Dune Core Modules (2.8.0)

affinegeometry.hh
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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_AFFINEGEOMETRY_HH
4#define DUNE_GEOMETRY_AFFINEGEOMETRY_HH
5
11#include <cmath>
12
13#include <dune/common/fmatrix.hh>
14#include <dune/common/fvector.hh>
15
16#include <dune/geometry/type.hh>
17
18namespace Dune
19{
20
21 // External Forward Declarations
22 // -----------------------------
23
24 namespace Geo
25 {
26
27 template< typename Implementation >
28 class ReferenceElement;
29
30 template< class ctype, int dim >
31 class ReferenceElementImplementation;
32
33 template< class ctype, int dim >
34 struct ReferenceElements;
35
36 }
37
38
39 namespace Impl
40 {
41
42 // FieldMatrixHelper
43 // -----------------
44
45 template< class ct >
46 struct FieldMatrixHelper
47 {
48 typedef ct ctype;
49
50 template< int m, int n >
51 static void Ax ( const FieldMatrix< ctype, m, n > &A, const FieldVector< ctype, n > &x, FieldVector< ctype, m > &ret )
52 {
53 for( int i = 0; i < m; ++i )
54 {
55 ret[ i ] = ctype( 0 );
56 for( int j = 0; j < n; ++j )
57 ret[ i ] += A[ i ][ j ] * x[ j ];
58 }
59 }
60
61 template< int m, int n >
62 static void ATx ( const FieldMatrix< ctype, m, n > &A, const FieldVector< ctype, m > &x, FieldVector< ctype, n > &ret )
63 {
64 for( int i = 0; i < n; ++i )
65 {
66 ret[ i ] = ctype( 0 );
67 for( int j = 0; j < m; ++j )
68 ret[ i ] += A[ j ][ i ] * x[ j ];
69 }
70 }
71
72 template< int m, int n, int p >
73 static void AB ( const FieldMatrix< ctype, m, n > &A, const FieldMatrix< ctype, n, p > &B, FieldMatrix< ctype, m, p > &ret )
74 {
75 for( int i = 0; i < m; ++i )
76 {
77 for( int j = 0; j < p; ++j )
78 {
79 ret[ i ][ j ] = ctype( 0 );
80 for( int k = 0; k < n; ++k )
81 ret[ i ][ j ] += A[ i ][ k ] * B[ k ][ j ];
82 }
83 }
84 }
85
86 template< int m, int n, int p >
87 static void ATBT ( const FieldMatrix< ctype, m, n > &A, const FieldMatrix< ctype, p, m > &B, FieldMatrix< ctype, n, p > &ret )
88 {
89 for( int i = 0; i < n; ++i )
90 {
91 for( int j = 0; j < p; ++j )
92 {
93 ret[ i ][ j ] = ctype( 0 );
94 for( int k = 0; k < m; ++k )
95 ret[ i ][ j ] += A[ k ][ i ] * B[ j ][ k ];
96 }
97 }
98 }
99
100 template< int m, int n >
101 static void ATA_L ( const FieldMatrix< ctype, m, n > &A, FieldMatrix< ctype, n, n > &ret )
102 {
103 for( int i = 0; i < n; ++i )
104 {
105 for( int j = 0; j <= i; ++j )
106 {
107 ret[ i ][ j ] = ctype( 0 );
108 for( int k = 0; k < m; ++k )
109 ret[ i ][ j ] += A[ k ][ i ] * A[ k ][ j ];
110 }
111 }
112 }
113
114 template< int m, int n >
115 static void ATA ( const FieldMatrix< ctype, m, n > &A, FieldMatrix< ctype, n, n > &ret )
116 {
117 for( int i = 0; i < n; ++i )
118 {
119 for( int j = 0; j <= i; ++j )
120 {
121 ret[ i ][ j ] = ctype( 0 );
122 for( int k = 0; k < m; ++k )
123 ret[ i ][ j ] += A[ k ][ i ] * A[ k ][ j ];
124 ret[ j ][ i ] = ret[ i ][ j ];
125 }
126
127 ret[ i ][ i ] = ctype( 0 );
128 for( int k = 0; k < m; ++k )
129 ret[ i ][ i ] += A[ k ][ i ] * A[ k ][ i ];
130 }
131 }
132
133 template< int m, int n >
