Detailed Description

template<class T, enum MaterialOutput out>
class gismo::gsMaterialMatrixIntegrateSingle< T, out >

This class serves as the integrator of material matrices, based on gsMaterialMatrixBase.

Template Parameters

T	Real type
out	Output type (see MaterialOutput)

Inheritance diagram for gsMaterialMatrixIntegrateSingle< T, out >:

Collaboration diagram for gsMaterialMatrixIntegrateSingle< T, out >:

Public Types
typedef memory::shared_ptr < gsFunction >	Ptr
	Shared pointer for gsFunction.

typedef memory::unique_ptr < gsFunction >	uPtr
	Unique pointer for gsFunction.

Public Member Functions
gsMatrix< index_t >	active (const gsMatrix< T > &u) const
	Returns the indices of active (nonzero) functions at points u, as a list of indices. More...

void	active_into (const gsMatrix< T > &u, gsMatrix< index_t > &result) const
	Indices of active (non-zero) function(s) for each point. More...

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.

uPtr	clone ()
	Clone methode. Produceds a deep copy inside a uPtr.

virtual void	compute (const gsMatrix< T > &in, gsFuncData< T > &out) const
	Computes function data. More...

virtual void	computeMap (gsMapData< T > &InOut) const
	Computes map function data. More...

gsFuncCoordinate< T >	coord (const index_t c) const
	Returns the scalar function giving the i-th coordinate of this function.

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 T	distanceL2 (gsFunction< T > const &) const
	Computes the L2-distance between this function and the field and a function func.

short_t	domainDim () const
	Domain dimension, always 2 for shells.

gsMatrix< T >	eval (const gsMatrix< T > &u) const
	Evaluate the function,. More...

void	eval_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Implementation of eval_into, see gsFunction. More...

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...

const gsFunction< T > &	function (const index_t k) const
	Helper which casts and returns the k-th piece of this function set as a gsFunction.

	gsMaterialMatrixIntegrateSingle (index_t patch, gsMaterialMatrixBase< T > materialMatrix, const gsFunctionSet< T > deformed)
	Constructor.

	gsMaterialMatrixIntegrateSingle (index_t patch, gsMaterialMatrixBase< T > materialMatrix, const gsFunctionSet< T > undeformed, const gsFunctionSet< T > *deformed)
	Constructor.

virtual void	invertPoints (const gsMatrix< T > &points, gsMatrix< T > &result, const T accuracy=1e-6, const bool useInitialPoint=false) const

int	newtonRaphson (const gsVector< T > &value, gsVector< T > &arg, bool withSupport=true, const T accuracy=1e-6, int max_loop=100, T damping_factor=1) const

virtual index_t	nPieces () const
	Number of pieces in the domain of definition.

virtual gsMatrix< T >	parameterCenter () const
	Returns a "central" point inside inside the parameter domain.

gsMatrix< T >	parameterCenter (const boxCorner &bc) const
	Get coordinates of the boxCorner bc in the parameter domain.

gsMatrix< T >	parameterCenter (const boxSide &bs) const
	Get coordinates of the midpoint of the boxSide bs in the parameter domain.

virtual const gsFunction &	piece (const index_t k) const
	Returns the piece(s) of the function(s) at subdomain k.

virtual std::ostream &	print (std::ostream &os) const
	Prints the object as a string.

void	recoverPoints (gsMatrix< T > &xyz, gsMatrix< T > &uv, index_t k, const T accuracy=1e-6) const

index_t	size () const
	size More...

short_t	targetDim () const
	Target dimension. More...

