def get_context(): """Set up context for solver""" # Get points and weights for Chebyshev weighted integrals assert params.Dquad == params.Bquad collapse_fourier = False if params.dealias == '3/2-rule' else True ST = Basis(params.N[2], 'C', bc=(0, 0), quad=params.Dquad) SB = Basis(params.N[2], 'C', bc='Biharmonic', quad=params.Bquad) CT = Basis(params.N[2], 'C', quad=params.Dquad) ST0 = Basis(params.N[2], 'C', bc=(0, 0), quad=params.Dquad) # For 1D problem K0 = Basis(params.N[0], 'F', domain=(0, params.L[0]), dtype='D') K1 = Basis(params.N[1], 'F', domain=(0, params.L[1]), dtype='d') kw0 = {'threads':params.threads, 'planner_effort':params.planner_effort["dct"], 'slab': (params.decomposition == 'slab'), 'collapse_fourier': collapse_fourier} FST = TensorProductSpace(comm, (K0, K1, ST), axes=(2, 0, 1), **kw0) # Dirichlet FSB = TensorProductSpace(comm, (K0, K1, SB), axes=(2, 0, 1), **kw0) # Biharmonic FCT = TensorProductSpace(comm, (K0, K1, CT), axes=(2, 0, 1), **kw0) # Regular Chebyshev VFS = MixedTensorProductSpace([FST, FST, FSB]) VFST = MixedTensorProductSpace([FST, FST, FST]) VUG = MixedTensorProductSpace([FST, FSB]) # Padded kw = {'padding_factor': 1.5 if params.dealias == '3/2-rule' else 1, 'dealias_direct': params.dealias == '2/3-rule'} if params.dealias == '3/2-rule': # Requires new bases due to planning and transforms on different size arrays STp = Basis(params.N[2], 'C', bc=(0, 0), quad=params.Dquad) SBp = Basis(params.N[2], 'C', bc='Biharmonic', quad=params.Bquad) CTp = Basis(params.N[2], 'C', quad=params.Dquad) else: STp, SBp, CTp = ST, SB, CT K0p = Basis(params.N[0], 'F', dtype='D', domain=(0, params.L[0]), **kw) K1p = Basis(params.N[1], 'F', dtype='d', domain=(0, params.L[1]), **kw) FSTp = TensorProductSpace(comm, (K0p, K1p, STp), axes=(2, 0, 1), **kw0) FSBp = TensorProductSpace(comm, (K0p, K1p, SBp), axes=(2, 0, 1), **kw0) FCTp = TensorProductSpace(comm, (K0p, K1p, CTp), axes=(2, 0, 1), **kw0) VFSp = MixedTensorProductSpace([FSTp, FSTp, FSBp]) Nu = params.N[2]-2 # Number of velocity modes in Shen basis Nb = params.N[2]-4 # Number of velocity modes in Shen biharmonic basis u_slice = slice(0, Nu) v_slice = slice(0, Nb) float, complex, mpitype = datatypes("double") # Mesh variables X = FST.local_mesh(True) x0, x1, x2 = FST.mesh() K = FST.local_wavenumbers(scaled=True) # Solution variables U = Array(VFS) U0 = Array(VFS) U_hat = Function(VFS) U_hat0 = Function(VFS) g = Function(FST) # primary variable u = (U_hat, g) H_hat = Function(VFST) H_hat0 = Function(VFST) H_hat1 = Function(VFST) dU = Function(VUG) hv = Function(FST) hg = Function(FST) Source = Array(VFS) Sk = Function(VFS) K2 = K[0]*K[0]+K[1]*K[1] K4 = K2**2 # Set Nyquist frequency to zero on K that is used for odd derivatives in nonlinear terms Kx = FST.local_wavenumbers(scaled=True, eliminate_highest_freq=True) K_over_K2 = np.zeros((2,)+g.shape) for i in range(2): K_over_K2[i] = K[i] / np.where(K2 == 0, 1, K2) work = work_arrays() u_dealias = Array(VFSp) u0_hat = np.zeros((2, params.N[2]), dtype=complex) h0_hat = np.zeros((2, params.N[2]), dtype=complex) w = np.zeros((params.N[2], ), dtype=complex) w1 = np.zeros((params.N[2], ), dtype=complex) nu, dt, N = params.nu, params.dt, params.N # Collect all matrices mat = config.AttributeDict( dict(CDD=inner_product((ST, 0), (ST, 1)), CTD=inner_product((CT, 0), (ST, 1)), BTT=inner_product((CT, 0), (CT, 0)), AB=HelmholtzCoeff(N[2], 1.0, -(K2 - 2.0/nu/dt), 2, ST.quad), AC=BiharmonicCoeff(N[2], nu*dt/2., (1. - nu*dt*K2), -(K2 - nu*dt/2.*K4), 2, 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)))) 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 = KMMFile(config.params.solver, checkpoint={'space': VFS, 'data': {'0': {'U': [U_hat]}, '1': {'U': [U_hat0]}}}, results={'space': VFS, 'data': {'U': [U]}}) return config.AttributeDict(locals())
# Solve and transform to real space u_hat = Function(T) # Solution spectral space u_hat = H(u_hat, f_hat) # Solve uq = T.backward(u_hat) # Compare with analytical solution uj = Array(T, buffer=ue) print(abs(uj-uq).max()) assert np.allclose(uj, uq) c = H.matvec(u_hat, Function(T)) assert np.allclose(c, f_hat) if 'pytest' not in os.environ: import matplotlib.pyplot as plt plt.figure() X = T.local_mesh(True) # With broadcasting=True the shape of X is local_shape, even though the number of datapoints are still the same as in 1D plt.contourf(X[0], X[1], uq) plt.colorbar() plt.figure() plt.contourf(X[0], X[1], uj) plt.colorbar() plt.figure() plt.contourf(X[0], X[1], uq-uj) plt.colorbar() plt.title('Error') plt.show()
def get_context(): """Set up context