1 // The libMesh Finite Element Library.
2 // Copyright (C) 2002-2020 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
3
4 // This library is free software; you can redistribute it and/or
5 // modify it under the terms of the GNU Lesser General Public
6 // License as published by the Free Software Foundation; either
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12 // Lesser General Public License for more details.
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15 // License along with this library; if not, write to the Free Software
16 // Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
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18
19
20 // <h1>Systems Example 3 - Navier-Stokes with SCALAR Lagrange Multiplier</h1>
21 // \author David Knezevic
22 // \date 2010
23 //
24 // This example shows how the transient Navier-Stokes problem from
25 // systems_of_equations_ex2 can be solved using a scalar Lagrange multiplier
26 // formulation to constrain the integral of the pressure variable,
27 // rather than pinning the pressure at a single point.
28
29 // C++ include files that we need
30 #include <iostream>
31 #include <algorithm>
32 #include <sstream>
33 #include <math.h>
34
35 // Basic include file needed for the mesh functionality.
36 #include "libmesh/libmesh.h"
37 #include "libmesh/mesh.h"
38 #include "libmesh/mesh_generation.h"
39 #include "libmesh/exodusII_io.h"
40 #include "libmesh/equation_systems.h"
41 #include "libmesh/fe.h"
42 #include "libmesh/quadrature_gauss.h"
43 #include "libmesh/dof_map.h"
44 #include "libmesh/sparse_matrix.h"
45 #include "libmesh/numeric_vector.h"
46 #include "libmesh/dense_matrix.h"
47 #include "libmesh/dense_vector.h"
48 #include "libmesh/linear_implicit_system.h"
49 #include "libmesh/transient_system.h"
50 #include "libmesh/perf_log.h"
51 #include "libmesh/boundary_info.h"
52 #include "libmesh/utility.h"
53 #include "libmesh/dirichlet_boundaries.h"
54 #include "libmesh/zero_function.h"
55 #include "libmesh/const_function.h"
56 #include "libmesh/enum_solver_package.h"
57
58 // For systems of equations the DenseSubMatrix
59 // and DenseSubVector provide convenient ways for
60 // assembling the element matrix and vector on a
61 // component-by-component basis.
62 #include "libmesh/dense_submatrix.h"
63 #include "libmesh/dense_subvector.h"
64
65 // The definition of a geometric element
66 #include "libmesh/elem.h"
67
68 // Bring in everything from the libMesh namespace
69 using namespace libMesh;
70
71 // Function prototype. This function will assemble the system
72 // matrix and right-hand-side.
73 void assemble_stokes (EquationSystems & es,
74 const std::string & system_name);
75
76 // Function which sets Dirichlet BCs for the lid-driven cavity.
77 void set_lid_driven_bcs(TransientLinearImplicitSystem & system);
78
79 // The main program.
main(int argc,char ** argv)80 int main (int argc, char ** argv)
81 {
82 // Initialize libMesh.
83 LibMeshInit init (argc, argv);
84
85 // This example requires a linear solver package.
86 libmesh_example_requires(libMesh::default_solver_package() != INVALID_SOLVER_PACKAGE,
87 "--enable-petsc, --enable-trilinos, or --enable-eigen");
88
89 // Skip this 2D example if libMesh was compiled as 1D-only.
90 libmesh_example_requires(2 <= LIBMESH_DIM, "2D support");
91
92 // We use Dirichlet boundary conditions here
93 #ifndef LIBMESH_ENABLE_DIRICHLET
94 libmesh_example_requires(false, "--enable-dirichlet");
95 #endif
96
97 // This example NaNs with the Eigen sparse linear solvers and
98 // Trilinos solvers, but should work OK with either PETSc or
99 // Laspack.
100 libmesh_example_requires(libMesh::default_solver_package() != EIGEN_SOLVERS, "--enable-petsc or --enable-laspack");
101 libmesh_example_requires(libMesh::default_solver_package() != TRILINOS_SOLVERS, "--enable-petsc or --enable-laspack");
102
103 // Create a mesh, with dimension to be overridden later, distributed
104 // across the default MPI communicator.
