Functions
void	mim_ip_span_nullspace (int nf, int nconn, int d, double C, double A, double X, double work, int lwork)

void	mim_ip_linpress_exact (int nf, int nconn, int d, double vol, double K, double N, double Binv, double work, int lwork)

void	mim_ip_simple (int nf, int nconn, int d, double v, double K, double C, double A, double N, double Binv, double work, int lwork)

void	mim_ip_simple_all (int ncells, int d, int max_ncf, int pconn, int conn, int fneighbour, double fcentroid, double fnormal, double farea, double ccentroid, double cvol, double perm, double Binv)

void	mim_ip_compute_gpress (int nc, int d, const double grav, const int pconn, const int conn, const double fcentroid, const double ccentroid, double gpress)

void	mim_ip_mobility_update (int nc, const int pconn, const double totmob, const double Binv0, double Binv)

void	mim_ip_density_update (int nc, const int pconn, const double omega, const double gpress0, double gpress)

Detailed Description

Routines to assist mimetic discretisations of the flow equation.

Function Documentation

void mim_ip_compute_gpress	(	int	nc,
		int	d,
		const double *	grav,
		const int *	pconn,
		const int *	conn,
		const double *	fcentroid,
		const double *	ccentroid,
		double *	gpress
	)

Compute local, static gravity pressure contributions to Darcy flow equation discretised using a mimetic finite-difference method.

The pressure contribution of local face $i$ in cell $c$ is

$\mathit{gpress}_{\mathit{pconn}_c + i} = \vec{g}\cdot (\Bar{x}_{\mathit{conn}_{\mathit{pconn}_c + i}} - \Bar{x}_c)$

Parameters

[in]	nc	Number of cells.
[in]	d	Number of physcial dimensions.
[in]	grav	Gravity vector. Array of size `d`.
[in]	pconn	Start pointers of cell-to-face topology mapping.
[in]	conn	Actual cell-to-face topology mapping.
[in]	fcentroid	Face centroids.
[in]	ccentroid	Cell centroids.
[out]	gpress	Gravity pressure result. Array of size at least `pconn[nc]`.

void mim_ip_density_update	(	int	nc,
		const int *	pconn,
		const double *	omega,
		const double *	gpress0,
		double *	gpress
	)

Incorporate effects of multiple fluid phases into existing, local, static mimetic discretisations of gravity pressure.

Specifically, update the result of mim_ip_compute_gpress() according to the rule

$\Tilde{G}_{\mathit{pconn}_c + i} = \omega_c\cdot G_{\mathit{pconn}_c + i}, \quad i=\mathit{pconn}_c, \dots, \mathit{pconn}_{c+1}-1, \quad c=0,\dots,\mathit{nc}-1$

in which $\omega_c = (\sum_\alpha \lambda_{\alpha,c} \rho_\alpha)/\lambda_{T,c}$ and $\Tilde{G}$ denotes the result of function mim_ip_compute_gpress().

Parameters

[in]	nc	Number of cells.
[in]	pconn	Start pointers of cell-to-face topology mapping.
[in]	omega	Sum of phase densities weighted by fractional flow.
[in]	gpress0	Result of mim_ip_compute_gpress().
[out]	gpress	Gravity pressure incorporating effects of multiple fluid phases.

void mim_ip_linpress_exact	(	int	nf,
		int	nconn,
		int	d,
		double	vol,
		double *	K,
		double *	N,
		double *	Binv,
		double *	work,
		int	lwork
	)

Form (inverse) mimetic inner product that reproduces linear pressure drops (constant velocity) on general polyhedral cells.

Specifically

$B^{-1} = \frac{1}{v} \big(NKN^\mathsf{T} + \frac{6t}{d}\,X\big)$

in which $t = \operatorname{tr}(K)$ is the trace of $K$ and $X$ is the result of function mim_ip_span_nullspace().

Parameters

[in]	nf	Number of faces connected to single grid cell.
[in]	nconn	Total number of grid cell connections. Typically equal to `nf`.
[in]	d	Number of physical dimensions. Assumed less than four.
[in]	vol	Cell volume.
[in]	K	Permeability. A $d\times d$ matrix in column major (Fortran) order.
[in]	N	Normal vectors. An $\mathit{nf}\times d$ matrix in column major (Fortran) order.
[in,out]	Binv	Inverse inner product result. An $\mathit{nconn}\times\mathit{nconn}$ matrix in column major format. On input, the result of mim_ip_span_nullspace(). On output, the upper left $\mathit{nf}\times\mathit{nf}$ sub-matrix will be overwritten with $B^{-1}$ .
[in,out]	work	Scratch array of size at least `nf * d`.
[in]	lwork	Actual size of scratch array.

void mim_ip_mobility_update	(	int	nc,
		const int *	pconn,
		const double *	totmob,
		const double *	Binv0,
		double *	Binv
	)

Incorporate effects of multiple phases in mimetic discretisation of flow equations.

