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153 | def massmatrix_intrinsic(l_sq,F,n=None,type='voronoi'):
"""FEM intrinsic mass matrix
Builds the finite elements mass matrix for a triangle mesh using a piecewise
linear hat function basis, using only intrinsic information (squared
halfedge edge lengths).
Parameters
----------
l_sq : (m,3) numpy double array
Vector of squared halfedge lengths as computed by halfedge_lengths_squared
F : (m,3) numpy int array
face index list of a triangle mesh (into a V assumed to exist)
n : int, optional (default: None)
Integer denoting the number of vertices in the mesh
type : str, optional (default: 'voronoi')
Type of mass matrix computation: 'voronoi' (default), 'full' or 'barycentric'
Returns
-------
M : (n,n) scipy sparse.csr_matrix
Intrinsicly computed mass matrix
See Also
--------
massmatrix.
Notes
-----
This implementation is lifted from https://github.com/alecjacobson/gptoolbox/blob/master/mesh/massmatrix_intrinsic.m
Examples
--------
TO-DO
"""
assert F.shape == l_sq.shape
assert F.shape[1]==3
assert np.all(l_sq >= 0)
dictionary ={
'voronoi' : 0,
'barycentric' : 1,
'full' : 2
}
massmatrix_type = dictionary.get(type,-1)
if n==None:
n = np.max(F)+1
dblA = doublearea_intrinsic(l_sq,F)
if massmatrix_type==0:
#Voronoi
l = np.sqrt(l_sq)
cosines = np.stack((
((l_sq[:,2]+l_sq[:,1]-l_sq[:,0])/(2.*l[:,2]*l[:,1])),
((l_sq[:,0]+l_sq[:,2]-l_sq[:,1])/(2.*l[:,0]*l[:,2])),
((l_sq[:,1]+l_sq[:,0]-l_sq[:,2])/(2.*l[:,1]*l[:,0]))
), axis=-1)
# cosines = [ ...
# (l(:,3).^2+l(:,2).^2-l(:,1).^2)./(2*l(:,2).*l(:,3)), ...
# (l(:,1).^2+l(:,3).^2-l(:,2).^2)./(2*l(:,1).*l(:,3)), ...
# (l(:,1).^2+l(:,2).^2-l(:,3).^2)./(2*l(:,1).*l(:,2))];
barycentric = cosines*l
normalized_barycentric = barycentric/np.hstack(( np.sum(barycentric,axis=1)[:,None], np.sum(barycentric,axis=1)[:,None], np.sum(barycentric,axis=1)[:,None] ))
# barycentric = cosines.*l;
# normalized_barycentric = barycentric./ ...
# [sum(barycentric')' sum(barycentric')' sum(barycentric')'];
partial_triangle_areas = normalized_barycentric * 0.5 * np.stack((dblA,dblA,dblA), axis=-1)
# partial_triangle_areas = normalized_barycentric.*[areas areas areas];
quads = np.stack((
((partial_triangle_areas[:,1]+ partial_triangle_areas[:,2])*0.5),
((partial_triangle_areas[:,0]+ partial_triangle_areas[:,2])*0.5),
((partial_triangle_areas[:,0]+ partial_triangle_areas[:,1])*0.5)
), axis=-1)
# quads = [ (partial_triangle_areas(:,2)+ partial_triangle_areas(:,3))*0.5 ...
# (partial_triangle_areas(:,1)+ partial_triangle_areas(:,3))*0.5 ...
# (partial_triangle_areas(:,1)+ partial_triangle_areas(:,2))*0.5];
c0s = cosines[:,0]<0
quads[c0s,:] = np.stack((
0.25*dblA[c0s], 0.125*dblA[c0s], 0.125*dblA[c0s]
), axis=-1)
c1s = cosines[:,1]<0
quads[c1s,:] = np.stack((
0.125*dblA[c1s], 0.25*dblA[c1s], 0.125*dblA[c1s]
), axis=-1)
c2s = cosines[:,2]<0
quads[c2s,:] = np.stack((
0.125*dblA[c2s], 0.125*dblA[c2s], 0.25*dblA[c2s]
), axis=-1)
# quads(cosines(:,1)<0,:) = [areas(cosines(:,1)<0,:)*0.5, ...
# areas(cosines(:,1)<0,:)*0.25, areas(cosines(:,1)<0,:)*0.25];
# quads(cosines(:,2)<0,:) = [areas(cosines(:,2)<0,:)*0.25, ...
# areas(cosines(:,2)<0,:)*0.5, areas(cosines(:,2)<0,:)*0.25];
# quads(cosines(:,3)<0,:) = [areas(cosines(:,3)<0,:)*0.25, ...
# areas(cosines(:,3)<0,:)*0.25, areas(cosines(:,3)<0,:)*0.5];
I = np.concatenate((F[:,0],F[:,1],F[:,2]))
J = I
vals = np.reshape(quads,(-1,1),order='F').squeeze()
# i = [i1 i2 i3];
# j = [i1 i2 i3];
# v = reshape(quads,size(quads,1)*3,1);
elif massmatrix_type==1:
#Barycentric
I = np.concatenate((F[:,0],F[:,1],F[:,2]))
J = I
vals = np.concatenate((dblA,dblA,dblA))/6.
elif massmatrix_type==2:
#Full
I = np.concatenate((F[:,0], F[:,1], F[:,1], F[:,2], F[:,2], F[:,0],
F[:,0], F[:,1], F[:,2]))
J = np.concatenate((F[:,1], F[:,0], F[:,2], F[:,1], F[:,0], F[:,2],
F[:,0], F[:,1], F[:,2]))
offd = dblA / 24.
diag = dblA / 12.
vals = np.concatenate((offd, offd, offd, offd, offd, offd,
diag, diag, diag))
# i = [i1 i2 i2 i3 i3 i1 i1 i2 i3];
# j = [i2 i1 i3 i2 i1 i3 i1 i2 i3];
# offd_v = dblA/24.;
# diag_v = dblA/12.;
# v = [offd_v,offd_v, offd_v,offd_v, offd_v,offd_v, diag_v,diag_v,diag_v];
else:
assert False, "invalid massmatrix type"
M = csr_matrix((vals,(I,J)),shape=(n,n))
return M
|