UFL Specification and User Manual 0.3 - FEniCS Project

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UFL Specification and User Manual 0.3Martin S. Alnæs, Anders LoggAlternatively, using the shorthand lambda notation, the strain operator maybe defined as follows:epsilon = lambda v: 0.5*(grad(v) + grad(v).T)2.13 Form TransformationsWhen you have defined a Form, you can derive new related forms from itautomatically. UFL defines a set of common form transformations describedin this section.2.13.1 Replacing arguments of a FormThe function replace lets you replace terminal objects with other values,using a mapping defined by a Python dict. This can be used for example toreplace a Function with a fixed value for optimized runtime evaluation.f = Function(element)g = Function(element)c = Constant(cell)a = f*g*v*dxb = replace(a, { f: 3.14, g: c })The replacement values must have the same basic properties as the originalvalues, in particular value shape and free indices.2.13.2 Action of a form on a functionThe action of a bilinear form a is defined asb(v;w) = a(v,w),50
UFL Specification and User Manual 0.3Martin S. Alnæs, Anders LoggThe action of a linear form L is defined asf(;w) = L(w)This operation is implemented in UFL simply by replacing the rightmostbasis function (trial function for a, test function for L) in a Form, and is usedlike this:L = action(a, w)f = action(L, w)To give a concrete example, these declarations are equivalent:a = inner(grad(u), grad(v))*dxL = action(a, w)a = inner(grad(u), grad(v))*dxL = inner(grad(w), grad(v))*dxIf a is a rank 2 form used to assemble the matrix A, L is a rank 1 formthat can be used to assemble the vector b = Ax directly. This can be usedto define both the form of a matrix and the form of its action without codeduplication, and for the action of a Jacobi matrix computed using derivative.If L is a rank 1 form used to assemble the vector b, f is a functional thatcan be used to assemble the scalar value f = b·w directly. This operationis sometimes used in, e.g., error control with L being the residual equationand w being the solution to the dual problem. (However, the discrete vectorfor the assembled residual equation will typically be available, so doing thedot product using linear algebra would be faster than using this feature.)FIXME: Is this right?2.13.3 Energy norm of a bilinear FormThe functional representing the energy norm |v| A = v T Av of a matrix Aassembled from a form a can be computed like this51
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 93: Bibliography[1] M. Alnæs and K.-A.
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UFL Specification and User Manual 0.3 - FEniCS Project

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