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GRASS Programmer's Manual
6.5.svn(2014)-r66266
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GRASS Programmer's Manual
6.5.svn(2014)-r66266
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Author: Soeren Gebbert
Overview
The GRASS partial differential equation library is designed to support the solution of PDE's within GRASS. Therefore it provides functions to create and solve linear equation systems. The library design is thread safe and supports threaded parallelism with OpenMP. The assembling of the linear equation systems is parallelized with OpenMP. The creation of a linear equation system can be done by using quadratic or sparse matrices. Sparse and quadratic matrices are supported by the iterative equation solvers of the gmath library, the direct equation solvers only support regular quadratic matrices.
To enable parallel computing, you will need a compiler with OpenMP capabilities and a computer with at least two cores or cpu's. Most proprietary compilers support OpenMP. The free gnu-C compiler supports OpenMP since version 4.2. More information about OpenMP are available at http://www.openmp.org .
Based on the finite volume discretization for structured grids, groundwater flow and solute transport in 2d and 3d are implemented. It is IMHO easy to add any partial differential equation which can be solved with the finite volume or the finite differences method based on structured grids. The basis for the discretisation is the geometrical structure of raster and volume maps.
Currently only planimetric projections are supported for numerical calculations.
The gpde library provides an easy to handle interface for reading raster and volume data into specific data arrays (in memory) and for writing data arrays to raster and volume maps. These data arrays supports all raster and volume datatypes, overlapping boundaries and provide a sophisticated null value support and array access functions. Basic mathematical operations are available for the arrays.
The principle to create and solve a linear equation systems with this library:
1.) Read all needed raster/volume data into the memory
- use the the two and three dimensional data arrays N_array_2d and N_array_3d
to manage the data
2.) Assemble the linear equation system
- use the sparse matrix structure to save lots of memory and cpu time
3.) Solve the linear equation system with the gauss, lu, cg, pcg or bicgstab method
- always prefer the iterative krylov-space (cg, pcg and bicgstab) methods for solving
- the available direct solvers don't support sparse matrices
4.) Write the result back as raster/volume map
The modules r.gwflow and r3.gwflow are examples how to use the gpde library.
All raster and volume maps which are used to create a linear equation system must be loaded completely into the memory.
Therefore 2d and 3d data structures are implemented to support this principle:
This structure manages two dimensional data
typedef struct
{
int type; /* which raster type CELL_TYPE, FCELL_TYPE, DCELL_TYPE */
int rows, cols;
int rows_intern, cols_intern;
int offset; /*number of cols/rows offset at each boundary */
CELL *cell_array;
FCELL *fcell_array;
DCELL *dcell_array;
} N_array_2d;
For performance reasons the data arrays are stored as a one dimensional array internally.
Use the following functions for memory allocation and value handling:
Memory allocation
N_array_2d *N_alloc_array_2d(int rows, int cols, int offset, int type);
void N_free_array_2d(N_array_2d * data_array);
Get the type of the array
int N_get_array_2d_type(N_array_2d * array2d);
To access the 2d arrays use the following functions for reading and writing of data values
void N_get_array_2d_value(N_array_2d * array2d, int row, int col, void *value);
CELL N_get_array_2d_c_value(N_array_2d * array2d, int row, int col);
FCELL N_get_array_2d_f_value(N_array_2d * array2d, int row, int col);
DCELL N_get_array_2d_d_value(N_array_2d * array2d, int row, int col);
void N_put_array_2d_value(N_array_2d * array2d, int row, int col, char *value);
void N_put_array_2d_c_value(N_array_2d * array2d, int row, int col, CELL value);
void N_put_array_2d_f_value(N_array_2d * array2d, int row, int col, FCELL value);
void N_put_array_2d_d_value(N_array_2d * array2d, int row, int col, DCELL value);
To copy one array to another use this function
void N_copy_array_2d(N_array_2d * source, N_array_2d * target);
Print the content of the array to stdout
void N_print_array_2d (N_array_2d * data);
Compute the norm of two arrays
double N_norm_array_2d (N_array_2d * array1, N_array_2d * array2, int type);
Performe some basic mathematical operations with two arrays
N_array_2d * N_math_array_2d (N_array_2d * array1, N_array_2d * array2, N_array_2d * result, int type);
Convert all null values to zero
int N_convert_array_2d_null_to_zero (N_array_2d * a);
Read a raster map into the memory
N_array_2d * N_read_rast_to_array_2d (char *name, N_array_2d * array);
Write a raster map to the disk
void N_write_array_2d_to_rast (N_array_2d * array, char *name);
Example implementation:
The GRASS module r.gwflow implements numerical calculation program for transient, confined and unconfined groundwater flow in two dimensions.