for classical (NS) solver""" float, complex, mpitype = datatypes(params.precision) collapse_fourier = False if params.dealias == '3/2-rule' else True dim = len(params.N) dtype = lambda d: float if d == dim-1 else complex V = [Basis(params.N[i], 'F', domain=(0, params.L[i]), dtype=dtype(i)) for i in range(dim)] kw0 = {'threads': params.threads, 'planner_effort': params.planner_effort['fft']} T = TensorProductSpace(comm, V, dtype=float, slab=(params.decomposition == 'slab'), collapse_fourier=collapse_fourier, **kw0) VT = VectorTensorProductSpace(T) # Different bases for nonlinear term, either 2/3-rule or 3/2-rule kw = {'padding_factor': 1.5 if params.dealias == '3/2-rule' else 1, 'dealias_direct': params.dealias == '2/3-rule'} Vp = [Basis(params.N[i], 'F', domain=(0, params.L[i]), dtype=dtype(i), **kw) for i in range(dim)] Tp = TensorProductSpace(comm, Vp, dtype=float, slab=(params.decomposition == 'slab'), collapse_fourier=collapse_fourier, **kw0) VTp = VectorTensorProductSpace(Tp) # Mesh variables X = T.local_mesh(True) K = T.local_wavenumbers(scaled=True) for i in range(dim): X[i] = X[i].astype(float) K[i] = K[i].astype(float) K2 = np.zeros(T.shape(True), dtype=float) for i in range(dim): K2 += K[i]*K[i] # Set Nyquist frequency to zero on K that is, from now on, used for odd derivatives Kx = T.local_wavenumbers(scaled=True, eliminate_highest_freq=True) for i in range(dim): Kx[i] = Kx[i].astype(float) K_over_K2 = np.zeros(VT.shape(True), dtype=float) for i in range(dim): K_over_K2[i] = K[i] / np.where(K2 == 0, 1, K2) # Velocity and pressure. Use ndarray view for efficiency U = Array(VT) U_hat = Function(VT) P = Array(T) P_hat = Function(T) u_dealias = Array(VTp) # Primary variable u = U_hat # RHS array dU = Function(VT) curl = Array(VT) Source = Function(VT) # Possible source term initialized to zero work = work_arrays() hdf5file = NSFile(config.params.solver, checkpoint={'space': VT, 'data': {'0': {'U': [U_hat]}}}, results={'space': VT, 'data': {'U': [U], 'P': [P]}}) return config.AttributeDict(locals())
def get_context(): """Set up context for classical (NS) solver""" float, complex, mpitype = datatypes(params.precision) collapse_fourier = False if params.dealias == '3/2-rule' else True dim = len(params.N) dtype = lambda d: float if d == dim - 1 else complex V = [ Basis(params.N[i], 'F', domain=(0, params.L[i]), dtype=dtype(i)) for i in range(dim) ] kw0 = { 'threads': params.threads, 'planner_effort': params.planner_effort['fft'] } T = TensorProductSpace(comm, V, dtype=float, slab=(params.decomposition == 'slab'), collapse_fourier=collapse_fourier, **kw0) VT = VectorTensorProductSpace(T) # Different bases for nonlinear term, either 2/3-rule or 3/2-rule kw = { 'padding_factor': 1.5 if params.dealias == '3/2-rule' else 1, 'dealias_direct': params.dealias == '2/3-rule' } Vp = [ Basis(params.N[i], 'F', domain=(0, params.L[i]), dtype=dtype(i), **kw) for i in range(dim) ] Tp = TensorProductSpace(comm, Vp, dtype=float, slab=(params.decomposition == 'slab'), collapse_fourier=collapse_fourier, **kw0) VTp = VectorTensorProductSpace(Tp) mask = T.get_mask_nyquist() if params.mask_nyquist else None # Mesh variables X = T.local_mesh(True) K = T.local_wavenumbers(scaled=True) for i in range(dim): X[i] = X[i].astype(float) K[i] = K[i].astype(float) K2 = np.zeros(T.shape(True), dtype=float) for i in range(dim): K2 += K[i] * K[i] K_over_K2 = np.zeros(VT.shape(True), dtype=float) for i in range(dim): K_over_K2[i] = K[i] / np.where(K2 == 0, 1, K2) # Velocity and pressure. Use ndarray view for efficiency U = Array(VT) U_hat = Function(VT) P = Array(T) P_hat = Function(T) u_dealias = Array(VTp) # Primary variable u = U_hat # RHS array dU = Function(VT) curl = Array(VT) Source = Function(VT) # Possible source term initialized to zero work = work_arrays() hdf5file = NSFile(config.params.solver, checkpoint={ 'space': VT, 'data': { '0': { 'U': [U_hat] } } }, results={ 'space': VT, 'data': { 'U': [U], 'P': [P] } }) return config.AttributeDict(locals())
uq = T.backward(u_hat, uq, fast_transform=True) uj = Array(T, buffer=ue) assert np.allclose(uj, uq) # Test eval at point point = np.array([[0.1, 0.5], [0.5, 0.6]]) p = T.eval(point, u_hat) ul = lambdify((x, y), ue) assert np.allclose(p, ul(*point)) p2 = u_hat.eval(point) assert np.allclose(p2, ul(*point)) print(np.sqrt(dx((uj - uq)**2))) if 'pytest' not in os.environ: import matplotlib.pyplot as plt plt.figure() X = T.local_mesh(True) plt.contourf(X[0], X[1], uq) plt.colorbar() plt.figure() plt.contourf(X[0], X[1], uj) plt.colorbar() plt.figure() plt.contourf(X[0], X[1], uq - uj) plt.colorbar() plt.title('Error') plt.show()
