def g(self, lam): amp_min = self.params['amp_min'] amp_max = self.params['amp_max'] lam_threslo = self.params['lam_threslo'] lam_maxlo = self.params['lam_maxlo'] lam_threshi = self.params['lam_threshi'] lam_maxhi = self.params['lam_maxhi'] # Diss Hirschvogel eq. 2.107 # TeX: g(\lambda_{\mathrm{myo}}) = \begin{cases} a_{\mathrm{min}}, & \lambda_{\mathrm{myo}} \leq \hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,lo}}, \\ a_{\mathrm{min}}+\frac{1}{2}\left(a_{\mathrm{max}}-a_{\mathrm{min}}\right)\left(1-\cos \frac{\pi(\lambda_{\mathrm{myo}}-\hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,lo}})}{\hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,lo}}-\hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,lo}}}\right), & \hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,lo}} \leq \lambda_{\mathrm{myo}} \leq \hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,lo}}, \\ a_{\mathrm{max}}, & \hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,lo}} \leq \lambda_{\mathrm{myo}} \leq \hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,hi}}, \\ a_{\mathrm{min}}+\frac{1}{2}\left(a_{\mathrm{max}}-a_{\mathrm{min}}\right)\left(1-\cos \frac{\pi(\lambda_{\mathrm{myo}}-\hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,hi}})}{\hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,hi}}-\hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,hi}}}\right), & \hat{\lambda}_{\mathrm{myo}}^{\mathrm{thres,hi}} \leq \lambda_{\mathrm{myo}} \leq \hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,hi}}, \\ a_{\mathrm{min}}, & \lambda_{\mathrm{myo}} \geq \hat{\lambda}_{\mathrm{myo}}^{\mathrm{max,hi}} \end{cases} return conditional( le(lam, lam_threslo), amp_min, conditional( And(ge(lam, lam_threslo), le(lam, lam_maxlo)), amp_min + 0.5 * (amp_max - amp_min) * (1. - cos(pi * (lam - lam_threslo) / (lam_maxlo - lam_threslo))), conditional( And(ge(lam, lam_maxlo), le(lam, lam_threshi)), amp_max, conditional( And(ge(lam, lam_threshi), le(lam, lam_maxhi)), amp_min + 0.5 * (amp_max - amp_min) * (1. - cos(pi * (lam - lam_maxhi) / (lam_maxhi - lam_threshi))), conditional(ge(lam, lam_maxhi), amp_min, as_ufl(0))))))
def rotation_matrix_2d(angle): """ Rotation matrix associated with some ``angle``, as a UFL matrix. """ return ufl.as_matrix( [[ufl.cos(angle), -ufl.sin(angle)], [ufl.sin(angle), ufl.cos(angle)]] )
def test_complex_assembly_solve(): """Solve a positive definite helmholtz problem and verify solution with the method of manufactured solutions """ degree = 3 mesh = dolfinx.generation.UnitSquareMesh(dolfinx.MPI.comm_world, 20, 20) P = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), degree) V = dolfinx.function.FunctionSpace(mesh, P) x = SpatialCoordinate(mesh) # Define source term A = 1.0 + 2.0 * (2.0 * np.pi)**2 f = (1. + 1j) * A * ufl.cos(2 * np.pi * x[0]) * ufl.cos(2 * np.pi * x[1]) # Variational problem u = ufl.TrialFunction(V) v = ufl.TestFunction(V) C = 1.0 + 1.0j a = C * inner(grad(u), grad(v)) * dx + C * inner(u, v) * dx L = inner(f, v) * dx # Assemble A = dolfinx.fem.assemble_matrix(a) A.assemble() b = dolfinx.fem.assemble_vector(L) b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE) # Create solver solver = PETSc.KSP().create(mesh.mpi_comm()) solver.setOptionsPrefix("test_lu_") opts = PETSc.Options("test_lu_") opts["ksp_type"] = "preonly" opts["pc_type"] = "lu" solver.setFromOptions() x = A.createVecRight() solver.setOperators(A) solver.solve(b, x) # Reference Solution def ref_eval(x): return np.cos(2 * np.pi * x[0]) * np.cos(2 * np.pi * x[1]) u_ref = dolfinx.function.Function(V) u_ref.interpolate(ref_eval) diff = (x - u_ref.vector).norm(PETSc.NormType.N2) assert diff == pytest.approx(0.0, abs=1e-1)
def test_complex_assembly_solve(): """Solve a positive definite helmholtz problem and verify solution with the method of manufactured solutions """ degree = 3 mesh = dolfin.generation.UnitSquareMesh(dolfin.MPI.comm_world, 20, 20) P = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), degree) V = dolfin.functionspace.FunctionSpace(mesh, P) x = SpatialCoordinate(mesh) # Define source term A = 1 + 2 * (2 * np.pi)**2 f = (1. + 1j) * A * ufl.cos(2 * np.pi * x[0]) * ufl.cos(2 * np.pi * x[1]) # Variational problem u = dolfin.function.TrialFunction(V) v = dolfin.function.TestFunction(V) C = 1 + 1j a = C * inner(grad(u), grad(v)) * dx + C * inner(u, v) * dx L = inner(f, v) * dx # Assemble A = dolfin.fem.assemble_matrix(a) A.assemble() b = dolfin.fem.assemble_vector(L) b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE) # Create solver solver = PETSc.KSP().create(mesh.mpi_comm()) opts = PETSc.Options() opts["ksp_type"] = "preonly" opts["pc_type"] = "lu" solver.setFromOptions() x = A.createVecRight() solver.setOperators(A) solver.solve(b, x) # Reference Solution def ref_eval(values, x): values[:, 0] = np.cos(2 * np.pi * x[:, 0]) * np.cos(2 * np.pi * x[:, 1]) u_ref = dolfin.interpolate(ref_eval, V) xnorm = x.norm(PETSc.NormType.N2) x_ref_norm = u_ref.vector.norm(PETSc.NormType.N2) assert np.isclose(xnorm, x_ref_norm)
def eigenstate_legacy(A): """Eigenvalues and eigenprojectors of the 3x3 (real-valued) tensor A. Provides the spectral decomposition A = sum_{a=0}^{2} λ_a * E_a with eigenvalues λ_a and their associated eigenprojectors E_a = n_a^R x n_a^L ordered by magnitude. The eigenprojectors of eigenvalues with multiplicity n are returned as 1/n-fold projector. Note: Tensor A must not have complex eigenvalues! """ if ufl.shape(A) != (3, 3): raise RuntimeError( f"Tensor A of shape {ufl.shape(A)} != (3, 3) is not supported!") # eps = 1.0e-10 # A = ufl.variable(A) # # --- determine eigenvalues λ0, λ1, λ2 # # additively decompose: A = tr(A) / 3 * I + dev(A) = q * I + B q = ufl.tr(A) / 3 B = A - q * ufl.Identity(3) # observe: det(λI - A) = 0 with shift λ = q + ω --> det(ωI - B) = 0 = ω**3 - j * ω - b j = ufl.tr( B * B ) / 2 # == -I2(B) for trace-free B, j < 0 indicates A has complex eigenvalues b = ufl.tr(B * B * B) / 3 # == I3(B) for trace-free B # solve: 0 = ω**3 - j * ω - b by substitution ω = p * cos(phi) # 0 = p**3 * cos**3(phi) - j * p * cos(phi) - b | * 4 / p**3 # 0 = 4 * cos**3(phi) - 3 * cos(phi) - 4 * b / p**3 | --> p := sqrt(j * 4 / 3) # 0 = cos(3 * phi) - 4 * b / p**3 # 0 = cos(3 * phi) - r with -1 <= r <= +1 # phi_k = [acos(r) + (k + 1) * 2 * pi] / 3 for k = 0, 1, 2 p = 2 / ufl.sqrt(3) * ufl.sqrt(j + eps**2) # eps: MMM r = 4 * b / p**3 r = ufl.Max(ufl.Min(r, +1 - eps), -1 + eps) # eps: LMM, MMH phi = ufl.acos(r) / 3 # sorted eigenvalues: λ0 <= λ1 <= λ2 λ0 = q + p * ufl.cos(phi + 2 / 3 * ufl.pi) # low λ1 = q + p * ufl.cos(phi + 4 / 3 * ufl.pi) # middle λ2 = q + p * ufl.cos(phi) # high # # --- determine eigenprojectors E0, E1, E2 # E0 = ufl.diff(λ0, A).T E1 = ufl.diff(λ1, A).T E2 = ufl.diff(λ2, A).T # return [λ0, λ1, λ2], [E0, E1, E2]
