Dune Core Modules (2.4.1)

multilineargeometry.hh
1// -*- tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 2 -*-
2// vi: set et ts=4 sw=2 sts=2:
3#ifndef DUNE_GEOMETRY_MULTILINEARGEOMETRY_HH
4#define DUNE_GEOMETRY_MULTILINEARGEOMETRY_HH
5
6#include <cassert>
7#include <limits>
8#include <vector>
9
13
14#include <dune/geometry/referenceelements.hh>
15#include <dune/geometry/type.hh>
16#include <dune/geometry/genericgeometry/geometrytraits.hh>
17#include <dune/geometry/genericgeometry/matrixhelper.hh>
18
19namespace Dune
20{
21
22 // External Forward Declarations
23 // -----------------------------
24
25 template< class ctype, int dim >
26 class ReferenceElement;
27
28 template< class ctype, int dim >
29 struct ReferenceElements;
30
31
32
33 // MultiLinearGeometryTraits
34 // -------------------------
35
45 template< class ct >
47 {
66 typedef GenericGeometry::MatrixHelper< GenericGeometry::DuneCoordTraits< ct > > MatrixHelper;
67
69 static ct tolerance () { return ct( 16 ) * std::numeric_limits< ct >::epsilon(); }
70
94 template< int mydim, int cdim >
96 {
97 typedef std::vector< FieldVector< ct, cdim > > Type;
98 };
99
113 template< int dim >
115 {
116 static const bool v = false;
117 static const unsigned int topologyId = ~0u;
118 };
119 };
120
121
122
123 // MultiLinearGeometry
124 // -------------------
125
146 template< class ct, int mydim, int cdim, class Traits = MultiLinearGeometryTraits< ct > >
148 {
150
151 public:
153 typedef ct ctype;
154
156 static const int mydimension= mydim;
158 static const int coorddimension = cdim;
159
164
167
169 class JacobianInverseTransposed;
170
173
174 private:
175 static const bool hasSingleGeometryType = Traits::template hasSingleGeometryType< mydimension >::v;
176
177 protected:
178 typedef typename Traits::MatrixHelper MatrixHelper;
179 typedef typename conditional< hasSingleGeometryType, integral_constant< unsigned int, Traits::template hasSingleGeometryType< mydimension >::topologyId >, unsigned int >::type TopologyId;
180
182
183 private:
184 typedef typename Traits::template CornerStorage< mydimension, coorddimension >::Type::const_iterator CornerIterator;
185
186 public:
196 template< class Corners >
198 const Corners &corners )
199 : refElement_( &refElement ),
200 corners_( corners )
201 {}
202
212 template< class Corners >
214 const Corners &corners )
215 : refElement_( &ReferenceElements::general( gt ) ),
216 corners_( corners )
217 {}
218
220 bool affine () const
221 {
223 return affine( jt );
224 }
225
227 Dune::GeometryType type () const { return GeometryType( toUnsignedInt(topologyId()), mydimension ); }
228
230 int corners () const { return refElement().size( mydimension ); }
231
233 GlobalCoordinate corner ( int i ) const
234 {
235 assert( (i >= 0) && (i < corners()) );
236 return corners_[ i ];
237 }
238
240 GlobalCoordinate center () const { return global( refElement().position( 0, 0 ) ); }
241
249 {
250 CornerIterator cit = corners_.begin();
252 global< false >( topologyId(), integral_constant< int, mydimension >(), cit, ctype( 1 ), local, ctype( 1 ), y );
253 return y;
254 }
255
269 {
270 const ctype tolerance = Traits::tolerance();
271 LocalCoordinate x = refElement().position( 0, 0 );
273 do
274 {
275 // Newton's method: DF^n dx^n = F^n, x^{n+1} -= dx^n
276 const GlobalCoordinate dglobal = (*this).global( x ) - global;
277 MatrixHelper::template xTRightInvA< mydimension, coorddimension >( jacobianTransposed( x ), dglobal, dx );
278 x -= dx;
279 } while( dx.two_norm2() > tolerance );
280 return x;
281 }
282
298 {
