def shape_functions(self,xi1,xi2,number_of_derivatives=0):
if number_of_derivatives > 1: raise "Only coded for first derivative"
# Make sure values are floating point
xi1=np.asarray(xi1,np.float).reshape(-1,)
xi2=np.asarray(xi2,np.float).reshape(-1,)
num = np.size(xi1)
Bi = NURBSinC.multiAllBernsteinDers(self.n,xi1,number_of_derivatives)
Bj = NURBSinC.multiAllBernsteinDers(self.m,xi2,number_of_derivatives)
N = Bi[:,0].reshape(num,-1,1) * Bj[:,0].reshape(num,1,-1) * self.wmat
W = np.sum(np.sum(N,1),1).reshape(-1,1,1)
if number_of_derivatives==1:
R = np.zeros((num,(self.n+1)*(self.m+1),3))
Nu = Bi[:,1].reshape(num,-1,1) * Bj[:,0].reshape(num,1,-1) * self.wmat
Wu = np.sum(np.sum(Nu,1),1).reshape(-1,1,1)
Nv = Bi[:,0].reshape(num,-1,1) * Bj[:,1].reshape(num,1,-1) * self.wmat
Wv = np.sum(np.sum(Nv,1),1).reshape(-1,1,1)
R[:,:,0] = (N/W).reshape(num,-1)
# From formula S_alpha(u,v) in NURBS books pg 136
R[:,:,1] = ((Nu*W-N*Wu)/(W**2)).reshape(num,-1)
R[:,:,2] = ((Nv*W-N*Wv)/(W**2)).reshape(num,-1)
else:
R = np.zeros((num,(self.n+1)*(self.m+1),1))
R[:,:,0] = (N/W).reshape(num,-1)
return R
def NURBS_basis_functions(self,xi,num_derivatives=0):
xi = np.asarray(xi,np.float).reshape(-1,)
spans = NURBSinC.multiFindSpan(self.n,self.p,xi,self.U)
if num_derivatives==1:
N = NURBSinC.multidersNURBSbasis(spans,xi,self.p,1,self.U,self.w)
else:
N = np.zeros((xi.size,1,self.n+1))
N[:,0,:] = NURBSinC.multiNURBSbasis(spans,xi,self.p,self.U,self.w)
return N
def NURBS_basis_functions(self, xi, num_derivatives=0):
xi = np.asarray(xi, np.float).reshape(-1, )
spans = NURBSinC.multiFindSpan(self.n, self.p, xi, self.U)
if num_derivatives == 1:
N = NURBSinC.multidersNURBSbasis(spans, xi, self.p, 1, self.U,
self.w)
else:
N = np.zeros((xi.size, 1, self.n + 1))
N[:, 0, :] = NURBSinC.multiNURBSbasis(spans, xi, self.p, self.U,
self.w)
return N
def shape_functions(self,xi,number_of_derivatives=0):
"""Returns the basis functions on the element
in the form [i,j,d] where:
i: xi coordinate index
j: Bezier function index
d: derivative"""
if number_of_derivatives > 1: raise "Only coded for first derivative"
xi=np.asarray(xi,np.float).reshape(-1,)
if number_of_derivatives==1:
B = NURBSinC.multiRationalBernsteinDers(self.n,self.Pw.T,xi,1)
else:
B = np.zeros((xi.size,self.n+1,1))
B[:,:,0] = NURBSinC.multiRationalBernstein(self.n,self.Pw.T,xi)
return B
def shape_functions(self, xi, number_of_derivatives=0):
"""Returns the basis functions on the element
in the form [i,j,d] where:
i: xi coordinate index
j: Bezier function index
d: derivative"""
if number_of_derivatives > 1: raise "Only coded for first derivative"
xi = np.asarray(xi, np.float).reshape(-1, )
if number_of_derivatives == 1:
B = NURBSinC.multiRationalBernsteinDers(self.n, self.Pw.T, xi, 1)
else:
B = np.zeros((xi.size, self.n + 1, 1))
