Adaptive w-refinement: A new paradigm in isogeometric analysis
Adaptive w-refinement: A new paradigm in isogeometric analysis
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DOI：
10.1016/j.cma.2020.113180
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发表时间：
2020-08
影响因子：
7.2
通讯作者：
A. Taheri;K. Suresh
中科院分区：
文献类型：
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作者：
A. Taheri;K. Suresh
受作者最近提出的广义NURBS(GNURBS)概念的启发，我们设计了一种新的等距几何分析(IGA)中的自适应技术，称为自适应新求精。基于GNURBS的免疫遗传算法是免疫遗传算法的自然扩展，它将几何空间和解空间中的基函数的权重解耦。将解函数空间中的附加未知权重作为设计变量，通过求解一个无约束优化问题，提出了一种寻找这些未知量的自适应算法。由于权重的解耦，可以以较低的成本获得分析灵敏度；因此，基于梯度的算法可以有效地解决优化问题。该算法得到了与所研究问题相关的最优有理函数空间，同时保持了基本几何结构及其参数，研究了该算法在具有光滑解和粗糙解的椭圆问题上的性能。数值结果表明，与经典的基于NURBS的免疫遗传算法相比，该算法在精度和收敛速度上都有明显的提高。此外，该方法使等距方法能够准确地求解闭式解位于有理空间中的问题，揭示了用有理样条法进行分析的一个新的重要方面。本文提出的自适应新求精技术是免疫遗传算法中一种新的强大的自适应技术，也可能是一种具有竞争力的工具，可以用来缓解NURBS分析中的不足。
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 adaptivew-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 unknowncontrol weightsin 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 adaptivew-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.