AI 中文总结
本研究针对有限应变粘弹性的全离散耦合问题，提出带内变量的整体牛顿求解及零空间凝聚方法，在保留外层牛顿行为的同时降低计算成本，可扩展至多类热力学一致内变量模型。
AI 中文摘要
有限应变粘弹性通常从本构角度进行讨论，而由时间和空间离散化引发的非线性求解架构则较少被系统研究。本研究考虑一个带有应变类内变量的代表性粘弹性模型，重点关注变形与内状态的全离散耦合问题。从基础的能耗散结构出发，推导离散弱形式并得到具有天然非对称块切线的整体牛顿系统。通过舒尔补在线性化系统层面一致消去内变量增量，且该约化过程可基于内约束流形的切线空间零空间基给出几何解释，由此得到经典嵌套高斯点凝聚的真正整体对应方法，经典方法需在全局平衡步前单独求解局部本构方程。二维和三维Cook膜基准的数值结果表明，所提策略保留了与经典方法基本相同的外层牛顿行为，同时通过避免重复的局部牛顿求解大幅降低了计算成本，且在嵌套方案失效的载荷增量下仍保持收敛性。此外，数值研究显示，为内变量选择合适的近似空间可在精度无明显损失的前提下进一步节省计算量。尽管针对有限粘弹性展开，该构造可自然扩展至更广泛的热力学一致内变量模型类。
英文摘要
Finite-strain viscoelasticity is commonly discussed from a constitutive perspective, whereas the nonlinear solution architecture induced by time and space discretization is studied less systematically. In this work, we consider a representative viscoelastic model with a strain-like internal variable and focus on the fully discrete coupled problem in the deformation and the internal state. Starting from the underlying energy-dissipation structure, we derive the discrete weak forms and obtain a monolithic Newton system with a naturally non-symmetric block tangent. The increment of the internal variable is eliminated consistently at the level of the linearized system by a Schur complement, and the same reduction is shown to admit a geometric interpretation in terms of a null-space basis of the tangent space of the internal constraint manifold. This yields a genuinely monolithic counterpart to classical nested Gauss-point condensation, in which local constitutive equations are solved separately before the global equilibrium step. Numerical results for two- and three-dimensional Cook's membrane benchmarks show that the proposed strategy retains essentially the same outer Newton behavior as the classical approach while substantially reducing computational cost by avoiding repeated local Newton solves, it also remains convergent for load increments for which the nested scheme fails. Moreover, the numerical study suggests that suitably chosen approximation spaces for the internal variable can yield additional savings in computational effort without significant loss of accuracy. Although presented for finite viscoelasticity, the construction extends naturally to broader classes of thermodynamically consistent internal-variable models.
DOI：10.1007/s00707-026-04891-3
