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首页> 外文学位 >Adaptive Spline-based Finite Element Method with Application to Phase-field Models of Biomembranes.
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Adaptive Spline-based Finite Element Method with Application to Phase-field Models of Biomembranes.

机译：基于自适应样条的有限元方法在生物膜相场模型中的应用。

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摘要

Interfaces play a dominant role in governing the response of many biological systems and they pose many challenges to traditional finite element. For sharp-interface model, traditional finite element methods necessitate the finite element mesh to align with surfaces of discontinuities. Diffuse-interface model replaces the sharp interface with continuous variations of an order parameter resulting in significant computational effort. To overcome these difficulties, we focus on developing a computationally efficient spline-based finite element method for interface problems.;A key challenge while employing B-spline basis functions in finite-element methods is the robust imposition of Dirichlet boundary conditions. We begin by examining weak enforcement of such conditions for B-spline basis functions, with application to both second- and fourth-order problems based on Nitsche's approach. The use of spline-based finite elements is further examined along with a Nitsche technique for enforcing constraints on an embedded interface. We show that how the choice of weights and stabilization parameters in the Nitsche consistency terms has a great influence on the accuracy and robustness of the method. In the presence of curved interface, to obtain optimal rates of convergence we employ a hierarchical local refinement approach to improve the geometrical representation of interface.;In multiple dimensions, a spline basis is obtained as a tensor product of the one-dimensional basis. This necessitates a rectangular grid that cannot be refined locally in regions of embedded interfaces. To address this issue, we develop an adaptive spline-based finite element method that employs hierarchical refinement and coarsening techniques. The process of refinement and coarsening guarantees linear independence and remains the regularity of the basis functions. We further propose an efficient data transfer algorithm during both refinement and coarsening which yields to accurate results.;The adaptive approach is applied to vesicle modeling which allows three-dimensional simulation to proceed efficiently. In this work, we employ a continuum approach to model the evolution of microdomains on the surface of Giant Unilamellar Vesicles. The chemical energy is described by a Cahn-Hilliard type density functional that characterizes the line energy between domains of different species. The generalized Canham-Helfrich-Evans model provides a description of the mechanical energy of the vesicle membrane. This coupled model is cast in a diffuse-interface form using the phase-field framework. The effect of coupling is seen through several numerical examples of domain formation coupled to vesicle shape changes.
机译：接口在控制许多生物系统的响应中起着主导作用，它们对传统的有限元构成了许多挑战。对于锋利的界面模型，传统的有限元方法必须使有限元网格与不连续面对齐。漫射界面模型用顺序参数的连续变化代替了尖锐的界面，从而导致了大量的计算工作。为了克服这些困难，我们致力于开发一种计算效率高的基于样条的有限元方法来解决界面问题。在有限元方法中采用B样条基函数时的主要挑战是Dirichlet边界条件的强健施加。我们首先研究基于B样条函数的这种条件的弱执行，并基于Nitsche的方法将其应用于二阶和四阶问题。进一步检查了基于样条的有限元的使用以及用于在嵌入式接口上执行约束的Nitsche技术。我们表明，在Nitsche一致性项中权重和稳定参数的选择如何对方法的准确性和鲁棒性有很大影响。在存在弯曲界面的情况下，为了获得最佳收敛速度，我们采用了分层局部细化方法来改善界面的几何表示。在多维中，样条曲线基作为一维基数的张量积而获得。这就需要一个矩形网格，该矩形网格无法在嵌入式接口的区域中进行局部优化。为了解决这个问题，我们开发了一种基于自适应样条的有限元方法，该方法采用了层次化的细化和粗化技术。细化和粗化的过程保证了线性独立性，并保持了基函数的规律性。我们进一步提出了一种在细化和粗化过程中均能获得准确结果的有效数据传输算法。自适应方法被应用于囊泡建模，从而使三维模拟得以有效进行。在这项工作中，我们采用一种连续方法来模拟巨型单层囊泡表面微区的演化。化学能由Cahn-Hilliard类型的密度泛函描述，该函数表征了不同物种的畴之间的线能量。广义的Canham-Helfrich-Evans模型描述了囊泡膜的机械能。使用相场框架将该模型耦合为扩散接口形式。通过与囊泡形状变化耦合的域形成的几个数值示例可以看出耦合的效果。

著录项

  • 作者

    Jiang, Wen.;

  • 作者单位

    Duke University.;

  • 授予单位 Duke University.;
  • 学科 Applied Mechanics.;Engineering Mechanical.;Engineering Civil.
  • 学位 Ph.D.
  • 年度 2015
  • 页码 154 p.
  • 总页数 154
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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