This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes. It relies on a new developed technique for the evaluation of polynomial integrals over spline boundary representations that is exclusively based on analytical computations. First, through a consistent polynomial approximation step, the finite element operators of the Galerkin method are transformed into integrals involving only polynomial integrands. Then, by successive applications of the divergence theorem, those integrals over B-Reps are transformed into first surface and then line integrals with polynomials integrands. Eventually these line integrals are evaluated analytically with machine precision accuracy. The performance of the proposed method is demonstrated by means of numerical experiments in the context of 2D and 3D elliptic problems, retrieving optimal error convergence order in all cases. Finally, the methodology is illustrated for 3D CAD models with an industrial level of complexity.

中文翻译：

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无正交浸入式等几何分析

本文提出了一种通过浸入式等几何离散化不需要正交方案来求解三维 CAD 几何体上偏微分方程的新方法。它依赖于一种新开发的技术，用于评估样条边界表示上的多项式积分，该技术完全基于分析计算。首先，通过一致的多项式逼近步骤，Galerkin 方法的有限元算子被转换为仅涉及多项式被积函数的积分。然后，通过发散定理的连续应用，B-Reps 上的那些积分被转换为第一个曲面，然后是多项式被积函数的线积分。最终，这些线积分以机器精度精度进行分析评估。在 2D 和 3D 椭圆问题的背景下，通过数值实验证明了所提出方法的性能，在所有情况下检索最佳误差收敛顺序。最后，针对具有工业复杂性的 3D CAD 模型说明了该方法。