| 1 | """Classes implementing general 2D geometrical mesh of triangles. |
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| 2 | |
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| 3 | Ole Nielsen, Stephen Roberts, Duncan Gray, Christopher Zoppou |
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| 4 | Geoscience Australia, 2004 |
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| 5 | """ |
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| 6 | |
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| 7 | from general_mesh import General_mesh |
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| 8 | |
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| 9 | class Mesh(General_mesh): |
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| 10 | """Collection of triangular elements (purely geometric) |
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| 11 | |
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| 12 | A triangular element is defined in terms of three vertex ids, |
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| 13 | ordered counter clock-wise, |
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| 14 | each corresponding to a given coordinate set. |
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| 15 | Vertices from different elements can point to the same |
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| 16 | coordinate set. |
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| 17 | |
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| 18 | Coordinate sets are implemented as an N x 2 Numeric array containing |
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| 19 | x and y coordinates. |
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| 20 | |
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| 21 | |
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| 22 | To instantiate: |
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| 23 | Mesh(coordinates, triangles) |
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| 24 | |
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| 25 | where |
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| 26 | |
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| 27 | coordinates is either a list of 2-tuples or an Mx2 Numeric array of |
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| 28 | floats representing all x, y coordinates in the mesh. |
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| 29 | |
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| 30 | triangles is either a list of 3-tuples or an Nx3 Numeric array of |
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| 31 | integers representing indices of all vertices in the mesh. |
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| 32 | Each vertex is identified by its index i in [0, M-1]. |
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| 33 | |
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| 34 | |
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| 35 | Example: |
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| 36 | a = [0.0, 0.0] |
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| 37 | b = [0.0, 2.0] |
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| 38 | c = [2.0,0.0] |
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| 39 | e = [2.0, 2.0] |
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| 40 | |
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| 41 | points = [a, b, c, e] |
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| 42 | triangles = [ [1,0,2], [1,2,3] ] #bac, bce |
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| 43 | mesh = Mesh(points, triangles) |
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| 44 | |
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| 45 | #creates two triangles: bac and bce |
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| 46 | |
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| 47 | |
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| 48 | Mesh takes the optional third argument boundary which is a |
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| 49 | dictionary mapping from (element_id, edge_id) to boundary tag. |
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| 50 | The default value is None which will assign the default_boundary_tag |
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| 51 | as specified in config.py to all boundary edges. |
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| 52 | """ |
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| 53 | |
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| 54 | #FIXME: Maybe rename coordinates to points (as in a poly file) |
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| 55 | #But keep 'vertex_coordinates' |
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| 56 | |
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| 57 | #FIXME: Put in check for angles less than a set minimum |
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| 58 | |
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| 59 | def __init__(self, coordinates, triangles, boundary = None, |
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| 60 | tagged_elements = None, origin = None): |
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| 61 | """ |
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| 62 | Build triangles from x,y coordinates (sequence of 2-tuples or |
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| 63 | Mx2 Numeric array of floats) and triangles (sequence of 3-tuples |
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| 64 | or Nx3 Numeric array of non-negative integers). |
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| 65 | """ |
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| 66 | |
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| 67 | from Numeric import array, zeros, Int, Float, maximum, sqrt, sum |
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| 68 | |
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| 69 | General_mesh.__init__(self, coordinates, triangles, origin) |
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| 70 | |
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| 71 | N = self.number_of_elements |
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| 72 | |
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| 73 | #Allocate space for geometric quantities |
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| 74 | self.centroid_coordinates = zeros((N, 2), Float) |
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| 75 | |
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| 76 | self.radii = zeros(N, Float) |
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| 77 | |
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| 78 | self.neighbours = zeros((N, 3), Int) |
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| 79 | self.neighbour_edges = zeros((N, 3), Int) |
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| 80 | self.number_of_boundaries = zeros(N, Int) |
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| 81 | self.surrogate_neighbours = zeros((N, 3), Int) |
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| 82 | |
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| 83 | #Get x,y coordinates for all triangles and store |
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| 84 | V = self.vertex_coordinates |
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| 85 | |
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| 86 | #Initialise each triangle |
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| 87 | for i in range(N): |
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| 88 | #if i % (N/10) == 0: print '(%d/%d)' %(i, N) |
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| 89 | |
