Step 6 Computing stencils
In this tutorial we will learn how to gather a stencil consisting of data from the current element and its face neighbors.
You will find the code to this example in the tutorials/general/step6* files. The executable is named t8_step6_stencil.
In the last tutorials we learned how to create a forest, adapt it, pre-allocate element data arrays, activate the ghost layer and store custom data fields in .vtu files. In this tutorial we will start by performing all these operations in one go as shown in step 5. Then, when we have our forest and built a data array, we gather data for the local elements of our process. Next, we exchange the data values of the ghost elements and compute a stencil from face neighbors in each element. Finally, the output of several custom data fields is written to .vtu files.
In terms of interacting with t8code, this step mainly builds on the functionality already detailed in the steps 3 to 5. Thus, we focus on the new concept of accessing neighboring element data and doing computations with it by reference of simple finite difference computations.
For many scientific and industrial applications the access to neighboring data of an element is a crucial feature every mesh library has to provide. The collection of the element's data and its neighbors is called a stencil. Stencils can be seen as a compact data structure which allows for convenient and performant finite difference computations yielding numerical approximations of gradients, curvatures, curls, etc. of a data field. An exemplary visualization of a 3x3 cross stencil is shown in the figure below. The uniform grid is divided into three parallelization zones with the already introduced ghost layers (cf. step 5) at their interfaces.
The data field in our specific case is the physical quantity "height" pinned at the midpoint of each mesh element. As an example, we compute the schlieren and a rough measure for the curvature which we define as the norms of the central difference formulas shown in the figure above. A code snippet doing these calculations for our uniform 2D grid might look like this:
double stencil[3][3] = {/* ... */}; /* A 3x3 matrix with arbitrary height data. */
double dx = /* ... */; /* Length of an element in x-direction. */
double dy = /* ... */; /* Length of an element in y-direction. */
/* Approximation of the first derivates in x- and y-direction. */
double xslope = 0.5*(stencil[2][1] - stencil[0][1])/dx;
double yslope = 0.5*(stencil[1][2] - stencil[1][0])/dy;
/* Approximation of the second derivates in x- and y-direction. */
double xcurve = (stencil[2][1] - 2.0*stencil[1][1] + stencil[0][1])/(dx*dx);
double ycurve = (stencil[1][2] - 2.0*stencil[1][1] + stencil[1][0])/(dy*dy);
/* Stores the results in the `element_data` array. `current_index` is the
* position of the element in the space-filling curve. */
element_data[current_index].schlieren = sqrt(xslope*xslope + yslope*yslope);
element_data[current_index].curvature = sqrt(xcurve*xcurve + ycurve*ycurve);Note, the actual implementation in the demo also accounts for adapted meshes with stencils crossing different refinement levels. This makes the code more involved which was left out of this wiki article for the sake of clarity.
The gathering of the stencil data is described in the next section.
t8code provides an API function to access neighboring elements at a given face.
The excerpt from t8_forest.h reads:
/** Compute the leaf face neighbors of a forest.
* \param [in] forest The forest. Must have a valid ghost layer.
* \param [in] ltreeid A local tree id.
* \param [in] leaf A leaf in tree \a ltreeid of \a forest.
* \param [out] neighbor_leafs Unallocated on input. On output the neighbor
* leafs are stored here.
* \param [in] face The index of the face across which the face neighbors
* are searched.
* \param [out] dual_face On output the face id's of the neighboring elements' faces.
* \param [out] num_neighbors On output the number of neighbor leafs.
* \param [out] pelement_indices Unallocated on input. On output the element indices
* of the neighbor leafs are stored here.
* 0, 1, ... num_local_el - 1 for local leafs and
* num_local_el , ... , num_local_el + num_ghosts - 1 for ghosts.
* \param [out] pneigh_scheme On output the eclass scheme of the neighbor elements.
* \param [in] forest_is_balanced True if we know that \a forest is balanced, false
* otherwise.
* \note If there are no face neighbors, then *neighbor_leafs = NULL, num_neighbors = 0,
* and *pelement_indices = NULL on output.
* \note Currently \a forest must be balanced.
* \note \a forest must be committed before calling this function.
*/
void t8_forest_leaf_face_neighbors (t8_forest_t forest,
t8_locidx_t ltreeid,
const t8_element_t *leaf,
t8_element_t
**pneighbor_leafs[],
int face,
int *dual_faces[],
int *num_neighbors,
t8_locidx_t
**pelement_indices,
t8_eclass_scheme_c
**pneigh_scheme,
int forest_is_balanced);A common usage pattern of above routine can be found in the function body of
t8_step6_compute_stencil in the demo source file tutorials/general/t8_step6_stencil.cxx.
For our 2D uniform grid from above the code is as follows. Remark: The decision, where to store the neighboring data in the stencil, is done via a switch case construct.
/* Fill the center of the stencil with the data of the current element. */
stencil[1][1] = element_data[current_index].height;
/* Loop over all faces of an element. */
int num_faces = eclass_scheme->t8_element_num_faces (element);
for (int iface = 0; iface < num_faces; iface++) {
int num_neighbors; /**< Number of neighbors for each face */
int *dual_faces; /**< The face indices of the neighbor elements */
t8_locidx_t *neighids; /**< Indices of the neighbor elements */
t8_element_t **neighbors; /*< Neighboring elements. */
t8_eclass_scheme_c *neigh_scheme; /*< Neighboring elements scheme. */
/* Collect all neighbors at the current face. */
t8_forest_leaf_face_neighbors (forest, itree, element,
&neighbors, iface,
&dual_faces,
&num_neighbors,
&neighids,
&neigh_scheme, 1);
/* Retrieve the 'height' of the face neighbor. */
double height = element_data[neighids[0]].height;
/* Fill in the neighbor information of the 3x3 stencil. */
switch (iface) {
case 0: // NORTH
stencil[0][1] = height;
break;
case 1: // SOUTH
stencil[2][1] = height;
break;
case 2: // WEST
stencil[1][0] = height;
break;
case 3: // EAST
stencil[1][2] = height;
break;
}
/* Free allocated memory. Never forget this! */
T8_FREE(neighbors);
T8_FREE(dual_faces);
T8_FREE(neighids);
}Once the stencil is filled the finite difference computation from the previous section can be applied. Above code is just an excerpt with some details left out, especially the treatment at non-conforming interfaces with more than one neighboring element. Please examine the demo code in order to get the full picture.
If t8code was compiled with MPI support the demo can be executed with for example three processes.
$ mpirun -n 3 ${INSTALL_DIR_OF_T8CODE}/bin/t8_step6_stencilThis results in three .vtu files and a meta .pvtu file.
$ ls
t8_step6_stencil.pvtu t8_step6_stencil_0000.vtu t8_step6_stencil_0001.vtu t8_step6_stencil_0002.vtu
Open t8_step6_stencil.pvtu in ParaView and visualize the height (left figure) and schlieren (right figure) data fields.
Schlieren plots are the norm of the gradient of a data field (in our case the height). For smoother pictures the inclined reader is encouraged to increase the initial refinement level, recompile and rerun the demo. The visualization of
the curvature approximation is left as an exercise for the reader.