299 return MatrixHelper::template sqrtDetAAT< mydimension, coorddimension >( jacobianTransposed( local ) );
300 }
301
310 ctype volume () const
311 {
312 return integrationElement( refElement().position( 0, 0 ) ) * refElement().volume();
313 }
314
325 {
327 CornerIterator cit = corners_.begin();
328 jacobianTransposed< false >( topologyId(), integral_constant< int, mydimension >(), cit, ctype( 1 ), local, ctype( 1 ), jt );
329 return jt;
330 }
331
338 JacobianInverseTransposed jacobianInverseTransposed ( const LocalCoordinate &local ) const;
339
340 protected:
341 const ReferenceElement &refElement () const { return *refElement_; }
342
343 TopologyId topologyId () const
344 {
345 return topologyId( integral_constant< bool, hasSingleGeometryType >() );
346 }
347
348 template< bool add, int dim >
349 static void global ( TopologyId topologyId, integral_constant< int, dim >,
350 CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
351 const ctype &rf, GlobalCoordinate &y );
352 template< bool add >
353 static void global ( TopologyId topologyId, integral_constant< int, 0 >,
354 CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
355 const ctype &rf, GlobalCoordinate &y );
356
357 template< bool add, int rows, int dim >
358 static void jacobianTransposed ( TopologyId topologyId, integral_constant< int, dim >,
359 CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
360 const ctype &rf, FieldMatrix< ctype, rows, cdim > &jt );
361 template< bool add, int rows >
362 static void jacobianTransposed ( TopologyId topologyId, integral_constant< int, 0 >,
363 CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
364 const ctype &rf, FieldMatrix< ctype, rows, cdim > &jt );
365
366 template< int dim >
367 static bool affine ( TopologyId topologyId, integral_constant< int, dim >, CornerIterator &cit, JacobianTransposed &jt );
368 static bool affine ( TopologyId topologyId, integral_constant< int, 0 >, CornerIterator &cit, JacobianTransposed &jt );
369
371 {
372 CornerIterator cit = corners_.begin();
373 return affine( topologyId(), integral_constant< int, mydimension >(), cit, jacobianTransposed );
374 }
375
376 private:
377 // The following methods are needed to convert the return type of topologyId to
378 // unsigned int with g++-4.4. It has problems casting integral_constant to the
379 // integral type.
380 static unsigned int toUnsignedInt(unsigned int i) { return i; }
381 template<unsigned int v>
382 static unsigned int toUnsignedInt(std::integral_constant<unsigned int,v> i) { return v; }
383 TopologyId topologyId ( integral_constant< bool, true > ) const { return TopologyId(); }
384 unsigned int topologyId ( integral_constant< bool, false > ) const { return refElement().type().id(); }
385
386 const ReferenceElement *refElement_;
387 typename Traits::template CornerStorage< mydimension, coorddimension >::Type corners_;
388 };
389
390
391
392 // MultiLinearGeometry::JacobianInverseTransposed
393 // ----------------------------------------------
394
395 template< class ct, int mydim, int cdim, class Traits >
396 class MultiLinearGeometry< ct, mydim, cdim, Traits >::JacobianInverseTransposed
397 : public FieldMatrix< ctype, coorddimension, mydimension >
398 {
400
401 public:
402 void setup ( const JacobianTransposed &jt )
403 {
404 detInv_ = MatrixHelper::template rightInvA< mydimension, coorddimension >( jt, static_cast< Base & >( *this ) );
405 }
406
407 void setupDeterminant ( const JacobianTransposed &jt )
408 {
409 detInv_ = MatrixHelper::template sqrtDetAAT< mydimension, coorddimension >( jt );
410 }
411
