// Copyright (c) 2017-2022, Lawrence Livermore National Security, LLC and other CEED contributors. // All Rights Reserved. See the top-level LICENSE and NOTICE files for details. // // SPDX-License-Identifier: BSD-2-Clause // // This file is part of CEED: http://github.com/ceed #include #include #include #include #include #include #include /// @file /// Implementation of CeedBasis interfaces /// @cond DOXYGEN_SKIP static struct CeedBasis_private ceed_basis_collocated; /// @endcond /// @addtogroup CeedBasisUser /// @{ /// Indicate that the quadrature points are collocated with the nodes const CeedBasis CEED_BASIS_COLLOCATED = &ceed_basis_collocated; /// @} /// ---------------------------------------------------------------------------- /// CeedBasis Library Internal Functions /// ---------------------------------------------------------------------------- /// @addtogroup CeedBasisDeveloper /// @{ /** @brief Compute Chebyshev polynomial values at a point @param[in] x Coordinate to evaluate Chebyshev polynomials at @param[in] n Number of Chebyshev polynomials to evaluate, n >= 2 @param[out] chebyshev_x Array of Chebyshev polynomial values @return An error code: 0 - success, otherwise - failure @ref Developer **/ static int CeedChebyshevPolynomialsAtPoint(CeedScalar x, CeedInt n, CeedScalar *chebyshev_x) { chebyshev_x[0] = 1.0; chebyshev_x[1] = 2 * x; for (CeedInt i = 2; i < n; i++) chebyshev_x[i] = 2 * x * chebyshev_x[i - 1] - chebyshev_x[i - 2]; return CEED_ERROR_SUCCESS; } /** @brief Compute values of the derivative of Chebyshev polynomials at a point @param[in] x Coordinate to evaluate derivative of Chebyshev polynomials at @param[in] n Number of Chebyshev polynomials to evaluate, n >= 2 @param[out] chebyshev_x Array of Chebyshev polynomial derivative values @return An error code: 0 - success, otherwise - failure @ref Developer **/ static int CeedChebyshevDerivativeAtPoint(CeedScalar x, CeedInt n, CeedScalar *chebyshev_dx) { CeedScalar chebyshev_x[3]; chebyshev_x[1] = 1.0; chebyshev_x[2] = 2 * x; chebyshev_dx[0] = 0.0; chebyshev_dx[1] = 2.0; for (CeedInt i = 2; i < n; i++) { chebyshev_x[0] = chebyshev_x[1]; chebyshev_x[1] = chebyshev_x[2]; chebyshev_x[2] = 2 * x * chebyshev_x[1] - chebyshev_x[0]; chebyshev_dx[i] = 2 * x * chebyshev_dx[i - 1] + 2 * chebyshev_x[1] - chebyshev_dx[i - 2]; } return CEED_ERROR_SUCCESS; } /** @brief Compute Householder reflection Computes A = (I - b v v^T) A, where A is an mxn matrix indexed as A[i*row + j*col] @param[in,out] A Matrix to apply Householder reflection to, in place @param[in] v Householder vector @param[in] b Scaling factor @param[in] m Number of rows in A @param[in] n Number of columns in A @param[in] row Row stride @param[in] col Col stride @return An error code: 0 - success, otherwise - failure @ref Developer **/ static int CeedHouseholderReflect(CeedScalar *A, const CeedScalar *v, CeedScalar b, CeedInt m, CeedInt n, CeedInt row, CeedInt col) { for (CeedInt j = 0; j < n; j++) { CeedScalar w = A[0 * row + j * col]; for (CeedInt i = 1; i < m; i++) w += v[i] * A[i * row + j * col]; A[0 * row + j * col] -= b * w; for (CeedInt i = 1; i < m; i++) A[i * row + j * col] -= b * w * v[i]; } return CEED_ERROR_SUCCESS; } /** @brief Compute Givens rotation Computes A = G A (or G^T A in transpose mode), where A is an mxn matrix indexed as A[i*n + j*m] @param[in,out] A Row major matrix to apply Givens rotation to, in place @param[in] c Cosine factor @param[in] s Sine factor @param[in] t_mode @ref CEED_NOTRANSPOSE to rotate the basis counter-clockwise, which has the effect of rotating columns of A clockwise; @ref CEED_TRANSPOSE for the opposite rotation @param[in] i First row/column to apply rotation @param[in] k Second row/column to apply rotation @param[in] m Number of rows in A @param[in] n Number of columns in A @return An error code: 0 - success, otherwise - failure @ref Developer **/ static int CeedGivensRotation(CeedScalar *A, CeedScalar c, CeedScalar s, CeedTransposeMode t_mode, CeedInt i, CeedInt k, CeedInt m, CeedInt n) { CeedInt stride_j = 1, stride_ik = m, num_its = n; if (t_mode == CEED_NOTRANSPOSE) { stride_j = n; stride_ik = 1; num_its = m; } // Apply rotation for (CeedInt j = 0; j < num_its; j++) { CeedScalar tau1 = A[i * stride_ik + j * stride_j], tau2 = A[k * stride_ik + j * stride_j]; A[i * stride_ik + j * stride_j] = c * tau1 - s * tau2; A[k * stride_ik + j * stride_j] = s * tau1 + c * tau2; } return CEED_ERROR_SUCCESS; } /** @brief View an array stored in a CeedBasis @param[in] name Name of array @param[in] fp_fmt Printing format @param[in] m Number of rows in array @param[in] n Number of columns in array @param[in] a Array to be viewed @param[in] stream Stream to view to, e.g., stdout @return An error code: 0 - success, otherwise - failure @ref Developer **/ static int CeedScalarView(const char *name, const char *fp_fmt, CeedInt m, CeedInt n, const CeedScalar *a, FILE *stream) { if (m > 1) { fprintf(stream, " %s:\n", name); } else { char padded_name[12]; snprintf(padded_name, 11, "%s:", name); fprintf(stream, " %-10s", padded_name); } for (CeedInt i = 0; i < m; i++) { if (m > 1) fprintf(stream, " [%" CeedInt_FMT "]", i); for (CeedInt j = 0; j < n; j++) fprintf(stream, fp_fmt, fabs(a[i * n + j]) > 1E-14 ? a[i * n + j] : 0); fputs("\n", stream); } return CEED_ERROR_SUCCESS; } /** @brief Create the interpolation and gradient matrices for projection from the nodes of `basis_from` to the nodes of `basis_to`. The interpolation is given by `interp_project = interp_to^+ * interp_from`, where the pseudoinverse `interp_to^+` is given by QR factorization. The gradient is given by `grad_project = interp_to^+ * grad_from`, and is only computed for H^1 spaces otherwise it should not be used. Note: `basis_from` and `basis_to` must have compatible quadrature spaces. @param[in] basis_from CeedBasis to project from @param[in] basis_to CeedBasis to project to @param[out] interp_project Address of the variable where the newly created interpolation matrix will be stored. @param[out] grad_project Address of the variable where the newly created gradient matrix will be stored. @return An error code: 0 - success, otherwise - failure @ref Developer **/ static int CeedBasisCreateProjectionMatrices(CeedBasis basis_from, CeedBasis basis_to, CeedScalar **interp_project, CeedScalar **grad_project) { Ceed ceed; CeedCall(CeedBasisGetCeed(basis_to, &ceed)); // Check for compatible quadrature spaces CeedInt Q_to, Q_from; CeedCall(CeedBasisGetNumQuadraturePoints(basis_to, &Q_to)); CeedCall(CeedBasisGetNumQuadraturePoints(basis_from, &Q_from)); CeedCheck(Q_to == Q_from, ceed, CEED_ERROR_DIMENSION, "Bases must have compatible quadrature spaces"); // Check for matching tensor or non-tensor CeedInt P_to, P_from, Q = Q_to; bool is_tensor_to, is_tensor_from; CeedCall(CeedBasisIsTensor(basis_to, &is_tensor_to)); CeedCall(CeedBasisIsTensor(basis_from, &is_tensor_from)); CeedCheck(is_tensor_to == is_tensor_from, ceed, CEED_ERROR_MINOR, "Bases must both be tensor or non-tensor"); if (is_tensor_to) { CeedCall(CeedBasisGetNumNodes1D(basis_to, &P_to)); CeedCall(CeedBasisGetNumNodes1D(basis_from, &P_from)); CeedCall(CeedBasisGetNumQuadraturePoints1D(basis_from, &Q)); } else { CeedCall(CeedBasisGetNumNodes(basis_to, &P_to)); CeedCall(CeedBasisGetNumNodes(basis_from, &P_from)); } // Check for matching FE space CeedFESpace fe_space_to, fe_space_from; CeedCall(CeedBasisGetFESpace(basis_to, &fe_space_to)); CeedCall(CeedBasisGetFESpace(basis_from, &fe_space_from)); CeedCheck(fe_space_to == fe_space_from, ceed, CEED_ERROR_MINOR, "Bases must both be the same FE space type"); // Get source matrices CeedInt dim, q_comp = 1; const CeedScalar *interp_to_source = NULL, *interp_from_source = NULL; CeedScalar *interp_to, *interp_from, *tau; CeedCall(CeedBasisGetDimension(basis_to, &dim)); if (is_tensor_to) { CeedCall(CeedBasisGetInterp1D(basis_to, &interp_to_source)); CeedCall(CeedBasisGetInterp1D(basis_from, &interp_from_source)); } else { CeedCall(CeedBasisGetNumQuadratureComponents(basis_from, CEED_EVAL_INTERP, &q_comp)); CeedCall(CeedBasisGetInterp(basis_to, &interp_to_source)); CeedCall(CeedBasisGetInterp(basis_from, &interp_from_source)); } CeedCall(CeedMalloc(Q * P_from * q_comp, &interp_from)); CeedCall(CeedMalloc(Q * P_to * q_comp, &interp_to)); CeedCall(CeedCalloc(P_to * P_from, interp_project)); CeedCall(CeedMalloc(Q * q_comp, &tau)); // `grad_project = interp_to^+ * grad_from` is computed for the H^1 space case so the // projection basis will have a gradient operation (allocated even if not H^1 for the // basis construction later on) const CeedScalar *grad_from_source = NULL; if (fe_space_to == CEED_FE_SPACE_H1) { if (is_tensor_to) { CeedCall(CeedBasisGetGrad1D(basis_from, &grad_from_source)); } else { CeedCall(CeedBasisGetGrad(basis_from, &grad_from_source)); } } CeedCall(CeedCalloc(P_to * P_from * (is_tensor_to ? 1 : dim), grad_project)); // QR Factorization, interp_to = Q R memcpy(interp_to, interp_to_source, Q * P_to * q_comp * sizeof(interp_to_source[0])); CeedCall(CeedQRFactorization(ceed, interp_to, tau, Q * q_comp, P_to)); // Build matrices CeedInt num_matrices = 1 + (fe_space_to == CEED_FE_SPACE_H1) * (is_tensor_to ? 