Line data Source code
1 0 : // Distributed under the MIT License.
2 : // See LICENSE.txt for details.
3 :
4 : #pragma once
5 :
6 : #include <cstddef>
7 : #include <string>
8 : #include <tuple>
9 : #include <unordered_map>
10 : #include <utility>
11 :
12 : #include "DataStructures/DataBox/Tag.hpp"
13 : #include "DataStructures/DataVector.hpp"
14 : #include "DataStructures/SliceVariables.hpp"
15 : #include "DataStructures/Tensor/Tensor.hpp"
16 : #include "DataStructures/Variables.hpp"
17 : #include "DataStructures/Variables/FrameTransform.hpp"
18 : #include "Domain/Structure/Direction.hpp"
19 : #include "Domain/Structure/DirectionMap.hpp"
20 : #include "Domain/Structure/Element.hpp"
21 : #include "Domain/Structure/ElementId.hpp"
22 : #include "Domain/Structure/IndexToSliceAt.hpp"
23 : #include "Elliptic/Protocols/FirstOrderSystem.hpp"
24 : #include "Elliptic/Systems/GetSourcesComputer.hpp"
25 : #include "NumericalAlgorithms/DiscontinuousGalerkin/ApplyMassMatrix.hpp"
26 : #include "NumericalAlgorithms/DiscontinuousGalerkin/Formulation.hpp"
27 : #include "NumericalAlgorithms/DiscontinuousGalerkin/LiftFlux.hpp"
28 : #include "NumericalAlgorithms/DiscontinuousGalerkin/LiftFromBoundary.hpp"
29 : #include "NumericalAlgorithms/DiscontinuousGalerkin/MortarHelpers.hpp"
30 : #include "NumericalAlgorithms/DiscontinuousGalerkin/NormalDotFlux.hpp"
31 : #include "NumericalAlgorithms/DiscontinuousGalerkin/ProjectToBoundary.hpp"
32 : #include "NumericalAlgorithms/DiscontinuousGalerkin/SimpleBoundaryData.hpp"
33 : #include "NumericalAlgorithms/DiscontinuousGalerkin/SimpleMortarData.hpp"
34 : #include "NumericalAlgorithms/LinearOperators/Divergence.hpp"
35 : #include "NumericalAlgorithms/LinearOperators/Divergence.tpp"
36 : #include "NumericalAlgorithms/LinearOperators/WeakDivergence.hpp"
37 : #include "NumericalAlgorithms/Spectral/Basis.hpp"
38 : #include "NumericalAlgorithms/Spectral/Mesh.hpp"
39 : #include "NumericalAlgorithms/Spectral/Projection.hpp"
40 : #include "NumericalAlgorithms/Spectral/Quadrature.hpp"
41 : #include "Utilities/ErrorHandling/Assert.hpp"
42 : #include "Utilities/Gsl.hpp"
43 : #include "Utilities/ProtocolHelpers.hpp"
44 : #include "Utilities/TMPL.hpp"
45 : #include "Utilities/TaggedTuple.hpp"
46 :
47 : /*!
48 : * \brief Functionality related to discontinuous Galerkin discretizations of
49 : * elliptic equations
50 : *
51 : * The following is a brief overview of the elliptic DG schemes that are
52 : * implemented here. The scheme is described in detail in \cite Fischer2021voj.
53 : *
54 : * The DG schemes apply to any elliptic PDE that can be formulated in
55 : * first-order flux-form, as detailed by
56 : * `elliptic::protocols::FirstOrderSystem`.
57 : * The DG discretization of equations in this first-order form amounts to
58 : * projecting the equations on the set of basis functions that we also use to
59 : * represent the fields on the computational grid. The currently implemented DG
60 : * operator uses Lagrange interpolating polynomials w.r.t. Legendre-Gauss or
61 : * Legendre-Gauss-Lobatto collocation points as basis functions. Skipping all
62 : * further details here, the discretization results in a linear equation
63 : * \f$A(u)=b\f$ over all grid points and primal variables. Solving the elliptic
64 : * equations amounts to numerically inverting the DG operator \f$A\f$, typically
65 : * without ever constructing the full matrix but by employing an iterative
66 : * linear solver that repeatedly applies the DG operator to "test data". Note
67 : * that the DG operator applies directly to the primal variables. Auxiliary
68 : * variables are only computed temporarily and don't inflate the size of the
69 : * operator. This means the DG operator essentially computes second derivatives
70 : * of the primal variables, modified by the fluxes and sources of the system
71 : * as well as by DG boundary corrections that couple grid points across element
72 : * boundaries.
73 : *
74 : * \par Boundary corrections:
75 : * In this implementation we employ the "internal penalty" DG scheme that
76 : * couples grid points across nearest-neighbor elements through the fluxes:
77 : *
78 : * \f{align}
79 : * \label{eq:internal_penalty_auxiliary}
80 : * u^* &= \frac{1}{2} \left(u^\mathrm{int} + u^\mathrm{ext}\right) \\
81 : * \label{eq:internal_penalty_primal}
82 : * (n_i F^i)^* &= \frac{1}{2} n_i \left(
83 : * F^i_\mathrm{int} + F^i_\mathrm{ext} \right)
84 : * - \sigma n_i F^i(n_j (u^\mathrm{int} - u^\mathrm{ext}))
85 : * \f}
86 : *
87 : * Note that \f$n_i\f$ denotes the face normal on the "interior" side of the
88 : * element under consideration. We assume \f$n^\mathrm{ext}_i=-n_i\f$ in the
89 : * implementation, i.e. face normals don't depend on the dynamic variables
90 : * (which may be discontinuous on element faces). This is the case for the
91 : * problems we are expecting to solve, because those will be on fixed background
92 : * metrics (e.g. a conformal metric for the XCTS system). Numerically, the face
93 : * normals on either side of a mortar may nonetheless be different because the
94 : * two faces adjacent to the mortar may resolve them at different resolutions.
95 : *
96 : * Also note that the numerical fluxes intentionally don't depend on the
97 : * auxiliary field values \f$v\f$. This property allows us to communicate data
98 : * for both the primal and auxiliary boundary corrections together, instead of
99 : * communicating them in two steps. If we were to resort to a two-step
100 : * communication we could replace the derivatives in \f$(n_i F^i)^*\f$ with
101 : * \f$v\f$, which would result in a generalized "stabilized central flux" that
102 : * is slightly less sparse than the internal penalty flux (see e.g.
103 : * \cite HesthavenWarburton, section 7.2). We could also choose to ignore the
104 : * fluxes in the penalty term, but preliminary tests suggest that this may hurt
105 : * convergence.
106 : *
107 : * For a Poisson system (see `Poisson::FirstOrderSystem`) this numerical flux
108 : * reduces to the standard internal penalty flux (see e.g.
109 : * \cite HesthavenWarburton, section 7.2, or \cite Arnold2002):
110 : *
111 : * \f{align}
112 : * u^* &= \frac{1}{2} \left(u^\mathrm{int} + u^\mathrm{ext}\right) \\
113 : * (n_i F^i)^* &= n_i v_i^* = \frac{1}{2} n_i \left(
114 : * \partial_i u^\mathrm{int} + \partial_i u^\mathrm{ext}\right)
115 : * - \sigma \left(u^\mathrm{int} - u^\mathrm{ext}\right)
116 : * \f}
117 : *
118 : * where a sum over repeated indices is assumed, since the equation is
119 : * formulated on a Euclidean geometry.
