Gradient.h File Reference

FEM gradient operator. More...

#include "Concepts.h"
#include "ShapeFunctions.h"
#include "pbat/Aliases.h"
#include "pbat/common/Eigen.h"
#include "pbat/profiling/Profiling.h"
#include <exception>
#include <fmt/core.h>
#include <tbb/parallel_for.h>

Go to the source code of this file.

Namespaces
namespace	pbat
	The main namespace of the library.
namespace	pbat::fem
	Finite Element Method (FEM)

Functions
template<CElement TElement, int Dims, class TDerivedE, class TDerivedeg, class TDerivedGNeg, class TDerivedIn, class TDerivedOut>
void	pbat::fem::GemmGradient (Eigen::DenseBase< TDerivedE > const &E, Eigen::Index nNodes, Eigen::DenseBase< TDerivedeg > const &eg, Eigen::MatrixBase< TDerivedGNeg > const &GNeg, Eigen::MatrixBase< TDerivedIn > const &X, Eigen::DenseBase< TDerivedOut > &Y)
	Compute gradient matrix-vector multiply \( Y += \mathbf{G} X \).
template<CMesh TMesh, class TDerivedeg, class TDerivedGNeg, class TDerivedIn, class TDerivedOut>
void	pbat::fem::GemmGradient (TMesh const &mesh, Eigen::DenseBase< TDerivedeg > const &eg, Eigen::MatrixBase< TDerivedGNeg > const &GNeg, Eigen::MatrixBase< TDerivedIn > const &X, Eigen::DenseBase< TDerivedOut > &Y)
	Compute gradient matrix-vector multiply \( Y += \mathbf{G} X \) using mesh.
template<CElement TElement, int Dims, Eigen::StorageOptions Options, class TDerivedE, class TDerivedeg, class TDerivedGNeg>
auto	pbat::fem::GradientMatrix (Eigen::DenseBase< TDerivedE > const &E, Eigen::Index nNodes, Eigen::DenseBase< TDerivedeg > const &eg, Eigen::MatrixBase< TDerivedGNeg > const &GNeg) -> Eigen::SparseMatrix< typename TDerivedGNeg::Scalar, Options, typename TDerivedE::Scalar >
	Construct the gradient operator's sparse matrix representation.
template<Eigen::StorageOptions Options, CMesh TMesh, class TDerivedeg, class TDerivedGNeg>
auto	pbat::fem::GradientMatrix (TMesh const &mesh, Eigen::DenseBase< TDerivedeg > const &eg, Eigen::MatrixBase< TDerivedGNeg > const &GNeg) -> Eigen::SparseMatrix< typename TDerivedGNeg::Scalar, Options, typename TMesh::IndexType >
	Construct the gradient operator's sparse matrix representation using mesh.

Detailed Description

FEM gradient operator.

Author

Quoc-Minh Ton-That (tonth.nosp@m.at.q.nosp@m.uocmi.nosp@m.nh@g.nosp@m.mail..nosp@m.com)

Date

2025-02-11

Given a function \( u(X) \) discretized at mesh nodes \( X_i \),

\[\nabla u(X) = \sum_j u_j \nabla \phi_j(X) \; \forall \; e \in E \]

where \( \nabla \phi_j(X) \) is the gradient of the shape function \( \phi_j(X) \) at element \( e \). The gradient operator \( \mathbf{G} \in \mathbb{R}^{d|Q| \times n} \) maps the nodal values to the gradient at quadrature points \( Q \) in dimensions \( d \).

This file provides functions to compute gradient operator related quantities.

Copyright

Copyright (c) 2025