1 : dim); CeedScalar *input_from[num_matrices], *output_project[num_matrices]; input_from[0] = (CeedScalar *)interp_from_source; output_project[0] = *interp_project; for (CeedInt m = 1; m < num_matrices; m++) { input_from[m] = (CeedScalar *)&grad_from_source[(m - 1) * Q * P_from]; output_project[m] = &((*grad_project)[(m - 1) * P_to * P_from]); } for (CeedInt m = 0; m < num_matrices; m++) { // Apply Q^T, interp_from = Q^T interp_from memcpy(interp_from, input_from[m], Q * P_from * q_comp * sizeof(input_from[m][0])); CeedCall(CeedHouseholderApplyQ(interp_from, interp_to, tau, CEED_TRANSPOSE, Q * q_comp, P_from, P_to, P_from, 1)); // Apply Rinv, output_project = Rinv interp_from for (CeedInt j = 0; j < P_from; j++) { // Column j output_project[m][j + P_from * (P_to - 1)] = interp_from[j + P_from * (P_to - 1)] / interp_to[P_to * P_to - 1]; for (CeedInt i = P_to - 2; i >= 0; i--) { // Row i output_project[m][j + P_from * i] = interp_from[j + P_from * i]; for (CeedInt k = i + 1; k < P_to; k++) { output_project[m][j + P_from * i] -= interp_to[k + P_to * i] * output_project[m][j + P_from * k]; } output_project[m][j + P_from * i] /= interp_to[i + P_to * i]; } } } // Cleanup CeedCall(CeedFree(&tau)); CeedCall(CeedFree(&interp_to)); CeedCall(CeedFree(&interp_from)); return CEED_ERROR_SUCCESS; } /// @} /// ---------------------------------------------------------------------------- /// Ceed Backend API /// ---------------------------------------------------------------------------- /// @addtogroup CeedBasisBackend /// @{ /** @brief Return collocated grad matrix @param[in] basis CeedBasis @param[out] collo_grad_1d Row-major (Q_1d * Q_1d) matrix expressing derivatives of basis functions at quadrature points @return An error code: 0 - success, otherwise - failure @ref Backend **/ int CeedBasisGetCollocatedGrad(CeedBasis basis, CeedScalar *collo_grad_1d) { Ceed ceed; CeedInt P_1d = (basis)->P_1d, Q_1d = (basis)->Q_1d; CeedScalar *interp_1d, *grad_1d, *tau; CeedCall(CeedMalloc(Q_1d * P_1d, &interp_1d)); CeedCall(CeedMalloc(Q_1d * P_1d, &grad_1d)); CeedCall(CeedMalloc(Q_1d, &tau)); memcpy(interp_1d, (basis)->interp_1d, Q_1d * P_1d * sizeof(basis)->interp_1d[0]); memcpy(grad_1d, (basis)->grad_1d, Q_1d * P_1d * sizeof(basis)->interp_1d[0]); // QR Factorization, interp_1d = Q R CeedCall(CeedBasisGetCeed(basis, &ceed)); CeedCall(CeedQRFactorization(ceed, interp_1d, tau, Q_1d, P_1d)); // Note: This function is for backend use, so all errors are terminal and we do not need to clean up memory on failure. // Apply R_inv, collo_grad_1d = grad_1d R_inv for (CeedInt i = 0; i < Q_1d; i++) { // Row i collo_grad_1d[Q_1d * i] = grad_1d[P_1d * i] / interp_1d[0]; for (CeedInt j = 1; j < P_1d; j++) { // Column j collo_grad_1d[j + Q_1d * i] = grad_1d[j + P_1d * i]; for (CeedInt k = 0; k < j; k++) collo_grad_1d[j + Q_1d * i] -= interp_1d[j + P_1d * k] * collo_grad_1d[k + Q_1d * i]; collo_grad_1d[j + Q_1d * i] /= interp_1d[j + P_1d * j]; } for (CeedInt j = P_1d; j < Q_1d; j++) collo_grad_1d[j + Q_1d * i] = 0; } // Apply Q^T, collo_grad_1d = collo_grad_1d Q^T CeedCall(CeedHouseholderApplyQ(collo_grad_1d, interp_1d, tau, CEED_NOTRANSPOSE, Q_1d, Q_1d, P_1d, 1, Q_1d)); CeedCall(CeedFree(&interp_1d)); CeedCall(CeedFree(&grad_1d)); CeedCall(CeedFree(&tau)); return CEED_ERROR_SUCCESS; } /** @brief Get tensor status for given CeedBasis @param[in] basis CeedBasis @param[out] is_tensor Variable to store tensor status @return An error code: 0 - success, otherwise - failure @ref Backend **/ int CeedBasisIsTensor(CeedBasis basis, bool *is_tensor) { *is_tensor = basis->is_tensor_basis; return CEED_ERROR_SUCCESS; } /** @brief Get backend data of a CeedBasis @param[in] basis CeedBasis @param[out] data Variable to store data @return An error code: 0 - success, otherwise - failure @ref Backend **/ int CeedBasisGetData(CeedBasis basis, void *data) { *(void **)data = basis->data; return CEED_ERROR_SUCCESS; } /** @brief Set backend data of a CeedBasis @param[in,out] basis CeedBasis @param[in] data Data to set @return An error code: 0 - success, otherwise - failure @ref Backend **/ int CeedBasisSetData(CeedBasis basis, void *data) { basis->data = data; return CEED_ERROR_SUCCESS; } /** @brief Increment the reference counter for a CeedBasis @param[in,out] basis Basis to increment the reference counter @return An error code: 0 - success, otherwise - failure @ref Backend **/ int CeedBasisReference(CeedBasis basis) { basis->ref_count++; return CEED_ERROR_SUCCESS; } /** @brief Get number of Q-vector components for given CeedBasis @param[in] basis CeedBasis @param[in] eval_mode \ref CEED_EVAL_INTERP to use interpolated values, \ref CEED_EVAL_GRAD to use gradients, \ref CEED_EVAL_DIV to use divergence, \ref CEED_EVAL_CURL to use curl. @param[out] q_comp Variable to store number of Q-vector components of basis @return An error code: 0 - success, otherwise - failure @ref Backend **/ int CeedBasisGetNumQuadratureComponents(CeedBasis basis, CeedEvalMode eval_mode, CeedInt *q_comp) { switch (eval_mode) { case CEED_EVAL_INTERP: *q_comp = (basis->fe_space == CEED_FE_SPACE_H1) ? 1 : basis->dim; break; case CEED_EVAL_GRAD: *q_comp = basis->dim; break; case CEED_EVAL_DIV: *q_comp = 1; break; case CEED_EVAL_CURL: *q_comp = (basis->dim < 3) ? 1 : basis->dim; break; case CEED_EVAL_NONE: case CEED_EVAL_WEIGHT: *q_comp = 1; break; } return CEED_ERROR_SUCCESS; } /** @brief Estimate number of FLOPs required to apply CeedBasis in t_mode and eval_mode @param[in] basis Basis to estimate FLOPs for @param[in] t_mode Apply basis or transpose @param[in] eval_mode Basis evaluation mode @param[out] flops Address of variable to hold FLOPs estimate @ref Backend **/ int CeedBasisGetFlopsEstimate(CeedBasis basis, CeedTransposeMode t_mode, CeedEvalMode eval_mode, CeedSize *flops) { bool is_tensor; CeedCall(CeedBasisIsTensor(basis, &is_tensor)); if (is_tensor) { CeedInt dim, num_comp, P_1d, Q_1d; CeedCall(CeedBasisGetDimension(basis, &dim)); CeedCall(CeedBasisGetNumComponents(basis, &num_comp)); CeedCall(CeedBasisGetNumNodes1D(basis, &P_1d)); CeedCall(CeedBasisGetNumQuadraturePoints1D(basis, &Q_1d)); if (t_mode == CEED_TRANSPOSE) { P_1d = Q_1d; Q_1d = P_1d; } CeedInt tensor_flops = 0, pre = num_comp * CeedIntPow(P_1d, dim - 1), post = 1; for (CeedInt d = 0; d < dim; d++) { tensor_flops += 2 * pre * P_1d * post * Q_1d; pre /= P_1d; post *= Q_1d; } switch (eval_mode) { case CEED_EVAL_NONE: *flops = 0; break; case CEED_EVAL_INTERP: *flops = tensor_flops; break; case CEED_EVAL_GRAD: *flops = tensor_flops * 2; break; case CEED_EVAL_DIV: case CEED_EVAL_CURL: // LCOV_EXCL_START return CeedError(basis->ceed, CEED_ERROR_INCOMPATIBLE, "Tensor basis evaluation for %s not supported", CeedEvalModes[eval_mode]); break; // LCOV_EXCL_STOP case CEED_EVAL_WEIGHT: *flops = dim * CeedIntPow(Q_1d, dim); break; } } else { CeedInt dim, num_comp, q_comp, num_nodes, num_qpts; CeedCall(CeedBasisGetDimension(basis, &dim)); CeedCall(CeedBasisGetNumComponents(basis, &num_comp)); CeedCall(CeedBasisGetNumQuadratureComponents(basis, eval_mode, &q_comp)); CeedCall(CeedBasisGetNumNodes(basis, &num_nodes)); CeedCall(CeedBasisGetNumQuadraturePoints(basis, &num_qpts)); switch (eval_mode) { case CEED_EVAL_NONE: *flops = 0; break; case CEED_EVAL_INTERP: case CEED_EVAL_GRAD: case CEED_EVAL_DIV: case CEED_EVAL_CURL: *flops = num_nodes * num_qpts * num_comp * q_comp; break; case CEED_EVAL_WEIGHT: *flops = 0; break; } } return CEED_ERROR_SUCCESS; } /** @brief Get CeedFESpace for a CeedBasis @param[in] basis CeedBasis @param[out] fe_space Variable to store CeedFESpace @return An error code: 0 - success, otherwise - failure @ref Backend **/ int CeedBasisGetFESpace(CeedBasis basis, CeedFESpace *fe_space) { *fe_space = basis->fe_space; return CEED_ERROR_SUCCESS; } /** @brief Get dimension for given CeedElemTopology @param[in] topo CeedElemTopology @param[out] dim Variable to store dimension of topology @return An error code: 0 - success, otherwise - failure @ref Backend **/ int CeedBasisGetTopologyDimension(CeedElemTopology topo, CeedInt *dim) { *dim = (CeedInt)topo >> 16; return CEED_ERROR_SUCCESS; } /** @brief Get CeedTensorContract of a CeedBasis @param[in] basis CeedBasis @param[out] contract Variable to store CeedTensorContract @return An error code: 0 - success, otherwise - failure @ref Backend **/ int CeedBasisGetTensorContract(CeedBasis basis, CeedTensorContract *contract) { *contract = basis->contract; return CEED_ERROR_SUCCESS; } /** @brief Set CeedTensorContract of a CeedBasis @param[in,out] basis CeedBasis @param[in] contract CeedTensorContract to set @return An error code: 0 - success, otherwise - failure @ref Backend **/ int CeedBasisSetTensorContract(CeedBasis basis, CeedTensorContract contract) { basis->contract = contract; CeedCall(CeedTensorContractReference(contract)); return CEED_ERROR_SUCCESS; } /** @brief Return a reference implementation of matrix multiplication C = A B. Note: This is a reference implementation for CPU CeedScalar pointers that is not intended for high performance. @param[in] ceed Ceed context