def test_complex_assembly_solve(): """Solve a positive definite helmholtz problem and verify solution with the method of manufactured solutions """ degree = 3 mesh = dolfin.generation.UnitSquareMesh(dolfin.MPI.comm_world, 20, 20) P = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), degree) V = dolfin.function.functionspace.FunctionSpace(mesh, P) x = dolfin.SpatialCoordinate(mesh) # Define source term A = 1 + 2 * (2 * np.pi)**2 f = (1. + 1j) * A * ufl.cos(2 * np.pi * x[0]) * ufl.cos(2 * np.pi * x[1]) # Variational problem u = dolfin.function.argument.TrialFunction(V) v = dolfin.function.argument.TestFunction(V) C = 1 + 1j a = C * inner(grad(u), grad(v)) * dx + C * inner(u, v) * dx L = inner(f, v) * dx # Assemble A = dolfin.fem.assemble(a) b = dolfin.fem.assemble(L) # Create solver solver = dolfin.cpp.la.PETScKrylovSolver(mesh.mpi_comm()) dolfin.cpp.la.PETScOptions.set("ksp_type", "preonly") dolfin.cpp.la.PETScOptions.set("pc_type", "lu") solver.set_from_options() x = dolfin.cpp.la.PETScVector() solver.set_operator(A) solver.solve(x, b) # Reference Solution @dolfin.function.expression.numba_eval def ref_eval(values, x, cell_idx): values[:, 0] = np.cos(2 * np.pi * x[:, 0]) * np.cos(2 * np.pi * x[:, 1]) u_ref = dolfin.interpolate(dolfin.Expression(ref_eval), V) xnorm = x.norm(dolfin.cpp.la.Norm.l2) x_ref_norm = u_ref.vector().norm(dolfin.cpp.la.Norm.l2) assert np.isclose(xnorm, x_ref_norm)
def assemble_solution(self, t): # returns """ :param t: time :return: Womersley flow (analytic solution) at time t analytic solution at any time is a steady parabolic flow + linear combination of 8 modes modes were precomputed as 8 functions on given mesh and stored in hdf5 file """ if self.tc is not None: self.tc.start('assembleSol') sol = Function(self.solutionSpace) # analytic solution has zero x and y components dofs2 = self.solutionSpace.sub(2).dofmap().dofs( ) # gives field of indices corresponding to z axis sol.assign(Constant(("0.0", "0.0", "0.0"))) # QQ not needed sol.vector()[dofs2] += self.factor * self.bessel_parabolic.vector( ).array() # parabolic part of sol for idx in range(8): # add modes of Womersley sol sol.vector()[dofs2] += self.factor * cos( self.coefs_exp[idx] * pi * t) * self.bessel_real[idx].vector().array() sol.vector()[dofs2] += self.factor * -sin( self.coefs_exp[idx] * pi * t) * self.bessel_complex[idx].vector().array() if self.tc is not None: self.tc.end('assembleSol') return sol
def test_dolfin_expression_compilation_of_math_functions(dolfin): # Define some PyDOLFIN coefficients mesh = dolfin.UnitSquareMesh(3, 3) # Using quadratic element deliberately for accuracy V = dolfin.FunctionSpace(mesh, "CG", 2) u = dolfin.Function(V) u.interpolate(dolfin.Expression("x[0]*x[1]")) w0 = u # Define ufl expression with math functions v = abs(ufl.cos(u))/2 + 0.02 uexpr = ufl.sin(u) + ufl.tan(v) + ufl.exp(u) + ufl.ln(v) + ufl.atan(v) + ufl.acos(v) + ufl.asin(v) #print dolfin.assemble(uexpr**2*dolfin.dx, mesh=mesh) # 11.7846508409 # Define expected output from compilation ucode = 'v_w0[0]' vcode = '0.02 + fabs(cos(v_w0[0])) / 2' funcs = 'asin(%(v)s) + (acos(%(v)s) + (atan(%(v)s) + (log(%(v)s) + (exp(%(u)s) + (sin(%(u)s) + tan(%(v)s))))))' oneliner = funcs % {'u':ucode, 'v':vcode} # Oneliner version (ignoring reuse): expected_lines = ['double s[1];', 'Array<double> v_w0(1);', 'w0->eval(v_w0, x);', 's[0] = %s;' % oneliner, 'values[0] = s[0];'] #cppcode = format_dolfin_expression(classname="DebugExpression", shape=(), eval_body=expected_lines) #print '-'*100 #print cppcode #print '-'*100 #dolfin.plot(dolfin.Expression(cppcode=cppcode, mesh=mesh)) #dolfin.interactive() # Split version (handles reuse of v, no other reuse): expected_lines = ['double s[2];', 'Array<double> v_w0(1);', 'w0->eval(v_w0, x);', 's[0] = %s;' % (vcode,), 's[1] = %s;' % (funcs % {'u':ucode,'v':'s[0]'},), 'values[0] = s[1];'] # Define expected evaluation values: [(x,value), (x,value), ...] import math x, y = 0.6, 0.7 u = x*y v = abs(math.cos(u))/2 + 0.02 v0 = .52 expected0 = math.tan(v0) + 1 + math.log(v0) + math.atan(v0) + math.acos(v0) + math.asin(v0) expected = math.sin(u) + math.tan(v) + math.exp(u) + math.log(v) + math.atan(v) + math.acos(v) + math.asin(v) expected_values = [((0.0, 0.0), (expected0,)), ((x, y), (expected,)), ] # Execute all tests check_dolfin_expression_compilation(uexpr, expected_lines, expected_values, members={'w0':w0})
def test_latex_formatting_of_cmath(): x = ufl.SpatialCoordinate(ufl.triangle)[0] assert expr2latex(ufl.exp(x)) == r"e^{x_0}" assert expr2latex(ufl.ln(x)) == r"\ln(x_0)" assert expr2latex(ufl.sqrt(x)) == r"\sqrt{x_0}" assert expr2latex(abs(x)) == r"\|x_0\|" assert expr2latex(ufl.sin(x)) == r"\sin(x_0)" assert expr2latex(ufl.cos(x)) == r"\cos(x_0)" assert expr2latex(ufl.tan(x)) == r"\tan(x_0)" assert expr2latex(ufl.asin(x)) == r"\arcsin(x_0)" assert expr2latex(ufl.acos(x)) == r"\arccos(x_0)" assert expr2latex(ufl.atan(x)) == r"\arctan(x_0)"
def a(phi): # phi = 1e5*phi n = ufl.grad(phi) / ufl.sqrt(ufl.dot(ufl.grad(phi), ufl.grad(phi)) + 1e-5) theta = ufl.atan_2(n[1], n[0]) n = 1e5 * n + ufl.as_vector([1e-5, 1e-5]) xx = n[1] / n[0] # theta = ufl.asin(xx/ ufl.sqrt(1 + xx**2)) theta = ufl.atan(xx) return 1.0 + epsilon_4 * ufl.cos(m * (theta - theta_0))
