SyFi - A package for symbolic finite element ... - FEniCS Project

SyFi - A package for symbolic finite elementcomputationsKent-Andre MardalNovember 8, 2006Kent-Andre MardalSyFi - A package for symbolic finite element computations

Outline - Short Description of SyFiPurposeSyFi is a tool for defining polygons, polynomial spaces, degreesof freedom, and finite elementsSyFi makes it easy to define weak forms (differentiation andintegration of polynomials over polygons)SyFi generates efficient C++ code for the computation ofmatricesDependenciesSyFi relies heavily on GiNaC and Swiginac for the symboliccomputations and code generationSyFi can generate matrices based on either a Dolfin or aDiffpack mesh (we plan to include other meshes soon)SyFi can generate either Epetra or PyCC matrices (we plan toinclude other matrices soon)We are working on generating UFC compliant code (Alnæs’ talk)Kent-Andre MardalSyFi - A package for symbolic finite element computations

Short Description of GiNaC and SwiginacGiNaCGiNaC is a C++ library for symbolic mathematicsAuthors: C. Bauer, C. Dams A. Frink, V. Kisil, R. Kreckel, A.Sheplyakov, J. VollingaSwiginacURL: www.ginac.deLicense : GPLSwiginac is a Python interface to GiNaCAuthors: O. Skavhaug and O. CertikURL: http://swiginac.berlios.de/License: OpenKent-Andre MardalSyFi - A package for symbolic finite element computations

Swiginac Code Examplefrom swiginac import *x = symbol(’x’); y = symbol(’y’)f = x*x*y*yprint "f = ", fdfdx = diff(f,x)print "df/dx = ", dfdxintf = integral(x,0,1,f).eval_integ()print "integral of f from 0 to 1: ", intfg = cos(x)print "Taylor series of cos(x) around x == 0.5: ",print g.series(x == 0.5, 10)Kent-Andre MardalSyFi - A package for symbolic finite element computations

SyFi Extends GiNaC/SwiginacGiNaC/Swiginac supports:PolynomialsDifferentiation with respect to one variableIntegration with respect to one variableBasic Extensions in SyFi:Polynomial spaces (such as Legendre and Bernstein)Differentiations with respect to several variablesIntegration over polygons→ SyFi extends GiNaC/Swiginac with the ingredients typicallyneeded in finite element methodsKent-Andre MardalSyFi - A package for symbolic finite element computations

Demo: Bernstein polynomials on a Trianglefrom swiginac import *from SyFi import *x = cvar.x; y = cvar.yt = ReferenceTriangle()B = bernstein(2, t, ’a’)print "2. order Bernstein polynomial on a triangle ", BP = B[0]dPdx = diff(P, x)print "Derivative with respect to x ", dPdxint_dPdx = t.integrate(dPdx)print "Integral of dPdx over an triangle ", int_dPdxint_line1_dPdx = t.line(1).integrate(dPdx)print "Integral of dPdx over an edge", int_line1_dPdxKent-Andre MardalSyFi - A package for symbolic finite element computations

Elements currently implemented in SyFiFinite Elementscontinuous and discontinuous Lagrangian elements (arbitraryorder)Nedelec elements (arbitrary order)Nedelec H(div) elements (arbitrary order)Raviart-Thomas elements (arbitrary order)Crouzeix-Raviart elementsHermite elementsBubble elementsArnold-Falk-Winther elasticity element (weak symmetry)(arbitrary order)Kent-Andre MardalSyFi - A package for symbolic finite element computations

Evaluation of Weak Forms in SyFiWe will now demonstrate the computation of various elementmatrices in SyFiThe mass matrix on a reference triangle:∫M ij = N i N j dxThe stiffness matrix on a mapped tetrahedron:∫A ij = (J −1 ∇N i ) · (J −1 ∇N j ) dxTwhere J is the Jacobian of the geometry mappingTKent-Andre MardalSyFi - A package for symbolic finite element computations

Demo: Computing the Mass Matrix on the ReferenceTrianglefrom swiginac import *from SyFi import *t = ReferenceTriangle()fe = LagrangeFE(t, 3)for i in range(0, fe.nbf()):for j in range(0, fe.nbf()):Aij = t.integrate(fe.N(i)*fe.N(i))Kent-Andre MardalSyFi - A package for symbolic finite element computations

