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.

翻译：本文提出了一种基于偏微分方程（PDE）的参数化框架，用于解决平面域的面到体（StV）问题，即在仅给定边界样条曲线描述的情况下，求得域内部的有效描述。该框架适用于等几何分析（IGA）应用，其中物理域由多于四条边构成，因此需要多个片。我们采用调和映射的概念，提出了若干基于PDE的问题公式，能够在具有相同边数的凸参数化多片域与分段光滑物理域之间建立有效映射。遵循IGA的等参范式，我们使用分析步骤中特有的技术来处理StV问题。因此，本研究提出了几种基于IGA的数值算法来求解问题的控制方程，这些算法可以无缝集成到成熟的IGA软件套件中。我们通过引入机制来增强该框架，以控制结果的参数化特性。参数化控制通过多种技术实现，其中包括在凸参数域中引入曲线坐标系，根据具体应用，该坐标系可在计算所得的调和映射中构建期望的特征，如均匀单元尺寸或边界层。