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/TaggedTuple.hpp"
16 : #include "DataStructures/Tensor/Tensor.hpp"
17 : #include "DataStructures/Variables.hpp"
18 : #include "DataStructures/Variables/FrameTransform.hpp"
19 : #include "Domain/Structure/Direction.hpp"
20 : #include "Domain/Structure/DirectionMap.hpp"
21 : #include "Domain/Structure/Element.hpp"
22 : #include "Domain/Structure/ElementId.hpp"
23 : #include "Domain/Structure/IndexToSliceAt.hpp"
24 : #include "Elliptic/Protocols/FirstOrderSystem.hpp"
25 : #include "Elliptic/Systems/GetFluxesComputer.hpp"
26 : #include "Elliptic/Systems/GetSourcesComputer.hpp"
27 : #include "NumericalAlgorithms/DiscontinuousGalerkin/ApplyMassMatrix.hpp"
28 : #include "NumericalAlgorithms/DiscontinuousGalerkin/Formulation.hpp"
29 : #include "NumericalAlgorithms/DiscontinuousGalerkin/LiftFlux.hpp"
30 : #include "NumericalAlgorithms/DiscontinuousGalerkin/LiftFromBoundary.hpp"
31 : #include "NumericalAlgorithms/DiscontinuousGalerkin/MortarHelpers.hpp"
32 : #include "NumericalAlgorithms/DiscontinuousGalerkin/NormalDotFlux.hpp"
33 : #include "NumericalAlgorithms/DiscontinuousGalerkin/ProjectToBoundary.hpp"
34 : #include "NumericalAlgorithms/DiscontinuousGalerkin/SimpleBoundaryData.hpp"
35 : #include "NumericalAlgorithms/DiscontinuousGalerkin/SimpleMortarData.hpp"
36 : #include "NumericalAlgorithms/LinearOperators/Divergence.hpp"
37 : #include "NumericalAlgorithms/LinearOperators/Divergence.tpp"
38 : #include "NumericalAlgorithms/LinearOperators/WeakDivergence.hpp"
39 : #include "NumericalAlgorithms/Spectral/Basis.hpp"
40 : #include "NumericalAlgorithms/Spectral/Mesh.hpp"
41 : #include "NumericalAlgorithms/Spectral/Quadrature.hpp"
42 : #include "NumericalAlgorithms/Spectral/SegmentSize.hpp"
43 : #include "Utilities/ErrorHandling/Assert.hpp"
44 : #include "Utilities/Gsl.hpp"
45 : #include "Utilities/ProtocolHelpers.hpp"
46 : #include "Utilities/TMPL.hpp"
47 :
48 : /*!
49 : * \brief Functionality related to discontinuous Galerkin discretizations of
50 : * elliptic equations
51 : *
52 : * The following is a brief overview of the elliptic DG schemes that are
53 : * implemented here. The scheme is described in detail in \cite Fischer2021voj.
54 : *
55 : * The DG schemes apply to any elliptic PDE that can be formulated in
56 : * first-order flux-form, as detailed by
57 : * `elliptic::protocols::FirstOrderSystem`.
58 : * The DG discretization of equations in this first-order form amounts to
59 : * projecting the equations on the set of basis functions that we also use to
60 : * represent the fields on the computational grid. The currently implemented DG
61 : * operator uses Lagrange interpolating polynomials w.r.t. Legendre-Gauss or
62 : * Legendre-Gauss-Lobatto collocation points as basis functions. Skipping all
63 : * further details here, the discretization results in a linear equation
64 : * \f$A(u)=b\f$ over all grid points and primal variables. Solving the elliptic
65 : * equations amounts to numerically inverting the DG operator \f$A\f$, typically
66 : * without ever constructing the full matrix but by employing an iterative
67 : * linear solver that repeatedly applies the DG operator to "test data". Note
68 : * that the DG operator applies directly to the primal variables. Auxiliary
69 : * variables are only computed temporarily and don't inflate the size of the
70 : * operator. This means the DG operator essentially computes second derivatives
71 : * of the primal variables, modified by the fluxes and sources of the system
72 : * as well as by DG boundary corrections that couple grid points across element
73 : * boundaries.
74 : *
75 : * \par Boundary corrections:
76 : * In this implementation we employ the "internal penalty" DG scheme that
77 : * couples grid points across nearest-neighbor elements through the fluxes:
78 : *
79 : * \f{align}
80 : * \label{eq:internal_penalty_auxiliary}
81 : * u^* &= \frac{1}{2} \left(u^\mathrm{int} + u^\mathrm{ext}\right) \\
82 : * \label{eq:internal_penalty_primal}
83 : * (n_i F^i)^* &= \frac{1}{2} n_i \left(
84 : * F^i_\mathrm{int} + F^i_\mathrm{ext} \right)
85 : * - \sigma n_i F^i(n_j (u^\mathrm{int} - u^\mathrm{ext}))
86 : * \f}
87 : *
88 : * Note that \f$n_i\f$ denotes the face normal on the "interior" side of the
89 : * element under consideration. We assume \f$n^\mathrm{ext}_i=-n_i\f$ in the
90 : * implementation, i.e. face normals don't depend on the dynamic variables
91 : * (which may be discontinuous on element faces). This is the case for the
92 : * problems we are expecting to solve, because those will be on fixed background
93 : * metrics (e.g. a conformal metric for the XCTS system). Numerically, the face
94 : * normals on either side of a mortar may nonetheless be different because the
95 : * two faces adjacent to the mortar may resolve them at different resolutions.
96 : *
97 : * Also note that the numerical fluxes intentionally don't depend on the
98 : * auxiliary field values \f$v\f$. This property allows us to communicate data
99 : * for both the primal and auxiliary boundary corrections together, instead of
100 : * communicating them in two steps. If we were to resort to a two-step
101 : * communication we could replace the derivatives in \f$(n_i F^i)^*\f$ with
102 : * \f$v\f$, which would result in a generalized "stabilized central flux" that
103 : * is slightly less sparse than the internal penalty flux (see e.g.
104 : * \cite HesthavenWarburton, section 7.2). We could also choose to ignore the
105 : * fluxes in the penalty term, but preliminary tests suggest that this may hurt
106 : * convergence.
107 : *
108 : * For a Poisson system (see `Poisson::FirstOrderSystem`) this numerical flux
109 : * reduces to the standard internal penalty flux (see e.g.
110 : * \cite HesthavenWarburton, section 7.2, or \cite Arnold2002):
111 : *
112 : * \f{align}
113 : * u^* &= \frac{1}{2} \left(u^\mathrm{int} + u^\mathrm{ext}\right) \\
114 : * (n_i F^i)^* &= n_i v_i^* = \frac{1}{2} n_i \left(
115 : * \partial_i u^\mathrm{int} + \partial_i u^\mathrm{ext}\right)
116 : * - \sigma \left(u^\mathrm{int} - u^\mathrm{ext}\right)
117 : * \f}
118 : *
119 : * where a sum over repeated indices is assumed, since the equation is
120 : * formulated on a Euclidean geometry.
