def test_Mult_Div(): SD = ShenDirichletBasis(N, "GC") SN = ShenNeumannBasis(N, "GC") SD.plan(N, 0, np.complex, {}) SN.plan(N, 0, np.complex, {}) Cm = inner_product((SN, 0), (SD, 1)) Bm = inner_product((SN, 0), (SD, 0)) uk = np.random.randn((N)) + np.random.randn((N)) * 1j vk = np.random.randn((N)) + np.random.randn((N)) * 1j wk = np.random.randn((N)) + np.random.randn((N)) * 1j b = np.zeros(N, dtype=np.complex) uk0 = np.zeros(N, dtype=np.complex) vk0 = np.zeros(N, dtype=np.complex) wk0 = np.zeros(N, dtype=np.complex) uk0 = SD.forward(uk, uk0) uk = SD.backward(uk0, uk) uk0 = SD.forward(uk, uk0) vk0 = SD.forward(vk, vk0) vk = SD.backward(vk0, vk) vk0 = SD.forward(vk, vk0) wk0 = SD.forward(wk, wk0) wk = SD.backward(wk0, wk) wk0 = SD.forward(wk, wk0) LUsolve.Mult_Div_1D(N, 7, 7, uk0[:N - 2], vk0[:N - 2], wk0[:N - 2], b[1:N - 2]) uu = np.zeros_like(uk0) v0 = np.zeros_like(vk0) w0 = np.zeros_like(wk0) uu = Cm.matvec(uk0, uu) uu += 1j * 7 * Bm.matvec(vk0, v0) + 1j * 7 * Bm.matvec(wk0, w0) #from IPython import embed; embed() assert np.allclose(uu, b) uk0 = uk0.repeat(4 * 4).reshape( (N, 4, 4)) + 1j * uk0.repeat(4 * 4).reshape((N, 4, 4)) vk0 = vk0.repeat(4 * 4).reshape( (N, 4, 4)) + 1j * vk0.repeat(4 * 4).reshape((N, 4, 4)) wk0 = wk0.repeat(4 * 4).reshape( (N, 4, 4)) + 1j * wk0.repeat(4 * 4).reshape((N, 4, 4)) b = np.zeros((N, 4, 4), dtype=np.complex) m = np.zeros((4, 4)) + 7 n = np.zeros((4, 4)) + 7 LUsolve.Mult_Div_3D(N, m, n, uk0[:N - 2], vk0[:N - 2], wk0[:N - 2], b[1:N - 2]) uu = np.zeros_like(uk0) v0 = np.zeros_like(vk0) w0 = np.zeros_like(wk0) uu = Cm.matvec(uk0, uu) uu += 1j * 7 * Bm.matvec(vk0, v0) + 1j * 7 * Bm.matvec(wk0, w0) assert np.allclose(uu, b)
def test_Mult_CTD(quad): SD = ShenDirichletBasis(N, quad=quad) SD.plan(N, 0, np.complex, {}) C = inner_product((SD.CT, 0), (SD, 1)) B = inner_product((SD.CT, 0), (SD.CT, 0)) vk = np.random.randn((N))+np.random.randn((N))*1j wk = np.random.randn((N))+np.random.randn((N))*1j bv = np.zeros(N, dtype=np.complex) bw = np.zeros(N, dtype=np.complex) vk0 = np.zeros(N, dtype=np.complex) wk0 = np.zeros(N, dtype=np.complex) cv = np.zeros(N, dtype=np.complex) cw = np.zeros(N, dtype=np.complex) vk0 = SD.forward(vk, vk0) vk = SD.backward(vk0, vk) vk0 = SD.forward(vk, vk0) wk0 = SD.forward(wk, wk0) wk = SD.backward(wk0, wk) wk0 = SD.forward(wk, wk0) LUsolve.Mult_CTD_1D(N, vk0, wk0, bv, bw) cv = np.zeros_like(vk0) cw = np.zeros_like(wk0) cv = C.matvec(vk0, cv) cw = C.matvec(wk0, cw) cv /= B[0] cw /= B[0] assert np.allclose(cv, bv) assert np.allclose(cw, bw)
