static char help[] = "Solves biharmonic equation in 1d.\n"; /* Solves the equation biharmonic equation in split form w = -kappa \Delta u u_t = \Delta w -1 <= u <= 1 Periodic boundary conditions Evolve the biharmonic heat equation with bounds: (same as biharmonic) --------------- ./biharmonic2 -ts_monitor -snes_monitor -ts_monitor_draw_solution -pc_type lu -draw_pause .1 -snes_converged_reason -ts_type beuler -da_refine 5 -draw_fields 1 -ts_time_step 9.53674e-9 w = -kappa \Delta u + u^3 - u u_t = \Delta w -1 <= u <= 1 Periodic boundary conditions Evolve the Cahn-Hillard equations: (this fails after a few timesteps 12/17/2017) --------------- ./biharmonic2 -ts_monitor -snes_monitor -ts_monitor_draw_solution -pc_type lu -draw_pause .1 -snes_converged_reason -ts_type beuler -da_refine 6 -draw_fields 1 -kappa .00001 -ts_time_step 5.96046e-06 -cahn-hillard */ #include #include #include #include /* User-defined routines */ extern PetscErrorCode FormFunction(TS, PetscReal, Vec, Vec, Vec, void *), FormInitialSolution(DM, Vec, PetscReal); typedef struct { PetscBool cahnhillard; PetscReal kappa; PetscInt energy; PetscReal tol; PetscReal theta; PetscReal theta_c; } UserCtx; int main(int argc, char **argv) { TS ts; /* nonlinear solver */ Vec x, r; /* solution, residual vectors */ Mat J; /* Jacobian matrix */ PetscInt steps, Mx; DM da; MatFDColoring matfdcoloring; ISColoring iscoloring; PetscReal dt; PetscReal vbounds[] = {-100000, 100000, -1.1, 1.1}; SNES snes; UserCtx ctx; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Initialize program - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscFunctionBeginUser; PetscCall(PetscInitialize(&argc, &argv, NULL, help)); ctx.kappa = 1.0; PetscCall(PetscOptionsGetReal(NULL, NULL, "-kappa", &ctx.kappa, NULL)); ctx.cahnhillard = PETSC_FALSE; PetscCall(PetscOptionsGetBool(NULL, NULL, "-cahn-hillard", &ctx.cahnhillard, NULL)); PetscCall(PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 2, vbounds)); PetscCall(PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 600, 600)); ctx.energy = 1; /*PetscCall(PetscOptionsGetInt(NULL,NULL,"-energy",&ctx.energy,NULL));*/ PetscCall(PetscOptionsGetInt(NULL, NULL, "-energy", &ctx.energy, NULL)); ctx.tol = 1.0e-8; PetscCall(PetscOptionsGetReal(NULL, NULL, "-tol", &ctx.tol, NULL)); ctx.theta = .001; ctx.theta_c = 1.0; PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta", &ctx.theta, NULL)); PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta_c", &ctx.theta_c, NULL)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create distributed array (DMDA) to manage parallel grid and vectors - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10, 2, 2, NULL, &da)); PetscCall(DMSetFromOptions(da)); PetscCall(DMSetUp(da)); PetscCall(DMDASetFieldName(da, 0, "Biharmonic heat equation: w = -kappa*u_xx")); PetscCall(DMDASetFieldName(da, 1, "Biharmonic heat equation: u")); PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)); dt = 1.0 / (10. * ctx.kappa * Mx * Mx * Mx * Mx); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Extract global vectors from DMDA; then duplicate for remaining vectors that are the same types - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(DMCreateGlobalVector(da, &x)); PetscCall(VecDuplicate(x, &r)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create timestepping solver context - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSCreate(PETSC_COMM_WORLD, &ts)); PetscCall(TSSetDM(ts, da)); PetscCall(TSSetProblemType(ts, TS_NONLINEAR)); PetscCall(TSSetIFunction(ts, NULL, FormFunction, &ctx)); PetscCall(TSSetMaxTime(ts, .02)); PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_INTERPOLATE)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create matrix data structure; set Jacobian evaluation routine < Set Jacobian matrix data structure and default Jacobian evaluation routine. User can override with: -snes_mf : matrix-free Newton-Krylov method with no preconditioning (unless user explicitly sets preconditioner) -snes_mf_operator : form matrix used to construct the preconditioner as set by the user, but use matrix-free approx for Jacobian-vector products within Newton-Krylov method - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSGetSNES(ts, &snes)); PetscCall(DMCreateColoring(da, IS_COLORING_GLOBAL, &iscoloring)); PetscCall(DMSetMatType(da, MATAIJ)); PetscCall(DMCreateMatrix(da, &J)); PetscCall(MatFDColoringCreate(J, iscoloring, &matfdcoloring)); PetscCall(MatFDColoringSetFunction(matfdcoloring, (MatFDColoringFn *)SNESTSFormFunction, ts)); PetscCall(MatFDColoringSetFromOptions(matfdcoloring)); PetscCall(MatFDColoringSetUp(J, iscoloring, matfdcoloring)); PetscCall(ISColoringDestroy(&iscoloring)); PetscCall(SNESSetJacobian(snes, J, J, SNESComputeJacobianDefaultColor, matfdcoloring)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Customize nonlinear solver - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSSetType(ts, TSBEULER)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set initial conditions - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(FormInitialSolution(da, x, ctx.kappa)); PetscCall(TSSetTimeStep(ts, dt)); PetscCall(TSSetSolution(ts, x)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set runtime options - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSSetFromOptions(ts)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Solve nonlinear system - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(TSSolve(ts, x)); PetscCall(TSGetStepNumber(ts, &steps)); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Free work space. All PETSc objects should be destroyed when they are no longer needed. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscCall(MatDestroy(&J)); PetscCall(MatFDColoringDestroy(&matfdcoloring)); PetscCall(VecDestroy(&x)); PetscCall(VecDestroy(&r)); PetscCall(TSDestroy(&ts)); PetscCall(DMDestroy(&da)); PetscCall(PetscFinalize()); return 0; } typedef struct { PetscScalar w, u; } Field; /* ------------------------------------------------------------------- */ /* FormFunction - Evaluates nonlinear function, F(x). Input Parameters: . ts - the TS context . X - input vector . ptr - optional user-defined context, as set by SNESSetFunction() Output Parameter: . F - function vector */ PetscErrorCode FormFunction(TS ts, PetscReal ftime, Vec X, Vec Xdot, Vec F, void *ptr) { DM da; PetscInt i, Mx, xs, xm; PetscReal hx, sx; Field *x, *xdot, *f; Vec localX, localXdot; UserCtx *ctx = (UserCtx *)ptr; PetscFunctionBegin; PetscCall(TSGetDM(ts, &da)); PetscCall(DMGetLocalVector(da, &localX)); PetscCall(DMGetLocalVector(da, &localXdot)); PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE)); hx = 1.0 / (PetscReal)Mx; sx = 1.0 / (hx * hx); /* Scatter ghost points to local vector,using the 2-step process DMGlobalToLocalBegin(),DMGlobalToLocalEnd(). By placing code between these two statements, computations can be done while messages are in transition. */ PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX)); PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX)); PetscCall(DMGlobalToLocalBegin(da, Xdot, INSERT_VALUES, localXdot)); PetscCall(DMGlobalToLocalEnd(da, Xdot, INSERT_VALUES, localXdot)); /* Get pointers to vector data */ PetscCall(DMDAVecGetArrayRead(da, localX, &x)); PetscCall(DMDAVecGetArrayRead(da, localXdot, &xdot)); PetscCall(DMDAVecGetArray(da, F, &f)); /* Get local grid boundaries */ PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL)); /* Compute function over the locally owned part of the grid */ for (i = xs; i < xs + xm; i++) { f[i].w = x[i].w + ctx->kappa * (x[i - 1].u + x[i + 1].u - 2.0 * x[i].u) * sx; if (ctx->cahnhillard) { switch (ctx->energy) { case 1: /* double well */ f[i].w += -x[i].u * x[i].u * x[i].u + x[i].u; break; case 2: /* double obstacle */ f[i].w += x[i].u; break; case 3: /* logarithmic */ if (PetscRealPart(x[i].u) < -1.0 + 2.0 * ctx->tol) f[i].w += .5 * ctx->theta * (-PetscLogReal(ctx->tol) + PetscLogScalar((1.0 - x[i].u) / 2.0)) + ctx->theta_c * x[i].u; else if (PetscRealPart(x[i].u) > 1.0 - 2.0 * ctx->tol) f[i].w += .5 * ctx->theta * (-PetscLogScalar((1.0 + x[i].u) / 2.0) + PetscLogReal(ctx->tol)) + ctx->theta_c * x[i].u; else f[i].w += .5 * ctx->theta * (-PetscLogScalar((1.0 + x[i].u) / 2.0) + PetscLogScalar((1.0 - x[i].u) / 2.0)) + ctx->theta_c * x[i].u; break; } } f[i].u = xdot[i].u - (x[i - 1].w + x[i + 1].w - 2.0 * x[i].w) * sx; } /* Restore vectors */ PetscCall(DMDAVecRestoreArrayRead(da, localXdot, &xdot)); PetscCall(DMDAVecRestoreArrayRead(da, localX, &x)); PetscCall(DMDAVecRestoreArray(da, F, &f)); PetscCall(DMRestoreLocalVector(da, &localX)); PetscCall(DMRestoreLocalVector(da, &localXdot)); PetscFunctionReturn(PETSC_SUCCESS); } /* ------------------------------------------------------------------- */ PetscErrorCode FormInitialSolution(DM da, Vec X, PetscReal kappa) { PetscInt i, xs, xm, Mx, xgs, xgm; Field *x; PetscReal hx, xx, r, sx; Vec Xg; PetscFunctionBegin; PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE)); hx = 1.0 / (PetscReal)Mx; sx = 1.0 / (hx * hx); /* Get pointers to vector data */ PetscCall(DMCreateLocalVector(da, &Xg)); PetscCall(DMDAVecGetArray(da, Xg, &x)); /* Get local grid boundaries */ PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL)); PetscCall(DMDAGetGhostCorners(da, &xgs, NULL, NULL, &xgm, NULL, NULL)); /* Compute u function over the locally owned part of the grid including ghost points */ for (i = xgs; i < xgs + xgm; i++) { xx = i * hx; r = PetscSqrtReal((xx - .5) * (xx - .5)); if (r < .125) x[i].u = 1.0; else x[i].u = -.50; /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */ x[i].w = 0; } for (i = xs; i < xs + xm; i++) x[i].w = -kappa * (x[i - 1].u + x[i + 1].u - 2.0 * x[i].u) * sx; /* Restore vectors */ PetscCall(DMDAVecRestoreArray(da, Xg, &x)); /* Grab only the global part of the vector */ PetscCall(VecSet(X, 0)); PetscCall(DMLocalToGlobalBegin(da, Xg, ADD_VALUES, X)); PetscCall(DMLocalToGlobalEnd(da, Xg, ADD_VALUES, X)); PetscCall(VecDestroy(&Xg)); PetscFunctionReturn(PETSC_SUCCESS); } /*TEST build: requires: !complex !single test: args: -ts_monitor -snes_monitor -pc_type lu -snes_converged_reason -ts_type beuler -da_refine 5 -ts_time_step 9.53674e-9 -ts_max_steps 50 requires: x TEST*/