for error handling @param[in] mat_A Row-major matrix A @param[in] mat_B Row-major matrix B @param[out] mat_C Row-major output matrix C @param[in] m Number of rows of C @param[in] n Number of columns of C @param[in] kk Number of columns of A/rows of B @return An error code: 0 - success, otherwise - failure @ref Utility **/ int CeedMatrixMatrixMultiply(Ceed ceed, const CeedScalar *mat_A, const CeedScalar *mat_B, CeedScalar *mat_C, CeedInt m, CeedInt n, CeedInt kk) { for (CeedInt i = 0; i < m; i++) { for (CeedInt j = 0; j < n; j++) { CeedScalar sum = 0; for (CeedInt k = 0; k < kk; k++) sum += mat_A[k + i * kk] * mat_B[j + k * n]; mat_C[j + i * n] = sum; } } return CEED_ERROR_SUCCESS; } /** @brief Return QR Factorization of a matrix @param[in] ceed Ceed context for error handling @param[in,out] mat Row-major matrix to be factorized in place @param[in,out] tau Vector of length m of scaling factors @param[in] m Number of rows @param[in] n Number of columns @return An error code: 0 - success, otherwise - failure @ref Utility **/ int CeedQRFactorization(Ceed ceed, CeedScalar *mat, CeedScalar *tau, CeedInt m, CeedInt n) { CeedScalar v[m]; // Check matrix shape CeedCheck(n <= m, ceed, CEED_ERROR_UNSUPPORTED, "Cannot compute QR factorization with n > m"); for (CeedInt i = 0; i < n; i++) { if (i >= m - 1) { // last row of matrix, no reflection needed tau[i] = 0.; break; } // Calculate Householder vector, magnitude CeedScalar sigma = 0.0; v[i] = mat[i + n * i]; for (CeedInt j = i + 1; j < m; j++) { v[j] = mat[i + n * j]; sigma += v[j] * v[j]; } CeedScalar norm = sqrt(v[i] * v[i] + sigma); // norm of v[i:m] CeedScalar R_ii = -copysign(norm, v[i]); v[i] -= R_ii; // norm of v[i:m] after modification above and scaling below // norm = sqrt(v[i]*v[i] + sigma) / v[i]; // tau = 2 / (norm*norm) tau[i] = 2 * v[i] * v[i] / (v[i] * v[i] + sigma); for (CeedInt j = i + 1; j < m; j++) v[j] /= v[i]; // Apply Householder reflector to lower right panel CeedHouseholderReflect(&mat[i * n + i + 1], &v[i], tau[i], m - i, n - i - 1, n, 1); // Save v mat[i + n * i] = R_ii; for (CeedInt j = i + 1; j < m; j++) mat[i + n * j] = v[j]; } return CEED_ERROR_SUCCESS; } /** @brief Apply Householder Q matrix Compute mat_A = mat_Q mat_A, where mat_Q is mxm and mat_A is mxn. @param[in,out] mat_A Matrix to apply Householder Q to, in place @param[in] mat_Q Householder Q matrix @param[in] tau Householder scaling factors @param[in] t_mode Transpose mode for application @param[in] m Number of rows in A @param[in] n Number of columns in A @param[in] k Number of elementary reflectors in Q, k 1, ceed, CEED_ERROR_UNSUPPORTED, "Cannot compute symmetric Schur decomposition of scalars"); CeedScalar v[n - 1], tau[n - 1], mat_T[n * n]; // Copy mat to mat_T and set mat to I memcpy(mat_T, mat, n * n * sizeof(mat[0])); for (CeedInt i = 0; i < n; i++) { for (CeedInt j = 0; j < n; j++) mat[j + n * i] = (i == j) ? 1 : 0; } // Reduce to tridiagonal for (CeedInt i = 0; i < n - 1; i++) { // Calculate Householder vector, magnitude CeedScalar sigma = 0.0; v[i] = mat_T[i + n * (i + 1)]; for (CeedInt j = i + 1; j < n - 1; j++) { v[j] = mat_T[i + n * (j + 1)]; sigma += v[j] * v[j]; } CeedScalar norm = sqrt(v[i] * v[i] + sigma); // norm of v[i:n-1] CeedScalar R_ii = -copysign(norm, v[i]); v[i] -= R_ii; // norm of v[i:m] after modification above and scaling below // norm = sqrt(v[i]*v[i] + sigma) / v[i]; // tau = 2 / (norm*norm) tau[i] = i == n - 2 ? 2 : 2 * v[i] * v[i] / (v[i] * v[i] + sigma); for (CeedInt j = i + 1; j < n - 1; j++) v[j] /= v[i]; // Update sub and super diagonal for (CeedInt j = i + 2; j < n; j++) { mat_T[i + n * j] = 0; mat_T[j + n * i] = 0; } // Apply symmetric Householder reflector to lower right panel CeedHouseholderReflect(&mat_T[(i + 1) + n * (i + 1)], &v[i], tau[i], n - (i + 1), n - (i + 1), n, 1); CeedHouseholderReflect(&mat_T[(i + 1) + n * (i + 1)], &v[i], tau[i], n - (i + 1), n - (i + 1), 1, n); // Save v mat_T[i + n * (i + 1)] = R_ii; mat_T[(i + 1) + n * i] = R_ii; for (CeedInt j = i + 1; j < n - 1; j++) { mat_T[i + n * (j + 1)] = v[j]; } } // Backwards accumulation of Q for (CeedInt i = n - 2; i >= 0; i--) { if (tau[i] > 0.0) { v[i] = 1; for (CeedInt j = i + 1; j < n - 1; j++) { v[j] = mat_T[i + n * (j + 1)]; mat_T[i + n * (j + 1)] = 0; } CeedHouseholderReflect(&mat[(i + 1) + n * (i + 1)], &v[i], tau[i], n - (i + 1), n - (i + 1), n, 1); } } // Reduce sub and super diagonal CeedInt p = 0, q = 0, itr = 0, max_itr = n * n * n * n; CeedScalar tol = CEED_EPSILON; while (itr < max_itr) { // Update p, q, size of reduced portions of diagonal p = 0; q = 0; for (CeedInt i = n - 2; i >= 0; i--) { if (fabs(mat_T[i + n * (i + 1)]) < tol) q += 1; else break; } for (CeedInt i = 0; i < n - q - 1; i++) { if (fabs(mat_T[i + n * (i + 1)]) < tol) p += 1; else break; } if (q == n - 1) break; // Finished reducing // Reduce tridiagonal portion CeedScalar t_nn = mat_T[(n - 1 - q) + n * (n - 1 - q)], t_nnm1 = mat_T[(n - 2 - q) + n * (n - 1 - q)]; CeedScalar d = (mat_T[(n - 2 - q) + n * (n - 2 - q)] - t_nn) / 2; CeedScalar mu = t_nn - t_nnm1 * t_nnm1 / (d + copysign(sqrt(d * d + t_nnm1 * t_nnm1), d)); CeedScalar x = mat_T[p + n * p] - mu; CeedScalar z = mat_T[p + n * (p + 1)]; for (CeedInt k = p; k < n - q - 1; k++) { // Compute Givens rotation CeedScalar c = 1, s = 0; if (fabs(z) > tol) { if (fabs(z) > fabs(x)) { CeedScalar tau = -x / z; s = 1 / sqrt(1 + tau * tau), c = s * tau; } else { CeedScalar tau = -z / x; c = 1 / sqrt(1 + tau * tau), s = c * tau; } } // Apply Givens rotation to T CeedGivensRotation(mat_T, c, s, CEED_NOTRANSPOSE, k, k + 1, n, n); CeedGivensRotation(mat_T, c, s, CEED_TRANSPOSE, k, k + 1, n, n); // Apply Givens rotation to Q CeedGivensRotation(mat, c, s, CEED_NOTRANSPOSE, k, k + 1, n, n); // Update x, z if (k < n - q - 2) { x = mat_T[k + n * (k + 1)]; z = mat_T[k + n * (k + 2)]; } } itr++; } // Save eigenvalues for (CeedInt i = 0; i < n; i++) lambda[i] = mat_T[i + n * i]; // Check convergence CeedCheck(itr < max_itr || q > n, ceed, CEED_ERROR_MINOR, "Symmetric QR failed to converge"); return CEED_ERROR_SUCCESS; } CeedPragmaOptimizeOn /** @brief Return Simultaneous Diagonalization of two matrices. This solves the generalized eigenvalue problem A x = lambda B x, where A and B are symmetric and B is positive definite. We generate the matrix X and vector Lambda such that X^T A X = Lambda and X^T B X = I. This is equivalent to the LAPACK routine 'sygv' with TYPE = 1. @param[in] ceed Ceed context for error handling @param[in] mat_A Row-major matrix to be factorized with eigenvalues @param[in] mat_B Row-major matrix to be factorized to identity @param[out] mat_X Row-major orthogonal matrix @param[out] lambda Vector of length n of generalized eigenvalues @param[in] n Number of rows/columns @return An error code: 0 - success, otherwise - failure @ref Utility **/ CeedPragmaOptimizeOff int CeedSimultaneousDiagonalization(Ceed ceed, CeedScalar *mat_A, CeedScalar *mat_B, CeedScalar *mat_X, CeedScalar *lambda, CeedInt n) { CeedScalar *mat_C, *mat_G, *vec_D; CeedCall(CeedCalloc(n * n, &mat_C)); CeedCall(CeedCalloc(n * n, &mat_G)); CeedCall(CeedCalloc(n, &vec_D)); // Compute B = G D G^T memcpy(mat_G, mat_B, n * n * sizeof(mat_B[0])); CeedCall(CeedSymmetricSchurDecomposition(ceed, mat_G, vec_D, n)); // Sort eigenvalues for (CeedInt i = n - 1; i >= 0; i--) { for (CeedInt j = 0; j < i; j++) { if (fabs(vec_D[j]) > fabs(vec_D[j + 1])) { CeedScalar temp; temp = vec_D[j]; vec_D[j] = vec_D[j + 1]; vec_D[j + 1] = temp; for (CeedInt k = 0; k < n; k++) { temp = mat_G[k * n + j]; mat_G[k * n + j] = mat_G[k * n + j + 1]; mat_G[k * n + j + 1] = temp; } } } } // Compute C = (G D^1/2)^-1 A (G D^1/2)^-T // = D^-1/2 G^T A G D^-1/2 // -- D = D^-1/2 for (CeedInt i = 0; i < n; i++) vec_D[i] = 1. / sqrt(vec_D[i]); // -- G = G D^-1/2 // -- C = D^-1/2 G^T for (CeedInt i = 0; i < n; i++) { for (CeedInt j = 0; j < n; j++) { mat_G[i * n + j] *= vec_D[j]; mat_C[j * n + i] = mat_G[i * n + j]; } } // -- X = (D^-1/2 G^T) A CeedCall(CeedMatrixMatrixMultiply(ceed, (const CeedScalar *)mat_C, (const CeedScalar *)mat_A, mat_X, n, n, n)); // -- C = (D^-1/2 G^T A) (G D^-1/2) CeedCall(CeedMatrixMatrixMultiply(ceed, (const CeedScalar *)mat_X, (const CeedScalar *)mat_G, mat_C, n, n, n)); // Compute Q^T C Q = lambda CeedCall(CeedSymmetricSchurDecomposition(ceed, mat_C, lambda, n)); // Sort eigenvalues for (CeedInt i = n - 1; i >= 0; i--) { for (CeedInt j = 0; j < i; j++) { if (fabs(lambda[j]) > fabs(lambda[j + 1])) { CeedScalar temp; temp = lambda[j]; lambda[j] = lambda[j + 1]; lambda[j + 1] = temp; for (CeedInt k = 0; k < n; k++) { temp = mat_C[k * n + j]; mat_C[k * n + j] = mat_C[k * n + j + 1]; mat_C[k * n + j + 1] = temp; } } } } // Set X = (G D^1/2)^-T Q // = G D^-1/2 Q CeedCall(CeedMatrixMatrixMultiply(ceed, (const CeedScalar *)mat_G, (const CeedScalar *)mat_C, mat_X, n, n, n)); // Cleanup CeedCall(CeedFree(&mat_C)); CeedCall(CeedFree(&mat_G)); CeedCall(CeedFree(&vec_D)); return CEED_ERROR_SUCCESS; } CeedPragmaOptimizeOn /// @} /// ---------------------------------------------------------------------------- /// CeedBasis Public API /// ---------------------------------------------------------------------------- /// @addtogroup CeedBasisUser /// @{ /** @brief Create a tensor-product basis for H^1 discretizations @param[in] ceed Ceed object where the CeedBasis will be created @param[in] dim Topological dimension @param[in] num_comp Number of field components (1 for scalar fields) @param[in] P_1d Number of nodes in one dimension @param[in] Q_1d Number of quadrature points in one dimension @param[in] interp_1d Row-major (Q_1d * P_1d) matrix expressing the values of nodal basis functions at quadrature points @param[in] grad_1d Row-major (Q_1d * P_1d) matrix expressing derivatives of nodal basis functions at quadrature points @param[in] q_ref_1d Array of length Q_1d holding the locations of quadrature points on the 1D reference element [-1, 1] @param[in] q_weight_1d Array of length Q_1d holding the quadrature weights on the reference element @param[out] basis Address of the variable where the newly created CeedBasis will be stored. @return An error code: 0 - success, otherwise - failure @ref User **/ int CeedBasisCreateTensorH1(Ceed ceed, CeedInt dim, CeedInt num_comp, CeedInt P_1d, CeedInt Q_1d, const CeedScalar *interp_1d, const CeedScalar *grad_1d, const CeedScalar *q_ref_1d, const CeedScalar *q_weight_1d, CeedBasis *basis) { if (!ceed->BasisCreateTensorH1) { Ceed delegate; CeedCall(CeedGetObjectDelegate(ceed, &delegate, "Basis")); CeedCheck(delegate, ceed, CEED_ERROR_UNSUPPORTED, "Backend does not support BasisCreateTensorH1"); CeedCall(CeedBasisCreateTensorH1(delegate, dim, num_comp, P_1d, Q_1d, interp_1d, grad_1d, q_ref_1d, q_weight_1d, basis)); return CEED_ERROR_SUCCESS; } CeedCheck(dim > 0, ceed, CEED_ERROR_DIMENSION, "Basis dimension must be a positive value"); CeedCheck(num_comp > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 component"); CeedCheck(P_1d > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 node"); CeedCheck(Q_1d > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 quadrature point"); CeedElemTopology topo = dim == 1 ? CEED_TOPOLOGY_LINE : dim == 2 ? CEED_TOPOLOGY_QUAD : CEED_TOPOLOGY_HEX; CeedCall(CeedCalloc(1, basis)); CeedCall(CeedReferenceCopy(ceed, &(*basis)->ceed)); (*basis)->ref_count = 1; (*basis)->is_tensor_basis = true; (*basis)->dim = dim; (*basis)->topo = topo; (*basis)->num_comp = num_comp; (*basis)->P_1d = P_1d; (*basis)->Q_1d = Q_1d; (*basis)->P = CeedIntPow(P_1d, dim); (*basis)->Q = CeedIntPow(Q_1d, dim); (*basis)->fe_space = CEED_FE_SPACE_H1; CeedCall(CeedCalloc(Q_1d, &(*basis)->q_ref_1d)); CeedCall(CeedCalloc(Q_1d, &(*basis)->q_weight_1d)); if (q_ref_1d) memcpy((*basis)->q_ref_1d, q_ref_1d, Q_1d * sizeof(q_ref_1d[0])); if (q_weight_1d) memcpy((*basis)->q_weight_1d, q_weight_1d, Q_1d * sizeof(q_weight_1d[0])); CeedCall(CeedCalloc(Q_1d * P_1d, &(*basis)->interp_1d)); CeedCall(CeedCalloc(Q_1d * P_1d, &(*basis)->grad_1d)); if (interp_1d) memcpy((*basis)->interp_1d, interp_1d, Q_1d * P_1d * sizeof(interp_1d[0])); if (grad_1d) memcpy((*basis)->grad_1d, grad_1d, Q_1d * P_1d * sizeof(grad_1d[0])); CeedCall(ceed->BasisCreateTensorH1(dim, P_1d, Q_1d, interp_1d, grad_1d, q_ref_1d, q_weight_1d, *basis)); return CEED_ERROR_SUCCESS; } /** @brief Create a tensor-product Lagrange basis @param[in] ceed Ceed object where the CeedBasis will be created @param[in] dim Topological dimension of element @param[in] num_comp Number of field components (1 for scalar fields) @param[in] P Number of Gauss-Lobatto nodes in one dimension. The polynomial degree of the resulting Q_k element is k=P-1. @param[in] Q Number of quadrature points in one dimension. @param[in] quad_mode Distribution of the Q quadrature points (affects order of accuracy for the quadrature) @param[out] basis Address of the variable where the newly created CeedBasis will be stored. @return An error code: 0 - success, otherwise - failure @ref User **/ int CeedBasisCreateTensorH1Lagrange(Ceed ceed, CeedInt dim, CeedInt num_comp, CeedInt P, CeedInt Q, CeedQuadMode quad_mode, CeedBasis *basis) { // Allocate int ierr = CEED_ERROR_SUCCESS; CeedScalar c1, c2, c3, c4, dx, *nodes, *interp_1d, *grad_1d, *q_ref_1d, *q_weight_1d; CeedCheck(dim > 0, ceed, CEED_ERROR_DIMENSION, "Basis dimension must be a positive value"); CeedCheck(num_comp > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 component"); CeedCheck(P > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 node"); CeedCheck(Q > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 quadrature point"); // Get Nodes and Weights CeedCall(CeedCalloc(P * Q, &interp_1d)); CeedCall(CeedCalloc(P * Q, &grad_1d)); CeedCall(CeedCalloc(P, &nodes)); CeedCall(CeedCalloc(Q, &q_ref_1d)); CeedCall(CeedCalloc(Q, &q_weight_1d)); if (CeedLobattoQuadrature(P, nodes, NULL) != CEED_ERROR_SUCCESS) goto cleanup; switch (quad_mode) { case CEED_GAUSS: ierr = CeedGaussQuadrature(Q, q_ref_1d, q_weight_1d); break; case CEED_GAUSS_LOBATTO: ierr = CeedLobattoQuadrature(Q, q_ref_1d, q_weight_1d); break; } if (ierr != CEED_ERROR_SUCCESS) goto cleanup; // Build B, D matrix // Fornberg, 1998 for (CeedInt i = 0; i < Q; i++) { c1 = 1.0; c3 = nodes[0] - q_ref_1d[i]; interp_1d[i * P + 0] = 1.0; for (CeedInt j = 1; j < P; j++) { c2 = 1.0; c4 = c3; c3 = nodes[j] - q_ref_1d[i]; for (CeedInt k = 0; k < j; k++) { dx = nodes[j] - nodes[k]; c2 *= dx; if (k == j - 1) { grad_1d[i * P + j] = c1 * (interp_1d[i * P + k] - c4 * grad_1d[i * P + k]) / c2; interp_1d[i * P + j] = -c1 * c4 * interp_1d[i * P + k] / c2; } grad_1d[i * P + k] = (c3 * grad_1d[i * P + k] - interp_1d[i * P + k]) / dx; interp_1d[i * P + k] = c3 * interp_1d[i * P + k] / dx; } c1 = c2; } } // Pass to CeedBasisCreateTensorH1 CeedCall(CeedBasisCreateTensorH1(ceed, dim, num_comp, P, Q, interp_1d, grad_1d, q_ref_1d, q_weight_1d, basis)); cleanup: CeedCall(CeedFree(&interp_1d)); CeedCall(CeedFree(&grad_1d)); CeedCall(CeedFree(&nodes)); CeedCall(CeedFree(&q_ref_1d)); CeedCall(CeedFree(&q_weight_1d)); return CEED_ERROR_SUCCESS; } /** @brief Create a non tensor-product basis for H^1 discretizations @param[in] ceed Ceed object where the CeedBasis will be created @param[in] topo Topology of element, e.g. hypercube, simplex, ect @param[in] num_comp Number of field components (1 for scalar fields) @param[in] num_nodes Total number of nodes @param[in] num_qpts Total number of quadrature points @param[in] interp Row-major (num_qpts * num_nodes) matrix expressing the values of nodal basis functions at quadrature points @param[in] grad Row-major (dim * num_qpts * num_nodes) matrix expressing derivatives of nodal basis functions at quadrature points @param[in] q_ref Array of length num_qpts * dim holding the locations of quadrature points on the reference element @param[in] q_weight Array of length num_qpts holding the quadrature weights on the reference element @param[out] basis Address of the variable where the newly created CeedBasis will be stored. @return An error code: 0 - success, otherwise - failure @ref User **/ int CeedBasisCreateH1(Ceed ceed, CeedElemTopology topo, CeedInt num_comp, CeedInt num_nodes, CeedInt num_qpts, const CeedScalar *interp, const CeedScalar *grad, const CeedScalar *q_ref, const CeedScalar *q_weight, CeedBasis *basis) { CeedInt P = num_nodes, Q = num_qpts, dim = 0; if (!ceed->BasisCreateH1) { Ceed delegate; CeedCall(CeedGetObjectDelegate(ceed, &delegate, "Basis")); CeedCheck(delegate, ceed, CEED_ERROR_UNSUPPORTED, "Backend does not support BasisCreateH1"); CeedCall(CeedBasisCreateH1(delegate, topo, num_comp, num_nodes, num_qpts, interp, grad, q_ref, q_weight, basis)); return CEED_ERROR_SUCCESS; } CeedCheck(num_comp > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 component"); CeedCheck(num_nodes > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 node"); CeedCheck(num_qpts > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 quadrature point"); CeedCall(CeedBasisGetTopologyDimension(topo, &dim)); CeedCall(CeedCalloc(1, basis)); CeedCall(CeedReferenceCopy(ceed, &(*basis)->ceed)); (*basis)->ref_count = 1; (*basis)->is_tensor_basis = false; (*basis)->dim = dim; (*basis)->topo = topo; (*basis)->num_comp = num_comp; (*basis)->P = P; (*basis)->Q = Q; (*basis)->fe_space = CEED_FE_SPACE_H1; CeedCall(CeedCalloc(Q * dim, &(*basis)->q_ref_1d)); CeedCall(CeedCalloc(Q, &(*basis)->q_weight_1d)); if (q_ref) memcpy((*basis)->q_ref_1d, q_ref, Q * dim * sizeof(q_ref[0])); if (q_weight) memcpy((*basis)->q_weight_1d, q_weight, Q * sizeof(q_weight[0])); CeedCall(CeedCalloc(Q * P, &(*basis)->interp)); CeedCall(CeedCalloc(dim * Q * P, &(*basis)->grad)); if (interp) memcpy((*basis)->interp, interp, Q * P * sizeof(interp[0])); if (grad) memcpy((*basis)->grad, grad, dim * Q * P * sizeof(grad[0])); CeedCall(ceed->BasisCreateH1(topo, dim, P, Q, interp, grad, q_ref, q_weight, *basis)); return CEED_ERROR_SUCCESS; } /** @brief Create a non tensor-product basis for \f$H(\mathrm{div})\f$ discretizations @param[in] ceed Ceed object where the CeedBasis will be created @param[in] topo Topology of element (`CEED_TOPOLOGY_QUAD`, `CEED_TOPOLOGY_PRISM`, etc.), dimension of which is used in some array sizes below @param[in] num_comp Number of components (usually 1 