def assemble_solution(self, t): # returns Womersley sol for time t if self.tc is not None: self.tc.start('assembleSol') sol = Function(self.solutionSpace) dofs2 = self.solutionSpace.sub(2).dofmap().dofs() # gives field of indices corresponding to z axis sol.assign(Constant(("0.0", "0.0", "0.0"))) # QQ not needed sol.vector()[dofs2] += self.factor * self.bessel_parabolic.vector().array() # parabolic part of sol for idx in range(8): # add modes of Womersley sol sol.vector()[dofs2] += self.factor * cos(self.coefs_exp[idx] * pi * t) * self.bessel_real[idx].vector().array() sol.vector()[dofs2] += self.factor * -sin(self.coefs_exp[idx] * pi * t) * self.bessel_complex[idx].vector().array() if self.tc is not None: self.tc.end('assembleSol') return sol
def test_cpp_formatting_of_cmath(): x, y = ufl.SpatialCoordinate(ufl.triangle) # Test cmath functions assert expr2cpp(ufl.exp(x)) == "exp(x[0])" assert expr2cpp(ufl.ln(x)) == "log(x[0])" assert expr2cpp(ufl.sqrt(x)) == "sqrt(x[0])" assert expr2cpp(abs(x)) == "fabs(x[0])" assert expr2cpp(ufl.sin(x)) == "sin(x[0])" assert expr2cpp(ufl.cos(x)) == "cos(x[0])" assert expr2cpp(ufl.tan(x)) == "tan(x[0])" assert expr2cpp(ufl.asin(x)) == "asin(x[0])" assert expr2cpp(ufl.acos(x)) == "acos(x[0])" assert expr2cpp(ufl.atan(x)) == "atan(x[0])"
def eig(A): """Eigenvalues of 3x3 tensor""" eps = 1.0e-12 q = ufl.tr(A) / 3.0 p1 = 0.5 * (A[0, 1]**2 + A[1, 0]**2 + A[0, 2]**2 + A[2, 0]**2 + A[1, 2]**2 + A[2, 1]**2) p2 = (A[0, 0] - q)**2 + (A[1, 1] - q)**2 + (A[2, 2] - q)**2 + 2 * p1 p = ufl.sqrt(p2 / 6) B = (A - q * ufl.Identity(3)) r = ufl.det(B) / (2 * p**3) r = ufl.Max(ufl.Min(r, 1.0 - eps), -1.0 + eps) phi = ufl.acos(r) / 3.0 eig0 = ufl.conditional(p2 < eps, q, q + 2 * p * ufl.cos(phi)) eig2 = ufl.conditional(p2 < eps, q, q + 2 * p * ufl.cos(phi + (2 * numpy.pi / 3))) eig1 = ufl.conditional(p2 < eps, q, 3 * q - eig0 - eig2) # since trace(A) = eig1 + eig2 + eig3 return eig0, eig1, eig2
def test_is_zero_simple_scalar_expressions(): mesh = UnitSquareMesh(4, 4) V = FunctionSpace(mesh, 'CG', 1) v = TestFunction(V) u = TrialFunction(V) check_is_zero(Zero() * u * v, 0) check_is_zero(Constant(0) * u * v, 1) check_is_zero(Zero() * v, 0) check_is_zero(Constant(0) * v, 1) check_is_zero(0 * u * v, 0) check_is_zero(0 * v, 0) check_is_zero(ufl.sin(0) * v, 0) check_is_zero(ufl.cos(0) * v, 1)
def test_comparison_checker(self): cell = triangle element = FiniteElement("Lagrange", cell, 1) u = TrialFunction(element) v = TestFunction(element) a = conditional(ge(abs(u), imag(v)), u, v) b = conditional(le(sqrt(abs(u)), imag(v)), as_ufl(1), as_ufl(1j)) c = conditional(gt(abs(u), pow(imag(v), 0.5)), sin(u), cos(v)) d = conditional(lt(as_ufl(-1), as_ufl(1)), u, v) e = max_value(as_ufl(0), real(u)) f = min_value(sin(u), cos(v)) g = min_value(sin(pow(u, 3)), cos(abs(v))) assert do_comparison_check(a) == conditional(ge(real(abs(u)), real(imag(v))), u, v) with pytest.raises(ComplexComparisonError): b = do_comparison_check(b) with pytest.raises(ComplexComparisonError): c = do_comparison_check(c) assert do_comparison_check(d) == conditional(lt(real(as_ufl(-1)), real(as_ufl(1))), u, v) assert do_comparison_check(e) == max_value(real(as_ufl(0)), real(real(u))) assert do_comparison_check(f) == min_value(real(sin(u)), real(cos(v))) assert do_comparison_check(g) == min_value(real(sin(pow(u, 3))), real(cos(abs(v))))
def __init__(self, mesh, k: int, omega, c, c0, lumped): P = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), k) self.V = FunctionSpace(mesh, P) self.u, self.v = Function(self.V), Function(self.V) self.g1 = Function(self.V) self.g2 = Function(self.V) self.omega = omega self.c = c self.c0 = c0 n = FacetNormal(mesh) # Pieces for plane wave incident field x = ufl.geometry.SpatialCoordinate(mesh) cos_wave = ufl.cos(self.omega / self.c0 * x[0]) sin_wave = ufl.sin(self.omega / self.c0 * x[0]) plane_wave = self.g1 * cos_wave + self.g2 * sin_wave dv, p = TrialFunction(self.V), TestFunction(self.V) self.L1 = - inner(grad(self.u), grad(p)) * dx(degree=k) \ - (1 / self.c) * inner(self.v, p) * ds \ - (1 / self.c**2) * (-self.omega**2) * inner(plane_wave, p) * dx \ - inner(grad(plane_wave), grad(p)) * dx \ + inner(dot(grad(plane_wave), n), p) * ds # Vector to be re-used for assembly self.b = None # TODO: precompile/pre-process Form L self.lumped = lumped if self.lumped: a = (1 / self.c**2) * p * dx(degree=k) self.M = dolfinx.fem.assemble_vector(a) self.M.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE) else: a = (1 / self.c**2) * inner(dv, p) * dx(degree=k) M = dolfinx.fem.assemble_matrix(a) M.assemble() self.solver = PETSc.KSP().create(mesh.mpi_comm()) opts = PETSc.Options() opts["ksp_type"] = "cg" opts["ksp_rtol"] = 1.0e-8 self.solver.setFromOptions() self.solver.setOperators(M)
def assemble_solution(self, t): # returns """ :param t: time :return: Womersley flow (analytic solution) at time t analytic solution at any time is a steady parabolic flow + linear combination of 8 modes modes were precomputed as 8 functions on given mesh and stored in hdf5 file """ if self.tc is not None: self.tc.start('assembleSol') sol = Function(self.solutionSpace) # analytic solution has zero x and y components dofs2 = self.solutionSpace.sub(2).dofmap().dofs() # gives field of indices corresponding to z axis sol.assign(Constant(("0.0", "0.0", "0.0"))) # QQ not needed sol.vector()[dofs2] += self.factor * self.bessel_parabolic.vector().array() # parabolic part of sol for idx in range(8): # add modes of Womersley sol sol.vector()[dofs2] += self.factor * cos(self.coefs_exp[idx] * pi * t) * self.bessel_real[idx].vector().array() sol.vector()[dofs2] += self.factor * -sin(self.coefs_exp[idx] * pi * t) * self.bessel_complex[idx].vector().array() if self.tc is not None: self.tc.end('assembleSol') return sol
