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gsGenericTensorBasis< d, T > Class Template Referenceabstract

Detailed Description

template<short_t d, class T>
class gismo::gsGenericTensorBasis< d, T >

Class for a quasi-tensor B-spline basis.

Parameters
Tcoefficient type
ddimension of the parameter domain
+ Inheritance diagram for gsGenericTensorBasis< d, T >:
+ Collaboration diagram for gsGenericTensorBasis< d, T >:

Public Types

typedef gsTensorBasis< d, T > Base
 Base type.
 
typedef memory::shared_ptr
< gsGenericTensorBasis
Ptr
 Shared pointer for gsGenericTensorBasis.
 
typedef T Scalar_t
 Coefficient type.
 
typedef memory::unique_ptr
< gsGenericTensorBasis
uPtr
 Unique pointer for gsGenericTensorBasis.
 

Public Member Functions

gsMatrix< index_tactive (const gsMatrix< T > &u) const
 Returns the indices of active (nonzero) functions at points u, as a list of indices. More...
 
void active_cwise (const gsMatrix< T > &u, gsVector< index_t, d > &low, gsVector< index_t, d > &upp) const
 
virtual void active_into (const gsMatrix< T > &u, gsMatrix< index_t > &result) const
 Returns the indices of active basis functions at points u, as a list of indices, in result. A function is said to be active in a point if this point lies in the closure of the function's support. More...
 
gsMatrix< index_tallBoundary () const
 
void anchor_into (index_t i, gsMatrix< T > &result) const
 Returns the anchors (graville absissae) that represent the members of the basis.
 
gsMatrix< T > anchors () const
 Returns the anchor points that represent the members of the basis. There is exactly one anchor point per basis function. More...
 
void anchors_into (gsMatrix< T > &result) const
 Returns the anchors (graville absissae) that represent the members of the basis.
 
const gsBasis< T > & basis (const index_t k) const
 Helper which casts and returns the k-th piece of this function set as a gsBasis.
 
const_iterator begin () const
 
iterator begin ()
 
gsMatrix< index_tboundary (boxSide const &s) const
 Returns the indices of the basis functions that are nonzero at the domain boundary as single-column-matrix.
 
gsBasis< T >::uPtr boundaryBasis (boxSide const &s)
 Returns the boundary basis for side s.
 
gsMatrix< index_tboundaryOffset (boxSide const &s, index_t offset) const
 
uPtr clone ()
 Clone methode. Produceds a deep copy inside a uPtr.
 
gsMatrix< index_tcoefSlice (short_t dir, index_t k) const
 Returns all the basis functions with tensor-numbering. More...
 
gsSparseMatrix< T > collocationMatrix (gsMatrix< T > const &u) const
 Computes the collocation matrix w.r.t. points u. More...
 
Basis_tcomponent (short_t dir)
 For a tensor product basis, return the 1-d basis for the i-th parameter component.
 
const Basis_tcomponent (short_t dir) const
 For a tensor product basis, return the (const) 1-d basis for the i-th parameter component.
 
virtual uPtr componentBasis (boxComponent b) const
 Returns the basis that corresponds to the component.
 
virtual uPtr componentBasis_withIndices (boxComponent b, gsMatrix< index_t > &indices, bool noBoundary=true) const
 Returns the basis that corresponds to the component. More...
 
virtual void compute (const gsMatrix< T > &in, gsFuncData< T > &out) const
 Computes function data. More...
 
virtual void connectivity (const gsMatrix< T > &nodes, gsMesh< T > &mesh) const
 
virtual void connectivityAtAnchors (gsMesh< T > &mesh) const
 
virtual gsBasis::uPtr create () const
 Create an empty basis of the derived type and return a pointer to it.
 
short_t degree (short_t i) const
 Returns the degree of the basis wrt variable i.
 
virtual void degreeDecrease (short_t const &i=1, short_t const dir=-1)
 Lower the degree of the basis by the given amount, preserving knots multiplicity.
 
virtual void degreeElevate (short_t const &i=1, short_t const dir=-1)
 Elevate the degree of the basis by the given amount, preserve smoothness.
 
virtual void degreeIncrease (short_t const &i=1, short_t const dir=-1)
 Elevate the degree of the basis by the given amount, preserve knots multiplicity.
 
virtual void degreeReduce (short_t const &i=1, short_t const dir=-1)
 Reduce the degree of the basis by the given amount, preserve smoothness.
 
gsMatrix< T > deriv (const gsMatrix< T > &u) const
 Evaluate the derivatives,. More...
 
gsMatrix< T > deriv2 (const gsMatrix< T > &u) const
 Evaluates the second derivatives of active (i.e., non-zero) functions at points u. More...
 
virtual void deriv2_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
 Evaluate the second derivatives of all active basis function at points u. More...
 
virtual void deriv2Single_into (index_t i, const gsMatrix< T > &u, gsMatrix< T > &result) const
 Evaluate the (partial) derivatives of the i-th basis function at points u into result.
 
virtual void deriv_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
 Evaluates the first partial derivatives of the nonzero basis function. More...
 
virtual void derivSingle_into (index_t i, const gsMatrix< T > &u, gsMatrix< T > &result) const
 Evaluates the (partial) derivatives of the i-th basis function at points u into result. More...
 
virtual std::string detail () const
 Prints the object as a string with extended details.
 
virtual gsDomain< T > * domain () const
 
short_t domainDim () const
 Dimension of the (source) domain. More...
 
size_t elementIndex (const gsVector< T > &u) const
 Returns an index for the element which contains point u.
 
gsMatrix< T > elementInSupportOf (index_t j) const
 Returns (the coordinates of) an element in the support of basis function j.
 