def test_Mult_CTD_3D(quad): SD = ShenDirichletBasis(N, quad=quad) SD.plan((N, 4, 4), 0, np.complex, {}) C = inner_product((SD.CT, 0), (SD, 1)) B = inner_product((SD.CT, 0), (SD.CT, 0)) vk = np.random.random((N, 4, 4))+np.random.random((N, 4, 4))*1j wk = np.random.random((N, 4, 4))+np.random.random((N, 4, 4))*1j bv = np.zeros((N, 4, 4), dtype=np.complex) bw = np.zeros((N, 4, 4), dtype=np.complex) vk0 = np.zeros((N, 4, 4), dtype=np.complex) wk0 = np.zeros((N, 4, 4), dtype=np.complex) cv = np.zeros((N, 4, 4), dtype=np.complex) cw = np.zeros((N, 4, 4), dtype=np.complex) vk0 = SD.forward(vk, vk0) vk = SD.backward(vk0, vk) vk0 = SD.forward(vk, vk0) wk0 = SD.forward(wk, wk0) wk = SD.backward(wk0, wk) wk0 = SD.forward(wk, wk0) LUsolve.Mult_CTD_3D_ptr(N, vk0, wk0, bv, bw, 0) cv = np.zeros_like(vk0) cw = np.zeros_like(wk0) cv = C.matvec(vk0, cv) cw = C.matvec(wk0, cw) cv /= B[0].repeat(np.array(bv.shape[1:]).prod()).reshape(bv.shape) cw /= B[0].repeat(np.array(bv.shape[1:]).prod()).reshape(bv.shape) assert np.allclose(cv, bv) assert np.allclose(cw, bw)
def test_Mult_CTD_3D(quad): SD = ShenDirichletBasis(N, quad=quad) SD.plan((N, 4, 4), 0, np.complex, {}) C = inner_product((SD.CT, 0), (SD, 1)) B = inner_product((SD.CT, 0), (SD.CT, 0)) vk = np.random.random((N, 4, 4))+np.random.random((N, 4, 4))*1j wk = np.random.random((N, 4, 4))+np.random.random((N, 4, 4))*1j bv = np.zeros((N, 4, 4), dtype=np.complex) bw = np.zeros((N, 4, 4), dtype=np.complex) vk0 = np.zeros((N, 4, 4), dtype=np.complex) wk0 = np.zeros((N, 4, 4), dtype=np.complex) cv = np.zeros((N, 4, 4), dtype=np.complex) cw = np.zeros((N, 4, 4), dtype=np.complex) vk0 = SD.forward(vk, vk0) vk = SD.backward(vk0, vk) vk0 = SD.forward(vk, vk0) wk0 = SD.forward(wk, wk0) wk = SD.backward(wk0, wk) wk0 = SD.forward(wk, wk0) LUsolve.Mult_CTD_3D_ptr(N, vk0, wk0, bv, bw, 0) cv = np.zeros_like(vk0) cw = np.zeros_like(wk0) cv = C.matvec(vk0, cv) cw = C.matvec(wk0, cw) cv /= B[0].repeat(np.array(bv.shape[1:]).prod()).reshape(bv.shape) cw /= B[0].repeat(np.array(bv.shape[1:]).prod()).reshape(bv.shape) assert np.allclose(cv, bv) assert np.allclose(cw, bw)
def test_Mult_Div(): SD = ShenDirichletBasis(N, "GC") SN = ShenNeumannBasis(N, "GC") SD.plan(N, 0, np.complex, {}) SN.plan(N, 0, np.complex, {}) Cm = inner_product((SN, 0), (SD, 1)) Bm = inner_product((SN, 0), (SD, 0)) uk = np.random.randn((N))+np.random.randn((N))*1j vk = np.random.randn((N))+np.random.randn((N))*1j wk = np.random.randn((N))+np.random.randn((N))*1j b = np.zeros(N, dtype=np.complex) uk0 = np.zeros(N, dtype=np.complex) vk0 = np.zeros(N, dtype=np.complex) wk0 = np.zeros(N, dtype=np.complex) uk0 = SD.forward(uk, uk0) uk = SD.backward(uk0, uk) uk0 = SD.forward(uk, uk0) vk0 = SD.forward(vk, vk0) vk = SD.backward(vk0, vk) vk0 = SD.forward(vk, vk0) wk0 = SD.forward(wk, wk0) wk = SD.backward(wk0, wk) wk0 = SD.forward(wk, wk0) LUsolve.Mult_Div_1D(N, 7, 7, uk0[:N-2], vk0[:N-2], wk0[:N-2], b[1:N-2]) uu = np.zeros_like(uk0) v0 = np.zeros_like(vk0) w0 = np.zeros_like(wk0) uu = Cm.matvec(uk0, uu) uu += 1j*7*Bm.matvec(vk0, v0) + 1j*7*Bm.matvec(wk0, w0) #from IPython import embed; embed() assert np.allclose(uu, b) uk0 = uk0.repeat(4*4).reshape((N, 4, 4)) + 1j*uk0.repeat(4*4).reshape((N, 4, 4)) vk0 = vk0.repeat(4*4).reshape((N, 4, 4)) + 1j*vk0.repeat(4*4).reshape((N, 4, 4)) wk0 = wk0.repeat(4*4).reshape((N, 4, 4)) + 1j*wk0.repeat(4*4).reshape((N, 4, 4)) b = np.zeros((N, 4, 4), dtype=np.complex) m = np.zeros((4, 4))+7 n = np.zeros((4, 4))+7 LUsolve.Mult_Div_3D(N, m, n, uk0[:N-2], vk0[:N-2], wk0[:N-2], b[1:N-2]) uu = np.zeros_like(uk0) v0 = np.zeros_like(vk0) w0 = np.zeros_like(wk0) uu = Cm.matvec(uk0, uu) uu += 1j*7*Bm.matvec(vk0, v0) + 1j*7*Bm.matvec(wk0, w0) assert np.allclose(uu, b)
def test_Helmholtz_matvec(quad): M = 2*N SD = ShenDirichletBasis(M, quad=quad) kx = 11 uj = np.random.randn(M) u_hat = np.zeros(M) u_hat = SD.forward(uj, u_hat) uj = SD.backward(u_hat, uj) B = inner_product((SD, 0), (SD, 0)) A = inner_product((SD, 0), (SD, 2)) AB = HelmholtzCoeff(M, 1, kx**2, SD.quad) u1 = np.zeros(M) u1 = SD.forward(uj, u1) c0 = np.zeros_like(u1) c1 = np.zeros_like(u1) c = A.matvec(u1, c0)+kx**2*B.matvec(u1, c1) b = np.zeros(M) #LUsolve.Mult_Helmholtz_1D(M, SD.quad=="GL", 1, kx**2, u1, b) b = AB.matvec(u1, b) #from IPython import embed; embed() assert np.allclose(c, b) b = np.zeros((M, 4, 4), dtype=np.complex) u1 = u1.repeat(16).reshape((M, 4, 4)) +1j*u1.repeat(16).reshape((M, 4, 4)) kx = np.zeros((1, 4, 4))+kx #LUsolve.Mult_Helmholtz_3D_complex(M, SD.quad=="GL", 1.0, kx**2, u1, b) AB = HelmholtzCoeff(M, 1, kx**2, SD.quad) b = AB.matvec(u1, b) assert np.linalg.norm(b[:, 2, 2].real - c)/(M*16) < 1e-12 assert np.linalg.norm(b[:, 2, 2].imag - c)/(M*16) < 1e-12
def test_Helmholtz2(quad): M = 2 * N SD = ShenDirichletBasis(M, quad=quad, plan=True) kx = 12 points, weights = SD.points_and_weights(M) uj = np.random.randn(M) u_hat = np.zeros(M) u_hat = SD.forward(uj, u_hat) uj = SD.backward(u_hat, uj) #from IPython import embed; embed() A = inner_product((SD, 0), (SD, 2)) B = inner_product((SD, 0), (SD, 0)) s = SD.slice() u1 = np.zeros(M) u1 = SD.forward(uj, u1) c0 = np.zeros_like(u1) c1 = np.zeros_like(u1) c = A.matvec(u1, c0) + kx**2 * B.matvec(u1, c1) b = np.zeros(M) H = Helmholtz(M, kx, SD) b = H.matvec(u1, b) #LUsolve.Mult_Helmholtz_1D(M, SD.quad=="GL", 1, kx**2, u1, b) assert np.allclose(c, b) b = np.zeros((M, 4, 4), dtype=np.complex) u1 = u1.repeat(16).reshape((M, 4, 4)) + 1j * u1.repeat(16).reshape( (M, 4, 4)) kx = np.zeros((4, 4)) + kx H = Helmholtz(M, kx, SD) b = H.matvec(u1, b) #LUsolve.Mult_Helmholtz_3D_complex(M, SD.quad=="GL", 1.0, kx**2, u1, b) assert np.linalg.norm(b[:, 2, 2].real - c) / (M * 16) < 1e-12 assert np.linalg.norm(b[:, 2, 2].imag - c) / (M * 16) < 1e-12
class OrrSommerfeld(object): def __init__(self, **kwargs): self.par = {'alfa': 1., 'Re': 8000., 'N': 80, 'quad': 'GC'} self.par.update(**kwargs) for name, val in six.iteritems(self.par): setattr(self, name, val) self.P4 = np.zeros(0) self.T4x = np.zeros(0) self.SB, self.SD, self.CDB = (None, ) * 3 self.x, self.w = None, None def interp(self, y, eigvals, eigvectors, eigval=1, same_mesh=False, verbose=False): """Interpolate solution eigenvector and it's derivative onto y args: y Interpolation points eigvals All computed eigenvalues eigvectors All computed eigenvectors kwargs: eigval The chosen eigenvalue, ranked with descending imaginary part. The largest imaginary part is 1, the second largest is 2, etc. same_mesh Boolean. Whether or not to interpolate to the same quadrature points as used for computing the eigenvectors verbose Boolean. Print information or not """ N = self.N nx, eigval = self.get_eigval(eigval, eigvals, verbose) phi_hat = np.zeros(N, np.complex) phi_hat[:-4] = np.squeeze(eigvectors[:, nx]) if same_mesh: phi = np.zeros_like(phi_hat) dphidy = np.zeros_like(phi_hat) if self.SB is None: self.SB = ShenBiharmonicBasis(N, quad=self.quad, plan=True) self.SD = ShenDirichletBasis(N, quad=self.quad, plan=True) self.CDB = inner_product((self.SD, 0), (self.SB, 1)) phi = self.SB.ifst(phi_hat, phi) dphidy_hat = self.CDB.matvec(phi_hat) dphidy_hat = self.SD.apply_inverse_mass(dphidy_hat) dphidy = self.SD.backward(dphidy_hat, dphidy) else: # Recompute interpolation matrices if necessary if not len(self.P4) == len(y): SB = ShenBiharmonicBasis(N, quad=self.quad) V = SB.vandermonde(y) self.P4 = SB.get_vandermonde_basis(V) self.T4x = SB.get_vandermonde_basis_derivative(V, 1) phi = np.dot(self.P4, phi_hat) dphidy = np.dot(self.T4x, phi_hat) return eigval, phi, dphidy def assemble(self): N = self.N SB = ShenBiharmonicBasis(N, quad=self.quad) SB.plan((N, N), 0, np.float, {}) x, w = self.x, self.w = SB.points_and_weights(N) V = SB.vandermonde(x) # Trial function P4 = SB.get_vandermonde_basis(V) # Second derivatives T2x = SB.get_vandermonde_basis_derivative(V, 2) # (u'', v) K = np.zeros((N, N)) K[:-4, :-4] = inner_product((SB, 0), (SB, 2)).diags().toarray() # ((1-x**2)u, v) xx = np.broadcast_to((1 - x**2)[:, np.newaxis], (N, N)) #K1 = np.dot(w*P4.T, xx*P4) # Alternative: K1 = np.dot(w*P4.T, ((1-x**2)*P4.T).T) K1 = np.zeros((N, N)) K1 = SB.scalar_product(xx * P4, K1) K1 = extract_diagonal_matrix( K1).diags().toarray() # For improved roundoff # ((1-x**2)u'', v) K2 = np.zeros((N, N)) K2 = SB.scalar_product(xx * T2x, K2) K2 = extract_diagonal_matrix( K2).diags().toarray() # For improved roundoff # (u'''', v) Q = np.zeros((self.N, self.N)) Q[:-4, :-4] = inner_product((SB, 0), (SB, 4)).diags().toarray() # (u, v) M = np.zeros((self.N, self.N)) M[:-4, :-4] = inner_product((SB, 0), (SB, 0)).diags().toarray() Re = self.Re a = self.alfa B = -Re * a * 1j * (K - a**2 * M) A = Q - 2 * a**2 * K + a**4 * M - 2 * a * Re * 1j * M - 1j * a * Re * ( K2 - a**2 * K1) return A, B def solve(self, verbose=False): """Solve the Orr-Sommerfeld eigenvalue problem """ if verbose: print('Solving the Orr-Sommerfeld eigenvalue problem...') print('Re = ' + str(self.par['Re']) + ' and alfa = ' + str(self.par['alfa'])) A, B = self.assemble() return eig(A[:-4, :-4], B[:-4, :-4]) # return eig(np.dot(inv(B[:-4, :-4]), A[:-4, :-4])) @staticmethod def get_eigval(nx, eigvals, verbose=False): """Get the chosen eigenvalue Args: nx The chosen eigenvalue. nx=1 corresponds to the one with the largest imaginary part, nx=2 the second largest etc. eigvals Computed eigenvalues verbose Print the value of the chosen eigenvalue """ indices = np.argsort(np.imag(eigvals)) indi = indices[-1 * np.array(nx)] eigval = eigvals[indi] if verbose: ev = list(eigval) if np.ndim(eigval) else [eigval] indi = list(indi) if np.ndim(indi) else [indi] for i, (e, v) in enumerate(zip(ev, indi)): print('Eigenvalue {} ({}) = {:2.16e}'.format(i + 1, v, e)) return indi, eigval
from shenfun import inner, div, grad, TestFunction, TrialFunction from shenfun.chebyshev.bases import ShenDirichletBasis # Use sympy to compute a rhs, given an analytical solution x = Symbol("x") ue = sin(np.pi * x) * (1 - x**2) fe = ue.diff(x, 2) # Lambdify for faster evaluation ul = lambdify(x, ue, 'numpy') fl = lambdify(x, fe, 'numpy') N = 32 SD = ShenDirichletBasis(N, plan=True) X = SD.mesh(N) u = TrialFunction(SD) v = TestFunction(SD) fj = fl(X) # Compute right hand side of Poisson equation f_hat = inner(v, fj) # array # Get left hand side of Poisson equation and solve A = inner(v, div(grad(u))) # matrix f_hat = A.solve(f_hat) uj = SD.backward(f_hat) # Compare with analytical solution ue = ul(X) assert np.allclose(uj, ue)