121 : *
122 : * The penalty factor \f$\sigma\f$ is responsible for removing zero eigenmodes
123 : * and impacts the conditioning of the linear operator to be solved. See
124 : * `elliptic::dg::penalty` for details. For the element size that goes into
125 : * computing the penalty we choose
126 : * \f$h=\frac{J_\mathrm{volume}}{J_\mathrm{face}}\f$, i.e. the ratio of Jacobi
127 : * determinants from logical to inertial coordinates in the element volume and
128 : * on the element face, both evaluated on the face (see \cite Vincent2019qpd).
129 : * Since both \f$N_\mathrm{points}\f$ and \f$h\f$ can be different on either
130 : * side of the element boundary we take the maximum of \f$N_\mathrm{points}\f$
131 : * and the pointwise minimum of \f$h\f$ across the element boundary as is done
132 : * in \cite Vincent2019qpd. Note that we use the number of points
133 : * \f$N_\mathrm{points}\f$ where \cite Vincent2019qpd uses the polynomial degree
134 : * \f$N_\mathrm{points} - 1\f$ because we found unstable configurations on
135 : * curved meshes when using the polynomial degree. Optimizing the penalty on
136 : * curved meshes is subject to further investigation.
137 : *
138 : * \par Discontinuous fluxes:
139 : * The DG operator also supports systems with potentially discontinuous fluxes,
140 : * such as elasticity with layered materials. The way to handle the
141 : * discontinuous fluxes in the DG scheme is described in \cite Vu2023thn.
142 : * Essentially, we evaluate the penalty term in
143 : * Eq. $\ref{eq:internal_penalty_primal}$ on both sides of an element boundary
144 : * and take the average. The other terms in the numerical flux remain unchanged.
145 : */
146 : namespace elliptic::dg {
147 :
148 : /// Data that is projected to mortars and communicated across element
149 : /// boundaries
150 : template <typename PrimalFields, typename PrimalFluxes>
151 1 : using BoundaryData = ::dg::SimpleBoundaryData<
152 : tmpl::append<PrimalFields,
153 : db::wrap_tags_in<::Tags::NormalDotFlux, PrimalFields>>,
154 : tmpl::list<>>;
155 :
156 : /// Boundary data on both sides of a mortar.
157 : ///
158 : /// \note This is a struct (as opposed to a type alias) so it can be used to
159 : /// deduce the template parameters
160 : template <typename TemporalId, typename PrimalFields, typename PrimalFluxes>
161 1 : struct MortarData
162 : : ::dg::SimpleMortarData<TemporalId,
163 : BoundaryData<PrimalFields, PrimalFluxes>,
164 : BoundaryData<PrimalFields, PrimalFluxes>> {};
165 :
166 1 : namespace Tags {
167 : /// Holds `elliptic::dg::MortarData`, i.e. boundary data on both sides of a
168 : /// mortar
169 : template <typename TemporalId, typename PrimalFields, typename PrimalFluxes>
170 1 : struct MortarData : db::SimpleTag {
171 0 : using type = elliptic::dg::MortarData<TemporalId, PrimalFields, PrimalFluxes>;
172 : };
173 : } // namespace Tags
174 :
175 : namespace detail {
176 :
177 : // Mass-conservative restriction: R = M^{-1}_face P^T M_mortar
178 : //
179 : // Note that projecting the mortar data times the Jacobian using
180 : // `Spectral::projection_matrix_child_to_parent(operand_is_massive=false)` is
181 : // equivalent to this implementation on Gauss grids. However, on Gauss-Lobatto
182 : // grids we usually diagonally approximate the mass matrix ("mass lumping") but
183 : // `projection_matrix_child_to_parent(operand_is_massive=false)` uses the full
184 : // mass matrix. Therefore, the two implementations differ slightly on
185 : // Gauss-Lobatto grids. Furthermore, note that
186 : // `projection_matrix_child_to_parent(operand_is_massive=false)` already
187 : // includes the factors of two that account for the mortar size, so they must be
188 : // omitted from the mortar Jacobian when using that approach.
189 : template <typename TagsList, size_t FaceDim>
190 : Variables<TagsList> mass_conservative_restriction(
191 : Variables<TagsList> mortar_vars,
192 : [[maybe_unused]] const Mesh<FaceDim>& mortar_mesh,
193 : [[maybe_unused]] const ::dg::MortarSize<FaceDim>& mortar_size,
194 : [[maybe_unused]] const Scalar<DataVector>& mortar_jacobian,
195 : [[maybe_unused]] const Mesh<FaceDim> face_mesh,
196 : [[maybe_unused]] const Scalar<DataVector>& face_jacobian) {
197 : if constexpr (FaceDim == 0) {
198 : return mortar_vars;
199 : } else {
200 : mortar_vars *= get(mortar_jacobian);
201 : ::dg::apply_mass_matrix(make_not_null(&mortar_vars), mortar_mesh);
202 : auto face_vars = Spectral::project(
203 : mortar_vars, mortar_mesh, face_mesh, mortar_size,
204 : make_array<FaceDim>(Spectral::SegmentSize::Full), true);
205 : face_vars /= get(face_jacobian);
206 : ::dg::apply_inverse_mass_matrix(make_not_null(&face_vars), face_mesh);
207 : return face_vars;
208 : }
209 : }
210 :
211 : template <typename System, bool Linearized,
212 : typename PrimalFields = typename System::primal_fields,
213 : typename PrimalFluxes = typename System::primal_fluxes>
214 : struct DgOperatorImpl;
215 :
216 : template <typename System, bool Linearized, typename... PrimalFields,
217 : typename... PrimalFluxes>
218 : struct DgOperatorImpl<System, Linearized, tmpl::list<PrimalFields...>,
219 : tmpl::list<PrimalFluxes...>> {
220 : static_assert(
221 : tt::assert_conforms_to_v<System, elliptic::protocols::FirstOrderSystem>);
222 :
223 : static constexpr size_t Dim = System::volume_dim;
224 : using FluxesComputer = elliptic::get_fluxes_computer<System, Linearized>;
225 : using SourcesComputer = elliptic::get_sources_computer<System, Linearized>;
226 :