virtual void elevateContinuity (int const &i=1)
 Elevates the continuity of the basis along element boundaries.
 
const_iterator end () const
 
iterator end ()
 
gsMatrix< T > eval (const gsMatrix< T > &u) const
 Evaluate the function,. More...
 
virtual void eval_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
 Evaluates nonzero basis functions at point u into result. More...
 
virtual void eval_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
 Evaluate an element of the space given by coefs at points u.
 
std::vector< gsMatrix< T > > evalAllDers (const gsMatrix< T > &u, int n, bool sameElement=false) const
 Evaluate all derivatives upto order n,. More...
 
virtual void evalAllDers_into (const gsMatrix< T > &u, int n, std::vector< gsMatrix< T > > &result, bool sameElement=false) const
 Evaluate the nonzero functions and their derivatives up to order n at points u into result. More...
 
void evalSingle_into (index_t i, const gsMatrix< T > &u, gsMatrix< T > &result) const
 Evaluate the i-th basis function at points u into result.
 
gsBasisFun< T > function (index_t i) const
 Returns the i-th basis function as a gsFunction. More...
 
void getComponentsForSide (boxSide const &s, std::vector< Basis_t * > &rr) const
 Returns the components for a basis on the face s.
 
virtual T getMaxCellLength () const
 Get the maximum mesh size, as expected for approximation error estimates.
 
virtual T getMinCellLength () const
 Get the minimum mesh size, as expected for inverse inequalities.
 
unsigned index (unsigned i, unsigned j, unsigned k=0) const
 
index_t index (gsVector< index_t, d > const &v) const
 
bool indexOnBoundary (const gsVector< index_t, d > &ind) const
 Returns true iff the basis function with multi-index ind is on the boundary.
 
bool indexOnBoundary (const index_t m) const
 Returns true iff the basis function indexed m is on the boundary.
 
virtual gsGeometry< T >::uPtr interpolateAtAnchors (gsMatrix< T > const &vals) const
 Applies interpolation of values pts using the anchors as parameter points. May be reimplemented in derived classes with more efficient algorithms. (by default uses interpolateData(pts,vals)
 
memory::unique_ptr< gsGeometry
< T > > 
interpolateData (gsMatrix< T > const &vals, gsMatrix< T > const &pts) const
 Applies interpolation given the parameter values pts and values vals.
 
gsGeometry< T >::uPtr interpolateGrid (gsMatrix< T > const &vals, std::vector< gsMatrix< T > >const &grid) const
 
bool isActive (const index_t i, const gsVector< T > &u) const
 Returns true if there the point u with non-zero value or derivatives when evaluated at the basis function i.
 
virtual bool isRational () const
 Returns false, since all bases that inherit from gsBasis are not rational.
 
gsBasis< T >::domainIter makeDomainIterator () const
 Create a domain iterator for the computational mesh of this basis, that points to the first element of the domain.
 
gsBasis< T >::domainIter makeDomainIterator (const boxSide &s) const
 Create a boundary domain iterator for the computational mesh this basis, that points to the first element on the boundary of the domain.
 
virtual memory::unique_ptr
< gsGeometry< T > > 
makeGeometry (gsMatrix< T > coefs) const =0
 Create a gsGeometry of proper type for this basis with the given coefficient matrix.
 
virtual memory::unique_ptr
< gsBasis< T > > 
makeNonRational () const
 
void matchWith (const boundaryInterface &bi, const gsBasis< T > &other, gsMatrix< index_t > &bndThis, gsMatrix< index_t > &bndOther, index_t offset) const
 Computes the indices of DoFs that match on the interface bi. The interface is assumed to be a common face between this patch and other, with an offset offset. The output is two lists of indices bndThis and bndOther, with indices that match one-to-one on the boundary bi. More...
 
short_t maxDegree () const
 If the basis is of polynomial or piecewise polynomial type, then this function returns the maximum polynomial degree.
 
short_t minDegree () const
 If the basis is of polynomial or piecewise polynomial type, then this function returns the minimum polynomial degree.
 
virtual index_t nPieces () const
 Number of pieces in the domain of definition.
 
size_t numElements (boxSide const &s=boundary::none) const
 The number of elements on side s.
 
void numElements_cwise (gsVector< unsigned > &result) const
 Returns the number of elements (component wise)
 
const gsBasis< T > & piece (const index_t k) const
 Returns the piece(s) of the function(s) at subdomain k.
 
GISMO_MAKE_GEOMETRY_NEW
std::ostream & 
print (std::ostream &os) const
 Prints the object as a string, pure virtual function of gsTensorBasis.
 
virtual void reduceContinuity (int const &i=1)
 Reduces the continuity of the basis along element boundaries.
 
virtual void refine (gsMatrix< T > const &boxes, int refExt=0)
 Refine the basis on the area defined by the matrix boxes. More...
 
void refineElements (std::vector< index_t > const &elements)
 Refine elements defined by their tensor-index. More...
 
virtual void refineElements_withCoefs (gsMatrix< T > &coefs, std::vector< index_t > const &boxes)
 Refine basis and geometry coefficients to levels. More...
 
virtual void reverse ()
 Reverse the basis.
 
void setDegree (short_t const &i)
 Set the degree of the basis (either elevate or reduce) in order to have degree equal to i wrt to each variable.
 
void setDegreePreservingMultiplicity (short_t const &i)
 Set the degree of the basis (either increase or decrecee) in order to have degree equal to i.
 
index_t size () const
 Returns the number of elements in the basis.
 
index_t size (short_t k) const
 The number of basis functions in the direction of the k-th parameter component.
 