def get_context(): """Set up context for solver""" # Get points and weights for Chebyshev weighted integrals assert params.Dquad == params.Bquad ST = ShenDirichletBasis(params.N[0], quad=params.Dquad) SB = ShenBiharmonicBasis(params.N[0], quad=params.Bquad) CT = Basis(params.N[0], quad=params.Dquad) ST0 = ShenDirichletBasis(params.N[0], quad=params.Dquad, plan=True) # For 1D problem K0 = C2CBasis(params.N[1], domain=(0, params.L[1])) K1 = R2CBasis(params.N[2], domain=(0, params.L[2])) FST = TensorProductSpace(comm, (ST, K0, K1), **{'threads':params.threads, 'planner_effort':params.planner_effort["dct"]}) # Dirichlet FSB = TensorProductSpace(comm, (SB, K0, K1), **{'threads':params.threads, 'planner_effort':params.planner_effort["dct"]}) # Biharmonic FCT = TensorProductSpace(comm, (CT, K0, K1), **{'threads':params.threads, 'planner_effort':params.planner_effort["dct"]}) # Regular Chebyshev VFS = VectorTensorProductSpace([FSB, FST, FST]) # Padded kw = {'padding_factor': 1.5 if params.dealias == '3/2-rule' else 1, 'dealias_direct': params.dealias == '2/3-rule'} if params.dealias == '3/2-rule': # Requires new bases due to planning and transforms on different size arrays STp = ShenDirichletBasis(params.N[0], quad=params.Dquad) SBp = ShenBiharmonicBasis(params.N[0], quad=params.Bquad) CTp = Basis(params.N[0], quad=params.Dquad) else: STp, SBp, CTp = ST, SB, CT K0p = C2CBasis(params.N[1], domain=(0, params.L[1]), **kw) K1p = R2CBasis(params.N[2], domain=(0, params.L[2]), **kw) FSTp = TensorProductSpace(comm, (STp, K0p, K1p), **{'threads':params.threads, 'planner_effort':params.planner_effort["dct"]}) FSBp = TensorProductSpace(comm, (SBp, K0p, K1p), **{'threads':params.threads, 'planner_effort':params.planner_effort["dct"]}) FCTp = TensorProductSpace(comm, (CTp, K0p, K1p), **{'threads':params.threads, 'planner_effort':params.planner_effort["dct"]}) VFSp = VectorTensorProductSpace([FSBp, FSTp, FSTp]) 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) float, complex, mpitype = datatypes("double") # Mesh variables X = FST.local_mesh(True) x0, x1, x2 = FST.mesh() K = FST.local_wavenumbers(scaled=True) # Solution variables U = Array(VFS, False) U0 = Array(VFS, False) U_hat = Array(VFS) U_hat0 = Array(VFS) g = Array(FST) # primary variable u = (U_hat, g) H_hat = Array(VFS) H_hat0 = Array(VFS) H_hat1 = Array(VFS) dU = Array(VFS) hv = Array(FST) hg = Array(FST) Source = Array(VFS, False) Sk = Array(VFS) 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 in nonlinear terms Kx = FST.local_wavenumbers(scaled=True, eliminate_highest_freq=True) K_over_K2 = np.zeros((2,)+g.shape) for i in range(2): K_over_K2[i] = K[i+1] / np.where(K2 == 0, 1, K2) work = work_arrays() nu, dt, N = params.nu, params.dt, params.N alfa = K2[0] - 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)),)) ## Collect all linear algebra solvers #la = config.AttributeDict(dict( #HelmholtzSolverG = old_Helmholtz(N[0], np.sqrt(K2[0]+2.0/nu/dt), ST), #BiharmonicSolverU = old_Biharmonic(N[0], -nu*dt/2., 1.+nu*dt*K2[0], #-(K2[0] + nu*dt/2.*K4[0]), quad=SB.quad, #solver="cython"), #HelmholtzSolverU0 = old_Helmholtz(N[0], np.sqrt(2./nu/dt), ST), #TDMASolverD = TDMA(inner_product((ST, 0), (ST, 0))) #) #) mat.ADD.axis = 0 mat.BDD.axis = 0 mat.SBB.axis = 0 la = config.AttributeDict( dict(HelmholtzSolverG=Helmholtz(mat.ADD, mat.BDD, -np.ones((1, 1, 1)), (K2[0]+2.0/nu/dt)[np.newaxis, :, :]), BiharmonicSolverU=Biharmonic(mat.SBB, mat.ABB, mat.BBB, -nu*dt/2.*np.ones((1, 1, 1)), (1.+nu*dt*K2[0])[np.newaxis, :, :], (-(K2[0] + nu*dt/2.*K4[0]))[np.newaxis, :, :]), HelmholtzSolverU0=old_Helmholtz(N[0], np.sqrt(2./nu/dt), ST), 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())
def get_context(): """Set up context for solver""" # Get points and weights for Chebyshev weighted integrals assert params.Dquad == params.Bquad collapse_fourier = False if params.dealias == '3/2-rule' else True ST = Basis(params.N[2], 'C', bc=(0, 0), quad=params.Dquad) SB = Basis(params.N[2], 'C', bc='Biharmonic', quad=params.Bquad) CT = Basis(params.N[2], 'C', quad=params.Dquad) ST0 = Basis(params.N[2], 'C', bc=(0, 0), quad=params.Dquad) # For 1D problem K0 = Basis(params.N[0], 'F', domain=(0, params.L[0]), dtype='D') K1 = Basis(params.N[1], 'F', domain=(0, params.L[1]), dtype='d') kw0 = { 'threads': params.threads, 'planner_effort': params.planner_effort["dct"], 'slab': (params.decomposition == 'slab'), 'collapse_fourier': collapse_fourier } FST = TensorProductSpace(comm, (K0, K1, ST), axes=(2, 0, 1), **kw0) # Dirichlet