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Matrix.rst
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.. _Matrix:
******
Matrix
******
| :download:`GooseFEM/Matrix.h <../../include/GooseFEM/Matrix.h>`
| :download:`GooseFEM/Matrix.hpp <../../include/GooseFEM/Matrix.hpp>`
| :download:`GooseFEM/MatrixPartitioned.h <../../include/GooseFEM/MatrixPartitioned.h>`
| :download:`GooseFEM/MatrixPartitioned.hpp <../../include/GooseFEM/MatrixPartitioned.hpp>`
| :download:`GooseFEM/MatrixPartitionedTyings.h <../../include/GooseFEM/MatrixPartitionedTyings.h>`
| :download:`GooseFEM/MatrixPartitionedTyings.hpp <../../include/GooseFEM/MatrixPartitionedTyings.hpp>`
| :download:`GooseFEM/MatrixDiagonal.h <../../include/GooseFEM/MatrixDiagonal.h>`
| :download:`GooseFEM/MatrixDiagonal.hpp <../../include/GooseFEM/MatrixDiagonal.hpp>`
| :download:`GooseFEM/MatrixDiagonalPartitioned.h <../../include/GooseFEM/MatrixDiagonalPartitioned.h>`
| :download:`GooseFEM/MatrixDiagonalPartitioned.hpp <../../include/GooseFEM/MatrixDiagonalPartitioned.hpp>`
Matrix
======
Matrix definition.
.. note::
A solver has to be chosen, see :ref:`linear_solver`.
Matrix::nelem()
---------------
Return the number of elements.
Matrix::nne()
-------------
Return the number of nodes-per-element.
Matrix::nnode()
---------------
Return the number of nodes.
Matrix::ndim()
--------------
Return the number of dimensions.
Matrix::ndof()
--------------
Return the number of DOFs.
Matrix::dofs()
--------------
Return the DOF-numbers per node [nnode, ndim].
Matrix::assemble(...)
---------------------
Assemble matrix from element matrices stored as "elemmat".
Matrix::solve(...)
------------------
Solve linear system.
MatrixPartitioned
=================
Partitioned matrix definition.
.. note::
A solver has to be chosen, see :ref:`linear_solver`.
MatrixPartitioned::nelem()
--------------------------
Return the number of elements.
MatrixPartitioned::nne()
------------------------
Return the number of nodes-per-element.
MatrixPartitioned::nnode()
--------------------------
Return the number of nodes.
MatrixPartitioned::ndim()
-------------------------
Return the number of dimensions.
MatrixPartitioned::ndof()
-------------------------
Return the number of DOFs.
MatrixPartitioned::nnu()
------------------------
Return the number of unknown DOFs.
MatrixPartitioned::nnp()
------------------------
Return the number of prescribed DOFs.
MatrixPartitioned::dofs()
-------------------------
Return the DOF-numbers per node [nnode, ndim].
MatrixPartitioned::iiu()
------------------------
Return the unknown DOF-numbers per node [nnu].
MatrixPartitioned::iip()
------------------------
Return the prescribed DOF-numbers per node [nnp].
MatrixPartitioned::assemble(...)
--------------------------------
Assemble matrix from element matrices stored as "elemmat".
MatrixPartitioned::solve(...)
-----------------------------
Solve linear system.
MatrixPartitioned::solve_u(...)
-------------------------------
Solve linear system.
MatrixPartitioned::reaction(...)
--------------------------------
Compute reaction forces (part of "b" that corresponds to "x_p").
MatrixPartitioned::reaction_p(...)
----------------------------------
Compute reaction forces (part of "b" that corresponds to "x_p").
MatrixPartitionedTyings
=======================
Partitioned matrix definition with nodal tyings.
.. note::
A solver has to be chosen, see :ref:`linear_solver`.
MatrixPartitionedTyings::nelem()
--------------------------------
Return the number of elements.
MatrixPartitionedTyings::nne()
------------------------------
Return the number of nodes-per-element.
MatrixPartitionedTyings::nnode()
--------------------------------
Return the number of nodes.
MatrixPartitionedTyings::ndim()
-------------------------------
Return the number of dimensions.
MatrixPartitionedTyings::ndof()
-------------------------------
Return the number of DOFs.
MatrixPartitionedTyings::nnu()
------------------------------
Return the number of unknown DOFs.
MatrixPartitionedTyings::nnp()
------------------------------
Return the number of prescribed DOFs.
MatrixPartitionedTyings::nni()
------------------------------
Return the number of independent DOFs.
MatrixPartitionedTyings::nnd()
------------------------------
Return the number of dependent DOFs.
MatrixPartitionedTyings::dofs()
-------------------------------
Return the DOF-numbers per node [nnode, ndim].
MatrixPartitionedTyings::iiu()
------------------------------
Return the unknown DOF-numbers per node [nnu].
MatrixPartitionedTyings::iip()
------------------------------
Return the prescribed DOF-numbers per node [nnp].
MatrixPartitionedTyings::iii()
------------------------------
Return the independent DOF-numbers per node [nni].
MatrixPartitionedTyings::iid()
------------------------------
Return the dependent DOF-numbers per node [nnd].
MatrixPartitionedTyings::assemble(...)
--------------------------------------
Assemble matrix from element matrices stored as "elemmat".
MatrixPartitionedTyings::solve(...)
