UFL Specification and User Manual 0.3 - FEniCS Project

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UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Logg• "sfc": A dict with further SFC specific options, see the SFC manual.• Otherstring: Adictwithfurtheroptionsspecifictosomeotherexternalcode.Other standardized options may be added in later versions.metadata0 = {"ffc": {"representation": "quadrature"}}metadata1 = {"integration_order": 7,"ffc": {"representation": "tensor"}}a = v*u*dx(0, metadata1) + f*v*dx(0, metadata2)2.2 Finite Element SpacesBeforewecanexplainhowformargumentsaredeclared, weneedtoshowhowto define function spaces. UFL can represent very flexible general hierarchiesof mixed finite elements, and has predefined names for most common elementfamilies.2.2.1 CellsA polygonal cell is defined by a basic shape and a degree 1 , written likecell = Cell(shape, degree)Valid shapes are ”interval”, ”triangle”, ”tetrahedron”, ”quadrilateral”, and”hexahedron”. Some examples:1 Note that the other components of FEniCS does not yet handle cells of higher degree,so this will only be useful in the future.18
UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Logg# Cubic triangle cellcell = Cell("triangle", 3)# Quadratic tetrahedron cellcell = Cell("tetrahedron", 2)Objects for linear cells of all basic shapes are predefined:# Predefined linear cellscell = intervalcell = trianglecell = tetrahedroncell = quadrilateralcell = hexahedronIn the rest of this document, a variable namecell will be used where any cellis a valid argument, to make the examples dimension independent whereverpossible. Using a variable cell to hold the cell type used in a form is highlyrecommended,sincethismakesmostformdefinitionsdimensionindependent.2.2.2 Element FamiliesUFL predefines a set of names of known element families. When defining afinite element below, the argument family is a string and its possible valuesinclude:• "Lagrange" or "CG", representing standard scalar Lagrange finite elements(continuous piecewise polynomial functions);• "Discontinuous Lagrange" or "DG", representing scalar discontinuousLagrange finite elements (discontinuous piecewise polynomial functions);19
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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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