An n-cross field is a locally-defined orthogonal coordinate system invariant with respect to the cubic symmetry group. Cross fields are finding wide-spread use in mesh generation, computer graphics, and materials science among many applications. We consider the problem of generating an  n-cross field using a higher-order Q-tensor theory that is constructed out of tensored projection matrices. It is shown that by a Ginzburg-Landau relaxation  , one can reliably generate an n-cross field on arbitrary Lipschitz domains  .
An n-cross field is a locally-defined orthogonal coordinate system invariant with respect to the cubic symmetry group. Cross fields are finding wide-spread use in mesh generation, computer graphics, and materials science among many applications. It was recently by other authors that 3-cross fields can be embedded into the set of symmetric 4th-order tensors. Another concurrent work further develops a relaxation of this tensor field via a certain set of varieties. In this paper, we consider the problem of generating an arbitrary n-cross field using a fourth-order Q-tensor theory that is constructed out of tensored projection matrices. We establish that by a Ginzburg-Landau relaxation towards a global projection , one can reliably generate an n-cross field on arbitrary Lipschitz domains . Our work provides a rigorous approach that offers several new results including porting the tensor framework to arbitrary dimensions, providing a new relaxation method that embeds the problem into a global steepest descent, and offering a relaxation scheme for aligning the cross field with the boundary. Our approach is designed to fit within the classical Ginzburg-Landau PDE theory, offering a concrete road map for the future careful study of singularities of energy minimizers .
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We
It was recently by other authors that 3-cross fields can be embedded into the set of symmetric 4th-order tensors. Another concurrent work further develops a relaxation of this tensor field via a certain set of varieties. In this paper, we
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arbitrary
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higher-order
fourth-order
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It is shown
We establish
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towards a global projection
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. Our work provides a rigorous approach that offers several new results including porting the tensor framework to arbitrary dimensions, providing a new relaxation method that embeds the problem into a global steepest descent, and offering a relaxation scheme for aligning the cross field with the boundary. Our approach is designed to fit within the classical Ginzburg-Landau PDE theory, offering a concrete road map for the future careful study of singularities of energy minimizers