Specifically, update the (inverse) inner products $B^{-1}$ previously computed using function mim_ip_linpress_exact() according to the rule

$\Tilde{B}_c^{-1} = \frac{1}{\lambda_{T,c}} B_c^{-1}, \quad i=0,\dots,\mathit{nc}-1$

in which $B_c^{-1}$ denotes the result of mim_ip_linpress_exact() for cell $c$ and $\lambda_{T,c}$ denotes the total mobility of cell $c$ .

Parameters

[in]	nc	Number of cells.
[in]	pconn	Start pointers of cell-to-face topology mapping.
[in]	totmob	Total mobility for all cells. Array of size `nc`.
[in]	Binv0	Inverse inner product results for all cells.
[out]	Binv	Inverse inner product results incorporating effects of multiple fluid phases.

void mim_ip_simple	(	int	nf,
		int	nconn,
		int	d,
		double	v,
		double *	K,
		double *	C,
		double *	A,
		double *	N,
		double *	Binv,
		double *	work,
		int	lwork
	)

Convenience wrapper around the function pair mim_ip_span_nullspace() and mim_ip_linpress_exact().

Parameters

[in]	nf	Number of faces connected to single grid cell.
[in]	nconn	Total number of grid cell connections. Typically equal to `nf`.
[in]	d	Number of physical dimensions. Assumed less than four.
[in]	v	Cell volume.
[in]	K	Permeability. A $d\times d$ matrix in column major (Fortran) order.
[in,out]	C	Centroid vectors. Specifically, $c_{ij} = \Bar{x}_{ij} - \Bar{x}_{cj}$ . Array of size $\mathit{nf}\times d$ in column major (Fortran) order. Contents destroyed on output.
[in]	A	Interface areas.
[in]	N	Outward normal vectors. An $\mathit{nf}\times d$ matrix in column major (Fortran) order.
[out]	Binv	Inverse inner product result. An $\mathit{nconn}\times\mathit{nconn}$ matrix in column major format. On output, the upper left $\mathit{nf}\times\mathit{nf}$ sub-matrix will be overwritten with $B^{-1}$ defined by function mim_ip_linpress_exact().
[in,out]	work	Scratch array of size at least `nf * d`.
[in]	lwork	Actual size of scratch array.

void mim_ip_simple_all	(	int	ncells,
		int	d,
		int	max_ncf,
		int *	pconn,
		int *	conn,
		int *	fneighbour,
		double *	fcentroid,
		double *	fnormal,
		double *	farea,
		double *	ccentroid,
		double *	cvol,
		double *	perm,
		double *	Binv
	)

Compute the mimetic inner products given a grid and cell-wise permeability tensors.

This function applies mim_ip_simple() to all specified cells.

Parameters

[in]	ncells	Number of cells.
[in]	d	Number of physical dimensions.
[in]	max_ncf	Maximum number of connections (faces) of any individual cell.
[in]	pconn	Start pointers of cell-to-face topology mapping.
[in]	conn	Actual cell-to-face topology mapping.
[in]	fneighbour	Face-to-cell mapping.
[in]	fcentroid	Face centroids.
[in]	fnormal	Face normals.
[in]	farea	Face areas.
[in]	ccentroid	Cell centroids.
[in]	cvol	Cell volumes.
[in]	perm	Cell permeability.
[out]	Binv	Inverse inner product result. Must point to an array of size at least $\sum_c n_c^2$ when denotes the number of connections (faces) of cell .

void mim_ip_span_nullspace	(	int	nf,
		int	nconn,
		int	d,
		double *	C,
		double *	A,
		double *	X,
		double *	work,
		int	lwork
	)

Form linear operator to span the null space of the normal vectors of a grid cell.

Specifically,

$\begin{aligned} X &= \operatorname{diag}(A) (I - QQ^\mathsf{T}) \operatorname{diag}(A), \\ Q &= \operatorname{orth}(\operatorname{diag}(A) C) \end{aligned}$

in which $\operatorname{orth}(M)$ denotes an orthonormal basis for the colum space (range) of the matrix $M$ , represented as a matrix.

Parameters

[in]	nf	Number of faces connected to single grid cell.
[in]	nconn	Total number of grid cell connections. Typically equal to `nf`.
[in]	d	Number of physical dimensions. Assumed less than four.
[in,out]	C	Centroid vectors. Specifically, $c_{ij} = \Bar{x}_{ij} - \Bar{x}_{cj}$ . Array of size $\mathit{nf}\times d$ in column major (Fortran) order. Contents destroyed on output.
[in]	A	Interface areas.
[out]	X	Null space linear operator. Array of size $\mathit{nconn}\times\mathit{nconn}$ in column major (Fortran) order. On output, the upper left $\mathit{nf}\times\mathit{nf}$ sub-matrix contains the required null space linear operator.
[out]	work	Scratch array of size at least `nconn`.
[in]	lwork	Actual size of scratch array.

Functions

Detailed Description

Function Documentation