UFL Specification and User Manual 0.3 - FEniCS Project

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UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Loggx = cell.xL = sin(x[1])*v*dxAnother quantity is the (outwards pointing) facet normal cell.n. The normalvector is only defined on the boundary, so it can’t be used in a cellintegral.Example functional M, an integral of the normal component of a function gover the boundary:n = cell.ng = Function(VectorElement("CG", cell, 1))M = dot(n, g)*dsPython scalars (int, float) can be used anywhere a scalar expression is allowed.Another literal constant type is Identity which represents an n×nunit matrix of given size n, as in this example:# Geometric dimensiond = cell.d# d x d identiy matrixI = Identity(d)# Kronecker deltadelta_ij = I[i,j][Advanced] Note that there are some differences from FFC. In particular,using FacetNormal or cell.n does not implicitly add another coefficientFunction to the form, the normal should be automatically computed in UFCcode. Note also that MeshSize has been removed because the meaning isambiguous (does it mean min, max, avg, cell radius?), so use a Constantinstead.28
UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Logg2.5 Indexing and tensor componentsUFL supports index notation, which is often a convenient way to expressforms. The basic principle of index notation is that summation is implicitover indices repeated twice in each term of an expression. The followingexamples illustrate the index notation, assuming that each of the variables iand j have been declared as a free Index:v[i]*w[i] ↔Dx(v, i)*Dx(w, i) ↔Dx(v[i], i) ↔Dx(v[i], j)*Dx(w[i], j) ↔∑n−1v i w i = v·w, (2.2)i=0∑d−1∂v ∂w= ∇v ·∇w, (2.3)∂x i ∂x ii=0∑d−1∂v i= ∇·v, (2.4)∂x ii=0∑n−1∑d−1∂v i ∂w i= ∇v : ∇w. (2.5)∂x j ∂x ji=0j=0Here we’ll try to very briefly summarize the basic concepts of tensor algebraand index notation, just enough to express the operators in UFL.Assuming an Euclidean space in d dimensions with d = 1, 2, or 3, and aset of orthonormal basis vectors i i for i ∈ 0,...,d−1, we can define the dotproduct of any two basis functions asi i ·i j = δ ij , (2.6)where δ ij is the Kronecker deltaδ ij ≡{1, i = j,0, otherwise.(2.7)Arank1tensor(vector)quantityvcanberepresentedintermsofunitvectorsand its scalar components in that basis. In tensor algebra it is common to29
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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
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Bibliography[1] M. Alnæs and K.-A.
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