From 2713e577b131932f1dd02687e016af49374e48b5 Mon Sep 17 00:00:00 2001 From: Alessio Fumagalli Date: Tue, 8 Oct 2024 09:26:43 +0200 Subject: [PATCH] Cosserat (#82) * first commit * move a file to a dedicated repo * add lumped mass for vec bdm1and new vec rt0 class * add lumped mass for pw linears * add optional data in the computation of the l2 errors * in the rt0 mass matrix do the dummy data if data are not given * added spanning tree * fix flake8 complain * fix a - in the asym 2d * add specific computation of the L2 error for vecP0 * update l2 error computation * fix some orthographic mistakes * update the dummy data creation in the rt0 * update the unit data in the mdg * in the projection use unit instead of flow as standard keyword * update the test elasticity with the new definition of the asym operator in 2d * remove comment from test projection * add for the text vecbdm1 the check for the lump mass matrix and projection to and from rt0 * update the test vector pw constant in the computation of the error * add test for vec rt0 * add tutorial for the elasticity mixed * add tutorial for cosserat * update create grid to pass kwargs correctly to pp * update requirements dev to be compliant with pp * update file according to the new version of black * fix some spelling mistake * changed asym to pwLinears implemented VecPWLinears implemented eval_at_centers for PwLinears * implemented cosserat asymasym term * add small details in the comments * introduced cosserat matrix VecRT0 introduced lumped cosserat matrix VecBDM1 included tests added weight to PwLinear mass included test for pwlinear eval * add test for the vector pw linear space * add test for spanning tree Cosserat * remove slash from \G and \d * update the tutorial for cosserat * use the mass_cosserat in the tutorial * update cosserat tutorial * change from pop to get in kwargs in the create_grid file --------- Co-authored-by: Wietse Boon --- requirements-dev.txt | 4 +- src/pygeon/__init__.py | 10 +- src/pygeon/discretizations/discretization.py | 3 +- src/pygeon/discretizations/fem/hdiv.py | 436 ++++++++++++++++--- src/pygeon/discretizations/fem/l2.py | 120 ++++- src/pygeon/discretizations/vem/h1.py | 2 +- src/pygeon/discretizations/vem/hdiv.py | 4 +- src/pygeon/grids/create_grid.py | 6 +- src/pygeon/grids/graph.py | 12 +- src/pygeon/grids/md_grid.py | 9 +- src/pygeon/numerics/projections.py | 2 +- src/pygeon/numerics/spanningtree.py | 64 +++ tests/integration/test_elasticity.py | 6 +- tests/unit/test_projections.py | 1 - tests/unit/test_pwlinear.py | 6 +- tests/unit/test_spanningtree.py | 52 +++ tests/unit/test_vecbdm1.py | 183 +++++++- tests/unit/test_vecpwconstant.py | 9 +- tests/unit/test_vecpwlinear.py | 167 +++++++ tests/unit/test_vecrt0.py | 222 ++++++++++ tutorials/cosserat.ipynb | 310 +++++++++++++ tutorials/elasticity_mixed.ipynb | 294 +++++++++++++ tutorials/fig/cosserat.png | Bin 0 -> 516204 bytes 23 files changed, 1829 insertions(+), 93 deletions(-) create mode 100644 tests/unit/test_vecpwlinear.py create mode 100644 tests/unit/test_vecrt0.py create mode 100644 tutorials/cosserat.ipynb create mode 100644 tutorials/elasticity_mixed.ipynb create mode 100644 tutorials/fig/cosserat.png diff --git a/requirements-dev.txt b/requirements-dev.txt index 385dbdab..8dac3c99 100644 --- a/requirements-dev.txt +++ b/requirements-dev.txt @@ -1,8 +1,8 @@ -black == 23.* +black == 24.* deepdiff flake8 isort -mypy >= 0.981 +mypy pytest >= 4.6 pytest-cov pytest-runner diff --git a/src/pygeon/__init__.py b/src/pygeon/__init__.py index caca007a..8347e6cc 100644 --- a/src/pygeon/__init__.py +++ b/src/pygeon/__init__.py @@ -18,9 +18,14 @@ from pygeon.discretizations.vec_discretization import VecDiscretization from pygeon.discretizations.fem.hcurl import Nedelec0, Nedelec1 -from pygeon.discretizations.fem.hdiv import RT0, BDM1, VecBDM1 +from pygeon.discretizations.fem.hdiv import RT0, BDM1, VecBDM1, VecRT0 from pygeon.discretizations.fem.h1 import Lagrange1, VecLagrange1 -from pygeon.discretizations.fem.l2 import PwConstants, PwLinears, VecPwConstants +from pygeon.discretizations.fem.l2 import ( + PwConstants, + PwLinears, + VecPwConstants, + VecPwLinears, +) from pygeon.discretizations.vem.hdiv import VRT0, VBDM1 from pygeon.discretizations.vem.h1 import VLagrange1, VecVLagrange1 @@ -44,6 +49,7 @@ SpanningTree, SpanningWeightedTrees, SpanningTreeElasticity, + SpanningTreeCosserat, ) import pygeon.utils.bmat as bmat diff --git a/src/pygeon/discretizations/discretization.py b/src/pygeon/discretizations/discretization.py index 1fb87d1e..2da02f22 100644 --- a/src/pygeon/discretizations/discretization.py +++ b/src/pygeon/discretizations/discretization.py @@ -265,6 +265,7 @@ def error_l2( ana_sol: Callable[[np.ndarray], np.ndarray], relative: Optional[bool] = True, etype: Optional[str] = "standard", + data: dict = None, ) -> float: """ Returns the l2 error computed against an analytical solution given as a function. @@ -281,7 +282,7 @@ def error_l2( float: The computed error. """ int_sol = self.interpolate(sd, ana_sol) - mass = self.assemble_mass_matrix(sd) + mass = self.assemble_mass_matrix(sd, data) norm = (int_sol @ mass @ int_sol.T) if relative else 1 diff --git a/src/pygeon/discretizations/fem/hdiv.py b/src/pygeon/discretizations/fem/hdiv.py index fe6ca7d7..7d5c2bf9 100644 --- a/src/pygeon/discretizations/fem/hdiv.py +++ b/src/pygeon/discretizations/fem/hdiv.py @@ -41,7 +41,7 @@ class RT0(pg.Discretization, pp.RT0): assemble_nat_bc(sd: pg.Grid, func: Callable[[np.ndarray], np.ndarray], b_faces: np.ndarray) -> np.ndarray: - Assembles the natural boundary condition term (n dot q, func)_\Gamma + Assembles the natural boundary condition term (n dot q, func)_Gamma get_range_discr_class(dim: int) -> pg.Discretization: Returns the range discretization class for the given dimension. @@ -121,8 +121,9 @@ def assemble_mass_matrix( Returns: sps.csc_matrix: The mass matrix. """ - + # create dummy data, unitary permeability, in case not present data = self.create_dummy_data(sd, data) + # perform the rt0 discretization pp.RT0.discretize(self, sd, data) return data[pp.DISCRETIZATION_MATRICES][self.keyword][ self.mass_matrix_key @@ -211,7 +212,7 @@ def assemble_nat_bc( ) -> np.ndarray: """ Assembles the natural boundary condition term - (n dot q, func)_\Gamma + (n dot q, func)_Gamma Args: sd (pg.Grid): The grid object representing the computational domain. @@ -584,7 +585,7 @@ def assemble_nat_bc( ) -> np.ndarray: """ Assembles the natural boundary condition term - (n dot q, func)_\Gamma + (n dot q, func)_Gamma Args: sd (pg.Grid): The grid object representing the computational domain. @@ -606,7 +607,7 @@ def assemble_nat_bc( sign = np.sum(sd.cell_faces.tocsr()[face, :]) loc_vals = np.array( [func(sd.nodes[:, node]) for node in sd.face_nodes[:, face].indices] - ) + ).ravel() vals[face + np.arange(sd.dim) * sd.num_faces] = sign * local_mass @ loc_vals @@ -812,7 +813,7 @@ def assemble_mass_matrix(self, sd: pg.Grid, data: dict) -> sps.csc_matrix: # Assemble the trace part B = self.assemble_trace_matrix(sd) - # Assemble the piecewise linear mass mastrix, to assemble the term + # Assemble the piecewise linear mass matrix, to assemble the term # (Trace(sigma), Trace(tau)) discr = pg.PwLinears(self.keyword) M = discr.assemble_mass_matrix(sd, data_for_PwL) @@ -820,6 +821,46 @@ def assemble_mass_matrix(self, sd: pg.Grid, data: dict) -> sps.csc_matrix: # Compose all the parts and return them return D - B.T @ M @ B + def assemble_mass_matrix_cosserat(self, sd: pg.Grid, data: dict) -> sps.csc_matrix: + """ + Assembles and returns the mass matrix for vector BDM1 discretizing the Cosserat inner + product, which is given by (A sigma, tau) where + A sigma = (sym(sigma) - coeff * Trace(sigma) * I) / (2 mu) + skw(sigma) / (2 mu_c) + with mu and lambda the Lamé constants, coeff = lambda / (2*mu + dim*lambda), and + mu_c the coupling Lamé modulus. + + Args: + sd (pg.Grid): The grid. + data (dict): Data for the assembly. + + Returns: + sps.csc_matrix: The mass matrix obtained from the discretization. + """ + M = self.assemble_mass_matrix(sd, data) + + # Extract the data + mu = data[pp.PARAMETERS][self.keyword]["mu"] + mu_c = data[pp.PARAMETERS][self.keyword]["mu_c"] + + coeff = 0.25 * (1 / mu_c - 1 / mu) + + # If coeff is a scalar, replace it by a vector so that it can be accessed per cell + if isinstance(coeff, np.ScalarType): + coeff = np.full(sd.num_cells, coeff) + + data_for_R = pp.initialize_data(sd, {}, self.keyword, {"weight": coeff}) + + if sd.dim == 2: + R_space = pg.PwLinears(self.keyword) + elif sd.dim == 3: + R_space = pg.VecPwLinears(self.keyword) + + R_mass = R_space.assemble_mass_matrix(sd, data_for_R) + + asym = self.assemble_asym_matrix(sd, False) + + return M + asym.T @ R_mass @ asym + def assemble_trace_matrix(self, sd: pg.Grid) -> sps.csc_matrix: """ Assembles and returns the trace matrix for the vector BDM1. @@ -876,69 +917,200 @@ def assemble_trace_matrix(self, sd: pg.Grid) -> sps.csc_matrix: # Construct the global matrices return sps.csc_matrix((data_IJ[:idx], (rows_I[:idx], cols_J[:idx]))) - def assemble_asym_matrix(self, sd: pg.Grid) -> sps.csc_matrix: + def assemble_asym_matrix(self, sd: pg.Grid, as_pwconstant=True) -> sps.csc_matrix: """ Assembles and returns the asymmetric matrix for the vector BDM1. The asymmetric operator `as' for a tensor