UFL Specification and User Manual 0.3 - FEniCS Project

UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Logg2.1 Forms and IntegralsUFL is designed to express forms in the following generalized format:a(v 1 ,...,v r ;w 1 ,...,w n ) = (2.1)∑n c ∫Ik(v c 1 ,...,v r ;w 1 ,...w n )dxk=1Ω k∑n e ∫+ Ik(v e 1 ,...,v r ;w 1 ,...,w n )dsk=1∂Ω k∑n i∫+ Ik(v i 1 ,...,v r ;w 1 ,...,w n )dS.Γ kk=1Here the form a depends on the form arguments v 1 ,...,v r and the formcoefficients w 1 ,...,w n , and its expression is a sum of integrals. Each term ofa valid form expression must be a scalar-valued expression integrated exactlyonce. How to define form arguments and integrand expressions is detailed inthe rest of this chapter.Integrals are expressed through multiplication with a measure, representingan integral over either of• the interior of the domain Ω (dx, cell integral);• the boundary ∂Ω of Ω (ds, exterior facet integral);• the set of interior facets Γ (dS, interior facet integral).UFL declares the measures dx ↔ dx, ds ↔ ds, and dS ↔ dS.As a basic example, assume v is a scalar-valued expression and consider theintegral of v over the interior of Ω. This may be expressed asa = v*dxand the integral of v over ∂Ω is written as16