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，用于解决平面曲面到体积（StV）问题，即在仅给定边界曲线的样条描述的情况下，寻找域内部的有效描述。该框架面向等几何分析（IGA）应用，其中物理域包含多于四条边，因此需要多个块。我们采用调和映射的概念，并提出了几种基于 PDE 的问题公式，这些公式能够在具有相同边数的凸参数多块域和分段光滑物理域之间找到有效的映射。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，用于求解控制方程，这些算法可以轻松集成到成熟的 IGA 软件套件中。我们通过引入机制来增强该框架，使其能够控制结果的参数特性。参数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，该坐标系根据应用需求，在计算出的调和映射中构建所需特征，例如均匀单元尺寸或边界层。