is a scalar and it is defined in 2d as - as(tau) = tau_xy - tau_yx + as(tau) = tau_yx - tau_xy while for a tensor in 3d it is a vector and given by as(tau) = [tau_zy - tau_yz, tau_xz - tau_zx, tau_yx - tau_xy]^T - Note: We assume that the as(tau) is a cell variable. + Note: We assume that the as(tau) is a piecewise linear. Args: sd (pg.Grid): The grid. + as_pwconstant (bool): Compute the operator with the range on the piece-wise + constant (default), otherwise the mapping is on the piece-wise linears. Returns: sps.csc_matrix: The asymmetric matrix obtained from the discretization. """ - # We need to map the BDM1 to the cell center, thus we construct the proper operator - P = self.eval_at_cell_centers(sd) - nc, cv = sd.num_cells, sd.cell_volumes - # We consider a different approach if sd is 2d or 3d - if sd.dim == 2: - # Every cell has two entries, one for the t_xy and one for the t_yx - rows_I = np.tile(np.arange(nc), 2) - # Given a 2d tensor, represented as a vector, the t_xy component is the second - # and the t_yx is the third - cols_J = np.hstack( - ( - np.arange(nc, 2 * nc), - np.arange(3 * nc, 4 * nc), - ) + + # overestimate the size + size = np.square((sd.dim + 1) * sd.dim) * sd.num_cells + rows_I = np.empty(size, dtype=int) + cols_J = np.empty(size, dtype=int) + data_IJ = np.empty(size) + idx = 0 + + # Helper functions for inside the loop + negate_col = [2, 0, 1] + zeroed_col = [0, 1, 2] + + if sd.dim == 3: + ind_list = np.arange(3) + shift = ind_list + rot_space = pg.VecPwLinears(self.keyword) + scaling = sps.diags(np.tile(sd.cell_volumes, 3)) + elif sd.dim == 2: + ind_list = [2] + shift = [0, 0, 0] + rot_space = pg.PwLinears(self.keyword) + scaling = sps.diags(sd.cell_volumes) + else: + raise ValueError("The grid should be either two or three-dimensional") + + cell_nodes = sd.cell_nodes() + for c in np.arange(sd.num_cells): + # For the current cell retrieve its faces and + # determine the location of the dof + loc = slice(sd.cell_faces.indptr[c], sd.cell_faces.indptr[c + 1]) + faces_loc = sd.cell_faces.indices[loc] + dof_loc = np.reshape( + sd.face_nodes[:, faces_loc].indices, (sd.dim, -1), order="F" ) - # t_xy gets a +1 and t_yx a -1, represented as p0 dofs so with the cell volume - data_IJ = np.hstack((cv, -cv)) - # Assemble the matrix with a given shape, t_xx and t_yy are not considered - T = sps.csc_matrix((data_IJ, (rows_I, cols_J)), shape=(nc, P.shape[0])) - return T @ P - elif sd.dim == 3: - # Define an help function - enum = lambda i: np.arange(i * nc, (i + 1) * nc) - # Since as(tau) is a piecewise vector we need to arrange the entries accordingly - # to the second component of the tensor - # as(tau)_x = tau_zy - tau_yz -> z and y with -1 and +1 - # as(tau)_y = tau_xz - tau_zx -> z and x with +1 and -1 - # as(tau)_z = tau_yx - tau_xy -> x and y with -1 and +1 - rows_I = np.hstack((enum(2), enum(1), enum(2), enum(0), enum(1), enum(0))) - # Given a 3d tensor, represented as a vector, the involved components are - # (t_xy, t_xz, t_yx, t_yz, t_zx, t_zy) which are in positions (1, 2, 3) and - # (5, 6, 7) - cols_J = np.hstack( - ( - np.arange(nc, 4 * nc), - np.arange(5 * nc, 8 * nc), - ) + + cell_nodes_loc = cell_nodes[:, c].toarray() + Psi = self.scalar_discr.eval_basis_at_node( + sd, dof_loc, cell_nodes_loc, faces_loc ) - # Assording to the ordering of rows and cols given abose we have the following - # signs (-1, +1, +1, -1, -1, +1), weighted by the cell volumes since it is - # represented as a p0 vector element - data_IJ = np.hstack((-cv, cv, cv, -cv, -cv, cv)) - T = sps.csc_matrix((data_IJ, (rows_I, cols_J)), shape=(3 * nc, P.shape[0])) - return T @ P + + # Get all the components of the basis at node + Psi_i, Psi_j, Psi_v = sps.find(Psi) + + for ind in ind_list: + Psi_v_copy = Psi_v.copy() + Psi_v_copy[np.mod(Psi_j, 3) == negate_col[ind]] *= -1 + Psi_v_copy[np.mod(Psi_j, 3) == zeroed_col[ind]] *= 0 + + loc_ind = np.hstack([faces_loc] * sd.dim) + loc_ind += np.repeat(np.arange(sd.dim), sd.dim + 1) * sd.num_faces + + cols = np.tile(loc_ind, (3, 1)) + cols[0, :] += np.mod(-ind, 3) * self.scalar_discr.ndof(sd) + cols[1, :] += np.mod(-ind - 1, 3) * self.scalar_discr.ndof(sd) + cols[2, :] += np.mod(-ind - 2, 3) * self.scalar_discr.ndof(sd) + + cols = np.tile(cols, (sd.dim + 1, 1)).T + + cols = cols[Psi_i, Psi_j] + + nodes_loc = np.arange((sd.dim + 1) * c, (sd.dim + 1) * (c + 1)) + rows = np.repeat(nodes_loc, 3)[Psi_j] + + # Save values of the local matrix in the global structure + loc_idx = slice(idx, idx + cols.size) + rows_I[loc_idx] = rows + shift[ind] * (sd.dim + 1) * sd.num_cells + cols_J[loc_idx] = cols + data_IJ[loc_idx] = Psi_v_copy + idx += cols.size + + # Construct the global matrices + asym = sps.csc_matrix((data_IJ[:idx], (rows_I[:idx], cols_J[:idx]))) + + # Return the operator that maps to the piece-wise constant + if as_pwconstant: + return scaling @ rot_space.eval_at_cell_centers(sd) @ asym else: - raise ValueError("The grid should be either bi or three-dimensional") + return asym + + def assemble_lumped_matrix( + self, sd: pg.Grid, data: Optional[dict] = None + ) -> sps.csc_matrix: + """ + Assembles the lumped matrix for the given grid. + + Args: + sd (pg.Grid): The grid object. + data (Optional[dict]): Optional data dictionary. + + Returns: + sps.csc_matrix: The assembled lumped matrix. + """ + # Assemble the block diagonal mass matrix for the base discretization class + D = super().assemble_lumped_matrix(sd) + + # Assemble the trace part + B = self.assemble_trace_matrix(sd) + + # Assemble the piecewise linear mass matrix, to assemble the term + # (Trace(sigma), Trace(tau)) + discr = pg.PwLinears(self.keyword) + M = discr.assemble_lumped_matrix(sd) + + # Extract the data and compute the coefficient for the trace part + mu = data[pp.PARAMETERS][self.keyword]["mu"] + lambda_ = data[pp.PARAMETERS][self.keyword]["lambda"] + coeff = lambda_ / (2 * mu + sd.dim * lambda_) + + # Compose all the parts and return them + return (D - coeff * B.T @ M @ B) / (2 * mu) + + def assemble_lumped_matrix_cosserat( + self, sd: pg.Grid, data: dict + ) -> sps.csc_matrix: + """ + Assembles the lumped matrix with cosserat terms for the given grid. + + Args: + sd (pg.Grid): The grid object. + data (Optional[dict]): Optional data dictionary. + + Returns: + sps.csc_matrix: The assembled lumped matrix. + """ + + M = self.assemble_lumped_matrix(sd, data) + + # Extract the data + mu = data[pp.PARAMETERS][self.keyword]["mu"] + mu_c = data[pp.PARAMETERS][self.keyword]["mu_c"] + + coeff = 0.25 * (1 / mu_c - 1 / mu) + + # If coeff is a scalar, replace it by a vector so that it can be accessed per cell + if isinstance(coeff, np.ScalarType): + coeff = np.full(sd.num_cells, coeff) + + data_for_R = pp.initialize_data(sd, {}, self.keyword, {"weight": coeff}) + + if sd.dim == 2: + R_space = pg.PwLinears(self.keyword) + elif sd.dim == 3: + R_space = pg.VecPwLinears(self.keyword) + + R_mass = R_space.assemble_lumped_matrix(sd, data_for_R) + + asym = self.assemble_asym_matrix(sd, False) + + return M + asym.T @ R_mass @ asym + + def proj_to_RT0(self, sd: pg.Grid) -> sps.csc_matrix: + """ + Project the function space to the lowest order Raviart-Thomas (RT0) space. + + Args: + sd (pg.Grid): The grid object representing the computational domain. + + Returns: + sps.csc_matrix: The projection matrix to the RT0 space. + """ + proj = self.scalar_discr.proj_to_RT0(sd) + return sps.block_diag([proj] * sd.dim, format="csc") + + def proj_from_RT0(self, sd: pg.Grid) -> sps.csc_matrix: + """ + Project the RT0 finite element space onto the faces of the given grid. + + Args: + sd (pg.Grid): The grid on which the projection is performed. + + Returns: + sps.csc_matrix: The projection matrix. + """ + proj = self.scalar_discr.proj_from_RT0(sd) + return sps.block_diag([proj] * sd.dim, format="csc") def get_range_discr_class(self, dim: int) -> object: """ @@ -952,3 +1124,163 @@ def get_range_discr_class(self, dim: int) -> object: differential """ return pg.VecPwConstants + + +class VecRT0(pg.VecDiscretization): + def __init__(self, keyword: str) -> None: + """ + Initialize the vector RT0 discretization class. + The scalar discretization class is pg.RT0. + + We are considering the following structure of the stress tensor in 2d + + sigma = [[sigma_xx, sigma_xy], + [sigma_yx, sigma_yy]] + + which is represented in the code unrolled row-wise as a vector of length 4 + + sigma = [sigma_xx, sigma_xy, + sigma_yx, sigma_yy] + + While in 3d the stress tensor can be written as + + sigma = [[sigma_xx, sigma_xy, sigma_xz], + [sigma_yx, sigma_yy, sigma_yz], + [sigma_zx, sigma_zy, sigma_zz]] + + where its vectorized structure of length 9 is given by + + sigma = [sigma_xx, sigma_xy, sigma_xz, + sigma_yx, sigma_yy, sigma_yz, + sigma_zx, sigma_zy, sigma_zz] + + Args: + keyword (str): The keyword for the vector discretization class. + + Returns: + None + """ + super().