A mechanical model and numerical method for structural membranes implied by all isosurfaces of a level-set function in a three-dimensional bulk domain are proposed. The mechanical model covers large displacements in the context of the finite strain theory and is formulated based on the tangential differential calculus. Alongside curved two-dimensional membranes embedded in three dimensions, also the simpler case of curved ropes (cables) in two-dimensional bulk domains is covered. The implicit geometries (shapes) are implied by the level sets and the boundaries of the structures are given by the intersection of the level sets with the boundary of the bulk domain. For the numerical analysis, the bulk domain is discretized using a background mesh composed by (higher-order) elements with the dimensionality of the embedding space. The elements are by no means aligned to the level sets, i.e., the geometries of the structures, which resembles a fictitious domain method, most importantly the Trace FEM. The proposed numerical method is a hybrid of the classical FEM and fictitious domain methods which may be labeled as "Bulk Trace FEM". Numerical studies confirm higher-order convergence rates and the potential for new material models with continuously embedded sub-structures in bulk domains.

翻译：提出了一种机械模型和数值方法，用于求解三维体域中水平集函数所有等值面所隐含的结构薄膜。该机械模型涵盖有限应变理论下的大位移问题，并基于切向微分运算进行公式化表述。除了嵌入三维空间的弯曲二维薄膜外，还涵盖了更简单的二维体域中弯曲绳索（缆索）的情况。隐式几何形状由水平集隐含定义，结构边界由水平集与体域边界的交集确定。在数值分析中，体域采用嵌入空间维度的（高阶）单元组成的背景网格进行离散。网格单元完全不与水平集（即结构几何形状）对齐，这类似于虚拟域方法，特别是Trace FEM。所提出的数值方法是经典有限元法与虚拟域方法的混合，可称为“体域迹线有限元法”。数值研究证实了高阶收敛速率，以及实现体域内连续嵌入子结构的新型材料模型的潜力。