Evaluation functions
These functions allow one to evaluate the function as well as its derivatives at one or more points in the parameter space. See also Evaluation members.
virtual void	eval_component_into (const gsMatrix< T > &u, const index_t comp, gsMatrix< T > &result) const
	Evaluate the function for component comp in the target dimension at points u into result.

virtual void	deriv_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluate derivatives of the function $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ at points u into result. More...

virtual void	jacobian_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Computes for each point u a block of result containing the Jacobian matrix.

void	div_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Computes for each point u a block of result containing the divergence matrix.

gsMatrix< T >	jacobian (const gsMatrix< T > &u) const

virtual void	deriv2_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluate second derivatives of the function at points u into result. More...

virtual void	hessian_into (const gsMatrix< T > &u, gsMatrix< T > &result, index_t coord=0) const

virtual gsMatrix< T >	hessian (const gsMatrix< T > &u, index_t coord=0) const

virtual gsMatrix< T >	laplacian (const gsMatrix< T > &u) const
	Evaluate the Laplacian at points u. More...

Protected Member Functions
gsMatrix< T >	_eval (const gsMatrix< T > &u) const
	Integrateuates the base class in 3D, but on Z=0. More...

gsMatrix< T >	_eval3D (const gsMatrix< T > &u, const gsMatrix< T > &Z) const
	Integrateuates the base class in 3D. More...

T	getMoment () const
	Gets the moment based on the output type. More...

void	integrateZ_into (const gsMatrix< T > &u, const index_t moment, gsMatrix< T > &result) const
	Integrates through-thickness using Gauss integration. More...

void	multiplyLinZ_into (const gsMatrix< T > &u, const index_t moment, gsMatrix< T > &result) const
	Uses the top and bottom parts of the thickness to compute the integral exactly. More...

void	multiplyZ_into (const gsMatrix< T > &u, index_t moment, gsMatrix< T > &result) const
	Multiplies the evaluation at z=0 with 2.0/(moment+1) * Thalf^(moment + 1) More...

void	setPatch (index_t p)
	Sets the patch index.

Private Member Functions
template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::VectorN\|\|_out==MaterialOutput::VectorM, gsMatrix< T > >::type	eval3D_impl (const gsMatrix< T > &u, const gsMatrix< T > &Z) const
	Specialisation of eval3D for vectors.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::CauchyVectorN\|\|_out==MaterialOutput::CauchyVectorM, gsMatrix< T > >::type	eval3D_impl (const gsMatrix< T > &u, const gsMatrix< T > &Z) const
	Specialisation of eval3D for vectors.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::MatrixA\|\|_out==MaterialOutput::MatrixB\|\|_out==MaterialOutput::MatrixC\|\|_out==MaterialOutput::MatrixD, gsMatrix< T > >::type	eval3D_impl (const gsMatrix< T > &u, const gsMatrix< T > &Z) const
	Specialisation of eval3D for matrix.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::PStress\|\|_out==MaterialOutput::PStressN\|\|_out==MaterialOutput::PStressM, gsMatrix< T > >::type	eval3D_impl (const gsMatrix< T > &u, const gsMatrix< T > &Z) const
	Specialisation of eval3D for vectors.