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| 90 | x0 = V[i, 0]; y0 = V[i, 1] |
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| 91 | x1 = V[i, 2]; y1 = V[i, 3] |
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| 92 | x2 = V[i, 4]; y2 = V[i, 5] |
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| 93 | |
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| 94 | #Compute centroid |
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| 95 | centroid = array([(x0 + x1 + x2)/3, (y0 + y1 + y2)/3]) |
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| 96 | self.centroid_coordinates[i] = centroid |
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| 97 | |
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| 98 | #Midpoints |
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| 99 | m0 = array([(x1 + x2)/2, (y1 + y2)/2]) |
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| 100 | m1 = array([(x0 + x2)/2, (y0 + y2)/2]) |
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| 101 | m2 = array([(x1 + x0)/2, (y1 + y0)/2]) |
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| 102 | |
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| 103 | #The radius is the distance from the centroid of |
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| 104 | #a triangle to the midpoint of the side of the triangle |
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| 105 | #closest to the centroid |
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| 106 | d0 = sqrt(sum( (centroid-m0)**2 )) |
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| 107 | d1 = sqrt(sum( (centroid-m1)**2 )) |
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| 108 | d2 = sqrt(sum( (centroid-m2)**2 )) |
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| 109 | |
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| 110 | self.radii[i] = min(d0, d1, d2) |
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| 111 | |
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| 112 | #Initialise Neighbours (-1 means that it is a boundary neighbour) |
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| 113 | self.neighbours[i, :] = [-1, -1, -1] |
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| 114 | |
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| 115 | #Initialise edge ids of neighbours |
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| 116 | #In case of boundaries this slot is not used |
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| 117 | self.neighbour_edges[i, :] = [-1, -1, -1] |
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| 118 | |
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| 119 | |
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| 120 | #Build neighbour structure |
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| 121 | self.build_neighbour_structure() |
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| 122 | |
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| 123 | #Build surrogate neighbour structure |
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| 124 | self.build_surrogate_neighbour_structure() |
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| 125 | |
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| 126 | #Build boundary dictionary mapping (id, edge) to symbolic tags |
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| 127 | self.build_boundary_dictionary(boundary) |
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| 128 | |
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| 129 | #Build tagged element dictionary mapping (tag) to array of elements |
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| 130 | self.build_tagged_elements_dictionary(tagged_elements) |
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| 131 | |
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| 132 | #Update boundary indices FIXME: OBSOLETE |
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| 133 | #self.build_boundary_structure() |
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| 134 | |
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| 135 | |
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| 136 | def __repr__(self): |
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| 137 | return 'Mesh: %d triangles, %d elements, %d boundary segments'\ |
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| 138 | %(self.coordinates.shape[0], len(self), len(self.boundary)) |
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| 139 | |
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| 140 | |
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| 141 | def build_neighbour_structure(self): |
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| 142 | """Update all registered triangles to point to their neighbours. |
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| 143 | |
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| 144 | Also, keep a tally of the number of boundaries for each triangle |
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| 145 | |
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| 146 | Postconditions: |
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| 147 | neighbours and neighbour_edges is populated |
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| 148 | number_of_boundaries integer array is defined. |
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| 149 | """ |
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| 150 | |
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| 151 | #Step 1: |
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| 152 | #Build dictionary mapping from segments (2-tuple of points) |
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| 153 | #to left hand side edge (facing neighbouring triangle) |
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| 154 | |
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| 155 | N = self.number_of_elements |
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| 156 | neighbourdict = {} |
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| 157 | for i in range(N): |
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| 158 | |
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| 159 | #Register all segments as keys mapping to current triangle |
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| 160 | #and segment id |
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| 161 | a = self.triangles[i, 0] |
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| 162 | b = self.triangles[i, 1] |
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| 163 | c = self.triangles[i, 2] |
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| 164 | neighbourdict[a,b] = (i, 2) #(id, edge) |
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| 165 | neighbourdict[b,c] = (i, 0) #(id, edge) |
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| 166 | neighbourdict[c,a] = (i, 1) #(id, edge) |
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| 167 | |
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| 168 | |
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| 169 | #Step 2: |
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| 170 | #Go through triangles again, but this time |
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| 171 | #reverse direction of segments and lookup neighbours. |
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| 172 | for i in range(N): |
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| 173 | a = self.triangles[i, 0] |
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| 174 | b = self.triangles[i, 1] |
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| 175 | c = self.triangles[i, 2] |
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| 176 | |
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| 177 | self.number_of_boundaries[i] = 3 |