134 static void AAT_L ( const FieldMatrix< ctype, m, n > &A, FieldMatrix< ctype, m, m > &ret )
135 {
136 /*
137 if (m==2) {
138 ret[0][0] = A[0]*A[0];
139 ret[1][1] = A[1]*A[1];
140 ret[1][0] = A[0]*A[1];
141 }
142 else
143 */
144 for( int i = 0; i < m; ++i )
145 {
146 for( int j = 0; j <= i; ++j )
147 {
148 ctype &retij = ret[ i ][ j ];
149 retij = A[ i ][ 0 ] * A[ j ][ 0 ];
150 for( int k = 1; k < n; ++k )
151 retij += A[ i ][ k ] * A[ j ][ k ];
152 }
153 }
154 }
155
156 template< int m, int n >
157 static void AAT ( const FieldMatrix< ctype, m, n > &A, FieldMatrix< ctype, m, m > &ret )
158 {
159 for( int i = 0; i < m; ++i )
160 {
161 for( int j = 0; j < i; ++j )
162 {
163 ret[ i ][ j ] = ctype( 0 );
164 for( int k = 0; k < n; ++k )
165 ret[ i ][ j ] += A[ i ][ k ] * A[ j ][ k ];
166 ret[ j ][ i ] = ret[ i ][ j ];
167 }
168 ret[ i ][ i ] = ctype( 0 );
169 for( int k = 0; k < n; ++k )
170 ret[ i ][ i ] += A[ i ][ k ] * A[ i ][ k ];
171 }
172 }
173
174 template< int n >
175 static void Lx ( const FieldMatrix< ctype, n, n > &L, const FieldVector< ctype, n > &x, FieldVector< ctype, n > &ret )
176 {
177 for( int i = 0; i < n; ++i )
178 {
179 ret[ i ] = ctype( 0 );
180 for( int j = 0; j <= i; ++j )
181 ret[ i ] += L[ i ][ j ] * x[ j ];
182 }
183 }
184
185 template< int n >
186 static void LTx ( const FieldMatrix< ctype, n, n > &L, const FieldVector< ctype, n > &x, FieldVector< ctype, n > &ret )
187 {
188 for( int i = 0; i < n; ++i )
189 {
190 ret[ i ] = ctype( 0 );
191 for( int j = i; j < n; ++j )
192 ret[ i ] += L[ j ][ i ] * x[ j ];
193 }
194 }
195
196 template< int n >
197 static void LTL ( const FieldMatrix< ctype, n, n > &L, FieldMatrix< ctype, n, n > &ret )
198 {
199 for( int i = 0; i < n; ++i )
200 {
201 for( int j = 0; j < i; ++j )
202 {
203 ret[ i ][ j ] = ctype( 0 );
204 for( int k = i; k < n; ++k )
205 ret[ i ][ j ] += L[ k ][ i ] * L[ k ][ j ];
206 ret[ j ][ i ] = ret[ i ][ j ];
207 }
208 ret[ i ][ i ] = ctype( 0 );
209 for( int k = i; k < n; ++k )
210 ret[ i ][ i ] += L[ k ][ i ] * L[ k ][ i ];
211 }
212 }
213
214 template< int n >
215 static void LLT ( const FieldMatrix< ctype, n, n > &L, FieldMatrix< ctype, n, n > &ret )
216 {
217 for( int i = 0; i < n; ++i )
218 {
219 for( int j = 0; j < i; ++j )
220 {
221 ret[ i ][ j ] = ctype( 0 );
222 for( int k = 0; k <= j; ++k )
223 ret[ i ][ j ] += L[ i ][ k ] * L[ j ][ k ];
224 ret[ j ][ i ] = ret[ i ][ j ];
225 }
226 ret[ i ][ i ] = ctype( 0 );
227 for( int k = 0; k <= i; ++k )
228 ret[ i ][ i ] += L[ i ][ k ] * L[ i ][ k ];
229 }
230 }
231
232 template< int n >
233 static bool cholesky_L ( const FieldMatrix< ctype, n, n > &A, FieldMatrix< ctype, n, n > &ret, const bool checkSingular = false )
234 {
235 using std::sqrt;
236 for( int i = 0; i < n; ++i )
237 {
238 ctype &rii = ret[ i ][ i ];
239
240 ctype xDiag = A[ i ][ i ];
241 for( int j = 0; j < i; ++j )
242 xDiag -= ret[ i ][ j ] * ret[ i ][ j ];
243
244 // in some cases A can be singular, e.g. when checking local for
245 // outside points during checkInside
246 if( checkSingular && ! ( xDiag > ctype( 0 )) )
247 return false ;
248