template<enum MaterialOutput _out>
std::enable_if<!(_out==MaterialOutput::VectorN\|\|_out==MaterialOutput::VectorM\|\|_out==MaterialOutput::CauchyVectorN\|\|_out==MaterialOutput::CauchyVectorM\|\|_out==MaterialOutput::MatrixA\|\|_out==MaterialOutput::MatrixB\|\|_out==MaterialOutput::MatrixC\|\|_out==MaterialOutput::MatrixD\|\|_out==MaterialOutput::PStressN\|\|_out==MaterialOutput::PStressM), gsMatrix< T > >::type	eval3D_impl (const gsMatrix< T > &u, const gsMatrix< T > &Z) const
	Specialisation of eval3D for other types.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::Density, void >::type	eval_into_impl (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Specialisation of eval_into for densities.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::VectorN\|\|_out==MaterialOutput::CauchyVectorN, void >::type	eval_into_impl (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Specialisation of eval_into for the membrane stress tensor N.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::VectorM\|\|_out==MaterialOutput::CauchyVectorM, void >::type	eval_into_impl (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Specialisation of eval_into for the flexural stress tensor M.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::MatrixA\|\|_out==MaterialOutput::MatrixB\|\|_out==MaterialOutput::MatrixC\|\|_out==MaterialOutput::MatrixD, void >::type	eval_into_impl (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Specialisation of eval_into for the moments of the material matrices.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::PStressN\|\|_out==MaterialOutput::PStressM, void >::type	eval_into_impl (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Specialisation of eval_into for the membrane and flexural principle stresses.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::PStrainN\|\|_out==MaterialOutput::PStrainM, void >::type	eval_into_impl (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Specialisation of eval_into for the membrane and flexural principle stresses.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::Stretch, void >::type	eval_into_impl (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Specialisation of eval_into for the stretches.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::StretchDir, void >::type	eval_into_impl (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Specialisation of eval_into for the stretch directions.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::VectorN\|\|_out==MaterialOutput::PStressN\|\|_out==MaterialOutput::CauchyVectorN, T >::type	getMoment_impl () const
	Implementation of getMoment for MaterialOutput::VectorN; the moment is 0.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::VectorM\|\|_out==MaterialOutput::PStressM\|\|_out==MaterialOutput::CauchyVectorM, T >::type	getMoment_impl () const
	Implementation of getMoment for MaterialOutput::VectorM; the moment is 1.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::MatrixA, T >::type	getMoment_impl () const
	Implementation of getMoment for MaterialOutput::MatrixA; the moment is 0.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::MatrixB, T >::type	getMoment_impl () const
	Implementation of getMoment for MaterialOutput::MatrixB; the moment is 1.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::MatrixC, T >::type	getMoment_impl () const
	Implementation of getMoment for MaterialOutput::MatrixC; the moment is 1.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::MatrixD, T >::type	getMoment_impl () const
	Implementation of getMoment for MaterialOutput::MatrixD; the moment is 2.

template<enum MaterialOutput _out>
std::enable_if<!(_out==MaterialOutput::VectorN\|\|_out==MaterialOutput::VectorM\|\|_out==MaterialOutput::CauchyVectorN\|\|_out==MaterialOutput::CauchyVectorM\|\|_out==MaterialOutput::MatrixA\|\|_out==MaterialOutput::MatrixB\|\|_out==MaterialOutput::MatrixC\|\|_out==MaterialOutput::MatrixD\|\|_out==MaterialOutput::PStressN\|\|_out==MaterialOutput::PStressM), index_t >::type	getMoment_impl () const
	Implementation of getMoment for MaterialOutput other than VectorN, VectorM, MatrixA, MatrixB, MatrixC, MatrixD, PStressN, PStressM.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::Density, short_t >::type	targetDim_impl () const
	Implementation of targetDim for densities.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::VectorN\|\|_out==MaterialOutput::CauchyVectorN\|\|_out==MaterialOutput::VectorM\|\|_out==MaterialOutput::CauchyVectorM, short_t >::type	targetDim_impl () const
	Implementation of targetDim for stress tensors.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::MatrixA\|\|_out==MaterialOutput::MatrixB\|\|_out==MaterialOutput::MatrixC\|\|_out==MaterialOutput::MatrixD, short_t >::type	targetDim_impl () const
	Implementation of targetDim for material tensors.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::PStressN\|\|_out==MaterialOutput::PStressM\|\|_out==MaterialOutput::PStrainN\|\|_out==MaterialOutput::PStrainM, short_t >::type	targetDim_impl () const
	Implementation of targetDim for principal stress fields.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::Stretch, short_t >::type	targetDim_impl () const
	Implementation of targetDim for principal stretch fields.

template<enum MaterialOutput _out>
std::enable_if< _out==MaterialOutput::StretchDir, short_t >::type	targetDim_impl () const
	Implementation of targetDim for principal stress directions.

Member Function Documentation

gsMatrix< T > _eval ( const gsMatrix< T > & u ) const

protected

Integrateuates the base class in 3D, but on Z=0.

This function is primarily used in cases where the base class already integrates the material matrix or vectors

Parameters

[in] u The evaluation points (in plane)

Returns: Matrix with results

gsMatrix< T > _eval3D	(	const gsMatrix< T > &	u,
		const gsMatrix< T > &	Z
	)		const

protected

Integrateuates the base class in 3D.

