面向平面多面片域的基于PDE的参数化技术
PDE-Based Parameterisation Techniques for Planar Multipatch Domains
AI总结：
本研究提出基于PDE的多面片域参数化框架，结合调和映射与IGA数值算法解决平面StV问题，可集成至IGA软件并通过曲线坐标系调控参数化特性。
摘要：
本文提出了一种基于偏微分方程（PDE）的参数化框架，用于解决平面表面到体积（StV）问题：在仅给定域边界轮廓的样条描述的前提下，获取域内部的有效描述。该框架面向等几何分析（IGA）应用场景，这类场景中物理域的边数超过四条，因此需要多个面片来表示。本文采用调和映射的概念，提出了多种基于PDE的问题公式，能够在凸参数化多面片域与边数相同的分段光滑物理域之间构建有效映射。遵循IGA的等参范式，我们使用分析步骤的典型技术来处理StV问题。据此，本研究针对该问题的控制方程提出了多种基于IGA的数值算法，这些算法可轻松集成到成熟的IGA软件套件中。我们还为该框架补充了可控制输出参数化特性的机制。参数化控制的实现手段包括在凸参数域中引入曲线坐标系，该坐标系可根据应用需求为计算得到的调和映射构建所需特性，例如均匀单元尺寸或边界层。
英文摘要：
This paper presents a PDE-based parameterisation framework for addressing the planar surface-to-volume (StV) problem of finding a valid description of the domain's interior given no more than a spline-based description of its boundary contours. The framework is geared towards isogeometric analysis (IGA) applications wherein the physical domain is comprised of more than four sides, hence requiring more than one patch. We adopt the concept of harmonic maps and propose several PDE-based problem formulations capable of finding a valid map between a convex parametric multipatch domain and the piecewise-smooth physical domain with an equal number of sides. In line with the isoparametric paradigm of IGA, we treat the StV problem using techniques that are characteristic for the analysis step. As such, this study proposes several IGA-based numerical algorithms for the problem's governing equations that can be effortlessly integrated into a well-developed IGA software suite. We augment the framework with mechanisms that enable controlling the parametric properties of the outcome. Parametric control is accomplished by, among other techniques, the introduction of a curvilinear coordinate system in the convex parametric domain that, depending on the application, builds desired features into the computed harmonic map, such as homogeneous cell sizes or boundary layers.
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