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 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 testSymmetry(self): tag = "B1" for k in range(1,3): quadrule = pyramidquadrature(k+1) for N in range(1,3): for elements in [H1Elements(k), HcurlElements(k), HdivElements(k)]: system = SymmetricSystem(elements, quadrule, lambda m: buildcubemesh(N, m, tag), [tag]) for deriv in [False, True]: SM = system.systemMatrix(deriv) g = lambda x: np.zeros((len(x),1)) S, SIBs, Gs = system.processBoundary(SM, {tag:g}) np.testing.assert_array_almost_equal(SM.todense(), SM.transpose().todense()) np.testing.assert_array_almost_equal(S.todense(), S.transpose().todense())
def laplacedirichlet(k, N, g, points): tag = "B1" elements = H1Elements(k) quadrule = pyramidquadrature(k+1) system = SymmetricSystem(elements, quadrule, lambda m: buildcubemesh(N, m, tag), [tag]) SM = system.systemMatrix(True) S, SIBs, Gs = system.processBoundary(SM, {tag:g}) SG = SIBs[tag] * Gs[tag] print S.shape U = spsolve(S, -SG)[:,numpy.newaxis] return system.evaluate(points, U, Gs, False)
def laplaceeigs(k,N,n): import scipy.linalg as sl tag = "B1" elements = H1Elements(k) quadrule = pyramidquadrature(k+1) system = SymmetricSystem(elements, quadrule, lambda m: buildcubemesh(N, m, tag), [tag]) SM = system.systemMatrix(True) S, SIBs, Gs = system.processBoundary(SM, {tag:lambda p: numpy.zeros((len(p),1))}) print S.shape MM = system.systemMatrix(False) M, _, _ = system.processBoundary(MM, {tag:lambda p: numpy.zeros((len(p),1))}) MLU = ssl.splu(M) L = A = ssl.LinearOperator( M.shape, matvec=lambda x: MLU.solve(S* x), dtype=float) return ssl.eigen(L, k=n, which='SM', return_eigenvectors=False)
def poissondirichlet(k,N,g,f, points): tag = "B1" elements = H1Elements(k) quadrule = pyramidquadrature(k+1) system = SymmetricSystem(elements, quadrule, lambda m: buildcubemesh(N, m, tag), [tag]) SM = system.systemMatrix(True) S, SIBs, Gs = system.processBoundary(SM, {tag:g}) SG = SIBs[tag] * Gs[tag] if f: F = system.loadVector(f) else: F = 0 print SM.shape, S.shape, SIBs[tag].shape, Gs[tag].shape, F.shape, SG.shape t = Timer().start() U = spsolve(S, -F-SG)[:,numpy.newaxis] t.split("spsolve").show() return system.evaluate(points, U, Gs, False)