A Finite Volume PDE Solver Using Python...

A Finite Volume PDE Solver Using Python (FiPy)

Jonathan E. Guyer, Daniel Wheeler & James A. Warrenguyer@nist.gov daniel.wheeler@nist.gov jwarren@nist.gov

Metallurgy DivisionMaterials Science and Engineering Laboratory

Certain software packages are identified in this document in order to specify the experimental

procedure adequately. Such identification is not intended to imply recommendation or endorsement by the

National Institute of Standards and Technology, nor is it intended to imply that the software packages identified

are necessarily the best available for the purpose.

Attention, users of this electronic file: The information contained herein should not be altered in any way. Attempts to change the file will adversely affect the integrity of the information and its usefulness.

MotivationMany interface tracking codes for solving Materials Science problems

Commercial CFD codes are expensive and written in FORTRAN or C

Interface tracking requires unstructured meshes

Unstructured meshing requires Finite Element / Finite Volume

Difficult for end users to customize existing codes in FORTRAN/C

Python is the code and the user scriptGood Python libraries to exploit

NumPy for array manipulationPySparse for linear algebraGist for viewingSciPy for C inlining within Python (weave)Scientific Python for physical dimensionsProfiler - PyGTK GUI by NIST’s Steve Langer

Phase Field

Research BackgroundModeling Phase Transformations

concentration

distance

potential

electrode electrolyte

concentration

distance

potential

Phase Field

SolidificationDendritesGrain boundary motion, impingement, rotation

L. Gránásy, T. Börzsönyi, T. Pusztai, Phys. Rev. Lett. 88, 205105 (2002)

Phase Field

Electrochemistry

Phase Field

ElectrochemistryStresses and strains

σ12J. Slutsker, G. B. McFadden, A. L. Roytburd,

W. J. Boettinger, and J. A. Warren

Phase Field

SolidificationDendritesGrain boundaries

ElectrochemistryStresses and strains

Level Set Method

Electrodeposition

FiPy OverviewFinite Volume code for solving coupled sets of PDEs

Code currently addresses Phase Field models

Compatible with unstructured meshes

Code will be freely available (watch <http://www.ctcms.nist.gov>)

Large archive of test problems

User controls program flow

Finite Volume MethodSolve a general PDE on a given domain for a field

!("#)!t︸︷︷︸

transient

+$.("u#)︸︷︷︸convection

= $.(%$#)︸︷︷︸diffusion

+ S#︸︷︷︸source

a11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂t︸︷︷︸transient

+∇ · ($uφ)︸︷︷︸convection

= ∇ · (Γ∇φ)︸︷︷︸diffusion

+ Sφ︸︷︷︸source∫

∂(ρφ)∂t

dV︸︷︷︸transient

S($n · $u)φ dS︸︷︷︸convection

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

ρφV − (ρφV )old

∆t︸︷︷︸transient

+∑face

[($n · $u)Aφ]face︸︷︷︸convection

=∑face

[ΓA$n ·∇φ]face︸︷︷︸diffusion

+ V Sφ︸︷︷︸source

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

domain

Integrate PDE over arbitrary control volumes

!("#)!t︸︷︷︸

transient

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

controlvolume

Evaluate PDE over polyhedral control volumes !("#)!t︸︷︷︸

transient

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

vertex

Evaluate PDE over polyhedral control volumes

Obtain a large coupled set of linear equations in

!("#)!t︸︷︷︸

transient

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

FiPy Design - Objects

Equation Iterator

Solver

Variable Viewer

SparseMatrix

MeshCell Face Vertex

BoundaryCondition

adjacentcell

vertex

FiPy Design - Equations

Equation Iterator

Solver

Variable Viewer

SparseMatrix

BoundaryCondition

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

Iterator

Equation

AdvectionEquation DiffusionEquation

Solver Term Variable SparseMatrix

PhaseEquation TemperatureEquation

FiPy Design - Terms

Equation Iterator

Solver

Variable Viewer

SparseMatrix

BoundaryCondition

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

Equation

CellTerm FaceTerm

BoundaryCondition

TransientTerm SourceTerm DiffusionTerm ConvectionTerm

adjacentcell

vertex

FiPy Design - Variables

Equation Iterator

Solver

Variable Viewer

SparseMatrix

BoundaryCondition

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

Equation

Variable

CellVariable FaceVariable GradVariable

Viewer Mesh

CellGradVariable FaceGradVariable

Either:

solution variables (evaluated by Equation)set by intermediate calculation

Lazy evaluation

Physical dimensions

Standard operators apply (__add__, __div__, etc.)

