1 static char help[] = "A Chebyshev spectral method for the compressible Blasius boundary layer equations.\n\n"; 2 3 /* 4 Include "petscsnes.h" so that we can use SNES solvers. Note that this 5 file automatically includes: 6 petscsys.h - base PETSc routines petscvec.h - vectors 7 petscmat.h - matrices 8 petscis.h - index sets petscksp.h - Krylov subspace methods 9 petscviewer.h - viewers petscpc.h - preconditioners 10 petscksp.h - linear solvers 11 Include "petscdt.h" so that we can have support for use of Quadrature formulas 12 */ 13 /*F 14 This examples solves the compressible Blasius boundary layer equations 15 2(\rho\muf'')' + ff'' = 0 16 (\rho\muh')' + Prfh' + Pr(\gamma-1)Ma^{2}\rho\muf''^{2} = 0 17 following Howarth-Dorodnitsyn transformation with boundary conditions 18 f(0) = f'(0) = 0, f'(\infty) = 1, h(\infty) = 1, h = \theta(0). Where \theta = T/T_{\infty} 19 Note: density (\rho) and viscosity (\mu) are treated as constants in this example 20 F*/ 21 #include <petscsnes.h> 22 #include <petscdt.h> 23 24 /* 25 User-defined routines 26 */ 27 28 extern PetscErrorCode FormFunction(SNES, Vec, Vec, void *); 29 30 typedef struct { 31 PetscReal Ma, Pr, h_0; 32 PetscInt N; 33 PetscReal dx_deta; 34 PetscReal *x; 35 PetscReal gamma; 36 } Blasius; 37 38 int main(int argc, char **argv) 39 { 40 SNES snes; /* nonlinear solver context */ 41 Vec x, r; /* solution, residual vectors */ 42 PetscMPIInt size; 43 Blasius *blasius; 44 PetscReal L, *weight; /* L is size of the domain */ 45 46 PetscFunctionBeginUser; 47 PetscCall(PetscInitialize(&argc, &argv, NULL, help)); 48 PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size)); 49 PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "Example is only for sequential runs"); 50 51 // Read command-line arguments 52 PetscCall(PetscCalloc1(1, &blasius)); 53 blasius->Ma = 2; /* Mach number */ 54 blasius->Pr = 0.7; /* Prandtl number */ 55 blasius->h_0 = 2.; /* relative temperature at the wall */ 56 blasius->N = 10; /* Number of Chebyshev terms */ 57 blasius->gamma = 1.4; /* specific heat ratio */ 58 L = 5; 59 PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Compressible Blasius boundary layer equations", ""); 60 PetscCall(PetscOptionsReal("-mach", "Mach number at freestream", "", blasius->Ma, &blasius->Ma, NULL)); 61 PetscCall(PetscOptionsReal("-prandtl", "Prandtl number", "", blasius->Pr, &blasius->Pr, NULL)); 62 PetscCall(PetscOptionsReal("-h_0", "Relative enthalpy at wall", "", blasius->h_0, &blasius->h_0, NULL)); 63 PetscCall(PetscOptionsReal("-gamma", "Ratio of specific heats", "", blasius->gamma, &blasius->gamma, NULL)); 64 PetscCall(PetscOptionsInt("-N", "Number of Chebyshev terms for f", "", blasius->N, &blasius->N, NULL)); 65 PetscCall(PetscOptionsReal("-L", "Extent of the domain", "", L, &L, NULL)); 66 PetscOptionsEnd(); 67 blasius->dx_deta = 2 / L; /* this helps to map [-1,1] to [0,L] */ 68 PetscCall(PetscMalloc2(blasius->N - 3, &blasius->x, blasius->N - 3, &weight)); 69 PetscCall(PetscDTGaussQuadrature(blasius->N - 3, -1., 1., blasius->x, weight)); 70 71 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 72 Create nonlinear solver context 73 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 74 PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes)); 75 76 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 77 Create matrix and vector data structures; set corresponding routines 78 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 79 /* 80 Create vectors for solution and nonlinear function 81 */ 82 PetscCall(VecCreate(PETSC_COMM_WORLD, &x)); 83 PetscCall(VecSetSizes(x, PETSC_DECIDE, 2 * blasius->N - 1)); 84 PetscCall(VecSetFromOptions(x)); 85 PetscCall(VecDuplicate(x, &r)); 86 87 /* 88 Set function evaluation routine and vector. 89 */ 90 PetscCall(SNESSetFunction(snes, r, FormFunction, blasius)); 91 { 92 KSP ksp; 93 PC pc; 94 PetscCall(SNESGetKSP(snes, &ksp)); 95 PetscCall(KSPSetType(ksp, KSPPREONLY)); 96 PetscCall(KSPGetPC(ksp, &pc)); 97 PetscCall(PCSetType(pc, PCLU)); 98 } 99 /* 100 Set SNES/KSP/KSP/PC runtime options, e.g., 101 -snes_view -snes_monitor -ksp_type <ksp> -pc_type <pc> 102 These options will override those specified above as long as 103 SNESSetFromOptions() is called _after_ any other customization 104 routines. 