FSB = TensorProductSpace(comm, (K0, K1, SB), axes=(2, 0, 1), **kw0) # Biharmonic FCT = TensorProductSpace(comm, (K0, K1, CT), axes=(2, 0, 1), **kw0) # Regular Chebyshev VFS = MixedTensorProductSpace([FST, FST, FSB]) VFST = MixedTensorProductSpace([FST, FST, FST]) VUG = MixedTensorProductSpace([FST, FSB]) mask = FST.get_mask_nyquist() if params.mask_nyquist else None # Padded kw = { 'padding_factor': 1.5 if params.dealias == '3/2-rule' else 1, 'dealias_direct': params.dealias == '2/3-rule' } if params.dealias == '3/2-rule': # Requires new bases due to planning and transforms on different size arrays STp = Basis(params.N[2], 'C', bc=(0, 0), quad=params.Dquad) SBp = Basis(params.N[2], 'C', bc='Biharmonic', quad=params.Bquad) CTp = Basis(params.N[2], 'C', quad=params.Dquad) else: STp, SBp, CTp = ST, SB, CT K0p = Basis(params.N[0], 'F', dtype='D', domain=(0, params.L[0]), **kw) K1p = Basis(params.N[1], 'F', dtype='d', domain=(0, params.L[1]), **kw) FSTp = TensorProductSpace(comm, (K0p, K1p, STp), axes=(2, 0, 1), **kw0) FSBp = TensorProductSpace(comm, (K0p, K1p, SBp), axes=(2, 0, 1), **kw0) FCTp = TensorProductSpace(comm, (K0p, K1p, CTp), axes=(2, 0, 1), **kw0) VFSp = MixedTensorProductSpace([FSTp, FSTp, FSBp]) float, complex, mpitype = datatypes("double") # Mesh variables X = FST.local_mesh(True) x0, x1, x2 = FST.mesh() K = FST.local_wavenumbers(scaled=True) # Solution variables U = Array(VFS) U0 = Array(VFS) U_hat = Function(VFS) U_hat0 = Function(VFS) g = Function(FST) # primary variable u = (U_hat, g) H_hat = Function(VFST) H_hat0 = Function(VFST) H_hat1 = Function(VFST) dU = Function(VUG) hv = Function(FST) hg = Function(FST) Source = Array(VFS) Sk = Function(VFS) K2 = K[0] * K[0] + K[1] * K[1] K4 = K2**2 # Set Nyquist frequency to zero on K that is used for odd derivatives in nonlinear terms Kx = FST.local_wavenumbers(scaled=True, eliminate_highest_freq=True) K_over_K2 = np.zeros((2, ) + g.shape) for i in range(2): K_over_K2[i] = K[i] / np.where(K2 == 0, 1, K2) for i in range(3): K[i] = K[i].astype(float) Kx[i] = Kx[i].astype(float) Kx2 = Kx[1] * Kx[1] + Kx[2] * Kx[2] work = work_arrays() u_dealias = Array(VFSp) u0_hat = np.zeros((2, params.N[2]), dtype=complex) h0_hat = np.zeros((2, params.N[2]), dtype=complex) w = np.zeros((params.N[2], ), dtype=complex) w1 = np.zeros((params.N[2], ), dtype=complex) nu, dt, N = params.nu, params.dt, params.N # Collect all matrices mat = config.AttributeDict( dict( CDD=inner_product((ST, 0), (ST, 1)), CTD=inner_product((CT, 0), (ST, 1)), BTT=inner_product((CT, 0), (CT, 0)), AB=HelmholtzCoeff(N[2], 1.0, -(K2 - 2.0 / nu / dt), 2, ST.quad), AC=BiharmonicCoeff(N[2], nu * dt / 2., (1. - nu * dt * K2), -(K2 - nu * dt / 2. * K4), 2, 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)))) 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 = KMMFile(config.params.solver, checkpoint={ 'space': VFS, 'data': { '0': { 'U': [U_hat] }, '1': { 'U': [U_hat0] } } }, results={ 'space': VFS, 'data': { 'U': [U] } }) return config.AttributeDict(locals())
def get_context(): """Set up context for solver""" # Get points and weights for Chebyshev weighted integrals assert params.Dquad == params.Bquad collapse_fourier = False if params.dealias == '3/2-rule' else True ST = FunctionSpace(params.N[0], 'C', bc=(0, 0), quad=params.Dquad) CT = FunctionSpace(params.N[0], 'C', quad=params.Dquad) CP = FunctionSpace(params.N[0], 'C', quad=params.Dquad) K0 = FunctionSpace(params.N[1], 'F', domain=(0, params.L[1]), dtype='D') K1 = FunctionSpace(params.N[2], 'F', domain=(0, params.L[2]), dtype='d') CP.slice = lambda: slice(0, CT.N) kw0 = { 'threads': params.threads, 'planner_effort': params.planner_effort["dct"], 'slab': (params.decomposition == 'slab'), 'collapse_fourier': collapse_fourier } FST = TensorProductSpace(comm, (ST, K0, K1), **kw0) # Dirichlet FCT = TensorProductSpace(comm, (CT, K0, K1), **kw0) # Regular Chebyshev N FCP = TensorProductSpace(comm, (CP, K0, K1), **kw0) # Regular Chebyshev N-2 VFS = VectorTensorProductSpace(FST) VCT = VectorTensorProductSpace(FCT) VQ = MixedTensorProductSpace([VFS, FCP]) mask = FST.get_mask_nyquist() if params.mask_nyquist else None # Padded kw = { 'padding_factor': 1.5 if params.dealias == '3/2-rule' else 1, 'dealias_direct': params.dealias == '2/3-rule' } if params.dealias == '3/2-rule': # Requires new bases due to planning and transforms on different size arrays STp = FunctionSpace(params.N[0], 'C', bc=(0, 0), quad=params.Dquad) CTp = FunctionSpace(params.N[0], 'C', quad=params.Dquad) else: STp, CTp = ST, CT K0p = FunctionSpace(params.N[1], 