__init__(keyword, pg.RT0) + + def assemble_mass_matrix(self, sd: pg.Grid, data: dict) -> sps.csc_matrix: + """ + Assembles and returns the mass matrix for vector RT0, which is given by + (A sigma, tau) where A sigma = (sigma - coeff * Trace(sigma) * I) / (2 mu) + with mu and lambda the Lamé constants and coeff = lambda / (2*mu + dim*lambda) + + Args: + sd (pg.Grid): The grid. + data (dict): Data for the assembly. + + Returns: + sps.csc_matrix: The mass matrix obtained from the discretization. + """ + # Assemble the block diagonal mass matrix for the base discretization class, + # with unitary data. The data are handled afterwards + D = super().assemble_mass_matrix(sd) + + # Assemble the trace part + B = self.assemble_trace_matrix(sd) + + # Assemble the piecewise linear mass matrix, to assemble the term + # (Trace(sigma), Trace(tau)) + discr = pg.PwLinears(self.keyword) + M = discr.assemble_mass_matrix(sd) + + # Extract the data and compute the coefficient for the trace part + mu = data[pp.PARAMETERS][self.keyword]["mu"] + lambda_ = data[pp.PARAMETERS][self.keyword]["lambda"] + coeff = lambda_ / (2 * mu + sd.dim * lambda_) + + # Compose all the parts and return them + return (D - coeff * B.T @ M @ B) / (2 * mu) + + def assemble_mass_matrix_cosserat(self, sd: pg.Grid, data: dict) -> sps.csc_matrix: + """ + Assembles and returns the mass matrix for vector BDM1 discretizing the Cosserat inner + product, which is given by (A sigma, tau) where + A sigma = (sym(sigma) - coeff * Trace(sigma) * I) / (2 mu) + skw(sigma) / (2 mu_c) + with mu and lambda the Lamé constants, coeff = lambda / (2*mu + dim*lambda), and + mu_c the coupling Lamé modulus. + + Args: + sd (pg.Grid): The grid. + data (dict): Data for the assembly. + + Returns: + sps.csc_matrix: The mass matrix obtained from the discretization. + + TODO: Consider using inheritance from VecBDM1.assemble_mass_matrix_cosserat + """ + + M = self.assemble_mass_matrix(sd, data) + + # Extract the data + mu = data[pp.PARAMETERS][self.keyword]["mu"] + mu_c = data[pp.PARAMETERS][self.keyword]["mu_c"] + + coeff = 0.25 * (1 / mu_c - 1 / mu) + + # If coeff is a scalar, replace it by a vector so that it can be accessed per cell + if isinstance(coeff, np.ScalarType): + coeff = np.full(sd.num_cells, coeff) + + data_for_R = pp.initialize_data(sd, {}, self.keyword, {"weight": coeff}) + + if sd.dim == 2: + R_space = pg.PwLinears(self.keyword) + elif sd.dim == 3: + R_space = pg.VecPwLinears(self.keyword) + + R_mass = R_space.assemble_mass_matrix(sd, data_for_R) + + asym = self.assemble_asym_matrix(sd, False) + + return M + asym.T @ R_mass @ asym + + def assemble_trace_matrix(self, sd: pg.Grid) -> sps.csc_matrix: + """ + Assembles and returns the trace matrix for the vector BDM1. + + Args: + sd (pg.Grid): The grid. + + Returns: + sps.csc_matrix: The trace matrix obtained from the discretization. + """ + vec_bdm1 = VecBDM1(self.keyword) + proj = vec_bdm1.proj_from_RT0(sd) + return vec_bdm1.assemble_trace_matrix(sd) @ proj + + def assemble_asym_matrix(self, sd: pg.Grid, as_pwconstant=True) -> sps.csc_matrix: + """ + Assembles and returns the asymmetric matrix for the vector RT0. + + The asymmetric operator `as' for a tensor is a scalar and it is defined in 2d as + as(tau) = tau_xy - tau_yx + while for a tensor in 3d it is a vector and given by + as(tau) = [tau_zy - tau_yz, tau_xz - tau_zx, tau_yx - tau_xy]^T + + Note: We assume that the as(tau) is a cell variable. + + Args: + sd (pg.Grid): The grid. + + Returns: + sps.csc_matrix: The asymmetric matrix obtained from the discretization. + """ + vec_bdm1 = VecBDM1(self.keyword) + proj = vec_bdm1.proj_from_RT0(sd) + return vec_bdm1.assemble_asym_matrix(sd, as_pwconstant) @ proj + + def get_range_discr_class(self, dim: int) -> object: + """ + Returns the range discretization class for the given dimension. + + Args: + dim (int): The dimension of the range space. + + Returns: + pg.Discretization: The range discretization class. + """ + return pg.VecPwConstants diff --git a/src/pygeon/discretizations/fem/l2.py b/src/pygeon/discretizations/fem/l2.py index bf851c9f..ced9a6f2 100644 --- a/src/pygeon/discretizations/fem/l2.py +++ b/src/pygeon/discretizations/fem/l2.py @@ -341,6 +341,27 @@ def assemble_mass_matrix( # Construct the global matrix return sps.csc_matrix((data_IJ, (rows_I, cols_J))) + def assemble_lumped_matrix( + self, sd: pg.Grid, data: Optional[dict] = None + ) -> sps.csc_matrix: + """ + Assembles the lumped matrix for the given grid. + + Args: + sd (pg.Grid): The grid object. + data (Optional[dict]): Optional data dictionary. + + Returns: + sps.csc_matrix: The assembled lumped matrix. + """ + try: + weight = data[pp.PARAMETERS][self.keyword]["weight"] + except Exception: + weight = np.ones(sd.num_cells) + + diag = np.repeat(weight * sd.cell_volumes, sd.dim + 1) / (sd.dim + 1) + return sps.diags(diag, format="csc") + def assemble_diff_matrix(self, sd: pg.Grid) -> sps.csc_matrix: """ Assembles the matrix corresponding to the differential operator. @@ -393,7 +414,12 @@ def eval_at_cell_centers(self, sd: pg.Grid) -> np.ndarray: Returns: np.ndarray: The evaluation matrix. """ - raise NotImplementedError + + rows = np.repeat(np.arange(sd.num_cells), sd.dim + 1) + cols = np.arange(self.ndof(sd)) + data = np.ones(self.ndof(sd)) / (sd.dim + 1) + + return sps.csc_matrix((data, (rows, cols))) def interpolate( self, sd: pg.Grid, func: Callable[[np.ndarray], np.ndarray] @@ -468,3 +494,95 @@ def assemble_nat_bc( np.ndarray: The assembled natural boundary condition vector. """ return np.zeros(self.ndof(sd)) + + def error_l2( + self, + sd: pg.Grid, + num_sol: np.ndarray, + ana_sol: Callable[[np.ndarray], np.ndarray], + relative: Optional[bool] = True, + etype: Optional[str] = "specific", + ) -> float: + """ + Returns the l2 error computed against an analytical solution given as a function. + + Args: + sd (pg.Grid): Grid, or a subclass. + num_sol (np.ndarray): Vector of the numerical solution. + ana_sol (Callable[[np.ndarray], np.ndarray]): Function that represents the + analytical solution. + relative (Optional[bool], optional): Compute the relative error or not. + Defaults to True. + etype (Optional[str], optional): Type of error computed. Defaults to "specific". + + Returns: + float: The computed error. + """ + + err2 = 0 + num_sol = num_sol.reshape((sd.dim, -1)) + for d in np.arange(sd.dim): + ana_sol_dim = lambda x: ana_sol(x)[d] + num_sol_dim = num_sol[d] + + err2_dim = self.scalar_discr.error_l2( + sd, num_sol_dim, ana_sol_dim, relative, etype + ) + err2 += err2_dim**2 + return np.sqrt(err2) + + +class VecPwLinears(pg.VecDiscretization): + """ + A class representing the discretization using vector piecewise linear functions. + + Attributes: + keyword (str): The keyword for the vector discretization class. + + Methods: + get_range_discr_class(self, dim: int) -> pg.Discretization: + Returns the discretization class for the range of the differential. + + """ + + def __init__(self, keyword: str) -> None: + """ + Initialize the vector discretization class. + The scalar discretization class is pg.PwLinears. + + Args: + keyword (str): The keyword for the vector discretization class. + + Returns: + None + """ + super().__init__(keyword, pg.PwLinears) + + def get_range_discr_class(self, dim: int) -> pg.Discretization: + """ + Returns the discretization class for the range of the differential. + + Args: + dim (int): The dimension of the range space. + + Returns: + pg.Discretization: The discretization class for the range of the differential. + """ + return self.scalar_discr.get_range_discr_class(dim) + + def assemble_nat_bc( + self, sd: pg.Grid, func: Callable[[np.ndarray], np.ndarray], b_faces: np.ndarray + ) -> np.ndarray: + """ + Assembles the natural boundary condition vector, equal to zero. + + Args: + sd (pg.Grid): The grid object. + func (Callable[[np.ndarray], np.ndarray]): The function defining the + natural boundary condition. + b_faces (np.ndarray): The array of boundary faces. + + Returns: + np.ndarray: The assembled natural boundary condition vector. + """ + return np.zeros(self.ndof(sd)) diff --git a/src/pygeon/discretizations/vem/h1.py b/src/pygeon/discretizations/vem/h1.py index a5561b9a..955f8679 100644 --- a/src/pygeon/discretizations/vem/h1.py +++ b/src/pygeon/discretizations/vem/h1.py @@ -150,7 +150,7 @@ def assemble_loc_proj_to_mon( ) -> np.ndarray: """ Computes the local projection onto the monomials - Returns the coefficients {a_i} in a_0 + [a_1, a_2] \dot (x - c) / d + Returns the coefficients {a_i} in a_0 + [a_1, a_2] dot (x - c) / d for each VL1 basis function. Args: diff --git a/src/pygeon/discretizations/vem/hdiv.py b/src/pygeon/discretizations/vem/hdiv.py index 645e3dc6..846ebb9c 100644 --- a/src/pygeon/discretizations/vem/hdiv.py +++ b/src/pygeon/discretizations/vem/hdiv.py @@ -167,7 +167,7 @@ def assemble_nat_bc( ) -> np.ndarray: """ Assembles the natural boundary condition term - (n dot q, func)_\Gamma. + (n dot q, func)_Gamma. Args: sd (pg.Grid): The grid object representing the computational domain. @@ -408,7 +408,7 @@ def assemble_nat_bc( ) -> np.ndarray: """ Assembles the natural boundary condition term - (n dot q, func)_\Gamma + (n dot q, func)_Gamma Args: sd (pg.Grid): The grid object representing the computational domain. diff --git a/src/pygeon/grids/create_grid.py b/src/pygeon/grids/create_grid.py index 5d9e66b2..f75a5254 100644 --- a/src/pygeon/grids/create_grid.py +++ b/src/pygeon/grids/create_grid.py @@ -25,12 +25,14 @@ def grid_from_domain( Returns: Either a pg.MixedDimensionalGrid or a pg.Grid, depending on the value of as_mdg. """ + as_mdg = kwargs.get("as_mdg", True) mesh_size_min = kwargs.get("mesh_size_min", mesh_size / 10) + mesh_kwargs = {"mesh_size_frac": mesh_size, "mesh_size_min": mesh_size_min} - mdg = pp.create_fracture_network(domain=domain).mesh(mesh_kwargs) + mdg = pp.create_fracture_network(domain=domain).mesh(mesh_kwargs, **kwargs) pg.convert_from_pp(mdg) - if kwargs.get("as_mdg", True): + if as_mdg: return mdg else: return mdg.subdomains(dim=mdg.dim_max())[0] diff --git a/src/pygeon/grids/graph.py b/src/pygeon/grids/graph.py index e61e6d70..b9c0c122 100644 --- a/src/pygeon/grids/graph.py +++ b/src/pygeon/grids/graph.py @@ -12,7 +12,7 @@ class Graph(pp.Grid): def __init__(self, graph, dim=2): """Construct a pp.Grid like representation of a graph. - The following informations are stored + The following information are stored dim: the dimension of the graph, always nodes: the coordinate of the nodes cell_faces: the map from the cells (graph nodes) to the faces (graph edges) @@ -89,7 +89,7 @@ def compute_ridges(self): This function create mapping between the faces and the ridges in the graph, the following data are created - num_ridges: the number of rideges (graph cicles) + num_ridges: the number of ridges (graph cycles) num_peaks: the number of peaks (always 0) face_ridges: the map from the faces to the ridges ridge_peaks: an empty matrix that maps the ridges to peaks @@ -182,7 +182,7 @@ def edges_of_nodes(self, nodes): def collapse(self, dim): """Collapse the graph removing the nodes with a given dimension. - From a bipartite graph it is possible to create a dim = 2 graph (only fracures) + From a bipartite graph it is possible to create a dim = 2 graph (only fractures) or a dim = 1 (only fracture intersections) graph. """ @@ -224,7 +224,7 @@ def shortest_paths(self, start, end): return np.array(list(sp), dtype=np.object) def not_shortest_paths(self, start, end, sp=None, cutoff=None): - """Compyte all the not shortest paths between two graph nodes. + """Compute all the not shortest paths between two graph nodes. It is possible to set a cutoff to make the computation faster leaving out long paths. """ @@ -291,8 +291,8 @@ def to_file(self, file_name): """ # make sure that an edge is sorted by dimension - sort = ( - lambda e: e + sort = lambda e: ( + e if self.graph.nodes[e[0]]["dim"] > self.graph.nodes[e[1]]["dim"] else np.flip(e) ) diff --git a/src/pygeon/grids/md_grid.py b/src/pygeon/grids/md_grid.py index 565af939..91bba4e4 100644 --- a/src/pygeon/grids/md_grid.py +++ b/src/pygeon/grids/md_grid.py @@ -89,26 +89,27 @@ def initialize_data(self) -> None: Returns: None """ + unit_keyword = "unit" for sd, data in self.subdomains(return_data=True): perm = pp.SecondOrderTensor(np.ones(sd.num_cells)) data[pp.PARAMETERS] = pp.Parameters( sd, - ["unit"], + [unit_keyword], [ {"second_order_tensor": perm}, ], ) - data[pp.DISCRETIZATION_MATRICES] = {} + data[pp.DISCRETIZATION_MATRICES] = {unit_keyword: {}} for _, data in self.interfaces(return_data=True): data[pp.PARAMETERS] = pp.Parameters( sd, - ["unit"], + [unit_keyword], [ {"normal_diffusivity": 1.0}, ], ) - data[pp.DISCRETIZATION_MATRICES] = {} + data[pp.DISCRETIZATION_MATRICES] = {unit_keyword: {}} def num_subdomain_faces( self, cond: Optional[Callable[[pg.Grid], bool]] = None diff --git a/src/pygeon/numerics/projections.py b/src/pygeon/numerics/projections.py index e739725f..fd4e4247 100644 --- a/src/pygeon/numerics/projections.py +++ b/src/pygeon/numerics/projections.py @@ -72,7 +72,7 @@ def proj_faces_to_cells( ) if discr is None: - discr = pg.RT0("flow") + discr = pg.RT0("unit") # Local mass matrices for nn_sd, (sd, d_sd) in enumerate(mdg.subdomains(return_data=True)): diff --git a/src/pygeon/numerics/spanningtree.py b/src/pygeon/numerics/spanningtree.py index 7899e4b5..99dcb364 100644 --- a/src/pygeon/numerics/spanningtree.py +++ b/src/pygeon/numerics/spanningtree.py @@ -497,6 +497,70 @@ def compute_system(self, sd: pg.Grid) -> sps.csc_matrix: return sps.vstack((-div, -asym)) @ self.expand +class SpanningTreeCosserat(SpanningTreeElasticity): + """ + Represents a class for computing the spanning tree for the Cosserat problem. + + Attributes: + expand (sps.csc_matrix): The expanded matrix for spanning tree computation. + system (sps.csc_matrix): The computed system matrix. + + Methods: + setup_system(self, mdg: pg.MixedDimensionalGrid, flagged_faces: np.ndarray) -> None: + Set up the system for the spanning tree algorithm. + + compute_expand(self, sd: pg.Grid, flagged_faces: np.ndarray) -> sps.csc_matrix: + Compute the expanded matrix for spanning tree computation. + + compute_system(self, sd: pg.Grid) -> sps.csc_matrix: + Computes the system matrix for the given grid. + """ + + def compute_expand(self, sd: pg.Grid, flagged_faces: np.ndarray) -> sps.csc_matrix: + """ + Compute the expanded matrix for spanning tree computation. + + Args: + sd (pg.Grid): The grid object. + flagged_faces (np.ndarray): Array of flagged faces. + + Returns: + sps.csc_matrix: The expanded matrix for spanning tree computation. + """ + dim_sig_omega = sd.dim * (sd.dim + 1) // 2 + + expand = pg.numerics.linear_system.create_restriction(flagged_faces).T.tocsc() + + return sps.block_diag([expand] * dim_sig_omega, format="csc") + + def compute_system(self, sd: pg.Grid) -> sps.csc_matrix: + """ + Computes the system matrix for the given grid. + + Args: + sd (pg.Grid): The grid object representing the domain. + + Returns: + sps.csc_matrix: The computed system matrix. + """ + assert sd.dim == 3, "Only implemented the 3D Raviart-Thomas version." + + # first we assemble the B matrix + key = "tree" + vec_rt0 = pg.VecRT0(key) + vec_p0 = pg.VecPwConstants(key) + + M = vec_p0.assemble_mass_matrix(sd) + + div = M @ vec_rt0.assemble_diff_matrix(sd) + asym = M @ vec_rt0.assemble_asym_matrix(sd) + + B = sps.bmat([[-div, None], [-asym, -div]], format="csc") + + # create the solution operator + return B @ self.expand + + class SpanningWeightedTrees: """ Class that can perform a spanning weighted trees solve, based on one of diff --git a/tests/integration/test_elasticity.py b/tests/integration/test_elasticity.py index 8c8dfc26..bf8d135d 100644 --- a/tests/integration/test_elasticity.py +++ b/tests/integration/test_elasticity.py @@ -180,9 +180,9 @@ def run_elasticity_2d(self, u_boundary, N): asym = Mr @ vec_bdm1.assemble_asym_matrix(sd) # fmt: off - spp = sps.bmat([[ Ms, div.T, asym.T], - [ -div, None, None], - [-asym, None, None]], format = "csc") + spp = sps.bmat([[ Ms, div.T, -asym.T], + [-div, None, None], + [asym, None, None]], format = "csc") # fmt: on b_faces = sd.tags["domain_boundary_faces"] diff --git a/tests/unit/test_projections.py b/tests/unit/test_projections.py index 5763f5bd..83cce460 100644 --- a/tests/unit/test_projections.py +++ b/tests/unit/test_projections.py @@ -294,5 +294,4 @@ def test3(self): if __name__ == "__main__": - # ProjectionsUnitTest().test3() unittest.main() diff --git a/tests/unit/test_pwlinear.py b/tests/unit/test_pwlinear.py index db457936..f63056b5 100644 --- a/tests/unit/test_pwlinear.py +++ b/tests/unit/test_pwlinear.py @@ -137,7 +137,11 @@ def test_eval_at_cell_centers(self): discr = pg.PwLinears("P1b") - self.assertRaises(NotImplementedError, discr.eval_at_cell_centers, sd) + known_func = np.ones(discr.ndof(sd)) + + P = discr.eval_at_cell_centers(sd) + + self.assertTrue(np.allclose(P @ known_func, np.ones(sd.num_cells))) def test_assemble_nat_bc(self): dim = 2 diff --git a/tests/unit/test_spanningtree.py b/tests/unit/test_spanningtree.py index 4208c8bd..a52fe34f 100644 --- a/tests/unit/test_spanningtree.py +++ b/tests/unit/test_spanningtree.py @@ -230,5 +230,57 @@ def test_for_errors(self): self.assertRaises(NotImplementedError, pg.SpanningTreeElasticity, mdg) +class SpanningTreeCosseratTest(unittest.TestCase): + def check(self, sd): + mdg = pg.as_mdg(sd) + pg.convert_from_pp(mdg) + mdg.compute_geometry() + + B = self.assemble_B(mdg) + f = np.random.rand(B.shape[0]) + + sptr = pg.SpanningTreeCosserat(mdg) + + s_f = sptr.solve(f) + self.assertTrue(np.allclose(B @ s_f, f)) + + def assemble_B(self, mdg): + sd = mdg.subdomains(dim=mdg.dim_max())[0] + + key = "tree" + vec_rt0 = pg.VecRT0(key) + vec_p0 = pg.VecPwConstants(key) + + M = vec_p0.assemble_mass_matrix(sd) + + div = M @ vec_rt0.assemble_diff_matrix(sd) + asym = M @ vec_rt0.assemble_asym_matrix(sd) + + return sps.bmat([[-div, None], [-asym, -div]]) + + def test_elasticity_struct_tet_grid(self): + sd = pp.StructuredTetrahedralGrid([1] * 3) + self.check(sd) + + def test_elasticity_unstruct_tet_grid(self): + sd = pg.unit_grid(3, 1.0) + self.check(sd) + + def test_assemble_SI(self): + N, dim = 3, 3 + sd = pp.StructuredTetrahedralGrid([N] * dim, [1] * dim) + mdg = pg.as_mdg(sd) + pg.convert_from_pp(mdg) + mdg.compute_geometry() + + sptr = pg.SpanningTreeCosserat(mdg) + + SI = sptr.assemble_SI() + B = self.assemble_B(mdg) + check = sps.eye_array(B.shape[0]) - B @ SI + + self.assertTrue(np.allclose(check.data, 0)) + + if __name__ == "__main__": unittest.main() diff --git a/tests/unit/test_vecbdm1.py b/tests/unit/test_vecbdm1.py index 07e36705..c20b8802 100644 --- a/tests/unit/test_vecbdm1.py +++ b/tests/unit/test_vecbdm1.py @@ -6,6 +6,15 @@ class VecBDM1Test(unittest.TestCase): + + def test_1d(self): + sd = pp.CartGrid([1]) + pg.convert_from_pp(sd) + sd.compute_geometry() + + vec_bdm1 = pg.VecBDM1("vecbdm1") + self.assertRaises(ValueError, vec_bdm1.assemble_asym_matrix, sd) + def test_trace_2d(self): sd = pp.StructuredTriangleGrid([1] * 2, [1] * 2) pg.convert_from_pp(sd) @@ -52,6 +61,61 @@ def test_assemble_mass_matrix_2d(self): M = vec_bdm1.assemble_mass_matrix(sd, data) self.assertAlmostEqual(u.T @ M @ u, 30) + def test_assemble_mass_matrix_cosserat_2d(self): + N = 10 + sd = pp.StructuredTriangleGrid([N] * 2, [1] * 2) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecbdm1" + vec_bdm1 = pg.VecBDM1(key) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0.5, "mu_c": 0.25}}} + M = vec_bdm1.assemble_mass_matrix_cosserat(sd, data) + + fun = lambda _: np.array([[1, 2, 0], [4, 3, 0]]) + u = vec_bdm1.interpolate(sd, fun) + + self.assertAlmostEqual(u.T @ M @ u, 28) + + def test_assemble_lumped_matrix_2d(self): + N = 10 + sd = pp.StructuredTriangleGrid([N] * 2, [1] * 2) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecbdm1" + vec_bdm1 = pg.VecBDM1(key) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0.5}}} + M = vec_bdm1.assemble_lumped_matrix(sd, data) + + fun = lambda _: np.array([[1, 2, 0], [4, 3, 0]]) + u = vec_bdm1.interpolate(sd, fun) + + self.assertAlmostEqual(u.T @ M @ u, 26) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0}}} + M = vec_bdm1.assemble_lumped_matrix(sd, data) + self.assertAlmostEqual(u.T @ M @ u, 30) + + def test_assemble_lumped_matrix_cosserat_2d(self): + N = 10 + sd = pp.StructuredTriangleGrid([N] * 2, [1] * 2) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecbdm1" + vec_bdm1 = pg.VecBDM1(key) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0.5, "mu_c": 0.25}}} + M = vec_bdm1.assemble_lumped_matrix_cosserat(sd, data) + + fun = lambda _: np.array([[1, 2, 0], [4, 3, 0]]) + u = vec_bdm1.interpolate(sd, fun) + + self.assertAlmostEqual(u.T @ M @ u, 28) + def test_eval_at_cell_centers_2d(self): N = 1 sd = pp.StructuredTriangleGrid([N] * 2, [1] * 2) @@ -72,6 +136,24 @@ def linear(x): self.assertAlmostEqual(np.linalg.norm(eval - known), 0) + def test_proj_to_and_from_rt0_2d(self): + N = 1 + sd = pp.StructuredTriangleGrid([N] * 2, [1] * 2) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecbdm1" + vec_bdm1 = pg.VecBDM1(key) + + def linear(x): + return np.array([x, 2 * x]) + + interp = vec_bdm1.interpolate(sd, linear) + interp_to_rt0 = vec_bdm1.proj_to_RT0(sd) @ interp + interp_from_rt0 = vec_bdm1.proj_from_RT0(sd) @ interp_to_rt0 + + self.assertAlmostEqual(np.linalg.norm(interp - interp_from_rt0), 0) + def test_range(self): key = "vecbdm1" vec_bdm1 = pg.VecBDM1(key) @@ -88,19 +170,19 @@ def test_assemble_asym_matrix_2d(self): fun = lambda _: np.array([[1, 2, 0], [4, 3, 0]]) u = vec_bdm1.interpolate(sd, fun) - asym = vec_bdm1.assemble_asym_matrix(sd) + asym = vec_bdm1.assemble_asym_matrix(sd, False) - p0 = pg.PwConstants("p0") - cell_asym_u = p0.eval_at_cell_centers(sd) @ (asym @ u) + p1 = pg.PwLinears("p1") + cell_asym_u = p1.eval_at_cell_centers(sd) @ (asym @ u) - self.assertTrue(np.allclose(cell_asym_u, -2)) + self.assertTrue(np.allclose(cell_asym_u, 2)) def test_trace_3d(self): sd = pp.StructuredTetrahedralGrid([1] * 3, [1] * 3) pg.convert_from_pp(sd) sd.compute_geometry() - vec_bdm1 = pg.VecBDM1("vec_bdm1") + vec_bdm1 = pg.VecBDM1("vecbdm1") B = vec_bdm1.assemble_trace_matrix(sd) @@ -118,7 +200,7 @@ def test_ndof_3d(self): pg.convert_from_pp(sd) sd.compute_geometry() - vec_bdm1 = pg.VecBDM1("vec_bdm1") + vec_bdm1 = pg.VecBDM1("vecbdm1") self.assertEqual(vec_bdm1.ndof(sd), 162) @@ -128,7 +210,7 @@ def test_assemble_mass_matrix_3d(self): pg.convert_from_pp(sd) sd.compute_geometry() - key = "vec_bdm1" + key = "vecbdm1" vec_bdm1 = pg.VecBDM1(key) data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0.5}}} @@ -143,13 +225,72 @@ def test_assemble_mass_matrix_3d(self): M = vec_bdm1.assemble_mass_matrix(sd, data) self.assertAlmostEqual(u.T @ M @ u, 32) + def test_assemble_mass_matrix_cosserat_3d(self): + N = 1 + sd = pp.StructuredTetrahedralGrid([N] * 3, [1] * 3) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecbdm1" + vec_bdm1 = pg.VecBDM1(key) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0.5, "mu_c": 0.25}}} + M = vec_bdm1.assemble_mass_matrix_cosserat(sd, data) + + fun = lambda _: np.array([[1, 2, 0], [4, 3, 0], [0, 1, 1]]) + u = vec_bdm1.interpolate(sd, fun) + + self.assertAlmostEqual(u.T @ M @ u, 29.5) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0, "mu_c": 0.25}}} + M = vec_bdm1.assemble_mass_matrix_cosserat(sd, data) + self.assertAlmostEqual(u.T @ M @ u, 34.5) + + def test_assemble_lumped_matrix_3d(self): + N = 1 + sd = pp.StructuredTetrahedralGrid([N] * 3, [1] * 3) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecbdm1" + vec_bdm1 = pg.VecBDM1(key) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0.5}}} + M = vec_bdm1.assemble_lumped_matrix(sd, data) + + fun = lambda _: np.array([[1, 2, 0], [4, 3, 0], [0, 1, 1]]) + u = vec_bdm1.interpolate(sd, fun) + + self.assertAlmostEqual(u.T @ M @ u, 27) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0}}} + M = vec_bdm1.assemble_lumped_matrix(sd, data) + self.assertAlmostEqual(u.T @ M @ u, 32) + + def test_assemble_lumped_matrix_cosserat_3d(self): + N = 1 + sd = pp.StructuredTetrahedralGrid([N] * 3, [1] * 3) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecbdm1" + vec_bdm1 = pg.VecBDM1(key) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0.5, "mu_c": 0.25}}} + M = vec_bdm1.assemble_lumped_matrix_cosserat(sd, data) + + fun = lambda _: np.array([[1, 2, 0], [4, 3, 0], [0, 1, 1]]) + u = vec_bdm1.interpolate(sd, fun) + + self.assertAlmostEqual(u.T @ M @ u, 29.5) + def test_eval_at_cell_centers_3d(self): N = 1 sd = pp.StructuredTetrahedralGrid([N] * 3, [1] * 3) pg.convert_from_pp(sd) sd.compute_geometry() - key = "vec_bdm1" + key = "vecbdm1" vec_bdm1 = pg.VecBDM1(key) def linear(x): @@ -168,21 +309,39 @@ def test_assemble_asym_matrix_3d(self): pg.convert_from_pp(sd) sd.compute_geometry() - key = "vec_bdm1" + key = "vecbdm1" vec_bdm1 = pg.VecBDM1(key) fun = lambda _: np.array([[1, 2, -1], [4, 3, 2], [1, 1, 1]]) u = vec_bdm1.interpolate(sd, fun) - asym = vec_bdm1.assemble_asym_matrix(sd) + asym = vec_bdm1.assemble_asym_matrix(sd, False) - p0 = pg.VecPwConstants("p0") - cell_asym_u = p0.eval_at_cell_centers(sd) @ (asym @ u) + p1 = pg.VecPwLinears("p1") + cell_asym_u = p1.eval_at_cell_centers(sd) @ (asym @ u) cell_asym_u = cell_asym_u.reshape((3, -1)) self.assertTrue(np.allclose(cell_asym_u[0], -1)) self.assertTrue(np.allclose(cell_asym_u[1], -2)) self.assertTrue(np.allclose(cell_asym_u[2], 2)) + def test_proj_to_and_from_rt0_3d(self): + N = 1 + sd = pp.StructuredTetrahedralGrid([N] * 3, [1] * 3) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecbdm1" + vec_bdm1 = pg.VecBDM1(key) + + def linear(x): + return np.array([x, 2 * x, 3 * x]) + + interp = vec_bdm1.interpolate(sd, linear) + interp_to_rt0 = vec_bdm1.proj_to_RT0(sd) @ interp + interp_from_rt0 = vec_bdm1.proj_from_RT0(sd) @ interp_to_rt0 + + self.assertAlmostEqual(np.linalg.norm(interp - interp_from_rt0), 0) + if __name__ == "__main__": unittest.main() diff --git a/tests/unit/test_vecpwconstant.py b/tests/unit/test_vecpwconstant.py index ee4b0f73..58a74714 100644 --- a/tests/unit/test_vecpwconstant.py +++ b/tests/unit/test_vecpwconstant.py @@ -13,7 +13,7 @@ def test_ndof(self): sd.compute_geometry() discr = pg.VecPwConstants("P0") - assert discr.ndof(sd) == sd.num_cells * dim + self.assertTrue(discr.ndof(sd) == sd.num_cells * dim) def test_assemble_mass_matrix(self): dim = 2 @@ -179,8 +179,13 @@ def test_error_l2(self): # fmt: on ana_sol = lambda x: np.sin(x) + err = discr.error_l2(sd, num_sol, ana_sol, etype="standard") + err_known = 7.951651025069553e-08 + + self.assertTrue(np.allclose(err, err_known)) + err = discr.error_l2(sd, num_sol, ana_sol) - err_known = 5.622666361455642e-08 + err_known = 0.5254595621486393 self.assertTrue(np.allclose(err, err_known)) diff --git a/tests/unit/test_vecpwlinear.py b/tests/unit/test_vecpwlinear.py new file mode 100644 index 00000000..90cdbc1d --- /dev/null +++ b/tests/unit/test_vecpwlinear.py @@ -0,0 +1,167 @@ +import unittest +import numpy as np +import scipy.sparse as sps + +import pygeon as pg +import porepy as pp + + +class VecPwLinearsTest(unittest.TestCase): + def test_ndof(self): + dim = 2 + sd = pp.StructuredTriangleGrid([2] * dim, [1] * dim) + sd.compute_geometry() + + discr = pg.VecPwLinears("P1") + self.assertTrue(discr.ndof(sd) == sd.num_cells * dim * (dim + 1)) + + def test_assemble_mass_matrix(self): + dim = 2 + sd = pp.StructuredTriangleGrid([1] * dim, [1] * dim) + sd.compute_geometry() + + discr = pg.VecPwLinears("P1") + M = discr.assemble_mass_matrix(sd) + + # fmt: off + M_known_data = np.array( + [0.08333333, 0.04166667, 0.04166667, 0.04166667, 0.08333333, + 0.04166667, 0.04166667, 0.04166667, 0.08333333, 0.08333333, + 0.04166667, 0.04166667, 0.04166667, 0.08333333, 0.04166667, + 0.04166667, 0.04166667, 0.08333333, 0.08333333, 0.04166667, + 0.04166667, 0.04166667, 0.08333333, 0.04166667, 0.04166667, + 0.04166667, 0.08333333, 0.08333333, 0.04166667, 0.04166667, + 0.04166667, 0.08333333, 0.04166667, 0.04166667, 0.04166667, + 0.08333333] + ) + + M_known_indices = np.array( + [ 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 4, 5, 3, 4, 5, 3, 4, + 5, 6, 7, 8, 6, 7, 8, 6, 7, 8, 9, 10, 11, 9, 10, 11, 9, + 10, 11] + ) + + M_known_indptr = np.array( + [ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36] + ) + # fmt: on + + self.assertTrue(np.allclose(M.data, M_known_data)) + self.assertTrue(np.allclose(M.indptr, M_known_indptr)) + self.assertTrue(np.allclose(M.indices, M_known_indices)) + + def