Parameters

[in]	u	The evaluation points (in plane)
[in]	Z	The through-thickness coordinate

Returns: Matrix ordered over Z and over u within

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_into	(	const gsMatrix< T > &	u,
		gsMatrix< index_t > &	result
	)		const

inlinevirtualinherited

Indices of active (non-zero) function(s) for each point.

The columns are sorted in increasing order, if on a point there are less active then the number of rows in the result matrix (some other point has more actives) then the rest of the column is filled with 0s.

Parameters

u
result

Reimplemented from gsFunctionSet< T >.

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 computeMap ( gsMapData< T > & InOut ) const

virtualinherited

Computes map function data.

This function evaluates the functions and their derivatives at the points InOut.points and writes them in the corresponding fields of InOut. Which field to write (and what to compute) is controlled by the InOut.flags (see also gsMapData). This is intended for parametrizations only and it works on functions sets of cardinality 1 only.

Parameters

[in,out] InOut

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] u Evaluation 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 second derivatives of the function at points u into result.

Let n be the dimension of the source space ( n = domainDim() ).
Let m be the dimension of the image/target space ( m = targetDim() ).
Let N denote the number of evaluation points.

Parameters

[in]	u	gsMatrix of size n x N, where each column of u represents one evaluation point.
[out]	result	gsMatrix of size (Sm)* x N, where S=n(n+1)/2. Each column in result* corresponds to one point (i.e., one column in u) and contains the following values (for n=3, m=3): $ (\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)},\ldots,\partial_{yz} f^{(3)} )^T$

Warning: By default uses central finite differences with h=0.00001! One must override this function in derived classes to get proper results.

Reimplemented from gsFunctionSet< T >.

Reimplemented in gsGeometry< T >, gsFunctionExpr< T >, gsConstantFunction< T >, gsPreCICEFunction< T >, gsAffineFunction< T >, gsMappedSingleSpline< d, T >, gsGeometryTransform< T >, and gsFuncCoordinate< T >.

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

virtualinherited

Evaluate derivatives of the function $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ at points u into result.

Let n be the dimension of the source space ( n = domainDim() ).
Let m be the dimension of the image/target space ( m = targetDim() ).
Let N denote the number of evaluation points.

Let $ f:\mathbb R^2 \rightarrow \mathbb R^3 $, i.e., $ f(x,y) = ( f^{(1)}(x,y), f^{(2)}(x,y), f^{(3)}(x,y) )^T$,
and let $ u = ( u_1, \ldots, u_N) = ( (x_1,y_1)^T, \ldots, (x_N, y_N)^T )$.
Then, result is of the form

\[ \left[ \begin{array}{cccc} \partial_x f^{(1)}(u_1) & \partial_x f^{(1)}(u_2) & \ldots & \partial_x f^{(1)}(u_N) \\ \partial_y f^{(1)}(u_1) & \partial_y f^{(1)}(u_2) & \ldots & \partial_y f^{(1)}(u_N) \\ \partial_x f^{(2)}(u_1) & \partial_x f^{(2)}(u_2) & \ldots & \partial_x f^{(2)}(u_N) \\ \partial_y f^{(2)}(u_1) & \partial_y f^{(2)}(u_2) & \ldots & \partial_x f^{(2)}(u_N) \\ \partial_x f^{(3)}(u_1) & \partial_x f^{(3)}(u_2) & \ldots & \partial_x f^{(3)}(u_N)\\ \partial_y f^{(3)}(u_1) & \partial_y f^{(3)}(u_2) & \ldots & \partial_y f^{(3)}(u_N) \end{array} \right] \]

Parameters

[in]	u	gsMatrix of size n x N, where each column of u represents one evaluation point.
[out]	result	gsMatrix of size (n m)* x N. Each row of result corresponds to one component in the target space and contains the gradients for each evaluation point, as row vectors, one after the other (see above for details on the format).

Warning: By default, gsFunction uses central finite differences with h=0.00001! One must override this function in derived classes to get proper results.

