/*! \page gpdelib GRASS partial differential equations Library (GPDE)

GRASS Partial Differential Equations Library (GPDE)

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. Most of the available solvers (expect the gauss seidel and jacobi solver) and the assembling of the linear equation systems are 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, 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.

There are plans to implement heat flow in two and three dimensions.

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. \section PDE The Creation of a lineare equation systems The principle to create and solve a linear equation systems with this library: \verbatim 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, jacobi, sor, cg or bicgstab method - always prefer the iterative krylov-space methods for solving - if the linear equation systems are in a bad condition try the jacobian solver - the available direct solvers don't support sparse matrices 4.) Write the result back as raster/volume map \endverbatim The following code shows how to implement this principle with the GRASS PDE lib: \verbatim /* *************************************************************** */ /* ****** calculate 2d dimensional groundwater flow ************** */ /* *************************************************************** */ int main() { int i, j; N_gwflow_data2d *data = NULL; /* data structure with the groundwater data */ N_geom_data *geom = NULL; /* geometry of the calculation area (region) */ N_les *les = NULL; /* the linear equation system structure */ N_les_callback_2d *call = NULL; /* the callback for the groundwater flow calulation */ /* allocate the callback structure */ call = N_alloc_les_callback_2d(); /* set the callback to groundwater flow calculation */ N_set_les_callback_2d_func(call, (*N_callback_gwflow_2d)); /* allocate the groundwater data structure with one million cells */ data = N_alloc_gwflow_data2d(1000, 1000); /* get the current region */ G_get_set_window(®ion); /* allocate and initiate the geometry structure for geometry and area calculation needed by the groundwater calculation callback */ geom = N_init_geom_data_2d(®ion, geom); /*fill the data arrays with raster maps*/ N_read_rast_to_array_2d("phead_map_name", data->phead); N_read_rast_to_array_2d("phead_map_name", data->phead_start); N_read_rast_to_array_2d("status_map_name", data->status); N_read_rast_to_array_2d("hc_x_map_name", data->hc_x); N_read_rast_to_array_2d("hc_y_map_name", data->hc_y); N_read_rast_to_array_2d("q_map_name", data->q); N_read_rast_to_array_2d("s_map_name", data->s); N_read_rast_to_array_2d("top_map_name", data->top); N_read_rast_to_array_2d("bottom_map_name", data->bottom); N_read_rast_to_array_2d("r_map_name", data->r); /* Set the inactive cells to zero, to assure a no flow boundary */ /* notice: the data arrays are of type DCELL, so the put_*_d_value functions are used */ for (y = 0; y < geom->rows; y++) { for (x = 0; x < geom->cols; x++) { stat = (int)N_get_array_2d_d_value(data->status, x, y); if (stat == N_CELL_INACTIVE) { /*only inactive cells */ N_put_array_2d_d_value(data->hc_x, x, y, 0); N_put_array_2d_d_value(data->hc_y, x, y, 0); N_put_array_2d_d_value(data->s, x, y, 0); N_put_array_2d_d_value(data->q, x, y, 0); } } } /*Assemble the sparse matrix */ les = N_assemble_les_2d(N_SPARSE_LES, geom, data->status, data->phead_start, (void *)data, call); /*Solve the linear equation system with the cg method*/ N_solver_cg(les, 100000, 0.1e-8); /* now copy the result from the x vector of the linear equation system into a data array, this function is not provided by the gpde lib, you have to write your own: take a look at r.gwflow */ copy_x_vect_to_data_array(les->x, data->phead, geom); /*write the x vector of the les back into a raster map*/ N_write_array_2d_to_rast(data->phead, "output_map_name"); /*free the memory*/ N_free_les(les); N_free_gwflow_data2d(data); N_free_geom_data(geom); G_free(call); return 0; } \endverbatim The assembling of lineare equation systems is based on a 5 point star for raster maps and a 7 point star for volume maps, implemented as finite volume/differences discretization.

