Creating domain mappings
This work addresses a fundamental problem in geometric modeling and mesh generation, but the results are theoretical and incremental without concrete performance numbers.
The paper tackles the problem of constructing a bijective, continuously differentiable extension of a mapping from the unit sphere to the boundary of a simply-connected region in R^d. It proposes methods for obtaining initial guesses and an iterative improvement scheme.
Consider being given a mapping ϕfrom the unit sphere S^{d-1}, d>2, to the smooth boundary of a simply-connected region Ωin R^d. We consider the problem of constructing an extension Φfrom the unit ball B_d to Ω. The mapping is required to be 1-1 and continuously differentiable with a nonsingular Jacobian matrix. We discuss ways of obtaining initial guesses for such a mapping Φand of then improving it by an iteration method.