Reimplemented from gsFunctionSet< T >.

Reimplemented in gsGeometry< T >, gsFunctionExpr< T >, gsConstantFunction< T >, gsPreCICEFunction< T >, gsAffineFunction< T >, gsMappedSingleSpline< d, T >, gsGeometrySlice< T >, gsSquaredDistance< T >, gsBasisFun< T >, gsGeometryTransform< T >, and gsFuncCoordinate< T >.

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

virtual

Implementation of eval_into, see gsFunction.

Non-parallel evaluation

Implements gsFunction< 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]	u	Evaluation points, each column corresponds to one evaluation point.
[in]	n	All derivatives up to order n are computed and stored in result.
[in,out]	result	See above for format.

Reimplemented in gsTensorBSplineBasis< 1, T >, gsTensorBasis< d, T >, gsTensorBasis< 1, T >, gsGeometry< T >, gsMappedSingleBasis< d, T >, gsConstantFunction< T >, gsTHBSplineBasis< d, T >, gsMappedSingleSpline< d, T >, and gsSquaredDistance< T >.

T getMoment ( ) const

inlineprotected

Gets the moment based on the output type.

for VectorN, the moment is 0 VectorM, the moment is 1 (unless MatIntegrated==Constant or MatIntegrated==Integrated) MatrixA, the moment is 0 MatrixB, the moment is 1 MatrixC, the moment is 1 MatrixD, the moment is 2 PStressN,the moment is 0 PStressM,the moment is 1

Returns: The moment.

virtual gsMatrix<T> hessian	(	const gsMatrix< T > &	u,
		index_t	coord = `0`
	)		const

inlinevirtualinherited

Evaluates the Hessian (matrix of second partial derivatives) of coordinate coord at points u.

void integrateZ_into	(	const gsMatrix< T > &	u,
		const index_t	moment,
		gsMatrix< T > &	result
	)		const

protected

Integrates through-thickness using Gauss integration.

Parameters

[in]	u	evaluation points (in plane)
[in]	moment	moment to be taken
	result	the result

void invertPoints	(	const gsMatrix< T > &	points,
		gsMatrix< T > &	result,
		const T	accuracy = `1e-6`,
		const bool	useInitialPoint = `false`
	)		const

virtualinherited

Takes the physical points and computes the corresponding parameter values. If the point cannot be inverted (eg. is not part of the geometry) the corresponding parameter values will be undefined

gsMatrix< T > laplacian ( const gsMatrix< T > & u ) const

virtualinherited

Evaluate the Laplacian at points u.

By default uses central finite differences with h=0.00001

Reimplemented in gsFunctionExpr< T >.

void multiplyLinZ_into	(	const gsMatrix< T > &	u,
		const index_t	moment,
		gsMatrix< T > &	result
	)		const

protected

Uses the top and bottom parts of the thickness to compute the integral exactly.

This function assumes that the function $ h(z,...) $ to be integrated is of the form $ h(z,...) = f(...) + z g(...)$ (no higher-order terms!). This implies that $f(...)$ is the even part and $z g(...)$ is the odd part of h. Writing out the thickness integral for a moment $\alpha$ gives:

\begin{eqnarray*} \int_{-t/2}^{t/2} z^\alpha h(z,...)\:\text{d}\z &= \int_{-t/2}^{t/2} z^\alpha f(...) + z^{\alpha+1} g(...)\:\text{d}\z \\ & = \frac{z^{\alpha+1}}{\alpha+1} f(...) + \frac{z^{\alpha+2}}{\alpha+2} g(...) \bigg\vert_{t/2}^{t/2} \\ & = \frac{z^{\alpha+1}}{\alpha+1} f(...) + \frac{z^{\alpha+1}}{\alpha+2} zg(...) \bigg\vert_{t/2}^{t/2} \\ & = \frac{1}{\alpha+1} \left[ \left(\frac{t}{2}\right)^{\alpha+1} - \left(-\frac{t}{2}\right)^{\alpha+1} \right]f(...) + \frac{1}{\alpha+2}\left[ \left(\frac{t}{2}\right)^{\alpha+2} - \left(-\frac{t}{2}\right)^{\alpha+2} \right] g(...) \bigg\vert_{t/2}^{t/2} \\ \end{eqnarray*}

