UFL Specification and User Manual 0.3 - FEniCS Project

More documents

Recommendations

Info

UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Loggand a linear form 5 can be differentiated to obtain the bilinear form correspondingto its Jacobi matrix:J(v,u;w) = ddw F(v;w).The UFL code to express this is (for a simple functional f(w) = ∫ Ω12 w2 dx)f = (w**2)/2 * dxF = derivative(f, w, v)J = derivative(F, w, u)which is equivalent to:f = (w**2)/2 * dxF = w*v*dxJ = u*v*dxAssume in the following examples that:v = TestFunction(element)u = TrialFunction(element)w = Function(element)The stiffness matrix can be computed from the functional ∫ ∇w : ∇wdx,Ωby the linesf = inner(grad(w), grad(w))/2 * dxF = derivative(f, w, v)J = derivative(F, w, u)5 Note that by “linear form” we only mean a form that is linear in its test function, notin the function you differentiate with respect to.54
UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Loggwhich is equivalent to:f = inner(grad(w), grad(w))/2 * dxF = inner(grad(w), grad(v)) * dxJ = inner(grad(u), grad(v)) * dxNotethatherethebasisfunctionsareprovidedexplicitly, whichissometimesnecessary, e.g., if part of the form is linearlized manually like in (TODO: Anexample that makes sense would be nicer, this is just a random form.)g = Function(element)f = inner(grad(w), grad(w))*dxF = derivative(f, w, v) + dot(w-g,v)*dxJ = derivative(F, w, u)Derivatives can also be computed w.r.t. functions in mixed spaces. Considerthis example, an implementation of the harmonic map equations usingautomatic differentiation.X = VectorElement("Lagrange", cell, 1)Y = FiniteElement("Lagrange", cell, 1)x = Function(X)y = Function(Y)L = inner(grad(x), grad(x))*dx + dot(x,x)*y*dxF = derivative(L, (x,y))J = derivative(F, (x,y))Here L is defined as a functional with two coefficient functions x and y fromseparate finite element spaces. However, F and J become linear and bilinearforms respectively with basis functions defined on the mixed finite element55
Page 1 and 2:
UFL Specification and User Manual 0
Page 3 and 4: ContentsAbout this manual 111 Intro
Page 5 and 6: 2.8.10 skew . . . . . . . . . . . .
Page 7 and 8: 4 Internal Representation Details 7
Page 9: B Installation 97B.1 Installing fro
Page 12 and 13: UFL Specification and User Manual 0
Page 93: Bibliography[1] M. Alnæs and K.-A.
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118:
show all

UFL Specification and User Manual 0.3 - FEniCS Project

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?