for vectors in H(div) bases) @param[in] num_nodes Total number of nodes (dofs per element) @param[in] num_qpts Total number of quadrature points @param[in] interp Row-major (dim * num_qpts * num_nodes) matrix expressing the values of basis functions at quadrature points @param[in] div Row-major (num_qpts * num_nodes) matrix expressing divergence of basis functions at quadrature points @param[in] q_ref Array of length num_qpts * dim holding the locations of quadrature points on the reference element @param[in] q_weight Array of length num_qpts holding the quadrature weights on the reference element @param[out] basis Address of the variable where the newly created CeedBasis will be stored. @return An error code: 0 - success, otherwise - failure @ref User **/ int CeedBasisCreateHdiv(Ceed ceed, CeedElemTopology topo, CeedInt num_comp, CeedInt num_nodes, CeedInt num_qpts, const CeedScalar *interp, const CeedScalar *div, const CeedScalar *q_ref, const CeedScalar *q_weight, CeedBasis *basis) { CeedInt Q = num_qpts, P = num_nodes, dim = 0; if (!ceed->BasisCreateHdiv) { Ceed delegate; CeedCall(CeedGetObjectDelegate(ceed, &delegate, "Basis")); CeedCheck(delegate, ceed, CEED_ERROR_UNSUPPORTED, "Backend does not implement BasisCreateHdiv"); CeedCall(CeedBasisCreateHdiv(delegate, topo, num_comp, num_nodes, num_qpts, interp, div, q_ref, q_weight, basis)); return CEED_ERROR_SUCCESS; } CeedCheck(num_comp > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 component"); CeedCheck(num_nodes > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 node"); CeedCheck(num_qpts > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 quadrature point"); CeedCall(CeedBasisGetTopologyDimension(topo, &dim)); CeedCall(CeedCalloc(1, basis)); CeedCall(CeedReferenceCopy(ceed, &(*basis)->ceed)); (*basis)->ref_count = 1; (*basis)->is_tensor_basis = false; (*basis)->dim = dim; (*basis)->topo = topo; (*basis)->num_comp = num_comp; (*basis)->P = P; (*basis)->Q = Q; (*basis)->fe_space = CEED_FE_SPACE_HDIV; CeedCall(CeedMalloc(Q * dim, &(*basis)->q_ref_1d)); CeedCall(CeedMalloc(Q, &(*basis)->q_weight_1d)); if (q_ref) memcpy((*basis)->q_ref_1d, q_ref, Q * dim * sizeof(q_ref[0])); if (q_weight) memcpy((*basis)->q_weight_1d, q_weight, Q * sizeof(q_weight[0])); CeedCall(CeedMalloc(dim * Q * P, &(*basis)->interp)); CeedCall(CeedMalloc(Q * P, &(*basis)->div)); if (interp) memcpy((*basis)->interp, interp, dim * Q * P * sizeof(interp[0])); if (div) memcpy((*basis)->div, div, Q * P * sizeof(div[0])); CeedCall(ceed->BasisCreateHdiv(topo, dim, P, Q, interp, div, q_ref, q_weight, *basis)); return CEED_ERROR_SUCCESS; } /** @brief Create a non tensor-product basis for \f$H(\mathrm{curl})\f$ discretizations @param[in] ceed Ceed object where the CeedBasis will be created @param[in] topo Topology of element (`CEED_TOPOLOGY_QUAD`, `CEED_TOPOLOGY_PRISM`, etc.), dimension of which is used in some array sizes below @param[in] num_comp Number of components (usually 1 for vectors in H(curl) bases) @param[in] num_nodes Total number of nodes (dofs per element) @param[in] num_qpts Total number of quadrature points @param[in] interp Row-major (dim * num_qpts * num_nodes) matrix expressing the values of basis functions at quadrature points @param[in] curl Row-major (curl_comp * num_qpts * num_nodes, curl_comp = 1 if dim < 3 else dim) matrix expressing curl of basis functions at quadrature points @param[in] q_ref Array of length num_qpts * dim holding the locations of quadrature points on the reference element @param[in] q_weight Array of length num_qpts holding the quadrature weights on the reference element @param[out] basis Address of the variable where the newly created CeedBasis will be stored. @return An error code: 0 - success, otherwise - failure @ref User **/ int CeedBasisCreateHcurl(Ceed ceed, CeedElemTopology topo, CeedInt num_comp, CeedInt num_nodes, CeedInt num_qpts, const CeedScalar *interp, const CeedScalar *curl, const CeedScalar *q_ref, const CeedScalar *q_weight, CeedBasis *basis) { CeedInt Q = num_qpts, P = num_nodes, dim = 0, curl_comp = 0; if (!ceed->BasisCreateHdiv) { Ceed delegate; CeedCall(CeedGetObjectDelegate(ceed, &delegate, "Basis")); CeedCheck(delegate, ceed, CEED_ERROR_UNSUPPORTED, "Backend does not implement BasisCreateHcurl"); CeedCall(CeedBasisCreateHcurl(delegate, topo, num_comp, num_nodes, num_qpts, interp, curl, q_ref, q_weight, basis)); return CEED_ERROR_SUCCESS; } CeedCheck(num_comp > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 component"); CeedCheck(num_nodes > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 node"); CeedCheck(num_qpts > 0, ceed, CEED_ERROR_DIMENSION, "Basis must have at least 1 quadrature point"); CeedCall(CeedBasisGetTopologyDimension(topo, &dim)); curl_comp = (dim < 3) ? 1 : dim; CeedCall(CeedCalloc(1, basis)); CeedCall(CeedReferenceCopy(ceed, &(*basis)->ceed)); (*basis)->ref_count = 1; (*basis)->is_tensor_basis = false; (*basis)->dim = dim; (*basis)->topo = topo; (*basis)->num_comp = num_comp; (*basis)->P = P; (*basis)->Q = Q; (*basis)->fe_space = CEED_FE_SPACE_HCURL; CeedCall(CeedMalloc(Q * dim, &(*basis)->q_ref_1d)); CeedCall(CeedMalloc(Q, &(*basis)->q_weight_1d)); if (q_ref) memcpy((*basis)->q_ref_1d, q_ref, Q * dim * sizeof(q_ref[0])); if (q_weight) memcpy((*basis)->q_weight_1d, q_weight, Q * sizeof(q_weight[0])); CeedCall(CeedMalloc(dim * Q * P, &(*basis)->interp)); CeedCall(CeedMalloc(curl_comp * Q * P, &(*basis)->curl)); if (interp) memcpy((*basis)->interp, interp, dim * Q * P * sizeof(interp[0])); if (curl) memcpy((*basis)->curl, curl, curl_comp * Q * P * sizeof(curl[0])); CeedCall(ceed->BasisCreateHcurl(topo, dim, P, Q, interp, curl, q_ref, q_weight, *basis)); return CEED_ERROR_SUCCESS; } /** @brief Create a CeedBasis for projection from the nodes of `basis_from` to the nodes of `basis_to`. Only `CEED_EVAL_INTERP` will be valid for the new basis, `basis_project`. For H^1 spaces, `CEED_EVAL_GRAD` will also be valid. The interpolation is given by `interp_project = interp_to^+ * interp_from`, where the pseudoinverse `interp_to^+` is given by QR factorization. The gradient (for the H^1 case) is given by `grad_project = interp_to^+ * grad_from`. Note: `basis_from` and `basis_to` must have compatible quadrature spaces. Note: `basis_project` will have the same number of components as `basis_from`, regardless of the number of components that `basis_to` has. If `basis_from` has 3 components and `basis_to` has 5 components, then `basis_project` will have 3 components. @param[in] basis_from CeedBasis to prolong from @param[in] basis_to CeedBasis to prolong to @param[out] basis_project Address of the variable where the newly created CeedBasis will be stored. @return An error code: 0 - success, otherwise - failure @ref User **/ int CeedBasisCreateProjection(CeedBasis basis_from, CeedBasis basis_to, CeedBasis *basis_project) { Ceed ceed; CeedCall(CeedBasisGetCeed(basis_to, &ceed)); // Create projection matrix CeedScalar *interp_project, *grad_project; CeedCall(CeedBasisCreateProjectionMatrices(basis_from, basis_to, &interp_project, &grad_project)); // Build basis bool is_tensor; CeedInt dim, num_comp; CeedScalar *q_ref, *q_weight; CeedCall(CeedBasisIsTensor(basis_to, &is_tensor)); CeedCall(CeedBasisGetDimension(basis_to, &dim)); CeedCall(CeedBasisGetNumComponents(basis_from, &num_comp)); if (is_tensor) { CeedInt P_1d_to, P_1d_from; CeedCall(CeedBasisGetNumNodes1D(basis_from, &P_1d_from)); CeedCall(CeedBasisGetNumNodes1D(basis_to, &P_1d_to)); CeedCall(CeedCalloc(P_1d_to, &q_ref)); CeedCall(CeedCalloc(P_1d_to, &q_weight)); CeedCall(CeedBasisCreateTensorH1(ceed, dim, num_comp, P_1d_from, P_1d_to, interp_project, grad_project, q_ref, q_weight, basis_project)); } else { // Even if basis_to and basis_from are not H1, the resulting basis is H1 for interpolation to work CeedElemTopology topo; CeedCall(CeedBasisGetTopology(basis_to, &topo)); CeedInt num_nodes_to, num_nodes_from; CeedCall(CeedBasisGetNumNodes(basis_from, &num_nodes_from)); CeedCall(CeedBasisGetNumNodes(basis_to, &num_nodes_to)); CeedCall(CeedCalloc(num_nodes_to * dim, &q_ref)); CeedCall(CeedCalloc(num_nodes_to, &q_weight)); CeedCall(CeedBasisCreateH1(ceed, topo, num_comp, num_nodes_from, num_nodes_to, interp_project, grad_project, q_ref, q_weight, basis_project)); } // Cleanup CeedCall(CeedFree(&interp_project)); CeedCall(CeedFree(&grad_project)); CeedCall(CeedFree(&q_ref)); CeedCall(CeedFree(&q_weight)); return CEED_ERROR_SUCCESS; } /** @brief Copy the pointer to a CeedBasis. Note: If the value of `basis_copy` passed into this function is non-NULL, then it is assumed that `basis_copy` is a pointer to a CeedBasis. This CeedBasis will be destroyed if `basis_copy` is the only reference to this CeedBasis. @param[in] basis CeedBasis to copy reference to @param[in,out] basis_copy Variable to store copied reference @return An error code: 0 - success, otherwise - failure @ref User **/ int CeedBasisReferenceCopy(CeedBasis basis, CeedBasis *basis_copy) { if (basis != CEED_BASIS_COLLOCATED) CeedCall(CeedBasisReference(basis)); CeedCall(CeedBasisDestroy(basis_copy)); *basis_copy = basis; return