120 : *
121 : * The penalty factor \f$\sigma\f$ is responsible for removing zero eigenmodes
122 : * and impacts the conditioning of the linear operator to be solved. See
123 : * `elliptic::dg::penalty` for details. For the element size that goes into
124 : * computing the penalty we choose
125 : * \f$h=\frac{J_\mathrm{volume}}{J_\mathrm{face}}\f$, i.e. the ratio of Jacobi
126 : * determinants from logical to inertial coordinates in the element volume and
127 : * on the element face, both evaluated on the face (see \cite Vincent2019qpd).
128 : * Since both \f$N_\mathrm{points}\f$ and \f$h\f$ can be different on either
129 : * side of the element boundary we take the maximum of \f$N_\mathrm{points}\f$
130 : * and the pointwise minimum of \f$h\f$ across the element boundary as is done
131 : * in \cite Vincent2019qpd. Note that we use the number of points
132 : * \f$N_\mathrm{points}\f$ where \cite Vincent2019qpd uses the polynomial degree
133 : * \f$N_\mathrm{points} - 1\f$ because we found unstable configurations on
134 : * curved meshes when using the polynomial degree. Optimizing the penalty on
135 : * curved meshes is subject to further investigation.
136 : *
137 : * \par Discontinuous fluxes:
138 : * The DG operator also supports systems with potentially discontinuous fluxes,
139 : * such as elasticity with layered materials. The way to handle the
140 : * discontinuous fluxes in the DG scheme is described in \cite Vu2023thn.
141 : * Essentially, we evaluate the penalty term in
142 : * Eq. $\ref{eq:internal_penalty_primal}$ on both sides of an element boundary
143 : * and take the average. The other terms in the numerical flux remain unchanged.
144 : */
145 : namespace elliptic::dg {
146 :
147 : /// Data that is projected to mortars and communicated across element
148 : /// boundaries
149 : template <typename PrimalFields, typename PrimalFluxes>
150 1 : using BoundaryData = ::dg::SimpleBoundaryData<
151 : tmpl::append<PrimalFields,
152 : db::wrap_tags_in<::Tags::NormalDotFlux, PrimalFields>>,
153 : tmpl::list<>>;
154 :
155 : /// Boundary data on both sides of a mortar.
156 : ///
157 : /// \note This is a struct (as opposed to a type alias) so it can be used to
158 : /// deduce the template parameters
159 : template <typename TemporalId, typename PrimalFields, typename PrimalFluxes>
160 1 : struct MortarData
161 : : ::dg::SimpleMortarData<TemporalId,
162 : BoundaryData<PrimalFields, PrimalFluxes>,
163 : BoundaryData<PrimalFields, PrimalFluxes>> {};
164 :
165 1 : namespace Tags {
166 : /// Holds `elliptic::dg::MortarData`, i.e. boundary data on both sides of a
167 : /// mortar
168 : template <typename TemporalId, typename PrimalFields, typename PrimalFluxes>
169 1 : struct MortarData : db::SimpleTag {
170 0 : using type = elliptic::dg::MortarData<TemporalId, PrimalFields, PrimalFluxes>;
171 : };
172 : } // namespace Tags
173 :
174 : namespace detail {
175 :
176 : // Mass-conservative restriction: R = M^{-1}_face P^T M_mortar
177 : //
178 : // Note that projecting the mortar data times the Jacobian using
179 : // `Spectral::projection_matrix_child_to_parent(operand_is_massive=false)` is
180 : // equivalent to this implementation on Gauss grids. However, on Gauss-Lobatto
181 : // grids we usually diagonally approximate the mass matrix ("mass lumping") but
182 : // `projection_matrix_child_to_parent(operand_is_massive=false)` uses the full
183 : // mass matrix. Therefore, the two implementations differ slightly on
184 : // Gauss-Lobatto grids. Furthermore, note that
185 : // `projection_matrix_child_to_parent(operand_is_massive=false)` already
186 : // includes the factors of two that account for the mortar size, so they must be
187 : // omitted from the mortar Jacobian when using that approach.
188 : template <typename TagsList, size_t FaceDim>
189 : Variables<TagsList> mass_conservative_restriction(
190 : Variables<TagsList> mortar_vars,
191 : [[maybe_unused]] const Mesh<FaceDim>& mortar_mesh,
192 : [[maybe_unused]] const ::dg::MortarSize<FaceDim>& mortar_size,
193 : [[maybe_unused]] const Scalar<DataVector>& mortar_jacobian,
194 : [[maybe_unused]] const Mesh<FaceDim> face_mesh,
195 : [[maybe_unused]] const Scalar<DataVector>& face_jacobian) {
196 : if constexpr (FaceDim == 0) {
197 : return mortar_vars;
198 : } else {
199 : const auto projection_matrices =
200 : Spectral::projection_matrix_child_to_parent(mortar_mesh, face_mesh,
201 : mortar_size, true);
202 : mortar_vars *= get(mortar_jacobian);
203 : ::dg::apply_mass_matrix(make_not_null(&mortar_vars), mortar_mesh);
204 : auto face_vars =
205 : apply_matrices(projection_matrices, mortar_vars, mortar_mesh.extents());
206 : face_vars /= get(face_jacobian);
207 : ::dg::apply_inverse_mass_matrix(make_not_null(&face_vars), face_mesh);
208 : return face_vars;
209 : }
210 : }
211 :
212 : template <typename System, bool Linearized,
213 : typename PrimalFields = typename System::primal_fields,
214 : typename PrimalFluxes = typename System::primal_fluxes>
215 : struct DgOperatorImpl;
216 :
217 : template <typename System, bool Linearized, typename... PrimalFields,
218 : typename... PrimalFluxes>
219 : struct DgOperatorImpl<System, Linearized, tmpl::list<PrimalFields...>,
220 : tmpl::list<PrimalFluxes...>> {
221 : static_assert(
222 : tt::assert_conforms_to_v<System, elliptic::protocols::FirstOrderSystem>);
223 :
224 : static constexpr size_t Dim = System::volume_dim;
225 : using FluxesComputer = typename System::fluxes_computer;
226 : using SourcesComputer = elliptic::get_sources_computer<System, Linearized>;
227 :
228 : struct AllDirections {
229 : bool operator()(const Direction<Dim>& /*unused*/) const { return true; }
230 : };
231 :
232 : struct NoDataIsZero {
233 : bool operator()(const ElementId<Dim>& /*unused*/) const { return false; }
234 : };
235 :
236 : static constexpr auto full_mortar_size =
237 : make_array<Dim - 1>(Spectral::MortarSize::Full);
238 :
239 : template <bool AllDataIsZero, typename... DerivTags, typename... PrimalVars,
240 : typename... PrimalFluxesVars, typename... PrimalMortarVars,
241 : typename... PrimalMortarFluxes, typename TemporalId,
242 : typename ApplyBoundaryCondition, typename... FluxesArgs,
243 : typename... SourcesArgs, typename DataIsZero = NoDataIsZero,
244 : typename DirectionsPredicate = AllDirections>
245 : static void prepare_mortar_data(
246 : const gsl::not_null<Variables<tmpl::list<DerivTags...>>*> deriv_vars,
247 : const gsl::not_null<Variables<tmpl::list<PrimalFluxesVars...>>*>
248 : primal_fluxes,
249 : const gsl::not_null<::dg::MortarMap<
250 : Dim, MortarData<TemporalId, tmpl::list<PrimalMortarVars...>,
251 : tmpl::list<PrimalMortarFluxes...>>>*>
252 : all_mortar_data,
253 : const Variables<tmpl::list<PrimalVars...>>& primal_vars,
254 : const Element<Dim>& element, const Mesh<Dim>& mesh,
255 : const InverseJacobian<DataVector, Dim, Frame::ElementLogical,
256 : Frame::Inertial>& inv_jacobian,
257 : const DirectionMap<Dim, tnsr::i<DataVector, Dim>>& face_normals,
258 : const ::dg::MortarMap<Dim, Mesh<Dim - 1>>& all_mortar_meshes,
259 : const ::dg::MortarMap<Dim, ::dg::MortarSize<Dim - 1>>& all_mortar_sizes,