227 : struct AllDirections {
228 : bool operator()(const Direction<Dim>& /*unused*/) const { return true; }
229 : };
230 :
231 : struct NoDataIsZero {
232 : bool operator()(const ElementId<Dim>& /*unused*/) const { return false; }
233 : };
234 :
235 : static constexpr auto full_mortar_size =
236 : make_array<Dim - 1>(Spectral::SegmentSize::Full);
237 :
238 : template <bool AllDataIsZero, typename... DerivVars, typename... PrimalVars,
239 : typename... PrimalFluxesVars, typename... PrimalMortarVars,
240 : typename... PrimalMortarFluxes, typename TemporalId,
241 : typename ApplyBoundaryCondition, typename... FluxesArgs,
242 : typename DataIsZero = NoDataIsZero,
243 : typename DirectionsPredicate = AllDirections>
244 : static void prepare_mortar_data(
245 : const gsl::not_null<Variables<tmpl::list<DerivVars...>>*> deriv_vars,
246 : const gsl::not_null<Variables<tmpl::list<PrimalFluxesVars...>>*>
247 : primal_fluxes,
248 : const gsl::not_null<::dg::MortarMap<
249 : Dim, MortarData<TemporalId, tmpl::list<PrimalMortarVars...>,
250 : tmpl::list<PrimalMortarFluxes...>>>*>
251 : all_mortar_data,
252 : const Variables<tmpl::list<PrimalVars...>>& primal_vars,
253 : const Element<Dim>& element, const Mesh<Dim>& mesh,
254 : const InverseJacobian<DataVector, Dim, Frame::ElementLogical,
255 : Frame::Inertial>& inv_jacobian,
256 : const DirectionMap<Dim, tnsr::i<DataVector, Dim>>& face_normals,
257 : const ::dg::MortarMap<Dim, Mesh<Dim - 1>>& all_mortar_meshes,
258 : const ::dg::MortarMap<Dim, ::dg::MortarSize<Dim - 1>>& all_mortar_sizes,
259 : const TemporalId& temporal_id,
260 : const ApplyBoundaryCondition& apply_boundary_condition,
261 : const std::tuple<FluxesArgs...>& fluxes_args,
262 : const DataIsZero& data_is_zero = NoDataIsZero{},
263 : const DirectionsPredicate& directions_predicate = AllDirections{}) {
264 : static_assert(
265 : sizeof...(PrimalVars) == sizeof...(PrimalFields) and
266 : sizeof...(PrimalFluxesVars) == sizeof...(PrimalFluxes),
267 : "The number of variables must match the number of system fields.");
268 : static_assert(
269 : (std::is_same_v<typename PrimalVars::type,
270 : typename PrimalFields::type> and
271 : ...) and
272 : (std::is_same_v<typename PrimalFluxesVars::type,
273 : typename PrimalFluxes::type> and
274 : ...),
275 : "The variables must have the same tensor types as the system fields.");
276 : #ifdef SPECTRE_DEBUG
277 : for (size_t d = 0; d < Dim; ++d) {
278 : ASSERT(mesh.basis(d) == Spectral::Basis::Legendre and
279 : (mesh.quadrature(d) == Spectral::Quadrature::GaussLobatto or
280 : mesh.quadrature(d) == Spectral::Quadrature::Gauss),
281 : "The elliptic DG operator is currently only implemented for "
282 : "Legendre-Gauss(-Lobatto) grids. Found basis '"
283 : << mesh.basis(d) << "' and quadrature '" << mesh.quadrature(d)
284 : << "' in dimension " << d << ".");
285 : }
286 : #endif // SPECTRE_DEBUG
287 : const auto& element_id = element.id();
288 : const bool local_data_is_zero = data_is_zero(element_id);
289 : ASSERT(Linearized or not local_data_is_zero,
290 : "Only a linear operator can take advantage of the knowledge that "
291 : "the operand is zero. Don't return 'true' in 'data_is_zero' unless "
292 : "you also set 'Linearized' to 'true'.");
293 : const size_t num_points = mesh.number_of_grid_points();
294 :
295 : // This function and the one below allocate various Variables to compute
296 : // intermediate quantities. It could be a performance optimization to reduce
297 : // the number of these allocations and/or move some of the memory buffers
298 : // into the DataBox to keep them around permanently. The latter should be
299 : // informed by profiling.
300 :
301 : // Compute partial derivatives grad(u) of the system variables, and from
302 : // those the fluxes F^i(grad(u)) in the volume. We will take the divergence
303 : // of the fluxes in the `apply_operator` function below to compute the full
304 : // elliptic equation -div(F) + S = f(x).
305 : if (AllDataIsZero or local_data_is_zero) {
306 : deriv_vars->initialize(num_points, 0.);
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<DerivVars>(*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<DerivVars>(*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<DerivVars...>> 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<DerivVars>(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 <typename... OperatorTags, typename... PrimalVars,
520 : typename... DerivVars, typename... PrimalFluxesVars,
521 : typename... PrimalMortarVars, typename... PrimalMortarFluxes,
522 : typename TemporalId, typename... FluxesArgs,
523 : typename... SourcesArgs, typename... ModifyBoundaryDataArgs,
524 : typename DataIsZero = NoDataIsZero,
525 : typename DirectionsPredicate = AllDirections>
526 : static void apply_operator(
527 : const gsl::not_null<Variables<tmpl::list<OperatorTags...>>*>
528 : operator_applied_to_vars,
529 : const gsl::not_null<::dg::MortarMap<
530 : Dim, MortarData<TemporalId, tmpl::list<PrimalMortarVars...>,
531 : tmpl::list<PrimalMortarFluxes...>>>*>
532 : all_mortar_data,
533 : const Variables<tmpl::list<PrimalVars...>>& primal_vars,
534 : const Variables<tmpl::list<DerivVars...>>& deriv_vars,
535 : // Taking the primal fluxes computed in the `prepare_mortar_data` function
536 : // by const-ref here because other code might use them and so we don't
537 : // want to modify them by adding boundary corrections. E.g. linearized
538 : // sources use the nonlinear fields and fluxes as background fields.
