable code.

. . .

batteries included
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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
http://wiki.python.org/moin/NumericAndScientifichttp://wiki.python.org/moin/NumericAndScientifichttp://find/


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Software Dependencies

1 libraries & modules

SciPy: free (BSD license) collection of numerical computinglibraries for Pythonenables Matlab-like array/matrix manipulations and indexing

UMFPACK: very fast direct solver for sparse systemsPython 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.. .v iaPyTables

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 in

Tcl/Tk MayaVi/MayaVi2: VTK-based advanced mesh and data viewer

in Python

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in Python

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in Python


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in Python

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in Python

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Shape Optimization in Incompressible Flow Problems

Objective Function

(u)

2

c|u|2 min (1)

minimize gradients of solution (e.g. losses) in c

by movingdesign boundary

perturbation of by vector field V(t) = + {tV(x)}x where V= 0 in c \ (2)

in outD

D

C C

D

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Objective Function

(u)

2

c|u|2 min (1)



perturbation of by vector field V(t) = + {tV(x)}x where V= 0 in c \ (2)

in outD

D

C C

D


S O
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Objective Function

(u)

2

c|u|2 min (1)



perturbation of by vector fieldV

(t) = + {tV(x)}x where V= 0 in c \ (2)

in outD

D

C C

D


A i
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Appetizer

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

connectivity of spline boxes (6 boxes in different colours)


C i F l i
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Continuous Formulation

laminar steady state incompressibleNavier-Stokes equations

2u+u u+ p = 0 in (3)

u = 0 in (4)

boundary conditions

u=u0 on in

pn+u

n =pn on out

u= 0 on walls

(5)

adjoint equationsfor computing the adjoint solution w:

2w+ u w u w r=c 2u in

w= 0 in (6)

w= 0 on in

walls

and c

is the characteristic function of c


C ti F l ti
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Continuous Formulation

laminar steady state incompressibleNavier-Stokes equations

2u+u u+ p = 0 in (3)

u = 0 in (4)

boundary conditions

u=u0 on in

pn+u

n =pn on out

u= 0 on walls

(5)

adjoint equationsfor computing the adjoint solution w:

2w+ u w u w r=c 2u in

w= 0 in (6)

w= 0 on in

walls

and c

is the characteristic function of c


W k F l ti
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Weak Formulation

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

v :u+

(u u) v

p v=

out

pv n vV0 ,

q u= 0 qQ

(7)

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

v :w+

(v u) w+

(u v) w+

r v

=

c

v :u vV0 ,

q w= 0 qQ

(8)

+ proper BC ...


W k F l ti


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Weak Formulation


v :u+

(u u) v

p v=

out

pv n vV0 ,

q u= 0 qQ

(7)


v :w+

(v u) w+

(u v) w+

r v

=

c

v :u vV0 ,

q w= 0 qQ

(8)

+ proper BC ...


Weak Formulation
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Weak Formulation


v :u+

(u u) v

p v=

out

pv n vV0 ,

q u= 0 qQ

(7)


v :w+

(v u) w+

(u v) w+

r v

=

c

v :u vV0 ,

q w= 0 qQ

(8)

+ proper BC ...


Problem Description File I
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a regular python file all python machinery is accessible!

requires/recognizes severalkeywordsto definemeshregions subdomains, boundaries, characteristic functions, . . .boundary conditions Dirichletfields field approximation per subdomainvariables unknown fields, test fields, parameters

equationsmaterials constitutive relations, e.g. newton fluid, elastic solid, . . .solver parameters nonlinear solver, flow solver, optimizationsolver, .... . .


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a regular python file all python machinery is accessible!

requires/recognizes severalkeywordsto definemeshregions subdomains, boundaries, characteristic functions, . . .boundary conditions Dirichletfields field approximation per subdomainvariables unknown fields, test fields, parameters

equationsmaterials constitutive relations, e.g. newton fluid, elastic solid, . . .solver parameters nonlinear solver, flow solver, optimizationsolver, .... . .