105 Mesh mesh(init.comm());
106
107 // Use the MeshTools::Generation mesh generator to create a uniform
108 // 2D grid on the square [-1,1]^2. We instruct the mesh generator
109 // to build a mesh of 8x8 Quad9 elements in 2D. Building these
110 // higher-order elements allows us to use higher-order
111 // approximation, as in example 3.
112 MeshTools::Generation::build_square (mesh,
113 20, 20,
114 0., 1.,
115 0., 1.,
116 QUAD9);
117
118 // Print information about the mesh to the screen.
119 mesh.print_info();
120
121 // Create an equation systems object.
122 EquationSystems equation_systems (mesh);
123
124 // Declare the system and its variables.
125 // Creates a transient system named "Navier-Stokes"
126 TransientLinearImplicitSystem & system =
127 equation_systems.add_system<TransientLinearImplicitSystem> ("Navier-Stokes");
128
129 // Add the variables "vel_x" & "vel_y" to "Navier-Stokes". They
130 // will be approximated using second-order approximation.
131 system.add_variable ("vel_x", SECOND);
132 system.add_variable ("vel_y", SECOND);
133
134 // Add the variable "p" to "Navier-Stokes". This will
135 // be approximated with a first-order basis,
136 // providing an LBB-stable pressure-velocity pair.
137 system.add_variable ("p", FIRST);
138
139 // Add a scalar Lagrange multiplier to constrain the
140 // pressure to have zero mean.
141 system.add_variable ("alpha", FIRST, SCALAR);
142
143 // Give the system a pointer to the matrix assembly
144 // function.
145 system.attach_assemble_function (assemble_stokes);
146
147 // Set Dirichlet boundary conditions.
148 set_lid_driven_bcs(system);
149
150 // Initialize the data structures for the equation system.
151 equation_systems.init ();
152
153 // Prints information about the system to the screen.
154 equation_systems.print_info();
155
156 // Create a performance-logging object for this example
157 PerfLog perf_log("Systems Example 3");
158
159 // Get a reference to the Stokes system to use later.
160 TransientLinearImplicitSystem & navier_stokes_system =
161 equation_systems.get_system<TransientLinearImplicitSystem>("Navier-Stokes");
162
163 // Now we begin the timestep loop to compute the time-accurate
164 // solution of the equations.
165 const Real dt = 0.1;
166 navier_stokes_system.time = 0.0;
167 const unsigned int n_timesteps = 15;
168
169 // The number of steps and the stopping criterion are also required
170 // for the nonlinear iterations.
171 const unsigned int n_nonlinear_steps = 15;
172 const Real nonlinear_tolerance = TOLERANCE*10;
173
174 // We also set a standard linear solver flag in the EquationSystems object
175 // which controls the maximum number of linear solver iterations allowed.
176 equation_systems.parameters.set<unsigned int>("linear solver maximum iterations") = 250;
177
178 // Tell the system of equations what the timestep is by using
179 // the set_parameter function. The matrix assembly routine can
180 // then reference this parameter.
181 equation_systems.parameters.set<Real> ("dt") = dt;
182
183 // The kinematic viscosity, nu = mu/rho, units of length**2/time.
184 equation_systems.parameters.set<Real> ("nu") = .007;
185
186 // The first thing to do is to get a copy of the solution at
187 // the current nonlinear iteration. This value will be used to
188 // determine if we can exit the nonlinear loop.
189 std::unique_ptr<NumericVector<Number>>
190 last_nonlinear_soln (navier_stokes_system.solution->clone());
191
192 #ifdef LIBMESH_HAVE_EXODUS_API
193 // Since we are not doing adaptivity, write all solutions to a single Exodus file.
194 ExodusII_IO exo_io(mesh);
195
196 // Write out the initial condition
197 exo_io.write_equation_systems ("out.e", equation_systems);
198 #endif
199
200 for (unsigned int t_step=1; t_step<=n_timesteps; ++t_step)
201 {
202 // Increment the time counter, set the time and the
203 // time step size as parameters in the EquationSystem.
