UFL Specification and User Manual 0.3 - FEniCS Project

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UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Loggf = w**4/4*dx(0) + inner(grad(w), grad(w))*dx(1)F = derivative(f, w)J = derivative(F, w)Ja = action(J, w)Jp = adjoint(J)Jpa = action(Jp, w)g = Function(element)Jnorm = energy_norm(J, g)TODO: Find some more examples, e.g. from error control!2.14 Tuple NotationIn addition to the standard integrand notation described above, UFL supportsa simplified tuple notation by which L 2 inner products may be expressedas tuples. Consider for example the following bilinear form as partof a variational problem for a reaction–diffusion problem:∫a(v,u) = ∇v ·∇u+vudxΩ= (∇v,∇u)+(v,u)In standard UFL notation, this bilinear form may be expressed asa = inner(grad(v), grad(u))*dx + v*u*dxIn tuple notation, this may alternatively be expressed asa = (grad(v), grad(u)) + (v, u)In general, a form may be expressed as a sum of tuples or triples of the form58
UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Logg(v, w)(v, w, dm)where v and w are expressions of matching rank (so that inner(v, w) makessense), and dm is a measure. If the measure is left out, it is assumed that itis dx.The following example illustrates how to express a form containing integralsover subdomains and facets:a = (grad(v), grad(u)) + (v, b*grad(u), dx(2))+ (v, u, ds) + (jump(v), jump(u), dS)The following caveats should be noted:• The only operation allowed on a tuple is addition. In particular, tuplesmay not subtracted. Thus, a = (grad(v), grad(u)) - (v, u) mustbe expressed as a = (grad(v), grad(u)) + (-v, u).• Tuple notation may not be mixed with standard UFL integrand notation.Thus, a = (grad(v), grad(u)) + inner(v, u)*dx is notvalid.[Advanced] Tuple notation is strictly speaking not a part of the formlanguage, but tuples may be converted to UFL forms using the functiontuple2form available from the module ufl.algorithms. This is normallyhandled automatically by form compilers, but the tuple2form utility mayuseful when working with UFL from a Python script. Automatic conversionis also carried out by UFL form operators such as lhs and rhs.2.15 Form FilesUFL forms and elements can be collected in a form file with the extension.ufl. Form compilers will typically execute this file with the global UFL59
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UFL Specification and User Manual 0
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ContentsAbout this manual 111 Intro
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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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UFL Specification and User Manual 0.3 - FEniCS Project

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