Computing IGA-suitable Planar Parameterizations

Isogeometric analysis(IGA)has been a new approach for solving Partial Differen-tial Equations by using the smooth spline basis that defines the geometry as the basis for analysis.IGA has great applications in many fields.A fundamental and crucial prob-lem in IGA is to compute a valid parametric spline representation of the computational domain from the given CAD model.This problem is called parameterization.Many approaches have been proposed to solve the planar parameterization problem in the past decade.Some methosd can not work for the complex and high genus do-mains.The partition-driven techniques were presented to decompose the input domains into coarse quad patches.However,these methods usually generate too many patches for the complex domains to be efficient for the subsequent matrix assembly.There are many methods focusing on generating quad meshes or quad patches.These methods either usually generate narrow rectangles,not suitable for IGA,or the high quality of the quad patch layout is highly dependent on the situations of the singularities of the input cross field.We propose an algorithm for computing injective IGA-suitable planar parameter-izations with as uniform and orthogonal as possible iso-parametric structure.Central to this approach is a PolySquare-enhanced domain partition procedure that decomposes the input complex planar domain into coarse and square-like quad patches.First,we triangulate the input planar domain that may be multiply connected,and deform it to be a PolySquare-like structure by optimizing a boundary alignment energy.Secondly,the PolySquare-like structure is pixelated to make the input domain be a quadrilateral mesh that is decimated to generate a coarse patch layout,where each patch is an approxi-mately squared quadrilateral.Finally,the sparse patches are subdivided to represent the input domain and produce the resulting partition.We parameterize each patch with continuous constraints to find the parameterization of the input domain.Compared with existing IGA-suitable planar parameterization methods,our method produces better pa-rameterizations and fewer patches than the ones that also use domain partition strategy.We demonstrate the superiority of our method over various complex domains,including an example containing 30 holes.Since the energies involved in this paper are nonlinear energies,the optimization of these energies using the traditional optimization methods,such as Newton method and L-BFGS method,can either be more computationally expensive or slow to converge.Therefore,we propose a preconditioned accelerated proxy gradient method(PAPG).It turns out that PAPG performs efficiently and effectively when optimizing these nonlin-ear energies.