249 // otherwise this should be true always
250 assert( xDiag > ctype( 0 ) );
251 rii = sqrt( xDiag );
252
253 ctype invrii = ctype( 1 ) / rii;
254 for( int k = i+1; k < n; ++k )
255 {
256 ctype x = A[ k ][ i ];
257 for( int j = 0; j < i; ++j )
258 x -= ret[ i ][ j ] * ret[ k ][ j ];
259 ret[ k ][ i ] = invrii * x;
260 }
261 }
262
263 // return true for meaning A is non-singular
264 return true;
265 }
266
267 template< int n >
268 static ctype detL ( const FieldMatrix< ctype, n, n > &L )
269 {
270 ctype det( 1 );
271 for( int i = 0; i < n; ++i )
272 det *= L[ i ][ i ];
273 return det;
274 }
275
276 template< int n >
277 static ctype invL ( FieldMatrix< ctype, n, n > &L )
278 {
279 ctype det( 1 );
280 for( int i = 0; i < n; ++i )
281 {
282 ctype &lii = L[ i ][ i ];
283 det *= lii;
284 lii = ctype( 1 ) / lii;
285 for( int j = 0; j < i; ++j )
286 {
287 ctype &lij = L[ i ][ j ];
288 ctype x = lij * L[ j ][ j ];
289 for( int k = j+1; k < i; ++k )
290 x += L[ i ][ k ] * L[ k ][ j ];
291 lij = (-lii) * x;
292 }
293 }
294 return det;
295 }
296
297 // calculates x := L^{-1} x
298 template< int n >
299 static void invLx ( FieldMatrix< ctype, n, n > &L, FieldVector< ctype, n > &x )
300 {
301 for( int i = 0; i < n; ++i )
302 {
303 for( int j = 0; j < i; ++j )
304 x[ i ] -= L[ i ][ j ] * x[ j ];
305 x[ i ] /= L[ i ][ i ];
306 }
307 }
308
309 // calculates x := L^{-T} x
310 template< int n >
311 static void invLTx ( FieldMatrix< ctype, n, n > &L, FieldVector< ctype, n > &x )
312 {
313 for( int i = n; i > 0; --i )
314 {
315 for( int j = i; j < n; ++j )
316 x[ i-1 ] -= L[ j ][ i-1 ] * x[ j ];
317 x[ i-1 ] /= L[ i-1 ][ i-1 ];
318 }
319 }
320
321 template< int n >
322 static ctype spdDetA ( const FieldMatrix< ctype, n, n > &A )
323 {
324 // return A[0][0]*A[1][1]-A[1][0]*A[1][0];
325 FieldMatrix< ctype, n, n > L;
326 cholesky_L( A, L );
327 return detL( L );
328 }
329
330 template< int n >
331 static ctype spdInvA ( FieldMatrix< ctype, n, n > &A )
332 {
333 FieldMatrix< ctype, n, n > L;
334 cholesky_L( A, L );
335 const ctype det = invL( L );
336 LTL( L, A );
337 return det;
338 }
339
340 // calculate x := A^{-1} x
341 template< int n >
342 static bool spdInvAx ( FieldMatrix< ctype, n, n > &A, FieldVector< ctype, n > &x, const bool checkSingular = false )
343 {
344 FieldMatrix< ctype, n, n > L;
345 const bool invertible = cholesky_L( A, L, checkSingular );
346 if( ! invertible ) return invertible ;
347 invLx( L, x );
348 invLTx( L, x );
349 return invertible;
350 }
351
352 template< int m, int n >
353 static ctype detATA ( const FieldMatrix< ctype, m, n > &A )
354 {
355 if( m >= n )
356 {
357 FieldMatrix< ctype, n, n > ata;
358 ATA_L( A, ata );
359 return spdDetA( ata );
360 }
361 else
362 return ctype( 0 );
363 }
364
370 template< int m, int n >
371 static ctype sqrtDetAAT ( const FieldMatrix< ctype, m, n > &A )
372 {
373 using std::abs;
374 using std::sqrt;
375 // These special cases are here not only for speed reasons:
376 // The general implementation aborts if the matrix is almost singular,
377 // and the special implementation provide a stable way to handle that case.