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| 178 | if neighbourdict.has_key((b,a)): |
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| 179 | self.neighbours[i, 2] = neighbourdict[b,a][0] |
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| 180 | self.neighbour_edges[i, 2] = neighbourdict[b,a][1] |
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| 181 | self.number_of_boundaries[i] -= 1 |
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| 182 | |
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| 183 | if neighbourdict.has_key((c,b)): |
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| 184 | self.neighbours[i, 0] = neighbourdict[c,b][0] |
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| 185 | self.neighbour_edges[i, 0] = neighbourdict[c,b][1] |
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| 186 | self.number_of_boundaries[i] -= 1 |
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| 187 | |
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| 188 | if neighbourdict.has_key((a,c)): |
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| 189 | self.neighbours[i, 1] = neighbourdict[a,c][0] |
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| 190 | self.neighbour_edges[i, 1] = neighbourdict[a,c][1] |
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| 191 | self.number_of_boundaries[i] -= 1 |
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| 192 | |
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| 193 | |
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| 194 | def build_surrogate_neighbour_structure(self): |
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| 195 | """Build structure where each triangle edge points to its neighbours |
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| 196 | if they exist. Otherwise point to the triangle itself. |
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| 197 | |
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| 198 | The surrogate neighbour structure is useful for computing gradients |
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| 199 | based on centroid values of neighbours. |
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| 200 | |
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| 201 | Precondition: Neighbour structure is defined |
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| 202 | Postcondition: |
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| 203 | Surrogate neighbour structure is defined: |
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| 204 | surrogate_neighbours: i0, i1, i2 where all i_k >= 0 point to |
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| 205 | triangles. |
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| 206 | |
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| 207 | """ |
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| 208 | |
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| 209 | N = self.number_of_elements |
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| 210 | for i in range(N): |
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| 211 | #Find all neighbouring volumes that are not boundaries |
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| 212 | for k in range(3): |
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| 213 | if self.neighbours[i, k] < 0: |
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| 214 | self.surrogate_neighbours[i, k] = i #Point this triangle |
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| 215 | else: |
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| 216 | self.surrogate_neighbours[i, k] = self.neighbours[i, k] |
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| 217 | |
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| 218 | |
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| 219 | |
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| 220 | def build_boundary_dictionary(self, boundary = None): |
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| 221 | """Build or check the dictionary of boundary tags. |
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| 222 | self.boundary is a dictionary of tags, |
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| 223 | keyed by volume id and edge: |
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| 224 | { (id, edge): tag, ... } |
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| 225 | |
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| 226 | Postconditions: |
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| 227 | self.boundary is defined. |
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| 228 | """ |
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| 229 | |
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| 230 | from config import default_boundary_tag |
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| 231 | |
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| 232 | if boundary is None: |
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| 233 | boundary = {} |
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| 234 | for vol_id in range(self.number_of_elements): |
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| 235 | for edge_id in range(0, 3): |
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| 236 | if self.neighbours[vol_id, edge_id] < 0: |
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| 237 | boundary[(vol_id, edge_id)] = default_boundary_tag |
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| 238 | else: |
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| 239 | #Check that all keys in given boundary exist |
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| 240 | for vol_id, edge_id in boundary.keys(): |
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| 241 | msg = 'Segment (%d, %d) does not exist' %(vol_id, edge_id) |
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| 242 | a, b = self.neighbours.shape |
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| 243 | assert vol_id < a and edge_id < b, msg |
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| 244 | |
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| 245 | #FIXME: This assert violates internal boundaries (delete it) |
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| 246 | #msg = 'Segment (%d, %d) is not a boundary' %(vol_id, edge_id) |
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| 247 | #assert self.neighbours[vol_id, edge_id] < 0, msg |
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| 248 | |
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| 249 | #Check that all boundary segments are assigned a tag |
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| 250 | for vol_id in range(self.number_of_elements): |
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| 251 | for edge_id in range(0, 3): |
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| 252 | if self.neighbours[vol_id, edge_id] < 0: |
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| 253 | if not boundary.has_key( (vol_id, edge_id) ): |
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| 254 | msg = 'WARNING: Given boundary does not contain ' |
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| 255 | msg += 'tags for edge (%d, %d). '\ |
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| 256 | %(vol_id, edge_id) |
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| 257 | msg += 'Assigning default tag (%s).'\ |
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| 258 | %default_boundary_tag |
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| 259 | |
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| 260 | #FIXME: Print only as per verbosity |
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| 261 | #print msg |