260 : const TemporalId& temporal_id,
261 : const ApplyBoundaryCondition& apply_boundary_condition,
262 : const std::tuple<FluxesArgs...>& fluxes_args,
263 : const DataIsZero& data_is_zero = NoDataIsZero{},
264 : const DirectionsPredicate& directions_predicate = AllDirections{}) {
265 : static_assert(
266 : sizeof...(PrimalVars) == sizeof...(PrimalFields) and
267 : sizeof...(PrimalFluxesVars) == sizeof...(PrimalFluxes),
268 : "The number of variables must match the number of system fields.");
269 : static_assert(
270 : (std::is_same_v<typename PrimalVars::type,
271 : typename PrimalFields::type> and
272 : ...) and
273 : (std::is_same_v<typename PrimalFluxesVars::type,
274 : typename PrimalFluxes::type> and
275 : ...),
276 : "The variables must have the same tensor types as the system fields.");
277 : #ifdef SPECTRE_DEBUG
278 : for (size_t d = 0; d < Dim; ++d) {
279 : ASSERT(mesh.basis(d) == Spectral::Basis::Legendre and
280 : (mesh.quadrature(d) == Spectral::Quadrature::GaussLobatto or
281 : mesh.quadrature(d) == Spectral::Quadrature::Gauss),
282 : "The elliptic DG operator is currently only implemented for "
283 : "Legendre-Gauss(-Lobatto) grids. Found basis '"
284 : << mesh.basis(d) << "' and quadrature '" << mesh.quadrature(d)
285 : << "' in dimension " << d << ".");
286 : }
287 : #endif // SPECTRE_DEBUG
288 : const auto& element_id = element.id();
289 : const bool local_data_is_zero = data_is_zero(element_id);
290 : ASSERT(Linearized or not local_data_is_zero,
291 : "Only a linear operator can take advantage of the knowledge that "
292 : "the operand is zero. Don't return 'true' in 'data_is_zero' unless "
293 : "you also set 'Linearized' to 'true'.");
294 : const size_t num_points = mesh.number_of_grid_points();
295 :
296 : // This function and the one below allocate various Variables to compute
297 : // intermediate quantities. It could be a performance optimization to reduce
298 : // the number of these allocations and/or move some of the memory buffers
299 : // into the DataBox to keep them around permanently. The latter should be
300 : // informed by profiling.
301 :
302 : // Compute partial derivatives grad(u) of the system variables, and from
303 : // those the fluxes F^i(grad(u)) in the volume. We will take the divergence
304 : // of the fluxes in the `apply_operator` function below to compute the full
305 : // elliptic equation -div(F) + S = f(x).
306 : if (AllDataIsZero or local_data_is_zero) {
307 : primal_fluxes->initialize(num_points, 0.);
308 : } else {
309 : // Compute partial derivatives of the variables
310 : partial_derivatives(deriv_vars, primal_vars, mesh, inv_jacobian);
311 : // Compute the fluxes
312 : primal_fluxes->initialize(num_points);
313 : std::apply(
314 : [&primal_fluxes, &primal_vars, &deriv_vars,
315 : &element_id](const auto&... expanded_fluxes_args) {
316 : if constexpr (FluxesComputer::is_discontinuous) {
317 : FluxesComputer::apply(
318 : make_not_null(&get<PrimalFluxesVars>(*primal_fluxes))...,
319 : expanded_fluxes_args..., element_id,
320 : get<PrimalVars>(primal_vars)...,
321 : get<DerivTags>(*deriv_vars)...);
322 : } else {
323 : (void)element_id;
324 : FluxesComputer::apply(
325 : make_not_null(&get<PrimalFluxesVars>(*primal_fluxes))...,
326 : expanded_fluxes_args..., get<PrimalVars>(primal_vars)...,
327 : get<DerivTags>(*deriv_vars)...);
328 : }
329 : },
330 : fluxes_args);
331 : }
332 :
333 : // Populate the mortar data on this element's side of the boundary so it's
334 : // ready to be sent to neighbors.
335 : for (const auto& direction : [&element]() -> const auto& {
336 : if constexpr (AllDataIsZero) {
337 : // Skipping internal boundaries for all-zero data because they
338 : // won't contribute boundary corrections anyway (data on both sides
339 : // of the boundary is the same). For all-zero data we are
340 : // interested in external boundaries, to extract inhomogeneous
341 : // boundary conditions from a non-linear operator.
342 : return element.external_boundaries();
343 : } else {
344 : (void)element;
345 : return Direction<Dim>::all_directions();
346 : };
347 : }()) {
348 : if (not directions_predicate(direction)) {
349 : continue;
350 : }
351 : const bool is_internal = element.neighbors().contains(direction);
352 : // Skip directions altogether when both this element and all neighbors in
353 : // the direction have zero data. These boundaries won't contribute
354 : // corrections, because the data is the same on both sides. External
355 : // boundaries also count as zero data here, because they are linearized
356 : // (see assert above).
357 : if (local_data_is_zero and
358 : (not is_internal or
359 : alg::all_of(element.neighbors().at(direction), data_is_zero))) {
360 : continue;
361 : }
362 : const auto face_mesh = mesh.slice_away(direction.dimension());
363 : const size_t face_num_points = face_mesh.number_of_grid_points();
364 : const auto& face_normal = face_normals.at(direction);
365 : Variables<tmpl::list<PrimalFluxesVars...>> primal_fluxes_on_face{};
366 : BoundaryData<tmpl::list<PrimalMortarVars...>,
367 : tmpl::list<PrimalMortarFluxes...>>
368 : boundary_data{};
369 : if (AllDataIsZero or local_data_is_zero) {
370 : if (is_internal) {
371 : // We manufacture zero boundary data directly on the mortars below.
372 : // Nothing to do here.
373 : } else {
374 : boundary_data.field_data.initialize(face_num_points, 0.);
375 : }
376 : } else {
377 : boundary_data.field_data.initialize(face_num_points);
378 : primal_fluxes_on_face.initialize(face_num_points);
379 : // Project fields to faces
380 : // Note: need to convert tags of `Variables` because
381 : // `project_contiguous_data_to_boundary` requires that the face and
382 : // volume tags are subsets.
383 : Variables<
384 : tmpl::list<PrimalVars..., ::Tags::NormalDotFlux<PrimalVars>...>>
385 : boundary_data_ref{};
386 : boundary_data_ref.set_data_ref(boundary_data.field_data.data(),
387 : boundary_data.field_data.size());
388 : ::dg::project_contiguous_data_to_boundary(
389 : make_not_null(&boundary_data_ref), primal_vars, mesh, direction);
390 : // Compute n_i F^i on faces
391 : ::dg::project_contiguous_data_to_boundary(
392 : make_not_null(&primal_fluxes_on_face), *primal_fluxes, mesh,
393 : direction);
394 : EXPAND_PACK_LEFT_TO_RIGHT(normal_dot_flux(
395 : make_not_null(&get<::Tags::NormalDotFlux<PrimalMortarVars>>(
396 : boundary_data.field_data)),
397 : face_normal, get<PrimalFluxesVars>(primal_fluxes_on_face)));
398 : }
399 :
400 : if (is_internal) {
401 : if constexpr (not AllDataIsZero) {
402 : // Project boundary data on internal faces to mortars
403 : for (const auto& neighbor_id : element.neighbors().at(direction)) {
404 : if (local_data_is_zero and data_is_zero(neighbor_id)) {
405 : continue;
406 : }
407 : const ::dg::MortarId<Dim> mortar_id{direction, neighbor_id};
408 : const auto& mortar_mesh = all_mortar_meshes.at(mortar_id);
409 : const auto& mortar_size = all_mortar_sizes.at(mortar_id);
410 : if (local_data_is_zero) {
411 : // No need to project anything. We just manufacture zero boundary
412 : // data on the mortar.
