pbat::fem::SymmetricLaplacianMatrix< TMesh > Struct Template Reference

A matrix-free representation of the symmetric part of the Laplacian \(\Delta u\) of a finite element discretized function \( u(X) \) under Galerkin projection. More...

#include <LaplacianMatrix.h>

Public Types
using	SelfType = SymmetricLaplacianMatrix<TMesh>
	Self type.
using	MeshType = TMesh
	Mesh type.
using	ElementType = typename TMesh::ElementType
	Element type.

Public Member Functions
	SymmetricLaplacianMatrix (MeshType const &mesh, Eigen::Ref< IndexVectorX const > const &eg, Eigen::Ref< VectorX const > const &wg, Eigen::Ref< MatrixX const > const &GNegg, int dims=1)
	Construct a new symmetric Laplacian operator.
SelfType &	operator= (SelfType const &)=delete
template<class TDerivedIn, class TDerivedOut>
void	Apply (Eigen::MatrixBase< TDerivedIn > const &x, Eigen::DenseBase< TDerivedOut > &y) const
	Applies this matrix as a linear operator on x, adding result to y.
CSCMatrix	ToMatrix () const
	Transforms this matrix-free matrix representation into sparse compressed format.
Index	InputDimensions () const
	Number of columns.
Index	OutputDimensions () const
	Number of rows.
void	ComputeElementLaplacians ()
	Compute and store the element laplacians.
void	CheckValidState () const
	Check if the state of this Laplacian matrix is valid.

Public Attributes
MeshType const &	mesh
	The finite element mesh.
Eigen::Ref< IndexVectorX const >	eg
	|# quad.pts.|x1 array of elements associated with quadrature points
Eigen::Ref< VectorX const >	wg
	|# quad.pts.|x1 array of quadrature weights
Eigen::Ref< MatrixX const >	GNeg
MatrixX	deltag
int	dims

Static Public Attributes
static int constexpr	kOrder
	Polynomial order of the Laplacian matrix.

Detailed Description

template<CMesh TMesh>
struct pbat::fem::SymmetricLaplacianMatrix< TMesh >

A matrix-free representation of the symmetric part of the Laplacian \(\Delta u\) of a finite element discretized function \( u(X) \) under Galerkin projection.

The precise definition of the Laplacian matrix's symmetric part is given by \( \mathbf{L}_{ij} = \int_\Omega -\nabla \phi_i \cdot \nabla \phi_j d\Omega \).

Todo

Explain the Laplacian matrix and its construction and link to my higher-level FEM crash course doc.

This matrix-free Laplacian requires the following inputs:

• A finite element mesh satisfying concept CMesh
• A vector of element indices eg, associating each quadrature point with an element
• A vector of quadrature weights wg
• A matrix GNegg of element shape function gradients at quadrature points (see ShapeFunctionGradients())
• An integer dims specifying the dimensionality of the image of the FEM function space

Note

The user-provided quadrature rule is injected into the Laplacian operator, allowing for arbitrary quadrature rules to be used. Since Laplacians of higher-dimensional functions are only one-dimensional Laplacian 'kroneckered' with the identity matrix, the Laplacian matrix is actually \( L \otimes I_{d \times d} \) where \( d \) is the function's dimensionality, but we need not store the duplicate entries.

Template Parameters

TMesh

Type satisfying concept CMesh

Constructor & Destructor Documentation

SymmetricLaplacianMatrix()

template<CMesh TMesh>

pbat::fem::SymmetricLaplacianMatrix< TMesh >::SymmetricLaplacianMatrix ( MeshType const & mesh, Eigen::Ref< IndexVectorX const > const & eg, Eigen::Ref< VectorX const > const & wg, Eigen::Ref< MatrixX const > const & GNegg, int dims = 1 )

inline

Construct a new symmetric Laplacian operator.

Parameters

mesh	Finite element mesh
eg	Element indices associating each quadrature point with an element
wg	Quadrature weights
GNegg	Element shape function gradients at quadrature points
dims	Dimensionality of the image of the FEM function space

Precondition

eg.size() == wg.size() and GNegg.rows() == mesh.E.rows() and dims >= 1

Member Function Documentation

Apply()

template<CMesh TMesh>

template<class TDerivedIn, class TDerivedOut>

void pbat::fem::SymmetricLaplacianMatrix< TMesh >::Apply ( Eigen::MatrixBase< TDerivedIn > const & x, Eigen::DenseBase< TDerivedOut > & y ) const

inline

Applies this matrix as a linear operator on x, adding result to y.

Template Parameters

TDerivedIn	Input matrix type
TDerivedOut	Output matrix type

Parameters

x	Input matrix
y	Output matrix

Precondition

x.rows() == InputDimensions() and y.rows() == OutputDimensions() and y.cols() == x.cols()

InputDimensions()

template<CMesh TMesh>

Index pbat::fem::SymmetricLaplacianMatrix< TMesh >::InputDimensions ( ) const

inline

Number of columns.

Returns

Number of columns

OutputDimensions()

template<CMesh TMesh>

Index pbat::fem::SymmetricLaplacianMatrix< TMesh >::OutputDimensions ( ) const

inline

Number of rows.

Returns

Number of rows

ToMatrix()

template<CMesh TMesh>

CSCMatrix pbat::fem::SymmetricLaplacianMatrix< TMesh >::ToMatrix ( ) const

inline

Transforms this matrix-free matrix representation into sparse compressed format.

Returns

Sparse compressed column matrix representation of this Laplacian matrix

Member Data Documentation

deltag

template<CMesh TMesh>

MatrixX pbat::fem::SymmetricLaplacianMatrix< TMesh >::deltag

|# element nodes| x |# element nodes * # quad.pts.| matrix of element laplacians at quadrature points

dims

template<CMesh TMesh>

int pbat::fem::SymmetricLaplacianMatrix< TMesh >::dims

Dimensionality of image of FEM function space, i.e. this Laplacian matrix is actually \( L \otimes I_{d} \). Must have dims >= 1.

GNeg

template<CMesh TMesh>

Eigen::Ref<MatrixX const> pbat::fem::SymmetricLaplacianMatrix< TMesh >::GNeg

|# element nodes|x|# dims * # quad.pts. * # elements| matrix of element shape function gradients at quadrature points

kOrder

template<CMesh TMesh>

int constexpr pbat::fem::SymmetricLaplacianMatrix< TMesh >::kOrder

staticconstexpr

Initial value:

=
        2 * (ElementType::kOrder - 1)

Polynomial order of the Laplacian matrix.

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The documentation for this struct was generated from the following file:

• C:/git/PhysicsBasedAnimationToolkit/source/pbat/fem/LaplacianMatrix.h