An Unconditionally Energy Stable Penalty Immersed Boundary Method for Simulating the Dynamics of an Inextensible Interface Interacting with a Solid Particle

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• 作者：Po-Wen Hsieh ; Ming-Chih Lai ; Suh-Yuh Yang ; Cheng-Shu You
• 关键词：Immersed boundary method ; Penalty method ; Stokes flow ; Inextensible interface ; Solid particle ; Stability ; 65M06 ; 65M12 ; 76D07 ; 76M20
• 刊名：Journal of Scientific Computing
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：64
• 期：2
• 页码：289-316
• 全文大小：3,376 KB
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• 作者单位：Po-Wen Hsieh (1)
  Ming-Chih Lai (2)
  Suh-Yuh Yang (3)
  Cheng-Shu You (3)
  
  1. Department of Applied Mathematics, Chung Yuan Christian University, Jhongli City, 32023, Taoyuan County, Taiwan
  2. Center of Mathematical Modeling and Scientific Computing, Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 30010, Taiwan
  3. Department of Mathematics, National Central University, Jhongli City, 32001, Taoyuan County, Taiwan
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691

文摘

In this paper, a novel penalty method based on the immersed boundary formulation is proposed for simulating the transient Stokes flow with an inextensible interface enclosing a suspended solid particle. The main idea of this approach relies on the penalty techniques by modifying the constitutive equation of Stokes flow to weaken the incompressibility condition, relating the surface divergence to the elastic tension \(\sigma \) to relax the interface’s inextensibility, and connecting the particle surface-velocity with the particle surface force \({\varvec{F}}\) to regularize the particle’s rigid motion. The advantage of these regularized governing equations is that when they are discretized by the standard centered difference scheme on a staggered grid, the resulting linear system can easily be reduced by eliminating the unknowns \(p_h, \sigma _h\) and \({\varvec{F}}_h\) directly, so that we just need to solve a smaller linear system of the velocity approximation \({\varvec{u}}_h\). This advantage is preserved and even enhanced when such approach is applied to the transient Stokes flow with multiple compound vesicles. Moreover, this smaller linear system is symmetric and negative-definite, which enables us to use efficient linear solvers. Another important feature of the proposed method is that the discretization scheme is unconditionally stable in the sense that an appropriately defined energy functional associated with the discrete system is decreasing and hence bounded in time. We numerically test the accuracy and stability of the immersed boundary discretization scheme. The tank-treading and tumbling motions of inextensible interface with a suspended solid particle in the simple shear flow will be studied extensively. The simulation of the motion of multiple compound vesicles will be performed as well. Numerical results illustrate the superior performance of the proposed penalty method.