539 : const Variables<tmpl::list<PrimalFluxesVars...>>& primal_fluxes,
540 : const Element<Dim>& element, const Mesh<Dim>& mesh,
541 : const InverseJacobian<DataVector, Dim, Frame::ElementLogical,
542 : Frame::Inertial>& inv_jacobian,
543 : const Scalar<DataVector>& det_inv_jacobian,
544 : const Scalar<DataVector>& det_jacobian,
545 : const InverseJacobian<DataVector, Dim, Frame::ElementLogical,
546 : Frame::Inertial>& det_times_inv_jacobian,
547 : const DirectionMap<Dim, tnsr::i<DataVector, Dim>>& face_normals,
548 : const DirectionMap<Dim, tnsr::I<DataVector, Dim>>& face_normal_vectors,
549 : const DirectionMap<Dim, Scalar<DataVector>>& face_normal_magnitudes,
550 : const DirectionMap<Dim, Scalar<DataVector>>& face_jacobians,
551 : const DirectionMap<Dim,
552 : InverseJacobian<DataVector, Dim, Frame::ElementLogical,
553 : Frame::Inertial>>&
554 : face_jacobian_times_inv_jacobians,
555 : const ::dg::MortarMap<Dim, Mesh<Dim - 1>>& all_mortar_meshes,
556 : const ::dg::MortarMap<Dim, ::dg::MortarSize<Dim - 1>>& all_mortar_sizes,
557 : const ::dg::MortarMap<Dim, Scalar<DataVector>>& mortar_jacobians,
558 : const ::dg::MortarMap<Dim, Scalar<DataVector>>& penalty_factors,
559 : const bool massive, const ::dg::Formulation formulation,
560 : const TemporalId& /*temporal_id*/,
561 : const DirectionMap<Dim, std::tuple<FluxesArgs...>>& fluxes_args_on_faces,
562 : const std::tuple<SourcesArgs...>& sources_args,
563 : const std::tuple<ModifyBoundaryDataArgs...>& modify_boundary_data_args,
564 : const DataIsZero& data_is_zero = NoDataIsZero{},
565 : const DirectionsPredicate& directions_predicate = AllDirections{}) {
566 : static_assert(
567 : sizeof...(PrimalVars) == sizeof...(PrimalFields) and
568 : sizeof...(PrimalFluxesVars) == sizeof...(PrimalFluxes) and
569 : sizeof...(PrimalMortarVars) == sizeof...(PrimalFields) and
570 : sizeof...(PrimalMortarFluxes) == sizeof...(PrimalFluxes) and
571 : sizeof...(OperatorTags) == sizeof...(PrimalFields),
572 : "The number of variables must match the number of system fields.");
573 : static_assert(
574 : (std::is_same_v<typename PrimalVars::type,
575 : typename PrimalFields::type> and
576 : ...) and
577 : (std::is_same_v<typename PrimalFluxesVars::type,
578 : typename PrimalFluxes::type> and
579 : ...) and
580 : (std::is_same_v<typename PrimalMortarVars::type,
581 : typename PrimalFields::type> and
582 : ...) and
583 : (std::is_same_v<typename PrimalMortarFluxes::type,
584 : typename PrimalFluxes::type> and
585 : ...) and
586 : (std::is_same_v<typename OperatorTags::type,
587 : typename PrimalFields::type> and
588 : ...),
589 : "The variables must have the same tensor types as the system fields.");
590 : #ifdef SPECTRE_DEBUG
591 : for (size_t d = 0; d < Dim; ++d) {
592 : ASSERT(mesh.basis(d) == Spectral::Basis::Legendre and
593 : (mesh.quadrature(d) == Spectral::Quadrature::GaussLobatto or
594 : mesh.quadrature(d) == Spectral::Quadrature::Gauss),
595 : "The elliptic DG operator is currently only implemented for "
596 : "Legendre-Gauss(-Lobatto) grids. Found basis '"
597 : << mesh.basis(d) << "' and quadrature '" << mesh.quadrature(d)
598 : << "' in dimension " << d << ".");
599 : }
600 : #endif // SPECTRE_DEBUG
601 : const auto& element_id = element.id();
602 : const bool local_data_is_zero = data_is_zero(element_id);
603 : ASSERT(Linearized or not local_data_is_zero,
604 : "Only a linear operator can take advantage of the knowledge that "
605 : "the operand is zero. Don't return 'true' in 'data_is_zero' unless "
606 : "you also set 'Linearized' to 'true'.");
607 : const size_t num_points = mesh.number_of_grid_points();
608 :
609 : // This function and the one above allocate various Variables to compute
610 : // intermediate quantities. It could be a performance optimization to reduce
611 : // the number of these allocations and/or move some of the memory buffers
612 : // into the DataBox to keep them around permanently. The latter should be
613 : // informed by profiling.
614 :
615 : // Compute volume terms: -div(F) + S
616 : if (local_data_is_zero) {
617 : operator_applied_to_vars->initialize(num_points, 0.);
618 : } else {
619 : // "Massive" operators retain the factors from the volume integral:
620 : // \int_volume div(F) \phi_p = w_p det(J)_p div(F)_p
621 : // Here, `w` are the quadrature weights (the diagonal logical mass matrix
622 : // with mass-lumping) and det(J) is the Jacobian determinant. The
623 : // quantities are evaluated at the grid point `p`.
624 : if (formulation == ::dg::Formulation::StrongInertial) {
625 : // Compute strong divergence:
626 : // div(F) = (J^\hat{i}_i)_p \sum_q (D_\hat{i})_pq (F^i)_q.
627 : divergence(operator_applied_to_vars, primal_fluxes, mesh,
628 : massive ? det_times_inv_jacobian : inv_jacobian);
629 : // This is the sign flip that makes the operator _minus_ the Laplacian
630 : // for a Poisson system
631 : *operator_applied_to_vars *= -1.;
632 : } else if (formulation == ::dg::Formulation::StrongLogical) {
633 : // Strong divergence but with the Jacobian moved into the divergence:
634 : // div(F) = 1/J_p \sum_q (D_\hat{i})_pq J_q (J^\hat{i}_i)_q (F^i)_q.
635 : const auto logical_fluxes = transform::first_index_to_different_frame(
636 : primal_fluxes, det_times_inv_jacobian);
637 : logical_divergence(operator_applied_to_vars, logical_fluxes, mesh);
638 : if (massive) {
639 : *operator_applied_to_vars *= -1.;
640 : } else {
641 : *operator_applied_to_vars *= -1. * get(det_inv_jacobian);
642 : }
643 : } else if (formulation == ::dg::Formulation::WeakInertial) {
644 : // Compute weak divergence:
645 : // F^i \partial_i \phi = 1/w_p \sum_q
646 : // (D^T_\hat{i})_pq (w det(J) J^\hat{i}_i F^i)_q
647 : weak_divergence(operator_applied_to_vars, primal_fluxes, mesh,
648 : det_times_inv_jacobian);
649 : if (not massive) {
650 : *operator_applied_to_vars *= get(det_inv_jacobian);
651 : }
652 : } else {
653 : ERROR("Unsupported DG formulation: "
654 : << formulation
655 : << "\nSupported formulations are: StrongInertial, WeakInertial, "
656 : "StrongLogical.");
657 : }
658 : if constexpr (not std::is_same_v<SourcesComputer, void>) {
659 : Variables<tmpl::list<OperatorTags...>> sources{num_points, 0.};
660 : std::apply(
661 : [&sources, &primal_vars, &deriv_vars,
662 : &primal_fluxes](const auto&... expanded_sources_args) {
663 : SourcesComputer::apply(
664 : make_not_null(&get<OperatorTags>(sources))...,
665 : expanded_sources_args..., get<PrimalVars>(primal_vars)...,
666 : get<DerivVars>(deriv_vars)...,
667 : get<PrimalFluxesVars>(primal_fluxes)...);
668 : },
669 : sources_args);
670 : if (massive) {
671 : sources *= get(det_jacobian);
672 : }
673 : *operator_applied_to_vars += sources;
674 : }
675 : }
676 : if (massive) {
677 : ::dg::apply_mass_matrix(operator_applied_to_vars, mesh);
678 : }
679 :
680 : // Add boundary corrections
681 : // Keeping track if any corrections were applied here, for an optimization
682 : // below
683 : bool has_any_boundary_corrections = false;
684 : Variables<tmpl::list<transform::Tags::TransformedFirstIndex<
685 : PrimalFluxesVars, Frame::ElementLogical>...>>
686 : lifted_logical_aux_boundary_corrections{num_points, 0.};
687 : for (auto& [mortar_id, mortar_data] : *all_mortar_data) {
688 : const auto& direction = mortar_id.direction();
689 : const auto& neighbor_id = mortar_id.id();
690 : const bool is_internal =
691 : (neighbor_id != ElementId<Dim>::external_boundary_id());
692 : if (not directions_predicate(direction)) {
693 : continue;
694 : }
695 : // When the data on both sides of the mortar is zero then we don't need to
696 : // handle this mortar at all.
