发表机构
Ruprecht-Karls-Universität Heidelberg（海德堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种完全变分、模型无关的CutFEM公式用于有限应变弹性，通过自动微分从能量密度导出应力和切线，实现最优h收敛，并分析角点奇异性对收敛率的限制。
AI 中文摘要
我们提出了一种完全变分、模型无关的切割有限元方法（CutFEM）公式，用于有限应变弹性。离散问题是增广能量泛函的驻点条件，该泛函由体超弹性能量、弱施加边界条件的Nitsche项以及鬼罚稳定项组成。残差和（对称化）切线通过该泛函的逐次变分得到。自动微分（AD）直接从标量能量密度生成第一Piola–Kirchhoff应力张量和弹性张量，避免了在交换超弹性模型时的手动重新推导。据我们所知，这是第一个将纯能量、模型无关构造与AD以及非拟合边界精度分析相结合的非拟合有限应变方案。对每个牛顿步求解的线性化问题的分析建立了切割无关的强制性、连续性和$O(h^{-2})$条件数界，通过Brezzi–Rappaz–Raviart框架为正则解提供了拟最优收敛定理。数值上，该方法在光滑测试例上对线性、二次和三次单元实现了最优h收敛。此外，我们使用Kolosov–Muskhelishvili特征方程量化了方法在混合Dirichlet–Neumann交界处的精度极限。精确解的角点奇异性对拟合和非拟合方法相同地限制了收敛速率。我们证明局部网格细化可以消除这一界限，非拟合离散化继承了恢复的最优速率和切割无关常数。
英文摘要
We present a fully variational, model-independent formulation of the Cut Finite Element Method (CutFEM) for finite-strain elasticity. The discrete problem is derived from a single augmented energy functional consisting of the bulk hyperelastic energy, the Nitsche terms that impose the boundary conditions weakly, and the ghost-penalty stabilisation. At each nonlinear iterate, the residual is the exact first variation of this functional with the adaptive Nitsche weight frozen, while the correction uses a symmetric, coercive approximation of its Hessian. Automatic differentiation (AD) generates the first Piola--Kirchhoff stress tensor and the elasticity tensor directly from the scalar energy density, avoiding manual re-derivation when exchanging hyperelastic models. To our knowledge, this is the first unfitted finite-strain scheme combining an energy-only, model-independent construction with AD and an accuracy analysis at unfitted boundaries. Analysis of the linearised problem solved at each quasi-Newton step establishes cut-independent coercivity, continuity, and an $O(h^{-2})$ condition number bound, yielding a quasi-optimal convergence theorem for regular solutions through the Brezzi--Rappaz--Raviart framework. Numerically, the method attains optimal $h$-convergence for quadratic and cubic elements on a smooth test case. Furthermore, we quantify the method's accuracy limit at mixed Dirichlet--Neumann junctions using the Kolosov--Muskhelishvili characteristic equation. The exact solution's corner singularity caps the convergence rate identically for fitted and unfitted methods. Local mesh refinement removes this bound: we verify the recovery of optimal rates numerically for first-order elements, and prove that the unfitted discretisation inherits the rate the underlying refinement attains, with cut-independent constants.
