def mixedpoissondual(k,N, g, f, points):
tag = "B1"
hdiveltsA = HdivElements(k)
hdiveltsB = HdivElements(k)
hdiveltsBt = HdivElements(k)
l2eltsB = L2Elements(k)
l2eltsBt = L2Elements(k)
quadrule = pyramidquadrature(k+1)
meshevents = lambda m: buildcubemesh(N,m,tag)
Asystem = SymmetricSystem(hdiveltsA, quadrule, meshevents, [])
Bsystem = AsymmetricSystem(l2eltsB, hdiveltsB, quadrule, meshevents, [],[])
Btsystem = AsymmetricSystem(hdiveltsBt, l2eltsBt, quadrule, meshevents, [],[])
A = Asystem.systemMatrix(False)
# Bt = Btsystem.systemMatrix(True, False)
B = Bsystem.systemMatrix(False, True)
print A.shape, B.shape
F = Bsystem.loadVector(f, False)
gn = lambda x,n: g(x).reshape(-1,1,1) * n.reshape(-1,1,3)
G = Btsystem.boundaryLoad({tag:gn}, squarequadrature(k+1), trianglequadrature(k+1), False)
u,p = ps.directsolve(A, B, G[tag], F)
um,pm = ps.mixedcg(A, B, G[tag], F)
# print numpy.hstack((UU[:len(u)], u))
print math.sqrt(numpy.sum((u - um)**2)/len(u))
print math.sqrt(numpy.sum((p - pm)**2)/len(p))
return Btsystem.evaluate(points, p, {}, False)
def stokes2(k, meshevents, v, points):
vortelts1 = pe.HcurlElements(k)
vortelts2 = pe.HcurlElements(k)
velelts1 = pe.HdivElements(k)
velelts2 = pe.HdivElements(k)
pressureelts1 = pe.L2Elements(k)
quadrule = pu.pyramidquadrature(k+1)
Asys = pa.SymmetricSystem(vortelts1, quadrule, meshevents, [])
# Bsys = pa.AsymmetricSystem(velelts1, vortelts2, quadrule, meshevents, [bdytag], [])
BsysT = pa.AsymmetricSystem(vortelts2, velelts1, quadrule, meshevents, [], [bdytag])
Csys = pa.AsymmetricSystem(pressureelts1,velelts2, quadrule, meshevents, [], [bdytag])
A = Asys.systemMatrix(False)
BT = BsysT.systemMatrix(True, False)
C = Csys.systemMatrix(False, True)
vv = lambda x: np.tile(v,(len(x), 1))[:,np.newaxis,:]
vn = lambda x,n: np.tensordot(n,v,([1],[1]))
# vt = lambda x,n: (v - vn(x,n)*n)[:,np.newaxis,:]
vc = lambda x,n: np.cross(v, n)[:, np.newaxis, :]
BTI, BTE, BTGs = BsysT.processBoundary(BT, {bdytag:vv})
CI, CE, CGs = Csys.processBoundary(C, {bdytag:vv})
P = Csys.loadVector(lambda x: np.ones((len(x),1,1)))
# print "P ",P
Gt = Asys.boundaryLoad({bdytag: vc}, pu.squarequadrature(k+1), pu.trianglequadrature(k+1), False)
# print "Gt ",Gt
print A.shape, BT.shape, C.shape, BTI.shape, BTE[bdytag].shape, BTGs[bdytag].shape, CI.shape
AL = Gt[bdytag] + BTE[bdytag] * BTGs[bdytag]
# print "AL ",AL
CL = -CE[bdytag] * CGs[bdytag]
nvort = A.get_shape()[0]
nvel = BTI.get_shape()[1]
# S = ss.bmat([[A, -BTI, None],[-BTI.transpose(), None, CI.transpose()],[None, CI, None]])
# L = np.vstack((AL,np.zeros((nvel,1)), CL))
S = ss.bmat([[A, -BTI, None, None],[-BTI.transpose(), None, CI.transpose(), None],[None, CI, None, P], [None,None,P.transpose(), None]])
L = np.vstack((AL,np.zeros((nvel,1)), CL, np.zeros((1,1))))
print "solving"
X = ps.solve(S, L)
U = X[nvort:(nvort + nvel)]
# print "X",X
# print "U", U
# print "BTGs", BTGs
#
u = BsysT.evaluate(points, U, BTGs, False)
# uu = Asys.evaluate(points, np.eye(nvort)[-2], {}, False)
# uu = BsysT.evaluate(points, U, {}, False)
uu = BsysT.evaluate(points, np.zeros_like(U), BTGs, False)
# print np.hstack((points, u))
# print u
return u, uu
def poissonneumann(k,N,g,f, points):
tag = "B1"
elements = H1Elements(k)
quadrule = pyramidquadrature(k+1)
system = SymmetricSystem(elements, quadrule, lambda m: buildcubemesh(N,m,tag), [])
SM = system.systemMatrix(True)
S = SM[1:,:][:,1:] # the first basis fn is guaranteed to be associated with an external degree, so is a linear comb of all the others
F = 0 if f is None else system.loadVector(f)
G = 0 if g is None else system.boundaryLoad({tag:g}, squarequadrature(k+1), trianglequadrature(k+1), False)