template<int s>
void size_cwise (gsVector< index_t, s > &result) const
 The number of basis functions in the direction of the k-th parameter component.
 
virtual const gsBasissource () const
 
virtual gsBasissource ()
 
unsigned stride (short_t dir) const
 Returns the stride for dimension dir.
 
void stride_cwise (gsVector< index_t, d > &result) const
 Returns the strides for all dimensions.
 
gsMatrix< T > support () const
 Returns (a bounding box for) the domain of the whole basis. More...
 
gsMatrix< T > support (const index_t &i) const
 Returns (a bounding box for) the support of the i-th basis function. More...
 
gsMatrix< T > supportInterval (index_t dir) const
 Returns an interval that contains the parameter values in direction dir. More...
 
virtual short_t targetDim () const
 Dimension of the target space. More...
 
gsVector< index_t, d > tensorIndex (const index_t &m) const
 Returns the tensor index of the basis function with global index m.
 
virtual gsBasis::uPtr tensorize (const gsBasis &other) const
 Return a tensor basis of this and other.
 
short_t totalDegree () const
 If the basis is of polynomial or piecewise polynomial type, then this function returns the total polynomial degree.
 
virtual void uniformCoarsen (int numKnots=1)
 Coarsen the basis uniformly by removing groups of numKnots consecutive knots, each knot removed mul times. More...
 
virtual void uniformCoarsen_withCoefs (gsMatrix< T > &coefs, int numKnots=1)
 Coarsen the basis uniformly. More...
 
void uniformCoarsen_withTransfer (gsSparseMatrix< T, RowMajor > &transfer, int numKnots=1)
 Coarsen the basis uniformly and produce a sparse matrix which maps coarse coefficient vectors to refined ones. More...
 
virtual void uniformRefine (int numKnots=1, int mul=1, int dir=-1)
 Refine the basis uniformly by inserting numKnots new knots with multiplicity mul on each knot span.
 
void uniformRefine_withCoefs (gsMatrix< T > &coefs, int numKnots=1, int mul=1, int dir=-1)
 
void uniformRefine_withTransfer (gsSparseMatrix< T, RowMajor > &transfer, int numKnots=1, int mul=1)
 
virtual const gsMatrix< T > & weights () const
 Only for compatibility reasons, with gsRationalBasis. It returns an empty matrix.
 
virtual gsMatrix< T > & weights ()
 Only for compatibility reasons, with gsRationalBasis. It returns an empty matrix.
 
Evaluation functions
gsMatrix< T > evalSingle (index_t i, const gsMatrix< T > &u) const
 Evaluate a single basis function i at points u.
 
gsMatrix< T > derivSingle (index_t i, const gsMatrix< T > &u) const
 Evaluate a single basis function i derivative at points u.
 
gsMatrix< T > deriv2Single (index_t i, const gsMatrix< T > &u) const
 Evaluate the second derivative of a single basis function i at points u.
 
gsVector< index_tnumActive (const gsMatrix< T > &u) const
 Number of active basis functions at an arbitrary parameter value. More...
 
virtual void numActive_into (const gsMatrix< T > &u, gsVector< index_t > &result) const
 Returns the number of active (nonzero) basis functions at points u in result.
 
virtual void activeCoefs_into (const gsVector< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
 Returns the matrix result of active coefficients at points u, each row being one coefficient. The order of the rows is the same as active_into and eval_into functions. More...
 
virtual void evalAllDersSingle_into (index_t i, const gsMatrix< T > &u, int n, gsMatrix< T > &result) const
 Evaluate the basis function i and its derivatives up to order n at points u into result.
 
virtual void evalDerSingle_into (index_t i, const gsMatrix< T > &u, int n, gsMatrix< T > &result) const
 Evaluate the (partial) derivative(s) of order n the i-th basis function at points u into result.
 
virtual gsMatrix< T > laplacian (const gsMatrix< T > &u) const
 Compute the Laplacian of all nonzero basis functions at points u.
 

Static Public Attributes

static const short_t Dim
 Dimension of the parameter domain.
 

Geometry evaluation functions

These functions evaluate not the individual basis functions of the basis, but a geometry object which is represented by a coefficient matrix w.r.t. this basis object. For the format of the coefficient matrix, see gsGeometry.

These functions have default implementations which simply compute the basis function values and perform linear combination, but they may be overridden in derived classes if a higher-performance implementation is possible.

gsMatrix< T > evalFunc (const gsMatrix< T > &u, const gsMatrix< T > &coefs) const
 Evaluate the function described by coefs at points u. More...
 
virtual void evalFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
 Evaluate the function described by coefs at points u, i.e., evaluates a linear combination of coefs x BasisFunctions, into result. More...
 
gsMatrix< T > derivFunc (const gsMatrix< T > &u, const gsMatrix< T > &coefs) const
 Evaluate the derivatives of the function described by coefs at points u. More...
 
virtual void derivFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
 Evaluate the derivatives of the function described by coefs at points u. More...
 
virtual void jacobianFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
 Evaluate the Jacobian of the function described by coefs at points u. Jacobian matrices are stacked in blocks.
 
gsMatrix< T > deriv2Func (const gsMatrix< T > &u, const gsMatrix< T > &coefs) const
 Evaluates the second derivatives of the function described by coefs at points u. More...
 
virtual void deriv2Func_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
 Evaluates the second derivatives of the function described by coefs at points u. More...
 
virtual void evalAllDersFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, const unsigned n, std::vector< gsMatrix< T > > &result, bool sameElement=false) const
 Evaluates all derivatives up to order n of the function described by coefs at points u. More...
 
static void linearCombination_into (const gsMatrix< T > &coefs, const gsMatrix< index_t > &actives, const gsMatrix< T > &values, gsMatrix< T > &result, bool sameElement=false)
 Computes the linear combination coefs * values( actives ) More...
 