697 : if (local_data_is_zero and
698 : (not is_internal or data_is_zero(neighbor_id))) {
699 : continue;
700 : }
701 : has_any_boundary_corrections = true;
702 :
703 : const auto face_mesh = mesh.slice_away(direction.dimension());
704 : auto [local_data, remote_data] = mortar_data.extract();
705 :
706 : if (is_internal) {
707 : if constexpr (Linearized and
708 : not std::is_same_v<typename System::modify_boundary_data,
709 : void>) {
710 : // Apply a linearized modification to received boundary data.
711 : // This allows modifications to depend linearly on the variables.
712 : std::apply(
713 : [&rem_data = remote_data, &loc_data = local_data,
714 : m_id = mortar_id](const auto&... args) {
715 : System::modify_boundary_data::apply_linearized(
716 : make_not_null(
717 : &get<PrimalMortarVars>(rem_data.field_data))...,
718 : make_not_null(&get<::Tags::NormalDotFlux<PrimalMortarVars>>(
719 : rem_data.field_data))...,
720 : get<PrimalMortarVars>(loc_data.field_data)...,
721 : get<::Tags::NormalDotFlux<PrimalMortarVars>>(
722 : loc_data.field_data)...,
723 : m_id, args...);
724 : },
725 : modify_boundary_data_args);
726 : }
727 : }
728 :
729 : const size_t face_num_points = face_mesh.number_of_grid_points();
730 : const auto& face_normal = face_normals.at(direction);
731 : const auto& face_normal_vector = face_normal_vectors.at(direction);
732 : const auto& fluxes_args_on_face = fluxes_args_on_faces.at(direction);
733 : const auto& face_normal_magnitude = face_normal_magnitudes.at(direction);
734 : const auto& face_jacobian = face_jacobians.at(direction);
735 : const auto& face_jacobian_times_inv_jacobian =
736 : face_jacobian_times_inv_jacobians.at(direction);
737 : const auto& mortar_mesh =
738 : is_internal ? all_mortar_meshes.at(mortar_id) : face_mesh;
739 : const auto& mortar_size =
740 : is_internal ? all_mortar_sizes.at(mortar_id) : full_mortar_size;
741 :
742 : // This is the strong auxiliary boundary correction:
743 : // G^i = F^i(n_j (avg(u) - u))
744 : // where
745 : // avg(u) - u = -0.5 * (u_int - u_ext)
746 : auto avg_vars_on_mortar = Variables<tmpl::list<PrimalMortarVars...>>(
747 : local_data.field_data
748 : .template extract_subset<tmpl::list<PrimalMortarVars...>>());
749 : const auto add_remote_contribution = [](auto& lhs, const auto& rhs) {
750 : for (size_t i = 0; i < lhs.size(); ++i) {
751 : lhs[i] -= rhs[i];
752 : }
753 : };
754 : EXPAND_PACK_LEFT_TO_RIGHT(add_remote_contribution(
755 : get<PrimalMortarVars>(avg_vars_on_mortar),
756 : get<PrimalMortarVars>(remote_data.field_data)));
757 : avg_vars_on_mortar *= -0.5;
758 :
759 : // Project from the mortar back down to the face if needed
760 : const auto avg_vars_on_face =
761 : Spectral::needs_projection(face_mesh, mortar_mesh, mortar_size)
762 : ? mass_conservative_restriction(
763 : std::move(avg_vars_on_mortar), mortar_mesh, mortar_size,
764 : mortar_jacobians.at(mortar_id), face_mesh, face_jacobian)
765 : : std::move(avg_vars_on_mortar);
766 :
767 : // Apply fluxes to get G^i
768 : Variables<tmpl::list<PrimalFluxesVars...>> auxiliary_boundary_corrections{
769 : face_num_points};
770 : std::apply(
771 : [&auxiliary_boundary_corrections, &face_normal, &face_normal_vector,
772 : &avg_vars_on_face,
773 : &element_id](const auto&... expanded_fluxes_args_on_face) {
774 : if constexpr (FluxesComputer::is_discontinuous) {
775 : FluxesComputer::apply(make_not_null(&get<PrimalFluxesVars>(
776 : auxiliary_boundary_corrections))...,
777 : expanded_fluxes_args_on_face..., element_id,
778 : face_normal, face_normal_vector,
779 : get<PrimalMortarVars>(avg_vars_on_face)...);
780 : } else {
781 : (void)element_id;
782 : FluxesComputer::apply(make_not_null(&get<PrimalFluxesVars>(
783 : auxiliary_boundary_corrections))...,
784 : expanded_fluxes_args_on_face...,
785 : face_normal, face_normal_vector,
786 : get<PrimalMortarVars>(avg_vars_on_face)...);
787 : }
788 : },
789 : fluxes_args_on_face);
790 :
791 : // Lifting for the auxiliary boundary correction:
792 : // \int_face G^i \partial_i \phi
793 : // We first transform the flux index to the logical frame, apply the
794 : // quadrature weights and the Jacobian for the face integral, then take
795 : // the logical weak divergence in the volume after lifting (below the loop
796 : // over faces).
797 : auto logical_aux_boundary_corrections =
798 : transform::first_index_to_different_frame(
799 : auxiliary_boundary_corrections, face_jacobian_times_inv_jacobian);
800 : ::dg::apply_mass_matrix(make_not_null(&logical_aux_boundary_corrections),
801 : face_mesh);
802 : if (mesh.quadrature(0) == Spectral::Quadrature::GaussLobatto) {
803 : add_slice_to_data(
804 : make_not_null(&lifted_logical_aux_boundary_corrections),
805 : logical_aux_boundary_corrections, mesh.extents(),
806 : direction.dimension(),
807 : index_to_slice_at(mesh.extents(), direction));
808 : } else {
809 : ::dg::lift_boundary_terms_gauss_points(
810 : make_not_null(&lifted_logical_aux_boundary_corrections),
811 : logical_aux_boundary_corrections, mesh, direction);
812 : }
813 :
814 : // This is the strong primal boundary correction:
815 : // -n.H = -avg(n.F) + n.F + penalty * n.F(n_j jump(u))
816 : // Note that the "internal penalty" numerical flux
817 : // (as opposed to the LLF flux) uses the raw field derivatives without
818 : // boundary corrections in the average, which is why we can communicate
819 : // the data so early together with the auxiliary boundary data. In this
820 : // case the penalty needs to include a factor N_points^2 / h (see the
821 : // `penalty` function).