204 navier_stokes_system.time += dt;
205
206 // A pretty update message
207 libMesh::out << "\n\n*** Solving time step "
208 << t_step
209 << ", time = "
210 << navier_stokes_system.time
211 << " ***"
212 << std::endl;
213
214 // Now we need to update the solution vector from the
215 // previous time step. This is done directly through
216 // the reference to the Stokes system.
217 *navier_stokes_system.old_local_solution = *navier_stokes_system.current_local_solution;
218
219 // At the beginning of each solve, reset the linear solver tolerance
220 // to a "reasonable" starting value.
221 const Real initial_linear_solver_tol = 1.e-6;
222 equation_systems.parameters.set<Real> ("linear solver tolerance") = initial_linear_solver_tol;
223
224 // We'll set this flag when convergence is (hopefully) achieved.
225 bool converged = false;
226
227 // Now we begin the nonlinear loop
228 for (unsigned int l=0; l<n_nonlinear_steps; ++l)
229 {
230 // Update the nonlinear solution.
231 last_nonlinear_soln->zero();
232 last_nonlinear_soln->add(*navier_stokes_system.solution);
233
234 // Assemble & solve the linear system.
235 perf_log.push("linear solve");
236 equation_systems.get_system("Navier-Stokes").solve();
237 perf_log.pop("linear solve");
238
239 // Compute the difference between this solution and the last
240 // nonlinear iterate.
241 last_nonlinear_soln->add (-1., *navier_stokes_system.solution);
242
243 // Close the vector before computing its norm
244 last_nonlinear_soln->close();
245
246 // Compute the l2 norm of the difference
247 const Real norm_delta = last_nonlinear_soln->l2_norm();
248
249 // How many iterations were required to solve the linear system?
250 const unsigned int n_linear_iterations = navier_stokes_system.n_linear_iterations();
251
252 // What was the final residual of the linear system?
253 const Real final_linear_residual = navier_stokes_system.final_linear_residual();
254
255 // If the solver did no work (sometimes -ksp_converged_reason
256 // says "Linear solve converged due to CONVERGED_RTOL
257 // iterations 0") but the nonlinear residual norm is above
258 // the tolerance, we need to pick an even lower linear
259 // solver tolerance and try again. Note that the tolerance
260 // is relative to the norm of the RHS, which for this
261 // particular problem does not go to zero, since we are
262 // solving for the full solution rather than the update.
263 //
264 // Similarly, if the solver did no work and this is the 0th
265 // nonlinear step, it means that the delta between solutions
266 // is being inaccurately measured as "0" since the solution
267 // did not change. Decrease the tolerance and try again.
268 if (n_linear_iterations == 0 &&
269 (navier_stokes_system.final_linear_residual() >= nonlinear_tolerance || l==0))
270 {
271 Real old_linear_solver_tolerance = equation_systems.parameters.get<Real> ("linear solver tolerance");
272 equation_systems.parameters.set<Real> ("linear solver tolerance") = 1.e-3 * old_linear_solver_tolerance;
273 continue;
274 }
275
276 // Print out convergence information for the linear and
277 // nonlinear iterations.
278 libMesh::out << "Linear solver converged at step: "
279 << n_linear_iterations
280 << ", final residual: "
281 << final_linear_residual
282 << " Nonlinear convergence: ||u - u_old|| = "
283 << norm_delta
284 << std::endl;
285
286 // Terminate the solution iteration if the difference between
287 // this nonlinear iterate and the last is sufficiently small, AND
288 // if the most recent linear system was solved to a sufficient tolerance.
289 if ((norm_delta < nonlinear_tolerance) &&
290 (navier_stokes_system.final_linear_residual() < nonlinear_tolerance))
291 {
292 libMesh::out << " Nonlinear solver converged at step "
293 << l
294 << std::endl;
295 converged = true;
296 break;
297 }
298
299 // Otherwise, decrease the linear system tolerance. For the inexact Newton
300 // method, the linear solver tolerance needs to decrease as we get closer to
301 // the solution to ensure quadratic convergence. The new linear solver tolerance
302 // is chosen (heuristically) as the square of the previous linear system residual norm.
