一种用于动态线性粘弹性的稳定DG-POD降阶方法
A stable DG-POD reduced-order method for dynamic linear viscoelasticity
AI总结：
本文提出一种稳定DG-POD降阶方法，用于动态线性粘弹性问题，可高效实现长时间模拟，数值实验验证其收敛性与精度，且计算成本显著降低。
AI中文摘要：
本文针对由广义Maxwell模型控制的动态线性粘弹性问题，提出了一种基于本征正交分解（POD）的降阶建模框架。采用内部变量重构遗传本构关系，所得问题在空间上通过对称内罚不连续Galerkin（DG）方法离散，在时间上采用向后欧拉或Crank-Nicolson格式离散。通过将全离散系统投影到位移快照构造的低维空间，得到降阶模型。针对全阶不连续Galerkin格式，证明了其适定性，并在不使用Gronwall不等式的情况下推导了先验稳定性界，该界随最终时间的增长最多为线性而非指数增长。进一步推导了空间误差估计和误差分解，将空间离散、时间离散及本征正交分解截断的影响分离开来。降阶格式用低维问题替代大规模全阶系统，实现高效的长时间模拟。数值实验验证了预测的收敛行为，评估了构造降阶基的不同内积，证明其能准确恢复瞬态振荡和粘弹性松弛，且计算成本显著降低，即使考虑快照生成和基构造的开销亦是如此。
英文摘要：
We present a reduced-order modeling framework based on proper orthogonal decomposition for dynamic linear viscoelasticity governed by generalized Maxwell models. The hereditary constitutive law is reformulated using internal variables, and the resulting problem is discretized in space by a symmetric interior penalty discontinuous Galerkin method and in time by either the backward Euler or Crank--Nicolson scheme. The reduced model is obtained by projecting the fully discrete system onto low-dimensional spaces constructed from displacement snapshots. For the full-order discontinuous Galerkin formulation, we show well-posedness and derive an a priori stability bound without using Gronwall's inequality. Consequently, the bound grows at most linearly, rather than exponentially, with the final time. We further derive spatial error estimates and an error decomposition that separates the effects of spatial discretization, temporal discretization, and proper orthogonal decomposition truncation. The reduced formulation replaces the large full-order systems with low-dimensional problems, enabling efficient long-time simulations. Numerical experiments verify the predicted convergence behavior, assess different inner products for constructing the reduced basis, and demonstrate accurate recovery of transient oscillations and viscoelastic relaxation with substantial computational savings, even after accounting for snapshot generation and basis construction.