822 : const auto& penalty_factor = penalty_factors.at(mortar_id);
823 : // Compute jump on mortar:
824 : // penalty * jump(u) = penalty * (u_int - u_ext)
825 : const auto add_remote_jump_contribution =
826 : [&penalty_factor](auto& lhs, const auto& rhs) {
827 : for (size_t i = 0; i < lhs.size(); ++i) {
828 : lhs[i] -= rhs[i];
829 : lhs[i] *= get(penalty_factor);
830 : }
831 : };
832 : EXPAND_PACK_LEFT_TO_RIGHT(add_remote_jump_contribution(
833 : get<PrimalMortarVars>(local_data.field_data),
834 : get<PrimalMortarVars>(remote_data.field_data)));
835 : // Compute average on mortar:
836 : // (strong) -avg(n.F) + n.F = 0.5 * (n.F)_int + 0.5 * (n.F)_ext
837 : // (weak) -avg(n.F) = -0.5 * (n.F)_int + 0.5 * (n.F)_ext
838 : const auto add_avg_contribution = [](auto& lhs, const auto& rhs,
839 : const double factor) {
840 : for (size_t i = 0; i < lhs.size(); ++i) {
841 : lhs[i] *= factor;
842 : lhs[i] += 0.5 * rhs[i];
843 : }
844 : };
845 : const bool is_strong_formulation =
846 : formulation == ::dg::Formulation::StrongInertial or
847 : formulation == ::dg::Formulation::StrongLogical;
848 : EXPAND_PACK_LEFT_TO_RIGHT(add_avg_contribution(
849 : get<::Tags::NormalDotFlux<PrimalMortarVars>>(local_data.field_data),
850 : get<::Tags::NormalDotFlux<PrimalMortarVars>>(remote_data.field_data),
851 : is_strong_formulation ? 0.5 : -0.5));
852 :
853 : // Project from the mortar back down to the face if needed, lift and add
854 : // to operator. See auxiliary boundary corrections above for details.
855 : auto primal_boundary_corrections_on_face =
856 : Spectral::needs_projection(face_mesh, mortar_mesh, mortar_size)
857 : ? mass_conservative_restriction(
858 : std::move(local_data.field_data), mortar_mesh, mortar_size,
859 : mortar_jacobians.at(mortar_id), face_mesh, face_jacobian)
860 : : std::move(local_data.field_data);
861 :
862 : // Compute fluxes for jump term: n.F(n_j jump(u))
863 : // If the fluxes are trivial (just the spatial metric), we can skip this
864 : // step because the face normal is normalized.
865 : if constexpr (not FluxesComputer::is_trivial) {
866 : // We reuse the memory buffer from above for the result.
867 : std::apply(
868 : [&auxiliary_boundary_corrections, &face_normal, &face_normal_vector,
869 : &primal_boundary_corrections_on_face,
870 : &element_id](const auto&... expanded_fluxes_args_on_face) {
871 : if constexpr (FluxesComputer::is_discontinuous) {
872 : FluxesComputer::apply(
873 : make_not_null(&get<PrimalFluxesVars>(
874 : auxiliary_boundary_corrections))...,
875 : expanded_fluxes_args_on_face..., element_id, face_normal,
876 : face_normal_vector,
877 : get<PrimalMortarVars>(
878 : primal_boundary_corrections_on_face)...);
879 : } else {
880 : (void)element_id;
881 : FluxesComputer::apply(
882 : make_not_null(&get<PrimalFluxesVars>(
883 : auxiliary_boundary_corrections))...,
884 : expanded_fluxes_args_on_face..., face_normal,
885 : face_normal_vector,
886 : get<PrimalMortarVars>(
887 : primal_boundary_corrections_on_face)...);
888 : }
889 : },
890 : fluxes_args_on_face);
891 : if constexpr (FluxesComputer::is_discontinuous) {
892 : if (is_internal) {
893 : // For penalty term with discontinuous fluxes: evaluate the fluxes
894 : // on the other side of the boundary as well and take average
895 : Variables<tmpl::list<PrimalFluxesVars...>> fluxes_other_side{
896 : face_num_points};
897 : std::apply(
898 : [&fluxes_other_side, &face_normal, &face_normal_vector,
899 : &primal_boundary_corrections_on_face,
900 : &local_neighbor_id =
901 : neighbor_id](const auto&... expanded_fluxes_args_on_face) {
902 : FluxesComputer::apply(
903 : make_not_null(
904 : &get<PrimalFluxesVars>(fluxes_other_side))...,
905 : expanded_fluxes_args_on_face..., local_neighbor_id,
906 : face_normal, face_normal_vector,
907 : get<PrimalMortarVars>(
908 : primal_boundary_corrections_on_face)...);
909 : },
910 : fluxes_args_on_face);
911 : auxiliary_boundary_corrections += fluxes_other_side;
912 : auxiliary_boundary_corrections *= 0.5;
913 : }
914 : }
915 : EXPAND_PACK_LEFT_TO_RIGHT(normal_dot_flux(
916 : make_not_null(
917 : &get<PrimalMortarVars>(primal_boundary_corrections_on_face)),
918 : face_normal,
919 : get<PrimalFluxesVars>(auxiliary_boundary_corrections)));
920 : }
921 :
922 : // Add penalty term to average term
923 : Variables<tmpl::list<PrimalMortarVars...>> primal_boundary_corrections{};
924 : // First half of the memory allocated above is filled with the penalty
925 : // term, so just use that memory here.
926 : primal_boundary_corrections.set_data_ref(
927 : primal_boundary_corrections_on_face.data(), avg_vars_on_face.size());
928 : // Second half of the memory is filled with the average term. Add that to
929 : // the penalty term.
930 : const auto add_avg_term = [](auto& lhs, const auto& rhs) {
931 : for (size_t i = 0; i < lhs.size(); ++i) {
932 : lhs[i] += rhs[i];
933 : }
934 : };
935 : EXPAND_PACK_LEFT_TO_RIGHT(
936 : add_avg_term(get<PrimalMortarVars>(primal_boundary_corrections),
937 : get<::Tags::NormalDotFlux<PrimalMortarVars>>(
938 : primal_boundary_corrections_on_face)));
939 :
940 : // Lifting for the primal boundary correction:
941 : // \int_face n.H \phi
942 : if (massive) {
943 : // We apply the quadrature weights and Jacobian for the face integral,
944 : // then lift to the volume
945 : primal_boundary_corrections *= get(face_jacobian);
946 : ::dg::apply_mass_matrix(make_not_null(&primal_boundary_corrections),
947 : face_mesh);
948 : if (mesh.quadrature(0) == Spectral::Quadrature::GaussLobatto) {
949 : add_slice_to_data(operator_applied_to_vars,
950 : primal_boundary_corrections, mesh.extents(),
951 : direction.dimension(),
952 : index_to_slice_at(mesh.extents(), direction));
953 : } else {
954 : ::dg::lift_boundary_terms_gauss_points(operator_applied_to_vars,
955 : primal_boundary_corrections,
956 : mesh, direction);
957 : }
958 : } else {
959 : // Apply an extra inverse mass matrix to the boundary corrections (with
960 : // mass lumping, so it's diagonal).
