AI 中文总结
针对三维流体-二维板耦合的流固相互作用问题，开发Schur补区域分解方法，推导界面Schur补公式并设计界面预条件子，大幅降低系统条件数与GMRES迭代次数，具备良好鲁棒性。
AI 中文摘要
涉及三维流体与弹性薄板耦合的流固相互作用问题广泛存在于诸多工程应用中，由于流固界面处的强耦合作用，这类问题面临显著的计算挑战。本研究针对三维流体-二维板相互作用问题开发了一种Schur补区域分解方法，其中流体采用非定常Stokes方程建模，结构采用重新表述的Kirchhoff板模型建模。从采用拉格朗日乘子施加界面条件的混合有限元离散出发，推导得到一种界面Schur补公式，该公式可解耦流体和板子问题，同时保留界面处的强耦合作用且无需子迭代。随后分析所得Schur补系统矩阵的条件数，表明其条件数会随网格加密而恶化。基于该分析，引入一种基于界面的预条件子，并建立理论界，证明所提预条件子可显著改善所得系统的条件数。数值实验验证了预期的空间和时间收敛率，表明与未预条件系统及块Jacobi预条件子相比，所提预条件子可大幅降低条件数和GMRES迭代次数，且证明了所提方法在挑战性附加质量工况下的鲁棒性。
英文摘要
Fluid-structure interaction problems involving three-dimensional fluids coupled with thin elastic plates arise in many engineering applications and present significant computational challenges due to the strong coupling across the fluid-structure interface. In this work, we develop a Schur complement domain decomposition method for a 3D fluid-2D plate interaction problem in which the fluid is modeled by the unsteady Stokes equations and the structure by a reformulated Kirchhoff plate model. Starting from a mixed finite element discretization with Lagrange multipliers enforcing the interface conditions, we derive an interface Schur complement formulation that decouples the fluid and plate subproblems while preserving the strong interface coupling without requiring subiterations. We then analyze the conditioning of the resulting Schur complement system matrix and show how its condition number deteriorates under mesh refinement. Motivated by this analysis, we introduce an interface-based preconditioner and establish theoretical bounds showing that the proposed preconditioner significantly improves the conditioning of the resulting system. Numerical experiments verify the expected spatial and temporal convergence rates, demonstrate significant reductions in condition numbers and GMRES iteration counts for the proposed preconditioner compared with both the unpreconditioned system and a block Jacobi preconditioner, and illustrate the robustness of the proposed method for challenging added-mass regimes.
Comments27 pages, 7 figures
