UFL Specification and User Manual 0.3 - FEniCS Project

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UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Loggf = energy_norm(a, w)which is equivalent tof = action(action(a, w), w)2.13.4 Adjoint of a bilinear FormThe adjoint a ′ of a bilinear form a is defined asa ′ (u,v) = a(v,u).This operation is implemented in UFL simply by swapping test and trialfunctions in a Form, and is used like this:aprime = adjoint(a)2.13.5 Linear and bilinear parts of a FormSome times it is useful to write an equation on the formata(v,u)−L(v) = 0.Before we can assemble the linear equationAu = b,we need to extract the forms corresponding to the left hand side and righthand side. This corresponds to extracting the bilinear and linear terms ofthe form respectively, or the terms that depend on both a test and a trialfunction on one side and the terms that depend on only a test function onthe other.This is easily done in UFL using lhs and rhs:52
UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Loggb = u*v*dx - f*v*dxa, L = lhs(b), rhs(b)Notethatrhsmultipliestheextractedtermsby−1, correspondingtomovingthem from left to right, so this is equivalent toa = u*v*dxL = f*v*dxAs a slightly more complicated example, this formulationF = v*(u - w)*dx + k*dot(grad(v), grad(0.5*(w + u)))*dxa, L = lhs(F), rhs(F)is equivalent toa = v*u*dx + k*dot(grad(v), 0.5*grad(u))*dxL = v*w*dx - k*dot(grad(v), 0.5*grad(w))*dx2.13.6 Automatic Functional DifferentiationUFL can compute derivatives of functionals or forms w.r.t. to a Function.This functionality can be used for example to linearize your nonlinear residualequation automatically, or derive a linear system from a functional, orcompute sensitivity vectors w.r.t. some coefficient.A functional can be differentiated to obtain a linear form,F(v;w) = ddw f(;w)53
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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 93: Bibliography[1] M. Alnæs and K.-A.
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