961 : // For Gauss-Lobatto grids this divides out the quadrature weights and
962 : // Jacobian on the face, leaving only factors perpendicular to the face.
963 : // Those are handled by `dg::lift_flux` (with an extra minus sign since
964 : // the function was written for evolution systems).
965 : // For Gauss grids the quadrature weights and Jacobians are handled
966 : // by `::dg::lift_boundary_terms_gauss_points` (which was also written
967 : // for evolution systems, hence the extra minus sign).
968 : primal_boundary_corrections *= -1.;
969 : if (mesh.quadrature(0) == Spectral::Quadrature::GaussLobatto) {
970 : ::dg::lift_flux(make_not_null(&primal_boundary_corrections),
971 : mesh.extents(direction.dimension()),
972 : face_normal_magnitude);
973 : add_slice_to_data(operator_applied_to_vars,
974 : primal_boundary_corrections, mesh.extents(),
975 : direction.dimension(),
976 : index_to_slice_at(mesh.extents(), direction));
977 : } else {
978 : // We already have the `face_jacobian = det(J) * magnitude(n)` here,
979 : // so just pass a constant 1 for `magnitude(n)`. This could be
980 : // optimized to avoid allocating the vector of ones.
981 : ::dg::lift_boundary_terms_gauss_points(
982 : operator_applied_to_vars, det_inv_jacobian, mesh, direction,
983 : primal_boundary_corrections,
984 : Scalar<DataVector>{face_mesh.number_of_grid_points(), 1.},
985 : face_jacobian);
986 : }
987 : }
988 : } // loop over all mortars
989 :
990 : if (not has_any_boundary_corrections) {
991 : // No need to handle auxiliary boundary corrections; return early
992 : return;
993 : }
994 :
995 : // Apply weak divergence to lifted auxiliary boundary corrections and add to
996 : // operator
997 : if (massive) {
998 : logical_weak_divergence(operator_applied_to_vars,
999 : lifted_logical_aux_boundary_corrections, mesh,
1000 : true);
1001 : } else {
1002 : // Possible optimization: eliminate this allocation by building the
1003 : // inverse mass matrix into `logical_weak_divergence`
1004 : Variables<tmpl::list<OperatorTags...>> massless_aux_boundary_corrections{
1005 : num_points};
1006 : logical_weak_divergence(make_not_null(&massless_aux_boundary_corrections),
1007 : lifted_logical_aux_boundary_corrections, mesh);
1008 : massless_aux_boundary_corrections *= get(det_inv_jacobian);
1009 : ::dg::apply_inverse_mass_matrix(
1010 : make_not_null(&massless_aux_boundary_corrections), mesh);
1011 : *operator_applied_to_vars += massless_aux_boundary_corrections;
1012 : }
1013 : }
1014 :
1015 : template <typename... FixedSourcesTags, typename ApplyBoundaryCondition,
1016 : typename... FluxesArgs, typename... SourcesArgs,
1017 : typename... ModifyBoundaryDataArgs>
1018 : static void impose_inhomogeneous_boundary_conditions_on_source(
1019 : const gsl::not_null<Variables<tmpl::list<FixedSourcesTags...>>*>
1020 : fixed_sources,
1021 : const Element<Dim>& element, const Mesh<Dim>& mesh,
1022 : const InverseJacobian<DataVector, Dim, Frame::ElementLogical,
1023 : Frame::Inertial>& inv_jacobian,
1024 : const Scalar<DataVector>& det_inv_jacobian,
1025 : const Scalar<DataVector>& det_jacobian,
1026 : const InverseJacobian<DataVector, Dim, Frame::ElementLogical,
1027 : Frame::Inertial>& det_times_inv_jacobian,
1028 : const DirectionMap<Dim, tnsr::i<DataVector, Dim>>& face_normals,
1029 : const DirectionMap<Dim, tnsr::I<DataVector, Dim>>& face_normal_vectors,
1030 : const DirectionMap<Dim, Scalar<DataVector>>& face_normal_magnitudes,
1031 : const DirectionMap<Dim, Scalar<DataVector>>& face_jacobians,
1032 : const DirectionMap<Dim,
1033 : InverseJacobian<DataVector, Dim, Frame::ElementLogical,
1034 : Frame::Inertial>>&
1035 : face_jacobian_times_inv_jacobians,
1036 : const ::dg::MortarMap<Dim, Mesh<Dim - 1>>& all_mortar_meshes,
1037 : const ::dg::MortarMap<Dim, ::dg::MortarSize<Dim - 1>>& all_mortar_sizes,
1038 : const ::dg::MortarMap<Dim, Scalar<DataVector>>& mortar_jacobians,
1039 : const ::dg::MortarMap<Dim, Scalar<DataVector>>& penalty_factors,
1040 : const bool massive, const ::dg::Formulation formulation,
1041 : const ApplyBoundaryCondition& apply_boundary_condition,
1042 : const std::tuple<FluxesArgs...>& fluxes_args,
1043 : const std::tuple<SourcesArgs...>& sources_args,
1044 : const DirectionMap<Dim, std::tuple<FluxesArgs...>>& fluxes_args_on_faces,
1045 : const std::tuple<ModifyBoundaryDataArgs...>& modify_boundary_data_args)
1046 : // This function adds nothing to the fixed sources if the operator is
1047 : // linearized, so it shouldn't be used in that case
1048 : requires(not Linearized)
1049 : {
1050 : // We just feed zero variables through the nonlinear operator to extract the
1051 : // constant contribution at external boundaries. Since the variables are
1052 : // zero the operator simplifies quite a lot. The simplification is probably
1053 : // not very important for performance because this function will only be
1054 : // called when solving a linear elliptic system and only once during
1055 : // initialization, but we specialize the operator for zero data nonetheless
1056 : // just so we can ignore internal boundaries. Only when the system modifies
1057 : // boundary data do we need to handle internal boundaries (see below),
1058 : // otherwise we can skip them.
