main_eigen.cpp 14.7 KB
 Pietro Incardona committed Jul 30, 2016 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 /*! * \page Stokes_1_3D Stokes incompressible 3D eigen * * # Stokes incompressible 3D # {#num_sk_inc_3D} * * In this example we try to solve the 3D stokes equation for incompressible * fluid with Reynold number = 0 * * Such equation require the inversion of a system. We will * show how to produce such system and solve it using Finite differences with * staggered grid. The system of equation to solve is the following * * * \f[ \eta \partial^{2}_{x} v_x + \eta \partial^{2}_{y} v_x + \eta \partial^{2}_{z} v_x - \partial_{x} P = 0 \f] * \f[ \eta \partial^{2}_{x} v_y + \eta \partial^{2}_{y} v_y + \eta \partial^{2}_{z} v_y - \partial_{y} P = 0 \f] * \f[ \eta \partial^{2}_{x} v_z + \eta \partial^{2}_{y} v_z + \eta \partial^{2}_{z} v_z - \partial_{y} P = 0 \f] * \f[ \partial_{x} v_x + \partial_{y} v_y + \partial_{z} v_z = 0 \f] * * \f$v_x = 0 \quad v_y = 1 \quad v_z = 1 \f$ at x = L * * \f$v_x = 0 \quad v_y = 0 \quad v_z = 0 \f$ otherwise * * ## General Model properties ## {#num_sk_inc_3D_gmp} * * In order to do this we have to define a structure that contain the main information * about the system * * \snippet Numerics/Stoke_flow/1_3D_incompressible/main_eigen.cpp def system * */ //! \cond [def system] \endcond #include "Grid/grid_dist_id.hpp" #include "Matrix/SparseMatrix.hpp" #include "Vector/Vector.hpp" #include "FiniteDifference/FDScheme.hpp" #include "FiniteDifference/util/common.hpp" #include "FiniteDifference/eq.hpp" #include "Solvers/umfpack_solver.hpp" #include "Solvers/petsc_solver.hpp" struct lid_nn { // dimensionaly of the equation ( 3D problem ...) static const unsigned int dims = 3; // number of fields in the system v_x, v_y, v_z, P so a total of 4 static const unsigned int nvar = 4; // boundary conditions PERIODIC OR NON_PERIODIC static const bool boundary[]; // type of space float, double, ... typedef float stype; // type of base grid, it is the distributed grid that will store the result // Note the first property is a 3D vector (velocity), the second is a scalar (Pressure) typedef grid_dist_id<3,float,aggregate,CartDecomposition<3,float>> b_grid; // type of SparseMatrix, for the linear system, this parameter is bounded by the solver // that you are using, in case of umfpack using it is the only possible choice // By default SparseMatrix is EIGEN based typedef SparseMatrix SparseMatrix_type; // type of Vector for the linear system, this parameter is bounded by the solver // that you are using, in case of umfpack using it is the only possible choice typedef Vector Vector_type; // Define that the underline grid where we discretize the system of equation is staggered static const int grid_type = STAGGERED_GRID; }; const bool lid_nn::boundary[] = {NON_PERIODIC,NON_PERIODIC,NON_PERIODIC}; //! \cond [def system] \endcond /*! * * \page Stokes_1_3D Stokes incompressible 3D eigen * * ## System of equation modeling ## {#num_sk_inc_3D_sem} * * We model the equations. This part will change in the near future to use template expression * parsing and simplify the process of defining equations. * * \f$\eta v_x \nabla v_x = eta\_lap\_vx \quad \nabla = \partial^{2}_{x} + \partial^{2}_{y} + \partial^{2}_{z}\f$ Step1 * * \f$\partial_{x} P = p\_x \f$ Step 2 * * \f$-p\_x = m\_ p\_ x \f$ Step 3 * * \f$eta\_lap\_vx - m\_p\_x \f$ Step4. This is the first equation in the system * * The second equation definition is similar to the first one * * \f$\partial^{forward}_{x} v_x = dx\_vx \f$ Step5 * * \f$\partial^{forward}_{y} v_y = dy\_vy \f$ Step6 * * \f$\partial^{forward}_{z} v_z = dz\_vz \f$ Step7 * * \f$dx\_vx + dy\_vy + dz_vz \f$ Step 8. This is the third equation in the system * * * \snippet Numerics/Stoke_flow/1_3D_incompressible/main_eigen.cpp def equation * */ //! \cond [def equation] \endcond // Constant Field struct eta { typedef void const_field; static float val() {return 1.0;} }; // Model the equations constexpr unsigned int v[] = {0,1,2}; constexpr unsigned int P = 3; constexpr unsigned int ic = 3; constexpr int x = 0; constexpr int y = 1; constexpr int z = 2; typedef Field v_x; typedef Field v_y; typedef Field v_z; typedef Field Prs; // Eq1 V_x typedef mul,lid_nn> eta_lap_vx; // Step1 typedef D p_x; // Step 2 typedef minus m_p_x; // Step3 typedef sum vx_eq; // Step4 // Eq2 V_y typedef mul,lid_nn> eta_lap_vy; typedef D p_y; typedef minus m_p_y; typedef sum vy_eq; // Eq3 V_z typedef mul,lid_nn> eta_lap_vz; typedef D p_z; typedef minus m_p_z; typedef sum vz_eq; // Eq4 Incompressibility typedef D dx_vx; // Step5 typedef D dy_vy; // Step6 typedef D dz_vz; // Step 7 typedef sum ic_eq; // Step8 //! \cond [def equation] \endcond /*! * \page Stokes_1_3D Stokes incompressible 3D eigen * * In case of boundary conditions and staggered grid we need a particular set of equations * at boundaries. Explain in detail is out of the scope of this example, but to qualitatively * have an idea consider the following staggered cell * \verbatim +--$--+ | | # * # | | 0--$--+ # = velocity(x) $= velocity(y) * = pressure \endverbatim * * As we can see several properties has different position in the cell. * Consider we want to impose \f$v_y = 0\f$on \f$x=0\f$. Because there are not * points at \f$x = 0\f$we have to interpolate between$ of this cell * and $of the previous cell on y Averaging using the Avg operator. The same apply for * \f$v_x\f$on \f$y=0\f$. Similar arguments can be done for other boundaries in * order to finally reach a well defined system. Such observation can be extended In 3D * leading to the following equations * * \snippet Numerics/Stoke_flow/1_3D_incompressible/main_eigen.cpp bond def eq * */ //! \cond [bond def eq] \endcond // Directional Avg typedef Avg avg_x_vy; typedef Avg avg_z_vy; typedef Avg avg_y_vx; typedef Avg avg_z_vx; typedef Avg avg_y_vz; typedef Avg avg_x_vz; typedef Avg avg_x_vy_f; typedef Avg avg_z_vy_f; typedef Avg avg_y_vx_f; typedef Avg avg_z_vx_f; typedef Avg avg_y_vz_f; typedef Avg avg_x_vz_f; #define EQ_1 0 #define EQ_2 1 #define EQ_3 2 #define EQ_4 3 //! \cond [bond def eq] \endcond #include "Vector/vector_dist.hpp" #include "data_type/aggregate.hpp" int main(int argc, char* argv[]) { /*! * \page Stokes_1_3D Stokes incompressible 3D eigen * * ## Initialization ## {#num_sk_inc_3D_init} * * After model our equation we: * * Initialize the library * * Define some useful constants * * define Ghost size * * Non-periodic boundary conditions * * Padding domain expansion * * Padding and Ghost differ in the fact the padding extend the domain. * Ghost is an extension for each sub-domain * * \snippet Numerics/Stoke_flow/0_2D_incompressible/main_eigen.cpp init * */ //! \cond [init] \endcond // Initialize openfpm_init(&argc,&argv); // velocity in the grid is the property 0, pressure is the property 1 constexpr int velocity = 0; constexpr int pressure = 1; // Domain Box<3,float> domain({0.0,0.0,0.0},{3.0,1.0,1.0}); // Ghost (Not important in this case but