413 : BoundaryData<tmpl::list<PrimalMortarVars...>,
414 : tmpl::list<PrimalMortarFluxes...>>
415 : zero_boundary_data{};
416 : zero_boundary_data.field_data.initialize(
417 : mortar_mesh.number_of_grid_points(), 0.);
418 : (*all_mortar_data)[mortar_id].local_insert(
419 : temporal_id, std::move(zero_boundary_data));
420 : continue;
421 : }
422 : // When no projection is necessary we can safely move the boundary
423 : // data from the face as there is only a single neighbor in this
424 : // direction
425 : auto projected_boundary_data =
426 : Spectral::needs_projection(face_mesh, mortar_mesh, mortar_size)
427 : // NOLINTNEXTLINE
428 : ? boundary_data.project_to_mortar(face_mesh, mortar_mesh,
429 : mortar_size)
430 : : std::move(boundary_data); // NOLINT
431 : (*all_mortar_data)[mortar_id].local_insert(
432 : temporal_id, std::move(projected_boundary_data));
433 : }
434 : }
435 : } else {
436 : // No need to do projections on external boundaries
437 : const ::dg::MortarId<Dim> mortar_id{
438 : direction, ElementId<Dim>::external_boundary_id()};
439 : (*all_mortar_data)[mortar_id].local_insert(temporal_id, boundary_data);
440 :
441 : // -------------------------
442 : // Apply boundary conditions
443 : // -------------------------
444 : //
445 : // To apply boundary conditions we fill the boundary data with
446 : // "exterior" or "ghost" data and set it as remote mortar data, so
447 : // external boundaries behave just like internal boundaries when
448 : // applying boundary corrections.
449 : //
450 : // The `apply_boundary_conditions` invocable is expected to impose
451 : // boundary conditions by modifying the fields and fluxes that are
452 : // passed by reference. Dirichlet-type boundary conditions are imposed
453 : // by modifying the fields, and Neumann-type boundary conditions are
454 : // imposed by modifying the interior n.F. Note that all data passed to
455 : // the boundary conditions is taken from the "interior" side of the
456 : // boundary, i.e. with a normal vector that points _out_ of the
457 : // computational domain.
458 : Variables<tmpl::list<DerivTags...>> deriv_vars_on_boundary{};
459 : if (AllDataIsZero or local_data_is_zero) {
460 : deriv_vars_on_boundary.initialize(face_num_points, 0.);
461 : } else {
462 : deriv_vars_on_boundary.initialize(face_num_points);
463 : ::dg::project_contiguous_data_to_boundary(
464 : make_not_null(&deriv_vars_on_boundary), *deriv_vars, mesh,
465 : direction);
466 : }
467 : apply_boundary_condition(
468 : direction,
469 : make_not_null(&get<PrimalMortarVars>(boundary_data.field_data))...,
470 : make_not_null(&get<::Tags::NormalDotFlux<PrimalMortarVars>>(
471 : boundary_data.field_data))...,
472 : get<DerivTags>(deriv_vars_on_boundary)...);
473 :
474 : // Invert the sign of the fluxes to account for the inverted normal on
475 : // exterior faces. Also multiply by 2 and add the interior fluxes to
476 : // impose the boundary conditions on the _average_ instead of just
477 : // setting the fields on the exterior:
478 : // (Dirichlet) u_D = avg(u) = 1/2 (u_int + u_ext)
479 : // => u_ext = 2 u_D - u_int
480 : // (Neumann) (n.F)_N = avg(n.F) = 1/2 [(n.F)_int - (n.F)_ext]
481 : // => (n.F)_ext = -2 (n.F)_N + (n.F)_int]
482 : const auto impose_on_average = [](const auto exterior_field,
483 : const auto& interior_field) {
484 : for (size_t i = 0; i < interior_field.size(); ++i) {
485 : (*exterior_field)[i] *= 2.;
486 : (*exterior_field)[i] -= interior_field[i];
487 : }
488 : };
489 : EXPAND_PACK_LEFT_TO_RIGHT(impose_on_average(
490 : make_not_null(&get<PrimalMortarVars>(boundary_data.field_data)),
491 : get<PrimalMortarVars>(all_mortar_data->at(mortar_id)
492 : .local_data(temporal_id)
493 : .field_data)));
494 : const auto invert_sign_and_impose_on_average =
495 : [](const auto exterior_n_dot_flux,
496 : const auto& interior_n_dot_flux) {
497 : for (size_t i = 0; i < interior_n_dot_flux.size(); ++i) {
498 : (*exterior_n_dot_flux)[i] *= -2.;
499 : (*exterior_n_dot_flux)[i] += interior_n_dot_flux[i];
500 : }
501 : };
502 : EXPAND_PACK_LEFT_TO_RIGHT(invert_sign_and_impose_on_average(
503 : make_not_null(&get<::Tags::NormalDotFlux<PrimalMortarVars>>(
504 : boundary_data.field_data)),
505 : get<::Tags::NormalDotFlux<PrimalMortarVars>>(
506 : all_mortar_data->at(mortar_id)
507 : .local_data(temporal_id)
508 : .field_data)));
509 :
510 : // Store the exterior boundary data on the mortar
511 : all_mortar_data->at(mortar_id).remote_insert(temporal_id,
512 : std::move(boundary_data));
513 : } // if (is_internal)
514 : } // loop directions
515 : }
516 :
517 : // --- This is essentially a break to communicate the mortar data ---
518 :
519 : template <bool AllDataIsZero, typename... OperatorTags,
520 : typename... PrimalVars, typename... PrimalFluxesVars,
521 : typename... PrimalMortarVars, typename... PrimalMortarFluxes,
522 : typename TemporalId, typename... FluxesArgs,
523 : typename... SourcesArgs, typename DataIsZero = NoDataIsZero,
524 : typename DirectionsPredicate = AllDirections>
525 : static void apply_operator(
526 : const gsl::not_null<Variables<tmpl::list<OperatorTags...>>*>
527 : operator_applied_to_vars,
528 : const gsl::not_null<::dg::MortarMap<
529 : Dim, MortarData<TemporalId, tmpl::list<PrimalMortarVars...>,
530 : tmpl::list<PrimalMortarFluxes...>>>*>
531 : all_mortar_data,
532 : const Variables<tmpl::list<PrimalVars...>>& primal_vars,
533 : // Taking the primal fluxes computed in the `prepare_mortar_data` function
534 : // by const-ref here because other code might use them and so we don't
535 : // want to modify them by adding boundary corrections. E.g. linearized
536 : // sources use the nonlinear fields and fluxes as background fields.
