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# Written by Robert C. Kirby
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# Copyright 2005 by The University of Chicago
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# Distributed under the LGPL license
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# This work is partially supported by the US Department of Energy
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# under award number DE-FG02-04ER25650
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# Last modified 16 May 2005
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"""This module defines the standard Lagrange finite element over
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any shape for which an appropriate expansion basis is defined. The
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module simply provides the class Lagrange, which takes a shape and a
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import dualbasis, functional, points, shapes, polynomial, functionalset
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class LagrangeDual( dualbasis.DualBasis ):
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"""Dual basis for the plain vanilla Lagrange finite element."""
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def __init__( self , shape , n , U ):
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# Nodes are point evaluation on the lattice, which here is
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# ordered by topological dimension and entity within that
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# dimension. For triangles we have vertex points plus
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# n - 1 points per edge. We then have max(0,(n-2)(n-1)/2)
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# points on the interior of the triangle, or that many points
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# per face of tetrahedra plus max(0,(n-2)(n-1)n/6 points
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pts = [ [ points.make_points(shape,d,e,n) \
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for e in shapes.entity_range(shape,d) ] \
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for d in shapes.dimension_range(shape) ]
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pts_flat = reduce( lambda a,b:a+b , \
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reduce( lambda a,b:a+b , pts ) )
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# get node location for each node i by self.pts[i]
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ls = functional.make_point_evaluations( U , pts_flat )
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# entity_ids is a dictionary whose keys are the topological
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# dimensions (0,1,2,3) and values are dictionaries mapping
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# the ids of entities of that dimension to the list of
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# ids of nodes associated with that entity.
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# see FIAT.base.dualbasis for description of entity_ids
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for d in shapes.dimension_range( shape ):
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for v in shapes.entity_range( shape, d ):
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num_nods = len( pts[d][v] )
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entity_ids[d][v] = range(id_cur,id_cur + num_nods)
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dualbasis.DualBasis.__init__( self , \
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functionalset.FunctionalSet( U , ls ) , \
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class Lagrange( polynomial.FiniteElement ):
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def __init__( self , shape , n ):
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"Lagrange elements are only defined for n >= 1"
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U = polynomial.OrthogonalPolynomialSet( shape , n )
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Udual = LagrangeDual( shape , n , U )
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polynomial.FiniteElement.__init__( self , \
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class VectorLagrangeDual( dualbasis.DualBasis ):
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def __init__( self , shape , n , U ):
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space_dim = shapes.dimension( shape )
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nc = U.tensor_shape()[0]
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pts = [ [ points.make_points(shape,d,e,n) \
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for e in shapes.entity_range(shape,d) ] \
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for d in shapes.dimension_range(shape) ]
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pts_flat = reduce( lambda a,b:a+b , \
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reduce( lambda a,b:a+b , pts ) )
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ls = [ functional.ComponentPointEvaluation( U , c , pt ) \
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# see FIAT.base.dualbasis for description of entity_ids
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for d in shapes.dimension_range( shape ):
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for v in shapes.entity_range( shape, d ):
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num_nods = len( pts[d][v] )
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entity_ids[d][v] = range(id_cur,id_cur + num_nods)
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fset = functionalset.FunctionalSet( U , ls )
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dualbasis.DualBasis.__init__( self , fset , entity_ids , nc )
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class VectorLagrange( polynomial.FiniteElement ):
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def __init__( self , shape , n , nc = None ):
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raise RuntimeError, \
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"Lagrange elements are only defined for n >= 1"
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U = polynomial.OrthogonalPolynomialArraySet( shape , n , nc )
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Udual = VectorLagrangeDual( shape , n , U )
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polynomial.FiniteElement.__init__( self , Udual , U )
added
removed
FIAT/Lagrange.py