378 if( (n == 2) && (m == 2) )
379 {
380 // Special implementation for 2x2 matrices: faster and more stable
381 return abs( A[ 0 ][ 0 ]*A[ 1 ][ 1 ] - A[ 1 ][ 0 ]*A[ 0 ][ 1 ] );
382 }
383 else if( (n == 3) && (m == 3) )
384 {
385 // Special implementation for 3x3 matrices
386 const ctype v0 = A[ 0 ][ 1 ] * A[ 1 ][ 2 ] - A[ 1 ][ 1 ] * A[ 0 ][ 2 ];
387 const ctype v1 = A[ 0 ][ 2 ] * A[ 1 ][ 0 ] - A[ 1 ][ 2 ] * A[ 0 ][ 0 ];
388 const ctype v2 = A[ 0 ][ 0 ] * A[ 1 ][ 1 ] - A[ 1 ][ 0 ] * A[ 0 ][ 1 ];
389 return abs( v0 * A[ 2 ][ 0 ] + v1 * A[ 2 ][ 1 ] + v2 * A[ 2 ][ 2 ] );
390 }
391 else if ( (n == 3) && (m == 2) )
392 {
393 // Special implementation for 2x3 matrices
394 const ctype v0 = A[ 0 ][ 0 ] * A[ 1 ][ 1 ] - A[ 0 ][ 1 ] * A[ 1 ][ 0 ];
395 const ctype v1 = A[ 0 ][ 0 ] * A[ 1 ][ 2 ] - A[ 1 ][ 0 ] * A[ 0 ][ 2 ];
396 const ctype v2 = A[ 0 ][ 1 ] * A[ 1 ][ 2 ] - A[ 0 ][ 2 ] * A[ 1 ][ 1 ];
397 return sqrt( v0*v0 + v1*v1 + v2*v2);
398 }
399 else if( n >= m )
400 {
401 // General case
402 FieldMatrix< ctype, m, m > aat;
403 AAT_L( A, aat );
404 return spdDetA( aat );
405 }
406 else
407 return ctype( 0 );
408 }
409
410 // A^{-1}_L = (A^T A)^{-1} A^T
411 // => A^{-1}_L A = I
412 template< int m, int n >
413 static ctype leftInvA ( const FieldMatrix< ctype, m, n > &A, FieldMatrix< ctype, n, m > &ret )
414 {
415 static_assert((m >= n), "Matrix has no left inverse.");
416 FieldMatrix< ctype, n, n > ata;
417 ATA_L( A, ata );
418 const ctype det = spdInvA( ata );
419 ATBT( ata, A, ret );
420 return det;
421 }
422
423 template< int m, int n >
424 static void leftInvAx ( const FieldMatrix< ctype, m, n > &A, const FieldVector< ctype, m > &x, FieldVector< ctype, n > &y )
425 {
426 static_assert((m >= n), "Matrix has no left inverse.");
427 FieldMatrix< ctype, n, n > ata;
428 ATx( A, x, y );
429 ATA_L( A, ata );
430 spdInvAx( ata, y );
431 }
432
434 template< int m, int n >
435 static ctype rightInvA ( const FieldMatrix< ctype, m, n > &A, FieldMatrix< ctype, n, m > &ret )
436 {
437 static_assert((n >= m), "Matrix has no right inverse.");
438 using std::abs;
439 if( (n == 2) && (m == 2) )
440 {
441 const ctype det = (A[ 0 ][ 0 ]*A[ 1 ][ 1 ] - A[ 1 ][ 0 ]*A[ 0 ][ 1 ]);
442 const ctype detInv = ctype( 1 ) / det;
443 ret[ 0 ][ 0 ] = A[ 1 ][ 1 ] * detInv;
444 ret[ 1 ][ 1 ] = A[ 0 ][ 0 ] * detInv;
445 ret[ 1 ][ 0 ] = -A[ 1 ][ 0 ] * detInv;
446 ret[ 0 ][ 1 ] = -A[ 0 ][ 1 ] * detInv;
447 return abs( det );
448 }
449 else
450 {
451 FieldMatrix< ctype, m , m > aat;
452 AAT_L( A, aat );
453 const ctype det = spdInvA( aat );
454 ATBT( A , aat , ret );
455 return det;
456 }
457 }
458
459 template< int m, int n >
460 static bool xTRightInvA ( const FieldMatrix< ctype, m, n > &A, const FieldVector< ctype, n > &x, FieldVector< ctype, m > &y )
461 {
462 static_assert((n >= m), "Matrix has no right inverse.");
463 FieldMatrix< ctype, m, m > aat;
464 Ax( A, x, y );
465 AAT_L( A, aat );
466 // check whether aat is singular and return true if non-singular
