Adaptive w-refinement: A new paradigm in isogeometric analysis

摘要

Motivated by the concept of generalized NURBS (GNURBS), recently introduced by the authors, we devise a novel adaptivity technique in isogeometric analysis (IGA), referred to as adaptive w-refinement. GNURBS-based IGA is a natural extension of IGA where the weights of the basis functions in geometry and solution space are decoupled. Considering the additional unknown control weights in the solution function space as design variables, we develop an adaptive algorithm to find these unknowns by solving an unconstrained optimization problem. Due to the decoupling of the weights, the analytical sensitivities can be derived cost effectively; consequently, the optimization problem can be solved efficiently by a gradient-based algorithm. This procedure leads to the optimal rational function space associated with the problem under study, while preserving the underlying geometry as well as its parameterization.We study the performance of this algorithm on elliptic problems with both smooth and rough solutions. Numerical results demonstrate significant improvement of accuracy as well as the convergence rate compared to classic NURBS-based IGA. Moreover, the proposed method enables the isogeometric method to solve problems, whose closed-form solutions lie in rational space, exactly, revealing a new crucial aspect of employing rational splines for analysis. The proposed adaptive w-refinement technique serves as a new powerful adaptive technique in IGA, and perhaps a competitive tool with hierarchical splines for alleviating the deficiencies of NURBS for analysis. (C) 2020 Elsevier B.V. All rights reserved.