test_assemble_lumped_matrix(self): + dim = 2 + sd = pp.StructuredTriangleGrid([1] * dim, [1] * dim) + sd.compute_geometry() + + discr = pg.VecPwLinears("P1") + + M = discr.assemble_lumped_matrix(sd) + + # fmt: off + M_known_data = np.array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]) / 6 + + M_known_indices = np.array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]) + + M_known_indptr = np.array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]) + # fmt: on + + self.assertTrue(np.allclose(M.data, M_known_data)) + self.assertTrue(np.allclose(M.indptr, M_known_indptr)) + self.assertTrue(np.allclose(M.indices, M_known_indices)) + + def test_assemble_diff_matrix(self): + dim = 2 + sd = pp.StructuredTriangleGrid([1] * dim, [1] * dim) + sd.compute_geometry() + + discr = pg.VecPwLinears("P1") + + self.assertRaises(NotImplementedError, discr.assemble_diff_matrix, sd) + + def test_assemble_stiff_matrix(self): + dim = 2 + sd = pp.StructuredTriangleGrid([1] * dim, [1] * dim) + sd.compute_geometry() + + discr = pg.VecPwLinears("P1") + + self.assertRaises(NotImplementedError, discr.assemble_stiff_matrix, sd) + + def test_interpolate(self): + dim = 2 + sd = pp.StructuredTriangleGrid([1] * dim, [1] * dim) + sd.compute_geometry() + + discr = pg.VecPwLinears("P1") + + func = lambda x: np.sin(x) # Example function + self.assertRaises(NotImplementedError, discr.interpolate, sd, func) + + def test_eval_at_cell_centers(self): + dim = 2 + sd = pp.StructuredTriangleGrid([1] * dim, [1] * dim) + sd.compute_geometry() + + discr = pg.VecPwLinears("P1") + + P = discr.eval_at_cell_centers(sd) + + # fmt: off + P_known_data = np.ones(discr.ndof(sd)) / (sd.dim + 1) + + P_known_indices = np.array([0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3]) + + P_known_indptr = np.arange(discr.ndof(sd)+1) + # fmt: on + + self.assertTrue(np.allclose(P.data, P_known_data)) + self.assertTrue(np.allclose(P.indptr, P_known_indptr)) + self.assertTrue(np.allclose(P.indices, P_known_indices)) + + def test_assemble_nat_bc(self): + dim = 2 + sd = pp.StructuredTriangleGrid([1] * dim, [1] * dim) + sd.compute_geometry() + + discr = pg.VecPwLinears("P1") + + func = lambda x: np.sin(x[0]) # Example function + + b_faces = np.array([0, 1, 3]) # Example boundary faces + vals = discr.assemble_nat_bc(sd, func, b_faces) + + vals_known = np.zeros(discr.ndof(sd)) + + self.assertTrue(np.allclose(vals, vals_known)) + + def test_get_range_discr_class(self): + dim = 2 + sd = pp.StructuredTriangleGrid([1] * dim, [1] * dim) + sd.compute_geometry() + + discr = pg.VecPwLinears("P1") + + self.assertRaises( + NotImplementedError, + discr.get_range_discr_class, + dim, + ) + + def test_error_l2(self): + dim = 2 + sd = pp.StructuredTriangleGrid([1] * dim, [1] * dim) + sd.compute_geometry() + + discr = pg.VecPwLinears("P1") + # fmt: off + num_sol = np.empty(0) + # fmt: on + ana_sol = lambda x: np.sin(x) + + self.assertRaises(NotImplementedError, discr.error_l2, sd, num_sol, ana_sol) + + +if __name__ == "__main__": + unittest.main() diff --git a/tests/unit/test_vecrt0.py b/tests/unit/test_vecrt0.py new file mode 100644 index 00000000..9b8bbabe --- /dev/null +++ b/tests/unit/test_vecrt0.py @@ -0,0 +1,222 @@ +import unittest +import numpy as np + +import porepy as pp +import pygeon as pg + + +class VecRT0Test(unittest.TestCase): + def test_trace_2d(self): + sd = pp.StructuredTriangleGrid([1] * 2, [1] * 2) + pg.convert_from_pp(sd) + sd.compute_geometry() + + vec_rt0 = pg.VecRT0("vecrt0") + + B = vec_rt0.assemble_trace_matrix(sd) + + fun = lambda _: np.array([[1, 2, 0], [3, 4, 0]]) + u = vec_rt0.interpolate(sd, fun) + + trace = B @ u + + self.assertTrue(np.allclose(trace, 5)) + + def test_ndof_2d(self): + sd = pp.StructuredTriangleGrid([1] * 2, [1] * 2) + pg.convert_from_pp(sd) + sd.compute_geometry() + + vec_rt0 = pg.VecRT0("vecrt0") + + self.assertEqual(vec_rt0.ndof(sd), 10) + + def test_assemble_mass_matrix_2d(self): + N = 10 + sd = pp.StructuredTriangleGrid([N] * 2, [1] * 2) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecrt0" + vec_rt0 = pg.VecRT0(key) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0.5}}} + M = vec_rt0.assemble_mass_matrix(sd, data) + + fun = lambda _: np.array([[1, 2, 0], [4, 3, 0]]) + u = vec_rt0.interpolate(sd, fun) + + self.assertAlmostEqual(u.T @ M @ u, 26) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0}}} + M = vec_rt0.assemble_mass_matrix(sd, data) + self.assertAlmostEqual(u.T @ M @ u, 30) + + def test_assemble_mass_matrix_cosserat_2d(self): + N = 10 + sd = pp.StructuredTriangleGrid([N] * 2, [1] * 2) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecrt0" + vec_rt0 = pg.VecRT0(key) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0.5, "mu_c": 0.25}}} + M = vec_rt0.assemble_mass_matrix_cosserat(sd, data) + + fun = lambda _: np.array([[1, 2, 0], [4, 3, 0]]) + u = vec_rt0.interpolate(sd, fun) + + self.assertAlmostEqual(u.T @ M @ u, 28) + + def test_eval_at_cell_centers_2d(self): + N = 1 + sd = pp.StructuredTriangleGrid([N] * 2, [1] * 2) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecrt0" + vec_rt0 = pg.VecRT0(key) + + def linear(x): + return np.array([x, 2 * x]) + + interp = vec_rt0.interpolate(sd, linear) + eval = vec_rt0.eval_at_cell_centers(sd) @ interp + eval = np.reshape(eval, (3, -1), order="F") + eval = np.vstack(np.split(eval, sd.dim, axis=1)) + + known = np.array([linear(x).ravel() for x in sd.cell_centers.T]).T + + self.assertAlmostEqual(np.linalg.norm(eval - known), 0) + + def test_range(self): + key = "vecrt0" + vec_rt0 = pg.VecRT0(key) + self.assertTrue(vec_rt0.get_range_discr_class(2) is pg.VecPwConstants) + + def test_assemble_asym_matrix_2d(self): + N = 1 + sd = pp.StructuredTriangleGrid([N] * 2, [1] * 2) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecrt0" + vec_rt0 = pg.VecRT0(key) + + fun = lambda _: np.array([[1, 2, 0], [4, 3, 0]]) + u = vec_rt0.interpolate(sd, fun) + asym = vec_rt0.assemble_asym_matrix(sd) + + p0 = pg.PwConstants("p0") + cell_asym_u = p0.eval_at_cell_centers(sd) @ (asym @ u) + + self.assertTrue(np.allclose(cell_asym_u, 2)) + + def test_trace_3d(self): + sd = pp.StructuredTetrahedralGrid([1] * 3, [1] * 3) + pg.convert_from_pp(sd) + sd.compute_geometry() + + vec_rt0 = pg.VecRT0("vecrt0") + + B = vec_rt0.assemble_trace_matrix(sd) + + fun = lambda _: np.array([[1, 2, 3], [4, 5, 6], [0, 0, 7]]) + u = vec_rt0.interpolate(sd, fun) + + trace = B @ u + + self.assertTrue(np.allclose(trace, 13)) + + def test_ndof_3d(self): + sd = pp.StructuredTetrahedralGrid([1] * 3, [1] * 3) + pg.convert_from_pp(sd) + sd.compute_geometry() + + vec_rt0 = pg.VecRT0("vecrt0") + + self.assertEqual(vec_rt0.ndof(sd), 54) + + def test_assemble_mass_matrix_3d(self): + N = 1 + sd = pp.StructuredTetrahedralGrid([N] * 3, [1] * 3) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecrt0" + vec_rt0 = pg.VecRT0(key) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0.5}}} + M = vec_rt0.assemble_mass_matrix(sd, data) + + fun = lambda _: np.array([[1, 2, 0], [4, 3, 0], [0, 1, 1]]) + u = vec_rt0.interpolate(sd, fun) + + self.assertAlmostEqual(u.T @ M @ u, 27) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0}}} + M = vec_rt0.assemble_mass_matrix(sd, data) + self.assertAlmostEqual(u.T @ M @ u, 32) + + def test_assemble_mass_matrix_cosserat_3d(self): + N = 1 + sd = pp.StructuredTetrahedralGrid([N] * 3, [1] * 3) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecrt0" + vec_rt0 = pg.VecRT0(key) + + data = {pp.PARAMETERS: {key: {"mu": 0.5, "lambda": 0.5, "mu_c": 0.25}}} + M = vec_rt0.assemble_mass_matrix_cosserat(sd, data) + + fun = lambda _: np.array([[1, 2, 0], [4, 3, 0], [0, 1, 1]]) + u = vec_rt0.interpolate(sd, fun) + + self.assertAlmostEqual(u.T @ M @ u, 29.5) + + def test_eval_at_cell_centers_3d(self): + N = 1 + sd = pp.StructuredTetrahedralGrid([N] * 3, [1] * 3) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecrt0" + vec_rt0 = pg.VecRT0(key) + + def linear(x): + return np.array([x, 2 * x, -x]) + + interp = vec_rt0.interpolate(sd, linear) + eval = vec_rt0.eval_at_cell_centers(sd) @ interp + eval = np.reshape(eval, (3, -1), order="F") + eval = np.vstack(np.split(eval, sd.dim, axis=1)) + + known = np.array([linear(x).ravel() for x in sd.cell_centers.T]).T + self.assertAlmostEqual(np.linalg.norm(eval - known), 0) + + def test_assemble_asym_matrix_3d(self): + N = 1 + sd = pp.StructuredTetrahedralGrid([N] * 3, [1] * 3) + pg.convert_from_pp(sd) + sd.compute_geometry() + + key = "vecrt0" + vec_rt0 = pg.VecRT0(key) + + fun = lambda _: np.array([[1, 2, -1], [4, 3, 2], [1, 1, 1]]) + u = vec_rt0.interpolate(sd, fun) + asym = vec_rt0.assemble_asym_matrix(sd) + + p0 = pg.VecPwConstants("p0") + cell_asym_u = p0.eval_at_cell_centers(sd) @ (asym @ u) + cell_asym_u = cell_asym_u.reshape((3, -1)) + + self.assertTrue(np.allclose(cell_asym_u[0], -1)) + self.assertTrue(np.allclose(cell_asym_u[1], -2)) + self.assertTrue(np.allclose(cell_asym_u[2], 2)) + + +if __name__ == "__main__": + unittest.main() diff --git a/tutorials/cosserat.ipynb b/tutorials/cosserat.ipynb new file mode 100644 index 00000000..dff6017f --- /dev/null +++ b/tutorials/cosserat.ipynb @@ -0,0 +1,310 @@ +{ + "cells": [ + { + "attachments": {}, + "cell_type": "markdown", + "id": "essential-american", + "metadata": {}, + "source": [ + "# Cosserat equation in mixed form\n", + "\n", + "In this tutorial we present how to solve the Cosserat equation with [PyGeoN](https://github.com/compgeo-mox/pygeon) in mixed form. The unknowns are the stress $\\sigma$, the micro-stress $\\omega$, the displacement $u$, and the rotation $r$.