412 ctype det () const { return ctype( 1 ) / detInv_; }
413 ctype detInv () const { return detInv_; }
414
415 private:
416 ctype detInv_;
417 };
418
419
420
433 template< class ct, int mydim, int cdim, class Traits = MultiLinearGeometryTraits< ct > >
435 : public MultiLinearGeometry< ct, mydim, cdim, Traits >
436 {
439
440 protected:
441 typedef typename Base::MatrixHelper MatrixHelper;
442
443 public:
445
446 typedef typename Base::ctype ctype;
447
448 using Base::mydimension;
450
451 typedef typename Base::LocalCoordinate LocalCoordinate;
453
455 typedef typename Base::JacobianInverseTransposed JacobianInverseTransposed;
456
457 template< class CornerStorage >
458 CachedMultiLinearGeometry ( const ReferenceElement &refElement, const CornerStorage &cornerStorage )
459 : Base( refElement, cornerStorage ),
460 affine_( Base::affine( jacobianTransposed_ ) ),
461 jacobianInverseTransposedComputed_( false ),
462 integrationElementComputed_( false )
463 {}
464
465 template< class CornerStorage >
466 CachedMultiLinearGeometry ( Dune::GeometryType gt, const CornerStorage &cornerStorage )
467 : Base( gt, cornerStorage ),
468 affine_( Base::affine( jacobianTransposed_ ) ),
469 jacobianInverseTransposedComputed_( false ),
470 integrationElementComputed_( false )
471 {}
472
474 bool affine () const { return affine_; }
475
476 using Base::corner;
477
479 GlobalCoordinate center () const { return global( refElement().position( 0, 0 ) ); }
480
488 {
489 if( affine() )
490 {
492 jacobianTransposed_.umtv( local, global );
493 return global;
494 }
495 else
496 return Base::global( local );
497 }
498
512 {
513 if( affine() )
514 {
515 LocalCoordinate local;
516 if( jacobianInverseTransposedComputed_ )
517 jacobianInverseTransposed_.mtv( global - corner( 0 ), local );
518 else
519 MatrixHelper::template xTRightInvA< mydimension, coorddimension >( jacobianTransposed_, global - corner( 0 ), local );
520 return local;
521 }
522 else
523 return Base::local( global );
524 }
525
540 ctype integrationElement ( const LocalCoordinate &local ) const
541 {
542 if( affine() )
543 {
544 if( !integrationElementComputed_ )
545 {
546 jacobianInverseTransposed_.setupDeterminant( jacobianTransposed_ );
547 integrationElementComputed_ = true;
548 }
549 return jacobianInverseTransposed_.detInv();
550 }
551 else
552 return Base::integrationElement( local );
553 }
554
556 ctype volume () const
557 {
558 if( affine() )
559 return integrationElement( refElement().position( 0, 0 ) ) * refElement().volume();
560 else
561 return Base::volume();
562 }
563
574 {
575 if( affine() )
576 return jacobianTransposed_;
577 else
578 return Base::jacobianTransposed( local );
579 }
580
587 JacobianInverseTransposed jacobianInverseTransposed ( const LocalCoordinate &local ) const
588 {
589 if( affine() )
590 {
591 if( !jacobianInverseTransposedComputed_ )
592 {
593 jacobianInverseTransposed_.setup( jacobianTransposed_ );
594 jacobianInverseTransposedComputed_ = true;
595 integrationElementComputed_ = true;
596 }
597 return jacobianInverseTransposed_;
598 }
599 else
600 return Base::jacobianInverseTransposed( local );
601 }
602
603 protected:
604 using Base::refElement;
605
606 private:
607 mutable JacobianTransposed jacobianTransposed_;
608 mutable JacobianInverseTransposed jacobianInverseTransposed_;
609
610 mutable bool affine_ : 1;
611
612 mutable bool jacobianInverseTransposedComputed_ : 1;
613 mutable bool integrationElementComputed_ : 1;
614 };
615
616
617
618 // Implementation of MultiLinearGeometry
619 // -------------------------------------
620
