PDE-Based and Solution-Dependent Parameterization for Isogeometric Analysis

This paper presents some results for PDE-based and solution-dependent parameterization of computational domains for isogeometric analysis (IGA) using non-uniform rational B-splines (NURBS). The final parameterization is produced based on the solution of a partial differential equation (PDE) that is solved using isogeometric collocation method (IGA-C) with Dirichlet boundary condition being the input boundary of the final desired computational domain for IGA, namely the IGA-C-PDE method for domain parameterization. The theory of PDE guarantees that the mapping between physical and transformed region will be one-to-one. In addition, we also apply intuitive position and ratio constraints while solving the PDE to achieve solution-dependent parameterization. While one may use any general PDE with any constraint, the PDEs and additional constraints selected in our case are such that the resulting solution can be efficiently solved through a system of linear equations with or without additional linear constraints. This approach is different from typical existing parameterization methods in IGA that are often solved through an expensive nonlinear optimization processes. The results show that the proposed method can efficiently produce satisfactory analysis-suitable parameterization.