303 //Real flr2 = final_linear_residual*final_linear_residual;
304 Real new_linear_solver_tolerance = std::min(Utility::pow<2>(final_linear_residual), initial_linear_solver_tol);
305 equation_systems.parameters.set<Real> ("linear solver tolerance") = new_linear_solver_tolerance;
306 } // end nonlinear loop
307
308 // Don't keep going if we failed to converge.
309 libmesh_error_msg_if(!converged, "Error: Newton iterations failed to converge!");
310
311 #ifdef LIBMESH_HAVE_EXODUS_API
312 // Write out every nth timestep to file.
313 const unsigned int write_interval = 1;
314
315 if ((t_step+1)%write_interval == 0)
316 {
317 exo_io.write_timestep("out.e",
318 equation_systems,
319 t_step+1, // we're off by one since we wrote the IC and the Exodus numbering is 1-based.
320 navier_stokes_system.time);
321 }
322 #endif // #ifdef LIBMESH_HAVE_EXODUS_API
323 } // end timestep loop.
324
325 // All done.
326 return 0;
327 }
328
329
330
331
332
333
334 // The matrix assembly function to be called at each time step to
335 // prepare for the linear solve.
assemble_stokes(EquationSystems & es,const std::string & libmesh_dbg_var (system_name))336 void assemble_stokes (EquationSystems & es,
337 const std::string & libmesh_dbg_var(system_name))
338 {
339 // It is a good idea to make sure we are assembling
340 // the proper system.
341 libmesh_assert_equal_to (system_name, "Navier-Stokes");
342
343 #if LIBMESH_DIM > 1
344 // Get a constant reference to the mesh object.
345 const MeshBase & mesh = es.get_mesh();
346
347 // The dimension that we are running
348 const unsigned int dim = mesh.mesh_dimension();
349
350 // Get a reference to the Stokes system object.
351 TransientLinearImplicitSystem & navier_stokes_system =
352 es.get_system<TransientLinearImplicitSystem> ("Navier-Stokes");
353
354 // Numeric ids corresponding to each variable in the system
355 const unsigned int u_var = navier_stokes_system.variable_number ("vel_x");
356 const unsigned int v_var = navier_stokes_system.variable_number ("vel_y");
357 const unsigned int p_var = navier_stokes_system.variable_number ("p");
358 const unsigned int alpha_var = navier_stokes_system.variable_number ("alpha");
359
360 // Get the Finite Element type for "vel_x". Note this will be
361 // the same as the type for "vel_y".
362 FEType fe_vel_type = navier_stokes_system.variable_type(u_var);
363
364 // Get the Finite Element type for "p".
365 FEType fe_pres_type = navier_stokes_system.variable_type(p_var);
366
367 // Build a Finite Element object of the specified type for
368 // the velocity variables.
369 std::unique_ptr<FEBase> fe_vel (FEBase::build(dim, fe_vel_type));
370
371 // Build a Finite Element object of the specified type for
372 // the pressure variables.
373 std::unique_ptr<FEBase> fe_pres (FEBase::build(dim, fe_pres_type));
374
375 // A Gauss quadrature rule for numerical integration.
376 // Let the FEType object decide what order rule is appropriate.
377 QGauss qrule (dim, fe_vel_type.default_quadrature_order());
378
379 // Tell the finite element objects to use our quadrature rule.
380 fe_vel->attach_quadrature_rule (&qrule);
381 fe_pres->attach_quadrature_rule (&qrule);
382
383 // Here we define some references to cell-specific data that
384 // will be used to assemble the linear system.
385 //
386 // The element Jacobian * quadrature weight at each integration point.
387 const std::vector<Real> & JxW = fe_vel->get_JxW();
388
389 // The element shape functions evaluated at the quadrature points.