def test_cpp_formatting_precedence_handling(): x, y = ufl.SpatialCoordinate(ufl.triangle) # Test precedence handling with sums # Note that the automatic sorting is reflected in formatting! assert expr2cpp(y + (2 + x)) == "x[1] + (2 + x[0])" assert expr2cpp((x + 2) + y) == "x[1] + (2 + x[0])" assert expr2cpp((2 + x) + (3 + y)) == "(2 + x[0]) + (3 + x[1])" assert expr2cpp((x + 3) + 2 + y) == "x[1] + (2 + (3 + x[0]))" assert expr2cpp(2 + (x + 3) + y) == "x[1] + (2 + (3 + x[0]))" assert expr2cpp(2 + (3 + x) + y) == "x[1] + (2 + (3 + x[0]))" assert expr2cpp(y + (2 + (3 + x))) == "x[1] + (2 + (3 + x[0]))" assert expr2cpp(2 + x + 3 + y) == "x[1] + (3 + (2 + x[0]))" assert expr2cpp(2 + x + 3 + y) == "x[1] + (3 + (2 + x[0]))" # Test precedence handling with divisions # This is more stable than sums since there is no sorting. assert expr2cpp((x / 2) / 3) == "(x[0] / 2) / 3" assert expr2cpp(x / (y / 3)) == "x[0] / (x[1] / 3)" assert expr2cpp((x / 2) / (y / 3)) == "(x[0] / 2) / (x[1] / 3)" assert expr2cpp(x / (2 / y) / 3) == "(x[0] / (2 / x[1])) / 3" # Test precedence handling with highest level types assert expr2cpp(ufl.sin(x)) == "sin(x[0])" assert expr2cpp(ufl.cos(x + 2)) == "cos(2 + x[0])" assert expr2cpp(ufl.tan(x / 2)) == "tan(x[0] / 2)" assert expr2cpp(ufl.acos(x + 3 * y)) == "acos(x[0] + 3 * x[1])" assert expr2cpp(ufl.asin(ufl.atan(x**4))) == "asin(atan(pow(x[0], 4)))" assert expr2cpp(ufl.sin(y) + ufl.tan(x)) == "sin(x[1]) + tan(x[0])" # Test precedence handling with mixed types assert expr2cpp(3 * (2 + x)) == "3 * (2 + x[0])" assert expr2cpp((2 * x) + (3 * y)) == "2 * x[0] + 3 * x[1]" assert expr2cpp(2 * (x + 3) * y) == "x[1] * (2 * (3 + x[0]))" assert expr2cpp(2 * (x + 3)**4 * y) == "x[1] * (2 * pow(3 + x[0], 4))"
def test_latex_formatting_precedence_handling(): x, y = ufl.SpatialCoordinate(ufl.triangle) # Test precedence handling with sums # Note that the automatic sorting is reflected in formatting! assert expr2latex(y + (2 + x)) == "x_1 + (2 + x_0)" assert expr2latex((x + 2) + y) == "x_1 + (2 + x_0)" assert expr2latex((2 + x) + (3 + y)) == "(2 + x_0) + (3 + x_1)" assert expr2latex((x + 3) + 2 + y) == "x_1 + (2 + (3 + x_0))" assert expr2latex(2 + (x + 3) + y) == "x_1 + (2 + (3 + x_0))" assert expr2latex(2 + (3 + x) + y) == "x_1 + (2 + (3 + x_0))" assert expr2latex(y + (2 + (3 + x))) == "x_1 + (2 + (3 + x_0))" assert expr2latex(2 + x + 3 + y) == "x_1 + (3 + (2 + x_0))" assert expr2latex(2 + x + 3 + y) == "x_1 + (3 + (2 + x_0))" # Test precedence handling with divisions # This is more stable than sums since there is no sorting. assert expr2latex((x / 2) / 3) == r"\frac{(\frac{x_0}{2})}{3}" assert expr2latex(x / (y / 3)) == r"\frac{x_0}{(\frac{x_1}{3})}" assert expr2latex((x / 2) / (y / 3)) == r"\frac{(\frac{x_0}{2})}{(\frac{x_1}{3})}" assert expr2latex(x / (2 / y) / 3) == r"\frac{(\frac{x_0}{(\frac{2}{x_1})})}{3}" # Test precedence handling with highest level types assert expr2latex(ufl.sin(x)) == r"\sin(x_0)" assert expr2latex(ufl.cos(x + 2)) == r"\cos(2 + x_0)" assert expr2latex(ufl.tan(x / 2)) == r"\tan(\frac{x_0}{2})" assert expr2latex(ufl.acos(x + 3 * y)) == r"\arccos(x_0 + 3 x_1)" assert expr2latex(ufl.asin(ufl.atan(x**4))) == r"\arcsin(\arctan({x_0}^{4}))" assert expr2latex(ufl.sin(y) + ufl.tan(x)) == r"\sin(x_1) + \tan(x_0)" # Test precedence handling with mixed types assert expr2latex(3 * (2 + x)) == "3 (2 + x_0)" assert expr2latex((2 * x) + (3 * y)) == "2 x_0 + 3 x_1" assert expr2latex(2 * (x + 3) * y) == "x_1 (2 (3 + x_0))"
from __future__ import print_function, division from ufl import as_vector, dot, grad, cos, pi, SpatialCoordinate, triangle from dune.grid import structuredGrid, gridFunction from dune.fem.space import lagrange, combined, product x = SpatialCoordinate(triangle) exact = as_vector([cos(2. * pi * x[0]) * cos(2. * pi * x[1]), dot(x, x)]) grid = structuredGrid([0, 0], [1, 1], [16, 16]) spc1 = lagrange(grid, dimRange=1, order=1) spc2 = lagrange(grid, dimRange=1, order=2) test1 = spc1.interpolate(exact[0], name="test") test2 = spc2.interpolate(exact[1], name="test") spc = combined(spc1, spc2) solution = spc.interpolate(exact, name="solution") space = product(spc1, spc2, components=["p", "s"]) df = space.interpolate(exact, name="df") # print(df.dofVector.size,solution.dofVector.size, # df.components[0].dofVector.size,df.p.dofVector.size,test1.dofVector.size) assert df.components[0].dofVector.size == test1.dofVector.size assert df.s.dofVector.size == test2.dofVector.size assert df.dofVector.size == solution.dofVector.size df.interpolate(solution) solution.interpolate(df) test1.interpolate(df.p) df.s.interpolate(test2) df.components[0].interpolate(solution[0]) df.p.interpolate(solution[0])
from ufl import * import math import dune.fem from dune.fem.function import integrate import dune.create as create from dune.ufl import DirichletBC, Space dimRange = 12 # needs to be >= 4, test with 4,8,11 grid = create.grid("ALUConform", "../data/mixed.dgf", dimgrid=2) from ufl import SpatialCoordinate uflSpace = dune.ufl.Space(2, dimRange) x = SpatialCoordinate(uflSpace.cell()) from math import pi, log, sqrt from ufl import cos, sin, as_vector exact = as_vector( [sin(3 * pi * x[0]), x[1] * x[1], x[0] * x[0], cos(3. * pi * x[1])] + [0] * (dimRange - 4)) v1 = integrate(grid, exact, 5).two_norm space = create.space("Lagrange", grid, dimRange=dimRange, order=1) u = space.interpolate(exact, name="u") v2 = integrate(grid, u, 5).two_norm print(v1, v2, v1 - v2) v3 = integrate(grid, inner(grad(u[11]), grad(u[11])), 5)
def eigenstate(A): """Eigenvalues and eigenprojectors of the 3x3 (real-valued) tensor A. Provides the spectral decomposition A = sum_{a=0}^{2} λ_a * E_a with (ordered) eigenvalues λ_a and their associated eigenprojectors E_a = n_a^R x n_a^L. Note: Tensor A must not have complex eigenvalues! """ if ufl.shape(A) != (3, 3): raise RuntimeError( f"Tensor A of shape {ufl.shape(A)} != (3, 3) is not supported!") # eps = 3.0e-16 # slightly above 2**-(53 - 1), see https://en.wikipedia.org/wiki/IEEE_754 # A = ufl.variable(A) # # --- determine eigenvalues λ0, λ1, λ2 # I1, I2, I3 = invariants_principal(A) dq = 2 * I1**3 - 9 * I1 * I2 + 27 * I3 # Δx = [ A[0, 1] * A[1, 2] * A[2, 0] - A[0, 2] * A[1, 0] * A[2, 1], A[0, 1]**2 * A[1, 2] - A[0, 1] * A[0, 2] * A[1, 1] + A[0, 1] * A[0, 2] * A[2, 2] - A[0, 2]**2 * A[2, 1], A[0, 0] * A[0, 1] * A[2, 1] - A[0, 1]**2 * A[2, 0] - A[0, 1] * A[2, 1] * A[2, 2] + A[0, 2] * A[2, 1]**2, A[0, 0] * A[0, 2] * A[1, 2] + A[0, 1] * A[1, 2]**2 - A[0, 2]**2 * A[1, 0] - A[0, 2] * A[1, 1] * A[1, 2], A[0, 0] * A[0, 1] * A[1, 2] - A[0, 1] * A[0, 2] * A[1, 0] - A[0, 1] * A[1, 2] * A[2, 2] + A[0, 2] * A[1, 2] * A[2, 1], # noqa: E501 A[0, 0] * A[0, 2] * A[2, 1] - A[0, 1] * A[0, 2] * A[2, 0] + A[0, 1] * A[1, 2] * A[2, 1] - A[0, 2] * A[1, 1] * A[2, 