def assemble(self): N = self.N SB = Basis(N, 'C', bc='Biharmonic', quad=self.quad) SB.plan((N, N), 0, np.float, {}) # (u'', v) K = inner_product((SB, 0), (SB, 2)) # ((1-x**2)u, v) x = sp.symbols('x', real=True) K1 = inner_product((SB, 0), (SB, 0), measure=(1-x**2)) # ((1-x**2)u'', v) K2 = inner_product((SB, 0), (SB, 2), measure=(1-x**2)) # (u'''', v) Q = inner_product((SB, 0), (SB, 4)) # (u, v) M = inner_product((SB, 0), (SB, 0)) 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.diags().toarray(), B.diags().toarray()
def assemble(self): N = self.N SB = Basis(N, 'C', bc='Biharmonic', quad=self.quad) SB.plan((N, N), 0, np.float, {}) x, _ = self.x, self.w = SB.points_and_weights(N) # Trial function P4 = SB.evaluate_basis_all(x=x) # Second derivatives T2x = SB.evaluate_basis_derivative_all(x=x, k=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 assemble(self): N = self.N SB = Basis(N, 'C', bc='Biharmonic', quad=self.quad) SB.plan((N, N), 0, np.float, {}) x, _ = self.x, self.w = SB.points_and_weights(N) # Trial function P4 = SB.evaluate_basis_all(x=x) # Second derivatives T2x = SB.evaluate_basis_derivative_all(x=x, k=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 get_context(): """Set up context for solver""" # Get points and weights for Chebyshev weighted integrals ST = Basis(params.N[0], 'C', bc=(0, 0), quad=params.Dquad) SB = Basis(params.N[0], 'C', bc='Biharmonic', quad=params.Bquad) ST0 = Basis(params.N[0], 'C', bc=(0, 0), quad=params.Dquad) # For 1D problem CT = ST.CT # Chebyshev transform Nu = params.N[0] - 2 # Number of velocity modes in Shen basis Nb = params.N[0] - 4 # Number of velocity modes in Shen biharmonic basis u_slice = slice(0, Nu) v_slice = slice(0, Nb) FST = SlabShen_R2C(params.N, params.L, comm, threads=params.threads, communication=params.communication, planner_effort=params.planner_effort, dealias_cheb=params.dealias_cheb) float, complex, mpitype = datatypes("double") ST.plan(FST.complex_shape(), 0, complex, { 'threads': params.threads, 'planner_effort': params.planner_effort["dct"] }) SB.plan(FST.complex_shape(), 0, complex, { 'threads': params.threads, 'planner_effort': params.planner_effort["dct"] }) # Mesh variables X = FST.get_local_mesh(ST) x0, x1, x2 = FST.get_mesh_dims(ST) K = FST.get_local_wavenumbermesh(scaled=True) K2 = K[1] * K[1] + K[2] * K[2] K4 = K2**2 # Set Nyquist frequency to zero on K that is used for odd derivatives Kx = FST.get_local_wavenumbermesh(scaled=True, eliminate_highest_freq=True) K_over_K2 = zeros((2, ) + FST.complex_shape()) for i in range(2): K_over_K2[i] = K[i + 1] / np.where(K2 == 0, 1, K2) # Solution variables U = zeros((3, ) + FST.real_shape(), dtype=float) U0 = zeros((3, ) + FST.real_shape(), dtype=float) U_hat = zeros((3, ) + FST.complex_shape(), dtype=complex) U_hat0 = zeros((3, ) + FST.complex_shape(), dtype=complex) g = zeros(FST.complex_shape(), dtype=complex) # primary variable u = (U_hat, g) H_hat = zeros((3, ) + FST.complex_shape(), dtype=complex) H_hat0 = zeros((3, ) + FST.complex_shape(), dtype=complex) H_hat1 = zeros((3, ) + FST.complex_shape(), dtype=complex) dU = zeros((3, ) + FST.complex_shape(), dtype=complex) hv = zeros(FST.complex_shape(), dtype=complex) hg = zeros(FST.complex_shape(), dtype=complex) Source = zeros((3, ) + FST.real_shape(), dtype=float) Sk = zeros((3, ) + FST.complex_shape(), dtype=complex) work = work_arrays() nu, dt, N = params.nu, params.dt, params.N kx = K[0][:, 0, 0] alfa = K2 - 2.0 / nu / dt # Collect all matrices mat = config.AttributeDict( dict( CDD=inner_product((ST, 0), (ST, 1)), AB=HelmholtzCoeff(N[0], 1.0, -alfa, ST.quad), AC=BiharmonicCoeff(N[0], nu * dt / 2., (1. - nu * dt * K2[0]), -(K2[0] - nu * dt / 2. * K4[0]), quad=SB.quad), # Matrices for biharmonic equation CBD=inner_product((SB, 0), (ST, 1)), ABB=inner_product((SB, 0), (SB, 2)), BBB=inner_product((SB, 0), (SB, 0)), SBB=inner_product((SB, 0), (SB, 4)), # Matrices for Helmholtz equation ADD=inner_product((ST, 0), (ST, 2)), BDD=inner_product((ST, 0), (ST, 0)), BBD=inner_product((SB, 0), (ST, 0)), CDB=inner_product((ST, 0), (SB, 1)), ADD0=inner_product((ST0, 0), (ST0, 2)), BDD0=inner_product((ST0, 0), (ST0, 0)))) mat.ADD.axis = 0 mat.BDD.axis = 0 mat.SBB.axis = 0 # Collect all linear algebra solvers la = config.AttributeDict( dict(HelmholtzSolverG=Helmholtz(mat.ADD, mat.BDD, -np.ones((1, 1, 1)), (K2 + 2.0 / nu / dt)), BiharmonicSolverU=Biharmonic(mat.SBB, mat.ABB, mat.BBB, -nu * dt / 2. * np.ones( (1, 1, 1)), (1. + nu * dt * K2), (-(K2 + nu * dt / 2. * K4))), HelmholtzSolverU0=Helmholtz(mat.ADD0, mat.BDD0, np.array([-1.]), np.array([2. / nu / dt])), TDMASolverD=TDMA(inner_product((ST, 0), (ST, 0))))) hdf5file = KMMWriter({ "U": U[0], "V": U[1], "W": U[2] }, chkpoint={ 'current': { 'U': U }, 'previous': { 'U': U0 } }, filename=params.solver + ".h5", mesh={ "x": x0, "y": x1, "z": x2 }) return config.AttributeDict(locals())