Before revision:  It is shown that by a Ginzburg-Landau relaxation  , one can reliably generate an n-cross field on arbitrary Lipschitz domains  .
After revision:  It is shown that by a Ginzburg-Landau relaxation , 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 .