1], # noqa: E501 A[0, 1] * A[1, 0] * A[1, 2] - A[0, 2] * A[1, 0] * A[1, 1] + A[0, 2] * A[1, 0] * A[2, 2] - A[0, 2] * A[1, 2] * A[2, 0], # noqa: E501 A[0, 0]**2 * A[1, 2] - A[0, 0] * A[0, 2] * A[1, 0] - A[0, 0] * A[1, 1] * A[1, 2] - A[0, 0] * A[1, 2] * A[2, 2] + A[0, 1] * A[1, 0] * A[1, 2] + A[0, 2] * A[1, 0] * A[2, 2] + A[1, 1] * A[1, 2] * A[2, 2] - A[1, 2]**2 * A[2, 1], # noqa: E501 A[0, 0]**2 * A[1, 2] - A[0, 0] * A[0, 2] * A[1, 0] - A[0, 0] * A[1, 1] * A[1, 2] - A[0, 0] * A[1, 2] * A[2, 2] + A[0, 2] * A[1, 0] * A[1, 1] + A[0, 2] * A[1, 2] * A[2, 0] + A[1, 1] * A[1, 2] * A[2, 2] - A[1, 2]**2 * A[2, 1], # noqa: E501 A[0, 0] * A[0, 1] * A[1, 1] - A[0, 0] * A[0, 1] * A[2, 2] - A[0, 1]**2 * A[1, 0] + A[0, 1] * A[0, 2] * A[2, 0] - A[0, 1] * A[1, 1] * A[2, 2] + A[0, 1] * A[2, 2]**2 + A[0, 2] * A[1, 1] * A[2, 1] - A[0, 2] * A[2, 1] * A[2, 2], # noqa: E501 A[0, 0] * A[0, 1] * A[1, 1] - A[0, 0] * A[0, 1] * A[2, 2] + A[0, 0] * A[0, 2] * A[2, 1] - A[0, 1]**2 * A[1, 0] - A[0, 1] * A[1, 1] * A[2, 2] + A[0, 1] * A[1, 2] * A[2, 1] + A[0, 1] * A[2, 2]**2 - A[0, 2] * A[2, 1] * A[2, 2], # noqa: E501 A[0, 0] * A[0, 1] * A[1, 2] - A[0, 0] * A[0, 2] * A[1, 1] + A[0, 0] * A[0, 2] * A[2, 2] - A[0, 1] * A[1, 1] * A[1, 2] - A[0, 2]**2 * A[2, 0] + A[0, 2] * A[1, 1]**2 - A[0, 2] * A[1, 1] * A[2, 2] + A[0, 2] * A[1, 2] * A[2, 1], # noqa: E501 A[0, 0] * A[0, 2] * A[1, 1] - A[0, 0] * A[0, 2] * A[2, 2] - A[0, 1] * A[0, 2] * A[1, 0] + A[0, 1] * A[1, 1] * A[1, 2] - A[0, 1] * A[1, 2] * A[2, 2] + A[0, 2]**2 * A[2, 0] - A[0, 2] * A[1, 1]**2 + A[0, 2] * A[1, 1] * A[2, 2], # noqa: E501 A[0, 0]**2 * A[1, 1] - A[0, 0]**2 * A[2, 2] - A[0, 0] * A[0, 1] * A[1, 0] + A[0, 0] * A[0, 2] * A[2, 0] - A[0, 0] * A[1, 1]**2 + A[0, 0] * A[2, 2]**2 + A[0, 1] * A[1, 0] * A[1, 1] - A[0, 2] * A[2, 0] * A[2, 2] + A[1, 1]**2 * A[2, 2] - A[1, 1] * A[1, 2] * A[2, 1] - A[1, 1] * A[2, 2]**2 + A[1, 2] * A[2, 1] * A[2, 2] ] # noqa: E501 Δy = [ A[0, 2] * A[1, 0] * A[2, 1] - A[0, 1] * A[1, 2] * A[2, 0], A[1, 0]**2 * A[2, 1] - A[1, 0] * A[1, 1] * A[2, 0] + A[1, 0] * A[2, 0] * A[2, 2] - A[1, 2] * A[2, 0]**2, A[0, 0] * A[1, 0] * A[1, 2] - A[0, 2] * A[1, 0]**2 - A[1, 0] * A[1, 2] * A[2, 2] + A[1, 2]**2 * A[2, 0], A[0, 0] * A[2, 0] * A[2, 1] - A[0, 1] * A[2, 0]**2 + A[1, 0] * A[2, 1]**2 - A[1, 1] * A[2, 0] * A[2, 1], A[0, 0] * A[1, 0] * A[2, 1] - A[0, 1] * A[1, 0] * A[2, 0] - A[1, 0] * A[2, 1] * A[2, 2] + A[1, 2] * A[2, 0] * A[2, 1], # noqa: E501 A[0, 0] * A[1, 2] * A[2, 0] - A[0, 2] * A[1, 0] * A[2, 0] + A[1, 0] * A[1, 2] * A[2, 1] - A[1, 1] * A[1, 2] * A[2, 0], # noqa: E501 A[0, 1] * A[1, 0] * A[2, 1] - A[0, 1] * A[1, 1] * A[2, 0] + A[0, 1] * A[2, 0] * A[2, 2] - A[0, 2] * A[2, 0] * A[2, 1], # noqa: E501 A[0, 0]**2 * A[2, 1] - A[0, 0] * A[0, 1] * A[2, 0] - A[0, 0] * A[1, 1] * A[2, 1] - A[0, 0] * A[2, 1] * A[2, 2] + A[0, 1] * A[1, 0] * A[2, 1] + A[0, 1] * A[2, 0] * A[2, 2] + A[1, 1] * A[2, 1] * A[2, 2] - A[1, 2] * A[2, 1]**2, # noqa: E501 A[0, 0]**2 * A[2, 1] - A[0, 0] * A[0, 1] * A[2, 0] - A[0, 0] * A[1, 1] * A[2, 1] - A[0, 0] * A[2, 1] * A[2, 2] + A[0, 1] * A[1, 1] * A[2, 0] + A[0, 2] * A[2, 0] * A[2, 1] + A[1, 1] * A[2, 1] * A[2, 2] - A[1, 2] * A[2, 1]**2, # noqa: E501 A[0, 0] * A[1, 0] * A[1, 1] - A[0, 0] * A[1, 0] * A[2, 2] - A[0, 1] * A[1, 0]**2 + A[0, 2] * A[1, 0] * A[2, 0] - A[1, 0] * A[1, 1] * A[2, 2] + A[1, 0] * A[2, 2]**2 + A[1, 1] * A[1, 2] * A[2, 0] - A[1, 2] * A[2, 0] * A[2, 2], # noqa: E501 A[0, 0] * A[1, 0] * A[1, 1] - A[0, 0] * A[1, 0] * A[2, 2] + A[0, 0] * A[1, 2] * A[2, 0] - A[0, 1] * A[1, 0]**2 - A[1, 0] * A[1, 1] * A[2, 2] + A[1, 0] * A[1, 2] * A[2, 1] + A[1, 0] * A[2, 2]**2 - A[1, 2] * A[2, 0] * A[2, 2], # noqa: E501 A[0, 0] * A[1, 0] * A[2, 1] - A[0, 0] * A[1, 1] * A[2, 0] + A[0, 0] * A[2, 0] * A[2, 2] - A[0, 2] * A[2, 0]**2 - A[1, 0] * A[1, 1] * A[2, 1] + A[1, 1]**2 * A[2, 0] - A[1, 1] * A[2, 0] * A[2, 2] + A[1, 2] * A[2, 0] * A[2, 1], # noqa: E501 A[0, 0] * A[1, 1] * A[2, 0] - A[0, 0] * A[2, 0] * A[2, 2] - A[0, 1] * A[1, 0] * A[2, 0] + A[0, 2] * A[2, 0]**2 + A[1, 0] * A[1, 1] * A[2, 1] - A[1, 0] * A[2, 1] * A[2, 2] - A[1, 1]**2 * A[2, 0] + A[1, 1] * A[2, 0] * A[2, 2], # noqa: E501 A[0, 0]**2 * A[1, 1] - A[0, 0]**2 * A[2, 2] - A[0, 0] * A[0, 1] * A[1, 0] + A[0, 0] * A[0, 2] * A[2, 0] - A[0, 0] * A[1, 1]**2 + A[0, 0] * A[2, 2]**2 + A[0, 1] * A[1, 0] * A[1, 1] - A[0, 2] * A[2, 0] * A[2, 2] + A[1, 1]**2 * A[2, 2] - A[1, 1] * A[1, 2] * A[2, 1] - A[1, 1] * A[2, 2]**2 + A[1, 2] * A[2, 1] * A[2, 2] ] # noqa: E501 Δd = [9, 6, 6, 6, 8, 8, 8, 2, 2, 2, 2, 2, 2, 1] Δ = 0 for i in range(len(Δd)): Δ += Δx[i] * Δd[i] * Δy[i] Δxp = [ A[1, 0], A[2, 0], A[2, 1], -A[0, 0] + A[1, 1], -A[0, 0] + A[2, 2], -A[1, 1] + A[2, 2] ] Δyp = [ A[0, 1], A[0, 2], A[1, 2], -A[0, 0] + A[1, 1], -A[0, 0] + A[2, 2], -A[1, 1] + A[2, 2] ] Δdp = [6, 6, 6, 1, 1, 1] dp = 0 for i in range(len(Δdp)): dp += 1 / 2 * Δxp[i] * Δdp[i] * Δyp[i] # Avoid dp = 0 and disc = 0, both are known with absolute error of ~eps**2 # Required to avoid sqrt(0) derivatives and negative square roots dp += eps**2 Δ += eps**2 phi3 = ufl.atan_2(ufl.sqrt(27) * ufl.sqrt(Δ), dq) # sorted eigenvalues: λ0 <= λ1 <= λ2 λ = [(I1 + 2 * ufl.sqrt(dp) * ufl.cos((phi3 + 2 * ufl.pi * k) / 3)) / 3 for k in range(1, 4)] # # --- determine eigenprojectors E0, E1, E2 # E = [ufl.diff(λk, A).T for λk in λ] return λ, E
