1 // MFEM Example 10
2 //
3 // Compile with: make ex10
4 //
5 // Sample runs:
6 // ex10 -m ../data/beam-quad.mesh -s 3 -r 2 -o 2 -dt 3
7 // ex10 -m ../data/beam-tri.mesh -s 3 -r 2 -o 2 -dt 3
8 // ex10 -m ../data/beam-hex.mesh -s 2 -r 1 -o 2 -dt 3
9 // ex10 -m ../data/beam-tet.mesh -s 2 -r 1 -o 2 -dt 3
10 // ex10 -m ../data/beam-wedge.mesh -s 2 -r 1 -o 2 -dt 3
11 // ex10 -m ../data/beam-quad.mesh -s 14 -r 2 -o 2 -dt 0.03 -vs 20
12 // ex10 -m ../data/beam-hex.mesh -s 14 -r 1 -o 2 -dt 0.05 -vs 20
13 // ex10 -m ../data/beam-quad-amr.mesh -s 3 -r 2 -o 2 -dt 3
14 //
15 // Description: This examples solves a time dependent nonlinear elasticity
16 // problem of the form dv/dt = H(x) + S v, dx/dt = v, where H is a
17 // hyperelastic model and S is a viscosity operator of Laplacian
18 // type. The geometry of the domain is assumed to be as follows:
19 //
20 // +---------------------+
21 // boundary --->| |
22 // attribute 1 | |
23 // (fixed) +---------------------+
24 //
25 // The example demonstrates the use of nonlinear operators (the
26 // class HyperelasticOperator defining H(x)), as well as their
27 // implicit time integration using a Newton method for solving an
28 // associated reduced backward-Euler type nonlinear equation
29 // (class ReducedSystemOperator). Each Newton step requires the
30 // inversion of a Jacobian matrix, which is done through a
31 // (preconditioned) inner solver. Note that implementing the
32 // method HyperelasticOperator::ImplicitSolve is the only
33 // requirement for high-order implicit (SDIRK) time integration.
34 //
35 // We recommend viewing examples 2 and 9 before viewing this
36 // example.
37
38 #include "mfem.hpp"
39 #include <memory>
40 #include <iostream>
41 #include <fstream>
42
43 using namespace std;
44 using namespace mfem;
45
46 class ReducedSystemOperator;
47
48 /** After spatial discretization, the hyperelastic model can be written as a
49 * system of ODEs:
50 * dv/dt = -M^{-1}*(H(x) + S*v)
51 * dx/dt = v,
52 * where x is the vector representing the deformation, v is the velocity field,
53 * M is the mass matrix, S is the viscosity matrix, and H(x) is the nonlinear
54 * hyperelastic operator.
55 *
56 * Class HyperelasticOperator represents the right-hand side of the above
57 * system of ODEs. */
58 class HyperelasticOperator : public TimeDependentOperator
59 {
60 protected:
61 FiniteElementSpace &fespace;
62
63 BilinearForm M, S;
64 NonlinearForm H;
65 double viscosity;
66 HyperelasticModel *model;
67
68 CGSolver M_solver; // Krylov solver for inverting the mass matrix M
69 DSmoother M_prec; // Preconditioner for the mass matrix M
70
71 /** Nonlinear operator defining the reduced backward Euler equation for the
72 velocity. Used in the implementation of method ImplicitSolve. */
73 ReducedSystemOperator *reduced_oper;
74
75 /// Newton solver for the reduced backward Euler equation
76 NewtonSolver newton_solver;
77
78 /// Solver for the Jacobian solve in the Newton method
79 Solver *J_solver;
80 /// Preconditioner for the Jacobian solve in the Newton method
81 Solver *J_prec;
82
83 mutable Vector z; // auxiliary vector
84
85 public:
86 HyperelasticOperator(FiniteElementSpace &f, Array<int> &ess_bdr,
87 double visc, double mu, double K);
88
89 /// Compute the right-hand side of the ODE system.
90 virtual void Mult(const Vector &vx, Vector &dvx_dt) const;
91 /** Solve the Backward-Euler equation: k = f(x + dt*k, t), for the unknown k.