Member Function Documentation

gsMatrix<index_t> active ( const gsMatrix< T > &  u) const
inlineinherited

Returns the indices of active (nonzero) functions at points u, as a list of indices.

See Also
active_into()
void active_cwise ( const gsMatrix< T > &  u,
gsVector< index_t, d > &  low,
gsVector< index_t, d > &  upp 
) const
inherited

Returns a box with the coordinate-wise active functions

Parameters
uevaluation points
lowlower left corner of the box
uppupper right corner of the box
void active_into ( const gsMatrix< T > &  u,
gsMatrix< index_t > &  result 
) const
virtualinherited

Returns the indices of active basis functions at points u, as a list of indices, in result. A function is said to be active in a point if this point lies in the closure of the function's support.

Tensor indexing in result
Assume that the parameter domain is three dimensional. Let n1, n2, and n3 denote the number of univariate basis functions in the first, second and third coordinate direction, respectively.
Let the trivariate tensor product basis function B_I be defined by
B_I(x,y,z) = B_i(x) * B_j(y) * B_k(z).
Then, the index I, which is returned in result, is computed as
I = i + j * n1 + k * n1*n2.
Examples:
I <-> (i,j,k)
0 <-> (0,0,0)
1 <-> (1,0,0)
2 <-> (2,0,0)
...
(n1-1) <-> (n1-1,0,0)
n1 <-> (0,1,0)
n1+1 <-> (1,1,0)
n1+2 <-> (2,1,0)
...
n1*n2-1 <-> (n1,n2,0)
n1*n2 <-> (0,0,1)
n1*n2+1 <-> (1,0,1)
...
n1*n2*n3-1 <-> (n1,n2,n3)
Parameters
[in]ugsMatrix containing evaluation points. Each column represents one evaluation point.
[out]resultFor every column i of u, a column containing the active basis functions at evaluation point u.col(i)

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >, gsTensorBSplineBasis< d, T >, and gsTensorBSplineBasis< domainDim+1, T >.

void activeCoefs_into ( const gsVector< T > &  u,
const gsMatrix< T > &  coefs,
gsMatrix< T > &  result 
) const
virtualinherited

Returns the matrix result of active coefficients at points u, each row being one coefficient. The order of the rows is the same as active_into and eval_into functions.

Parameters
[in]ugsVector containing an evaluation point.
[in]coefsgsMatrix is a coefficient matrix with as many rows as the size of the basis
[out]resultFor every column i of u, a column containing the indices of the active basis functions at evaluation point u.col(i).
gsMatrix< index_t > allBoundary ( ) const
virtualinherited

Returns the indices of the basis functions that touch the domain boundary

Reimplemented from gsBasis< T >.

gsMatrix<T> anchors ( ) const
inlineinherited

Returns the anchor points that represent the members of the basis. There is exactly one anchor point per basis function.

The exact definition of the anchor points depends on the particular basis. For instance, for a Bspline basis these are the Greville abscissae. In general, evaluating a function at the anchor points should provide enough information to interpolate that function using this basis.

const_iterator begin ( ) const
inlineinherited

Get a const-iterator to the beginning of the bases vector

Returns
an iterator to the beginning of the bases vector
iterator begin ( )
inlineinherited

Get an iterator to the beginning of the bases vector

Returns
an iterator to the beginning of the bases vector
gsMatrix< index_t > boundaryOffset ( boxSide const &  s,
index_t  offset 
) const
virtualinherited

Returns the indices of the basis functions that touch the domain boundary

Reimplemented from gsBasis< T >.

gsMatrix< index_t > coefSlice ( short_t  dir,
index_t  k 
) const
inherited

Returns all the basis functions with tensor-numbering.

Parameters
kin direction
dir

Detailed explanation:

Tensor-numbering in N-variate tensor-product basis means that each basis function is assigned an identifier (i_0, i_1, ..., i_{N-1}). This function returns indices of basis functions with i_dir = k and the returned indices are numbering of the basis functions in the basis (i.e., 0,1, ..., basis.size() ).

Example:

Bivariate tensor-product basis functions have tensor numbering (a,b). Calling dir=0, k=1 gives all functions with tensor-numbering (1,b). Calling dir=1, k=3 gives all functions with tensor-numbering (a,3).

gsSparseMatrix< T > collocationMatrix ( gsMatrix< T > const &  u) const
inlineinherited

Computes the collocation matrix w.r.t. points u.

The collocation matrix is a sparse matrix with u.cols rows and size() columns. The entry (i,j) is the value of basis function j at evaluation point i.

gsBasis< T >::uPtr componentBasis_withIndices ( boxComponent  b,
gsMatrix< index_t > &  indices,
bool  noBoundary = true 
) const
virtualinherited

Returns the basis that corresponds to the component.

Parameters
bThe component
indicesThe row vector where the indices are stored to
noBoundaryIf true, the transfer matrix does not include parts belonging to lower-order components (i.e., edges without corners or faces without corners and edges)
void compute ( const gsMatrix< T > &  in,
gsFuncData< T > &  out 
) const
virtualinherited

Computes function data.

This function evaluates the functions and their derivatives at the points in and writes them in the corresponding fields of out. Which field to write (and what to compute) is controlled by the out.flags (see also gsFuncData).