B[:, :, 0] = NURBSinC.multiRationalBernstein(self.n, self.Pw.T, xi)
return B
def NURBSbasis(self,uv,i=None,j=None):
"""Evaluate one basis function given uv, a (2,-1) matrix"""
if uv.shape[1]>1:
if i!=None and j!=None:
uspan = NURBSinC.multiFindSpan(self.n,self.p,uv[0],self.U)
Nu = NURBSinC.multiBasisFuns(uspan,uv[0],self.p,self.U)
vspan = NURBSinC.multiFindSpan(self.m,self.q,uv[1],self.V)
Nv = NURBSinC.multiBasisFuns(vspan,uv[1],self.q,self.V)
denominator = np.zeros(uv.shape[1],)
for k in xrange(self.n+1):
for l in xrange(self.m+1):
denominator += Nu[:,k]*Nv[:,l]*self.w[k,l]
return Nu[:,i]*Nv[:,j]*self.w[i,j] / denominator
else:
return NURBSinC.multiSurfaceBasis(uv,self.p,self.n,self.U,self.q,self.m,self.V,self.w)
else:
print "Need to 'NURBSbasis' for single point"
def NURBSbasis(self, uv, i=None, j=None):
"""Evaluate one basis function given uv, a (2,-1) matrix"""
if uv.shape[1] > 1:
if i != None and j != None:
uspan = NURBSinC.multiFindSpan(self.n, self.p, uv[0], self.U)
Nu = NURBSinC.multiBasisFuns(uspan, uv[0], self.p, self.U)
vspan = NURBSinC.multiFindSpan(self.m, self.q, uv[1], self.V)
Nv = NURBSinC.multiBasisFuns(vspan, uv[1], self.q, self.V)
denominator = np.zeros(uv.shape[1], )
for k in xrange(self.n + 1):
for l in xrange(self.m + 1):
denominator += Nu[:, k] * Nv[:, l] * self.w[k, l]
return Nu[:, i] * Nv[:, j] * self.w[i, j] / denominator
else:
return NURBSinC.multiSurfaceBasis(uv, self.p, self.n, self.U,
self.q, self.m, self.V,
self.w)
else:
print "Need to 'NURBSbasis' for single point"
def refineknotvector(self, XI, knotvec_choice):
"""Insert the knots contained in XI into U or V"""
XI = np.asarray(XI, np.float).reshape(-1, )
if knotvec_choice == 'U':
if np.min(XI) <= np.min(self.U):
raise "Knots cannot be inserted: some elements too small"
if np.max(XI) >= np.max(self.U):
raise "Knots cannot be inserted: some elements too large"
if np.any([
np.equal(xi, self.U).sum() + np.equal(xi, XI).sum() >
self.p for xi in np.unique(XI)
]):
raise "Knots cannot be inserted: some multiplicities too large"
self.U, self.V, self.Pw = NURBSinC.RefineKnotVectSurface(
self.n, self.p, self.U, self.m, self.q, self.V, self.Pw, XI, 0)
self.r = self.U.shape[0] - 1
self.n = self.r - self.p - 1
self.pspans = np.unique(self.U)
if knotvec_choice == 'V':
if np.min(XI) <= np.min(self.V):
raise "Knots cannot be inserted: some elements too small"
if np.max(XI) >= np.max(self.V):
raise "Knots cannot be inserted: some elements too large"
if np.any([
np.equal(xi, self.V).sum() + np.equal(xi, XI).sum() >
self.q for xi in np.unique(XI)
]):
raise "Knots cannot be inserted: some multiplicities too large"
self.V, self.U, self.Pw = NURBSinC.RefineKnotVectSurface(
self.m, self.q, self.V, self.n, self.p, self.U,
np.transpose(self.Pw, (1, 0, 2)), XI, 0)