621 template< class ct, int mydim, int cdim, class Traits >
622 inline typename MultiLinearGeometry< ct, mydim, cdim, Traits >::JacobianInverseTransposed
624 {
625 JacobianInverseTransposed jit;
626 jit.setup( jacobianTransposed( local ) );
627 return jit;
628 }
629
630
631 template< class ct, int mydim, int cdim, class Traits >
632 template< bool add, int dim >
634 ::global ( TopologyId topologyId, integral_constant< int, dim >,
635 CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
636 const ctype &rf, GlobalCoordinate &y )
637 {
638 const ctype xn = df*x[ dim-1 ];
639 const ctype cxn = ctype( 1 ) - xn;
640
641 if( GenericGeometry::isPrism( toUnsignedInt(topologyId), mydimension, mydimension-dim ) )
642 {
643 // apply (1-xn) times mapping for bottom
644 global< add >( topologyId, integral_constant< int, dim-1 >(), cit, df, x, rf*cxn, y );
645 // apply xn times mapping for top
646 global< true >( topologyId, integral_constant< int, dim-1 >(), cit, df, x, rf*xn, y );
647 }
648 else
649 {
650 assert( GenericGeometry::isPyramid( toUnsignedInt(topologyId), mydimension, mydimension-dim ) );
651 // apply (1-xn) times mapping for bottom (with argument x/(1-xn))
652 if( cxn > Traits::tolerance() || cxn < -Traits::tolerance() )
653 global< add >( topologyId, integral_constant< int, dim-1 >(), cit, df/cxn, x, rf*cxn, y );
654 else
655 global< add >( topologyId, integral_constant< int, dim-1 >(), cit, df, x, ctype( 0 ), y );
656 // apply xn times the tip
657 y.axpy( rf*xn, *cit );
658 ++cit;
659 }
660 }
661
662 template< class ct, int mydim, int cdim, class Traits >
663 template< bool add >
665 ::global ( TopologyId topologyId, integral_constant< int, 0 >,
666 CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
667 const ctype &rf, GlobalCoordinate &y )
668 {
669 const GlobalCoordinate &origin = *cit;
670 ++cit;
671 for( int i = 0; i < coorddimension; ++i )
672 y[ i ] = (add ? y[ i ] + rf*origin[ i ] : rf*origin[ i ]);
673 }
674
675
676 template< class ct, int mydim, int cdim, class Traits >
677 template< bool add, int rows, int dim >
679 ::jacobianTransposed ( TopologyId topologyId, integral_constant< int, dim >,
680 CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
681 const ctype &rf, FieldMatrix< ctype, rows, cdim > &jt )
682 {
683 assert( rows >= dim );
684
685 const ctype xn = df*x[ dim-1 ];
686 const ctype cxn = ctype( 1 ) - xn;
687
688 CornerIterator cit2( cit );
689 if( GenericGeometry::isPrism( toUnsignedInt(topologyId), mydimension, mydimension-dim ) )
690 {
691 // apply (1-xn) times Jacobian for bottom
692 jacobianTransposed< add >( topologyId, integral_constant< int, dim-1 >(), cit2, df, x, rf*cxn, jt );
693 // apply xn times Jacobian for top
694 jacobianTransposed< true >( topologyId, integral_constant< int, dim-1 >(), cit2, df, x, rf*xn, jt );
695 // compute last row as difference between top value and bottom value
696 global< add >( topologyId, integral_constant< int, dim-1 >(), cit, df, x, -rf, jt[ dim-1 ] );
697 global< true >( topologyId, integral_constant< int, dim-1 >(), cit, df, x, rf, jt[ dim-1 ] );
698 }
699 else
700 {
701 assert( GenericGeometry::isPyramid( toUnsignedInt(topologyId), mydimension, mydimension-dim ) );
702 /*
703 * In the pyramid case, we need a transformation Tb: B -> R^n for the
704 * base B \subset R^{n-1}. The pyramid transformation is then defined as
705 * T: P \subset R^n -> R^n
706 * (x, xn) |-> (1-xn) Tb(x*) + xn t (x \in R^{n-1}, xn \in R)
707 * with the tip of the pyramid mapped to t and x* = x/(1-xn)
708 * the projection of (x,xn) onto the base.