Example Problem - Grain ImpingementGoverning Equations

Phase field variable

Orientation variable

0 ≤ φ ≤ 1

−π ≤ θ ≤ π

0 ≤ φ ≤ 1

−π ≤ θ ≤ π

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

N FORTRAN (s) Python (s) Penalty100 0.0619 12.8 ×207400 0.188 40.2 ×214

1600 0.573 157 ×2736400 3.06 707 ×231

25600 12.0 4060 ×338102400 37.3 ??? ×???409600 148 ??? ×???

N FORTRAN (s) Python (s) Penalty Inlined (s) Penalty100 0.0619 12.8 ×207 2.45 ×39.6400 0.188 40.2 ×214 2.95 ×15.3

1600 0.573 157 ×273 5.22 × 9.106400 3.06 707 ×231 15.3 × 4.99

25600 12.0 4060 ×338 66.0 × 5.53102400 37.3 ??? ×??? 243 × 6.50409600 148 ??? ×??? 1510 ×10.2

Q(φ,∇θ)∂φ

= α2∇2φ︸︷︷︸diffusion

− ∂f

∂φ− ∂g

∂φs|∇φ|− ∂h

2|∇φ|2︸︷︷︸

source

P (φ,∇θ)∂θ

= ∇ ·[hε2∇θ + gs

|∇θ|]

︸︷︷︸diffusion

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

N FORTRAN (s) Python (s) Penalty100 0.0619 12.8 ×207400 0.188 40.2 ×214

1600 0.573 157 ×2736400 3.06 707 ×231

25600 12.0 4060 ×338102400 37.3 ??? ×???409600 148 ??? ×???

N FORTRAN (s) Python (s) Penalty Inlined (s) Penalty100 0.0619 12.8 ×207 2.45 ×39.6400 0.188 40.2 ×214 2.95 ×15.3

1600 0.573 157 ×273 5.22 × 9.106400 3.06 707 ×231 15.3 × 4.99

25600 12.0 4060 ×338 66.0 × 5.53102400 37.3 ??? ×??? 243 × 6.50409600 148 ??? ×??? 1510 ×10.2

Q(φ,∇θ)∂φ

− ∂f

∂φ− ∂g

∂φs|∇φ|− ∂h

2|∇φ|2︸︷︷︸

source

P (φ,∇θ)∂θ

= ∇ ·[hε2∇θ + gs

|∇θ|]

R. Koybayashi, W. C. Carter, and J. A. Warren

Example Problem - Input Fileclass ImpingementSystem: def __init__(self, n = 100, dx = 0.01, steps = 10, plotsteps = 5, timeStepDuration = 0.02):

# create meshmesh = Grid2D(dx, dy, nx, ny)

# create variables::

# create viewersself.viewers = [Grid2DGistViewer(var = field, palette = 'rainbow.gp') for field in (phase,theta)]

# create equations::

# create iteratorself.it = Iterator(equations = (thetaEq, phaseEq))

def run(self):::

# create meshmesh = Grid2D(dx, dy, nx, ny)

# create variablesphase = CellVariable(name = 'PhaseField', mesh = mesh, value = 0.)theta = ModularVariable(name = 'Theta', mesh = mesh, value = 0.)

setCells = mesh.getCells(lambda cell: cell.getCenter()[0] > L/2.)phase.setValue(1.,setCells)

# create viewersself.viewers = [Grid2DGistViewer(var = field, palette = 'rainbow.gp') for field in (phase,theta)]

# create equations::

def run(self):::