CEED_ERROR_SUCCESS; } /** @brief View a CeedBasis @param[in] basis CeedBasis to view @param[in] stream Stream to view to, e.g., stdout @return An error code: 0 - success, otherwise - failure @ref User **/ int CeedBasisView(CeedBasis basis, FILE *stream) { CeedElemTopology topo = basis->topo; CeedFESpace fe_space = basis->fe_space; CeedInt q_comp = 0; // Print FE space and element topology of the basis fprintf(stream, "CeedBasis in a %s on a %s element\n", CeedFESpaces[fe_space], CeedElemTopologies[topo]); if (basis->is_tensor_basis) { fprintf(stream, " P: %" CeedInt_FMT "\n Q: %" CeedInt_FMT "\n", basis->P_1d, basis->Q_1d); } else { fprintf(stream, " P: %" CeedInt_FMT "\n Q: %" CeedInt_FMT "\n", basis->P, basis->Q); } fprintf(stream, " dimension: %" CeedInt_FMT "\n field components: %" CeedInt_FMT "\n", basis->dim, basis->num_comp); // Print quadrature data, interpolation/gradient/divergence/curl of the basis if (basis->is_tensor_basis) { // tensor basis CeedCall(CeedScalarView("qref1d", "\t% 12.8f", 1, basis->Q_1d, basis->q_ref_1d, stream)); CeedCall(CeedScalarView("qweight1d", "\t% 12.8f", 1, basis->Q_1d, basis->q_weight_1d, stream)); CeedCall(CeedScalarView("interp1d", "\t% 12.8f", basis->Q_1d, basis->P_1d, basis->interp_1d, stream)); CeedCall(CeedScalarView("grad1d", "\t% 12.8f", basis->Q_1d, basis->P_1d, basis->grad_1d, stream)); } else { // non-tensor basis CeedCall(CeedScalarView("qref", "\t% 12.8f", 1, basis->Q * basis->dim, basis->q_ref_1d, stream)); CeedCall(CeedScalarView("qweight", "\t% 12.8f", 1, basis->Q, basis->q_weight_1d, stream)); CeedCall(CeedBasisGetNumQuadratureComponents(basis, CEED_EVAL_INTERP, &q_comp)); CeedCall(CeedScalarView("interp", "\t% 12.8f", q_comp * basis->Q, basis->P, basis->interp, stream)); if (basis->grad) { CeedCall(CeedBasisGetNumQuadratureComponents(basis, CEED_EVAL_GRAD, &q_comp)); CeedCall(CeedScalarView("grad", "\t% 12.8f", q_comp * basis->Q, basis->P, basis->grad, stream)); } if (basis->div) { CeedCall(CeedBasisGetNumQuadratureComponents(basis, CEED_EVAL_DIV, &q_comp)); CeedCall(CeedScalarView("div", "\t% 12.8f", q_comp * basis->Q, basis->P, basis->div, stream)); } if (basis->curl) { CeedCall(CeedBasisGetNumQuadratureComponents(basis, CEED_EVAL_CURL, &q_comp)); CeedCall(CeedScalarView("curl", "\t% 12.8f", q_comp * basis->Q, basis->P, basis->curl, stream)); } } return CEED_ERROR_SUCCESS; } /** @brief Apply basis evaluation from nodes to quadrature points or vice versa @param[in] basis CeedBasis to evaluate @param[in] num_elem The number of elements to apply the basis evaluation to; the backend will specify the ordering in CeedElemRestrictionCreateBlocked() @param[in] t_mode \ref CEED_NOTRANSPOSE to evaluate from nodes to quadrature points; \ref CEED_TRANSPOSE to apply the transpose, mapping from quadrature points to nodes @param[in] eval_mode \ref CEED_EVAL_NONE to use values directly, \ref CEED_EVAL_INTERP to use interpolated values, \ref CEED_EVAL_GRAD to use gradients, \ref CEED_EVAL_DIV to use divergence, \ref CEED_EVAL_CURL to use curl, \ref CEED_EVAL_WEIGHT to use quadrature weights. @param[in] u Input CeedVector @param[out] v Output CeedVector @return An error code: 0 - success, otherwise - failure @ref User **/ int CeedBasisApply(CeedBasis basis, CeedInt num_elem, CeedTransposeMode t_mode, CeedEvalMode eval_mode, CeedVector u, CeedVector v) { CeedSize u_length = 0, v_length; CeedInt dim, num_comp, q_comp, num_nodes, num_qpts; CeedCall(CeedBasisGetDimension(basis, &dim)); CeedCall(CeedBasisGetNumComponents(basis, &num_comp)); CeedCall(CeedBasisGetNumQuadratureComponents(basis, eval_mode, &q_comp)); CeedCall(CeedBasisGetNumNodes(basis, &num_nodes)); CeedCall(CeedBasisGetNumQuadraturePoints(basis, &num_qpts)); CeedCall(CeedVectorGetLength(v, &v_length)); if (u) CeedCall(CeedVectorGetLength(u, &u_length)); CeedCheck(basis->Apply, basis->ceed, CEED_ERROR_UNSUPPORTED, "Backend does not support BasisApply"); // Check compatibility of topological and geometrical dimensions CeedCheck((t_mode == CEED_TRANSPOSE && v_length % num_nodes == 0 && u_length % num_qpts == 0) || (t_mode == CEED_NOTRANSPOSE && u_length % num_nodes == 0 && v_length % num_qpts == 0), basis->ceed, CEED_ERROR_DIMENSION, "Length of input/output vectors incompatible with basis dimensions"); // Check vector lengths to prevent out of bounds issues bool good_dims = true; switch (eval_mode) { case CEED_EVAL_NONE: case CEED_EVAL_INTERP: case CEED_EVAL_GRAD: case CEED_EVAL_DIV: case CEED_EVAL_CURL: good_dims = ((t_mode == CEED_TRANSPOSE && u_length >= num_elem * num_comp * num_qpts * q_comp && v_length >= num_elem * num_comp * num_nodes) || (t_mode == CEED_NOTRANSPOSE && v_length >= num_elem * num_qpts * num_comp * q_comp && u_length >= num_elem * num_comp * num_nodes)); break; case CEED_EVAL_WEIGHT: good_dims = v_length >= num_elem * num_qpts; break; } CeedCheck(good_dims, basis->ceed, CEED_ERROR_DIMENSION, "Input/output vectors too short for basis and evaluation mode"); CeedCall(basis->Apply(basis, num_elem, t_mode, eval_mode, u, v)); return CEED_ERROR_SUCCESS; } /** @brief Apply basis evaluation from nodes to arbitrary points @param[in] basis CeedBasis to evaluate @param[in] num_points The number of points to apply the basis evaluation to @param[in] t_mode \ref CEED_NOTRANSPOSE to evaluate from nodes to points; \ref CEED_TRANSPOSE to apply the transpose, mapping from points to nodes @param[in] eval_mode \ref CEED_EVAL_INTERP to use interpolated values, \ref CEED_EVAL_GRAD to use gradients @param[in] x_ref CeedVector holding reference coordinates of each point @param[in] u Input CeedVector, of length `num_nodes * num_comp` for `CEED_NOTRANSPOSE` @param[out] v Output CeedVector, of length `num_points * num_q_comp` for `CEED_NOTRANSPOSE` with `CEED_EVAL_INTERP` @return An error code: 0 - success, otherwise - failure @ref User **/ int CeedBasisApplyAtPoints(CeedBasis basis, CeedInt num_points, CeedTransposeMode t_mode, CeedEvalMode eval_mode, CeedVector x_ref, CeedVector u, CeedVector v) { CeedSize x_length = 0, u_length = 0, v_length; CeedInt dim, num_comp, num_q_comp, num_nodes, P_1d = 1, Q_1d = 1; CeedCall(CeedBasisGetDimension(basis, &dim)); CeedCall(CeedBasisGetNumNodes1D(basis, &P_1d)); CeedCall(CeedBasisGetNumQuadraturePoints1D(basis, &Q_1d)); CeedCall(CeedBasisGetNumComponents(basis, &num_comp)); CeedCall(CeedBasisGetNumQuadratureComponents(basis, eval_mode, &num_q_comp)); CeedCall(CeedBasisGetNumNodes(basis, &num_nodes)); CeedCall(CeedVectorGetLength(x_ref, &x_length)); CeedCall(CeedVectorGetLength(v, &v_length)); CeedCall(CeedVectorGetLength(u, &u_length)); // Check compatibility of topological and geometrical dimensions CeedCheck((t_mode == CEED_TRANSPOSE && v_length % num_nodes == 0) || (t_mode == CEED_NOTRANSPOSE && u_length % num_nodes == 0), basis->ceed, CEED_ERROR_DIMENSION, "Length of input/output vectors incompatible with basis dimensions and number of points"); // Check compatibility coordinates vector CeedCheck(x_length >= num_points * dim, basis->ceed, CEED_ERROR_DIMENSION, "Length of reference coordinate vector incompatible with basis dimension and number of points"); // Check vector lengths to prevent out of bounds issues bool good_dims = false; switch (eval_mode) { case CEED_EVAL_INTERP: good_dims = ((t_mode == CEED_TRANSPOSE && (u_length >= num_points * num_q_comp || v_length >= num_nodes * num_comp)) || (t_mode == CEED_NOTRANSPOSE && (v_length >= num_points * num_q_comp || u_length >= num_nodes * num_comp))); break; case CEED_EVAL_GRAD: good_dims = ((t_mode == CEED_TRANSPOSE && (u_length >= num_points * num_q_comp * dim || v_length >= num_nodes * num_comp)) || (t_mode == CEED_NOTRANSPOSE && (v_length >= num_points * num_q_comp * dim || u_length >= num_nodes * num_comp))); break; case CEED_EVAL_NONE: case CEED_EVAL_WEIGHT: case CEED_EVAL_DIV: case CEED_EVAL_CURL: // LCOV_EXCL_START return CeedError(basis->ceed, CEED_ERROR_UNSUPPORTED, "Evaluation at arbitrary points not supported for %s", CeedEvalModes[eval_mode]); // LCOV_EXCL_STOP } CeedCheck(good_dims, basis->ceed, CEED_ERROR_DIMENSION, "Input/output vectors too short for basis and evaluation mode"); // Backend method if (basis->ApplyAtPoints) { CeedCall(basis->ApplyAtPoints(basis, num_points, t_mode, eval_mode, x_ref, u, v)); return CEED_ERROR_SUCCESS; } // Default implementation CeedCheck(basis->is_tensor_basis, basis->ceed, CEED_ERROR_UNSUPPORTED, "Evaluation at arbitrary points only supported for tensor product bases"); CeedCheck(eval_mode == CEED_EVAL_INTERP || t_mode == CEED_NOTRANSPOSE, basis->ceed, CEED_ERROR_UNSUPPORTED, "%s evaluation only supported for %s", CeedEvalModes[eval_mode], CeedTransposeModes[CEED_NOTRANSPOSE]); if (!basis->basis_chebyshev) { // Build matrix mapping from quadrature point values to Chebyshev coefficients CeedScalar *tau, *C, *I, *chebyshev_coeffs_1d; const CeedScalar *q_ref_1d; // Build coefficient matrix // -- Note: Clang-tidy needs this check because it does not understand the is_tensor_basis check above CeedCheck(P_1d > 0 && Q_1d > 0, basis->ceed, CEED_ERROR_INCOMPATIBLE, "Basis dimensions are malformed"); CeedCall(CeedCalloc(Q_1d * Q_1d, &C)); CeedCall(CeedBasisGetQRef(basis, &q_ref_1d)); for (CeedInt i = 0; i < Q_1d; i++) CeedCall(CeedChebyshevPolynomialsAtPoint(q_ref_1d[i], Q_1d, &C[i * Q_1d])); // Inverse of coefficient matrix CeedCall(CeedCalloc(Q_1d * Q_1d, &chebyshev_coeffs_1d)); CeedCall(CeedCalloc(Q_1d * Q_1d, &I)); CeedCall(CeedCalloc(Q_1d, &tau)); // -- QR Factorization, C = Q R CeedCall(CeedQRFactorization(basis->ceed, C, tau, Q_1d, Q_1d)); // -- chebyshev_coeffs_1d = R_inv Q^T for (CeedInt i = 0; i < Q_1d; i++) I[i * Q_1d + i] = 1.0; // ---- Apply R_inv, chebyshev_coeffs_1d = I R_inv for (CeedInt i = 0; i < Q_1d; i++) { // Row i chebyshev_coeffs_1d[Q_1d * i] = I[Q_1d * i] / C[0]; for (CeedInt j = 1; j < Q_1d; j++) { // Column j chebyshev_coeffs_1d[j + Q_1d * i] = I[j + Q_1d * i]; for (CeedInt k = 0; k < j; k++) chebyshev_coeffs_1d[j + Q_1d * i] -= C[j + Q_1d * k] * chebyshev_coeffs_1d[k + Q_1d * i]; chebyshev_coeffs_1d[j + Q_1d * i] /= C[j + Q_1d * j]; } } // ---- Apply Q^T, chebyshev_coeffs_1d = R_inv Q^T CeedCall(CeedHouseholderApplyQ(chebyshev_coeffs_1d, C, tau, CEED_NOTRANSPOSE, Q_1d, Q_1d, Q_1d, 1, Q_1d)); // Build basis mapping from nodes to Chebyshev coefficients CeedScalar *chebyshev_interp_1d, *chebyshev_grad_1d, *chebyshev_q_weight_1d; const CeedScalar *interp_1d; CeedCall(CeedCalloc(Q_1d * Q_1d, &chebyshev_interp_1d)); CeedCall(CeedCalloc(Q_1d * Q_1d, &chebyshev_grad_1d)); CeedCall(CeedCalloc(Q_1d, &chebyshev_q_weight_1d)); CeedCall(CeedBasisGetInterp1D(basis, &interp_1d)); CeedCall(CeedMatrixMatrixMultiply(basis->ceed, chebyshev_coeffs_1d, interp_1d, chebyshev_interp_1d, Q_1d, P_1d, Q_1d)); CeedCall(CeedVectorCreate(basis->ceed, num_comp * CeedIntPow(Q_1d, dim), &basis->vec_chebyshev)); CeedCall(CeedBasisCreateTensorH1(basis->ceed, dim, num_comp, Q_1d, Q_1d, chebyshev_interp_1d, chebyshev_grad_1d, q_ref_1d, chebyshev_q_weight_1d, &basis->basis_chebyshev)); // Cleanup CeedCall(CeedFree(&C)); CeedCall(CeedFree(&chebyshev_coeffs_1d)); CeedCall(CeedFree(&I)); CeedCall(CeedFree(&tau)); CeedCall(CeedFree(&chebyshev_interp_1d)); CeedCall(CeedFree(&chebyshev_grad_1d)); CeedCall(CeedFree(&chebyshev_q_weight_1d)); } // Create TensorContract object if needed, such as a basis from the GPU backends if (!basis->contract) { Ceed ceed_ref; CeedBasis basis_ref; CeedCall(CeedInit("/cpu/self", &ceed_ref)); // Only need matching tensor contraction dimensions, any type of basis will work CeedCall(CeedBasisCreateTensorH1Lagrange(ceed_ref, dim, num_comp, Q_1d, Q_1d, CEED_GAUSS, &basis_ref)); CeedCall(CeedTensorContractReference(basis_ref->contract)); basis->contract = basis_ref->contract; CeedCall(CeedBasisDestroy(&basis_ref)); CeedCall(CeedDestroy(&ceed_ref)); } // Basis evaluation switch (t_mode) { case CEED_NOTRANSPOSE: { // Nodes to arbitrary points CeedScalar *v_array; const CeedScalar *chebyshev_coeffs, *x_array_read; // -- Interpolate to Chebyshev coefficients CeedCall(CeedBasisApply(basis->basis_chebyshev, 1, CEED_NOTRANSPOSE, CEED_EVAL_INTERP, u, basis->vec_chebyshev)); // -- Evaluate Chebyshev polynomials at arbitrary points CeedCall(CeedVectorGetArrayRead(basis->vec_chebyshev, CEED_MEM_HOST, &chebyshev_coeffs)); CeedCall(CeedVectorGetArrayRead(x_ref, CEED_MEM_HOST, &x_array_read)); CeedCall(CeedVectorGetArrayWrite(v, CEED_MEM_HOST, &v_array)); switch (eval_mode) { case CEED_EVAL_INTERP: { CeedScalar tmp[2][num_comp * CeedIntPow(Q_1d, dim)], chebyshev_x[Q_1d]; // ---- Values at point for (CeedInt p = 0; p < num_points; p++) { CeedInt pre = num_comp * CeedIntPow(Q_1d, dim - 1), post = 1; // Note: stepping "backwards" through the tensor contractions to agree with the ordering of the Chebyshev coefficients for (CeedInt d = dim - 1; d >= 0; d--) { // ------ Tensor contract with current Chebyshev polynomial values CeedCall(CeedChebyshevPolynomialsAtPoint(x_array_read[p * dim + d], Q_1d, chebyshev_x)); CeedCall(CeedTensorContractApply(basis->contract, pre, Q_1d, post, 1, chebyshev_x, t_mode, false, d == (dim - 1) ? chebyshev_coeffs : tmp[d % 2], d == 0 ? &v_array[p * num_comp] : tmp[(d + 1) % 2])); pre /= Q_1d; post *= 1; } } break; } case CEED_EVAL_GRAD: { CeedScalar tmp[2][num_comp * CeedIntPow(Q_1d, dim)], chebyshev_x[Q_1d]; // ---- Values at point for (CeedInt p = 0; p < num_points; p++) { // Note: stepping "backwards" through the tensor contractions to agree with the ordering of the Chebyshev coefficients // Dim**2 contractions, apply grad when pass == dim for (CeedInt pass = dim - 1; pass >= 0; pass--) { CeedInt pre = num_comp * CeedIntPow(Q_1d, dim - 1), post = 1; for (CeedInt d = dim - 1; d >= 0; d--) { // ------ Tensor contract with current Chebyshev polynomial values if (pass == d) CeedCall(CeedChebyshevDerivativeAtPoint(x_array_read[p * dim + d], Q_1d, chebyshev_x)); else CeedCall(CeedChebyshevPolynomialsAtPoint(x_array_read[p * dim + d], Q_1d, chebyshev_x)); CeedCall(CeedTensorContractApply(basis->contract, pre, Q_1d, post, 1, chebyshev_x, t_mode, false, d == (dim - 1) ? chebyshev_coeffs : tmp[d % 2], d == 0 ? &v_array[p * num_comp * dim + pass] : tmp[(d + 1) % 2])); pre /= Q_1d; post *= 1; } } } break; } default: // Nothing to do, this won't occur break; } CeedCall(CeedVectorRestoreArrayRead(basis->vec_chebyshev, &chebyshev_coeffs)); CeedCall(CeedVectorRestoreArrayRead(x_ref, &x_array_read)); CeedCall(CeedVectorRestoreArray(v, &v_array)); break; } case CEED_TRANSPOSE: { // Note: No switch on e_mode here because only CEED_EVAL_INTERP is supported at this time // Arbitrary points to nodes CeedScalar *chebyshev_coeffs; const CeedScalar *u_array, *x_array_read; // -- Transpose of evaluaton of Chebyshev polynomials at arbitrary points CeedCall(CeedVectorGetArrayWrite(basis->vec_chebyshev, CEED_MEM_HOST, &chebyshev_coeffs)); CeedCall(CeedVectorGetArrayRead(x_ref, CEED_MEM_HOST, &x_array_read)); CeedCall(CeedVectorGetArrayRead(u, CEED_MEM_HOST, &u_array)); { CeedScalar tmp[2][num_comp * CeedIntPow(Q_1d, dim)], chebyshev_x[Q_1d]; // ---- Values at point for (CeedInt p = 0; p < num_points; p++) { CeedInt pre = num_comp * 1, post = 1; // Note: stepping "backwards" through the tensor contractions to agree with the ordering of the Chebyshev coefficients for (CeedInt d = dim - 1; d >= 0; d--) { // ------ Tensor contract with current Chebyshev polynomial values CeedCall(CeedChebyshevPolynomialsAtPoint(x_array_read[p * dim + d], Q_1d, chebyshev_x)); CeedCall(CeedTensorContractApply(basis->contract, pre, 1, post, Q_1d, chebyshev_x, t_mode, p > 0 && d == 0, d == (dim - 1) ? &u_array[p * num_comp] : tmp[d % 2], d == 0 ? chebyshev_coeffs : tmp[(d + 1) % 2])); pre /= 1; post *= Q_1d; } } } CeedCall(CeedVectorRestoreArray(basis->vec_chebyshev, &chebyshev_coeffs)); CeedCall(CeedVectorRestoreArrayRead(x_ref, &x_array_read)); CeedCall(CeedVectorRestoreArrayRead(u, &u_array)); // -- Interpolate transpose from Chebyshev coefficients CeedCall(CeedBasisApply(basis->basis_chebyshev, 1, CEED_TRANSPOSE, CEED_EVAL_INTERP, basis->vec_chebyshev, v)); break; } } return CEED_ERROR_SUCCESS; } /** @brief Get Ceed associated with a CeedBasis @param[in] basis CeedBasis @param[out] ceed Variable to store Ceed @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetCeed(CeedBasis basis, Ceed *ceed) { *ceed = basis->ceed; return CEED_ERROR_SUCCESS; } /** @brief Get dimension for given CeedBasis @param[in] basis CeedBasis @param[out] dim Variable to store dimension of basis @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetDimension(CeedBasis basis, CeedInt *dim) { *dim = basis->dim; return CEED_ERROR_SUCCESS; } /** @brief Get topology for given CeedBasis @param[in] basis CeedBasis @param[out] topo Variable to store topology of basis @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetTopology(CeedBasis basis, CeedElemTopology *topo) { *topo = basis->topo; return CEED_ERROR_SUCCESS; } /** @brief Get number of components for given CeedBasis @param[in] basis CeedBasis @param[out] num_comp Variable to store number of components of basis @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetNumComponents(CeedBasis basis, CeedInt *num_comp) { *num_comp = basis->num_comp; return CEED_ERROR_SUCCESS; } /** @brief Get total number of nodes (in dim dimensions) of a CeedBasis @param[in] basis CeedBasis @param[out] P Variable to store number of nodes @return An error code: 0 - success, otherwise - failure @ref Utility **/ int CeedBasisGetNumNodes(CeedBasis basis, CeedInt *P) { *P = basis->P; return CEED_ERROR_SUCCESS; } /** @brief Get total number of nodes (in 1 dimension) of a CeedBasis @param[in] basis CeedBasis @param[out] P_1d Variable to store number of nodes @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetNumNodes1D(CeedBasis basis, CeedInt *P_1d) { CeedCheck(basis->is_tensor_basis, basis->ceed, CEED_ERROR_MINOR, "Cannot supply P_1d for non-tensor basis"); *P_1d = basis->P_1d; return CEED_ERROR_SUCCESS; } /** @brief Get total number of quadrature points (in dim dimensions) of a CeedBasis @param[in] basis CeedBasis @param[out] Q Variable to store number of quadrature points @return An error code: 0 - success, otherwise - failure @ref Utility **/ int CeedBasisGetNumQuadraturePoints(CeedBasis basis, CeedInt *Q) { *Q = basis->Q; return CEED_ERROR_SUCCESS; } /** @brief Get total number of quadrature points (in 1 dimension) of a CeedBasis @param[in] basis CeedBasis @param[out] Q_1d Variable to store number of quadrature points @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetNumQuadraturePoints1D(CeedBasis basis, CeedInt *Q_1d) { CeedCheck(basis->is_tensor_basis, basis->ceed, CEED_ERROR_MINOR, "Cannot supply Q_1d for non-tensor basis"); *Q_1d = basis->Q_1d; return CEED_ERROR_SUCCESS; } /** @brief Get reference coordinates of quadrature points (in dim dimensions) of a CeedBasis @param[in] basis CeedBasis @param[out] q_ref Variable to store reference coordinates of quadrature points @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetQRef(CeedBasis basis, const CeedScalar **q_ref) { *q_ref = basis->q_ref_1d; return CEED_ERROR_SUCCESS; } /** @brief Get quadrature weights of quadrature points (in dim dimensions) of a CeedBasis @param[in] basis CeedBasis @param[out] q_weight Variable to store quadrature weights @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetQWeights(CeedBasis basis, const CeedScalar **q_weight) { *q_weight = basis->q_weight_1d; return CEED_ERROR_SUCCESS; } /** @brief Get interpolation matrix of a CeedBasis @param[in] basis CeedBasis @param[out] interp Variable to store interpolation matrix @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetInterp(CeedBasis basis, const CeedScalar **interp) { if (!basis->interp && basis->is_tensor_basis) { // Allocate CeedCall(CeedMalloc(basis->Q * basis->P, &basis->interp)); // Initialize for (CeedInt i = 0; i < basis->Q * basis->P; i++) basis->interp[i] = 1.0; // Calculate for (CeedInt d = 0; d < basis->dim; d++) { for (CeedInt qpt = 0; qpt < basis->Q; qpt++) { for (CeedInt node = 0; node < basis->P; node++) { CeedInt p = (node / CeedIntPow(basis->P_1d, d)) % basis->P_1d; CeedInt q = (qpt / CeedIntPow(basis->Q_1d, d)) % basis->Q_1d; basis->interp[qpt * (basis->P) + node] *= basis->interp_1d[q * basis->P_1d + p]; } } } } *interp = basis->interp; return CEED_ERROR_SUCCESS; } /** @brief Get 1D interpolation matrix of a tensor product CeedBasis @param[in] basis CeedBasis @param[out] interp_1d Variable to store interpolation matrix @return An error code: 0 - success, otherwise - failure @ref Backend **/ int CeedBasisGetInterp1D(CeedBasis basis, const CeedScalar **interp_1d) { CeedCheck(basis->is_tensor_basis, basis->ceed, CEED_ERROR_MINOR, "CeedBasis is not a tensor product basis."); *interp_1d = basis->interp_1d; return CEED_ERROR_SUCCESS; } /** @brief Get gradient matrix of a CeedBasis @param[in] basis CeedBasis @param[out] grad Variable to store gradient matrix @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetGrad(CeedBasis basis, const CeedScalar **grad) { if (!basis->grad && basis->is_tensor_basis) { // Allocate CeedCall(CeedMalloc(basis->dim * basis->Q * basis->P, &basis->grad)); // Initialize for (CeedInt i = 0; i < basis->dim * basis->Q * basis->P; i++) basis->grad[i] = 1.0; // Calculate for (CeedInt d = 0; d < basis->dim; d++) { for (CeedInt i = 0; i < basis->dim; i++) { for (CeedInt qpt = 0; qpt < basis->Q; qpt++) { for (CeedInt node = 0; node < basis->P; node++) { CeedInt p = (node / CeedIntPow(basis->P_1d, d)) % basis->P_1d; CeedInt q = (qpt / CeedIntPow(basis->Q_1d, d)) % basis->Q_1d; if (i == d) basis->grad[(i * basis->Q + qpt) * (basis->P) + node] *= basis->grad_1d[q * basis->P_1d + p]; else basis->grad[(i * basis->Q + qpt) * (basis->P) + node] *= basis->interp_1d[q * basis->P_1d + p]; } } } } } *grad = basis->grad; return CEED_ERROR_SUCCESS; } /** @brief Get 1D gradient matrix of a tensor product CeedBasis @param[in] basis CeedBasis @param[out] grad_1d Variable to store gradient matrix @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetGrad1D(CeedBasis basis, const CeedScalar **grad_1d) { CeedCheck(basis->is_tensor_basis, basis->ceed, CEED_ERROR_MINOR, "CeedBasis is not a tensor product basis."); *grad_1d = basis->grad_1d; return CEED_ERROR_SUCCESS; } /** @brief Get divergence matrix of a CeedBasis @param[in] basis CeedBasis @param[out] div Variable to store divergence matrix @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetDiv(CeedBasis basis, const CeedScalar **div) { CeedCheck(basis->div, basis->ceed, CEED_ERROR_MINOR, "CeedBasis does not have divergence matrix."); *div = basis->div; return CEED_ERROR_SUCCESS; } /** @brief Get curl matrix of a CeedBasis @param[in] basis CeedBasis @param[out] curl Variable to store curl matrix @return An error code: 0 - success, otherwise - failure @ref Advanced **/ int CeedBasisGetCurl(CeedBasis basis, const CeedScalar **curl) { CeedCheck(basis->curl, basis->ceed, CEED_ERROR_MINOR, "CeedBasis does not have curl matrix."); *curl = basis->curl; return CEED_ERROR_SUCCESS; } /** @brief Destroy a CeedBasis @param[in,out] basis CeedBasis to destroy @return An error code: 0 - success, otherwise - failure @ref User **/ int CeedBasisDestroy(CeedBasis *basis) { if (!*basis || *basis == CEED_BASIS_COLLOCATED || --(*basis)->ref_count > 0) { *basis = NULL; return CEED_ERROR_SUCCESS; } if ((*basis)->Destroy) CeedCall((*basis)->Destroy(*basis)); CeedCall(CeedTensorContractDestroy(&(*basis)->contract)); CeedCall(CeedFree(&(*basis)->q_ref_1d)); CeedCall(CeedFree(&(*basis)->q_weight_1d)); CeedCall(CeedFree(&(*basis)->interp)); CeedCall(CeedFree(&(*basis)->interp_1d)); CeedCall(CeedFree(&(*basis)->grad)); CeedCall(CeedFree(&(*basis)->grad_1d)); CeedCall(CeedFree(&(*basis)->div)); CeedCall(CeedFree(&(*basis)->curl)); CeedCall(CeedVectorDestroy(&(*basis)->vec_chebyshev)); CeedCall(CeedBasisDestroy(&(*basis)->basis_chebyshev)); CeedCall(CeedDestroy(&(*basis)->ceed)); CeedCall(CeedFree(basis)); return CEED_ERROR_SUCCESS; } /** @brief Construct a Gauss-Legendre quadrature @param[in] Q Number of quadrature points (integrates polynomials of degree 2*Q-1 exactly) @param[out] q_ref_1d Array of length Q to hold the abscissa on [-1, 1] @param[out] q_weight_1d Array of length Q to hold the weights @return An error code: 0 - success, otherwise - failure @ref Utility **/ int CeedGaussQuadrature(CeedInt Q, CeedScalar *q_ref_1d, CeedScalar *q_weight_1d) { // Allocate CeedScalar P0, P1, P2, dP2, xi, wi, PI = 4.0 * atan(1.0); // Build q_ref_1d, q_weight_1d for (CeedInt i = 0; i <= Q / 2; i++) { // Guess xi = cos(PI * (CeedScalar)(2 * i + 1) / ((CeedScalar)(2 * Q))); // Pn(xi) P0 = 1.0; P1 = xi; P2 = 0.0; for (CeedInt j = 2; j <= Q; j++) { P2 = (((CeedScalar)(2 * j - 1)) * xi * P1 - ((CeedScalar)(j - 1)) * P0) / ((CeedScalar)(j)); P0 = P1; P1 = P2; } // First Newton Step dP2 = (xi * P2 - P0) * (CeedScalar)Q / (xi * xi - 1.0); xi = xi - P2 / dP2; // Newton to convergence for (CeedInt k = 0; k < 100 && fabs(P2) > 10 * CEED_EPSILON; k++) { P0 = 1.0; P1 = xi; for (CeedInt j = 2; j <= Q; j++) { P2 = (((CeedScalar)(2 * j - 1)) * xi * P1 - ((CeedScalar)(j - 1)) * P0) / ((CeedScalar)(j)); P0 = P1; P1 = P2; } dP2 = (xi * P2 - P0) * (CeedScalar)Q / (xi * xi - 1.0); xi = xi - P2 / dP2; } // Save xi, wi wi = 2.0 / ((1.0 - xi * xi) * dP2 * dP2); q_weight_1d[i] = wi; q_weight_1d[Q - 1 - i] = wi; q_ref_1d[i] = -xi; q_ref_1d[Q - 1 - i] = xi; } return CEED_ERROR_SUCCESS; } /** @brief Construct a Gauss-Legendre-Lobatto quadrature @param[in] Q Number of quadrature points (integrates polynomials of degree 2*Q-3 exactly) @param[out] q_ref_1d Array of length Q to hold the abscissa on [-1, 1] @param[out] q_weight_1d Array of length Q to hold the weights @return An error code: 0 - success, otherwise - failure @ref Utility **/ int CeedLobattoQuadrature(CeedInt Q, CeedScalar *q_ref_1d, CeedScalar *q_weight_1d) { // Allocate CeedScalar P0, P1, P2, dP2, d2P2, xi, wi, PI = 4.0 * atan(1.0); // Build q_ref_1d, q_weight_1d // Set endpoints CeedCheck(Q > 1, NULL, CEED_ERROR_DIMENSION, "Cannot create Lobatto quadrature with Q=%" CeedInt_FMT " < 2 points", Q); wi = 2.0 / ((CeedScalar)(Q * (Q - 1))); if (q_weight_1d) { q_weight_1d[0] = wi; q_weight_1d[Q - 1] = wi; } q_ref_1d[0] = -1.0; q_ref_1d[Q - 1] = 1.0; // Interior for (CeedInt i = 1; i <= (Q - 1) / 2; i++) { // Guess xi = cos(PI * (CeedScalar)(i) / (CeedScalar)(Q - 1)); // Pn(xi) P0 = 1.0; P1 = xi; P2 = 0.0; for (CeedInt j = 2; j < Q; j++) { P2 = (((CeedScalar)(2 * j - 1)) * xi * P1 - ((CeedScalar)(j - 1)) * P0) / ((CeedScalar)(j)); P0 = P1; P1 = P2; } // First Newton step dP2 = (xi * P2 - P0) * (CeedScalar)Q / (xi * xi - 1.0); d2P2 = (2 * xi * dP2 - (CeedScalar)(Q * (Q - 1)) * P2) / (1.0 - xi * xi); xi = xi - dP2 / d2P2; // Newton to convergence for (CeedInt k = 0; k < 100 && fabs(dP2) > 10 * CEED_EPSILON; k++) { P0 = 1.0; P1 = xi; for (CeedInt j = 2; j < Q; j++) { P2 = (((CeedScalar)(2 * j - 1)) * xi * P1 - ((CeedScalar)(j - 1)) * P0) / ((CeedScalar)(j)); P0 = P1; P1 = P2; } dP2 = (xi * P2 - P0) * (CeedScalar)Q / (xi * xi - 1.0); d2P2 = (2 * xi * dP2 - (CeedScalar)(Q * (Q - 1)) * P2) / (1.0 - xi * xi); xi = xi - dP2 / d2P2; } // Save xi, wi wi = 2.0 / (((CeedScalar)(Q * (Q - 1))) * P2 * P2); if (q_weight_1d) { q_weight_1d[i] = wi; q_weight_1d[Q - 1 - i] = wi; } q_ref_1d[i] = -xi; q_ref_1d[Q - 1 - i] = xi; } return CEED_ERROR_SUCCESS; } /// @}