709 *
710 * For the Jacobi matrix DT we get
711 * DT = ( A | b )
712 * with A = DTb(x*) (n x n-1 matrix)
713 * and b = dT/dxn (n-dim column vector).
714 * Furthermore
715 * b = -Tb(x*) + t + \sum_i dTb/dx_i(x^*) x_i/(1-xn)
716 *
717 * Note that both A and b are not defined in the pyramid tip (x=0, xn=1)!
718 * Indeed for B the unit square, Tb mapping B to the quadrilateral given
719 * by the vertices (0,0,0), (2,0,0), (0,1,0), (1,1,0) and t=(0,0,1), we get
720 *
721 * T(x,y,xn) = ( x(2-y/(1-xn)), y, xn )
722 * / 2-y/(1-xn) -x 0 \
723 * DT(x,y,xn) = | 0 1 0 |
724 * \ 0 0 1 /
725 * which is not continuous for xn -> 1, choose for example
726 * x=0, y=1-xn, xn -> 1 --> DT -> diag(1,1,1)
727 * x=1-xn, y=0, xn -> 1 --> DT -> diag(2,1,1)
728 *
729 * However, for Tb affine-linear, Tb(y) = My + y0, DTb = M:
730 * A = M
731 * b = -M x* - y0 + t + \sum_i M_i x_i/(1-xn)
732 * = -M x* - y0 + t + M x*
733 * = -y0 + t
734 * which is continuous for xn -> 1. Note that this b is also given by
735 * b = -Tb(0) + t + \sum_i dTb/dx_i(0) x_i/1
736 * that is replacing x* by 1 and 1-xn by 1 in the formular above.
737 *
738 * For xn -> 1, we can thus set x*=0, "1-xn"=1 (or anything != 0) and get
739 * the right result in case Tb is affine-linear.
740 */
741
742 /* The second case effectively results in x* = 0 */
743 ctype dfcxn = (cxn > Traits::tolerance() || cxn < -Traits::tolerance()) ? ctype(df / cxn) : ctype(0);
744
745 // initialize last row
746 // b = -Tb(x*)
747 // (b = -Tb(0) = -y0 in case xn -> 1 and Tb affine-linear)
748 global< add >( topologyId, integral_constant< int, dim-1 >(), cit, dfcxn, x, -rf, jt[ dim-1 ] );
749 // b += t
750 jt[ dim-1 ].axpy( rf, *cit );
751 ++cit;
752 // apply Jacobian for bottom (with argument x/(1-xn)) and correct last row
753 if( add )
754 {
755 FieldMatrix< ctype, dim-1, coorddimension > jt2;
756 // jt2 = dTb/dx_i(x*)
757 jacobianTransposed< false >( topologyId, integral_constant< int, dim-1 >(), cit2, dfcxn, x, rf, jt2 );
758 // A = dTb/dx_i(x*) (jt[j], j=0..dim-1)
759 // b += \sum_i dTb/dx_i(x*) x_i/(1-xn) (jt[dim-1])
760 // (b += 0 in case xn -> 1)
761 for( int j = 0; j < dim-1; ++j )
762 {
763 jt[ j ] += jt2[ j ];
764 jt[ dim-1 ].axpy( dfcxn*x[ j ], jt2[ j ] );
765 }
766 }
767 else
768 {
769 // jt = dTb/dx_i(x*)
770 jacobianTransposed< false >( topologyId, integral_constant< int, dim-1 >(), cit2, dfcxn, x, rf, jt );
771 // b += \sum_i dTb/dx_i(x*) x_i/(1-xn)
772 for( int j = 0; j < dim-1; ++j )
773 jt[ dim-1 ].axpy( dfcxn*x[ j ], jt[ j ] );
774 }
775 }
776 }
777
778 template< class ct, int mydim, int cdim, class Traits >
779 template< bool add, int rows >
781 ::jacobianTransposed ( TopologyId topologyId, integral_constant< int, 0 >,
782 CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
783 const ctype &rf, FieldMatrix< ctype, rows, cdim > &jt )