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| 262 | |
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| 263 | #FIXME: Make this situation an error in the future |
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| 264 | #and make another function which will |
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| 265 | #enable default boundary-tags where |
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| 266 | #tags a not specified |
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| 267 | boundary[ (vol_id, edge_id) ] =\ |
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| 268 | default_boundary_tag |
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| 269 | |
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| 270 | |
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| 271 | |
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| 272 | self.boundary = boundary |
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| 273 | |
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| 274 | |
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| 275 | def build_tagged_elements_dictionary(self, tagged_elements = None): |
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| 276 | """Build the dictionary of element tags. |
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| 277 | self.tagged_elements is a dictionary of element arrays, |
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| 278 | keyed by tag: |
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| 279 | { (tag): [e1, e2, e3..] } |
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| 280 | |
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| 281 | Postconditions: |
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| 282 | self.element_tag is defined |
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| 283 | """ |
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| 284 | from Numeric import array, Int |
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| 285 | |
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| 286 | if tagged_elements is None: |
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| 287 | tagged_elements = {} |
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| 288 | else: |
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| 289 | #Check that all keys in given boundary exist |
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| 290 | for tag in tagged_elements.keys(): |
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| 291 | tagged_elements[tag] = array(tagged_elements[tag]).astype(Int) |
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| 292 | |
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| 293 | msg = 'Not all elements exist. ' |
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| 294 | assert max(tagged_elements[tag]) < self.number_of_elements, msg |
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| 295 | #print "tagged_elements", tagged_elements |
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| 296 | self.tagged_elements = tagged_elements |
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| 297 | |
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| 298 | def build_boundary_structure(self): |
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| 299 | """Traverse boundary and |
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| 300 | enumerate neighbour indices from -1 and |
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| 301 | counting down. |
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| 302 | |
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| 303 | Precondition: |
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| 304 | self.boundary is defined. |
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| 305 | Post condition: |
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| 306 | neighbour array has unique negative indices for boundary |
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| 307 | boundary_segments array imposes an ordering on segments |
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| 308 | (not otherwise available from the dictionary) |
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| 309 | |
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| 310 | Note: If a segment is listed in the boundary dictionary |
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| 311 | it *will* become a boundary - even if there is a neighbouring triangle. |
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| 312 | This would be the case for internal boundaries |
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| 313 | """ |
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| 314 | |
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| 315 | #FIXME: Now Obsolete - maybe use some comments from here in |
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| 316 | #domain.set_boundary |
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| 317 | |
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| 318 | if self.boundary is None: |
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| 319 | msg = 'Boundary dictionary must be defined before ' |
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| 320 | msg += 'building boundary structure' |
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| 321 | raise msg |
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| 322 | |
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| 323 | |
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| 324 | self.boundary_segments = self.boundary.keys() |
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| 325 | self.boundary_segments.sort() |
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| 326 | |
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| 327 | index = -1 |
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| 328 | for id, edge in self.boundary_segments: |
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| 329 | |
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| 330 | #FIXME: One would detect internal boundaries as follows |
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| 331 | #if self.neighbours[id, edge] > -1: |
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| 332 | # print 'Internal boundary' |
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| 333 | |
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| 334 | self.neighbours[id, edge] = index |
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| 335 | index -= 1 |
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| 336 | |
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| 337 | |
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| 338 | def get_boundary_tags(self): |
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| 339 | """Return list of available boundary tags |
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| 340 | """ |
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| 341 | |
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| 342 | tags = {} |
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| 343 | for v in self.boundary.values(): |
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| 344 | tags[v] = 1 |
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| 345 | |
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| 346 | return tags.keys() |
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| 347 | |
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| 348 | |
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| 349 | def get_boundary_polygon(self): |
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| 350 | """Return bounding polygon as a list of points |
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| 351 | """ |
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| 352 | from Numeric import allclose, sqrt, array, minimum, maximum |