92 This is the only requirement for high-order SDIRK implicit integration.*/
93 virtual void ImplicitSolve(const double dt, const Vector &x, Vector &k);
94
95 double ElasticEnergy(const Vector &x) const;
96 double KineticEnergy(const Vector &v) const;
97 void GetElasticEnergyDensity(const GridFunction &x, GridFunction &w) const;
98
99 virtual ~HyperelasticOperator();
100 };
101
102 /** Nonlinear operator of the form:
103 k --> (M + dt*S)*k + H(x + dt*v + dt^2*k) + S*v,
104 where M and S are given BilinearForms, H is a given NonlinearForm, v and x
105 are given vectors, and dt is a scalar. */
106 class ReducedSystemOperator : public Operator
107 {
108 private:
109 BilinearForm *M, *S;
110 NonlinearForm *H;
111 mutable SparseMatrix *Jacobian;
112 double dt;
113 const Vector *v, *x;
114 mutable Vector w, z;
115
116 public:
117 ReducedSystemOperator(BilinearForm *M_, BilinearForm *S_, NonlinearForm *H_);
118
119 /// Set current dt, v, x values - needed to compute action and Jacobian.
120 void SetParameters(double dt_, const Vector *v_, const Vector *x_);
121
122 /// Compute y = H(x + dt (v + dt k)) + M k + S (v + dt k).
123 virtual void Mult(const Vector &k, Vector &y) const;
124
125 /// Compute J = M + dt S + dt^2 grad_H(x + dt (v + dt k)).
126 virtual Operator &GetGradient(const Vector &k) const;
127
128 virtual ~ReducedSystemOperator();
129 };
130
131
132 /** Function representing the elastic energy density for the given hyperelastic
133 model+deformation. Used in HyperelasticOperator::GetElasticEnergyDensity. */
134 class ElasticEnergyCoefficient : public Coefficient
135 {
136 private:
137 HyperelasticModel &model;
138 const GridFunction &x;
139 DenseMatrix J;
140
141 public:
ElasticEnergyCoefficient(HyperelasticModel & m,const GridFunction & x_)142 ElasticEnergyCoefficient(HyperelasticModel &m, const GridFunction &x_)
143 : model(m), x(x_) { }
144 virtual double Eval(ElementTransformation &T, const IntegrationPoint &ip);
~ElasticEnergyCoefficient()145 virtual ~ElasticEnergyCoefficient() { }
146 };
147
148 void InitialDeformation(const Vector &x, Vector &y);
149
150 void InitialVelocity(const Vector &x, Vector &v);
151
152 void visualize(ostream &out, Mesh *mesh, GridFunction *deformed_nodes,
153 GridFunction *field, const char *field_name = NULL,
154 bool init_vis = false);
155
156
main(int argc,char * argv[])157 int main(int argc, char *argv[])
158 {
159 // 1. Parse command-line options.
160 const char *mesh_file = "../data/beam-quad.mesh";
161 int ref_levels = 2;
162 int order = 2;
163 int ode_solver_type = 3;
164 double t_final = 300.0;
165 double dt = 3.0;
166 double visc = 1e-2;
167 double mu = 0.25;
168 double K = 5.0;
169 bool visualization = true;
170 int vis_steps = 1;
171
172 OptionsParser args(argc, argv);
173 args.AddOption(&mesh_file, "-m", "--mesh",
174 "Mesh file to use.");
175 args.AddOption(&ref_levels, "-r", "--refine",
176 "Number of times to refine the mesh uniformly.");
177 args.AddOption(&order, "-o", "--order",
178 "Order (degree) of the finite elements.");
179 args.AddOption(&ode_solver_type, "-s", "--ode-solver",
180 "ODE solver: 1 - Backward Euler, 2 - SDIRK2, 3 - SDIRK3,\n\t"
181 " 11 - Forward Euler, 12 - RK2,\n\t"
182 " 13 - RK3 SSP, 14 - RK4."
183 " 22 - Implicit Midpoint Method,\n\t"
184 " 23 - SDIRK23 (A-stable), 24 - SDIRK34");
185 args.AddOption(&t_final, "-tf", "--t-final",
186 "Final time; start time is 0.");
187 args.AddOption(&dt, "-dt", "--time-step",
188 "Time step.");
189 args.AddOption(&visc, "-v", "--viscosity",
190 "Viscosity coefficient.");
191 args.AddOption(&mu, "-mu", "--shear-modulus",
192 "Shear modulus in the Neo-Hookean hyperelastic model.");
193 args.AddOption(&K, "-K", "--bulk-modulus",
194 "Bulk modulus in the Neo-Hookean hyperelastic model.");
195 args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
196 "--no-visualization",
197 "Enable or disable GLVis visualization.");
198 args.AddOption(&vis_steps, "-vs", "--visualization-steps",
199 "Visualize every n-th timestep.");
200 args.Parse();
201 if (!args.Good())
202 {
203 args.PrintUsage(cout);
204 return 1;
205 }
206 args.PrintOptions(cout);
207
208 // 2. Read the mesh from the given mesh file. We can handle triangular,
209 // quadrilateral, tetrahedral and hexahedral meshes with the same code.
