MatrixEquation Class Reference

The MatrixEquation class sets up the matrix equation from the system of equations found from the governing equations. More...

#include <matrixequation.hpp>

Public Member Functions
	~MatrixEquation ()
	Declared to avoid the inline compiler-generated default destructor.
void	fillCoefficientMatrixAndConstantsVector (const arma::uword nGridPoints, const arma::uword nEquationsAndVariables, const BoundaryConditions &boundaryConditions, const arma::cube &term_i, const arma::cube &term_ipp, const arma::mat &boundaryTerms)
	Fill in the coefficient matrix A (MatrixEquation::m_coefficients) and constants vector b (MatrixEquation::m_constants) in the matrix equation Ax = b. More...
arma::mat	solve (const arma::uword nGridPoints, const arma::uword nEquationsAndVariables, const BoundaryConditions &boundaryConditions) const
	Solve the matrix equation Ax = b. More...
const arma::sp_mat &	coefficients () const
	Get coefficient matrix A. For testing purposes.
const arma::vec &	constants () const
	Get constants vector b. For testing purposes.

Private Member Functions
arma::mat	reshapeSolverOutput (const arma::vec &x, const BoundaryConditions &boundaryConditions, const arma::uword nGridPoints, const arma::uword nVariables) const
	Reshape output from solving the matrix equation into a matrix containing flow, pressure and temperature as columns. More...
arma::vec	solveMatrixEquation () const
	Internal method that solves the matrix equation. More...

Private Attributes
arma::sp_mat	m_coefficients
	Coefficient matrix A.
arma::vec	m_constants
	Constants vector b.

Detailed Description

The MatrixEquation class sets up the matrix equation from the system of equations found from the governing equations.

The governing equations are discretized using a implicit backward difference method, as detailed in

• Gas composition tracking in transient pipeline flow, Chaczykowski et. al., Journal of Natural Gas Science and Engineering 55 (2018)

This class supports all combinations of inlet and outlet boundary conditions, so we can for example use flow at the inlet, pressure at the outlet, and temperature at the inlet, or any other combination. Also supports over-determined systems.

This class is designed for use together with a Discretizer instance, which sets up the individual elements of the matrix. MatrixEquation just fills in the matrix using results from Discretizer, and solves the matrix equation.

The matrix equation is solved using the sparse matrix solver arma::spsolve, which uses superlu internally. This solver only works for critically determined systems, not for over-determined systems. In that case the regular solver is used, and the sparse matrix has to be converted to a regular matrix.

Most of the elements in the matrix are zero, so we use sparse matrices. The non-zero elements are all located near the diagonal.

See also

Discretizer

Member Function Documentation

fillCoefficientMatrixAndConstantsVector()

void MatrixEquation::fillCoefficientMatrixAndConstantsVector	(	const arma::uword	nGridPoints,
		const arma::uword	nEquationsAndVariables,
		const BoundaryConditions &	boundaryConditions,
		const arma::cube &	term_i,
		const arma::cube &	term_ipp,
		const arma::mat &	boundaryTerms
	)

Fill in the coefficient matrix A (MatrixEquation::m_coefficients) and constants vector b (MatrixEquation::m_constants) in the matrix equation Ax = b.

Here term_i and term_ipp refer to the coefficients of the variables in the discretized governing equations (flow, pressure and temperature – \(y_i\) in general). See for example eq. (14)-(16) in Gas composition tracking in transient pipeline flow. These coefficients are calculated by the Discretizer class.

The boundary terms are the constant terms, the elements of the vector \(b\) in the matrix equation \(Ax = b\).

MatrixEquation supports all combinations of inlet and outlet boundary conditions, but giving more than 3 boundary conditions lead to an over-determined system, and giving less than 3 leads to a under-determined system.

See also

Discretizer::discretize()

Parameters

nGridPoints	Number of grid points
nEquationsAndVariables	Number of equations and variables
boundaryConditions	The boundary conditions
term_i	Matrix coefficients at point i (see above)
term_ipp	Matrix coefficients at point i+1 (see above)
boundaryTerms	The boundary terms (see above)

reshapeSolverOutput()

arma::mat MatrixEquation::reshapeSolverOutput ( const arma::vec &  x, const BoundaryConditions &  boundaryConditions, const arma::uword  nGridPoints, const arma::uword  nVariables  ) const

private

Reshape output from solving the matrix equation into a matrix containing flow, pressure and temperature as columns.

Reshapes the vector x, the solution of the matrix equation Ax = b, into a matrix with three columns. Also inserts the boundary conditions where at the correct locations.

Supports all combinations of inlet/outlet boundary settings.

Parameters

x	Solution of matrix equation
boundaryConditions	Boundary conditions
nGridPoints	Number of grid points
nVariables	Number of variables

Returns

Matrix containing flow, pressure and temperature columns.

solve()

arma::mat MatrixEquation::solve	(	const arma::uword	nGridPoints,
		const arma::uword	nEquationsAndVariables,
		const BoundaryConditions &	boundaryConditions
	)		const

Solve the matrix equation Ax = b.

This solves the matrix equation, after the coefficient matrix MatrixEquation::m_coefficients and constant vector MatrixEquation::m_coefficients has been filled in by fillCoefficientMatrixAndConstantsVector().

Parameters

nGridPoints	Number of grid points
nEquationsAndVariables	Number of equations and variables
boundaryConditions	The boundary conditions

Returns

The solution of the matrix equation (x), reshaped to contain flow, pressure and temperature in separate columns.

solveMatrixEquation()

arma::vec MatrixEquation::solveMatrixEquation ( ) const

private

Internal method that solves the matrix equation.

See also

solve()

Returns

Vector x, solution of the matrix equation Ax = b.

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

• D:/Code/transient-pipeline-flow-gh-pages/src/solver/matrixequation.hpp