784 {
785 ++cit;
786 }
787
788
789
790 template< class ct, int mydim, int cdim, class Traits >
791 template< int dim >
793 ::affine ( TopologyId topologyId, integral_constant< int, dim >, CornerIterator &cit, JacobianTransposed &jt )
794 {
795 const GlobalCoordinate &orgBottom = *cit;
796 if( !affine( topologyId, integral_constant< int, dim-1 >(), cit, jt ) )
797 return false;
798 const GlobalCoordinate &orgTop = *cit;
799
800 if( GenericGeometry::isPrism( toUnsignedInt(topologyId), mydimension, mydimension-dim ) )
801 {
802 JacobianTransposed jtTop;
803 if( !affine( topologyId, integral_constant< int, dim-1 >(), cit, jtTop ) )
804 return false;
805
806 // check whether both jacobians are identical
807 ctype norm( 0 );
808 for( int i = 0; i < dim-1; ++i )
809 norm += (jtTop[ i ] - jt[ i ]).two_norm2();
810 if( norm >= Traits::tolerance() )
811 return false;
812 }
813 else
814 ++cit;
815 jt[ dim-1 ] = orgTop - orgBottom;
816 return true;
817 }
818
819 template< class ct, int mydim, int cdim, class Traits >
821 ::affine ( TopologyId topologyId, integral_constant< int, 0 >, CornerIterator &cit, JacobianTransposed &jt )
822 {
823 ++cit;
824 return true;
825 }
826
827} // namespace Dune
828
829#endif // #ifndef DUNE_GEOMETRY_MULTILINEARGEOMETRY_HH
Implement a MultiLinearGeometry with additional caching.
Definition: multilineargeometry.hh:436
GlobalCoordinate global(const LocalCoordinate &local) const
evaluate the mapping
Definition: multilineargeometry.hh:487
bool affine() const
is this mapping affine?
Definition: multilineargeometry.hh:474
JacobianTransposed jacobianTransposed(const LocalCoordinate &local) const
obtain the transposed of the Jacobian
Definition: multilineargeometry.hh:573
GlobalCoordinate corner(int i) const
obtain coordinates of the i-th corner
Definition: multilineargeometry.hh:233
ctype volume() const
obtain the volume of the mapping's image
Definition: multilineargeometry.hh:556
ctype integrationElement(const LocalCoordinate &local) const
obtain the integration element
Definition: multilineargeometry.hh:540
GlobalCoordinate center() const
obtain the centroid of the mapping's image
Definition: multilineargeometry.hh:479
JacobianInverseTransposed jacobianInverseTransposed(const LocalCoordinate &local) const
obtain the transposed of the Jacobian's inverse
Definition: multilineargeometry.hh:587
void umtv(const X &x, Y &y) const
y += A^T x
Definition: densematrix.hh:441
FieldTraits< value_type >::real_type two_norm2() const
square of two norm (sum over squared values of entries), need for block recursion
Definition: densevector.hh:598
vector space out of a tensor product of fields.
Definition: fvector.hh:94
Unique label for each type of entities that can occur in DUNE grids.
Definition: type.hh:25
unsigned int id() const
Return the topology id the type.