210 Mesh *mesh = new Mesh(mesh_file, 1, 1);
211 int dim = mesh->Dimension();
212
213 // 3. Define the ODE solver used for time integration. Several implicit
214 // singly diagonal implicit Runge-Kutta (SDIRK) methods, as well as
215 // explicit Runge-Kutta methods are available.
216 ODESolver *ode_solver;
217 switch (ode_solver_type)
218 {
219 // Implicit L-stable methods
220 case 1: ode_solver = new BackwardEulerSolver; break;
221 case 2: ode_solver = new SDIRK23Solver(2); break;
222 case 3: ode_solver = new SDIRK33Solver; break;
223 // Explicit methods
224 case 11: ode_solver = new ForwardEulerSolver; break;
225 case 12: ode_solver = new RK2Solver(0.5); break; // midpoint method
226 case 13: ode_solver = new RK3SSPSolver; break;
227 case 14: ode_solver = new RK4Solver; break;
228 case 15: ode_solver = new GeneralizedAlphaSolver(0.5); break;
229 // Implicit A-stable methods (not L-stable)
230 case 22: ode_solver = new ImplicitMidpointSolver; break;
231 case 23: ode_solver = new SDIRK23Solver; break;
232 case 24: ode_solver = new SDIRK34Solver; break;
233 default:
234 cout << "Unknown ODE solver type: " << ode_solver_type << '\n';
235 delete mesh;
236 return 3;
237 }
238
239 // 4. Refine the mesh to increase the resolution. In this example we do
240 // 'ref_levels' of uniform refinement, where 'ref_levels' is a
241 // command-line parameter.
242 for (int lev = 0; lev < ref_levels; lev++)
243 {
244 mesh->UniformRefinement();
245 }
246
247 // 5. Define the vector finite element spaces representing the mesh
248 // deformation x, the velocity v, and the initial configuration, x_ref.
249 // Define also the elastic energy density, w, which is in a discontinuous
250 // higher-order space. Since x and v are integrated in time as a system,
251 // we group them together in block vector vx, with offsets given by the
252 // fe_offset array.
253 H1_FECollection fe_coll(order, dim);
254 FiniteElementSpace fespace(mesh, &fe_coll, dim);
255
256 int fe_size = fespace.GetTrueVSize();
257 cout << "Number of velocity/deformation unknowns: " << fe_size << endl;
258 Array<int> fe_offset(3);
259 fe_offset[0] = 0;
260 fe_offset[1] = fe_size;
261 fe_offset[2] = 2*fe_size;
262
263 BlockVector vx(fe_offset);
264 GridFunction v, x;
265 v.MakeTRef(&fespace, vx.GetBlock(0), 0);
266 x.MakeTRef(&fespace, vx.GetBlock(1), 0);
267
268 GridFunction x_ref(&fespace);
269 mesh->GetNodes(x_ref);
270
271 L2_FECollection w_fec(order + 1, dim);
272 FiniteElementSpace w_fespace(mesh, &w_fec);
273 GridFunction w(&w_fespace);
274
275 // 6. Set the initial conditions for v and x, and the boundary conditions on
276 // a beam-like mesh (see description above).
277 VectorFunctionCoefficient velo(dim, InitialVelocity);
278 v.ProjectCoefficient(velo);
279 v.SetTrueVector();
280 VectorFunctionCoefficient deform(dim, InitialDeformation);
281 x.ProjectCoefficient(deform);
282 x.SetTrueVector();
283
284 Array<int> ess_bdr(fespace.GetMesh()->bdr_attributes.Max());
285 ess_bdr = 0;
286 ess_bdr[0] = 1; // boundary attribute 1 (index 0) is fixed
287
288 // 7. Initialize the hyperelastic operator, the GLVis visualization and print
289 // the initial energies.