\n", + "\n", + "Let $\\Omega=(0,1)^2$ with boundary $\\partial \\Omega$ and outward unit normal ${\\nu}$. Given the macroscopic \n", + "$\\lambda_\\sigma$ Lamé constant and $\\mu_\\sigma$ the Kirchhoff modulus and their microscopic version $\\lambda_\\omega$ and $\\mu_\\omega$, \n", + "we want to solve the following problem: find $(\\sigma, \\omega, u, r)$ such that\n", + "\\begin{align*}\n", + "&A_\\sigma \\sigma - \\epsilon(u) + {\\rm asym}^* r = 0\\\\\n", + "&A_\\omega \\omega - \\nabla r = 0\\\\\n", + "&\\nabla \\cdot \\sigma = - b\\\\\n", + "&{\\rm asym}\\, \\sigma - \\nabla \\cdot \\omega = 0\n", + "\\end{align*}\n", + "with $\\epsilon$ the symmetric gradient and $b$ a body force, which is set to $0$ in this example.\n", + "The operator $A_\\sigma$ is defined as\n", + "$$\n", + "A_\\sigma \\sigma = \\frac1{2\\mu_\\sigma} \\left( \\mathrm{sym} \\sigma - \\frac{\\lambda_\\sigma}{2\\mu_\\sigma + d \\lambda_\\sigma} \\mathrm{Tr} (\\sigma) I\\right) + \\frac1{2\\mu_\\sigma^c} \\mathrm{skw} \\sigma\n", + "$$\n", + "with $d$ the dimension of the domain, here $d=2$. While the operator $A_\\omega$ depends on the dimension \n", + "and is defined as \n", + "$$\n", + "A_\\omega \\omega = \\frac1{2\\mu_\\omega} \\left( \\mathrm{sym} \\omega - \\frac{\\lambda_\\omega}{2\\mu_\\omega + d \\lambda_\\omega} \\mathrm{Tr} (\\omega) I\\right) + \\frac1{2\\mu_\\omega^c} \\mathrm{skw} \\omega\n", + " \\qquad d = 3\\\\\n", + "A_\\omega \\omega = \\frac{1}{2\\mu_\\omega}\\omega \\qquad d = 2\n", + "$$\n", + "Finally, ${\\rm asym}$ is the asymmetric operator, which is defined as followed\n", + "$$\n", + " {\\rm asym }\\, \\sigma = \\begin{bmatrix}\n", + " \\sigma_{32} - \\sigma_{23} \\\\\n", + " \\sigma_{13} - \\sigma_{31} \\\\\n", + " \\sigma_{21} - \\sigma_{12}\n", + " \\end{bmatrix} \n", + " \\qquad\n", + " {\\rm asym}^* r =\n", + " \\begin{bmatrix}\n", + " 0 & -r_3 & r_2 \\\\\n", + " r_3 & 0 & -r_1 \\\\\n", + " -r_2 & r_1 & 0\n", + " \\end{bmatrix}\n", + " \\qquad\n", + " d = 3\\\\\n", + " {\\rm asym} \\, \\sigma = \\sigma_{21} - \\sigma_{12} \\qquad\n", + " {\\rm asym}^* r =\n", + " \\begin{bmatrix}\n", + " 0 & -r \\\\\n", + " r & 0\n", + " \\end{bmatrix} \\qquad\n", + " d = 2.\n", + "$$\n", + "For this test case we set the following boundary conditions related to the so-called footstep problem:\n", + "$$ u = 0 \\text{ on } \\partial_{bottom} \\Omega \\qquad \\nu \\cdot \\sigma = [0, 0]^\\top \\text{ on } \\partial_{left} \\Omega \\cup \\partial_{right} \\Omega \\qquad \\nu \\cdot \\sigma = [0, -1]^\\top \\text{ on } \\partial_{top} \\Omega$$\n", + "\n", + "We present *step-by-step* how to create the grid, declare the problem data, and finally solve the problem." + ] + }, + { + "cell_type": "markdown", + "id": "planned-danger", + "metadata": {}, + "source": [ + "First we import some of the standard modules, like `numpy` and `scipy.sparse`. Since PyGeoN is based on [PorePy](https://github.com/pmgbergen/porepy) we import both modules." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "dietary-perth", + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import scipy.sparse as sps\n", + "\n", + "import porepy as pp\n", + "import pygeon as pg" + ] + }, + { + "cell_type": "markdown", + "id": "roman-glossary", + "metadata": {}, + "source": [ + "We create now the grid, since we use vector RT0 for ${\\sigma}$, RT0 for $\\omega$, vector piece-wise constant for $u$ and piece-wise constant for $r$, we are restricted to simplices. In this example we consider a 2-dimensional structured grid, but the presented code will work also in 3d with the only attention that in 3d the micro-stress is a vector RT0 and the rotations is a vector piece-wise constant." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "spectacular-saturn", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\n" + ] + } + ], + "source": [ + "mesh_size = 0.05\n", + "dim = 2\n", + "\n", + "sd = pg.unit_grid(dim, mesh_size, as_mdg=False)\n", + "sd.compute_geometry()" + ] + }, + { + "cell_type": "markdown", + "id": "d625fca2", + "metadata": {}, + "source": [ + "We define now the finite element spaces for all the variables." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "52400a88", + "metadata": {}, + "outputs": [], + "source": [ + "key = \"cosserat\"\n", + "\n", + "# finite element spaces\n", + "vec_rt0 = pg.VecRT0(key)\n", + "rt0 = pg.RT0(key)\n", + "vec_p0 = pg.VecPwConstants(key)\n", + "p0 = pg.PwConstants(key)\n", + "\n", + "# build the degrees of freedom\n", + "dofs = np.array([vec_rt0.ndof(sd), rt0.ndof(sd), vec_p0.ndof(sd), p0.ndof(sd)])" + ] + }, + { + "cell_type": "markdown", + "id": "precious-belle", + "metadata": {}, + "source": [ + "With the following code we set the data, in particular the Lamé and the Kirchhoff modulus, and the boundary conditions. Since we need to identify each side of $\\partial \\Omega$ we need few steps." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "spare-person", + "metadata": {}, + "outputs": [], + "source": [ + "# data for sigma\n", + "data_sigma = {pp.PARAMETERS: {key: {\"lambda\": 1, \"mu\": 0.5, \"mu_c\": 0.25}}}\n", + "\n", + "# data for omega\n", + "sot = pp.SecondOrderTensor(np.ones(sd.num_cells))\n", + "data_omega = {pp.PARAMETERS: {key: {\"second_order_tensor\": sot}}}\n", + "\n", + "# select the portions of the boundary where we want to apply the boundary conditions\n", + "bottom = np.isclose(sd.face_centers[1, :], 0)\n", + "top = np.isclose(sd.face_centers[1, :], 1)\n", + "left = np.isclose(sd.face_centers[0, :], 0)\n", + "right = np.isclose(sd.face_centers[0, :], 1)\n", + "\n", + "# select the part of the boundary where we apply the essential boundary conditions\n", + "b_faces = np.logical_or.reduce((top, left, right))\n", + "ess_dof = np.tile(b_faces, sd.dim)\n", + "\n", + "# function for the essential boundary conditions\n", + "val = np.array([[0, 0, 0], [0, -1, 0]])\n", + "fct = lambda pt: val if np.isclose(pt[1], 1) else 0 * val\n", + "\n", + "# interpolate the essential boundary conditions\n", + "ess_val = vec_rt0.interpolate(sd, fct)" + ] + }, + { + "attachments": {}, + "cell_type": "markdown", + "id": "secure-flesh", + "metadata": {}, + "source": [ + "Once the data are assigned to the grid, we construct the matrices. Once the latter is created, we also construct the right-hand side containing the boundary conditions." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "romance-findings", + "metadata": {}, + "outputs": [], + "source": [ + "# assemble mass matrices\n", + "Ms = vec_rt0.assemble_mass_matrix_cosserat(sd, data_sigma)\n", + "Mw = rt0.assemble_mass_matrix(sd, data_omega)\n", + "Mu = vec_p0.assemble_mass_matrix(sd)\n", + "Mr = p0.assemble_mass_matrix(sd)\n", + "\n", + "# assemble matrices associated with the divergence and asymmetry operators\n", + "div_s = Mu @ vec_rt0.assemble_diff_matrix(sd)\n", + "asym = Mr @ vec_rt0.assemble_asym_matrix(sd)\n", + "div_w = Mr @ rt0.assemble_diff_matrix(sd)\n", + "\n", + "# assemble the saddle point problem matrix\n", + "# fmt: off\n", + "A = sps.block_diag([Ms, Mw])\n", + "B = sps.bmat([[-div_s, None],\n", + " [ asym, -div_w]])\n", + "spp = sps.bmat([[A, -B.T],\n", + " [B, None]], format=\"csc\")\n", + "# fmt: on" + ] + }, + { + "cell_type": "markdown", + "id": "mobile-nirvana", + "metadata": {}, + "source": [ + "We need to solve the linear system, PyGeoN provides a framework for that. The actual imposition of essential boundary conditions (stress boundary conditions) might change the symmetry of the global system, the class `pg.LinearSystem` preserves this structure by internally eliminating these degrees of freedom." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "subtle-wonder", + "metadata": {}, + "outputs": [], + "source": [ + "# solve the linear system\n", + "ls = pg.LinearSystem(spp, np.zeros(spp.shape[0]))\n", + "ls.flag_ess_bc(np.where(ess_dof)[0], ess_val)\n", + "x = ls.solve()\n", + "\n", + "# split the solution into the components\n", + "idx = np.cumsum(dofs[:-1])\n", + "sigma, omega, u, r = np.split(x, idx)" + ] + }, + { + "cell_type": "markdown", + "id": "pacific-alpha", + "metadata": {}, + "source": [ + "We finally export the solution to be visualized by [ParaView](https://www.paraview.org/)." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "satisfactory-jerusalem", + "metadata": {}, + "outputs": [], + "source": [ + "# post process of the displacement\n", + "proj_u = vec_p0.eval_at_cell_centers(sd)\n", + "cell_u = (proj_u @ u).reshape((sd.dim, -1), order=\"C\")\n", + "# since we are in 2d we need to add the z component for the exporting\n", + "cell_u = np.vstack((cell_u, np.zeros(cell_u.shape[1])))\n", + "\n", + "# post process of the rotation\n", + "proj_r = p0.eval_at_cell_centers(sd)\n", + "cell_r = proj_r @ r\n", + "\n", + "save = pp.Exporter(sd, \"sol\", folder_name=key)\n", + "save.write_vtu([(\"cell_u\", cell_u), (\"cell_r\", cell_r)])" + ] + }, + { + "cell_type": "markdown", + "id": "developing-mobile", + "metadata": {}, + "source": [ + "A representation of the computed solution is given below.