From this we observe that for $\alpha$ odd, the part of $f$ contributes, whereas for $\alpha$ even, the $g$ part contributes. Remember our assumption on the form of $h(z,...)$, then we can integrate this function by evaluating the following:

\begin{eqnarray*} \int_{-t/2}^{t/2} z^\alpha h(z,...)\:\text{d}\z = \begin{dcases} \frac{1}{\alpha+1} \left[ \left(\frac{t}{2}\right)^{\alpha+1} - \left(-\frac{t}{2}\right)^{\alpha+1} \right]h(...) & \text{} if $\alpha$ is odd} \\ \frac{1}{\alpha+2}\left[ \left(\frac{t}{2}\right)^{\alpha+1} - \left(-\frac{t}{2}\right)^{\alpha+1} \right] h(...) & \text{} if $\alpha$ is even} \end{dcases} \end{eqnarray*}

Parameters

[in]	u	evaluation points (in plane)
[in]	moment	moment to be taken
	result	the result

void multiplyZ_into	(	const gsMatrix< T > &	u,
		index_t	moment,
		gsMatrix< T > &	result
	)		const

protected

Multiplies the evaluation at z=0 with 2.0/(moment+1) * Thalf^(moment + 1)

WARNING: Recommended use only for constant functions over thickness.

The function assumes that moments are handled inside the evaluation. For example, when we want to integrate $f(z,...) = g(...) + zh(...)$ with different moments, then for moment 0, the function assumes that $g(...)$ is evaluated, for moment 1, the function assumes that nothing is returned and for moment 2, it assumes that $h$ is returned.

The function is equivalent to multiplyLinZ_into when moment = 2 here and moment = 1 in multiplyLinZ_into (given that the correct function is returned).

Parameters

[in]	u	evaluation points (in plane)
[in]	moment	moment to be taken
	result	the result

int newtonRaphson	(	const gsVector< T > &	value,
		gsVector< T > &	arg,
		bool	withSupport = `true`,
		const T	accuracy = `1e-6`,
		int	max_loop = `100`,
		T	damping_factor = `1`
	)		const

inherited

Newton-Raphson method to find a solution of the equation f(arg) = value with starting vector arg. If the point cannot be inverted the corresponding parameter values will be undefined

void recoverPoints	(	gsMatrix< T > &	xyz,
		gsMatrix< T > &	uv,
		index_t	k,
		const T	accuracy = `1e-6`
	)		const

inherited

Recovers a point on the (geometry) together with its parameters uv, assuming that the k-th coordinate of the point xyz is not known (and has a random value as input argument).

index_t size ( ) const

inlinevirtualinherited

size

Warning: gsFunction and gsGeometry have size() == 1. This should not be confused with the size eg. of gsGeometry::basis(), which is the number of basis functions in the basis

Returns: the size of the function set: the total number of functions

Reimplemented from gsFunctionSet< T >.

Reimplemented in gsPiecewiseFunction< T >.

short_t targetDim ( ) const

inlinevirtual

Target dimension.

For a scalar (e.g. density) the target dimension is 1, for a vector (e.g. stress tensor in Voight notation) the target dimension is 3 and for a matrix (e.g. the material matrix) the target dimension is 9, which can be reshaped to a 3x3 matrix.

Returns: Returns the target dimension depending on the specified type (scalar, vector, matrix etc.)

Reimplemented from gsFunctionSet< T >.

Detailed Description

template<class T, enum MaterialOutput out> class gismo::gsMaterialMatrixIntegrateSingle< T, out >

Public Types

Public Member Functions

Protected Member Functions

Private Member Functions

Member Function Documentation

template<class T, enum MaterialOutput out>
class gismo::gsMaterialMatrixIntegrateSingle< T, out >