def test_div_grad_then_integrate_over_cells_and_boundary(): # Define 2D geometry n = 10 mesh = RectangleMesh(Point(0.0, 0.0), Point(2.0, 3.0), 2 * n, 3 * n) x, y = SpatialCoordinate(mesh) xs = 0.1 + 0.8 * x / 2 # scaled to be within [0.1,0.9] # ys = 0.1 + 0.8 * y / 3 # scaled to be within [0.1,0.9] n = FacetNormal(mesh) # Define list of expressions to test, and configure accuracies # these expressions are known to pass with. The reason some # functions are less accurately integrated is likely that the # default choice of quadrature rule is not perfect F_list = [] def reg(exprs, acc=10): for expr in exprs: F_list.append((expr, acc)) # FIXME: 0*dx and 1*dx fails in the ufl-ffc-jit framework somewhere # reg([Constant(0.0, cell=cell)]) # reg([Constant(1.0, cell=cell)]) monomial_list = [x**q for q in range(2, 6)] reg(monomial_list) reg([2.3 * p + 4.5 * q for p in monomial_list for q in monomial_list]) reg([xs**xs]) reg( [xs**(xs**2)], 8 ) # Note: Accuracies here are from 1D case, not checked against 2D results. reg([xs**(xs**3)], 6) reg([xs**(xs**4)], 2) # Special functions: reg([atan(xs)], 8) reg([sin(x), cos(x), exp(x)], 5) reg([ln(xs), pow(x, 2.7), pow(2.7, x)], 3) reg([asin(xs), acos(xs)], 1) reg([tan(xs)], 7) # To handle tensor algebra, make an x dependent input tensor # xx and square all expressions def reg2(exprs, acc=10): for expr in exprs: F_list.append((inner(expr, expr), acc)) xx = as_matrix([[2 * x**2, 3 * x**3], [11 * x**5, 7 * x**4]]) xxs = as_matrix([[2 * xs**2, 3 * xs**3], [11 * xs**5, 7 * xs**4]]) x3v = as_vector([3 * x**2, 5 * x**3, 7 * x**4]) cc = as_matrix([[2, 3], [4, 5]]) reg2( [xx] ) # TODO: Make unit test for UFL from this, results in listtensor with free indices reg2([x3v]) reg2([cross(3 * x3v, as_vector([-x3v[1], x3v[0], x3v[2]]))]) reg2([xx.T]) reg2([tr(xx)]) reg2([det(xx)]) reg2([dot(xx, 0.1 * xx)]) reg2([outer(xx, xx.T)]) reg2([dev(xx)]) reg2([sym(xx)]) reg2([skew(xx)]) reg2([elem_mult(7 * xx, cc)]) reg2([elem_div(7 * xx, xx + cc)]) reg2([elem_pow(1e-3 * xxs, 1e-3 * cc)]) reg2([elem_pow(1e-3 * cc, 1e-3 * xx)]) reg2([elem_op(lambda z: sin(z) + 2, 0.03 * xx)], 2) # pretty inaccurate... # FIXME: Add tests for all UFL operators: # These cause discontinuities and may be harder to test in the # above fashion: # 'inv', 'cofac', # 'eq', 'ne', 'le', 'ge', 'lt', 'gt', 'And', 'Or', 'Not', # 'conditional', 'sign', # 'jump', 'avg', # 'LiftingFunction', 'LiftingOperator', # FIXME: Test other derivatives: (but algorithms for operator # derivatives are the same!): # 'variable', 'diff', # 'Dx', 'grad', 'div', 'curl', 'rot', 'Dn', 'exterior_derivative', # Run through all operators defined above and compare integrals debug = 0 if debug: F_list = F_list[1:] for F, acc in F_list: if debug: print('\n', "F:", str(F)) # Integrate over domain and its boundary int_dx = assemble(div(grad(F)) * dx(mesh)) # noqa int_ds = assemble(dot(grad(F), n) * ds(mesh)) # noqa if debug: print(int_dx, int_ds) # Compare results. Using custom relative delta instead of # decimal digits here because some numbers are >> 1. delta = min(abs(int_dx), abs(int_ds)) * 10**-acc assert int_dx - int_ds <= delta
def test_diff_then_integrate(): # Define 1D geometry n = 21 mesh = UnitIntervalMesh(MPI.comm_world, n) # Shift and scale mesh x0, x1 = 1.5, 3.14 mesh.coordinates()[:] *= (x1 - x0) mesh.coordinates()[:] += x0 x = SpatialCoordinate(mesh)[0] xs = 0.1 + 0.8 * x / x1 # scaled to be within [0.1,0.9] # Define list of expressions to test, and configure # accuracies these expressions are known to pass with. # The reason some functions are less accurately integrated is # likely that the default choice of quadrature rule is not perfect F_list = [] def reg(exprs, acc=10): for expr in exprs: F_list.append((expr, acc)) # FIXME: 0*dx and 1*dx fails in the ufl-ffc-jit framework somewhere # reg([Constant(0.0, cell=cell)]) # reg([Constant(1.0, cell=cell)]) monomial_list = [x**q for q in range(2, 6)] reg(monomial_list) reg([2.3 * p + 4.5 * q for p in monomial_list for q in monomial_list]) reg([x**x]) reg([x**(x**2)], 8) reg([x**(x**3)], 6) reg([x**(x**4)], 2) # Special functions: reg([atan(xs)], 8) reg([sin(x), cos(x), exp(x)], 5) reg([ln(xs), pow(x, 2.7), pow(2.7, x)], 3) reg([asin(xs), acos(xs)], 1) reg([tan(xs)], 7) try: import scipy except ImportError: scipy = None if hasattr(math, 'erf') or scipy is not None: reg([erf(xs)]) else: print( "Warning: skipping test of erf, old python version and no scipy.") # if 0: # print("Warning: skipping tests of bessel functions, doesn't build on all platforms.") # elif scipy is None: # print("Warning: skipping tests of bessel functions, missing scipy.") # else: # for nu in (0, 1, 2): # # Many of these are possibly more accurately integrated, # # but 4 covers all and is sufficient for this test # reg([bessel_J(nu, xs), bessel_Y(nu, xs), bessel_I(nu, xs), bessel_K(nu, xs)], 4) # To handle tensor algebra, make an x dependent input tensor # xx and square all expressions def reg2(exprs, acc=10): for expr in exprs: F_list.append((inner(expr, expr), acc)) xx = as_matrix([[2 * x**2, 3 * x**3], [11 * x**5, 7 * x**4]]) x3v = as_vector([3 * x**2, 5 * x**3, 7 * x**4]) cc = as_matrix([[2, 3], [4, 5]]) reg2([xx]) reg2([x3v]) reg2([cross(3 * x3v, as_vector([-x3v[1], x3v[0], x3v[2]]))]) reg2([xx.T]) reg2([tr(xx)]) reg2([det(xx)]) reg2([dot(xx, 0.1 * xx)]) reg2([outer(xx, xx.T)]) reg2([dev(xx)]) reg2([sym(xx)]) reg2([skew(xx)]) reg2([elem_mult(7 * xx, cc)]) reg2([elem_div(7 * xx, xx + cc)]) reg2([elem_pow(1e-3 * xx, 1e-3 * cc)]) reg2([elem_pow(1e-3 * cc, 1e-3 * xx)]) reg2([elem_op(lambda z: sin(z) + 2, 0.03 * xx)], 2) # pretty inaccurate... # FIXME: Add tests for all UFL operators: # These cause discontinuities and may be harder to test in the # above fashion: # 'inv', 'cofac', # 'eq', 'ne', 'le', 'ge', 'lt', 'gt', 'And', 'Or', 'Not', # 'conditional', 'sign', # 'jump', 'avg', # 'LiftingFunction', 'LiftingOperator', # FIXME: Test other derivatives: (but algorithms for operator # derivatives are the same!): # 'variable', 'diff', # 'Dx', 'grad', 'div', 'curl', 'rot', 'Dn', 'exterior_derivative', # Run through all operators defined above and compare integrals debug = 0 for F, acc in F_list: # Apply UFL differentiation f = diff(F, SpatialCoordinate(mesh))[..., 0] if debug: print(F) print(x) print(f) # Apply integration with DOLFIN # (also passes through form compilation and jit) M = f * dx f_integral = assemble_scalar(M) # noqa f_integral = MPI.sum(mesh.mpi_comm(), f_integral) # Compute integral of f manually from anti-derivative F # (passes through PyDOLFIN interface and uses UFL evaluation) F_diff = F((x1, )) - F((x0, )) # Compare results. Using custom relative delta instead # of decimal digits here because some numbers are >> 1. delta = min(abs(f_integral), abs(F_diff)) * 10**-acc assert f_integral - F_diff <= delta
def CDelta(psi, eps): return conditional(lt(abs(psi), eps), 1.0 / (2.0 * eps) * (1.0 + ufl.cos(np.pi * psi / eps)), 0.0)
