UFL Specification and User Manual 0.3 - FEniCS Project

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UFL Specification and User Manual 0.3Martin S. Alnæs, Anders LoggM = X + YThere is a subtle difference between defining x and y separately and thisalternative implementation (reusing the elements X,Y,M):u = Function(M)x, y = split(u)L = inner(grad(x), grad(x))*dx + dot(x,x)*y*dxF = derivative(L, u)J = derivative(F, u)The difference is that the forms here have one coefficient function u in themixed space, and the forms above have two coefficient functions x and y.TODO: Move this to implementation part? If you wonder how this is alldone, a brief explanation follows. Recall that a Function represents a sumof unknown coefficients multiplied with unknown basis functions in somefinite element space.w(x) = ∑ kw k φ k (x) (2.50)Also recall that a BasisFunction represents any (unknown) basis functionin some finite element space.v(x) = φ k (x), φ k ∈ V h . (2.51)A form L(v;w) implemented in UFL is intended for discretization likeb i = L(φ i ; ∑ kw k φ k ), ∀φ i ∈ V h . (2.52)The Jacobi matrix A ij of this vector can be obtained by differentiation of b iw.r.t. w j , which can be writtenA ij = db idw j= a(φ i ,φ j ; ∑ kw k φ k ), ∀φ i ∈ V h , ∀φ j ∈ V h , (2.53)56
UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Loggfor some form a. In UFL, the form a can be obtained by differentiating L.To manage this, we note that as long as the domain Ω is independent ofw j , ∫ dcommutes withΩ dw j, and we can differentiate the integrand expressioninstead, e.g.,∫ ∫L(v;w) = I c (v;w)dx+ I e (v;w)ds, (2.54)Ω ∂Ω∫ ∫d dI c dI eL(v;w) = dx+ ds. (2.55)dw j dw j dw jIn addition, we need thatwhich in UFL can be represented asΩ∂Ωdwdw j= φ j , ∀φ j ∈ V h , (2.56)w = Function(element), (2.57)v = BasisFunction(element), (2.58)dw= v,dw j(2.59)since w represents the sum and v represents any and all basis functions inV h .Other operators have well defined derivatives, and by repeatedly applyingthe chain rule we can differentiate the integrand automatically.The notation here has potential for improvement, feel free to ask if somethingis unclear, or suggest improvements.2.13.7 Combining form transformationsForm transformations can be combined freely. Note that to do this, derivativesare usually be evaluated before applying e.g. the action of a form,because derivative changes the arity of the form.element = FiniteElement("CG", cell, 1)w = Function(element)57
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UFL Specification and User Manual 0
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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
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Page 93: Bibliography[1] M. Alnæs and K.-A.
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