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| 353 | |
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| 354 | |
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| 355 | |
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| 356 | V = self.get_vertex_coordinates() |
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| 357 | segments = {} |
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| 358 | |
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| 359 | #pmin = (min(self.coordinates[:,0]), min(self.coordinates[:,1])) |
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| 360 | #pmax = (max(self.coordinates[:,0]), max(self.coordinates[:,1])) |
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| 361 | |
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| 362 | #FIXME:Can this be written more compactly, e.g. |
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| 363 | #using minimum and maximium? |
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| 364 | pmin = array( [min(self.coordinates[:,0]), |
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| 365 | min(self.coordinates[:,1]) ] ) |
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| 366 | |
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| 367 | pmax = array( [max(self.coordinates[:,0]), |
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| 368 | max(self.coordinates[:,1]) ] ) |
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| 369 | |
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| 370 | mindist = sqrt(sum( (pmax-pmin)**2 )) |
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| 371 | for i, edge_id in self.boundary.keys(): |
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| 372 | #Find vertex ids for boundary segment |
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| 373 | if edge_id == 0: a = 1; b = 2 |
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| 374 | if edge_id == 1: a = 2; b = 0 |
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| 375 | if edge_id == 2: a = 0; b = 1 |
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| 376 | |
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| 377 | A = tuple(self.get_vertex_coordinate(i, a)) |
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| 378 | B = tuple(self.get_vertex_coordinate(i, b)) |
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| 379 | |
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| 380 | #Take the point closest to pmin as starting point |
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| 381 | #Note: Could be arbitrary, but nice to have |
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| 382 | #a unique way of selecting |
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| 383 | dist_A = sqrt(sum( (A-pmin)**2 )) |
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| 384 | dist_B = sqrt(sum( (B-pmin)**2 )) |
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| 385 | |
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| 386 | #Find minimal point |
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| 387 | if dist_A < mindist: |
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| 388 | mindist = dist_A |
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| 389 | p0 = A |
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| 390 | if dist_B < mindist: |
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| 391 | mindist = dist_B |
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| 392 | p0 = B |
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| 393 | |
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| 394 | |
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| 395 | if p0 is None: |
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| 396 | raise 'Weird' |
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| 397 | p0 = A #We need a starting point (FIXME) |
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| 398 | |
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| 399 | print 'A', A |
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| 400 | print 'B', B |
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| 401 | print 'pmin', pmin |
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| 402 | print |
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| 403 | |
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| 404 | segments[A] = B |
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| 405 | |
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| 406 | |
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| 407 | #Start with smallest point and follow boundary (counter clock wise) |
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| 408 | polygon = [p0] |
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| 409 | while len(polygon) < len(self.boundary): |
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| 410 | p1 = segments[p0] |
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| 411 | polygon.append(p1) |
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| 412 | p0 = p1 |
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| 413 | |
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| 414 | return polygon |
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| 415 | |
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| 416 | |
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| 417 | def check_integrity(self): |
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| 418 | """Check that triangles are internally consistent e.g. |
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| 419 | that area corresponds to edgelengths, that vertices |
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| 420 | are arranged in a counter-clockwise order, etc etc |
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| 421 | Neighbour structure will be checked by class Mesh |
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| 422 | """ |
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| 423 | |
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| 424 | from config import epsilon |
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| 425 | from math import pi |
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| 426 | from util import anglediff |
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| 427 | |
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| 428 | N = self.number_of_elements |
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| 429 | |
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| 430 | #Get x,y coordinates for all vertices for all triangles |
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| 431 | V = self.get_vertex_coordinates() |
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| 432 | |
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| 433 | #Check each triangle |
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| 434 | for i in range(N): |
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| 435 | x0 = V[i, 0]; y0 = V[i, 1] |
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| 436 | x1 = V[i, 2]; y1 = V[i, 3] |
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| 437 | x2 = V[i, 4]; y2 = V[i, 5] |
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| 438 | |
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| 439 | #Check that area hasn't been compromised |
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| 440 | area = self.areas[i] |
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| 441 | ref = abs((x1*y0-x0*y1)+(x2*y1-x1*y2)+(x0*y2-x2*y0))/2 |
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| 442 | msg = 'Wrong area for vertex coordinates: %f %f %f %f %f %f'\ |
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| 443 | %(x0,y0,x1,y1,x2,y2) |