467 return spdInvAx( aat, y, true );
468 }
469 };
470
471 } // namespace Impl
472
473
474
480 template< class ct, int mydim, int cdim>
481 class AffineGeometry
482 {
483 public:
484
486 typedef ct ctype;
487
489 static const int mydimension= mydim;
490
492 static const int coorddimension = cdim;
493
495 typedef FieldVector< ctype, mydimension > LocalCoordinate;
496
498 typedef FieldVector< ctype, coorddimension > GlobalCoordinate;
499
501 typedef ctype Volume;
502
504 typedef FieldMatrix< ctype, mydimension, coorddimension > JacobianTransposed;
505
507 typedef FieldMatrix< ctype, coorddimension, mydimension > JacobianInverseTransposed;
508
509 private:
511 typedef Geo::ReferenceElement< Geo::ReferenceElementImplementation< ctype, mydimension > > ReferenceElement;
512
513 typedef Geo::ReferenceElements< ctype, mydimension > ReferenceElements;
514
515 // Helper class to compute a matrix pseudo inverse
516 typedef Impl::FieldMatrixHelper< ct > MatrixHelper;
517
518 public:
520 AffineGeometry ( const ReferenceElement &refElement, const GlobalCoordinate &origin,
521 const JacobianTransposed &jt )
522 : refElement_(refElement), origin_(origin), jacobianTransposed_(jt)
523 {
524 integrationElement_ = MatrixHelper::template rightInvA< mydimension, coorddimension >( jacobianTransposed_, jacobianInverseTransposed_ );
525 }
526
528 AffineGeometry ( Dune::GeometryType gt, const GlobalCoordinate &origin,
529 const JacobianTransposed &jt )
530 : AffineGeometry(ReferenceElements::general( gt ), origin, jt)
531 { }
532
534 template< class CoordVector >
535 AffineGeometry ( const ReferenceElement &refElement, const CoordVector &coordVector )
536 : refElement_(refElement), origin_(coordVector[0])
537 {
538 for( int i = 0; i < mydimension; ++i )
539 jacobianTransposed_[ i ] = coordVector[ i+1 ] - origin_;
540 integrationElement_ = MatrixHelper::template rightInvA< mydimension, coorddimension >( jacobianTransposed_, jacobianInverseTransposed_ );
541 }
542
544 template< class CoordVector >
545 AffineGeometry ( Dune::GeometryType gt, const CoordVector &coordVector )
546 : AffineGeometry(ReferenceElements::general( gt ), coordVector)
547 { }
548
550 bool affine () const { return true; }
551
553 Dune::GeometryType type () const { return refElement_.type(); }
554
556 int corners () const { return refElement_.size( mydimension ); }
557
559 GlobalCoordinate corner ( int i ) const
560 {
561 return global( refElement_.position( i, mydimension ) );
562 }
563
565 GlobalCoordinate center () const { return global( refElement_.position( 0, 0 ) ); }
566
573 GlobalCoordinate global ( const LocalCoordinate &local ) const
574 {
575 GlobalCoordinate global( origin_ );
576 jacobianTransposed_.umtv( local, global );
577 return global;
578 }
579
593 LocalCoordinate local ( const GlobalCoordinate &global ) const
594 {
595 LocalCoordinate local;
596 jacobianInverseTransposed_.mtv( global - origin_, local );
597 return local;