\n", + "![](fig/cosserat.png)" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.12.4" + }, + "vscode": { + "interpreter": { + "hash": "e4cc1db98167c7fd7d55a1da8057731abc6cd6fe154328a2ae319df8aab4e24d" + } + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/tutorials/elasticity_mixed.ipynb b/tutorials/elasticity_mixed.ipynb new file mode 100644 index 00000000..40ec2f36 --- /dev/null +++ b/tutorials/elasticity_mixed.ipynb @@ -0,0 +1,294 @@ +{ + "cells": [ + { + "attachments": {}, + "cell_type": "markdown", + "id": "essential-american", + "metadata": {}, + "source": [ + "# Elasticity equation in mixed form\n", + "\n", + "In this tutorial we present how to solve the elasticity equation with [PyGeoN](https://github.com/compgeo-mox/pygeon) in mixed form. The unknowns are the stress $\\sigma$, the displacement $u$, and the rotation $r$.\n", + "\n", + "Let $\\Omega=(0,1)^2$ with boundary $\\partial \\Omega$ and outward unit normal ${\\nu}$. Given \n", + "$\\lambda$ Lamé constant and $\\mu$ the Kirchhoff modulus, we want to solve the following problem: find $(\\sigma, u, r)$ such that\n", + "\\begin{align*}\n", + "&A \\sigma - \\epsilon(u) - {\\rm asym}^* r = 0\\\\\n", + "&\\nabla \\cdot \\sigma = - b\\\\\n", + "&{\\rm asym}\\, \\sigma = 0\n", + "\\end{align*}\n", + "with $\\epsilon$ the symmetric gradient and $b$ a body force, which is set to $0$ in this example.\n", + "The operator $A$ is defined as\n", + "$$\n", + "A \\sigma = \\frac{1}{2\\mu}\\left( \\sigma -\\frac{\\lambda}{2\\mu + d \\lambda} {\\rm Tr}(\\sigma) I\\right)\n", + "$$\n", + "with $d$ the dimension of the domain, here $d=2$.\n", + "Finally, ${\\rm asym}$ is the asymmetric operator, which is defined as followed\n", + "$$\n", + " {\\rm asym }\\, \\sigma = \\begin{bmatrix}\n", + " \\sigma_{32} - \\sigma_{23} \\\\\n", + " \\sigma_{13} - \\sigma_{31} \\\\\n", + " \\sigma_{21} - \\sigma_{12}\n", + " \\end{bmatrix} \n", + " \\qquad\n", + " {\\rm asym}^* r =\n", + " \\begin{bmatrix}\n", + " 0 & -r_3 & r_2 \\\\\n", + " r_3 & 0 & -r_1 \\\\\n", + " -r_2 & r_1 & 0\n", + " \\end{bmatrix}\n", + " \\qquad\n", + " d = 3\\\\\n", + " {\\rm asym} \\, \\sigma = \\sigma_{21} - \\sigma_{12} \\qquad\n", + " {\\rm asym}^* r =\n", + " \\begin{bmatrix}\n", + " 0 & -r \\\\\n", + " r & 0\n", + " \\end{bmatrix} \\qquad\n", + " d = 2.\n", + "$$\n", + "For this test case we set the following boundary conditions related to the so-called footstep problem:\n", + "$$ u = 0 \\text{ on } \\partial_{bottom} \\Omega \\qquad \\nu \\cdot \\sigma = [0, 0]^\\top \\text{ on } \\partial_{left} \\Omega \\cup \\partial_{right} \\Omega \\qquad \\nu \\cdot \\sigma = [0, -1]^\\top \\text{ on } \\partial_{top} \\Omega$$\n", + "\n", + "We present *step-by-step* how to create the grid, declare the problem data, and finally solve the problem." + ] + }, + { + "cell_type": "markdown", + "id": "planned-danger", + "metadata": {}, + "source": [ + "First we import some of the standard modules, like `numpy` and `scipy.sparse`. Since PyGeoN is based on [PorePy](https://github.com/pmgbergen/porepy) we import both modules." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "dietary-perth", + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import scipy.sparse as sps\n", + "\n", + "import porepy as pp\n", + "import pygeon as pg" + ] + }, + { + "cell_type": "markdown", + "id": "roman-glossary", + "metadata": {}, + "source": [ + "We create now the grid, since we use vector BDM1 for ${\\sigma}$, vector piece-wise constant for $u$ and piece-wise constant for $r$, we are restricted to simplices. In this example we consider a 2-dimensional structured grid, but the presented code will work also in 3d with the only attention that in 3d the rotations should be discretized with vector piece-wise constant finite elements." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "spectacular-saturn", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\n" + ] + } + ], + "source": [ + "mesh_size = 0.05\n", + "dim = 2\n", + "\n", + "sd = pg.unit_grid(dim, mesh_size, as_mdg=False)\n", + "sd.compute_geometry()" + ] + }, + { + "cell_type": "markdown", + "id": "d625fca2", + "metadata": {}, + "source": [ + "We define now the finite element spaces for all the variables." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "52400a88", + "metadata": {}, + "outputs": [], + "source": [ + "key = \"elasticity\"\n", + "\n", + "# finite element spaces\n", + "vec_bdm1 = pg.VecBDM1(key)\n", + "vec_p0 = pg.VecPwConstants(key)\n", + "p0 = pg.PwConstants(key)\n", + "\n", + "# build the degrees of freedom\n", + "dofs = np.array([vec_bdm1.ndof(sd), vec_p0.ndof(sd), p0.ndof(sd)])" + ] + }, + { + "cell_type": "markdown", + "id": "precious-belle", + "metadata": {}, + "source": [ + "With the following code we set the data, in particular the Lamé and the Kirchhoff modulus, and the boundary conditions. Since we need to identify each side of $\\partial \\Omega$ we need few steps." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "spare-person", + "metadata": {}, + "outputs": [], + "source": [ + "data = {pp.PARAMETERS: {key: {\"lambda\": 1, \"mu\": 0.5}}}\n", + "\n", + "# select the portions of the boundary where we want to apply the boundary conditions\n", + "bottom = np.isclose(sd.face_centers[1, :], 0)\n", + "top = np.isclose(sd.face_centers[1, :], 1)\n", + "left = np.isclose(sd.face_centers[0, :], 0)\n", + "right = np.isclose(sd.face_centers[0, :], 1)\n", + "\n", + "# select the part of the boundary where we apply the essential boundary conditions\n", + "b_faces = np.logical_or.reduce((top, left, right))\n", + "ess_dof = np.tile(b_faces, sd.dim**2)\n", + "\n", + "# function for the essential boundary conditions\n", + "val = np.array([[0, 0, 0], [0, -1, 0]])\n", + "fct = lambda pt: val if np.isclose(pt[1], 1) else 0 * val\n", + "\n", + "# interpolate the essential boundary conditions\n", + "ess_val = vec_bdm1.interpolate(sd, fct)" + ] + }, + { + "attachments": {}, + "cell_type": "markdown", + "id": "secure-flesh", + "metadata": {}, + "source": [ + "Once the data are assigned to the grid, we construct the matrices. Once the latter is created, we also construct the right-hand side containing the boundary conditions." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "romance-findings", + "metadata": {}, + "outputs": [], + "source": [ + "# assemble mass matrices\n", + "Ms = vec_bdm1.assemble_mass_matrix(sd, data)\n", + "Mu = vec_p0.assemble_mass_matrix(sd)\n", + "Mr = p0.assemble_mass_matrix(sd)\n", + "\n", + "# assemble matrices associated with the divergence and asymmetry operators\n", + "div = Mu @ vec_bdm1.assemble_diff_matrix(sd)\n", + "asym = Mr @ vec_bdm1.assemble_asym_matrix(sd)\n", + "\n", + "# assemble the saddle point problem matrix\n", + "# fmt: off\n", + "spp = sps.bmat([[ Ms, div.T, -asym.T],\n", + " [-div, None, None],\n", + " [asym, None, None]], format = \"csc\")\n", + "# fmt: on" + ] + }, + { + "cell_type": "markdown", + "id": "mobile-nirvana", + "metadata": {}, + "source": [ + "We need to solve the linear system, PyGeoN provides a framework for that. The actual imposition of essential boundary conditions (stress boundary conditions) might change the symmetry of the global system, the class `pg.LinearSystem` preserves this structure by internally eliminating these degrees of freedom." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "subtle-wonder", + "metadata": {}, + "outputs": [], + "source": [ + "# solve the linear system\n", + "ls = pg.LinearSystem(spp, np.zeros(spp.shape[0]))\n", + "ls.flag_ess_bc(np.where(ess_dof)[0], ess_val)\n", + "x = ls.solve()\n", + "\n", + "# split the solution into the components\n", + "idx = np.cumsum(dofs[:-1])\n", + "sigma, u, r = np.split(x, idx)" + ] + }, + { + "cell_type": "markdown", + "id": "pacific-alpha", + "metadata": {}, + "source": [ + "We finally export the solution to be visualized by [ParaView](https://www.paraview.org/)." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "satisfactory-jerusalem", + "metadata": {}, + "outputs": [], + "source": [ + "# post process of the displacement\n", + "proj_u = vec_p0.eval_at_cell_centers(sd)\n", + "cell_u = (proj_u @ u).reshape((sd.dim, -1), order=\"C\")\n", + "# since we are in 2d we need to add the z component for the exporting\n", + "cell_u = np.vstack((cell_u, np.zeros(cell_u.shape[1])))\n", + "\n", + "# post process of the rotation\n", + "proj_r = p0.eval_at_cell_centers(sd)\n", + "cell_r = proj_r @ r\n", + "\n", + "save = pp.Exporter(sd, \"sol\")\n", + "save.write_vtu([(\"cell_u\", cell_u), (\"cell_r\", cell_r)])" + ] + }, + { + "cell_type": "markdown", + "id": "developing-mobile", + "metadata": {}, + "source": [ + "A representation of the computed solution is given below.
\n", + "![](fig/elasticity_mixed.png)\n", + "
\n", + "![](fig/elasticity_mixed1.png)" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.1" + }, + "vscode": { + "interpreter": { + "hash": "e4cc1db98167c7fd7d55a1da8057731abc6cd6fe154328a2ae319df8aab4e24d" + } + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/tutorials/fig/cosserat.png b/tutorials/fig/cosserat.png new file mode 100644 index 0000000000000000000000000000000000000000..1d764e3a524f18fc202d8552de6be18508b487f5 GIT binary patch literal 516204 zcmeFZbx@o^vp2f9F78WUv0wof-G$)p?!H)pE$;44upkM+Eocakz~W945)vc?*Wd)V zV0V*O&U@TM`c-`HCh?gw`FbaQZYu?KqxxZ8v6{T&?u0RLH6?KxmSnDFLFV(Kyu 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