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: \subsection rastermaps Raster maps \verbatim 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; \endverbatim 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);

\subsubsection highlevel Higher level array functions 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. \subsection g3dmaps Volume maps \verbatim 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; \endverbatim 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);

\subsubsection highlevel Higher level array functions 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. \subsection les Entries in the linear equation system 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: \verbatim 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; \endverbatim 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);

\subsection les functions to assemble the lineare equation system

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); \section solvers Available solvers \subsection direct_solvers Direct solvers Two direct solvers are implemented, the gauss elimination solver and the lu decomposition solver. The direct solvers only work with regular quadratic matrices, not with sparse matrices.

int #N_solver_gauss (N_les * les);

int #N_solver_lu (N_les * les); \subsection iterative_solvers Iterative solvers The iterative solvers work with regular quadartic and sparse matrices.

To solve symmetric and positive definite linear equation systems the iterative conjugated gradient method with additional diagonal preconditioning are implemented. int #N_solver_cg(N_les * les, int maxit, double error);

int #N_solver_pcg(N_les * les, int maxit, double error);

To solve unsymmetric non definite linear equation system the iterative BiCGSatb method is implemented

int #N_solver_bicgstab(N_les * les, int maxit, double error);

Additionally jacobi and Gauss-Seidel iterative solvers with relaxation are implemented

int #N_solver_jacobi (N_les * L, int maxit, double sor, double error);

int #N_solver_SOR (N_les * L, int maxit, double sor, double error); \section available_pdes Implemented PDE's 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 \verbatim 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; \endverbatim This data structure contains all data needed to compute the groundwater mass balance in two dimension \verbatim 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; \endverbatim

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);

\section geom_data Handling geometrical data To handle geometrical data a special data structure was implemented.
\verbatim 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; \endverbatim 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); \section mathtools Mathematical tools 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) \section gradient Calculating and managing gradient and vector field data 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: \verbatim 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 / | \endverbatim Gradient between the cells in X and Y direction \verbatim typedef struct { double NC, SC, WC, EC; } N_gradient_2d; \endverbatim Gradient between the cells in X, Y and Z direction \verbatim typedef struct { double NC, SC, WC, EC, TC, BC; } N_gradient_3d; \endverbatim The gradients between the neighbours of the center cell is needed for tensor calculation in the discretization of several partial differential equations. \verbatim 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| |____|____|____| /____/____/____/ \endverbatim Gradient between the cell neighbours in X direction \verbatim typedef struct { double NWN, NEN, WC, EC, SWS, SES; } N_gradient_neighbours_x; \endverbatim Gradient between the cell neighbours in Y direction \verbatim typedef struct { double NWW, NEE, NC, SC, SWW, SEE; } N_gradient_neighbours_y; \endverbatim Gradient between the cell neighbours in Z direction \verbatim typedef struct { double NWZ, NZ, NEZ, WZ, CZ, EZ, SWZ, SZ, SEZ; } N_gradient_neighbours_z; \endverbatim Gradient between the cell neighbours in X and Y direction \verbatim typedef struct { N_gradient_neighbours_x *x; N_gradient_neighbours_y *y; } N_gradient_neighbours_2d; \endverbatim Gradient between the cell neighbours in X, Y and Z direction \verbatim 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; \endverbatim Two dimensional gradient field \verbatim typedef struct { N_array_2d *x_array; N_array_2d *y_array; } N_gradient_field_2d; \endverbatim Three dimensional gradient field \verbatim typedef struct { N_array_3d *x_array; N_array_3d *y_array; N_array_3d *z_array; } N_gradient_field_3d; \endverbatim 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: \verbatim ______________ | | | | | | | | |----|-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 \endverbatim

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 \verbatim ______________ | | | | | | | | |----|-NC-|----| | | | | | WC EC | | | | | |----|-SC-|----| | | | | |____|____|____| x vector component: x = (WC + EC) / 2 y vector component: y = (NC + SC) / 2 \endverbatim

void #N_compute_gradient_field_components_2d(N_gradient_field_2d *field, N_array_2d *x_comp, N_array_2d *y_comp);

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: \verbatim | / 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/col \endverbatim

N_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 \verbatim | / 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 \endverbatim

N_compute_gradient_field_components_3d(N_gradient_field_3d * field, N_array_3d * x_comp, N_array_3d * y_comp, N_array_3d * z_comp) */