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Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach

机译：基于分层方法的二维修剪域自适应异步分析

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摘要

The focus of this work is on the development of an error-driven isogeometric framework, capable of automatically performing an adaptive simulation in the context of second- and fourth-order, elliptic partial differential equations defined on two-dimensional trimmed domains. The method is steered by an a posteriori error estimator, which is computed with the aid of an auxiliary residual-like problem formulated onto a space spanned by splines with single element support. The local refinement of the basis is achieved thanks to the use of truncated hierarchical B-splines. We prove numerically the applicability of the proposed estimator to various engineering-relevant problems, namely the Poisson problem, linear elasticity and Kirchhoff-Love shells, formulated on trimmed geometries. In particular, we study several benchmark problems which exhibit both smooth and singular solutions, where we recover optimal asymptotic rates of convergence for the error measured in the energy norm and we observe a substantial increase in accuracy per-degree-of-freedom compared to uniform refinement. Lastly, we show the applicability of our framework to the adaptive shell analysis of an industrial-like trimmed geometry modeled in the commercial software Rhinoceros, which represents the B-pillar of a car. (C) 2020 Elsevier B.V. All rights reserved.
机译：这项工作的重点是开发出错误驱动的异步框架，能够在二维修整域上定义的第二阶和四阶的椭圆部分微分方程的上下文中自动执行自适应仿真。该方法由后验误差估计器引导，该误差估计器借助于配制到具有单个元素支撑件跨越的空间的空间上的辅助残余问题来计算。由于使用截短的等级B样条曲线，因此实现了基础的当地改进。我们证明了拟议估计人对各种工程相关问题的适用性，即泊松问题，线性弹性和柯彻夫夫 - 爱贝壳，配制在修剪的几何形状上。特别是，我们研究了几种基准问题，其展示了光滑和奇异的解决方案，在那里我们回收了在能量规范中测量的误差的最佳渐近率，并且与均匀相比，我们观察了每自由度的精度大幅增加细化。最后，我们展示了我们的框架适用于商业软件犀牛模型的工业样修剪几何形状的自适应外壳分析，这代表了汽车的B柱。 （c）2020 Elsevier B.v.保留所有权利。

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