对称二嵌段共聚物Ohta-Kawasaki模型的锐界面问题
Sharp-interface problem of the Ohta-Kawasaki model for symmetric diblock copolymers
AI总结：
针对对称二嵌段共聚物Ohta-Kawasaki模型的锐界面问题，推导匹配渐近展开的锐界面极限，提出适用于二维多连通域的边界积分公式，数值验证其稳定高效且具空间谱精度。
摘要：
用于二嵌段共聚物的Ohta-Kawasaki模型在扩散界面方法科研领域广为人知。为准确捕捉运动界面的长时间演化，我们利用匹配渐近展开推导了对应的锐界面极限，并证明该极限过程导出一类Hele-Shaw型运动界面问题。由于方程的刚性，锐界面极限的数值处理更为复杂。为解决这一问题，我们提出了对应于Ohta-Kawasaki模型锐界面极限的边界积分公式。从定义在各相分离相域上的控制方程出发，我们推导了适用于二维平面多连通域的边界积分方程。出于数值简化考虑，我们假设问题由圆形远场边界上的均匀Dirichlet条件驱动。由于修正后的边界条件，该问题的积分公式同时涉及双层势和单层势。特别地，我们的公式可用于计算非平衡系统的非线性动力学以及平衡系统的图案形成。对演化中的微扰圆形界面（分隔两相）的数值测试与线性分析高度吻合，证明该方法稳定、高效且在空间上具有谱精度。
英文摘要：
The Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space.
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