390 const std::vector<std::vector<Real>> & phi = fe_vel->get_phi();
391
392 // The element shape function gradients for the velocity
393 // variables evaluated at the quadrature points.
394 const std::vector<std::vector<RealGradient>> & dphi = fe_vel->get_dphi();
395
396 // The element shape functions for the pressure variable
397 // evaluated at the quadrature points.
398 const std::vector<std::vector<Real>> & psi = fe_pres->get_phi();
399
400 // The value of the linear shape function gradients at the quadrature points
401 // const std::vector<std::vector<RealGradient>> & dpsi = fe_pres->get_dphi();
402
403 // A reference to the DofMap object for this system. The DofMap
404 // object handles the index translation from node and element numbers
405 // to degree of freedom numbers. We will talk more about the DofMap
406 // in future examples.
407 const DofMap & dof_map = navier_stokes_system.get_dof_map();
408
409 // Define data structures to contain the element matrix
410 // and right-hand-side vector contribution. Following
411 // basic finite element terminology we will denote these
412 // "Ke" and "Fe".
413 DenseMatrix<Number> Ke;
414 DenseVector<Number> Fe;
415
416 DenseSubMatrix<Number>
417 Kuu(Ke), Kuv(Ke), Kup(Ke),
418 Kvu(Ke), Kvv(Ke), Kvp(Ke),
419 Kpu(Ke), Kpv(Ke), Kpp(Ke);
420 DenseSubMatrix<Number> Kalpha_p(Ke), Kp_alpha(Ke);
421
422 DenseSubVector<Number>
423 Fu(Fe),
424 Fv(Fe),
425 Fp(Fe);
426
427 // This vector will hold the degree of freedom indices for
428 // the element. These define where in the global system
429 // the element degrees of freedom get mapped.
430 std::vector<dof_id_type> dof_indices;
431 std::vector<dof_id_type> dof_indices_u;
432 std::vector<dof_id_type> dof_indices_v;
433 std::vector<dof_id_type> dof_indices_p;
434 std::vector<dof_id_type> dof_indices_alpha;
435
436 // Find out what the timestep size parameter is from the system, and
437 // the value of theta for the theta method. We use implicit Euler (theta=1)
438 // for this simulation even though it is only first-order accurate in time.
439 // The reason for this decision is that the second-order Crank-Nicolson
440 // method is notoriously oscillatory for problems with discontinuous
441 // initial data such as the lid-driven cavity. Therefore,
442 // we sacrifice accuracy in time for stability, but since the solution
443 // reaches steady state relatively quickly we can afford to take small
444 // timesteps. If you monitor the initial nonlinear residual for this
445 // simulation, you should see that it is monotonically decreasing in time.
446 const Real dt = es.parameters.get<Real>("dt");
447 const Real theta = 1.;
448
449 // The kinematic viscosity, multiplies the "viscous" terms.
450 const Real nu = es.parameters.get<Real>("nu");
451
452 // Now we will loop over all the elements in the mesh that
453 // live on the local processor. We will compute the element
454 // matrix and right-hand-side contribution. Since the mesh
455 // will be refined we want to only consider the ACTIVE elements,
456 // hence we use a variant of the active_elem_iterator.
457 for (const auto & elem : mesh.active_local_element_ptr_range())
458 {
459 // Get the degree of freedom indices for the
460 // current element. These define where in the global
461 // matrix and right-hand-side this element will
462 // contribute to.
463 dof_map.dof_indices (elem, dof_indices);
464 dof_map.dof_indices (elem, dof_indices_u, u_var);
465 dof_map.dof_indices (elem, dof_indices_v, v_var);
466 dof_map.dof_indices (elem, dof_indices_p, p_var);
467 dof_map.dof_indices (elem, dof_indices_alpha, alpha_var);
468
469 const unsigned int n_dofs = dof_indices.size();
470 const unsigned int n_u_dofs = dof_indices_u.size();
471 const unsigned int n_v_dofs = dof_indices_v.size();
472 const unsigned int n_p_dofs = dof_indices_p.size();
473
474 // Compute the element-specific data for the current
475 // element. This involves computing the location of the
476 // quadrature points (q_point) and the shape functions
477 // (phi, dphi) for the current element.
