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/// can be efficiently assembled into a sparse matrix later.
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///
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/// @tparam IDContainer A container type that supports operator[] and size() to access vertex ids. Can be a std::vector, Eigen::VectorXi, etc.
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/// @tparam GlobalOrder The storage order of the global gradient. Must be either Eigen::RowMajor or Eigen::ColMajor. If RowMajor, the global gradient is assumed to be ordered as [x0, y0, z0, x1, y1, z1, ...]. If ColMajor, the global gradient is assumed to be ordered as [x0, x1, ..., y0, y1, ..., z0, z1, ...].
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/// @tparam GlobalOrder The storage order of the global Hessian. Must be either Eigen::RowMajor or Eigen::ColMajor. If RowMajor, the global Hessian is assumed to be ordered as [x0, y0, z0, x1, y1, z1, ...]. If ColMajor, the global Hessian is assumed to be ordered as [x0, x1, ..., y0, y1, ..., z0, z1, ...].
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/// @param local_hessian The local Hessian to be added to the global Hessian. Must be of size (n_verts * dim) x (n_verts * dim), where n_verts is the number of vertices in the local element and dim is the dimension of the problem.
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/// @param ids The vertex ids corresponding to the local gradient. Must be of size at least n_verts, but can be larger (e.g., if the local element has fewer vertices than the number of ids).
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/// @param dim The dimension of the problem (e.g., 2 for 2D, 3 for 3D).
/// @tparam Derived The derived type of the local Jacobian matrix. Must be a matrix expression that can be evaluated to an Eigen::MatrixXd. The local Jacobian must be of size (n_rows * dim) x (n_cols * dim), where n_rows and n_cols are the number of vertices in the local element for the row and column respectively, and dim is the dimension of the problem.
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/// @tparam IDContainer1 A container type that supports operator[] and size() to access vertex ids for the row indices. Can be a std::vector, Eigen::VectorXi, etc. Must be of size at least n_rows, but can be larger (e.g., if the local element has fewer vertices than the number of ids).
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/// @tparam IDContainer2 A container type that supports operator[] and size() to access vertex ids for the column indices. Can be a std::vector, Eigen::VectorXi, etc. Must be of size at least n_cols, but can be larger (e.g., if the local element has fewer vertices than the number of ids).
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/// @tparam GlobalOrder The storage order of the global gradient. Must be either Eigen::RowMajor or Eigen::ColMajor. If RowMajor, the global gradient is assumed to be ordered as [x0, y0, z0, x1, y1, z1, ...]. If ColMajor, the global gradient is assumed to be ordered as [x0, x1, ..., y0, y1, ..., z0, z1, ...].
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/// @tparam GlobalOrder The storage order of the global Jacobian. Must be either Eigen::RowMajor or Eigen::ColMajor. If RowMajor, the global Jacobian is assumed to be ordered as [x0, y0, z0, x1, y1, z1, ...]. If ColMajor, the global Jacobian is assumed to be ordered as [x0, x1, ..., y0, y1, ..., z0, z1, ...].
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/// @param local_jacobian The local Jacobian to be added to the global Jacobian. Must be of size (n_rows * dim) x (n_cols * dim), where n_rows and n_cols are the number of vertices in the local element for the row and column respectively, and dim is the dimension of the problem.
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/// @param row_ids The vertex ids corresponding to the row indices of the local Jacobian. Must be of size at least n_rows, but can be larger (e.g., if the local element has fewer vertices than the number of ids).
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/// @param col_ids The vertex ids corresponding to the column indices of the local Jacobian. Must be of size at least n_cols, but can be larger (e.g., if the local element has fewer vertices than the number of ids).
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