volume mismatch is unreliable
the volume of a deformed hexahedron is not properly defined, the approximation error is in the order of the deviation from 1.0 (for typical crystal plasticity cases)
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Martin Diehl
parent
5d2d92ff6b
commit
198736a859
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@ -13,85 +13,6 @@ import damask
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scriptName = os.path.splitext(os.path.basename(__file__))[0]
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scriptID = ' '.join([scriptName,damask.version])
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def volTetrahedron(coords):
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"""
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Return the volume of the tetrahedron with given vertices or sides.
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If vertices are given they must be in a NumPy array with shape (4,3): the
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position vectors of the 4 vertices in 3 dimensions; if the six sides are
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given, they must be an array of length 6. If both are given, the sides
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will be used in the calculation.
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This method implements
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Tartaglia's formula using the Cayley-Menger determinant:
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|0 1 1 1 1 |
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|1 0 s1^2 s2^2 s3^2|
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288 V^2 = |1 s1^2 0 s4^2 s5^2|
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|1 s2^2 s4^2 0 s6^2|
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|1 s3^2 s5^2 s6^2 0 |
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where s1, s2, ..., s6 are the tetrahedron side lengths.
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from http://codereview.stackexchange.com/questions/77593/calculating-the-volume-of-a-tetrahedron
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"""
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# The indexes of rows in the vertices array corresponding to all
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# possible pairs of vertices
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vertex_pair_indexes = np.array(((0, 1), (0, 2), (0, 3),
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(1, 2), (1, 3), (2, 3)))
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# Get all the squares of all side lengths from the differences between
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# the 6 different pairs of vertex positions
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vertices = np.concatenate((coords[0],coords[1],coords[2],coords[3])).reshape(4,3)
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vertex1, vertex2 = vertex_pair_indexes[:,0], vertex_pair_indexes[:,1]
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sides_squared = np.sum((vertices[vertex1] - vertices[vertex2])**2,axis=-1)
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# Set up the Cayley-Menger determinant
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M = np.zeros((5,5))
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# Fill in the upper triangle of the matrix
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M[0,1:] = 1
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# The squared-side length elements can be indexed using the vertex
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# pair indices (compare with the determinant illustrated above)
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M[tuple(zip(*(vertex_pair_indexes + 1)))] = sides_squared
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# The matrix is symmetric, so we can fill in the lower triangle by
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# adding the transpose
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M = M + M.T
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return np.sqrt(np.linalg.det(M) / 288)
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def volumeMismatch(size,F,nodes):
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"""
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Calculates the volume mismatch.
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volume mismatch is defined as the difference between volume of reconstructed
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(compatible) cube and determinant of deformation gradient at Fourier point.
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"""
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coords = np.empty([8,3])
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vMismatch = np.empty(F.shape[:3])
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#--------------------------------------------------------------------------------------------------
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# calculate actual volume and volume resulting from deformation gradient
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for k in range(grid[0]):
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for j in range(grid[1]):
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for i in range(grid[2]):
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coords[0,0:3] = nodes[k, j, i ,0:3]
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coords[1,0:3] = nodes[k ,j, i+1,0:3]
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coords[2,0:3] = nodes[k ,j+1,i+1,0:3]
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coords[3,0:3] = nodes[k, j+1,i ,0:3]
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coords[4,0:3] = nodes[k+1,j, i ,0:3]
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coords[5,0:3] = nodes[k+1,j, i+1,0:3]
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coords[6,0:3] = nodes[k+1,j+1,i+1,0:3]
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coords[7,0:3] = nodes[k+1,j+1,i ,0:3]
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vMismatch[k,j,i] = \
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( abs(volTetrahedron([coords[6,0:3],coords[0,0:3],coords[7,0:3],coords[3,0:3]])) \
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+ abs(volTetrahedron([coords[6,0:3],coords[0,0:3],coords[7,0:3],coords[4,0:3]])) \
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+ abs(volTetrahedron([coords[6,0:3],coords[0,0:3],coords[2,0:3],coords[3,0:3]])) \
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+ abs(volTetrahedron([coords[6,0:3],coords[0,0:3],coords[2,0:3],coords[1,0:3]])) \
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+ abs(volTetrahedron([coords[6,0:3],coords[4,0:3],coords[1,0:3],coords[5,0:3]])) \
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+ abs(volTetrahedron([coords[6,0:3],coords[4,0:3],coords[1,0:3],coords[0,0:3]]))) \
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/np.linalg.det(F[k,j,i,0:3,0:3])
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return vMismatch/(size.prod()/grid.prod())
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def shapeMismatch(size,F,nodes,centres):
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"""
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@ -155,10 +76,6 @@ parser.add_option('--no-shape','-s',
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dest = 'shape',
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action = 'store_false',
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help = 'omit shape mismatch')
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parser.add_option('--no-volume','-v',
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dest = 'volume',
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action = 'store_false',
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help = 'omit volume mismatch')
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parser.set_defaults(pos = 'pos',
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defgrad = 'f',
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shape = True,
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@ -185,10 +102,4 @@ for name in filenames:
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shapeMismatch.reshape(-1,1,order='F'),
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scriptID+' '+' '.join(sys.argv[1:]))
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if options.volume:
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volumeMismatch = volumeMismatch(size,F,nodes)
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table = table.add('volMismatch(({}))'.format(options.defgrad),
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volumeMismatch.reshape(-1,1,order='F'),
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scriptID+' '+' '.join(sys.argv[1:]))
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table.save((sys.stdout if name is None else name), legacy=True)
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