1059 : const size_t num_points = mesh.number_of_grid_points();
1060 : const Variables<tmpl::list<PrimalFields...>> zero_primal_vars{num_points,
1061 : 0.};
1062 : Variables<tmpl::list<PrimalFluxes...>> primal_fluxes_buffer{num_points};
1063 : Variables<tmpl::list<
1064 : ::Tags::deriv<PrimalFields, tmpl::size_t<Dim>, Frame::Inertial>...>>
1065 : zero_deriv_vars{num_points, 0.};
1066 : Variables<tmpl::list<FixedSourcesTags...>> operator_applied_to_zero_vars{
1067 : num_points};
1068 : // Set up data on mortars
1069 : ::dg::MortarMap<Dim, MortarData<size_t, tmpl::list<PrimalFields...>,
1070 : tmpl::list<PrimalFluxes...>>>
1071 : all_mortar_data{};
1072 : constexpr size_t temporal_id = std::numeric_limits<size_t>::max();
1073 : // Apply the operator to the zero variables, skipping internal boundaries
1074 : prepare_mortar_data<true>(
1075 : make_not_null(&zero_deriv_vars), make_not_null(&primal_fluxes_buffer),
1076 : make_not_null(&all_mortar_data), zero_primal_vars, element, mesh,
1077 : inv_jacobian, face_normals, all_mortar_meshes, all_mortar_sizes,
1078 : temporal_id, apply_boundary_condition, fluxes_args);
1079 : // Modify internal boundary data if needed, e.g. to transform from one
1080 : // variable to another when crossing element boundaries. This is a nonlinear
1081 : // operation in the sense that feeding zero through the operator is nonzero,
1082 : // so we evaluate it here and add the contribution to the fixed sources,
1083 : // just like inhomogeneous external boundary conditions.
1084 : if constexpr (not std::is_same_v<typename System::modify_boundary_data,
1085 : void>) {
1086 : for (const auto& [direction, neighbors] : element.neighbors()) {
1087 : for (const auto& neighbor_id : neighbors) {
1088 : const ::dg::MortarId<Dim> mortar_id{direction, neighbor_id};
1089 : ASSERT(not all_mortar_data.contains(mortar_id),
1090 : "Mortar data with ID " << mortar_id << " already exists.");
1091 : auto& mortar_data = all_mortar_data[mortar_id];
1092 : // Manufacture zero mortar data, store as local data, then treat as
1093 : // received from neighbor and apply modifications
1094 : BoundaryData<tmpl::list<PrimalFields...>, tmpl::list<PrimalFluxes...>>
1095 : remote_boundary_data_on_mortar{};
1096 : remote_boundary_data_on_mortar.field_data.initialize(
1097 : all_mortar_meshes.at(mortar_id).number_of_grid_points(), 0.);
1098 : mortar_data.local_insert(temporal_id, remote_boundary_data_on_mortar);
1099 : // Modify "received" mortar data
1100 : std::apply(
1101 : [&remote_boundary_data_on_mortar,
1102 : &mortar_id](const auto&... args) {
1103 : System::modify_boundary_data::apply(
1104 : make_not_null(&get<PrimalFields>(
1105 : remote_boundary_data_on_mortar.field_data))...,
1106 : make_not_null(&get<::Tags::NormalDotFlux<PrimalFields>>(
1107 : remote_boundary_data_on_mortar.field_data))...,
1108 : mortar_id, args...);
1109 : },
1110 : modify_boundary_data_args);
1111 : // Insert as remote mortar data
1112 : mortar_data.remote_insert(temporal_id,
1113 : std::move(remote_boundary_data_on_mortar));
1114 : }
1115 : }
1116 : }
1117 : apply_operator(
1118 : make_not_null(&operator_applied_to_zero_vars),
1119 : make_not_null(&all_mortar_data), zero_primal_vars, zero_deriv_vars,
1120 : primal_fluxes_buffer, element, mesh, inv_jacobian, det_inv_jacobian,
1121 : det_jacobian, det_times_inv_jacobian, face_normals, face_normal_vectors,
1122 : face_normal_magnitudes, face_jacobians,
1123 : face_jacobian_times_inv_jacobians, all_mortar_meshes, all_mortar_sizes,
1124 : mortar_jacobians, penalty_factors, massive, formulation, temporal_id,
1125 : fluxes_args_on_faces, sources_args, modify_boundary_data_args);
1126 : // Impose the nonlinear (constant) boundary contribution as fixed sources on
1127 : // the RHS of the equations
1128 : *fixed_sources -= operator_applied_to_zero_vars;
1129 : }
1130 : };
1131 :
1132 : } // namespace detail
1133 :
1134 : /*!
1135 : * \brief Prepare data on mortars so they can be communicated to neighbors
1136 : *
1137 : * Call this function on all elements and communicate the mortar data, then call
1138 : * `elliptic::dg::apply_operator`.
1139 : */
1140 : template <typename System, bool Linearized, typename... Args>
1141 1 : void prepare_mortar_data(Args&&... args) {
1142 : detail::DgOperatorImpl<System, Linearized>::template prepare_mortar_data<
1143 : false>(std::forward<Args>(args)...);
1144 : }
1145 :
1146 : /*!
1147 : * \brief Apply the elliptic DG operator
1148 : *
1149 : * This function applies the elliptic DG operator on an element, assuming all
1150 : * data on mortars is already available. Use the
1151 : * `elliptic::dg::prepare_mortar_data` function to prepare mortar data on
1152 : * neighboring elements, then communicate the data and insert them on the
1153 : * "remote" side of the mortars before calling this function.
1154 : */
1155 : template <typename System, bool Linearized, typename... Args>
1156 1 : void apply_operator(Args&&... args) {
1157 : detail::DgOperatorImpl<System, Linearized>::apply_operator(
1158 : std::forward<Args>(args)...);
1159 : }
1160 :
1161 : /*!
1162 : * \brief For linear systems, impose inhomogeneous boundary conditions as
1163 : * contributions to the fixed sources (i.e. the RHS of the equations).
1164 : *
1165 : * This function exists because the DG operator must typically be linear, but
1166 : * even for linear elliptic equations we typically apply boundary conditions
1167 : * with a constant, and therefore nonlinear, contribution. Standard examples are
1168 : * inhomogeneous (i.e. non-zero) Dirichlet or Neumann boundary conditions. This
1169 : * nonlinear contribution can be added to the fixed sources, leaving only the
1170 : * linearized boundary conditions in the DG operator. For standard constant
1171 : * Dirichlet or Neumann boundary conditions the linearization is of course just
1172 : * zero.
1173 : *
1174 : * This function essentially feeds zero variables through the nonlinear operator
1175 : * and subtracts the result from the fixed sources: `b -= A(x=0)`.
1176 : */
1177 : template <typename System, typename... Args>
1178 1 : void impose_inhomogeneous_boundary_conditions_on_source(Args&&... args) {
1179 : detail::DgOperatorImpl<System, false>::
1180 : impose_inhomogeneous_boundary_conditions_on_source(
1181 : std::forward<Args>(args)...);
1182 : }
1183 :
1184 : } // namespace elliptic::dg
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