478 fe_vel->reinit (elem);
479 fe_pres->reinit (elem);
480
481 // Zero the element matrix and right-hand side before
482 // summing them. We use the resize member here because
483 // the number of degrees of freedom might have changed from
484 // the last element. Note that this will be the case if the
485 // element type is different (i.e. the last element was a
486 // triangle, now we are on a quadrilateral).
487 Ke.resize (n_dofs, n_dofs);
488 Fe.resize (n_dofs);
489
490 // Reposition the submatrices... The idea is this:
491 //
492 // - - - -
493 // | Kuu Kuv Kup | | Fu |
494 // Ke = | Kvu Kvv Kvp |; Fe = | Fv |
495 // | Kpu Kpv Kpp | | Fp |
496 // - - - -
497 //
498 // The DenseSubMatrix.reposition () member takes the
499 // (row_offset, column_offset, row_size, column_size).
500 //
501 // Similarly, the DenseSubVector.reposition () member
502 // takes the (row_offset, row_size)
503 Kuu.reposition (u_var*n_u_dofs, u_var*n_u_dofs, n_u_dofs, n_u_dofs);
504 Kuv.reposition (u_var*n_u_dofs, v_var*n_u_dofs, n_u_dofs, n_v_dofs);
505 Kup.reposition (u_var*n_u_dofs, p_var*n_u_dofs, n_u_dofs, n_p_dofs);
506
507 Kvu.reposition (v_var*n_v_dofs, u_var*n_v_dofs, n_v_dofs, n_u_dofs);
508 Kvv.reposition (v_var*n_v_dofs, v_var*n_v_dofs, n_v_dofs, n_v_dofs);
509 Kvp.reposition (v_var*n_v_dofs, p_var*n_v_dofs, n_v_dofs, n_p_dofs);
510
511 Kpu.reposition (p_var*n_u_dofs, u_var*n_u_dofs, n_p_dofs, n_u_dofs);
512 Kpv.reposition (p_var*n_u_dofs, v_var*n_u_dofs, n_p_dofs, n_v_dofs);
513 Kpp.reposition (p_var*n_u_dofs, p_var*n_u_dofs, n_p_dofs, n_p_dofs);
514
515 // Also, add a row and a column to constrain the pressure
516 Kp_alpha.reposition (p_var*n_u_dofs, p_var*n_u_dofs+n_p_dofs, n_p_dofs, 1);
517 Kalpha_p.reposition (p_var*n_u_dofs+n_p_dofs, p_var*n_u_dofs, 1, n_p_dofs);
518
519
520 Fu.reposition (u_var*n_u_dofs, n_u_dofs);
521 Fv.reposition (v_var*n_u_dofs, n_v_dofs);
522 Fp.reposition (p_var*n_u_dofs, n_p_dofs);
523
524 // Now we will build the element matrix and right-hand-side.
525 // Constructing the RHS requires the solution and its
526 // gradient from the previous timestep. This must be
527 // calculated at each quadrature point by summing the
528 // solution degree-of-freedom values by the appropriate
529 // weight functions.
530 for (unsigned int qp=0; qp<qrule.n_points(); qp++)
531 {
532 // Values to hold the solution & its gradient at the previous timestep.
533 Number u = 0., u_old = 0.;
534 Number v = 0., v_old = 0.;
535 Number p_old = 0.;
536 Gradient grad_u, grad_u_old;
537 Gradient grad_v, grad_v_old;
538
539 // Compute the velocity & its gradient from the previous timestep
540 // and the old Newton iterate.
