This paper presents a new computational method for solving two-phase engineering problems. The novelty of this approach lies in its ability to perform the analysis over the exact two-phase geometry. The geometry is defined by a spline-based representation and constructed based on the level-set method. The level-set-based geometry is mapped into a spline-based geometry using an untrimming technique that is newly developed in this paper. The spline-based representation is defined over unstructured meshes enabling precise representations of geometrically complex domains. An analysis-suitable mechanical model that accurately replicates the geometry is constructed and the governing equations are solved using the isogeometric analysis method. The interface between phases is explicitly represented by cubic B-spline curves, allowing for the evaluation of the solution field and various physical measures over its exact geometry. Additionally, algorithms that allow for changing the continuity of the solution field between the phases while precisely preserving the discontinuous two-phase domain are developed. Consistent analytical sensitivity analysis is formulated for generating the geometrical and mechanical models to allow for solving engineering problems with moving boundaries following the gradient-based approach. The proposed approach is tested for solving Poisson’s equation, linear elasticity, topology optimization involving linear elasticity for compliance minimization, and two-phase flow problems. The results expose the capability of the proposed approach to solve two-phase engineering problems while accurately representing the interface.

本文提出了一种解决两相工程问题的新计算方法。这种方法的新颖之处在于它能够对精确的两相几何形状进行分析。几何形状由基于样条的表示定义，并基于水平集方法构造。使用本文新开发的未修剪技术将基于水平集的几何图形映射到基于样条线的几何图形。基于样条的表示是在非结构化网格上定义的，可以精确表示复杂的几何域。构建了精确复制几何形状的适合分析的力学模型，并使用等几何分析方法求解控制方程。相之间的界面由三次 B 样条曲线明确表示，允许评估解场以及对其精确几何形状的各种物理测量。此外，还开发了允许改变相之间解场的连续性同时精确保留不连续两相域的算法。制定一致的分析灵敏度分析，用于生成几何和机械模型，以便按照基于梯度的方法解决移动边界的工程问题。所提出的方法经过测试，可用于求解泊松方程、线性弹性、涉及柔量最小化的线性弹性的拓扑优化以及两相流问题。结果揭示了所提出的方法在准确表示界面的同时解决两阶段工程问题的能力。