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| 444 | assert abs((area - ref)/area) < epsilon, msg |
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| 445 | |
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| 446 | #Check that points are arranged in counter clock-wise order |
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| 447 | v0 = [x1-x0, y1-y0] |
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| 448 | v1 = [x2-x1, y2-y1] |
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| 449 | v2 = [x0-x2, y0-y2] |
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| 450 | |
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| 451 | a0 = anglediff(v1, v0) |
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| 452 | a1 = anglediff(v2, v1) |
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| 453 | a2 = anglediff(v0, v2) |
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| 454 | |
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| 455 | msg = '''Vertices (%s,%s), (%s,%s), (%s,%s) are not arranged |
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| 456 | in counter clockwise order''' %(x0, y0, x1, y1, x2, y2) |
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| 457 | assert a0 < pi and a1 < pi and a2 < pi, msg |
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| 458 | |
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| 459 | #Check that normals are orthogonal to edge vectors |
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| 460 | #Note that normal[k] lies opposite vertex k |
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| 461 | |
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| 462 | normal0 = self.normals[i, 0:2] |
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| 463 | normal1 = self.normals[i, 2:4] |
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| 464 | normal2 = self.normals[i, 4:6] |
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| 465 | |
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| 466 | for u, v in [ (v0, normal2), (v1, normal0), (v2, normal1) ]: |
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| 467 | assert u[0]*v[0] + u[1]*v[1] < epsilon |
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| 468 | |
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| 469 | |
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| 470 | #Check that all vertices have been registered |
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| 471 | for v in self.vertexlist: |
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| 472 | msg = 'Some points do not belong to an element.\n' |
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| 473 | msg += 'Make sure all points appear as element vertices!' |
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| 474 | if v is None: |
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| 475 | print 'WARNING (mesh.py): %s' %msg |
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| 476 | break |
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| 477 | |
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| 478 | |
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| 479 | |
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| 480 | #Check integrity of neighbour structure |
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| 481 | for i in range(N): |
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| 482 | for v in self.triangles[i, :]: |
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| 483 | #Check that all vertices have been registered |
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| 484 | assert self.vertexlist[v] is not None |
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| 485 | |
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| 486 | #Check that this triangle is listed with at least one vertex |
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| 487 | assert (i, 0) in self.vertexlist[v] or\ |
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| 488 | (i, 1) in self.vertexlist[v] or\ |
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| 489 | (i, 2) in self.vertexlist[v] |
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| 490 | |
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| 491 | |
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| 492 | |
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| 493 | #Check neighbour structure |
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| 494 | for k, neighbour_id in enumerate(self.neighbours[i,:]): |
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| 495 | |
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| 496 | #Assert that my neighbour's neighbour is me |
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| 497 | #Boundaries need not fulfill this |
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| 498 | if neighbour_id >= 0: |
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| 499 | edge = self.neighbour_edges[i, k] |
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| 500 | assert self.neighbours[neighbour_id, edge] == i |
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| 501 | |
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| 502 | |
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| 503 | |
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| 504 | #Check that all boundaries have |
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| 505 | # unique, consecutive, negative indices |
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| 506 | |
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| 507 | #L = len(self.boundary) |
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| 508 | #for i in range(L): |
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| 509 | # id, edge = self.boundary_segments[i] |
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| 510 | # assert self.neighbours[id, edge] == -i-1 |
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| 511 | |
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| 512 | |
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| 513 | #NOTE: This assert doesn't hold true if there are internal boundaries |
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| 514 | #FIXME: Look into this further. |
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| 515 | #FIXME (Ole): In pyvolution mark 3 this is OK again |
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| 516 | #NOTE: No longer works because neighbour structure is modified by |
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| 517 | # domain set_boundary. |
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| 518 | #for id, edge in self.boundary: |
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| 519 | # assert self.neighbours[id,edge] < 0 |
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| 520 | # |
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| 521 | #NOTE (Ole): I reckon this was resolved late 2004? |
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| 522 | # |
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| 523 | #See domain.set_boundary |
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| 524 | |
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| 525 | |
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| 526 | def get_centroid_coordinates(self): |
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| 527 | """Return all centroid coordinates. |
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| 528 | Return all centroid coordinates for all triangles as an Nx2 array |
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| 529 | (ordered as x0, y0 for each triangle) |
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| 530 | """ |
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| 531 | return self.centroid_coordinates |
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