537 : const Variables<tmpl::list<PrimalFluxesVars...>>& primal_fluxes,
538 : const Element<Dim>& element, const Mesh<Dim>& mesh,
539 : const InverseJacobian<DataVector, Dim, Frame::ElementLogical,
540 : Frame::Inertial>& inv_jacobian,
541 : const Scalar<DataVector>& det_inv_jacobian,
542 : const Scalar<DataVector>& det_jacobian,
543 : const InverseJacobian<DataVector, Dim, Frame::ElementLogical,
544 : Frame::Inertial>& det_times_inv_jacobian,
545 : const DirectionMap<Dim, tnsr::i<DataVector, Dim>>& face_normals,
546 : const DirectionMap<Dim, tnsr::I<DataVector, Dim>>& face_normal_vectors,
547 : const DirectionMap<Dim, Scalar<DataVector>>& face_normal_magnitudes,
548 : const DirectionMap<Dim, Scalar<DataVector>>& face_jacobians,
549 : const DirectionMap<Dim,
550 : InverseJacobian<DataVector, Dim, Frame::ElementLogical,
551 : Frame::Inertial>>&
552 : face_jacobian_times_inv_jacobians,
553 : const ::dg::MortarMap<Dim, Mesh<Dim - 1>>& all_mortar_meshes,
554 : const ::dg::MortarMap<Dim, ::dg::MortarSize<Dim - 1>>& all_mortar_sizes,
555 : const ::dg::MortarMap<Dim, Scalar<DataVector>>& mortar_jacobians,
556 : const ::dg::MortarMap<Dim, Scalar<DataVector>>& penalty_factors,
557 : const bool massive, const ::dg::Formulation formulation,
558 : const TemporalId& /*temporal_id*/,
559 : const DirectionMap<Dim, std::tuple<FluxesArgs...>>& fluxes_args_on_faces,
560 : const std::tuple<SourcesArgs...>& sources_args,
561 : const DataIsZero& data_is_zero = NoDataIsZero{},
562 : const DirectionsPredicate& directions_predicate = AllDirections{}) {
563 : static_assert(
564 : sizeof...(PrimalVars) == sizeof...(PrimalFields) and
565 : sizeof...(PrimalFluxesVars) == sizeof...(PrimalFluxes) and
566 : sizeof...(PrimalMortarVars) == sizeof...(PrimalFields) and
567 : sizeof...(PrimalMortarFluxes) == sizeof...(PrimalFluxes) and
568 : sizeof...(OperatorTags) == sizeof...(PrimalFields),
569 : "The number of variables must match the number of system fields.");
570 : static_assert(
571 : (std::is_same_v<typename PrimalVars::type,
572 : typename PrimalFields::type> and
573 : ...) and
574 : (std::is_same_v<typename PrimalFluxesVars::type,
575 : typename PrimalFluxes::type> and
576 : ...) and
577 : (std::is_same_v<typename PrimalMortarVars::type,
578 : typename PrimalFields::type> and
579 : ...) and
580 : (std::is_same_v<typename PrimalMortarFluxes::type,
581 : typename PrimalFluxes::type> and
582 : ...) and
583 : (std::is_same_v<typename OperatorTags::type,
584 : typename PrimalFields::type> and
585 : ...),
586 : "The variables must have the same tensor types as the system fields.");
587 : #ifdef SPECTRE_DEBUG
588 : for (size_t d = 0; d < Dim; ++d) {
589 : ASSERT(mesh.basis(d) == Spectral::Basis::Legendre and
590 : (mesh.quadrature(d) == Spectral::Quadrature::GaussLobatto or
591 : mesh.quadrature(d) == Spectral::Quadrature::Gauss),
592 : "The elliptic DG operator is currently only implemented for "
593 : "Legendre-Gauss(-Lobatto) grids. Found basis '"
594 : << mesh.basis(d) << "' and quadrature '" << mesh.quadrature(d)
595 : << "' in dimension " << d << ".");
596 : }
597 : #endif // SPECTRE_DEBUG
598 : const auto& element_id = element.id();
599 : const bool local_data_is_zero = data_is_zero(element_id);
600 : ASSERT(Linearized or not local_data_is_zero,
601 : "Only a linear operator can take advantage of the knowledge that "
602 : "the operand is zero. Don't return 'true' in 'data_is_zero' unless "
603 : "you also set 'Linearized' to 'true'.");
604 : const size_t num_points = mesh.number_of_grid_points();
605 :
606 : // This function and the one above allocate various Variables to compute
607 : // intermediate quantities. It could be a performance optimization to reduce
608 : // the number of these allocations and/or move some of the memory buffers
609 : // into the DataBox to keep them around permanently. The latter should be
610 : // informed by profiling.
611 :
612 : // Compute volume terms: -div(F) + S
613 : if (local_data_is_zero) {
614 : operator_applied_to_vars->initialize(num_points, 0.);
615 : } else {
616 : // "Massive" operators retain the factors from the volume integral:
617 : // \int_volume div(F) \phi_p = w_p det(J)_p div(F)_p
618 : // Here, `w` are the quadrature weights (the diagonal logical mass matrix
619 : // with mass-lumping) and det(J) is the Jacobian determinant. The
620 : // quantities are evaluated at the grid point `p`.
621 : if (formulation == ::dg::Formulation::StrongInertial) {
622 : // Compute strong divergence:
623 : // div(F) = (J^\hat{i}_i)_p \sum_q (D_\hat{i})_pq (F^i)_q.
624 : divergence(operator_applied_to_vars, primal_fluxes, mesh,
625 : massive ? det_times_inv_jacobian : inv_jacobian);
626 : // This is the sign flip that makes the operator _minus_ the Laplacian
627 : // for a Poisson system
628 : *operator_applied_to_vars *= -1.;
629 : } else {
630 : // Compute weak divergence:
631 : // F^i \partial_i \phi = 1/w_p \sum_q
632 : // (D^T_\hat{i})_pq (w det(J) J^\hat{i}_i F^i)_q
633 : weak_divergence(operator_applied_to_vars, primal_fluxes, mesh,
634 : det_times_inv_jacobian);
635 : if (not massive) {
636 : *operator_applied_to_vars *= get(det_inv_jacobian);
637 : }
638 : }
639 : if constexpr (not std::is_same_v<SourcesComputer, void>) {
640 : Variables<tmpl::list<OperatorTags...>> sources{num_points, 0.};
641 : std::apply(
642 : [&sources, &primal_vars,
643 : &primal_fluxes](const auto&... expanded_sources_args) {
644 : SourcesComputer::apply(
645 : make_not_null(&get<OperatorTags>(sources))...,
646 : expanded_sources_args..., get<PrimalVars>(primal_vars)...,
647 : get<PrimalFluxesVars>(primal_fluxes)...);
648 : },
649 : sources_args);
650 : if (massive) {
651 : sources *= get(det_jacobian);
652 : }
653 : *operator_applied_to_vars += sources;
654 : }
655 : }
656 : if (massive) {
657 : ::dg::apply_mass_matrix(operator_applied_to_vars, mesh);
658 : }
659 :
660 : // Add boundary corrections
661 : // Keeping track if any corrections were applied here, for an optimization
662 : // below
663 : bool has_any_boundary_corrections = false;
664 : Variables<tmpl::list<transform::Tags::TransformedFirstIndex<
665 : PrimalFluxesVars, Frame::ElementLogical>...>>
666 : lifted_logical_aux_boundary_corrections{num_points, 0.};
667 : for (auto& [mortar_id, mortar_data] : *all_mortar_data) {
668 : const auto& direction = mortar_id.direction();
669 : const auto& neighbor_id = mortar_id.id();
670 : const bool is_internal =
671 : (neighbor_id != ElementId<Dim>::external_boundary_id());
672 : if constexpr (AllDataIsZero) {
673 : if (is_internal) {
674 : continue;
675 : }
676 : }
677 : if (not directions_predicate(direction)) {
678 : continue;
679 : }
680 : // When the data on both sides of the mortar is zero then we don't need to
681 : // handle this mortar at all.