290 HyperelasticOperator oper(fespace, ess_bdr, visc, mu, K);
291
292 socketstream vis_v, vis_w;
293 if (visualization)
294 {
295 char vishost[] = "localhost";
296 int visport = 19916;
297 vis_v.open(vishost, visport);
298 vis_v.precision(8);
299 v.SetFromTrueVector(); x.SetFromTrueVector();
300 visualize(vis_v, mesh, &x, &v, "Velocity", true);
301 vis_w.open(vishost, visport);
302 if (vis_w)
303 {
304 oper.GetElasticEnergyDensity(x, w);
305 vis_w.precision(8);
306 visualize(vis_w, mesh, &x, &w, "Elastic energy density", true);
307 }
308 }
309
310 double ee0 = oper.ElasticEnergy(x.GetTrueVector());
311 double ke0 = oper.KineticEnergy(v.GetTrueVector());
312 cout << "initial elastic energy (EE) = " << ee0 << endl;
313 cout << "initial kinetic energy (KE) = " << ke0 << endl;
314 cout << "initial total energy (TE) = " << (ee0 + ke0) << endl;
315
316 double t = 0.0;
317 oper.SetTime(t);
318 ode_solver->Init(oper);
319
320 // 8. Perform time-integration (looping over the time iterations, ti, with a
321 // time-step dt).
322 bool last_step = false;
323 for (int ti = 1; !last_step; ti++)
324 {
325 double dt_real = min(dt, t_final - t);
326
327 ode_solver->Step(vx, t, dt_real);
328
329 last_step = (t >= t_final - 1e-8*dt);
330
331 if (last_step || (ti % vis_steps) == 0)
332 {
333 double ee = oper.ElasticEnergy(x.GetTrueVector());
334 double ke = oper.KineticEnergy(v.GetTrueVector());
335
336 cout << "step " << ti << ", t = " << t << ", EE = " << ee << ", KE = "
337 << ke << ", ΔTE = " << (ee+ke)-(ee0+ke0) << endl;
338
339 if (visualization)
340 {
341 v.SetFromTrueVector(); x.SetFromTrueVector();
342 visualize(vis_v, mesh, &x, &v);
343 if (vis_w)
344 {
345 oper.GetElasticEnergyDensity(x, w);
346 visualize(vis_w, mesh, &x, &w);
347 }
348 }
349 }
350 }
351
352 // 9. Save the displaced mesh, the velocity and elastic energy.
353 {
354 v.SetFromTrueVector(); x.SetFromTrueVector();
355 GridFunction *nodes = &x;
356 int owns_nodes = 0;
357 mesh->SwapNodes(nodes, owns_nodes);
358 ofstream mesh_ofs("deformed.mesh");
359 mesh_ofs.precision(8);
360 mesh->Print(mesh_ofs);
361 mesh->SwapNodes(nodes, owns_nodes);
362 ofstream velo_ofs("velocity.sol");
363 velo_ofs.precision(8);
364 v.Save(velo_ofs);
365 ofstream ee_ofs("elastic_energy.sol");
366 ee_ofs.precision(8);
367 oper.GetElasticEnergyDensity(x, w);
368 w.Save(ee_ofs);
369 }
370
371 // 10. Free the used memory.