'F', dtype='D', domain=(0, params.L[1]), **kw) K1p = FunctionSpace(params.N[2], 'F', dtype='d', domain=(0, params.L[2]), **kw) FSTp = TensorProductSpace(comm, (STp, K0p, K1p), **kw0) FCTp = TensorProductSpace(comm, (CTp, K0p, K1p), **kw0) VFSp = VectorTensorProductSpace(FSTp) VCp = MixedTensorProductSpace([FSTp, FCTp, FCTp]) float, complex, mpitype = datatypes("double") constraints = ((3, 0, 0), (3, params.N[0] - 1, 0)) # Mesh variables X = FST.local_mesh(True) x0, x1, x2 = FST.mesh() K = FST.local_wavenumbers(scaled=True) # Solution variables UP_hat = Function(VQ) UP_hat0 = Function(VQ) U_hat, P_hat = UP_hat U_hat0, P_hat0 = UP_hat0 UP = Array(VQ) UP0 = Array(VQ) U, P = UP U0, P0 = UP0 # primary variable u = UP_hat H_hat = Function(VFS) H_hat0 = Function(VFS) H_hat1 = Function(VFS) dU = Function(VQ) Source = Array( VFS) # Note - not using VQ. Only used for constant pressure gradient Sk = Function(VFS) K2 = K[1] * K[1] + K[2] * K[2] for i in range(3): K[i] = K[i].astype(float) work = work_arrays() u_dealias = Array(VFSp) curl_hat = Function(VCp) curl_dealias = Array(VCp) nu, dt, N = params.nu, params.dt, params.N up = TrialFunction(VQ) vq = TestFunction(VQ) ut, pt = up vt, qt = vq alfa = 2. / nu / dt a0 = inner(vt, (2. / nu / dt) * ut - div(grad(ut))) a1 = inner(vt, (2. / nu) * grad(pt)) a2 = inner(qt, (2. / nu) * div(ut)) M = BlockMatrix(a0 + a1 + a2) # Collect all matrices mat = config.AttributeDict( dict( CDD=inner_product((ST, 0), (ST, 1)), AB=HelmholtzCoeff(N[0], 1., alfa - K2, 0, ST.quad), )) la = None hdf5file = CoupledFile(config.params.solver, checkpoint={ 'space': VQ, 'data': { '0': { 'UP': [UP_hat] }, '1': { 'UP': [UP_hat0] } } }, results={ 'space': VFS, 'data': { 'U': [U] } }) return config.AttributeDict(locals())
fe = ue.diff(x, 2) + ue.diff(y, 2) + ue.diff(z, 2) # Lambdify for faster evaluation ul = lambdify((x, y, z), ue, 'numpy') fl = lambdify((x, y, z), fe, 'numpy') # Size of discretization N = int(sys.argv[-2]) N = [N, N + 1, N + 2] #N = (14, 15, 16) SD = Basis(N[0], family=family, bc=(a, b)) K1 = Basis(N[1], family='F', dtype='D') K2 = Basis(N[2], family='F', dtype='d') T = TensorProductSpace(comm, (K1, K2, SD), axes=(0, 1, 2), slab=True) X = T.local_mesh() u = TrialFunction(T) v = TestFunction(T) K = T.local_wavenumbers() # Get f on quad points fj = Array(T, buffer=fl(*X)) # Compute right hand side of Poisson equation f_hat = inner(v, fj) if family == 'legendre': f_hat *= -1. # Get left hand side of Poisson equation if family == 'chebyshev':
def get_context(): """Set up context for solver""" collapse_fourier = False if params.dealias == '3/2-rule' else True family = 'C' ST = Basis(params.N[0], family, bc=(0, 0), quad=params.Dquad) CT = Basis(params.N[0], family, quad=params.Dquad) CP = Basis(params.N[0], family, quad=params.Dquad) K0 = Basis(params.N[1], 'F', domain=(0, params.L[1]), dtype='D') K1 = Basis(params.N[2], 'F', domain=(0, params.L[2]), dtype='d') #CP.slice = lambda: slice(0, CP.N-2) constraints = ((3, 0, 0), (3, params.N[0]-1, 0)) kw0 = {'threads': params.threads, 'planner_effort': params.planner_effort["dct"], 'slab': (params.decomposition == 'slab'), 'collapse_fourier': collapse_fourier} FST = TensorProductSpace(comm, (ST, K0, K1), **kw0) # Dirichlet FCT = TensorProductSpace(comm, (CT, K0, K1), **kw0) # Regular Chebyshev N FCP = TensorProductSpace(comm, (CP, K0, K1), **kw0) # Regular Chebyshev N-2 VFS = VectorTensorProductSpace(FST) VCT = VectorTensorProductSpace(FCT) VQ = MixedTensorProductSpace([VFS, FCP]) mask = FST.get_mask_nyquist() if params.mask_nyquist else None # Padded kw = {'padding_factor': 1.5 if params.dealias == '3/2-rule' else 1, 'dealias_direct': params.dealias == '2/3-rule'} if params.dealias == '3/2-rule': # Requires new bases due to planning and transforms on different size arrays STp = Basis(params.N[0], family, bc=(0, 0), quad=params.Dquad) CTp = Basis(params.N[0], family, quad=params.Dquad) else: STp, CTp = ST, CT K0p = Basis(params.N[1], 'F', dtype='D', domain=(0, params.L[1]), **kw) K1p = Basis(params.N[2], 'F', dtype='d', domain=(0, params.L[2]), **kw) FSTp = TensorProductSpace(comm, (STp, K0p, K1p), **kw0) FCTp = TensorProductSpace(comm, (CTp, K0p, K1p), **kw0) VFSp = VectorTensorProductSpace(FSTp) VCp = MixedTensorProductSpace([FSTp, FCTp, FCTp]) float, complex, mpitype = datatypes("double") # Mesh variables X = FST.local_mesh(True) x0, x1, x2 = FST.mesh() K = FST.local_wavenumbers(scaled=True) # Solution variables UP_hat = Function(VQ) UP_hat0 = Function(VQ) U_hat, P_hat = UP_hat U_hat0, P_hat0 = UP_hat0 UP = Array(VQ) UP0 = Array(VQ) U, P = UP U0, P0 = UP0 # RK parameters a = (8./15., 5./12., 3./4.) b = (0.0, -17./60., -5./12.) # primary variable u = UP_hat H_hat = Function(VFS) dU = Function(VQ) hv = np.zeros((2,)+H_hat.shape, dtype=np.complex) Source = Array(VFS) # Note - not using VQ. Only used for constant pressure gradient Sk = Function(VFS) K2 = K[1]*K[1]+K[2]*K[2] for i in range(3): K[i] = K[i].astype(float) work = work_arrays() u_dealias = Array(VFSp) curl_hat = Function(VCp) curl_dealias = Array(VCp) nu, dt, N = params.nu, params.dt, params.N up = TrialFunction(VQ) vq = TestFunction(VQ) ut, pt = up vt, qt = vq M = [] for rk in range(3): a0 = inner(vt, (2./nu/dt/(a[rk]+b[rk]))*ut-div(grad(ut))) a1 = inner(vt, (2./nu/(a[rk]+b[rk]))*grad(pt)) a2 = inner(qt, (2./nu/(a[rk]+b[rk]))*div(ut)) M.append(BlockMatrix(a0+a1+a2)) # Collect all matrices if ST.family() == 'chebyshev': mat = config.AttributeDict( dict(AB=[HelmholtzCoeff(N[0], 1., -(K2 - 2./nu/dt/(a[rk]+b[rk])), 0, ST.quad) for rk in range(3)],)) else: mat = config.AttributeDict( dict(ADD=inner_product((ST, 0), (ST, 2)), BDD=inner_product((ST, 0), (ST, 0))) ) la = None hdf5file = CoupledRK3File(config.params.solver, checkpoint={'space': VQ, 'data': {'0': {'UP': [UP_hat]}}}, results={'space': VFS, 'data': {'U': [U]}}) del rk return config.AttributeDict(locals())
def test_eval_tensor(typecode, dim, ST, quad): # Using sympy to compute an analytical solution # Testing for Dirichlet and regular basis x, y, z = symbols("x,y,z") sizes = (16, 15) funcx = {'': (1-x**2)*sin(np.pi*x), 'Dirichlet': (1-x**2)*sin(np.pi*x), 'Neumann': (1-x**2)*sin(np.pi*x), 'Biharmonic': (1-x**2)*sin(2*np.pi*x), '6th order': (1-x**2)**3*sin(np.pi*x), 'BiPolar': (1-x**2)*sin(2*np.pi*x), 'BiPolar0': (1-x**2)*sin(2*np.pi*x), 'UpperDirichlet': (1-x)*sin(np.pi*x), 'DirichletNeumann': (1-x**2)*sin(2*np.pi*x), 'NeumannDirichlet': (1-x**2)*sin(2*np.pi*x)} funcy = {'': (1-y**2)*sin(np.pi*y), 'Dirichlet': (1-y**2)*sin(np.pi*y), 'Neumann': (1-y**2)*sin(np.pi*y), 'Biharmonic': (1-y**2)*sin(2*np.pi*y), '6th order': (1-y**2)**3*sin(np.pi*y), 'BiPolar': (1-y**2)*sin(2*np.pi*y), 'BiPolar0': (1-y**2)*sin(2*np.pi*y), 'UpperDirichlet': (1-y)*sin(np.pi*y), 'DirichletNeumann': (1-y**2)*sin(2*np.pi*y), 'NeumannDirichlet': (1-y**2)*sin(2*np.pi*y)} funcz = {'': (1-z**2)*sin(np.pi*z), 'Dirichlet': (1-z**2)*sin(np.pi*z), 'Neumann': (1-z**2)*sin(np.pi*z), 'Biharmonic': (1-z**2)*sin(2*np.pi*z), '6th order': (1-z**2)**3*sin(np.pi*z), 'BiPolar': (1-z**2)*sin(2*np.pi*z), 'BiPolar0': (1-z**2)*sin(2*np.pi*z), 'UpperDirichlet': (1-z)*sin(np.pi*z), 'DirichletNeumann': (1-z**2)*sin(2*np.pi*z), 'NeumannDirichlet': (1-z**2)*sin(2*np.pi*z)} funcs = { (1, 0): cos(2*y)*funcx[ST.boundary_condition()], (1, 1): cos(2*x)*funcy[ST.boundary_condition()], (2, 0): sin(3*z)*cos(4*y)*funcx[ST.boundary_condition()], (2, 1): sin(2*z)*cos(4*x)*funcy[ST.boundary_condition()], (2, 2): sin(2*x)*cos(4*y)*funcz[ST.boundary_condition()] } syms = {1: (x, y), 2:(x, y, z)} points = None if comm.Get_rank() == 0: points = np.random.random((dim+1, 4)) points = comm.bcast(points) t_0 = 0 t_1 = 0 t_2 = 0 for shape in product(*([sizes]*dim)): #for shape in ((64, 64),): bases = [] for n in shape[:-1]: bases.append(FunctionSpace(n, 'F', dtype=typecode.upper())) bases.append(FunctionSpace(shape[-1], 'F', dtype=typecode)) for axis in range(dim+1): #for axis in (0,): ST0 = ST(shape[-1], quad=quad) bases.insert(axis, ST0) # Spectral space must be aligned in nonperiodic direction, hence axes fft = TensorProductSpace(comm, bases, dtype=typecode, axes=axes[dim][axis]) #print('axes', axes[dim][axis]) #print('bases', bases) #print(bases[0].axis, bases[1].axis) X = fft.local_mesh(True) ue = funcs[(dim, axis)] ul = lambdify(syms[dim], ue, 'numpy') uq = ul(*points).astype(typecode) u_hat = Function(fft, buffer=ue) t0 = time() result = fft.eval(points, u_hat, method=0) t_0 += time()-t0 assert np.allclose(uq, result, 0, 1e-3) t0 = time() result = fft.eval(points, u_hat, method=1) t_1 += time()-t0 assert np.allclose(uq, result, 0, 1e-3) t0 = time() result = fft.eval(points, u_hat, method=2) t_2 += time()-t0 assert np.allclose(uq, result, 0, 1e-3), uq/result result = u_hat.eval(points) assert np.allclose(uq, result, 0, 1e-3) bases.pop(axis) fft.destroy() print('method=0', t_0) print('method=1', t_1) print('method=2', t_2)