598 }
599
610 ctype integrationElement ([[maybe_unused]] const LocalCoordinate &local) const
611 {
612 return integrationElement_;
613 }
614
616 Volume volume () const
617 {
618 return integrationElement_ * refElement_.volume();
619 }
620
627 const JacobianTransposed &jacobianTransposed ([[maybe_unused]] const LocalCoordinate &local) const
628 {
629 return jacobianTransposed_;
630 }
631
638 const JacobianInverseTransposed &jacobianInverseTransposed ([[maybe_unused]] const LocalCoordinate &local) const
639 {
640 return jacobianInverseTransposed_;
641 }
642
643 friend ReferenceElement referenceElement ( const AffineGeometry &geometry )
644 {
645 return geometry.refElement_;
646 }
647
648 private:
649 ReferenceElement refElement_;
650 GlobalCoordinate origin_;
651 JacobianTransposed jacobianTransposed_;
652 JacobianInverseTransposed jacobianInverseTransposed_;
653 ctype integrationElement_;
654 };
655
656} // namespace Dune
657
658#endif // #ifndef DUNE_GEOMETRY_AFFINEGEOMETRY_HH
Dune::AffineGeometry
Implementation of the Geometry interface for affine geometries.
Definition: affinegeometry.hh:482
Dune::AffineGeometry::AffineGeometry
AffineGeometry(const ReferenceElement &refElement, const CoordVector &coordVector)
Create affine geometry from reference element and a vector of vertex coordinates.
Definition: affinegeometry.hh:535
Dune::AffineGeometry::AffineGeometry
AffineGeometry(Dune::GeometryType gt, const GlobalCoordinate &origin, const JacobianTransposed &jt)
Create affine geometry from GeometryType, one vertex, and the Jacobian matrix.
Definition: affinegeometry.hh:528
Dune::AffineGeometry::LocalCoordinate
FieldVector< ctype, mydimension > LocalCoordinate
Type for local coordinate vector.
Definition: affinegeometry.hh:495
Dune::AffineGeometry::type
Dune::GeometryType type() const
Obtain the type of the reference element.
Definition: affinegeometry.hh:553
Dune::AffineGeometry::mydimension
static const int mydimension
Dimension of the geometry.
Definition: affinegeometry.hh:489
Dune::AffineGeometry::AffineGeometry
AffineGeometry(const ReferenceElement &refElement, const GlobalCoordinate &origin, const JacobianTransposed &jt)
Create affine geometry from reference element, one vertex, and the Jacobian matrix.
Definition: affinegeometry.hh:520
Dune::AffineGeometry::Volume
ctype Volume
Type used for volume.
Definition: affinegeometry.hh:501
Dune::AffineGeometry::AffineGeometry
AffineGeometry(Dune::GeometryType gt, const CoordVector &coordVector)
Create affine geometry from GeometryType and a vector of vertex coordinates.
Definition: affinegeometry.hh:545
Dune::AffineGeometry::integrationElement
ctype integrationElement(const LocalCoordinate &local) const
Obtain the integration element.
Definition: affinegeometry.hh:610
Dune::AffineGeometry::jacobianInverseTransposed
const JacobianInverseTransposed & jacobianInverseTransposed(const LocalCoordinate &local) const
Obtain the transposed of the Jacobian's inverse.