372 delete ode_solver;
373 delete mesh;
374
375 return 0;
376 }
377
378
visualize(ostream & out,Mesh * mesh,GridFunction * deformed_nodes,GridFunction * field,const char * field_name,bool init_vis)379 void visualize(ostream &out, Mesh *mesh, GridFunction *deformed_nodes,
380 GridFunction *field, const char *field_name, bool init_vis)
381 {
382 if (!out)
383 {
384 return;
385 }
386
387 GridFunction *nodes = deformed_nodes;
388 int owns_nodes = 0;
389
390 mesh->SwapNodes(nodes, owns_nodes);
391
392 out << "solution\n" << *mesh << *field;
393
394 mesh->SwapNodes(nodes, owns_nodes);
395
396 if (init_vis)
397 {
398 out << "window_size 800 800\n";
399 out << "window_title '" << field_name << "'\n";
400 if (mesh->SpaceDimension() == 2)
401 {
402 out << "view 0 0\n"; // view from top
403 out << "keys jl\n"; // turn off perspective and light
404 }
405 out << "keys cm\n"; // show colorbar and mesh
406 out << "autoscale value\n"; // update value-range; keep mesh-extents fixed
407 out << "pause\n";
408 }
409 out << flush;
410 }
411
412
ReducedSystemOperator(BilinearForm * M_,BilinearForm * S_,NonlinearForm * H_)413 ReducedSystemOperator::ReducedSystemOperator(
414 BilinearForm *M_, BilinearForm *S_, NonlinearForm *H_)
415 : Operator(M_->Height()), M(M_), S(S_), H(H_), Jacobian(NULL),
416 dt(0.0), v(NULL), x(NULL), w(height), z(height)
417 { }
418
SetParameters(double dt_,const Vector * v_,const Vector * x_)419 void ReducedSystemOperator::SetParameters(double dt_, const Vector *v_,
420 const Vector *x_)
421 {
422 dt = dt_; v = v_; x = x_;
423 }
424
Mult(const Vector & k,Vector & y) const425 void ReducedSystemOperator::Mult(const Vector &k, Vector &y) const
426 {
427 // compute: y = H(x + dt*(v + dt*k)) + M*k + S*(v + dt*k)
428 add(*v, dt, k, w);
429 add(*x, dt, w, z);
430 H->Mult(z, y);
431 M->AddMult(k, y);
432 S->AddMult(w, y);
433 }
434
GetGradient(const Vector & k) const435 Operator &ReducedSystemOperator::GetGradient(const Vector &k) const
436 {
437 delete Jacobian;
438 Jacobian = Add(1.0, M->SpMat(), dt, S->SpMat());
439 add(*v, dt, k, w);
440 add(*x, dt, w, z);
441 SparseMatrix *grad_H = dynamic_cast<SparseMatrix *>(&H->GetGradient(z));
442 Jacobian->Add(dt*dt, *grad_H);
443 return *Jacobian;
444 }
445
~ReducedSystemOperator()446 ReducedSystemOperator::~ReducedSystemOperator()
447 {
448 delete Jacobian;
449 }
450
451
HyperelasticOperator(FiniteElementSpace & f,Array<int> & ess_bdr,double visc,double mu,double K)452 HyperelasticOperator::HyperelasticOperator(FiniteElementSpace &f,
453 Array<int> &ess_bdr, double visc,
454 double mu, double K)
455 : TimeDependentOperator(2*f.GetTrueVSize(), 0.0), fespace(f),
456 M(&fespace), S(&fespace), H(&fespace),
457 viscosity(visc), z(height/2)
458 {
459 const double rel_tol = 1e-8;
460 const int skip_zero_entries = 0;
461
462 const double ref_density = 1.0; // density in the reference configuration
463 ConstantCoefficient rho0(ref_density);
464 M.AddDomainIntegrator(new VectorMassIntegrator(rho0));
465 M.Assemble(skip_zero_entries);
466 Array<int> ess_tdof_list;
467 fespace.GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
468 SparseMatrix tmp;
469 M.FormSystemMatrix(ess_tdof_list, tmp);
470
471 M_solver.iterative_mode = false;
472 M_solver.SetRelTol(rel_tol);
473 M_solver.SetAbsTol(0.0);
474 M_solver.SetMaxIter(30);
475 M_solver.SetPrintLevel(0);
476 M_solver.SetPreconditioner(M_prec);
477 M_solver.SetOperator(M.SpMat());
478
479 model = new NeoHookeanModel(mu, K);
480 H.AddDomainIntegrator(new HyperelasticNLFIntegrator(model));
481 H.SetEssentialTrueDofs(ess_tdof_list);
482
483 ConstantCoefficient visc_coeff(viscosity);
484 S.AddDomainIntegrator(new VectorDiffusionIntegrator(visc_coeff));
485 S.Assemble(skip_zero_entries);
486 S.FormSystemMatrix(ess_tdof_list, tmp);
487
488 reduced_oper = new ReducedSystemOperator(&M, &S, &H);
