This paper further develops the Method of Matched Sections (MMS), a robust numerical framework for the solution of boundary value problems governed by partial differential equations. It demonstrates its unique applicability to the challenges of surface modeling, which lie at the intersection of computational mechanics and computer graphics. This work shows how the MMS successfully bridges this gap. By decomposing the domain into an assembly of 1D directional components matched along their entire boundaries, the method inherently enforces the continuity of all variational parameters, including second-order (curvature) and third-order (shear) derivatives. We demonstrate the method's advanced capabilities in high-fidelity surface reconstruction and blending, showing that it consistently generates energetically optimal, fair surfaces even from complex boundary conditions or sparse internal data points. By advancing the application of the MMS, this research establishes it as a powerful, physics-informed geometric tool that satisfies the dual demands of rigorous numerical analysis and aesthetic computer-aided design.

翻译：本文进一步发展了匹配截面法（MMS），这是一种用于求解偏微分方程边值问题的稳健数值框架。该方法在计算力学与计算机图形学交叉领域的曲面建模挑战中展现出独特的适用性。本研究展示了MMS如何成功弥合两个学科间的鸿沟：通过将计算域分解为沿完整边界匹配的一维方向分量组合，该方法固有地保证了所有变分参数（包括二阶曲率与三阶剪切导数）的连续性。我们论证了该方法在高保真曲面重建与混合中的先进能力，证明其即使在复杂边界条件或稀疏内部数据点条件下，也能持续生成能量最优的公平曲面。通过推进MMS的应用，本研究将其确立为一种兼具严苛数值分析与美学计算机辅助设计双重需求的物理信息几何工具。