typedef struct
{
This structure manages three dimensional data
int type; /* which raster type FCELL_TYPE, DCELL_TYPE */
int rows, cols, depths;
int rows_intern, cols_intern, depths_intern;
int offset; /*number of cols/rows/depths offset at each boundary */
float *fcell_array;
double *dcell_array;
} N_array_3d;
For performance reasons the data arrays are stored as a one dimensional array internally.
The following functions should be used for memory allocation and value handling:
N_array_3d *N_alloc_array_3d(int depths, int rows, int cols, int offset, int type);
void N_free_array_3d(N_array_3d * data_array);
int N_get_array_3d_type(N_array_3d * array);
To access the 3d arrays use the following functions for reading and writing of data values
void N_get_array_3d_value(N_array_3d * array3d, int depth, int row, int col, void *value);
float N_get_array_3d_f_value(N_array_3d * array3d, int depth, int row, int col);
double N_get_array_3d_d_value(N_array_3d * array3d, int depth, int row, int col);
void N_put_array_3d_value(N_array_3d * array3d, int depth, int row, int col, char *value);
void N_put_array_3d_f_value(N_array_3d * array3d, int depth, int row, int col, float value);
void N_put_array_3d_d_value(N_array_3d * array3d, int depth, int row, int col, double value);
To copy one array to another use this function
void N_copy_array_3d(N_array_3d * source, N_array_3d * target);
Print the content of the array to stdout
void N_print_array_3d (N_array_3d * data);
Compute the norm of two arrays
double N_norm_array_3d (N_array_3d * array1, N_array_3d * array2, int type);
Performe some basic mathematical operations with two arrays
N_array_3d * N_math_array_3d (N_array_3d * array1, N_array_3d * array2, N_array_3d * result, int type);
Convert all null values to zero
int N_convert_array_3d_null_to_zero (N_array_3d * a);
Read a volume map into the memory
N_array_3d * N_read_rast3d_to_array_3d (char *name, N_array_3d * array, int mask);
Write a volume map to the disk
void N_write_array_3d_to_rast3d (N_array_3d * array, char *name, int mask);
Example implementation:
The GRASS module r3.gwflow implements numerical calculation program for transient, confined groundwater flow in three dimensions.
To make entries in the linear equation system, a special structure must be provided. Currently implemented structures includes the 5 point and 7 point star scheme:
Matrix entries for the mass balance of a 5 star system
The entries are center, east, west, north, south and the
right side vector b of Ax = b. This system is typical used in 2d.
N
|
W-- C --E
|
S
Matrix entries for the mass balance of a 7 star system
The entries are center, east, west, north, south, top, bottom and the
right side vector b of Ax = b. This system is typical used in 3d.
T N
|/
W-- C --E
/|
S B
Matrix entries for the mass balance of a 9 star system
The entries are center, east, west, north, south, north-east, south-east,
north-wast, south-west and the
right side vector b of Ax = b. This system is typical used in 2d.