The input points in are expected to be compatible with the implementation/representation of the function, i.e. they should be points inside the domain of definitition of the function

Parameters
[in]in
[out]out

Reimplemented in gsGeometry< T >, and gsConstantFunction< T >.

void connectivity ( const gsMatrix< T > &  nodes,
gsMesh< T > &  mesh 
) const
virtualinherited

Returns the connectivity structure of the basis The returned mesh has vertices the rows of matrix nodes

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

void connectivityAtAnchors ( gsMesh< T > &  mesh) const
inlinevirtualinherited

Returns the connectivity structure of the basis The returned mesh has the anchor points as vertices

gsMatrix< T > deriv ( const gsMatrix< T > &  u) const
inherited

Evaluate the derivatives,.

See Also
deriv_into()
gsMatrix< T > deriv2 ( const gsMatrix< T > &  u) const
inherited

Evaluates the second derivatives of active (i.e., non-zero) functions at points u.

See documentation for deriv2_into() (the one without input parameter coefs) for details.

See Also
deriv2_into()
Parameters
[in]uEvaluation points in columns.
Returns
For every column of u, a column containing the second derivatives. See documentation for deriv2_into() (the one without input parameter coefs) for details.
void deriv2_into ( const gsMatrix< T > &  u,
gsMatrix< T > &  result 
) const
virtualinherited

Evaluate the second derivatives of all active basis function at points u.

Input parameter u is a gsMatrix of size d x N, where
d is the dimension of the parameter domain and
N is the number of evaluation points.
Each column of u corresponds to the coordinates of one evaluation point.

result is a gsMatrix of size (K * d) x N, where
K is the number of active basis functions at the evaluation point.
Each column of result corresponds to a column of u. It contains the "pure" and the mixed derivatives for each active basis function, "above" each other.

Example (bivariate): Let \(B_i(x,y)\), d = 2 be bivariate basis functions, and let the functions with indices 3,4,7, and 8 (K = 4) be active at an evaluation point u. Then, the corresponding column of result represents:
\( ( \partial_{xx}\, B_3(u), \partial_{yy}\, B_3(u), \partial_{xy}\, B_3(u), \partial_{xx}\, B_4(u), \partial_{yy}\, B_4(u), \partial_{xy}\, B_4(u), \partial_{xx}\, B_7(u), ... , \partial_{xy}\, B_8(u) )^T \)

Example (trivariate): Let \(B_i(x,y,z)\), d = 3 be trivariate basis functions, and let the functions with indices 3,4,7, and 8 be active at an evaluation point u. Then, the corresponding column of result represents:
\(( \partial_{xx}\, B_3(u), \partial_{yy}\, B_3(u), \partial_{zz}\, B_3(u), \partial_{xy}\, B_3(u), \partial_{xz}\, B_3(u), \partial_{yz}\, B_3(u), \partial_{xx}\, B_4(u), ... , \partial_{yz}\, B_8(u) )^T \)

Parameters
[in]uEvaluation points in columns (see above for format).
[in,out]resultFor every column of u, a column containing the second derivatives as described above.

See also deriv2() (the one without input parameter coefs).

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

gsMatrix<T> deriv2Func ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs 
) const
inlineinherited

Evaluates the second derivatives of the function described by coefs at points u.

See documentation for deriv2_into() (the one with input parameter coefs) for details.

Parameters
[in]uEvaluation points in columns.
[in]coefsCoefficient matrix describing the geometry in this basis.
Returns
For every column of u, a column containing the second derivatives. See documentation for deriv2_into() (the one with input parameter coefs) for details.
void deriv2Func_into ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs,
gsMatrix< T > &  result 
) const
virtualinherited

Evaluates the second derivatives of the function described by coefs at points u.

...i.e., evaluates a linear combination of coefs * (2nd derivatives of basis functions), into result.

Evaluation points u are given as gsMatrix of size d x N, where
d is the dimension of the parameter domain and
N is the number of evaluation points.
Each column of u corresponds to the coordinates of one evaluation point.

The coefficients coefs are given as gsMatrix of size N x n, where
N is the number of points = number of basis functions and
n is the dimension of the physical domain.
Each row of coefs corresponds to the coordinates of one control point.

Let the function \( f: \mathbb R^3 \to \mathbb R^3\) be given by

\[ f = ( f_1, f_2, f_3)^T = \sum_{i=1}^N c_i B_i(x,y,z), \]

where \( B_i(x,y,z)\) are scalar basis functions and \(c_i\) are the corresponding (m-dimensional) control points. Then, for each column in u, the corresponding column in result represents

\[ ( \partial_{xx}\ f_1, \partial_{yy}\ f_1, \partial_{zz}\ f_1, \partial_{xy}\ f_1, \partial_{xz}\ f_1, \partial_{yz}\ f_1, \partial_{xx}\ f_2, \partial_{yy}\ f_2, \ldots , \partial_{xz}\ f_3, \partial_{yz}\ f_3)^T. \]

at the respective evaluation point.

Parameters
[in]uEvaluation points in columns (see above for format).
[in]coefsCoefficient matrix describing the geometry in this basis.
[out]resultFor every column of u, a column containing the second derivatives at the respective point in the format described above.

This function has a default implementation that may be overridden in derived classes for higher performance.
See also deriv2() (the one with input parameter coefs).

void deriv_into ( const gsMatrix< T > &  u,
gsMatrix< T > &  result 
) const
virtualinherited

Evaluates the first partial derivatives of the nonzero basis function.

Let
d denote the dimension of the parameter domain.
K denote the number of active (i.e., non-zero) basis functions (see active_into()).
N denote the number of evaluation points.
The N evaluation points u are given in a gsMatrix of size d x N. Each column of u represents one evaluation point.