Definition: affinegeometry.hh:638
Dune::AffineGeometry::JacobianTransposed
FieldMatrix< ctype, mydimension, coorddimension > JacobianTransposed
Type for the transposed Jacobian matrix.
Definition: affinegeometry.hh:504
Dune::AffineGeometry::corner
GlobalCoordinate corner(int i) const
Obtain coordinates of the i-th corner.
Definition: affinegeometry.hh:559
Dune::AffineGeometry::corners
int corners() const
Obtain number of corners of the corresponding reference element.
Definition: affinegeometry.hh:556
Dune::AffineGeometry::JacobianInverseTransposed
FieldMatrix< ctype, coorddimension, mydimension > JacobianInverseTransposed
Type for the transposed inverse Jacobian matrix.
Definition: affinegeometry.hh:507
Dune::AffineGeometry::coorddimension
static const int coorddimension
Dimension of the world space.
Definition: affinegeometry.hh:492
Dune::AffineGeometry::global
GlobalCoordinate global(const LocalCoordinate &local) const
Evaluate the mapping.
Definition: affinegeometry.hh:573
Dune::AffineGeometry::center
GlobalCoordinate center() const
Obtain the centroid of the mapping's image.
Definition: affinegeometry.hh:565
Dune::AffineGeometry::ctype
ct ctype
Type used for coordinates.
Definition: affinegeometry.hh:486
Dune::AffineGeometry::GlobalCoordinate
FieldVector< ctype, coorddimension > GlobalCoordinate
Type for coordinate vector in world space.
Definition: affinegeometry.hh:498
Dune::AffineGeometry::affine
bool affine() const
Always true: this is an affine geometry.
Definition: affinegeometry.hh:550
Dune::AffineGeometry::jacobianTransposed
const JacobianTransposed & jacobianTransposed(const LocalCoordinate &local) const
Obtain the transposed of the Jacobian.
Definition: affinegeometry.hh:627
Dune::AffineGeometry::volume
Volume volume() const
Obtain the volume of the element.
Definition: affinegeometry.hh:616
Dune::DenseMatrix::mtv
void mtv(const X &x, Y &y) const
y = A^T x
Definition: densematrix.hh:415
Dune::DenseMatrix::umtv
void umtv(const X &x, Y &y) const
y += A^T x
Definition: densematrix.hh:447
Dune::FieldVector
vector space out of a tensor product of fields.
Definition: fvector.hh:95
Dune::Geo::ReferenceElement
This class provides access to geometric and topological properties of a reference element.
Definition: referenceelement.hh:50
Dune::Geo::ReferenceElement::volume
CoordinateField volume() const
obtain the volume of the reference element
Definition: referenceelement.hh:239
Dune::Geo::ReferenceElement::type
decltype(auto) type(int i, int c) const
obtain the type of subentity (i,c)
Definition: referenceelement.hh:169
Dune::Geo::ReferenceElement::size
int size(int c) const
number of subentities of codimension c
Definition: referenceelement.hh:92
Dune::Geo::ReferenceElement::position
decltype(auto) position(int i, int c) const
position of the barycenter of entity (i,c)
Definition: referenceelement.hh:201
Dune::GeometryType
Unique label for each type of entities that can occur in DUNE grids.
Definition: type.hh:123
fmatrix.hh
Implements a matrix constructed from a given type representing a field and compile-time given number ...
fvector.hh
Implements a vector constructed from a given type representing a field and a compile-time given size.
Dune::FloatCmp::gt
bool gt(const T &first, const T &second, typename EpsilonType< T >::Type epsilon)
test if first greater than second
Definition: float_cmp.cc:156
Dune::ReferenceElement
unspecified-type ReferenceElement
Returns the type of reference element for the argument type T.
Definition: referenceelements.hh:495
Dune
Dune namespace.
Definition: alignedallocator.hh:11
Dune::Geo::ReferenceElements
Class providing access to the singletons of the reference elements.
Definition: referenceelements.hh:168
type.hh
A unique label for each type of element that can occur in a grid.
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