682 : if (local_data_is_zero and
683 : (not is_internal or data_is_zero(neighbor_id))) {
684 : continue;
685 : }
686 : has_any_boundary_corrections = true;
687 :
688 : const auto face_mesh = mesh.slice_away(direction.dimension());
689 : auto [local_data, remote_data] = mortar_data.extract();
690 : const size_t face_num_points = face_mesh.number_of_grid_points();
691 : const auto& face_normal = face_normals.at(direction);
692 : const auto& face_normal_vector = face_normal_vectors.at(direction);
693 : const auto& fluxes_args_on_face = fluxes_args_on_faces.at(direction);
694 : const auto& face_normal_magnitude = face_normal_magnitudes.at(direction);
695 : const auto& face_jacobian = face_jacobians.at(direction);
696 : const auto& face_jacobian_times_inv_jacobian =
697 : face_jacobian_times_inv_jacobians.at(direction);
698 : const auto& mortar_mesh =
699 : is_internal ? all_mortar_meshes.at(mortar_id) : face_mesh;
700 : const auto& mortar_size =
701 : is_internal ? all_mortar_sizes.at(mortar_id) : full_mortar_size;
702 :
703 : // This is the strong auxiliary boundary correction:
704 : // G^i = F^i(n_j (avg(u) - u))
705 : // where
706 : // avg(u) - u = -0.5 * (u_int - u_ext)
707 : auto avg_vars_on_mortar = Variables<tmpl::list<PrimalMortarVars...>>(
708 : local_data.field_data
709 : .template extract_subset<tmpl::list<PrimalMortarVars...>>());
710 : const auto add_remote_contribution = [](auto& lhs, const auto& rhs) {
711 : for (size_t i = 0; i < lhs.size(); ++i) {
712 : lhs[i] -= rhs[i];
713 : }
714 : };
715 : EXPAND_PACK_LEFT_TO_RIGHT(add_remote_contribution(
716 : get<PrimalMortarVars>(avg_vars_on_mortar),
717 : get<PrimalMortarVars>(remote_data.field_data)));
718 : avg_vars_on_mortar *= -0.5;
719 :
720 : // Project from the mortar back down to the face if needed
721 : const auto avg_vars_on_face =
722 : Spectral::needs_projection(face_mesh, mortar_mesh, mortar_size)
723 : ? mass_conservative_restriction(
724 : std::move(avg_vars_on_mortar), mortar_mesh, mortar_size,
725 : mortar_jacobians.at(mortar_id), face_mesh, face_jacobian)
726 : : std::move(avg_vars_on_mortar);
727 :
728 : // Apply fluxes to get G^i
729 : Variables<tmpl::list<PrimalFluxesVars...>> auxiliary_boundary_corrections{
730 : face_num_points};
731 : std::apply(
732 : [&auxiliary_boundary_corrections, &face_normal, &face_normal_vector,
733 : &avg_vars_on_face,
734 : &element_id](const auto&... expanded_fluxes_args_on_face) {
735 : if constexpr (FluxesComputer::is_discontinuous) {
736 : FluxesComputer::apply(make_not_null(&get<PrimalFluxesVars>(
737 : auxiliary_boundary_corrections))...,
738 : expanded_fluxes_args_on_face..., element_id,
739 : face_normal, face_normal_vector,
740 : get<PrimalMortarVars>(avg_vars_on_face)...);
741 : } else {
742 : (void)element_id;
743 : FluxesComputer::apply(make_not_null(&get<PrimalFluxesVars>(
744 : auxiliary_boundary_corrections))...,
745 : expanded_fluxes_args_on_face...,
746 : face_normal, face_normal_vector,
747 : get<PrimalMortarVars>(avg_vars_on_face)...);
748 : }
749 : },
750 : fluxes_args_on_face);
751 :
752 : // Lifting for the auxiliary boundary correction:
753 : // \int_face G^i \partial_i \phi
754 : // We first transform the flux index to the logical frame, apply the
755 : // quadrature weights and the Jacobian for the face integral, then take
756 : // the logical weak divergence in the volume after lifting (below the loop
757 : // over faces).
758 : auto logical_aux_boundary_corrections =
759 : transform::first_index_to_different_frame(
760 : auxiliary_boundary_corrections, face_jacobian_times_inv_jacobian);
761 : ::dg::apply_mass_matrix(make_not_null(&logical_aux_boundary_corrections),
762 : face_mesh);
763 : if (mesh.quadrature(0) == Spectral::Quadrature::GaussLobatto) {
764 : add_slice_to_data(
765 : make_not_null(&lifted_logical_aux_boundary_corrections),
766 : logical_aux_boundary_corrections, mesh.extents(),
767 : direction.dimension(),
768 : index_to_slice_at(mesh.extents(), direction));
769 : } else {
770 : ::dg::lift_boundary_terms_gauss_points(
771 : make_not_null(&lifted_logical_aux_boundary_corrections),
772 : logical_aux_boundary_corrections, mesh, direction);
773 : }
774 :
775 : // This is the strong primal boundary correction:
776 : // -n.H = -avg(n.F) + n.F + penalty * n.F(n_j jump(u))
777 : // Note that the "internal penalty" numerical flux
778 : // (as opposed to the LLF flux) uses the raw field derivatives without
779 : // boundary corrections in the average, which is why we can communicate
780 : // the data so early together with the auxiliary boundary data. In this
781 : // case the penalty needs to include a factor N_points^2 / h (see the
782 : // `penalty` function).
783 : const auto& penalty_factor = penalty_factors.at(mortar_id);
784 : // Compute jump on mortar:
785 : // penalty * jump(u) = penalty * (u_int - u_ext)
786 : const auto add_remote_jump_contribution =
787 : [&penalty_factor](auto& lhs, const auto& rhs) {
788 : for (size_t i = 0; i < lhs.size(); ++i) {
789 : lhs[i] -= rhs[i];
790 : lhs[i] *= get(penalty_factor);
791 : }
792 : };
793 : EXPAND_PACK_LEFT_TO_RIGHT(add_remote_jump_contribution(
794 : get<PrimalMortarVars>(local_data.field_data),
795 : get<PrimalMortarVars>(remote_data.field_data)));
796 : // Compute average on mortar:
797 : // (strong) -avg(n.F) + n.F = 0.5 * (n.F)_int + 0.5 * (n.F)_ext
798 : // (weak) -avg(n.F) = -0.5 * (n.F)_int + 0.5 * (n.F)_ext
799 : const auto add_avg_contribution = [](auto& lhs, const auto& rhs,
800 : const double factor) {
801 : for (size_t i = 0; i < lhs.size(); ++i) {
802 : lhs[i] *= factor;
803 : lhs[i] += 0.5 * rhs[i];
804 : }
805 : };
806 : EXPAND_PACK_LEFT_TO_RIGHT(add_avg_contribution(
807 : get<::Tags::NormalDotFlux<PrimalMortarVars>>(local_data.field_data),
808 : get<::Tags::NormalDotFlux<PrimalMortarVars>>(remote_data.field_data),
809 : formulation == ::dg::Formulation::StrongInertial ? 0.5 : -0.5));
810 :
811 : // Project from the mortar back down to the face if needed, lift and add
812 : // to operator. See auxiliary boundary corrections above for details.
813 : auto primal_boundary_corrections_on_face =
814 : Spectral::needs_projection(face_mesh, mortar_mesh, mortar_size)
815 : ? mass_conservative_restriction(
816 : std::move(local_data.field_data), mortar_mesh, mortar_size,
817 : mortar_jacobians.at(mortar_id), face_mesh, face_jacobian)
818 : : std::move(local_data.field_data);
819 :
820 : // Compute fluxes for jump term: n.F(n_j jump(u))
821 : // If the fluxes are trivial (just the spatial metric), we can skip this
822 : // step because the face normal is normalized.
