具有隐式定义、连续嵌入纤维的超弹性膜
Hyperelastic Membranes with Implicitly Defined, Continuously Embedded Fibers
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中文总结 AI 辅助
本文提出一种新型各向异性超弹性曲膜力学模型，采用切向微分演算实现纤维几何的半隐式描述，结合Surface FEM与虚构域方法，获得高阶收敛率，为相关领域提供了有效数值方案。
中文摘要 AI 辅助
本文针对各向异性超弹性曲膜，提出了一种新型力学模型及对应的有限元方法。超弹性纤维被嵌入原本各向同性的膜中，相关应用包括生物组织和纺织品的简化模型等。该几何非线性力学基于连续介质力学的第一性原理（有限应变理论）构建，所采用的微分算子以与坐标无关的方式表述，通过称为切向微分演算的框架实现。这使得纤维几何可通过标量函数的水平集与显式定义的膜表面的交集，实现（半）隐式描述。随后将隐式纤维的力学模型与膜的力学特性耦合。数值分析中应用有限元方法，所得方案是经典表面有限元法（Surface FEM）与虚构域方法的混合方案。对于光滑物理场，该方法可获得高阶收敛率，验证了该数值方法的有效性。
英文摘要
A novel mechanical model and corresponding finite element method for anisotropic, hyperelastic, curved membranes are proposed. Hyperelastic fibers are embedded into the otherwise isotropic membrane, being relevant, for example, in reduced models for biological tissues and textiles. The geometrically nonlinear mechanics is formulated based on first principles of continuum mechanics (finite strain theory). The employed differential operators are formulated in a coordinate-free manner, through a framework known as tangential differential calculus. This enables a (semi-)implicit description of the fiber geometry through the intersection of level sets of some scalar function with the explicitly defined membrane surface. The mechanical model of the implicit fibers is then coupled to the mechanics of the membrane. For the numerical analysis, finite elements are applied such that the resulting scheme is a hybrid between classical Surface FEM and fictitious domain methods. For smooth physical fields, higher-order convergence rates are obtained and confirm the success of the numerical method.