NW N NE
\ | /
W-- C --E
/ | \
SW S SE
typedef struct
{
int type;
int count;
double W, E, N, S, C, T, B, NE, NW, SE, SW, V;
} N_data_star;
The following functions should be used to create and handle the N_data_star structures:
Memory allocation
N_data_star *N_alloc_5star(void);
N_data_star *N_alloc_7star(void);
N_data_star *N_alloc_9star(void);
Memory allocation with initialisation
N_data_star *N_create_5star(double C, double W, double E, double N, double S, double V);
N_data_star *N_create_7star(double C, double W, double E, double N, double S, double T, double B, double V);
N_data_star *N_create_9star(double C, double W, double E, double N, double S, double NW, double SW, double NE, double SE, double V);
Setting the callback function which fills the 5 or 7 point structure for every row of the linear equation system (eg: for each cell of the raster or volume map)
void N_set_les_callback_3d_func(N_les_callback_3d * data, N_data_star * (*callback_func_3d) ());
void N_set_les_callback_2d_func(N_les_callback_2d * data, N_data_star * (*callback_func_2d) ());
Allocation the callback structure for 2d and 3d
N_les_callback_3d *N_alloc_les_callback_3d(void);
N_les_callback_2d *N_alloc_les_callback_2d(void);
Assemble the linear equation system in 2d or 3d
N_les *N_assemble_les_3d(int les_type, N_geom_data * geom, N_array_3d * status, N_array_3d * start_val, void *data, N_les_callback_3d * callback);
N_les *N_assemble_les_2d(int les_type, N_geom_data * geom, N_array_2d * status, N_array_2d * start_val, void *data, N_les_callback_2d * callback);
templates for callback functions
N_data_star *N_callback_template_3d(void *data, N_geom_data * geom, int depth, int row, int col);
N_data_star *N_callback_template_2d(void *data, N_geom_data * geom, int row, int col);
Groundwater flow in 2 and 3 dimensions are implemented.
This data structure contains all data needed to compute the groundwater mass balance in three dimension
typedef struct
{
N_array_3d *phead; /*!piezometric head */
N_array_3d *phead_start; /*!start conditions */
N_array_3d *hc_x; /*!x part of the hydraulic conductivity tensor */
N_array_3d *hc_y; /*!y part of the hydraulic conductivity tensor */
N_array_3d *hc_z; /*!z part of the hydraulic conductivity tensor */
N_array_3d *q; /*!sources and sinks */
N_array_2d *r; /*!recharge at the top of the gw leayer */
N_array_3d *s; /*!specific yield */
N_array_3d *nf; /*!effective porosity */
N_array_3d *status; /*!active/inactive/dirichlet cell status */
double dt; /*!calculation time */
} N_gwflow_data3d;
This data structure contains all data needed to compute the groundwater mass balance in two dimension
typedef struct
{
N_array_2d *phead; /*!piezometric head */
N_array_2d *phead_start; /*!start conditions */
N_array_2d *hc_x; /*!x part of the hydraulic conductivity tensor */
N_array_2d *hc_y; /*!y part of the hydraulic conductivity tensor */
N_array_2d *q; /*!sources and sinks */
N_array_2d *r; /*!recharge at the top of the gw leayer */
N_array_2d *s; /*!specific yield */
N_array_2d *nf; /*!effective porosity */
N_array_2d *top; /*!top surface of the quifer */
N_array_2d *bottom; /*!bottom of the aquifer */
N_array_2d *status; /*!active/inactive/dirichlet cell status */
double dt; /*!calculation time */
int gwtype; /*!Which type of groundwater, N_GW_CONFINED or N_GW_UNCONFIED */
} N_gwflow_data2d;
The callback to compute a les row entry in 3d dimesnions
N_data_star *N_callback_gwflow_3d (void *gwdata, N_geom_data * geom, int col, int row, int depth);
The callback to compute a les row entry in 2d dimesnions
N_data_star *N_callback_gwflow_2d (void *gwdata, N_geom_data * geom, int col, int row);
Allocation the 3d groundwater flow data structure
N_gwflow_data3d *N_alloc_gwflow_data3d (int cols, int rows, int depths);
Allocation the 2d groundwater flow data structure
N_gwflow_data2d *N_alloc_gwflow_data2d (int cols, int rows);
Releasing memory
void N_free_gwflow_data3d (N_gwflow_data3d * data);
Releasing memory
void N_free_gwflow_data2d (N_gwflow_data2d * data);
To handle geometrical data a special data structure was implemented.