U = numpy.concatenate((numpy.array([0]), spsolve(S, G[tag][1:]-F[1:])))[:,numpy.newaxis]
return system.evaluate(points, U, {}, False)
def convergence(N):
x,w= pu.squarequadrature(N*2)
Nv = poisson(N*2,x)[0].flatten()
Nvl2 = math.sqrt(sum(Nv**2 * w))
e = np.zeros(N)
for n in range(N):
nv = poisson(n,x)[0].flatten()
e[n] = math.sqrt(sum((Nv - nv)**2 * w)) / Nvl2
print e
p = mp.semilogy(np.arange(N)*2, e)
mp.xlabel("polynomial degree", fontsize=14)
mp.ylabel("relative error", fontsize=14)
def testBoundaryQuad(self):
from pypyr.utils import squarequadrature, trianglequadrature
tag = "TAG"
N = 1
n = 3
bq = BoundaryQuadrature()
buildcubemesh(N, bq, boundarytag = tag)
s = 0
sx = 0
for x,w, ns in bq.getQuadratures(tag, squarequadrature(n), trianglequadrature(n)):
if len(x):
s+=sum(w)
sx += numpy.dot(x[:,0], w)
x0 = x[0]
# print ns, len(x), len(w)
for xp,np in zip(x, ns):
self.assertEqual(numpy.dot(xp - x0, np), 0)
self.assertTrue(numpy.dot(xp - numpy.array([0.5,0.5,0.5]), np) > 0)
# print xp, np, numpy.dot(xp - numpy.array([0.5,0.5,0.5]), np)
self.assertAlmostEqual(s, 6)
self.assertAlmostEqual(sx, 3)
def stokespressure(k, meshevents, pressures, points, countdofs = False, avpressure = False):
vortelts1 = pe.HcurlElements(k)
vortelts2 = pe.HcurlElements(k)
velelts1 = pe.HdivElements(k)
velelts2 = pe.HdivElements(k)
pressureelts1 = pe.L2Elements(k)
quadrule = pu.pyramidquadrature(k+1)
Asys = pa.SymmetricSystem(vortelts1, quadrule, meshevents, [])
# Bsys = pa.AsymmetricSystem(velelts1, vortelts2, quadrule, meshevents, [bdytag], [])
BsysT = pa.AsymmetricSystem(vortelts2, velelts1, quadrule, meshevents, [], [closedbdytag])
Csys = pa.AsymmetricSystem(pressureelts1,velelts2, quadrule, meshevents, [], [closedbdytag])
A = Asys.systemMatrix(False)
BT = BsysT.systemMatrix(True, False)
C = Csys.systemMatrix(False, True)
v0 = lambda x: np.zeros_like(x)[:,np.newaxis,:]
vt = lambda x,n: np.zeros_like(x)[:,np.newaxis,:]
BTI, BTE, BTGs = BsysT.processBoundary(BT, {closedbdytag:v0})
CI, CE, CGs = Csys.processBoundary(C, {closedbdytag:v0})
Pav = Csys.loadVector(lambda x: np.ones((len(x),1,1)))
# print "P ",P
alltags = pressures.keys() + [closedbdytag]
Gt = Asys.boundaryLoad(dict([(tag,vt) for tag in alltags]), pu.squarequadrature(k+1), pu.trianglequadrature(k+1), False)
Pv = Csys.transpose().boundaryLoad(pressures, pu.squarequadrature(k+1), pu.trianglequadrature(k+1), False)
# print "Gt ",Gt
print A.shape, BT.shape, C.shape, BTI.shape, map(np.shape, BTE.values()), map(np.shape, BTGs.values()), map(np.shape, Gt.values()), map(np.shape, Pv.values()), CI.shape
AL = sum(Gt.values()) + BTE[closedbdytag] * BTGs[closedbdytag]
BL = sum(Pv.values())
# print "AL ",AL
CL = -CE[closedbdytag] * CGs[closedbdytag]
nvort = A.get_shape()[0]
nvel = BTI.get_shape()[1]
npress = C.get_shape()[0]
print nvel
if avpressure:
S = ss.bmat([[A, -BTI, None, None],[-BTI.transpose(), None, CI.transpose(), None],[None, CI, None, Pav], [None,None,Pav.transpose(), None]])
L = np.vstack((AL,BL, CL, np.zeros((1,1))))
else:
S = ss.bmat([[A, -BTI, None],[-BTI.transpose(), None, CI.transpose()],[None, CI, None]])
L = np.vstack((AL,BL, CL))
X = ps.solve(S, L)
U = X[nvort:(nvort + nvel)]
P = X[(nvort+nvel):(nvort+nvel+npress)]
# print "X",X
# print "U", U
# print "BTGs", BTGs
#
u = BsysT.evaluate(points, U, BTGs, False)
# uu = Asys.evaluate(points, np.eye(nvort)[-2], {}, False)
# uu = BsysT.evaluate(points, U, {}, False)
p = Csys.transpose().evaluate(points, P, {}, False)
# print np.hstack((points, u))
# print u
if countdofs:
return u, len(X)
return u, p