用于等几何分析的局部修改非钳制片段的可容许样条空间的混合重构
Hybrid Reconstruction of Admissible Spline Spaces from Locally Modified Unclamped Patches for Isogeometric Analysis
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中文总结 AI 辅助
提出一种解耦重构框架，通过混合重构算子实现局部样条编辑与全局连续性解耦，保持几何精确性，并验证了最优收敛速率，为自适应等几何分析提供高效工具。
中文摘要 AI 辅助
我们提出了一种解耦重构框架，该框架将NURBS表示临时分解为独立的局部活动段，从而在保持精确CAD几何形状的同时，允许任意局部节点插入、度数提升和基函数修改。然而，这些独立的修改不可避免地违反了片间连续性，需要一种稳健的代数恢复方法来恢复全局可容许性。我们引入了一种新颖的重构方法，构建一个正的混合重构算子：通过QR主元选择识别锚定自由度，随后一系列带距离正则化的线性规划问题生成严格非负的零空间向量，而后续的非负最小二乘求解则强制实现精确的单位分割。关键在于，为确保计算效率并保持局部性，我们实施了一种分层成对凝聚策略，该策略冻结已满足新界面约束的列，将优化限制在每一步合并时的活动自由度上。由此产生的混合基函数在构造上严格非负，张成精确的约束零空间，将单位分割保持到机器精度，并精确再现原始几何形状。数值基准测试，包括在具有异质局部多项式次数的多片曲线上的非线性扩散问题，确认了最优收敛速率，并展示了该框架将局部几何灵活性与全局一致的逼近空间无缝结合的能力。通过将局部样条编辑与连续性强制完全解耦，该方法为自适应等几何分析提供了一种高效、数学上合理且普遍适用的工具。
英文摘要
We propose a decoupled reconstruction framework that temporarily decomposes a NURBS representation into independent local Active Sections, enabling arbitrary local knot insertion, degree elevation, and basis modifications while preserving exact CAD geometry. However, these independent modifications inevitably violate inter-patch continuity, requiring a robust algebraic recovery of global admissibility. We introduce a novel reconstruction methodology that constructs a positive Hybrid Reconstruction Operator: anchor degrees of freedom are identified via QR pivoting, after which a sequence of linear programming problems with distance-based regularization generates strictly non-negative nullspace vectors, while a subsequent non-negative least-squares solve enforces a precise partition of unity. Crucially, to ensure computational efficiency and preserve locality, we implement a hierarchical pairwise condensation strategy that freezes columns already satisfying new interface constraints, confining the optimization exclusively to the active degrees of freedom at each merge step. The resulting hybrid basis is, by construction, strictly non-negative, spans the exact constrained nullspace, preserves partition of unity to machine precision, and reproduces the original geometry exactly. Numerical benchmarks, including a nonlinear diffusion problem on multi-patch curves with heterogeneous local polynomial degrees, confirm optimal convergence rates and demonstrate the framework's ability to seamlessly combine local geometric flexibility with globally consistent approximation spaces. By entirely decoupling local spline editing from the enforcement of continuity, this methodology provides a highly efficient, mathematically principled, and universally applicable tool for adaptive Isogeometric Analysis.
发表机构
- National Technical University of Athens（雅典国立技术大学）
机构由 AI 辅助整理，请以论文原文为准。