Geometric information about the structured grid is stored in this structure
typedef struct
{
int planimetric; /*If the projection is not planimetric (0), the array calculation is different for each row*/
double *area; /* the vector of area values for non-planimetric projection for each row*/
int dim; /* 2 or 3*/
double dx;
double dy;
double dz;
double Az;
int depths;
int rows;
int cols;
} N_geom_data;
Use the follwoing functions to handle the geometric data structures:
Creating a N_geom_data structure
N_geom_data *N_alloc_geom_data (void);
Releasing memory
void N_free_geom_data (N_geom_data *geodata);
Initialize the N_geom_data structure with a G3D_Region
N_geom_data *N_init_geom_data_3d(G3D_Region * region3d, N_geom_data * geodata);
Initialize the N_geom_data structure with a 2d region
N_geom_data *N_init_geom_data_2d(struct Cell_head * region, N_geom_data * geodata);
Get the area of a cell in row
double N_get_geom_data_area_of_cell(N_geom_data * geom, int row);
Several mean calculation algorithms are implemented. Two versions of each algorithm are available working with two values or a vector of values.
double N_calc_arith_mean(double a, double b)
double N_calc_arith_mean_n(double *a, int size)
double N_calc_geom_mean(double a, double b)
double N_calc_geom_mean_n(double *a, int size)
double N_calc_harmonic_mean(double a, double b)
double N_calc_harmonic_mean_n(double *a, int size)
double N_calc_quad_mean(double a, double b)
double N_calc_quad_mean_n(double *a, int size)
Two methods for upwinding stabilization are implemented.
double N_full_upwinding(double vector, double distance, double D)
double N_exp_upwinding(double vector, double distance, double D)
To compute and manage gradient and vector field data, specific data structures with access and management functions are implemented. Gradient and vector field data is often needed in transport calculation like solute/heat transport or navier stokes equations. The following structures and functions provide a concient way to performe gradient and vector field calculations.
The gradient of one cell to there neighbours in each direction has the following components:
The two dimensional gradient consits of 4 values between the neighbour cells
______________
| | | |
| | | |
|----|-NC-|----|
| | | |
| WC EC |
| | | |
|----|-SC-|----|
| | | |
|____|____|____|
The three dimensional gradient consits of 6 values between the neighbour cells
| /
TC NC
|/
--WC-----EC--
/|
SC BC
/ |Gradient between the cells in X and Y direction
typedef struct {
double NC, SC, WC, EC;
} N_gradient_2d;
Gradient between the cells in X, Y and Z direction
typedef struct {
double NC, SC, WC, EC, TC, BC;
} N_gradient_3d;
The gradients between the neighbours of the center cell is needed for tensor calculation in the discretization of several partial differential equations.