Problem Description File II
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Problem Description File IImesh

f i l e N a m e m e s h = s i m p l e . mes h

regionsr e gi o n 1 0 0 0 = {

name : Omega , s e l e c t : e l em e nt s o f g ro up 6 ,

}r e g i o n 0 = {

name : W a l l s ,

s e l e c t : n o d e s o f s u r f a c e n r . O u t l e t ,}r e g i o n 1 = {

name : I n l e t , s e l e c t : n o de s by c i n c ( x , y , z , 0 ) ,

}r e g i o n 2 = {

name : O u t l e t ,

s e l e c t : n o de s by c i n c ( x , y , z , 1 ) ,}r e g i o n 1 0 0 = {

name : Omega C , # C o n t r o l d om ai n . s e l e c t : n o de s i n ( x > 25.9999e 3 ) & ( x < 18.990e3 ) ,

}r e g i o n 1 0 1 = {

name : Omega D , # D e s i g n d om a in .

s e l e c t : a l l e r . Omega C ,}
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Problem Description File III


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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 , ) , b a s e s : ( ( Omega , 3 4 P 1B ) , )

}

f i e l d 2 = { name : p r e s s u r e , dim : ( 1 , 1 ) , f l a g s : ( ) , b a s e s : ( ( Omega , 3 4 P 1 ) , )

}

variablesv a r i a b l e s = {

u : ( f i e l d , p a ra m et e r , 3 v e l o c i t y , ( 3 , 4 , 5 ) ) , w 0 : ( f i e l d , p a ra me te r , 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 , p a ra me te r , p r e s s u r e , ( 9 , ) ) , r : ( f i e l d , unknown , p r e s s ur 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 , m e sh Ve lo ci t y , p r e s s ur e , ( 9 , ) ) ,

}
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Problem Description File IV


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Problem Description File IV

Dirichlet BC

e b c = { W a ll s : ( ( W a ll s , ( 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 q u a t i o n s = { b a l a n ce :+ d w d i v g r a d . Omega ( f l u i d , v , w )

+ d w c o n v e c t . Omega ( v , w ) dw g r ad . Omega ( v , r )= d w s u r f a c e l t r . O u t le t ( b p re s s , v ) ,

i n c o m p r e s s i b i l i t y : d w d i v . Omega ( q , w ) = 0 ,

}


Problem Description File V
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adjoint equations(compare with (8))

e q u a t i o n s a d j o i n t = { b a l a n ce :+ d w d i v g r a d . Omega ( f l u i d , v , w )

+ d w a d j c o n v e c t 1 . Omega ( v , w , u )+ d w a d j c o n v e c t 2 . Omega ( v , w , u )+ d w g r a d . Omega ( v , r )= d w a d j d i v g r a d 2 . Omega C ( one , f l u i d , v , u ) ,

i n c o m p r e s s i b i l i t y :

d w d i v . Omega ( q , w ) = 0 ,}

materialsm a t e r i a l 2 = {

name : f l u i d , mode : h e r e , r e g i o n : Omega ,

v i s c o s i t y : 1 . 2 5 e1,}m a t e r i a l 1 0 0 = {

name : b p r e s s , mode : h e r e , r e g i o n : Omega , v a l : 0 . 0 ,

}


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p

adjoint equations(compare with (8))

e q u a t i o n s a d j o i n t = { b a l a n ce :+ d w d i v g r a d . Omega ( f l u i d , v , w )

+ d w a d j c o n v e c t 1 . Omega ( v , w , u )+ d w a d j c o n v e c t 2 . Omega ( v , w , u )+ d w g r a d . Omega ( v , r )= d w a d j d i v g r a d 2 . Omega C ( one , f l u i d , v , u ) ,

i n c o m p r e s s i b i l i t y :

d w d i v . Omega ( q , w ) = 0 ,}

materialsm a t e r i a l 2 = {

name : f l u i d , mode : h e r e , r e g i o n : Omega ,

v i s c o s i t y : 1 . 2 5 e1,}m a t e r i a l 1 0 0 = {

name : b p r e s s , mode : h e r e , r e g i o n : Omega , v a l : 0 . 0 ,

}


Yes, the final slide!
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,

What is done

basic FE element engineapproximations up to P2 onsimplexes (possibly withbubble)Q1 tensor-product

approximation on rectangleshandling of fields and variables

boundary conditions, DOFhandling

FE assembling

equations, terms, regionsmaterials, material caches

generic solvers (Newton iteration,direct linear system solution)

What is not done

general FE engine, with symbolicevaluation (a la SyFi - merge it?)

documentation, unit tests

organization 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



What is not done






use PETSC?


GUI



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What is done




FE assembling



What is not done






use PETSC?


GUI



This is not a slide!
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Cimrman Python Fem 04-03-2007

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