发表机构
Eastern Institute of Technology; The Chinese University of Hong Kong（宁波东方理工大学; 香港中文大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对近不可压缩弹性混合离散的约化系统，提出结合位移与离散旋度粗空间的双网格预条件子，使条件数独立于Lamé比，实验验证其有效性。
AI 中文摘要
近不可压缩弹性问题通常采用混合方法离散以避免体积锁定，但由此产生的鞍点系统求解代价高昂。我们考虑一种弱对称多点应力离散，其应力与旋转未知量可在顶点相互作用区域上独立消除，仅留下以单元中心位移为未知量的对称正定系统。尽管该约化系统规模更小，但其中包含参数相关的Schur补，随着Lamé比增大，其预条件处理变得困难。我们证明该系统的能量由一致强制的剪切部分与占主导的半正定体积部分构成。标准单元中心插值不必保持体积核，并可能在近不可压缩情形下引入能量放大。受此结构启发，我们开发了一种双网格预条件子，结合两个互补的粗空间：常规位移子空间与精确位于细网格体积核中的离散旋度子空间。对称顶点块平滑完成仅含位移的求解器。能量分解与核相容分解对每个固定网格对给出独立于Lamé比的条件数界。二维与三维实验证实了局部消除离散的逼近性质，并表明核修正可防止仅位移粗子空间的退化，即使对不连续材料系数亦然。
英文摘要
Nearly incompressible elasticity is often discretized by mixed methods to avoid locking, but the resulting saddle-point systems can be expensive to solve. We consider a weakly symmetric multipoint stress discretization whose stress and rotation unknowns can be eliminated independently over vertex interaction regions, leaving a symmetric positive definite system for cell-centered displacement only. Although smaller, the reduced system contains a parameter-dependent Schur complement that becomes difficult to precondition as the Lamé ratio increases. We show that its energy consists of a uniformly coercive shear part and a dominant semidefinite volumetric part. Standard cell-centered interpolation need not preserve the volumetric kernel and can introduce energy amplified near incompressibility. Motivated by this structure, we develop a two-grid preconditioner combining two complementary coarse subspaces: a conventional displacement subspace and a discrete-curl subspace lying exactly in the fine-grid volumetric kernel. Symmetric vertex-patch smoothing completes the displacement-only solver. The energy splitting and kernel-compatible decompositions yield a condition-number bound independent of the Lamé ratio for each fixed grid pair. Two- and three-dimensional experiments confirm the approximation properties of the locally eliminated discretization and show that the kernel correction prevents the deterioration of the displacement coarse subspace alone, also for discontinuous material coefficients.