Gradient in X direction between the cell neighbours ____ ____ ____ | | | | | NWN NEN | |____|____|____| | | | | | WN EN | |____|____|____| | | | | | SWS SES | |____|____|____| Gradient in Y direction between the cell neighbours ______________ | | | | | | | | |NWW-|-NC-|-NEE| | | | | | | | | |SWW-|-SC-|-SEE| | | | | |____|____|____| Gradient in Z direction between the cell neighbours /______________/ /| | | | | NWZ| NZ | NEZ| |____|____|____| /| | | | | WZ | CZ | EZ | |____|____|____| /| | | | | SWZ| SZ | SEZ| |____|____|____| /____/____/____/
Gradient between the cell neighbours in X direction
typedef struct {
double NWN, NEN, WC, EC, SWS, SES;
} N_gradient_neighbours_x;
Gradient between the cell neighbours in Y direction
typedef struct {
double NWW, NEE, NC, SC, SWW, SEE;
} N_gradient_neighbours_y;
Gradient between the cell neighbours in Z direction
typedef struct {
double NWZ, NZ, NEZ, WZ, CZ, EZ, SWZ, SZ, SEZ;
} N_gradient_neighbours_z;
Gradient between the cell neighbours in X and Y direction
typedef struct {
N_gradient_neighbours_x *x;
N_gradient_neighbours_y *y;
} N_gradient_neighbours_2d;
Gradient between the cell neighbours in X, Y and Z direction
typedef struct {
N_gradient_neighbours_x *xt; /*top values*/
N_gradient_neighbours_x *xc; /*center values*/
N_gradient_neighbours_x *xb; /*bottom values*/
N_gradient_neighbours_y *yt; /*top values*/
N_gradient_neighbours_y *yc; /*center values*/
N_gradient_neighbours_y *yb; /*bottom values*/
N_gradient_neighbours_z *zt; /*top-center values*/
N_gradient_neighbours_z *zb; /*bottom-center values*/
} N_gradient_neighbours_3d;
Two dimensional gradient field
typedef struct {
N_array_2d *x_array;
N_array_2d *y_array;
} N_gradient_field_2d;
Three dimensional gradient field
typedef struct {
N_array_3d *x_array;
N_array_3d *y_array;
N_array_3d *z_array;
} N_gradient_field_3d;Use the following functions for allocation and data handling:
Functions to handle a 2d gradient
N_gradient_2d * N_alloc_gradient_2d(void);
void N_free_gradient_2d(N_gradient_2d * grad);
N_gradient_2d * N_create_gradient_2d(double NC, double SC, double WC, double EC);
int N_copy_gradient_2d(N_gradient_2d * source, N_gradient_2d *target);
N_gradient_2d * N_get_gradient_2d(N_gradient_field_2d *field, N_gradient_2d * gradient, int col, int row);
Functions to handle a 2d gradient
N_gradient_3d * N_alloc_gradient_3d(void);
void N_free_gradient_3d(N_gradient_3d * grad);
N_gradient_3d * N_create_gradient_3d(double NC, double SC, double WC, double EC, double TC, double BC);
int N_copy_gradient_3d(N_gradient_3d * source, N_gradient_3d *target);
N_gradient_3d * N_get_gradient_3d(N_gradient_field_3d *field, N_gradient_3d * gradient, int col, int row, int depth);
Functions to handle gradient neighbours in x direction of the center cell
N_gradient_neighbours_x * N_alloc_gradient_neighbours_x(void);
void N_free_gradient_neighbours_x(N_gradient_neighbours_x *grad);
N_gradient_neighbours_x * N_create_gradient_neighbours_x(double NWN, double NEN, double WC, double EC, double SWS, double SES);
int N_copy_gradient_neighbours_x(N_gradient_neighbours_x * source, N_gradient_neighbours_x *target);
Functions to handle gradient neighbours in y direction of the center cell
N_gradient_neighbours_y * N_alloc_gradient_neighbours_y(void);
void N_free_gradient_neighbours_y(N_gradient_neighbours_y *grad);
N_gradient_neighbours_y * N_create_gradient_neighbours_y(double NWW, double NEE, double NC, double SC, double SWW, double SEE);
int N_copy_gradient_neighbours_y(N_gradient_neighbours_y * source, N_gradient_neighbours_y *target);
Functions to handle gradient neighbours in z direction of the center cell
N_gradient_neighbours_z * N_alloc_gradient_neighbours_z(void);
void N_free_gradient_neighbours_z(N_gradient_neighbours_z *grad);