541 for (unsigned int l=0; l<n_u_dofs; l++)
542 {
543 // From the old timestep:
544 u_old += phi[l][qp]*navier_stokes_system.old_solution (dof_indices_u[l]);
545 v_old += phi[l][qp]*navier_stokes_system.old_solution (dof_indices_v[l]);
546 grad_u_old.add_scaled (dphi[l][qp], navier_stokes_system.old_solution (dof_indices_u[l]));
547 grad_v_old.add_scaled (dphi[l][qp], navier_stokes_system.old_solution (dof_indices_v[l]));
548
549 // From the previous Newton iterate:
550 u += phi[l][qp]*navier_stokes_system.current_solution (dof_indices_u[l]);
551 v += phi[l][qp]*navier_stokes_system.current_solution (dof_indices_v[l]);
552 grad_u.add_scaled (dphi[l][qp], navier_stokes_system.current_solution (dof_indices_u[l]));
553 grad_v.add_scaled (dphi[l][qp], navier_stokes_system.current_solution (dof_indices_v[l]));
554 }
555
556 // Compute the old pressure value at this quadrature point.
557 for (unsigned int l=0; l<n_p_dofs; l++)
558 p_old += psi[l][qp]*navier_stokes_system.old_solution (dof_indices_p[l]);
559
560 // Definitions for convenience. It is sometimes simpler to do a
561 // dot product if you have the full vector at your disposal.
562 const NumberVectorValue U_old (u_old, v_old);
563 const NumberVectorValue U (u, v);
564 const Number u_x = grad_u(0);
565 const Number u_y = grad_u(1);
566 const Number v_x = grad_v(0);
567 const Number v_y = grad_v(1);
568
569 // First, an i-loop over the velocity degrees of freedom.
570 // We know that n_u_dofs == n_v_dofs so we can compute contributions
571 // for both at the same time.
572 for (unsigned int i=0; i<n_u_dofs; i++)
573 {
574 Fu(i) += JxW[qp]*(u_old*phi[i][qp] - // mass-matrix term
575 (1.-theta)*dt*(U_old*grad_u_old)*phi[i][qp] + // convection term
576 (1.-theta)*dt*p_old*dphi[i][qp](0) - // pressure term on rhs
577 (1.-theta)*dt*nu*(grad_u_old*dphi[i][qp]) + // diffusion term on rhs
578 theta*dt*(U*grad_u)*phi[i][qp]); // Newton term
579
580
581 Fv(i) += JxW[qp]*(v_old*phi[i][qp] - // mass-matrix term
582 (1.-theta)*dt*(U_old*grad_v_old)*phi[i][qp] + // convection term
583 (1.-theta)*dt*p_old*dphi[i][qp](1) - // pressure term on rhs
584 (1.-theta)*dt*nu*(grad_v_old*dphi[i][qp]) + // diffusion term on rhs
585 theta*dt*(U*grad_v)*phi[i][qp]); // Newton term
586
587
588 // Note that the Fp block is identically zero unless we are using
589 // some kind of artificial compressibility scheme...
590
591 // Matrix contributions for the uu and vv couplings.
592 for (unsigned int j=0; j<n_u_dofs; j++)
593 {
594 Kuu(i,j) += JxW[qp]*(phi[i][qp]*phi[j][qp] + // mass matrix term
595 theta*dt*nu*(dphi[i][qp]*dphi[j][qp]) + // diffusion term
596 theta*dt*(U*dphi[j][qp])*phi[i][qp] + // convection term
597 theta*dt*u_x*phi[i][qp]*phi[j][qp]); // Newton term
598
599 Kuv(i,j) += JxW[qp]*theta*dt*u_y*phi[i][qp]*phi[j][qp]; // Newton term
600
601 Kvv(i,j) += JxW[qp]*(phi[i][qp]*phi[j][qp] + // mass matrix term
602 theta*dt*nu*(dphi[i][qp]*dphi[j][qp]) + // diffusion term
603 theta*dt*(U*dphi[j][qp])*phi[i][qp] + // convection term
604 theta*dt*v_y*phi[i][qp]*phi[j][qp]); // Newton term
605
606 Kvu(i,j) += JxW[qp]*theta*dt*v_x*phi[i][qp]*phi[j][qp]; // Newton term
607 }
608
609 // Matrix contributions for the up and vp couplings.