The gsMatrix result contains the computed derivatives in the following form:
Column j of result corresponds to one evaluation point (specified by the j-th column of u). The column contains the gradients of all active functions "above" each other.
For example, for scalar basis functions \(B_i : (x,y,z)-> R\), a column represents
\((dx B_1, dy B_1, dz B_1, dx B_2, dy B_2, dz B_2, ... , dx B_n, dy B_N, dz B_N)^T\),
where the order the basis functions \(B_i\) is as returned by active() and active_into().

Parameters
[in]uEvaluation points given as gsMatrix of size d x N. See above for details.
[in,out]resultgsMatrix of size (K*d) x N. See above for details.
Todo:
Rename to _ grad_into

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

gsMatrix<T> derivFunc ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs 
) const
inlineinherited

Evaluate the derivatives of the function described by coefs at points u.

Parameters
uevaluation points as N column vectors
coefscoefficient matrix describing the geometry in this basis, n columns
Returns
For every column of u, the result matrix will contain one Jacobian matrix of size d * n, such that the total size of the result is n x (d * n) x N
void derivFunc_into ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs,
gsMatrix< T > &  result 
) const
virtualinherited

Evaluate the derivatives of the function described by coefs at points u.

Evaluates a linear combination of coefs*BasisFunctionDerivatives, into result.

This function has a default implementation that may be overridden in derived classes for higher performance.

Let the function \(f: \mathbb R^d \to \mathbb R^m \) be described by the coefficients coefs, i.e.,
each evaluation point is in \(\mathbb R^d\), and
each coefficient is a point in \(\mathbb R^m\).

The N evaluation points u are given in a gsMatrix of size d x N. Each column of u represents one evaluation point.

The K coefficients coefs are given as a gsMatrix of size K x m. Each row of coefs represents one coefficient in \(\mathbb R^m\).

The gsMatrix result contains the following data:
For every column of u, the corresponding column in the matrix result contains the gradients of the m components of the function above each other. Hence, the size of result is (d*m) x N.

Example 1:
Let \(f(s,t)\) be a bivariate scalar function, \(f:\mathbb R^2 \to \mathbb R\) (i.e., d=2, m=1), and let the evaluation point \( u_i\) be represented by the i-th column of u.
Then, result has the form

\[ \left( \begin{array}{cccc} \partial_s f(u_1) & \partial_s f(u_2) & \ldots & \partial_t f(u_{N}) \\ \partial_t f(u_1) & \partial_t f(u_2) & \ldots & \partial_t f(u_{N}) \end{array} \right) \]

Example 2:
Let \(f(s,t) = ( f_1(s,t), f_2(s,t), f_3(s,t) )\) represent a surface in space, \(f:\mathbb R^2 \to \mathbb R^3\) (i.e., d=2, m=3), and let the evaluation point \( u_i\) be represented by the i-th column of u.
Then, result has the form

\[ \left( \begin{array}{ccccccc} \partial_s f_1(u_1) & \partial_s f_1(u_2) & \ldots & \partial_s f_1(u_N) \\ \partial_t f_1(u_1) & \partial_t f_1(u_2) & \ldots & \partial_t f_1(u_N) \\ \partial_s f_2(u_1) & \partial_s f_2(u_2) & \ldots & \partial_s f_2(u_N) \\ \partial_t f_2(u_1) & \partial_t f_2(u_2) & \ldots & \partial_t f_2(u_N) \\ \partial_s f_3(u_1) & \partial_s f_3(u_2) & \ldots & \partial_s f_3(u_N) \\ \partial_t f_3(u_1) & \partial_t f_3(u_2) & \ldots & \partial_t f_3(u_N) \\ \end{array} \right) \]

Parameters
[in]uEvaluation points as d x N-matrix.
[in]coefsCoefficient matrix describing the geometry in this basis as K x m-matrix.
K should equal the size() of the basis, i.e., the number basis functions.
[in,out]resultgsMatrix of size d*m x N, see above for format.

where
d is the dimension of the parameter domain
m is the dimension of the physical domain
N is the number of evaluation points
K is the number of coefficients

void derivSingle_into ( index_t  i,
const gsMatrix< T > &  u,
gsMatrix< T > &  result 
) const
virtualinherited

Evaluates the (partial) derivatives of the i-th basis function at points u into result.

See deriv_into() for detailed documentation.

Todo:
rename grad_into

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

gsDomain< T > * domain ( ) const
virtualinherited
short_t domainDim ( ) const
inlinevirtualinherited

Dimension of the (source) domain.

Returns
For \(f:\mathbb{R}^n\rightarrow\mathbb{R}^m\), returns \(n\).

Implements gsFunctionSet< T >.

const_iterator end ( ) const
inlineinherited

Get a const-iterator to the end of the bases vector

Returns
an iterator to the end of the bases vector
iterator end ( )
inlineinherited

Get an iterator to the end of the bases vector

Returns
an iterator to the end of the bases vector
gsMatrix< T > eval ( const gsMatrix< T > &  u) const
inherited

Evaluate the function,.

See Also
eval_into()
void eval_into ( const gsMatrix< T > &  u,
gsMatrix< T > &  result 
) const
virtualinherited

Evaluates nonzero basis functions at point u into result.

Let...
d denote the dimension of the parameter domain.
K denote the number of active (i.e., non-zero) basis functions (see active_into()). N denote the number of evaluation points.
The n evaluation points u are given in a gsMatrix of size d x N. Each column of u represents one evaluation point.

The gsMatrix result contains the computed function values in the following form:
Column j of result corresponds to one evaluation point (specified by the j-th column of u). The column contains the values of all active functions "above" each other.
For example, for scalar basis functions Bi : (x,y,z)-> R, a column represents
(B1, B2, ... , BN)^T,
where the order the basis functions Bi is as returned by active() and active_into().