N_gradient_neighbours_z * N_create_gradient_neighbours_z(double NWZ, double NZ, double NEZ, double WZ, double CZ, double EZ, double SWZ, double SZ, double SEZ);
int N_copy_gradient_neighbours_z(N_gradient_neighbours_z * source, N_gradient_neighbours_z *target);
Functions to handle a 2d gradient neighbour structure of the center cell
N_gradient_neighbours_2d * N_alloc_gradient_neighbours_2d(void);
void N_free_gradient_neighbours_2d(N_gradient_neighbours_2d *grad);
N_gradient_neighbours_2d * N_create_gradient_neighbours_2d(N_gradient_neighbours_x *x, N_gradient_neighbours_y *y);
int N_copy_gradient_neighbours_2d(N_gradient_neighbours_2d *source, N_gradient_neighbours_2d *target);
Functions to handle a 2d gradient neighbour structure of the center cell
N_gradient_neighbours_3d * N_alloc_gradient_neighbours_3d(void);
void N_free_gradient_neighbours_3d(N_gradient_neighbours_3d *grad);
int N_copy_gradient_neighbours_3d(N_gradient_neighbours_3d *source, N_gradient_neighbours_3d *target);
To compute and handle a 2d gradient field the following functions are implemented:
N_gradient_field_2d * N_alloc_gradient_field_2d(int cols, int rows);
void N_free_gradient_field_2d(N_gradient_field_2d *field);
int N_copy_gradient_field_2d(N_gradient_field_2d *source, N_gradient_field_2d *target);
The gradient is calculated between cells for each cell and direction. The creation of a 2d gradient field is based on the following scheme:
______________ | | | | | | | | |----|-NC-|----| | | | | | WC EC | | | | | |----|-SC-|----| | | | | |____|____|____| x - direction: r = 2 * weight[row][col]*weight[row][col + 1] / (weight[row][col]*weight[row][col + 1]) EC = r * (pot[row][col] - pot[row][col + 1])/dx y - direction: r = 2 * weight[row][col]*weight[row + 1][col] / (weight[row][col]*weight[row + 1][col]) SC = r * (pot[row][col] - pot[row + 1][col])/dy the values SC and EC are the values of the next row/col
N_gradient_field_2d * N_compute_gradient_field_2d(N_array_2d *pot, N_array_2d *weight_x, N_array_2d *weight_y, N_geom_data *geom, N_gradient_field_2d *gradfield);
The computation of the gradient vectors for each cell is based on this scheme
______________ | | | | | | | | |----|-NC-|----| | | | | | WC EC | | | | | |----|-SC-|----| | | | | |____|____|____| x vector component: x = (WC + EC) / 2 y vector component: y = (NC + SC) / 2
N_gradient_field_3d * N_alloc_gradient_field_3d(int cols, int rows, int depths);
void N_free_gradient_field_3d(N_gradient_field_3d *field);
int N_copy_gradient_field_3d(N_gradient_field_3d *source, N_gradient_field_3d *target);
The gradient is calculated between cells for each cell and direction. The creation of a 2d gradient field is based on the following scheme:
| /
TC NC
|/
--WC-----EC--
/|
SC BC
/ |
x - direction:
r = 2 * weight_x[depth][row][col]*weight_x[depth][row][col + 1] / (weight_X[depth][row][col]*weight_x[depth][row][col + 1])
EC = r * (pot[depth][row][col] - pot[depth][row][col + 1])/dx
y - direction:
r = 2 * weight_y[depth][row][col]*weight_y[depth][row + 1][col] / (weight_y[depth][row][col]*weight_y[depth][row + 1][col])
SC = r * (pot[depth][row][col] - pot[depth][row + 1][col])/dy
z - direction:
r = 2 * weight_z[depth][row][col]*weight_z[depth + 1][row][col] / (weight_z[depth][row][col]*weight_z[depth + 1][row][col])
TC = r * (pot[depth][row][col] - pot[depth + 1][row][col])/dy
the values BC, NC, WC are the values of the next depth/row/colN_gradient_field_3d *N_compute_gradient_field_3d(N_array_3d * pot, N_array_3d * weight_x, N_array_3d * weight_y, N_array_3d * weight_z, N_geom_data * geom, N_gradient_field_3d *gradfield)
Based on this storages scheme the gradient vector for each cell is calculated and stored in the provided N_array_3d structures
| /
TC NC
|/
--WC-----EC--
/|
SC BC
/ |
x vector component:
x = (WC + EC) / 2
y vector component:
y = (NC + SC) / 2
z vector component:
z = (TC + BC) / 2
1.8.5