# create equationsthetaEq = ThetaEquation(var = theta,

solver = LinearPCGSolver( tolerance = 1.e-15, steps = 2000 ),boundaryConditions = ( FixedFlux(mesh.getExteriorFaces(), 0.), ),

parameters = {'time step duration' : timeStepDuration, 'beta' : 1e5, 'mu' : 1e3

},fields = (phase,))

phaseEq = PhaseEquation(var = phase,mPhi = Type1MPhiVariable,solver = LinearPCGSolver(tolerance = 1.e-15, steps = 1000),boundaryConditions = ( FixedFlux(mesh.getExteriorFaces(), 0.), ),

parameters = { 'tau' : 0.1,

'time step duration' : timeStepDuration },

fields = (theta,))

def run(self):

for plotstep in range(self.plotsteps): for step in range(self.steps):

self.it.timestep() for viewer in self.viewers:

viewer.plot()

Example Problem - Grain Impingement

NumPy is much faster than plain Python, but still orders of magnitude slower than tailored FORTRAN code

NumArray is faster for large arrays?

Significant overhead in transition between Python and C for lots of simple operations

Example Problem - Efficiency

0 ≤ φ ≤ 1

−π ≤ θ ≤ π

$A = $B × $C + tan $D

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

N FORTRAN (s) Numeric (s) Penalty100 0.0619 12.8 ×207400 0.188 40.2 ×214

1600 0.573 157 ×2736400 3.06 707 ×231

25600 12.0 4060 ×338102400 37.3 ??? ×???409600 148 ??? ×???

N FORTRAN (s) Numeric (s) Penalty Inline C (s) Penalty100 0.0619 12.8 ×207 2.45 ×39.6400 0.188 40.2 ×214 2.95 ×15.3

1600 0.573 157 ×273 5.22 × 9.106400 3.06 707 ×231 15.3 × 4.99

25600 12.0 4060 ×338 66.0 × 5.53102400 37.3 ??? ×??? 243 × 6.50409600 148 ??? ×??? 1510 ×10.2

Q(φ,∇θ)∂φ

− ∂f

∂φ− ∂g

∂φs|∇φ|− ∂h

2|∇φ|2︸︷︷︸

source

P (φ,∇θ)∂θ

= ∇ ·[hε2∇θ + gs

|∇θ|]

Costly operations are not neatly encapsulated in a generalized library

C code is embedded within Python code using SciPy’s weave

Design in Python, optimize in C

Python and C variants are maintained in parallel (--inline at command line)

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

1600 0.573 157 ×2736400 3.06 707 ×231

25600 12.0 4060 ×338102400 37.3 ??? ×???409600 148 ??? ×???

1600 0.573 157 ×273 5.22 × 9.106400 3.06 707 ×231 15.3 × 4.99

25600 12.0 4060 ×338 66.0 × 5.53102400 37.3 ??? ×??? 243 × 6.50409600 148 ??? ×??? 1510 ×10.2

Q(φ,∇θ)∂φ

− ∂f

∂φ− ∂g

∂φs|∇φ|− ∂h

2|∇φ|2︸︷︷︸

source

P (φ,∇θ)∂θ

= ∇ ·[hε2∇θ + gs

|∇θ|]

Maintaining clean and fast code is challenging

blitz isn’t fast enoughweave is a bit clumsySwig?Pyrex? 10:30 am Thursday

φa11 a12

a21 a22. . .

. . . . . . . . .. . . ann

φ2...

∂(ρφ)

∂(ρφ)∂t

SΓ($n ·∇φ) dS︸︷︷︸

diffusion

VSφ dV︸︷︷︸

source

+∑face

=∑face

1600 0.573 157 ×2736400 3.06 707 ×231

25600 12.0 4060 ×338102400 37.3 ??? ×???409600 148 ??? ×???

1600 0.573 157 ×273 5.22 × 9.106400 3.06 707 ×231 15.3 × 4.99

25600 12.0 4060 ×338 66.0 × 5.53102400 37.3 ??? ×??? 243 × 6.50409600 148 ??? ×??? 1510 ×10.2

Q(φ,∇θ)∂φ

− ∂f

∂φ− ∂g

∂φs|∇φ|− ∂h

2|∇φ|2︸︷︷︸

source

P (φ,∇θ)∂θ

= ∇ ·[hε2∇θ + gs

|∇θ|]

Further WorkMeshing packages

DX viewer

Documentation

Simplify installation

Even more test cases

Level Sets

A Finite Volume PDE Solver Using Python...

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