-----------------------------------
Solve linear system.
MatrixPartitionedTyings::solve_u(...)
-------------------------------------
Solve linear system.
MatrixDiagonal
==============
Diagonal matrix definition.
MatrixDiagonal::nelem()
-----------------------
Return the number of elements.
MatrixDiagonal::nne()
---------------------
Return the number of nodes-per-element.
MatrixDiagonal::nnode()
-----------------------
Return the number of nodes.
MatrixDiagonal::ndim()
----------------------
Return the number of dimensions.
MatrixDiagonal::ndof()
----------------------
Return the number of DOFs.
MatrixDiagonal::dofs()
----------------------
Return the DOF-numbers per node [nnode, ndim].
MatrixDiagonal::assemble(...)
-----------------------------
Assemble matrix from element matrices stored as "elemmat".
MatrixDiagonal::dot(...)
------------------------
Dot-product:
.. math::
b_i = A_{ij} x_j
MatrixDiagonal::solve(...)
--------------------------
Solve linear system.
MatrixDiagonal::AsDiagonal(...)
-------------------------------
Return matrix as diagonal matrix (column)
MatrixDiagonalPartitioned
=========================
Diagonal and partitioned matrix definition.
MatrixDiagonalPartitioned::nelem()
----------------------------------
Return the number of elements.
MatrixDiagonalPartitioned::nne()
--------------------------------
Return the number of nodes-per-element.
MatrixDiagonalPartitioned::nnode()
----------------------------------
Return the number of nodes.
MatrixDiagonalPartitioned::ndim()
---------------------------------
Return the number of dimensions.
MatrixDiagonalPartitioned::ndof()
---------------------------------
Return the number of DOFs.
MatrixDiagonalPartitioned::nnu()
--------------------------------
Return the number of unknown DOFs.
MatrixDiagonalPartitioned::nnp()
--------------------------------
Return the number of prescribed DOFs.
MatrixDiagonalPartitioned::dofs()
---------------------------------
Return the DOF-numbers per node [nnode, ndim].
MatrixDiagonalPartitioned::iiu()
--------------------------------
Return the unknown DOF-numbers per node [nnu].
MatrixDiagonalPartitioned::iip()
--------------------------------
Return the prescribed DOF-numbers per node [nnp].
MatrixDiagonalPartitioned::assemble(...)
----------------------------------------
Assemble matrix from element matrices stored as "elemmat".
MatrixDiagonalPartitioned::dot(...)
-----------------------------------
Dot-product:
.. math::
b_i = A_{ij} x_j
MatrixDiagonalPartitioned::dot_u(...)
-------------------------------------
Dot-product:
.. math::
b_i = A_{ij} x_j
MatrixDiagonalPartitioned::dot_p(...)
-------------------------------------
Dot-product:
.. math::
b_i = A_{ij} x_j
MatrixDiagonalPartitioned::solve(...)
-------------------------------------
Solve linear system.
MatrixDiagonalPartitioned::solve_u(...)
---------------------------------------
Solve linear system.
MatrixDiagonalPartitioned::reaction(...)
----------------------------------------
Compute reaction forces (part of "b" that corresponds to "x_p").
MatrixDiagonalPartitioned::reaction_p(...)
------------------------------------------
Compute reaction forces (part of "b" that corresponds to "x_p").
MatrixDiagonalPartitioned::AsDiagonal(...)
------------------------------------------
Return matrix as diagonal matrix (column)
.. _linear_solver:
Linear solver
=============
The classes ``GooseFEM:::MatrixPartitioned`` and ``GooseFEM:::MatrixPartitionedTyings`` make use of a library to solver the linear system (stored as a sparse matrix). In particular the Eigen library and its plug-ins are used. To use the library's default solver:
.. code-block:: cpp
#include <Eigen/Eigen>
#include <GooseFEM/GooseFEM.h>
int main()
{
...
GooseFEM::MatrixPartitioned<> K(...);
...
return 0;
}
The default solver can, however, be quite slow. Therefore Eigen has quite some `plug-ins <http://eigen.tuxfamily.org/dox/group__TopicSparseSystems.html>`_ for the solver. GooseFEM allows the use of Eigen's Sparse Solver Concept to use such plug-ins. For example, to use SuiteSparse:
.. code-block:: cpp
#include <Eigen/Eigen>
#include <Eigen/CholmodSupport>
#include <GooseFEM/GooseFEM.h>
int main()
{
...
GooseFEM::MatrixPartitioned<Eigen::CholmodSupernodalLLT<Eigen::SparseMatrix<double>>> K(...);
...
return 0;
}
.. todo::
1. `Download SuiteSparse <http://faculty.cse.tamu.edu/davis/suitesparse.html>`_.
2. Extract the downloaded ``SuiteSparse-X.X.X.tar.gz```.
3. Compile the library by:
.. code-block:: bash
cd /path/to/SuiteSparse
make
.. code-block:: bash
clang++ -I/path/to/include/eigen3 -I/path/to/lapack/include -L/path/to/lapack/lib -I/path/to/openblas/include -L/path/to/openblas/lib -lopenblas -I/path/to/SuiteSparse/include -L/path/to/SuiteSparse/lib -lumfpack -lamd -lcholmod -lsuitesparseconfig -lm -std=c++14 -Wall -Wextra -pedantic -march=native -O3 -o example example.cpp
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