Demo: Computing the Stiffness Matrix on a MappedTethrahedronfrom swiginac import *from SyFi import *t = ReferenceTetrahedron()fe = LagrangeFE(t, 3)J = symbolic_matrix(3,3, ‘‘J’’)for i in range(0, fe.nbf()):for j in range(0, fe.nbf()):integrand = inner(grad(J, fe.N(i)),grad(J, fe.N(j)))Aij = t.integrate(integrand)Kent-Andre MardalSyFi - A package for symbolic finite element computations

The Computation of the Jacobian matrixWe will now demonstrate the computation the Jacobian matrix for anonlinear PDEu = ∑ j u jN jNonlinear convection diffusion equation∫F i = (u · ∇u) · N i + ∇u : ∇N i ) dxThe Jacobian matrixTJ ij = ∂F i∂u jKent-Andre MardalSyFi - A package for symbolic finite element computations

SyFi Code Example : Convection-Diffusion Matrix.. initialize elementfor i in range(0,fe.nbf()):# compute diffusion termfi_diffusion = inner(grad(u), grad(fe.N(i)))# compute convection termuxgradu = (u.transpose()*grad(u)).evalm()fi_convection = inner(uxgradu, fe.N(i), True)fi = fi_diffusion + fi_convectionFi = polygon.integrate(fi)for j in range(0,fe.nbf()):# differentiate to get the JacobianJij = diff(Fi, uj)print "J[%d,%d]=%s "%(i,j,Jij)Kent-Andre MardalSyFi - A package for symbolic finite element computations

Convection and Power-law DiffusionNow consider the Power-law modelNonlinear convection diffusion equation∫F i = (u · ∇u) · N i + µ(u)∇u : ∇N i ) dxTThe viscosity:µ = µ 0 ‖∇u‖ 2nKent-Andre MardalSyFi - A package for symbolic finite element computations

SyFi Code Example : Convection and Power-lawDiffusionn = symbol("n")for i in range(0,fe.nbf()):# nonlinear power-law diffusion termmu = mu0*inner(grad(u), grad(u))fi_diffusion = pow(mu,n)*inner(grad(u), grad(fe.N(i)))# nonlinear convection termuxgradu = (u.transpose()*grad(u)).evalm()fi_convection = inner(uxgradu, fe.N(i), True)fi = fi_diffusion + fi_convectionFi = polygon.integrate(fi)for j in range(0,fe.nbf()):# differentiate to get the JacobianJij = diff(Fi, uj)print "J[%d,%d]=%s\n"%(i,j, Jij.evalf().printc())Kent-Andre MardalSyFi - A package for symbolic finite element computations

Inlining of C++ in PythonThe module InstantWe have created a Python module for inlining of C++ code inPythonIt employs SWIG for the generation of wrapper code and distutilsfor compiling extenstion modulesfrom Instant import inlineadd_func = inline("double add(double a, double b){ return a+b; }")print "The sum of 3 and 4.5 is ", add_func(3, 4.5)Kent-Andre MardalSyFi - A package for symbolic finite element computations

UFC compliant code generationUFCUFC - Unified Form-assembly CodeLow level spesification of class declaration and functionsignatures for finite elements, element tensors etc.Together with Logg, Alnæs, Skavhaug and Langtangen we areworking on spesifing UFCAlnæs will talk more about it later todayKent-Andre MardalSyFi - A package for symbolic finite element computations

Using SyFi/UFC within PyCCimport stuff..n = 10m = Mesh.UnitSquare(n,n)mf = UMF.UserMapMatrixFactory()mf.select_mesh(m)polygon = SyFi.ReferenceTriangle()fe = SyFi.LagrangeFE(polygon, 1)def mass_integrand(u, v, G, Ginv):return inner(u, v)mass_form = UfcCG.BiLinearForm(mass_integrand, name = "mass" )compiled_fe, compiled_form = UfcCG.gen_extension_module(fe, mass_form)A = mf.compute_matrix(compiled_form)Kent-Andre MardalSyFi - A package for symbolic finite element computations

Concluding RemarksPresent Use of SyFiWe have implemented a set of simple test examples withinSyFi/PyCC for the Poisson problem, Stokes problem,convection-diffusion problems and Navier-Stokes equationsWe have used SyFi/PyCC in rather advanced applicationsconcerning the electrical activity of the heartFutureSupport other meshes and other matricesImplement more elementsContinue developing solvers for fluid and solid mechanicsKent-Andre MardalSyFi - A package for symbolic finite element computations