required) Ghost<3,float> g(0.01); // Grid points on x=36 and y=12 z=12 long int sz[] = {36,12,12}; size_t szu[3]; szu[0] = (size_t)sz[0]; szu[1] = (size_t)sz[1]; szu[2] = (size_t)sz[2]; // We need one more point on the left and down part of the domain // This is given by the boundary conditions that we impose. // Padding<3> pd({1,1,1},{0,0,0}); //! \cond [init] \endcond /*! * \page Stokes_1_3D Stokes incompressible 3D eigen * * Distributed grid that store the solution * * \see \ref e0_s_grid_inst * * \snippet Numerics/Stoke_flow/1_3D_incompressible/main_eigen.cpp grid inst * */ //! \cond [grid inst] \endcond grid_dist_id<3,float,aggregate> g_dist(szu,domain,g); //! \cond [grid inst] \endcond /*! * \page Stokes_1_3D Stokes incompressible 3D eigen * * Solving the system above require the solution of a system like that * * \f$ Ax = b \quad x = A^{-1}b\f\$ * * where A is the system the discretize the left hand side of the equations + boundary conditions * and b discretize the right hand size + boundary conditions * * FDScheme is the object that we use to produce the Matrix A and the vector b. * Such object require the maximum extension of the stencil * * \snippet Numerics/Stoke_flow/1_3D_incompressible/main_eigen.cpp fd scheme * */ //! \cond [fd scheme] \endcond // It is the maximum extension of the stencil (order 2 laplacian stencil has extension 1) Ghost<3,long int> stencil_max(1); // Finite difference scheme  incardon committed Jul 11, 2017 321  FDScheme fd(pd, stencil_max, domain, g_dist);  Pietro Incardona committed Jul 30, 2016 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492  //! \cond [fd scheme] \endcond /*! * \page Stokes_1_3D Stokes incompressible 3D eigen * * ## Impose the equation on the domain ## {#num_sk_inc_3D_ied} * * Here we impose the system of equation, we start from the incompressibility Eq imposed in the bulk with the * exception of the first point {0,0} and than we set P = 0 in {0,0}, why we are doing this is again * mathematical to have a well defined system, an intuitive explanation is that P and P + c are both * solution for the incompressibility equation, this produce an ill-posed problem to make it well posed * we set one point in this case {0,0} the pressure to a fixed constant for convenience P = 0 * * The best way to understand what we are doing is to draw a smaller example like 8x8. * Considering that we have one additional point on the left for padding we have a grid * 9x9. If on each point we have v_x v_y and P unknown we have * 9x9x3 = 243 unknown. In order to fully determine and unique solution we have to * impose 243 condition. The code under impose (in the case of 9x9) between domain * and bulk 243 conditions. * * \snippet Numerics/Stoke_flow/1_3D_incompressible/main_eigen.cpp impose eq dom * * */ //! \cond [impose eq dom] \endcond // start and end of the bulk fd.impose(ic_eq(),0.0, EQ_4, {0,0,0},{sz[0]-2,sz[1]-2,sz[2]-2},true); fd.impose(Prs(), 0.0, EQ_4, {0,0,0},{0,0,0}); fd.impose(vx_eq(),0.0, EQ_1, {1,0},{sz[0]-2,sz[1]-2,sz[2]-2}); fd.impose(vy_eq(),0.0, EQ_2, {0,1},{sz[0]-2,sz[1]-2,sz[2]-2}); fd.impose(vz_eq(),0.0, EQ_3, {0,0,1},{sz[0]-2,sz[1]-2,sz[2]-2}); // v_x // R L (Right,Left) fd.impose(v_x(),0.0, EQ_1, {0,0,0}, {0,sz[1]-2,sz[2]-2}); fd.impose(v_x(),0.0, EQ_1, {sz[0]-1,0,0},{sz[0]-1,sz[1]-2,sz[2]-2}); // T B (Top,Bottom) fd.impose(avg_y_vx_f(),0.0, EQ_1, {0,-1,0}, {sz[0]-1,-1,sz[2]-2}); fd.impose(avg_y_vx(),0.0, EQ_1, {0,sz[1]-1,0},{sz[0]-1,sz[1]-1,sz[2]-2}); // A F (Forward,Backward) fd.impose(avg_z_vx_f(),0.0, EQ_1, {0,-1,-1}, {sz[0]-1,sz[1]-1,-1}); fd.impose(avg_z_vx(),0.0, EQ_1, {0,-1,sz[2]-1},{sz[0]-1,sz[1]-1,sz[2]-1}); // v_y // R L fd.impose(avg_x_vy_f(),0.0, EQ_2, {-1,0,0}, {-1,sz[1]-1,sz[2]-2}); fd.impose(avg_x_vy(),1.0, EQ_2, {sz[0]-1,0,0},{sz[0]-1,sz[1]-1,sz[2]-2}); // T B fd.impose(v_y(), 0.0, EQ_2, {0,0,0}, {sz[0]-2,0,sz[2]-2}); fd.impose(v_y(), 0.0, EQ_2, {0,sz[1]-1,0},{sz[0]-2,sz[1]-1,sz[2]-2}); // F A fd.impose(avg_z_vy(),0.0, EQ_2, {-1,0,sz[2]-1}, {sz[0]-1,sz[1]-1,sz[2]-1}); fd.impose(avg_z_vy_f(),0.0, EQ_2, {-1,0,-1}, {sz[0]-1,sz[1]-1,-1}); // v_z // R L fd.impose(avg_x_vz_f(),0.0, EQ_3, {-1,0,0}, {-1,sz[1]-2,sz[2]-1}); fd.impose(avg_x_vz(),1.0, EQ_3, {sz[0]-1,0,0},{sz[0]-1,sz[1]-2,sz[2]-1}); // T B fd.impose(avg_y_vz(),0.0, EQ_3, {-1,sz[1]-1,0},{sz[0]-1,sz[1]-1,sz[2]-1}); fd.impose(avg_y_vz_f(),0.0, EQ_3, {-1,-1,0}, {sz[0]-1,-1,sz[2]-1}); // F A fd.impose(v_z(),0.0, EQ_3, {0,0,0}, {sz[0]-2,sz[1]-2,0}); fd.impose(v_z(),0.0, EQ_3, {0,0,sz[2]-1},{sz[0]-2,sz[1]-2,sz[2]-1}); // When we pad the grid, there are points of the grid that are not // touched by the previous condition. Mathematically this lead // to have too many variables for the conditions that we are imposing. // Here we are imposing variables that we do not touch to zero // // L R fd.impose(Prs(), 0.0, EQ_4, {-1,-1,-1},{-1,sz[1]-1,sz[2]-1}); fd.impose(Prs(), 0.0, EQ_4, {sz[0]-1,-1,-1},{sz[0]-1,sz[1]-1,sz[2]-1}); // T B fd.impose(Prs(), 0.0, EQ_4, {0,sz[1]-1,-1}, {sz[0]-2,sz[1]-1,sz[2]-1}); fd.impose(Prs(), 0.0, EQ_4, {0,-1 ,-1}, {sz[0]-2,-1, sz[2]-1}); // F A fd.impose(Prs(), 0.0, EQ_4, {0,0,sz[2]-1}, {sz[0]-2,sz[1]-2,sz[2]-1}); fd.impose(Prs(), 0.0, EQ_4, {0,0,-1}, {sz[0]-2,sz[1]-2,-1}); // Impose v_x v_y v_z padding fd.impose(v_x(), 0.0, EQ_1, {-1,-1,-1},{-1,sz[1]-1,sz[2]-1}); fd.impose(v_y(), 0.0, EQ_2, {-1,-1,-1},{sz[0]-1,-1,sz[2]-1}); fd.impose(v_z(), 0.0, EQ_3, {-1,-1,-1},{sz[0]-1,sz[1]-1,-1}); //! \cond [impose eq dom] \endcond /*! * \page Stokes_1_3D Stokes incompressible 3D eigen * * ## Solve the system of equation ## {#num_sk_inc_3D_sse} * * Once we imposed all the equations we can retrieve the Matrix A and the vector b * and pass these two element to the solver. In this example we are using a serial * direct solver Umfpack. * * \snippet Numerics/Stoke_flow/1_3D_incompressible/main_eigen.cpp solver * */ //! \cond [solver] \endcond // Create an UMFPACK solver umfpack_solver solver; // Give to the solver A and b, return x, the solution auto x = solver.solve(fd.getA(),fd.getB()); //! \cond [solver] \endcond /*! * \page Stokes_1_3D Stokes incompressible 3D eigen * * ## Copy the solution on the grid and write on VTK ## {#num_sk_inc_3D_csg} * * Once we have the solution we copy it on the grid * * \snippet Numerics/Stoke_flow/1_3D_incompressible/main_eigen.cpp copy write * */ //! \cond [copy write] \endcond // Bring the solution to grid fd.template copy(x,{0,0},{sz[0]-1,sz[1]-1,sz[2]-1},g_dist); g_dist.write("lid_driven_cavity_p_umfpack"); //! \cond [copy write] \endcond /*! * \page Stokes_1_3D Stokes incompressible 3D eigen * * ## Finalize ## {#num_sk_inc_3D_fin} * * At the very end of the program we have always to de-initialize the library * * \snippet Numerics/Stoke_flow/1_3D_incompressible/main_eigen.cpp fin lib * */ //! \cond [fin lib] \endcond openfpm_finalize(); //! \cond [fin lib] \endcond /*! * \page Stokes_1_3D Stokes incompressible 3D eigen * * # Full code # {#num_sk_inc_3D_code} * * \include Numerics/Stoke_flow/1_3D_incompressible/main_eigen.cpp * */ }