from dune.fem.view import geometryGridView from dune.fem.space import lagrange as solutionSpace from dune.fem.scheme import galerkin as solutionScheme endTime = 0.05 saveInterval = 0.05 # setup reference surface referenceView = leafGridView("sphere.dgf", dimgrid=2, dimworld=3) space = solutionSpace(referenceView, dimRange=referenceView.dimWorld, order=1) # setup deformed surface x = ufl.SpatialCoordinate(space) # positions = space.interpolate(x, name="position") positions = space.interpolate( x * (1 + 0.5 * sin(2 * pi * x[0] * x[1]) * cos(pi * x[2])), name="position") gridView = geometryGridView(positions) space = solutionSpace(gridView, dimRange=gridView.dimWorld, order=1) u = ufl.TrialFunction(space) phi = ufl.TestFunction(space) dt = dune.ufl.Constant(0.001, "timeStep") t = dune.ufl.Constant(0.0, "time") # define storage for discrete solutions uh = space.interpolate(x, name="uh") uh_old = uh.copy() # problem definition
from dune.fem.scheme import galerkin as solutionScheme order = 3 storage = "istl" # setup reference surface referenceView = leafGridView("sphere.dgf", dimgrid=2, dimworld=3) space = solutionSpace(referenceView, dimRange=referenceView.dimWorld, order=order, storage=storage) # setup deformed surface x = ufl.SpatialCoordinate(space) # positions = space.interpolate(x, name="position") positions = space.interpolate( x * (1 + 0.5 * sin(2 * pi * (x[0] + x[1])) * cos(0.25 * pi * x[2])), name="position") gridView = geometryGridView(positions) space = solutionSpace(gridView, dimRange=gridView.dimWorld, order=order, storage=storage) u = ufl.TrialFunction(space) phi = ufl.TestFunction(space) dt = dune.ufl.Constant(0.01, "timeStep") t = dune.ufl.Constant(0.0, "time") # define storage for discrete solutions uh = space.interpolate(x, name="uh") uh_old = uh.copy()
# ---------------------------------------------------------------------------- # DERIVATIVE with respect to arc-length coordinate s of straight reference configuration: du/ds = du/dx * dx/dr * dr/ds def GRAD(u): return ufl.dot(ufl.grad(u), J0[:, 0]) * 1 / ufl.geometry.JacobianDeterminant(mesh) # Undeformed configuration: stretch (at the principal axis) λ0 = ufl.sqrt(ufl.dot(GRAD(x0), GRAD(x0))) # from geometry (!= 1) # Undeformed configuration: curvature κ0 = -B0i[0, 0] # from curvature tensor B0i # Deformed configuration: stretch components (at the principal axis) λs = (1.0 + GRAD(x0[0]) * GRAD(u) + GRAD(x0[2]) * GRAD(w)) * ufl.cos(r) + \ (GRAD(x0[2]) * GRAD(u) - GRAD(x0[0]) * GRAD(w)) * ufl.sin(r) λξ = (1.0 + GRAD(x0[0]) * GRAD(u) + GRAD(x0[2]) * GRAD(w)) * ufl.sin(r) - \ (GRAD(x0[2]) * GRAD(u) - GRAD(x0[0]) * GRAD(w)) * ufl.cos(r) # Deformed configuration: curvature κ = GRAD(r) # Green-Lagrange strains (total): determined by deformation kinematics e_total = 1 / 2 * (λs**2 + λξ**2 - λ0**2) g_total = λξ k_total = λs * κ + (λs - λ0) * κ0 # Green-Lagrange strains (elastic): e_total = e_elast + e_presc e = e_elast = e_total g = g_elast = g_total k = k_elast = k_total
from __future__ import print_function, unicode_literals import time, math, numpy, ufl import dune, dune.create from dune.generator import algorithm from dune.grid import cartesianDomain, yaspGrid domain = cartesianDomain([0, 0], [1, 0.25], [12, 3]) yaspView = yaspGrid(domain) x = ufl.SpatialCoordinate(ufl.triangle) function = ufl.as_vector([ufl.cos(2 * ufl.pi / (0.3 + x[0] * x[1]))]) # function = dune.create.function("global",gridview=yaspView, name="gl",order=5, # value=lambda x: [math.cos(2*math.pi/(0.3+x[0]*x[1]))] ) space = dune.create.space("lagrange", yaspView, order=1, dimRange=1) uh = space.interpolate(function, name="uh") error = dune.create.function("ufl", gridView=yaspView, name="error", order=5, ufl=uh - function, virtualize=True) rules = dune.geometry.quadratureRules(5) if False: # dune.fem type grid functions don't yet have vectorization support start = time.time() l2norm2 = 0 for e in yaspView.elements: hatxs, hatws = rules(e.type).get() weights = hatws * e.geometry.integrationElement(hatxs)
import pyamg import ufl import numpy import scipy from minidolfin.meshing import read_meshio, write_meshio from minidolfin.dofmap import build_dofmap, interpolate_vertex_values from minidolfin.assembling import assemble from minidolfin.bcs import build_dirichlet_dofs, bc_apply mesh = read_meshio('tet_cube.xdmf') element = ufl.FiniteElement("P", ufl.tetrahedron, 1) u, v = ufl.TrialFunction(element), ufl.TestFunction(element) x = ufl.SpatialCoordinate(ufl.tetrahedron) a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx L = ufl.cos(x[1]) * v * ufl.dx dofmap = build_dofmap(element, mesh) def u_bound(x): return x[0] t = time.time() bc_dofs, bc_vals = build_dirichlet_dofs(dofmap, u_bound) bc_map = {i: v for i, v in zip(bc_dofs, bc_vals)} elapsed = time.time() - t print('BC time = ', elapsed) t = time.time() # A, b = symass(dofmap, a, L, bc_map, dtype=numpy.float32)
# This example is based on # http://www.dolfin-adjoint.org/en/latest/documentation/tutorial.html import firedrake as fd from firedrake.petsc import PETSc from firedrake import dmhooks import ufl import firedrake_ts equation = "burgers" # 'burgers' or 'heat' n = 50 mesh = fd.UnitSquareMesh(n, n) V = fd.VectorFunctionSpace(mesh, "CG", 2) x = ufl.SpatialCoordinate(mesh) expr = ufl.as_vector([ufl.sin(2 * ufl.pi * x[0]), ufl.cos(2 * ufl.pi * x[1])]) u = fd.interpolate(expr, V) u_dot = fd.Function(V) v = fd.TestFunction(V) nu = fd.Constant(0.0001) # for burgers if equation == "heat": nu = fd.Constant(0.1) # for heat M = fd.derivative(fd.inner(u, v) * fd.dx, u) R = -(fd.inner(fd.grad(u) * u, v) + nu * fd.inner(fd.grad(u), fd.grad(v))) * fd.dx if equation == "heat": R = -nu * fd.inner(fd.grad(u), fd.grad(v)) * fd.dx F = fd.action(M, u_dot) - R
from __future__ import print_function, division from ufl import as_vector, grad, cos, pi, SpatialCoordinate, triangle from dune.grid import structuredGrid, cartesianDomain from dune.fem.space import lagrange from dune.alugrid import aluConformGrid as gridManager grid = gridManager(cartesianDomain([0, 0], [1, 1], [16, 16])) # grid = structuredGrid([0, 0], [1, 1], [16, 16]) space = lagrange(grid, dimRange=1, order=1) x = SpatialCoordinate(triangle) exact = as_vector([cos(2. * pi * x[0]) * cos(2. * pi * x[1])]) solution = space.interpolate(exact, name="solution") test = space.interpolate(solution[0], name="tmp") test = space.interpolate(grad(solution)[0, 0], name="tmp")