Definition: type.hh:326
generic geometry implementation based on corner coordinates
Definition: multilineargeometry.hh:148
static const int mydimension
geometry dimension
Definition: multilineargeometry.hh:156
Dune::ReferenceElement< ctype, mydimension > ReferenceElement
type of reference element
Definition: multilineargeometry.hh:172
Dune::GeometryType type() const
obtain the name of the reference element
Definition: multilineargeometry.hh:227
FieldVector< ctype, coorddimension > GlobalCoordinate
type of global coordinates
Definition: multilineargeometry.hh:163
JacobianTransposed jacobianTransposed(const LocalCoordinate &local) const
obtain the transposed of the Jacobian
Definition: multilineargeometry.hh:324
GlobalCoordinate global(const LocalCoordinate &local) const
evaluate the mapping
Definition: multilineargeometry.hh:248
GlobalCoordinate center() const
obtain the centroid of the mapping's image
Definition: multilineargeometry.hh:240
GlobalCoordinate corner(int i) const
obtain coordinates of the i-th corner
Definition: multilineargeometry.hh:233
ct ctype
coordinate type
Definition: multilineargeometry.hh:153
static const int coorddimension
coordinate dimension
Definition: multilineargeometry.hh:158
int corners() const
obtain number of corners of the corresponding reference element
Definition: multilineargeometry.hh:230
FieldVector< ctype, mydimension > LocalCoordinate
type of local coordinates
Definition: multilineargeometry.hh:161
ctype volume() const
obtain the volume of the mapping's image
Definition: multilineargeometry.hh:310
MultiLinearGeometry(const ReferenceElement &refElement, const Corners &corners)
constructor
Definition: multilineargeometry.hh:197
MultiLinearGeometry(Dune::GeometryType gt, const Corners &corners)
constructor
Definition: multilineargeometry.hh:213
ctype integrationElement(const LocalCoordinate &local) const
obtain the integration element
Definition: multilineargeometry.hh:297
FieldMatrix< ctype, mydimension, coorddimension > JacobianTransposed
type of jacobian transposed
Definition: multilineargeometry.hh:166
JacobianInverseTransposed jacobianInverseTransposed(const LocalCoordinate &local) const
obtain the transposed of the Jacobian's inverse
Definition: multilineargeometry.hh:623
bool affine() const
is this mapping affine?
Definition: multilineargeometry.hh:220
This class provides access to geometric and topological properties of a reference element.
Definition: referenceelements.hh:55
int size(int c) const
number of subentities of codimension c
Definition: referenceelements.hh:82
const FieldVector< ctype, dim > & position(int i, int c) const
position of the barycenter of entity (i,c)
Definition: referenceelements.hh:150
ctype volume() const
obtain the volume of the reference element
Definition: referenceelements.hh:210
const GeometryType & type(int i, int c) const
obtain the type of subentity (i,c)
Definition: referenceelements.hh:132
Implements a matrix constructed from a given type representing a field and compile-time given number ...
Implements a vector constructed from a given type representing a field and a compile-time given size.
bool gt(const T &first, const T &second, typename EpsilonType< T >::Type epsilon)
test if first greater than second
Definition: float_cmp.cc:132
Dune namespace.
Definition: alignment.hh:10
template specifying the storage for the corners
Definition: multilineargeometry.hh:96
will there be only one geometry type for a dimension?
Definition: multilineargeometry.hh:115
default traits class for MultiLinearGeometry
Definition: multilineargeometry.hh:47
static ct tolerance()
tolerance to numerical algorithms
Definition: multilineargeometry.hh:69
GenericGeometry::MatrixHelper< GenericGeometry::DuneCoordTraits< ct > > MatrixHelper
helper structure containing some matrix routines
Definition: multilineargeometry.hh:66
Class providing access to the singletons of the reference elements.
Definition: referenceelements.hh:479
A unique label for each type of element that can occur in a grid.
Traits for type conversions and type information.
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