610 for (unsigned int j=0; j<n_p_dofs; j++)
611 {
612 Kup(i,j) += JxW[qp]*(-theta*dt*psi[j][qp]*dphi[i][qp](0));
613 Kvp(i,j) += JxW[qp]*(-theta*dt*psi[j][qp]*dphi[i][qp](1));
614 }
615 }
616
617 // Now an i-loop over the pressure degrees of freedom. This code computes
618 // the matrix entries due to the continuity equation. Note: To maintain a
619 // symmetric matrix, we may (or may not) multiply the continuity equation by
620 // negative one. Here we do not.
621 for (unsigned int i=0; i<n_p_dofs; i++)
622 {
623 Kp_alpha(i,0) += JxW[qp]*psi[i][qp];
624 Kalpha_p(0,i) += JxW[qp]*psi[i][qp];
625 for (unsigned int j=0; j<n_u_dofs; j++)
626 {
627 Kpu(i,j) += JxW[qp]*psi[i][qp]*dphi[j][qp](0);
628 Kpv(i,j) += JxW[qp]*psi[i][qp]*dphi[j][qp](1);
629 }
630 }
631 } // end of the quadrature point qp-loop
632
633 // Since we're using heterogeneous DirichletBoundary objects for
634 // the boundary conditions, we need to call a specific function
635 // to constrain the element stiffness matrix.
636 dof_map.heterogenously_constrain_element_matrix_and_vector (Ke, Fe, dof_indices);
637
638 // The element matrix and right-hand-side are now built
639 // for this element. Add them to the global matrix and
640 // right-hand-side vector. The SparseMatrix::add_matrix()
641 // and NumericVector::add_vector() members do this for us.
642 navier_stokes_system.matrix->add_matrix (Ke, dof_indices);
643 navier_stokes_system.rhs->add_vector (Fe, dof_indices);
644 } // end of element loop
645
646 // We can set the mean of the pressure by setting Falpha. Typically
647 // a value of zero is chosen, but the value should be arbitrary.
648 navier_stokes_system.rhs->add(navier_stokes_system.rhs->size()-1, 10.);
649 #else
650 libmesh_ignore(es);
651 #endif
652 }
653
654
655
set_lid_driven_bcs(TransientLinearImplicitSystem & system)656 void set_lid_driven_bcs(TransientLinearImplicitSystem & system)
657 {
658 #ifdef LIBMESH_ENABLE_DIRICHLET
659 unsigned short int
660 u_var = system.variable_number("vel_x"),
661 v_var = system.variable_number("vel_y");
662
663 // Get a convenient reference to the System's DofMap
664 DofMap & dof_map = system.get_dof_map();
665
666 {
667 // u=v=0 on bottom, left, right
668 std::set<boundary_id_type> boundary_ids;
669 boundary_ids.insert(0);
670 boundary_ids.insert(1);
671 boundary_ids.insert(3);
672
673 std::vector<unsigned int> variables;
674 variables.push_back(u_var);
675 variables.push_back(v_var);
676
677 dof_map.add_dirichlet_boundary(DirichletBoundary(boundary_ids,
678 variables,
679 ZeroFunction<Number>()));
680 }
681 {
682 // u=1 on top
683 std::set<boundary_id_type> boundary_ids;
684 boundary_ids.insert(2);
685
686 std::vector<unsigned int> variables;
687 variables.push_back(u_var);
688
689 dof_map.add_dirichlet_boundary(DirichletBoundary(boundary_ids,
690 variables,
691 ConstFunction<Number>(1.)));
692 }
693 {
694 // v=0 on top
695 std::set<boundary_id_type> boundary_ids;
696 boundary_ids.insert(2);
697
698 std::vector<unsigned int> variables;
699 variables.push_back(v_var);
700
701 dof_map.add_dirichlet_boundary(DirichletBoundary(boundary_ids,
702 variables,
703 ZeroFunction<Number>()));
704 }
705 #else
706 libmesh_ignore(system);
707 #endif
708 }
709