A First Step Towards Automatic PDE Code Ver ... - FEniCS Project

A First Step Towards Automatic PDE Code Verification
Ola Skavhaug
Kent-Andre Mardal
Hans Petter Langtangen
Simula Research Laboratory, Norway
University of Oslo, Norway
PDE Code Verification 1

Validation and Verification
Validation:
Is the PDE model appropriate?
Or: Do we solve the right equations?
Core interest among scientists and engineers
Verification:
Are the numerical methods correctly implemented?
Or: Do we solve the equations right?
Attracts much less interest than validation
Validation requires successful verification
PDE Code Verification 2

Generation of Analytical Solutions
Given a PDE or system of PDEs:
F(u) = 0 in Ω
(F and u may be scalar or vector)
Pick any v as analytical solution
v solves the problem
F(u) = b in Ω,
b = F(v)
e.g. u = v on ∂Ω
i.e., add a source term in the equation such that v is a
solution
Known as the method of manufactured solutions
PDE Code Verification 3

Verification Procedure
Make a sequence of refinements in space and time
Compute errors
Fit a convergence estimate
error = A∆x p + B∆y q + C∆z r + D∆t s
(by nonlinear least squares)
Does the method converge? With expected rate?
This is easy, but not a standard or required scientific
procedure in Computational Science
PDE Code Verification 4

Computing F(v) (Source Term)
Computing the source term F(v) is a matter of
differentiation
Calculating F(v) by hand is tedious and error-prone
This is critical for coupled systems of PDEs
PDE Code Verification 5

Example
∇ · [µ∇w] = −β
∇ 2 T = −κ −1 µ˙γ 2
µ = e −α(T−T 0) ˙γ n−1
√
˙γ = (w ,x ) 2 + (w ,y ) 2
w = cos(sin(x + y)) + tanx
T = x + y + e xy
PDE Code Verification 6

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Example cont.
PDE Code Verification 7

Automation Idea 1
Use Maple (or similar) to define F(u) and v. Then
compute F(v) and generate C/F77 code
Incorporate this C/F77 code in the simulator.
Easy in principle, but still somewhat tedious;
requires some manual work
The process must be simpler and safer if we want to make
extensive use of manufactured solutions!
PDE Code Verification 8

Automation Idea 2
Build a fully automatic “problem solving environment”
for this type of code verification
Need to glue symbolic package, source term F(v) code
generation, and PDE solver
Use a “scripting language” for gluing!
– we have chosen Python
PDE Code Verification 9

Our Approach
GiNaC: symbolic math engine (in C++)
Generate Python interface to GiNaC (using SWIG)
Extend Python-GiNaC with (e.g.)ÖandÚ
Diffpack: PDE solver library (in C++)
Generate Python interface to Diffpack PDE solver
(using SWIG/SIP)
Famms: home-made Python module to glue GiNaC
and solver; specify F(u) and v in Python, the rest is
automatic
PDE Code Verification 10

Example
ÐÐ×ØØݽŴµ ÖÓÑÐ×ØØݽÅÑÔÓÖØ ÔÈ×ÓÐÚÖ
∇ ((λ + µ)∇ · u) + ∇
È×ÓÐÚÖ×··»ÈÝØÓÒÓØ
· (µ∇u) = 0
ÖÓÑÑÑ×ÑÔÓÖØ ÖÓÑ×ÝÑÓÐÑÔÓÖØ
Ü¸ÝºÜ ÑÑ×´Ò×ÔѾµ Ú½×ҴܵھÓ״ݵ ÚÎØÓÖ´´Ü¸Ýµ¸´Ú½¸Ú¾µµ Ð××ÓÖÒÔÒÒØÚÖÐ× ÔÙÒØÓÒ×ÓÖÓÑÔÓÒÒØ
Äѽ¾¼ÑÙ¿ Ð×ØØÝÔÖÑØÖ× ×ÔÝÑÒÙØÙÖ×ÓÐÙØÓÒ
´Ùµ
º××Ò´ÕÙØÓÒ¸×ÓÐÙØÓÒÚ¸×ÑÙÐØÓÖе ÖØÙÖÒÖ´´ÄÑ·ÑÙµÚ´Ùµµ·Ú´ÑÙÖ´Ùµµ
PDE Code Verification 11