489
490 #ifndef MFEM_USE_SUITESPARSE
491 J_prec = new DSmoother(1);
492 MINRESSolver *J_minres = new MINRESSolver;
493 J_minres->SetRelTol(rel_tol);
494 J_minres->SetAbsTol(0.0);
495 J_minres->SetMaxIter(300);
496 J_minres->SetPrintLevel(-1);
497 J_minres->SetPreconditioner(*J_prec);
498 J_solver = J_minres;
499 #else
500 J_solver = new UMFPackSolver;
501 J_prec = NULL;
502 #endif
503
504 newton_solver.iterative_mode = false;
505 newton_solver.SetSolver(*J_solver);
506 newton_solver.SetOperator(*reduced_oper);
507 newton_solver.SetPrintLevel(1); // print Newton iterations
508 newton_solver.SetRelTol(rel_tol);
509 newton_solver.SetAbsTol(0.0);
510 newton_solver.SetMaxIter(10);
511 }
512
Mult(const Vector & vx,Vector & dvx_dt) const513 void HyperelasticOperator::Mult(const Vector &vx, Vector &dvx_dt) const
514 {
515 // Create views to the sub-vectors v, x of vx, and dv_dt, dx_dt of dvx_dt
516 int sc = height/2;
517 Vector v(vx.GetData() + 0, sc);
518 Vector x(vx.GetData() + sc, sc);
519 Vector dv_dt(dvx_dt.GetData() + 0, sc);
520 Vector dx_dt(dvx_dt.GetData() + sc, sc);
521
522 H.Mult(x, z);
523 if (viscosity != 0.0)
524 {
525 S.AddMult(v, z);
526 }
527 z.Neg(); // z = -z
528 M_solver.Mult(z, dv_dt);
529
530 dx_dt = v;
531 }
532
ImplicitSolve(const double dt,const Vector & vx,Vector & dvx_dt)533 void HyperelasticOperator::ImplicitSolve(const double dt,
534 const Vector &vx, Vector &dvx_dt)
535 {
536 int sc = height/2;
537 Vector v(vx.GetData() + 0, sc);
538 Vector x(vx.GetData() + sc, sc);
539 Vector dv_dt(dvx_dt.GetData() + 0, sc);
540 Vector dx_dt(dvx_dt.GetData() + sc, sc);
541
542 // By eliminating kx from the coupled system:
543 // kv = -M^{-1}*[H(x + dt*kx) + S*(v + dt*kv)]
544 // kx = v + dt*kv
545 // we reduce it to a nonlinear equation for kv, represented by the
546 // reduced_oper. This equation is solved with the newton_solver
547 // object (using J_solver and J_prec internally).
548 reduced_oper->SetParameters(dt, &v, &x);
549 Vector zero; // empty vector is interpreted as zero r.h.s. by NewtonSolver
550 newton_solver.Mult(zero, dv_dt);
551 MFEM_VERIFY(newton_solver.GetConverged(), "Newton solver did not converge.");
552 add(v, dt, dv_dt, dx_dt);
553 }
554
ElasticEnergy(const Vector & x) const555 double HyperelasticOperator::ElasticEnergy(const Vector &x) const
556 {
557 return H.GetEnergy(x);
558 }
559
KineticEnergy(const Vector & v) const560 double HyperelasticOperator::KineticEnergy(const Vector &v) const
561 {
562 return 0.5*M.InnerProduct(v, v);
563 }
564
GetElasticEnergyDensity(const GridFunction & x,GridFunction & w) const565 void HyperelasticOperator::GetElasticEnergyDensity(
566 const GridFunction &x, GridFunction &w) const
567 {
568 ElasticEnergyCoefficient w_coeff(*model, x);
569 w.ProjectCoefficient(w_coeff);
570 }
571
~HyperelasticOperator()572 HyperelasticOperator::~HyperelasticOperator()
573 {
574 delete J_solver;
575 delete J_prec;
576 delete reduced_oper;
577 delete model;
578 }
579
580
Eval(ElementTransformation & T,const IntegrationPoint & ip)581 double ElasticEnergyCoefficient::Eval(ElementTransformation &T,
582 const IntegrationPoint &ip)
583 {
584 model.SetTransformation(T);
585 x.GetVectorGradient(T, J);
586 // return model.EvalW(J); // in reference configuration
587 return model.EvalW(J)/J.Det(); // in deformed configuration
588 }
589
590
InitialDeformation(const Vector & x,Vector & y)591 void InitialDeformation(const Vector &x, Vector &y)
592 {
593 // set the initial configuration to be the same as the reference, stress
594 // free, configuration
595 y = x;
596 }
597
InitialVelocity(const Vector & x,Vector & v)598 void InitialVelocity(const Vector &x, Vector &v)
599 {
600 const int dim = x.Size();
601 const double s = 0.1/64.;
602
603 v = 0.0;
604 v(dim-1) = s*x(0)*x(0)*(8.0-x(0));
605 v(0) = -s*x(0)*x(0);
606 }
607