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trirefine.py
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""" Mesh refinement for triangular grids. """ import numpy as np from matplotlib import cbook from matplotlib.tri.triangulation import Triangulation import matplotlib.tri.triinterpolate class TriRefiner: """ Abstract base class for classes implementing mesh refinement. A TriRefiner encapsulates a Triangulation object and provides tools for mesh refinement and interpolation. Derived classes must implement: - ``refine_triangulation(return_tri_index=False, **kwargs)`` , where the optional keyword arguments *kwargs* are defined in each TriRefiner concrete implementation, and which returns: - a refined triangulation, - optionally (depending on *return_tri_index*), for each point of the refined triangulation: the index of the initial triangulation triangle to which it belongs. - ``refine_field(z, triinterpolator=None, **kwargs)``, where: - *z* array of field values (to refine) defined at the base triangulation nodes, - *triinterpolator* is an optional `~matplotlib.tri.TriInterpolator`, - the other optional keyword arguments *kwargs* are defined in each TriRefiner concrete implementation; and which returns (as a tuple) a refined triangular mesh and the interpolated values of the field at the refined triangulation nodes. """ def __init__(self, triangulation): cbook._check_isinstance(Triangulation, triangulation=triangulation) self._triangulation = triangulation class UniformTriRefiner(TriRefiner): """ Uniform mesh refinement by recursive subdivisions. Parameters ---------- triangulation : `~matplotlib.tri.Triangulation` The encapsulated triangulation (to be refined) """ # See Also # -------- # :class:`~matplotlib.tri.CubicTriInterpolator` and # :class:`~matplotlib.tri.TriAnalyzer`. # """ def __init__(self, triangulation): TriRefiner.__init__(self, triangulation) def refine_triangulation(self, return_tri_index=False, subdiv=3): """ Compute an uniformly refined triangulation *refi_triangulation* of the encapsulated :attr:`triangulation`. This function refines the encapsulated triangulation by splitting each father triangle into 4 child sub-triangles built on the edges midside nodes, recursing *subdiv* times. In the end, each triangle is hence divided into ``4**subdiv`` child triangles. Parameters ---------- return_tri_index : bool, default: False Whether an index table indicating the father triangle index of each point is returned. subdiv : int, default: 3 Recursion level for the subdivision. Each triangle is divided into ``4**subdiv`` child triangles; hence, the default results in 64 refined subtriangles for each triangle of the initial triangulation. Returns ------- refi_triangulation : `~matplotlib.tri.Triangulation` The refined triangulation. found_index : int array Index of the initial triangulation containing triangle, for each point of *refi_triangulation*. Returned only if *return_tri_index* is set to True. """ refi_triangulation = self._triangulation ntri = refi_triangulation.triangles.shape[0] # Computes the triangulation ancestors numbers in the reference # triangulation. ancestors = np.arange(ntri, dtype=np.int32) for _ in range(subdiv): refi_triangulation, ancestors = self._refine_triangulation_once( refi_triangulation, ancestors) refi_npts = refi_triangulation.x.shape[0] refi_triangles = refi_triangulation.triangles # Now we compute found_index table if needed if return_tri_index: # We have to initialize found_index with -1 because some nodes # may very well belong to no triangle at all, e.g., in case of # Delaunay Triangulation with DuplicatePointWarning. found_index = np.full(refi_npts, -1, dtype=np.int32) tri_mask = self._triangulation.mask if tri_mask is None: found_index[refi_triangles] = np.repeat(ancestors, 3).reshape(-1, 3) else: # There is a subtlety here: we want to avoid whenever possible # that refined points container is a masked triangle (which # would result in artifacts in plots). # So we impose the numbering from masked ancestors first, # then overwrite it with unmasked ancestor numbers. ancestor_mask = tri_mask[ancestors] found_index[refi_triangles[ancestor_mask, :] ] = np.repeat(ancestors[ancestor_mask], 3).reshape(-1, 3) found_index[refi_triangles[~ancestor_mask, :] ] = np.repeat(ancestors[~ancestor_mask], 3).reshape(-1, 3) return refi_triangulation, found_index else: return refi_triangulation def refine_field(self, z, triinterpolator=None, subdiv=3): """ Refine a field defined on the encapsulated triangulation. Parameters ---------- z : 1d-array-like of length ``n_points`` Values of the field to refine, defined at the nodes of the encapsulated triangulation. (``n_points`` is the number of points in the initial triangulation) triinterpolator : `~matplotlib.tri.TriInterpolator`, optional Interpolator used for field interpolation. If not specified, a `~matplotlib.tri.CubicTriInterpolator` will be used. subdiv : int, default: 3 Recursion level for the subdivision. Each triangle is divided into ``4**subdiv`` child triangles. Returns ------- refi_tri : `~matplotlib.tri.Triangulation` The returned refined triangulation. refi_z : 1d array of length: *refi_tri* node count. The returned interpolated field (at *refi_tri* nodes). """ if triinterpolator is None: interp = matplotlib.tri.CubicTriInterpolator( self._triangulation, z) else: cbook._check_isinstance(matplotlib.tri.TriInterpolator, triinterpolator=triinterpolator) interp = triinterpolator refi_tri, found_index = self.refine_triangulation( subdiv=subdiv, return_tri_index=True) refi_z = interp._interpolate_multikeys( refi_tri.x, refi_tri.y, tri_index=found_index)[0] return refi_tri, refi_z @staticmethod def _refine_triangulation_once(triangulation, ancestors=None): """ Refine a `.Triangulation` by splitting each triangle into 4 child-masked_triangles built on the edges midside nodes. Masked triangles, if present, are also split, but their children returned masked. If *ancestors* is not provided, returns only a new triangulation: child_triangulation. If the array-like key table *ancestor* is given, it shall be of shape (ntri,) where ntri is the number of *triangulation* masked_triangles. In this case, the function returns (child_triangulation, child_ancestors) child_ancestors is defined so that the 4 child masked_triangles share the same index as their father: child_ancestors.shape = (4 * ntri,). """ x = triangulation.x y = triangulation.y # According to tri.triangulation doc: # neighbors[i, j] is the triangle that is the neighbor # to the edge from point index masked_triangles[i, j] to point # index masked_triangles[i, (j+1)%3]. neighbors = triangulation.neighbors triangles = triangulation.triangles npts = np.shape(x)[0] ntri = np.shape(triangles)[0] if ancestors is not None: ancestors = np.asarray(ancestors) if np.shape(ancestors) != (ntri,): raise ValueError( "Incompatible shapes provide for triangulation" ".masked_triangles and ancestors: {0} and {1}".format( np.shape(triangles), np.shape(ancestors))) # Initiating tables refi_x and refi_y of the refined triangulation # points # hint: each apex is shared by 2 masked_triangles except the borders. borders = np.sum(neighbors == -1) added_pts = (3*ntri + borders) // 2 refi_npts = npts + added_pts refi_x = np.zeros(refi_npts) refi_y = np.zeros(refi_npts) # First part of refi_x, refi_y is just the initial points refi_x[:npts] = x refi_y[:npts] = y # Second part contains the edge midside nodes. # Each edge belongs to 1 triangle (if border edge) or is shared by 2 # masked_triangles (interior edge). # We first build 2 * ntri arrays of edge starting nodes (edge_elems, # edge_apexes); we then extract only the masters to avoid overlaps. # The so-called 'master' is the triangle with biggest index # The 'slave' is the triangle with lower index # (can be -1 if border edge) # For slave and master we will identify the apex pointing to the edge # start edge_elems = np.tile(np.arange(ntri, dtype=np.int32), 3) edge_apexes = np.repeat(np.arange(3, dtype=np.int32), ntri) edge_neighbors = neighbors[edge_elems, edge_apexes] mask_masters = (edge_elems > edge_neighbors) # Identifying the "masters" and adding to refi_x, refi_y vec masters = edge_elems[mask_masters] apex_masters = edge_apexes[mask_masters] x_add = (x[triangles[masters, apex_masters]] + x[triangles[masters, (apex_masters+1) % 3]]) * 0.5 y_add = (y[triangles[masters, apex_masters]] + y[triangles[masters, (apex_masters+1) % 3]]) * 0.5 refi_x[npts:] = x_add refi_y[npts:] = y_add # Building the new masked_triangles; each old masked_triangles hosts # 4 new masked_triangles # there are 6 pts to identify per 'old' triangle, 3 new_pt_corner and # 3 new_pt_midside new_pt_corner = triangles # What is the index in refi_x, refi_y of point at middle of apex iapex # of elem ielem ? # If ielem is the apex master: simple count, given the way refi_x was # built. # If ielem is the apex slave: yet we do not know; but we will soon # using the neighbors table. new_pt_midside = np.empty([ntri, 3], dtype=np.int32) cum_sum = npts for imid in range(3): mask_st_loc = (imid == apex_masters) n_masters_loc = np.sum(mask_st_loc) elem_masters_loc = masters[mask_st_loc] new_pt_midside[:, imid][elem_masters_loc] = np.arange( n_masters_loc, dtype=np.int32) + cum_sum cum_sum += n_masters_loc # Now dealing with slave elems. # for each slave element we identify the master and then the inode # once slave_masters is identified, slave_masters_apex is such that: # neighbors[slaves_masters, slave_masters_apex] == slaves mask_slaves = np.logical_not(mask_masters) slaves = edge_elems[mask_slaves] slaves_masters = edge_neighbors[mask_slaves] diff_table = np.abs(neighbors[slaves_masters, :] - np.outer(slaves, np.ones(3, dtype=np.int32))) slave_masters_apex = np.argmin(diff_table, axis=1) slaves_apex = edge_apexes[mask_slaves] new_pt_midside[slaves, slaves_apex] = new_pt_midside[ slaves_masters, slave_masters_apex] # Builds the 4 child masked_triangles child_triangles = np.empty([ntri*4, 3], dtype=np.int32) child_triangles[0::4, :] = np.vstack([ new_pt_corner[:, 0], new_pt_midside[:, 0], new_pt_midside[:, 2]]).T child_triangles[1::4, :] = np.vstack([ new_pt_corner[:, 1], new_pt_midside[:, 1], new_pt_midside[:, 0]]).T child_triangles[2::4, :] = np.vstack([ new_pt_corner[:, 2], new_pt_midside[:, 2], new_pt_midside[:, 1]]).T child_triangles[3::4, :] = np.vstack([ new_pt_midside[:, 0], new_pt_midside[:, 1], new_pt_midside[:, 2]]).T child_triangulation = Triangulation(refi_x, refi_y, child_triangles) # Builds the child mask if triangulation.mask is not None: child_triangulation.set_mask(np.repeat(triangulation.mask, 4)) if ancestors is None: return child_triangulation else: return child_triangulation, np.repeat(ancestors, 4)