823 : if constexpr (not FluxesComputer::is_trivial) {
824 : // We reuse the memory buffer from above for the result.
825 : std::apply(
826 : [&auxiliary_boundary_corrections, &face_normal, &face_normal_vector,
827 : &primal_boundary_corrections_on_face,
828 : &element_id](const auto&... expanded_fluxes_args_on_face) {
829 : if constexpr (FluxesComputer::is_discontinuous) {
830 : FluxesComputer::apply(
831 : make_not_null(&get<PrimalFluxesVars>(
832 : auxiliary_boundary_corrections))...,
833 : expanded_fluxes_args_on_face..., element_id, face_normal,
834 : face_normal_vector,
835 : get<PrimalMortarVars>(
836 : primal_boundary_corrections_on_face)...);
837 : } else {
838 : (void)element_id;
839 : FluxesComputer::apply(
840 : make_not_null(&get<PrimalFluxesVars>(
841 : auxiliary_boundary_corrections))...,
842 : expanded_fluxes_args_on_face..., face_normal,
843 : face_normal_vector,
844 : get<PrimalMortarVars>(
845 : primal_boundary_corrections_on_face)...);
846 : }
847 : },
848 : fluxes_args_on_face);
849 : if constexpr (FluxesComputer::is_discontinuous) {
850 : if (is_internal) {
851 : // For penalty term with discontinuous fluxes: evaluate the fluxes
852 : // on the other side of the boundary as well and take average
853 : Variables<tmpl::list<PrimalFluxesVars...>> fluxes_other_side{
854 : face_num_points};
855 : std::apply(
856 : [&fluxes_other_side, &face_normal, &face_normal_vector,
857 : &primal_boundary_corrections_on_face,
858 : &local_neighbor_id =
859 : neighbor_id](const auto&... expanded_fluxes_args_on_face) {
860 : FluxesComputer::apply(
861 : make_not_null(
862 : &get<PrimalFluxesVars>(fluxes_other_side))...,
863 : expanded_fluxes_args_on_face..., local_neighbor_id,
864 : face_normal, face_normal_vector,
865 : get<PrimalMortarVars>(
866 : primal_boundary_corrections_on_face)...);
867 : },
868 : fluxes_args_on_face);
869 : auxiliary_boundary_corrections += fluxes_other_side;
870 : auxiliary_boundary_corrections *= 0.5;
871 : }
872 : }
873 : EXPAND_PACK_LEFT_TO_RIGHT(normal_dot_flux(
874 : make_not_null(
875 : &get<PrimalMortarVars>(primal_boundary_corrections_on_face)),
876 : face_normal,
877 : get<PrimalFluxesVars>(auxiliary_boundary_corrections)));
878 : }
879 :
880 : // Add penalty term to average term
881 : Variables<tmpl::list<PrimalMortarVars...>> primal_boundary_corrections{};
882 : // First half of the memory allocated above is filled with the penalty
883 : // term, so just use that memory here.
884 : primal_boundary_corrections.set_data_ref(
885 : primal_boundary_corrections_on_face.data(), avg_vars_on_face.size());
886 : // Second half of the memory is filled with the average term. Add that to
887 : // the penalty term.
888 : const auto add_avg_term = [](auto& lhs, const auto& rhs) {
889 : for (size_t i = 0; i < lhs.size(); ++i) {
890 : lhs[i] += rhs[i];
891 : }
892 : };
893 : EXPAND_PACK_LEFT_TO_RIGHT(
894 : add_avg_term(get<PrimalMortarVars>(primal_boundary_corrections),
895 : get<::Tags::NormalDotFlux<PrimalMortarVars>>(
896 : primal_boundary_corrections_on_face)));
897 :
898 : // Lifting for the primal boundary correction:
899 : // \int_face n.H \phi
900 : if (massive) {
901 : // We apply the quadrature weights and Jacobian for the face integral,
902 : // then lift to the volume
903 : primal_boundary_corrections *= get(face_jacobian);
904 : ::dg::apply_mass_matrix(make_not_null(&primal_boundary_corrections),
905 : face_mesh);
906 : if (mesh.quadrature(0) == Spectral::Quadrature::GaussLobatto) {
907 : add_slice_to_data(operator_applied_to_vars,
908 : primal_boundary_corrections, mesh.extents(),
909 : direction.dimension(),
910 : index_to_slice_at(mesh.extents(), direction));
911 : } else {
912 : ::dg::lift_boundary_terms_gauss_points(operator_applied_to_vars,
913 : primal_boundary_corrections,
914 : mesh, direction);
915 : }
916 : } else {
917 : // Apply an extra inverse mass matrix to the boundary corrections (with
918 : // mass lumping, so it's diagonal).
919 : // For Gauss-Lobatto grids this divides out the quadrature weights and
920 : // Jacobian on the face, leaving only factors perpendicular to the face.
921 : // Those are handled by `dg::lift_flux` (with an extra minus sign since
922 : // the function was written for evolution systems).
923 : // For Gauss grids the quadrature weights and Jacobians are handled
924 : // by `::dg::lift_boundary_terms_gauss_points` (which was also written
925 : // for evolution systems, hence the extra minus sign).
926 : primal_boundary_corrections *= -1.;
927 : if (mesh.quadrature(0) == Spectral::Quadrature::GaussLobatto) {
928 : ::dg::lift_flux(make_not_null(&primal_boundary_corrections),
929 : mesh.extents(direction.dimension()),
930 : face_normal_magnitude);
931 : add_slice_to_data(operator_applied_to_vars,
932 : primal_boundary_corrections, mesh.extents(),
933 : direction.dimension(),
934 : index_to_slice_at(mesh.extents(), direction));
935 : } else {
936 : // We already have the `face_jacobian = det(J) * magnitude(n)` here,
937 : // so just pass a constant 1 for `magnitude(n)`. This could be
938 : // optimized to avoid allocating the vector of ones.