Parameters
[in]uEvaluation points given as gsMatrix of size d x N. See above for details.
[in,out]resultgsMatrix of size K x N. See above for details.
Todo:
more efficient ?

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

std::vector< gsMatrix< T > > evalAllDers ( const gsMatrix< T > &  u,
int  n,
bool  sameElement = false 
) const
inherited

Evaluate all derivatives upto order n,.

See Also
evalAllDers_into
void evalAllDers_into ( const gsMatrix< T > &  u,
int  n,
std::vector< gsMatrix< T > > &  result,
bool  sameElement = false 
) const
virtualinherited

Evaluate the nonzero functions and their derivatives up to order n at points u into result.

The derivatives (the 0-th derivative is the function value) are stored in a result. result is a std::vector, where result[i] is a gsMatrix which contains the i-th derivatives.

The entries in result[0], result[1], and result[2] are ordered as in eval_into(), deriv_into(), and deriv2_into(), respectively. For i > 2, the derivatives are stored in lexicographical order, e.g. for order i = 3 and dimension 2 the derivatives are stored as follows: \( \partial_{xxx}, \, \partial_{xxy}, \, \partial_{xyy}, \, \partial_{yyy}.\, \)

Parameters
[in]uEvaluation points, each column corresponds to one evaluation point.
[in]nAll derivatives up to order n are computed and stored in result.
[in,out]resultSee above for format.

Reimplemented from gsFunctionSet< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

void evalAllDersFunc_into ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs,
const unsigned  n,
std::vector< gsMatrix< T > > &  result,
bool  sameElement = false 
) const
virtualinherited

Evaluates all derivatives up to order n of the function described by coefs at points u.

Evaluation points u are given as gsMatrix of size d x N, where
d is the dimension of the parameter domain and
N is the number of evaluation points.
Each column of u corresponds to the coordinates of one evaluation point.

The coefficients coefs are given as gsMatrix of size K x n, where
K is the number of (active) basis functions (=size()) and
n is the dimension of the physical domain.
Each row of coefs corresponds to the coordinates of one control point.

result is a std::vector, where the entry result[i] contains the gsMatrix corresponding to the i-th derivatives. The format of the respective entry is as in
evalFunc_into()
derivFunc_into()
deriv2Func_into()

Todo:
finish documentation
Parameters
[in]u
[in]coefs
[in]n
[out]result
gsMatrix<T> evalFunc ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs 
) const
inlineinherited

Evaluate the function described by coefs at points u.

This function has a default implementation that may be overridden in derived classes for higher performance.

Parameters
uevaluation points as m column vectors
coefscoefficient matrix describing the geometry in this basis, n columns
Returns
a matrix of size n x m with one function value as a column vector per evaluation point
void evalFunc_into ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs,
gsMatrix< T > &  result 
) const
virtualinherited

Evaluate the function described by coefs at points u, i.e., evaluates a linear combination of coefs x BasisFunctions, into result.

This function has a default implementation that may be overridden in derived classes for higher performance.

Parameters
uevaluation points as N column vectors
coefscoefficient matrix describing the geometry in this basis, n columns
[out]resulta matrix of size n x N with one function value as a column vector per evaluation point

Reimplemented in gsRationalBasis< SrcT >, gsTensorBSplineBasis< 1, T >, and gsMonomialBasis< T >.

gsBasisFun< T > function ( index_t  i) const
inherited

Returns the i-th basis function as a gsFunction.

Note that the gsBasisFun object only holds a reference to the current basis, so it is invalidated when the basis is destroyed.

unsigned index ( unsigned  i,
unsigned  j,
unsigned  k = 0 
) const
inlineinherited

Returns the global index of the basis function created by components of indices i,j,k (for 2d or 3d only)

index_t index ( gsVector< index_t, d > const &  v) const
inlineinherited

Returns the global index of the basis function created by components of indices given in the vector v

gsGeometry< T >::uPtr interpolateGrid ( gsMatrix< T > const &  vals,
std::vector< gsMatrix< T > >const &  grid 
) const
inherited

Interpolates values on a tensor-grid of points, given in tensor form (d coordinate-wise vectors). Samples vals should be ordered as the tensor-basis coefficients

void linearCombination_into ( const gsMatrix< T > &  coefs,
const gsMatrix< index_t > &  actives,
const gsMatrix< T > &  values,
gsMatrix< T > &  result,
bool  sameElement = false 
)
staticinherited

Computes the linear combination coefs * values( actives )

Todo:
documentation
Parameters
[in]coefsgsMatrix of size K x m, where K should equal size() of the basis (i.e., the number of basis functions).
[in]activesgsMatrix of size numAct x numPts
[in]valuesgsMatrix of size stride*numAct x numPts
[out]resultgsMatrix of size stride x numPts
virtual memory::unique_ptr<gsBasis<T> > makeNonRational ( ) const
inlinevirtualinherited

Clone the source of this basis in case of rational basis, same as clone() otherwise

Reimplemented in gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, and gsRationalBasis< gsBSplineBasis< T > >.

void matchWith ( const boundaryInterface bi,
const gsBasis< T > &  other,
gsMatrix< index_t > &  bndThis,
gsMatrix< index_t > &  bndOther,
index_t  offset 
) const
virtualinherited

Computes the indices of DoFs that match on the interface bi. The interface is assumed to be a common face between this patch and other, with an offset offset. The output is two lists of indices bndThis and bndOther, with indices that match one-to-one on the boundary bi.

NOTE: bndThis will have offset but bndOther will NOT have an offset (hence offset 0)

Reimplemented from gsBasis< T >.

gsVector<index_t> numActive ( const gsMatrix< T > &  u) const
inlineinherited

Number of active basis functions at an arbitrary parameter value.

