一种基于多点应力离散的异构弹性波传播时间显式多尺度框架
A time explicit multiscale framework for heterogeneous elastic wave propagation based on multipoint stress discretization
- The Chinese University of Hong Kong（香港中文大学）
- Eastern Institute of Technology（宁波东方理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对异构介质弹性波模拟中细尺度混合求解与粗尺度质量求解的瓶颈，提出基于多点应力离散的时间显式多尺度方法，通过统一时空降阶消除瓶颈，实现显式中心差分，并验证了稳定性与收敛性。
AI中文摘要：
在异构介质中模拟弹性波传播带来了重大的数学和计算挑战，因为解析细尺度材料变化需要极其精细的空间离散化，而混合应力-位移公式通常会导致大型鞍点系统，这些系统不适合高效的时间积分。此外，传统的多尺度降阶通常会产生非平凡的粗尺度质量矩阵，从而需要在每个时间步进行额外的线性求解，削弱了显式格式的优势。为解决这些困难，我们开发了一种基于多点应力控制体积离散的时间显式多尺度方法。核心创新在于一种统一的空间-时间降阶策略，该策略同时消除了两个主要的代数瓶颈：复杂的全局细尺度混合求解以及传统多尺度时间步进中反复出现的粗尺度质量求解。具体而言，应力和辅助旋转变量被局部消除，得到一个对称正定的降阶刚度算子，在此基础上使用密度加权的局部谱问题构造多尺度空间。相应的投影质量双线性形式保留了物理质量内积，并在密度正交辅助基下给出单位质量矩阵，从而自然产生显式中心差分格式。我们在粗尺度CFL条件下建立了离散能量稳定性，并推导了密度加权位移和局部恢复应力的收敛性估计。异构介质中的数值实验验证了理论结果，并证明了所提方法在弹性波传播中的有效性。
英文摘要：
Simulating elastic wave propagation in heterogeneous media presents significant mathematical and computational challenges, since resolving fine-scale material variations requires extremely fine spatial discretizations, while mixed stress--displacement formulations typically lead to large saddle-point systems that are not well suited for efficient time integration. Moreover, conventional multiscale reductions generally produce nontrivial coarse mass matrices, thereby requiring additional linear solves at every time step and diminishing the advantages of explicit schemes. To address these difficulties, we develop a time explicit multiscale method based on a multipoint stress control volume discretization. The central innovation is a unified spatial--temporal reduction strategy that simultaneously removes two dominant algebraic bottlenecks: the complex global fine-scale mixed solve and the repeated coarse-scale mass solve arising in conventional multiscale time stepping. More specifically, the stress and auxiliary rotation variables are eliminated locally, leading to a symmetric positive definite reduced stiffness operator, upon which a multiscale space is constructed using a density-weighted local spectral problem. The associated projected mass bilinear form preserves the physical mass inner product and gives an identity mass matrix under a density-orthonormal auxiliary basis, leading naturally to an explicit central-difference scheme. We establish discrete energy stability under a coarse-scale CFL condition and derive convergence estimates for both the density-weighted displacement and the locally recovered stress. Numerical experiments in heterogeneous media validate the theoretical results and demonstrate the effectiveness of the proposed method for elastic wave propagation.