939 : ::dg::lift_boundary_terms_gauss_points(
940 : operator_applied_to_vars, det_inv_jacobian, mesh, direction,
941 : primal_boundary_corrections,
942 : Scalar<DataVector>{face_mesh.number_of_grid_points(), 1.},
943 : face_jacobian);
944 : }
945 : }
946 : } // loop over all mortars
947 :
948 : if (not has_any_boundary_corrections) {
949 : // No need to handle auxiliary boundary corrections; return early
950 : return;
951 : }
952 :
953 : // Apply weak divergence to lifted auxiliary boundary corrections and add to
954 : // operator
955 : if (massive) {
956 : logical_weak_divergence(operator_applied_to_vars,
957 : lifted_logical_aux_boundary_corrections, mesh,
958 : true);
959 : } else {
960 : // Possible optimization: eliminate this allocation by building the
961 : // inverse mass matrix into `logical_weak_divergence`
962 : Variables<tmpl::list<OperatorTags...>> massless_aux_boundary_corrections{
963 : num_points};
964 : logical_weak_divergence(make_not_null(&massless_aux_boundary_corrections),
965 : lifted_logical_aux_boundary_corrections, mesh);
966 : massless_aux_boundary_corrections *= get(det_inv_jacobian);
967 : ::dg::apply_inverse_mass_matrix(
968 : make_not_null(&massless_aux_boundary_corrections), mesh);
969 : *operator_applied_to_vars += massless_aux_boundary_corrections;
970 : }
971 : }
972 :
973 : template <typename... FixedSourcesTags, typename ApplyBoundaryCondition,
974 : typename... FluxesArgs, typename... SourcesArgs,
975 : bool LocalLinearized = Linearized,
976 : // This function adds nothing to the fixed sources if the operator
977 : // is linearized, so it shouldn't be used in that case
978 : Requires<not LocalLinearized> = nullptr>
979 : static void impose_inhomogeneous_boundary_conditions_on_source(
980 : const gsl::not_null<Variables<tmpl::list<FixedSourcesTags...>>*>
981 : fixed_sources,
982 : const Element<Dim>& element, const Mesh<Dim>& mesh,
983 : const InverseJacobian<DataVector, Dim, Frame::ElementLogical,
984 : Frame::Inertial>& inv_jacobian,
985 : const Scalar<DataVector>& det_inv_jacobian,
986 : const Scalar<DataVector>& det_jacobian,
987 : const InverseJacobian<DataVector, Dim, Frame::ElementLogical,
988 : Frame::Inertial>& det_times_inv_jacobian,
989 : const DirectionMap<Dim, tnsr::i<DataVector, Dim>>& face_normals,
990 : const DirectionMap<Dim, tnsr::I<DataVector, Dim>>& face_normal_vectors,
991 : const DirectionMap<Dim, Scalar<DataVector>>& face_normal_magnitudes,
992 : const DirectionMap<Dim, Scalar<DataVector>>& face_jacobians,
993 : const DirectionMap<Dim,
994 : InverseJacobian<DataVector, Dim, Frame::ElementLogical,
995 : Frame::Inertial>>&
996 : face_jacobian_times_inv_jacobians,
997 : const ::dg::MortarMap<Dim, Mesh<Dim - 1>>& all_mortar_meshes,
998 : const ::dg::MortarMap<Dim, ::dg::MortarSize<Dim - 1>>& all_mortar_sizes,
999 : const ::dg::MortarMap<Dim, Scalar<DataVector>>& penalty_factors,
1000 : const bool massive, const ::dg::Formulation formulation,
1001 : const ApplyBoundaryCondition& apply_boundary_condition,
1002 : const std::tuple<FluxesArgs...>& fluxes_args,
1003 : const std::tuple<SourcesArgs...>& sources_args,
1004 : const DirectionMap<Dim, std::tuple<FluxesArgs...>>&
1005 : fluxes_args_on_faces) {
1006 : // We just feed zero variables through the nonlinear operator to extract the
1007 : // constant contribution at external boundaries. Since the variables are
1008 : // zero the operator simplifies quite a lot. The simplification is probably
1009 : // not very important for performance because this function will only be
1010 : // called when solving a linear elliptic system and only once during
1011 : // initialization, but we specialize the operator for zero data nonetheless
1012 : // just so we can ignore internal boundaries. For internal boundaries we
1013 : // would unnecessarily have to copy mortar data around to emulate the
1014 : // communication step, so by just skipping internal boundaries we avoid
1015 : // that.
1016 : const size_t num_points = mesh.number_of_grid_points();
1017 : const Variables<tmpl::list<PrimalFields...>> zero_primal_vars{num_points,
1018 : 0.};
1019 : Variables<tmpl::list<PrimalFluxes...>> primal_fluxes_buffer{num_points};
1020 : Variables<tmpl::list<
1021 : ::Tags::deriv<PrimalFields, tmpl::size_t<Dim>, Frame::Inertial>...>>
1022 : unused_deriv_vars_buffer{};
1023 : Variables<tmpl::list<FixedSourcesTags...>> operator_applied_to_zero_vars{
1024 : num_points};
1025 : // Set up data on mortars. We only need them at external boundaries.
1026 : ::dg::MortarMap<Dim, MortarData<size_t, tmpl::list<PrimalFields...>,
1027 : tmpl::list<PrimalFluxes...>>>
1028 : all_mortar_data{};
1029 : constexpr size_t temporal_id = std::numeric_limits<size_t>::max();
1030 : // Apply the operator to the zero variables, skipping internal boundaries
1031 : prepare_mortar_data<true>(make_not_null(&unused_deriv_vars_buffer),
1032 : make_not_null(&primal_fluxes_buffer),
1033 : make_not_null(&all_mortar_data), zero_primal_vars,
1034 : element, mesh, inv_jacobian, face_normals,
1035 : all_mortar_meshes, all_mortar_sizes, temporal_id,
1036 : apply_boundary_condition, fluxes_args);
1037 : apply_operator<true>(
1038 : make_not_null(&operator_applied_to_zero_vars),
1039 : make_not_null(&all_mortar_data), zero_primal_vars, primal_fluxes_buffer,
1040 : element, mesh, inv_jacobian, det_inv_jacobian, det_jacobian,
1041 : det_times_inv_jacobian, face_normals, face_normal_vectors,
1042 : face_normal_magnitudes, face_jacobians,
1043 : face_jacobian_times_inv_jacobians, all_mortar_meshes, all_mortar_sizes,
1044 : {}, penalty_factors, massive, formulation, temporal_id,
1045 : fluxes_args_on_faces, sources_args);
1046 : // Impose the nonlinear (constant) boundary contribution as fixed sources on
1047 : // the RHS of the equations
1048 : *fixed_sources -= operator_applied_to_zero_vars;
1049 : }
1050 : };
1051 :
1052 : } // namespace detail
1053 :
1054 : /*!
1055 : * \brief Prepare data on mortars so they can be communicated to neighbors
1056 : *
1057 : * Call this function on all elements and communicate the mortar data, then call
1058 : * `elliptic::dg::apply_operator`.
1059 : */
1060 : template <typename System, bool Linearized, typename... Args>
1061 1 : void prepare_mortar_data(Args&&... args) {
1062 : detail::DgOperatorImpl<System, Linearized>::template prepare_mortar_data<
1063 : false>(std::forward<Args>(args)...);
1064 : }
1065 :
1066 : /*!
1067 : * \brief Apply the elliptic DG operator
1068 : *
1069 : * This function applies the elliptic DG operator on an element, assuming all
1070 : * data on mortars is already available. Use the
1071 : * `elliptic::dg::prepare_mortar_data` function to prepare mortar data on
1072 : * neighboring elements, then communicate the data and insert them on the
1073 : * "remote" side of the mortars before calling this function.
1074 : */
1075 : template <typename System, bool Linearized, typename... Args>
1076 1 : void apply_operator(Args&&... args) {
1077 : detail::DgOperatorImpl<System, Linearized>::template apply_operator<false>(
1078 : std::forward<Args>(args)...);
1079 : }
1080 :
1081 : /*!
1082 : * \brief For linear systems, impose inhomogeneous boundary conditions as
1083 : * contributions to the fixed sources (i.e. the RHS of the equations).
1084 : *
1085 : * This function exists because the DG operator must typically be linear, but
1086 : * even for linear elliptic equations we typically apply boundary conditions
1087 : * with a constant, and therefore nonlinear, contribution. Standard examples are
1088 : * inhomogeneous (i.e. non-zero) Dirichlet or Neumann boundary conditions. This
1089 : * nonlinear contribution can be added to the fixed sources, leaving only the
1090 : * linearized boundary conditions in the DG operator. For standard constant
1091 : * Dirichlet or Neumann boundary conditions the linearization is of course just
1092 : * zero.
1093 : *
1094 : * This function essentially feeds zero variables through the nonlinear operator
1095 : * and subtracts the result from the fixed sources: `b -= A(x=0)`.
1096 : */
1097 : template <typename System, typename... Args>
1098 1 : void impose_inhomogeneous_boundary_conditions_on_source(Args&&... args) {
1099 : detail::DgOperatorImpl<System, false>::
1100 : impose_inhomogeneous_boundary_conditions_on_source(
1101 : std::forward<Args>(args)...);
1102 : }
1103 :
1104 : } // namespace elliptic::dg
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