self.Pw = np.transpose(self.Pw, (1, 0, 2))
self.s = self.V.shape[0] - 1
self.m = self.s - self.q - 1
self.qspans = np.unique(self.V)
self.P = (self.Pw[:, :, :3].T / self.Pw[:, :, 3].T).T
self.w = self.Pw[:, :, 3]
def shape_functions(self, xi1, xi2, number_of_derivatives=0):
if number_of_derivatives > 1: raise "Only coded for first derivative"
# Make sure values are floating point
xi1 = np.asarray(xi1, np.float).reshape(-1, )
xi2 = np.asarray(xi2, np.float).reshape(-1, )
num = np.size(xi1)
Bi = NURBSinC.multiAllBernsteinDers(self.n, xi1, number_of_derivatives)
Bj = NURBSinC.multiAllBernsteinDers(self.m, xi2, number_of_derivatives)
N = Bi[:, 0].reshape(num, -1, 1) * Bj[:, 0].reshape(num, 1,
-1) * self.wmat
W = np.sum(np.sum(N, 1), 1).reshape(-1, 1, 1)
if number_of_derivatives == 1:
R = np.zeros((num, (self.n + 1) * (self.m + 1), 3))
Nu = Bi[:, 1].reshape(num, -1, 1) * Bj[:, 0].reshape(
num, 1, -1) * self.wmat
Wu = np.sum(np.sum(Nu, 1), 1).reshape(-1, 1, 1)
Nv = Bi[:, 0].reshape(num, -1, 1) * Bj[:, 1].reshape(
num, 1, -1) * self.wmat
Wv = np.sum(np.sum(Nv, 1), 1).reshape(-1, 1, 1)
R[:, :, 0] = (N / W).reshape(num, -1)
# From formula S_alpha(u,v) in NURBS books pg 136
R[:, :, 1] = ((Nu * W - N * Wu) / (W**2)).reshape(num, -1)
R[:, :, 2] = ((Nv * W - N * Wv) / (W**2)).reshape(num, -1)
else:
R = np.zeros((num, (self.n + 1) * (self.m + 1), 1))
R[:, :, 0] = (N / W).reshape(num, -1)
return R
def refineknotvector(self, XI):
XI = np.asarray(XI, np.float).reshape(-1, )
"""Refine the curve knot vector"""
if np.min(XI) <= np.min(self.U):
raise "Knots cannot be inserted: some values too small"
if np.max(XI) >= np.max(self.U):
raise "Knots cannot be inserted: some values too large"
if np.any([
np.equal(xi, self.U).sum() + np.equal(xi, XI).sum() > self.p
for xi in np.unique(XI)
]):
raise "Knots cannot be inserted: some multiplicities too high"
self.n, self.U, self.Pw = NURBSinC.RefineKnotVectCurve(
self.n, self.p, self.U, self.Pw, XI)
self.P, self.w = (self.Pw[:, :2].T / self.Pw[:, 2].T).T, self.Pw[:, 2]
def decompose(self):
"""Decompose NURBS surface into rational Bezier patches"""
return NURBSinC.DecomposeSurface(self.n, self.p, self.U, self.m,
self.q, self.V, self.Pw)
def vals(self, u, v):
"""Return coordinates of NURBS surface at (u,v)"""
u = np.asarray(u, np.float).reshape(-1, )
v = np.asarray(v, np.float).reshape(-1, )
return NURBSinC.multiSurfacePoint(self.n, self.p, self.U, self.m,
self.q, self.V, self.Pw, u, v)
def vals(self,u,v):
"""Return coordinates of NURBS surface at (u,v)"""
u=np.asarray(u,np.float).reshape(-1,)
v=np.asarray(v,np.float).reshape(-1,)
return NURBSinC.multiSurfacePoint(self.n,self.p,self.U,self.m,self.q,self.V,self.Pw,u,v)
def decompose_into_Bezier_segments(self):
nb, Qw = NURBSinC.DecomposeCurve(self.n, self.p, self.U, self.Pw)
return nb, Qw