dolfiny.interpolation.interpolate(gξ, n0i) # ---------------------------------------------------------------------------- # Orthogonal projection operator (assumes sufficient geometry approximation) P = ufl.Identity(mesh.geometry.dim) - ufl.outer(n0i, n0i) # Thickness variable X = dolfinx.FunctionSpace(mesh, ("DG", q)) ξ = dolfinx.Function(X, name='ξ') # Undeformed configuration: director d0 and placement b0 d0 = n0i # normal of manifold mesh, interpolated b0 = x0 + ξ * d0 # Deformed configuration: director d and placement b, assumed kinematics, director uses rotation matrix d = ufl.as_matrix([[ufl.cos(r), 0, ufl.sin(r)], [0, 1, 0], [-ufl.sin(r), 0, ufl.cos(r)]]) * d0 b = x0 + ufl.as_vector([u, 0, w]) + ξ * d # Configuration gradient, undeformed configuration J0 = ufl.grad(b0) - ufl.outer(d0, d0) # = P * ufl.grad(x0) + ufl.grad(ξ * d0) J0 = ufl.algorithms.apply_algebra_lowering.apply_algebra_lowering(J0) J0 = ufl.algorithms.apply_derivatives.apply_derivatives(J0) J0 = ufl.replace(J0, {ufl.grad(ξ): d0}) # Configuration gradient, deformed configuration J = ufl.grad(b) - ufl.outer( d0, d0) # = P * ufl.grad(x0) + ufl.grad(ufl.as_vector([u, 0, w]) + ξ * d) J = ufl.algorithms.apply_algebra_lowering.apply_algebra_lowering(J) J = ufl.algorithms.apply_derivatives.apply_derivatives(J) J = ufl.replace(J, {ufl.grad(ξ): d0})
dt = dune.ufl.Constant(5e-2, "timeStep") t = dune.ufl.Constant(0.0, "time") # define storage for discrete solutions uh = space.interpolate(0, name="uh") uh_old = uh.copy() # initial solution initial = 0 # problem definition # moving oven ROven = 0.6 omegaOven = 0.01 * pi * t P = ufl.as_vector([ROven*cos(omegaOven*t), ROven*sin(omegaOven*t)]) rOven = 0.2 ovenEnergy = 8 chiOven = ufl.conditional(dot(x-P, x-P) < rOven**2, 1, 0) ovenLoad = ovenEnergy * chiOven # desk in corner of room deskCenter = [-0.8, -0.8] deskSize = 0.2 chiDesk = ufl.conditional(abs(x[0]-deskCenter[0]) < deskSize, 1, 0)\ * ufl.conditional(abs(x[1] - deskCenter[1]) < deskSize, 1, 0) # Robin condition for window windowWidth = 0.5 transmissionCoefficient = 1.2 outerTemperature = -5.0
from minidolfin.meshing import read_mesh, write_meshio from minidolfin.dofmap import build_dofmap, interpolate_vertex_values from minidolfin.assembling import symass from minidolfin.bcs import build_dirichlet_dofs from minidolfin.plot import plot mesh = read_mesh( 'https://raw.githubusercontent.com/chrisrichardson/meshdata/master/data/rectangle_mesh.xdmf' ) # noqa element = ufl.FiniteElement("P", ufl.triangle, 1) u, v = ufl.TrialFunction(element), ufl.TestFunction(element) x = ufl.SpatialCoordinate(ufl.triangle) a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx L = 50.0 * ufl.cos(6.28 * x[0]) * v * ufl.dx dofmap = build_dofmap(element, mesh) def u_bound(x): return x[0] t = time.time() bc_dofs, bc_vals = build_dirichlet_dofs(dofmap, u_bound, dtype=numpy.float64) bc_map = {i: v for i, v in zip(bc_dofs, bc_vals)} elapsed = time.time() - t print('BC time = ', elapsed) t = time.time() A, b = symass(dofmap, a, L, bc_map, dtype=numpy.float64)
import ufl import numpy from minidolfin.meshing import read_mesh from minidolfin.dofmap import build_dofmap from minidolfin.assembling import symass from minidolfin.bcs import build_dirichlet_dofs mesh = read_mesh( 'https://raw.githubusercontent.com/chrisrichardson/meshdata/master/data/rectangle_mesh.xdmf' ) # noqa element = ufl.FiniteElement("P", ufl.triangle, 1) u, v = ufl.TrialFunction(element), ufl.TestFunction(element) f = ufl.Coefficient(element) a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx L = ufl.cos(1.0) * v * ufl.dx dofmap = build_dofmap(element, mesh) def u_bound(x): return x[0] t = time.time() bc_dofs, bc_vals = build_dirichlet_dofs(dofmap, u_bound) bc_map = {i: v for i, v in zip(bc_dofs, bc_vals)} elapsed = time.time() - t print('BC time = ', elapsed) t = time.time() A, b = symass(dofmap, a, L, bc_map, dtype=numpy.float64)
def ode_1st_nonlinear_odeint(a=1.0, b=1.0, c=1.0, nT=10, dt=0.1, **kwargs): """ Create 1st order ODE problem and solve with `ODEInt` time integrator. First order nonlinear non-autonomous ODE: t * dot u - a * cos(c*t) * u^2 - 2 * u - a * b^2 * t^4 * cos(c*t) = 0 with initial condition u(t=1) = 0 """ mesh = UnitCubeMesh(MPI.COMM_WORLD, 1, 1, 1) U = FunctionSpace(mesh, ("DG", 0)) u = Function(U, name="u") ut = Function(U, name="ut") u.vector.set(0.0) # initial condition ut.vector.set( a * b**2 * numpy.cos(c)) # exact initial rate of this ODE for generalised alpha u.vector.ghostUpdate() ut.vector.ghostUpdate() δu = ufl.TestFunction(U) dx = ufl.Measure("dx", domain=mesh) # Global time t = dolfinx.Constant(mesh, 1.0) # Time step size dt = dolfinx.Constant(mesh, dt) # Time integrator odeint = dolfiny.odeint.ODEInt(t=t, dt=dt, x=u, xt=ut, **kwargs) # Weak form (as one-form) f = δu * (t * ut - a * ufl.cos(c * t) * u**2 - 2 * u - a * b**2 * t**4 * ufl.cos(c * t)) * dx # Overall form (as one-form) F = odeint.discretise_in_time(f) # Overall form (as list of forms) F = dolfiny.function.extract_blocks(F, [δu]) # # Options for PETSc backend from petsc4py import PETSc opts = PETSc.Options() opts["snes_type"] = "newtonls" opts["snes_linesearch_type"] = "basic" opts["snes_atol"] = 1.0e-10 opts["snes_rtol"] = 1.0e-12 # Silence SNES monitoring during test dolfiny.snesblockproblem.SNESBlockProblem.print_norms = lambda self, it: 1 # Create nonlinear problem problem = dolfiny.snesblockproblem.SNESBlockProblem(F, [u]) # Book-keeping of results u_avg = numpy.zeros(nT + 1) u_avg[0] = u.vector.sum() / u.vector.getSize() dolfiny.utils.pprint(f"+++ Processing time steps = {nT}") # Process time steps for time_step in range(1, nT + 1): # Stage next time step odeint.stage() # Solve nonlinear problem u, = problem.solve() # Assert convergence of nonlinear solver assert problem.snes.getConvergedReason( ) > 0, "Nonlinear solver did not converge!" # Update solution states for time integration odeint.update() # Store result u_avg[time_step] = u.vector.sum() / u.vector.getSize() return u_avg