Usually, this is used for getting the active functions on one element, assuming that this number doesn't change for different parameters inside the element.

void refine ( gsMatrix< T > const &  boxes,
int  refExt = 0 
)
virtualinherited

Refine the basis on the area defined by the matrix boxes.

boxes is a d x n-matrix (n even), where d is the dimension of the parameter domain.
n must be even, and every 2 successive columns in the matrix define a box in the parameter domain (the first column represents the coordinates of the lower corner, the second column the coordinates of the upper corner).

Example: The input of the matrix

\[ \left[\begin{array}{cccc} 0 & 0.2 & 0.8 & 1 \\ 0.4 & 0.6 & 0.2 & 0.4 \end{array} \right] \]

results in refinement of the two boxes \([0,0.2]\times[0.4,0.6]\) and \([0.8,1]\times[0.2,0.4]\).

Parameters
[in]boxesgsMatrix of size d x n, see above for description of size and meaning.
[in]refExtExtension to be applied to the refinement boxes

Reimplemented in gsHTensorBasis< d, T >, gsTensorBSplineBasis< d, T >, and gsTensorBSplineBasis< domainDim+1, T >.

void refineElements ( std::vector< index_t > const &  elements)
virtualinherited

Refine elements defined by their tensor-index.

In a tensor mesh, each element has a unique index computed as follows:

Let \(n_i\) denote the number of basis functions in the i-th component, and let \(k_i\) denote the index of an element in the (1-dimensional) mesh of the i-th component. The global index of element \((a,b,c)\) is given by \(a + b \cdot n_1 + c\cdot n_1 \cdot n_2\).

Parameters
[in]elementsvector of unsigned containing the indices of the elements that should be refined (see above).

Reimplemented from gsBasis< T >.

void refineElements_withCoefs ( gsMatrix< T > &  coefs,
std::vector< index_t > const &  boxes 
)
virtualinherited

Refine basis and geometry coefficients to levels.

Refines the basis as well as the coefficients. The refinement and the format of the input depend on the implementation of refineElements().

Reimplemented in gsHTensorBasis< d, T >, and gsRationalBasis< SrcT >.

virtual const gsBasis& source ( ) const
inlinevirtualinherited

Applicable for rational bases: returns the underlying "source" (non-rational) basis

Reimplemented in gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, and gsRationalBasis< gsBSplineBasis< T > >.

virtual gsBasis& source ( )
inlinevirtualinherited

Applicable for rational bases: returns the underlying "source" (non-rational) basis

Reimplemented in gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, and gsRationalBasis< gsBSplineBasis< T > >.

gsMatrix< T > support ( ) const
virtualinherited

Returns (a bounding box for) the domain of the whole basis.

Returns a dx2 matrix, containing the two diagonally extreme corners of a hypercube.

Reimplemented from gsBasis< T >.

gsMatrix< T > support ( const index_t i) const
virtualinherited

Returns (a bounding box for) the support of the i-th basis function.

Returns a dx2 matrix, containing the two diagonally extreme corners of a hypercube.

Reimplemented from gsBasis< T >.

gsMatrix< T > supportInterval ( index_t  dir) const
inherited

Returns an interval that contains the parameter values in direction dir.

Returns a 1x2 matrix, containing the two endpoints of the interval.

virtual void uniformCoarsen ( int  numKnots = 1)
inlinevirtualinherited

Coarsen the basis uniformly by removing groups of numKnots consecutive knots, each knot removed mul times.

This function is the oposite of gsBasis::uniformRefine

The execution of

basis->uniformRefine (nKnots, mul)
basis->uniformCoarsen(nKnots);

results in no overall change in "basis". However,

basis->uniformCoarsen(nKnots);
basis->uniformRefine (nKnots, mul)

is not guaranteed to keep "basis" unchanged.

See Also
gsBasis::uniformRefine

Reimplemented from gsBasis< T >.

Reimplemented in gsTensorBSplineBasis< 1, T >.

void uniformCoarsen_withCoefs ( gsMatrix< T > &  coefs,
int  numKnots = 1 
)
virtualinherited

Coarsen the basis uniformly.

The function simultainously updates the vector coefs, representing a function in the bases, such that its new version represents the same function.

This function is equivalent to

gsSparseMatrix<T,RowMajor> transfer;
basis->uniformCoarsen_withTransfer(transfer, numKnots);
coefs = transfer * coefs;
See Also
gsBasis::uniformRefine

Reimplemented in gsHTensorBasis< d, T >.

void uniformCoarsen_withTransfer ( gsSparseMatrix< T, RowMajor > &  transfer,
int  numKnots = 1 
)
virtualinherited

Coarsen the basis uniformly and produce a sparse matrix which maps coarse coefficient vectors to refined ones.

The function writes a sparse matrix into the variable transfer that indicates how the functions on the coarse grid are represented as linear combinations as fine grid functions

See Also
gsBasis::uniformCoarsen

Reimplemented from gsBasis< T >.

void uniformRefine_withCoefs ( gsMatrix< T > &  coefs,
int  numKnots = 1,
int  mul = 1,
int  dir = -1 
)
virtualinherited

Refine the basis uniformly and perform knot refinement for the given coefficient vector

Reimplemented from gsBasis< T >.

void uniformRefine_withTransfer ( gsSparseMatrix< T, RowMajor > &  transfer,
int  numKnots = 1,
int  mul = 1 
)
virtualinherited

Refine the basis uniformly and produce a sparse matrix which maps coarse coefficient vectors to refined ones

Reimplemented from gsBasis< T >.