def get_context(): float, complex, mpitype = datatypes(params.precision) collapse_fourier = False if params.dealias == '3/2-rule' else True dim = len(params.N) dtype = lambda d: float if d == dim-1 else complex V = [Basis(params.N[i], 'F', domain=(0, params.L[i]), dtype=dtype(i)) for i in range(dim)] kw0 = {'threads': params.threads, 'planner_effort': params.planner_effort['fft']} T = TensorProductSpace(comm, V, dtype=float, slab=(params.decomposition == 'slab'), collapse_fourier=collapse_fourier, **kw0) VT = VectorTensorProductSpace(T) VM = MixedTensorProductSpace([T]*2*dim) kw = {'padding_factor': 1.5 if params.dealias == '3/2-rule' else 1, 'dealias_direct': params.dealias == '2/3-rule'} Vp = [Basis(params.N[i], 'F', domain=(0, params.L[i]), dtype=dtype(i), **kw) for i in range(dim)] Tp = TensorProductSpace(comm, Vp, dtype=float, slab=(params.decomposition == 'slab'), collapse_fourier=collapse_fourier, **kw0) VTp = VectorTensorProductSpace(Tp) VMp = MixedTensorProductSpace([Tp]*2*dim) # Mesh variables X = T.local_mesh(True) K = T.local_wavenumbers(scaled=True) for i in range(dim): X[i] = X[i].astype(float) K[i] = K[i].astype(float) K2 = np.zeros(T.shape(True), dtype=float) for i in range(dim): K2 += K[i]*K[i] # Set Nyquist frequency to zero on K that is, from now on, used for odd derivatives Kx = T.local_wavenumbers(scaled=True, eliminate_highest_freq=True) for i in range(dim): Kx[i] = Kx[i].astype(float) K_over_K2 = np.zeros(VT.shape(True), dtype=float) for i in range(dim): K_over_K2[i] = K[i] / np.where(K2 == 0, 1, K2) UB = Array(VM) P = Array(T) curl = Array(VT) UB_hat = Function(VM) P_hat = Function(T) dU = Function(VM) Source = Array(VM) ub_dealias = Array(VMp) ZZ_hat = np.zeros((3, 3) + Tp.shape(True), dtype=complex) # Work array # Create views into large data structures U = UB[:3] U_hat = UB_hat[:3] B = UB[3:] B_hat = UB_hat[3:] # Primary variable u = UB_hat hdf5file = MHDFile(config.params.solver, checkpoint={'space': VM, 'data': {'0': {'UB': [UB_hat]}}}, results={'space': VM, 'data': {'UB': [UB]}}) return config.AttributeDict(locals())
def get_context(): float, complex, mpitype = datatypes(params.precision) collapse_fourier = False if params.dealias == '3/2-rule' else True dim = len(params.N) dtype = lambda d: float if d == dim-1 else complex V = [FunctionSpace(params.N[i], 'F', domain=(0, params.L[i]), dtype=dtype(i)) for i in range(dim)] kw0 = {'threads': params.threads, 'planner_effort': params.planner_effort['fft']} T = TensorProductSpace(comm, V, dtype=float, slab=(params.decomposition == 'slab'), collapse_fourier=collapse_fourier, **kw0) VT = VectorTensorProductSpace(T) VM = MixedTensorProductSpace([T]*2*dim) mask = T.get_mask_nyquist() if params.mask_nyquist else None kw = {'padding_factor': 1.5 if params.dealias == '3/2-rule' else 1, 'dealias_direct': params.dealias == '2/3-rule'} Vp = [FunctionSpace(params.N[i], 'F', domain=(0, params.L[i]), dtype=dtype(i), **kw) for i in range(dim)] Tp = TensorProductSpace(comm, Vp, dtype=float, slab=(params.decomposition == 'slab'), collapse_fourier=collapse_fourier, **kw0) VTp = VectorTensorProductSpace(Tp) VMp = MixedTensorProductSpace([Tp]*2*dim) # Mesh variables X = T.local_mesh(True) K = T.local_wavenumbers(scaled=True) for i in range(dim): X[i] = X[i].astype(float) K[i] = K[i].astype(float) K2 = np.zeros(T.shape(True), dtype=float) for i in range(dim): K2 += K[i]*K[i] K_over_K2 = np.zeros(VT.shape(True), dtype=float) for i in range(dim): K_over_K2[i] = K[i] / np.where(K2 == 0, 1, K2) UB = Array(VM) P = Array(T) curl = Array(VT) UB_hat = Function(VM) P_hat = Function(T) dU = Function(VM) Source = Array(VM) ub_dealias = Array(VMp) ZZ_hat = np.zeros((3, 3) + Tp.shape(True), dtype=complex) # Work array # Create views into large data structures U = UB[:3] U_hat = UB_hat[:3] B = UB[3:] B_hat = UB_hat[3:] # Primary variable u = UB_hat hdf5file = MHDFile(config.params.solver, checkpoint={'space': VM, 'data': {'0': {'UB': [UB_hat]}}}, results={'space': VM, 'data': {'UB': [UB]}}) return config.AttributeDict(locals())
def get_context(): """Set up context for classical (NS) solver""" V0 = C2CBasis(params.N[0], domain=(0, params.L[0])) V1 = C2CBasis(params.N[1], domain=(0, params.L[1])) V2 = R2CBasis(params.N[2], domain=(0, params.L[2])) T = TensorProductSpace(comm, (V0, V1, V2), **{'threads': params.threads}) VT = VectorTensorProductSpace([T] * 3) kw = { 'padding_factor': 1.5 if params.dealias == '3/2-rule' else 1, 'dealias_direct': params.dealias == '2/3-rule' } V0p = C2CBasis(params.N[0], domain=(0, params.L[0]), **kw) V1p = C2CBasis(params.N[1], domain=(0, params.L[1]), **kw) V2p = R2CBasis(params.N[2], domain=(0, params.L[2]), **kw) Tp = TensorProductSpace(comm, (V0p, V1p, V2p), **{'threads': params.threads}) VTp = VectorTensorProductSpace([Tp] * 3) float, complex, mpitype = datatypes(params.precision) FFT = T # For compatibility - to be removed # Mesh variables X = T.local_mesh(True) K = T.local_wavenumbers(scaled=True) K2 = K[0] * K[0] + K[1] * K[1] + K[2] * K[2] # Set Nyquist frequency to zero on K that is, from now on, used for odd derivatives Kx = T.local_wavenumbers(scaled=True, eliminate_highest_freq=True) K_over_K2 = np.zeros((3, ) + VT.local_shape()) for i in range(3): K_over_K2[i] = K[i] / np.where(K2 == 0, 1, K2) # Velocity and pressure U = Array(VT, False) U_hat = Array(VT) P = Array(T, False) P_hat = Array(T) # Primary variable u = U_hat # RHS array dU = Array(VT) curl = Array(VT, False) Source = Array(VT) # Possible source term initialized to zero work = work_arrays() hdf5file = NSWriter({ "U": U[0], "V": U[1], "W": U[2], "P": P }, chkpoint={ "current": { "U": U, "P": P }, "previous": {} }, filename=params.h5filename + ".h5") return config.AttributeDict(locals())
def test_eval_tensor(typecode, dim, ST, quad): # Using sympy to compute an analytical solution # Testing for Dirichlet and regular basis x, y, z = symbols("x,y,z") sizes = (25, 24) funcx = (x**2 - 1) * cos(2 * np.pi * x) funcy = (y**2 - 1) * cos(2 * np.pi * y) funcz = (z**2 - 1) * cos(2 * np.pi * z) funcs = { (1, 0): cos(4 * y) * funcx, (1, 1): cos(4 * x) * funcy, (2, 0): (sin(6 * z) + cos(4 * y)) * funcx, (2, 1): (sin(2 * z) + cos(4 * x)) * funcy, (2, 2): (sin(2 * x) + cos(4 * y)) * funcz } syms = {1: (x, y), 2: (x, y, z)} points = np.array([[0.1] * (dim + 1), [0.01] * (dim + 1), [0.4] * (dim + 1), [0.5] * (dim + 1)]) for shape in product(*([sizes] * dim)): bases = [] for n in shape[:-1]: bases.append(Basis(n, 'F', dtype=typecode.upper())) bases.append(Basis(shape[-1], 'F', dtype=typecode)) if dim < 3: n = min(shape) if typecode in 'fdg': n //= 2 n += 1 if n < comm.size: continue for axis in range(dim + 1): ST0 = ST(shape[-1], quad=quad) bases.insert(axis, ST0) # Spectral space must be aligned in nonperiodic direction, hence axes fft = TensorProductSpace(comm, bases, dtype=typecode, axes=axes[dim][axis]) X = fft.local_mesh(True) ue = funcs[(dim, axis)] ul = lambdify(syms[dim], ue, 'numpy') uu = ul(*X).astype(typecode) uq = ul(*points.T).astype(typecode) u_hat = Function(fft) u_hat = fft.forward(uu, u_hat) result = fft.eval(points, u_hat, cython=True) assert np.allclose(uq, result, 0, 1e-6) result = fft.eval(points, u_hat, cython=False) assert np.allclose(uq, result, 0, 1e-6) result = u_hat.eval(points) assert np.allclose(uq, result, 0, 1e-6) ua = u_hat.backward() assert np.allclose(uu, ua, 0, 1e-6) ua = Array(fft) ua = u_hat.backward(ua) assert np.allclose(uu, ua, 0, 1e-6) bases.pop(axis) fft.destroy()
def test_project2(typecode, dim, ST, quad): # Using sympy to compute an analytical solution x, y, z = symbols("x,y,z") sizes = (25, 24) funcx = ((2 * np.pi**2 * (x**2 - 1) - 1) * cos(2 * np.pi * x) - 2 * np.pi * x * sin(2 * np.pi * x)) / (4 * np.pi**3) funcy = ((2 * np.pi**2 * (y**2 - 1) - 1) * cos(2 * np.pi * y) - 2 * np.pi * y * sin(2 * np.pi * y)) / (4 * np.pi**3) funcz = ((2 * np.pi**2 * (z**2 - 1) - 1) * cos(2 * np.pi * z) - 2 * np.pi * z * sin(2 * np.pi * z)) / (4 * np.pi**3) funcs = { (1, 0): cos(4 * y) * funcx, (1, 1): cos(4 * x) * funcy, (2, 0): sin(6 * z) * cos(4 * y) * funcx, (2, 1): sin(2 * z) * cos(4 * x) * funcy, (2, 2): sin(2 * x) * cos(4 * y) * funcz } syms = {1: (x, y), 2: (x, y, z)} xs = {0: x, 1: y, 2: z} for shape in product(*([sizes] * dim)): bases = [] for n in shape[:-1]: bases.append(Basis(n, 'F', dtype=typecode.upper())) bases.append(Basis(shape[-1], 'F', dtype=typecode)) if dim < 3: n = min(shape) if typecode in 'fdg': n //= 2 n += 1 if n < comm.size: continue for axis in range(dim + 1): ST0 = ST(shape[-1], quad=quad) bases.insert(axis, ST0) # Spectral space must be aligned in nonperiodic direction, hence axes fft = TensorProductSpace(comm, bases, dtype=typecode, axes=axes[dim][axis]) X = fft.local_mesh(True) ue = funcs[(dim, axis)] ul = lambdify(syms[dim], ue, 'numpy') uq = ul(*X).astype(typecode) uh = Function(fft) uh = fft.forward(uq, uh) due = ue.diff(xs[axis], 1) dul = lambdify(syms[dim], due, 'numpy') duq = dul(*X).astype(typecode) uf = project(Dx(uh, axis, 1), fft) uy = Array(fft) uy = fft.backward(uf, uy) assert np.allclose(uy, duq, 0, 1e-6) # Test also several derivatives for ax in (x for x in range(dim + 1) if x is not axis): due = ue.diff(xs[ax], 1, xs[axis], 1) dul = lambdify(syms[dim], due, 'numpy') duq = dul(*X).astype(typecode) uf = project(Dx(Dx(uh, ax, 1), axis, 1), fft) uy = Array(fft) uy = fft.backward(uf, uy) assert np.allclose(uy, duq, 0, 1e-6) bases.pop(axis) fft.destroy()