105 */ 106 PetscCall(SNESSetFromOptions(snes)); 107 108 PetscCall(SNESSolve(snes, NULL, x)); 109 //PetscCall(VecView(x,PETSC_VIEWER_STDOUT_WORLD)); 110 111 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 112 Free work space. All PETSc objects should be destroyed when they 113 are no longer needed. 114 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 115 116 PetscCall(PetscFree2(blasius->x, weight)); 117 PetscCall(PetscFree(blasius)); 118 PetscCall(VecDestroy(&x)); 119 PetscCall(VecDestroy(&r)); 120 PetscCall(SNESDestroy(&snes)); 121 PetscCall(PetscFinalize()); 122 return 0; 123 } 124 125 /* 126 Helper function to evaluate Chebyshev polynomials with a set of coefficients 127 with all their derivatives represented as a recurrence table 128 */ 129 static void ChebyshevEval(PetscInt N, const PetscScalar *Tf, PetscReal x, PetscReal dx_deta, PetscScalar *f) 130 { 131 PetscScalar table[4][3] = { 132 {1, x, 2 * x * x - 1}, 133 {0, 1, 4 * x }, 134 {0, 0, 4 }, 135 {0, 0, 0 } /* Chebyshev polynomials T_0, T_1, T_2 of the first kind in (-1,1) */ 136 }; 137 for (int i = 0; i < 4; i++) f[i] = table[i][0] * Tf[0] + table[i][1] * Tf[1] + table[i][2] * Tf[2]; /* i-th derivative of f */ 138 for (int i = 3; i < N; i++) { 139 table[0][i % 3] = 2 * x * table[0][(i - 1) % 3] - table[0][(i - 2) % 3]; /* T_n(x) = 2xT_{n-1}(x) - T_{n-2}(x) */ 140 /* Differentiate Chebyshev polynomials with the recurrence relation */ 141 for (int j = 1; j < 4; j++) table[j][i % 3] = i * (2 * table[j - 1][(i - 1) % 3] + table[j][(i - 2) % 3] / (i - 2)); /* T'_{n}(x)/n = 2T_{n-1}(x) + T'_{n-2}(x)/n-2 */ 142 for (int j = 0; j < 4; j++) f[j] += table[j][i % 3] * Tf[i]; 143 } 144 for (int i = 1; i < 4; i++) { 145 for (int j = 0; j < i; j++) f[i] *= dx_deta; /* Here happens the physics of the problem */ 146 } 147 } 148 149 /* 150 FormFunction - Evaluates nonlinear function, F(x). 151 152 Input Parameters: 153 . snes - the SNES context 154 . X - input vector 155 . ctx - optional user-defined context 156 157 Output Parameter: 158 . R - function vector 159 */ 160 PetscErrorCode FormFunction(SNES snes, Vec X, Vec R, void *ctx) 161 { 162 Blasius *blasius = (Blasius *)ctx; 163 const PetscScalar *Tf, *Th; /* Tf and Th are Chebyshev coefficients */ 164 PetscScalar *r, f[4], h[4]; 165 PetscInt N = blasius->N; 166 PetscReal Ma = blasius->Ma, Pr = blasius->Pr; 167 168 PetscFunctionBeginUser; 169 /* 170 Get pointers to vector data. 171 - For default PETSc vectors, VecGetArray() returns a pointer to 172 the data array. Otherwise, the routine is implementation dependent. 173 - You MUST call VecRestoreArray() when you no longer need access to 174 the array. 175 */ 176 PetscCall(VecGetArrayRead(X, &Tf)); 177 Th = Tf + N; 178 PetscCall(VecGetArray(R, &r)); 179 180 /* Compute function */ 181 ChebyshevEval(N, Tf, -1., blasius->dx_deta, f); 182 r[0] = f[0]; 183 r[1] = f[1]; 184 ChebyshevEval(N, Tf, 1., blasius->dx_deta, f); 185 r[2] = f[1] - 1; /* Right end boundary condition */ 186 for (int i = 0; i < N - 3; i++) { 187 ChebyshevEval(N, Tf, blasius->x[i], blasius->dx_deta, f); 188 r[3 + i] = 2 * f[3] + f[2] * f[0]; 189 ChebyshevEval(N - 1, Th, blasius->x[i], blasius->dx_deta, h); 190 r[N + 2 + i] = h[2] + Pr * f[0] * h[1] + Pr * (blasius->gamma - 1) * PetscSqr(Ma * f[2]); 191 } 192 ChebyshevEval(N - 1, Th, -1., blasius->dx_deta, h); 193 r[N] = h[0] - blasius->h_0; /* Left end boundary condition */ 194 ChebyshevEval(N - 1, Th, 1., blasius->dx_deta, h); 195 r[N + 1] = h[0] - 1; /* Left end boundary condition */ 196 197 /* Restore vectors */ 198 PetscCall(VecRestoreArrayRead(X, &Tf)); 199 PetscCall(VecRestoreArray(R, &r)); 200 PetscFunctionReturn(PETSC_SUCCESS); 201 } 202 203 /*TEST 204 205 test: 206 args: -snes_monitor -pc_type svd 207 requires: !single 208 209 TEST*/ 210