Example cont.
××ÒÒÔÙØØ ÑÅÒÙËÝ×ØÑ´µ
∇ ((λ + µ)∇ · u) + ∇
ÑÒÙÒØÖØÓÈ×ÓÐÚÖ
· (µ∇u) = 0
Ѻ×Ø´ÑÙÐØÐÚÐÑØÓ¸ÅÙÐØÖµ Ѻ×Ø´×ÑÓÓØÖ×ÑØÓ¸ËËÇʵ кҴѵ ÒØ×ØÖÙØ×ÓÖÒÔÙØ
Ѻ×Ø´ÓÖ×Ö×ÑØÓ¸Ù××Ðѵ Ѻ×Ø´ÄѸÄѵѺ×Ø´ÑÙ¸ÑÙµ ººº
к×ÓÐÚÈÖÓÐÑ´µ к×Ò´µ ×ÒÒÔÙØØÓ×ÓÐÚÖ
к×ÚÊ×ÙÐØ×´µ
PDE Code Verification 12

Another Example
∇ · [µ∇w] = −β
∇ 2 T = −κ −1 µ˙γ 2
µ = e −α(T−T 0) ˙γ n−1
ÛÓ×´×Ҵܷݵµ·ØҴܵ
√
ÌÜ·Ý·ÜÔ´Üݵ
˙γ = (w ,x ) 2 + (w ,y ) 2
ÑÑ´Ûµ ÑÝ̴̵ ÑÝÛ´Ñѵ ÖØÙÖÒÜÔ´¹ÐԴ̹̼µµ ÖØÙÖÒÑÑ´Ò¹½µ ÖØÙÖÒ×ÕÖØ´Ûº´Ü¸½µ¾·Ûº´Ý¸½µ¾µ
½´Û¸ÌµÖØÙÖÒÐÔд̵·ÔÔ´¹½µÑݴ̸ÑÑ´ÛµµÑÑ´Ûµ¾
Ñݴ̸Ñѵ ÖØÙÖÒÑÝ̴̵ÑÝÛ´Ñѵ
¾´Û¸ÌµÖØÙÖÒÚ´Ñݴ̸ÑÑ´ÛµµÖ´Ûµµ·Ø
PDE Code Verification 13

What Have We Done?
We have
“created” an engine for symbolic PDE math, with nice
input syntax
made a seamless integration of our native PDE solver
and this math engine such that the PDE solver solves a
new symbolically computed problem – no solver
modification required
incorporated empirical convergence estimation
incorporated visualization (Vtk/MayaVi)
...and this works for any Diffpack PDE solver
PDE Code Verification 14

Beyond Diffpack
How much of this is actually tied to Diffpack?
Famms is meant to be independent of the PDE package
Key mechanism: determine functions run–time
Functors (function objects) in C++
Function pointers in plain C
Should be “easy” to incorporate in other software
packages; we only need Python callback mechanisms
PDE Code Verification 15

The Glue
Ú×Ò´Üݵ ܸݸÚÆÓØ×
Easy to evaluate v pointwise in GiNaC
Ð××ÅÝÊÀËÔÙÐ×Ð××ÙÒØÓÖ
v and F(v) are expressions with GiNaC objects
Bind
ßÔÙÐ
generic Python pointers in the PDE solver to
Python functions for evaluating v and source term F(v)
ÈÝÇØ ÔÝÙÒ »»ÈÝØÓÒÙÒØÓÒØÓÐÐ
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ÖØÙÖÒÓÙÐÖ×ÙÐØ ÓÙÐÖ×ÙÐØÈÝÐÓØ×ÓÙдÔÝÖ×ÙÐص ÔÝÖ×ÙÐØÈÝÚÐÐÐÇØ´ÔÝÙÒ¸Ö×µ
Ð
PDE Code Verification 16