【作者】 李明辉；

【导师】 许传炬； Mejdi Azaiez；

【作者基本信息】 厦门大学 ， 计算数学， 2022， 博士

【摘要】 本文主要研究了两类多相问题相场模型的高效数值方法,其中一类为Cahn-Hiliard型的两相流模型。另一类为Allen-Cahn型的枝晶生长模型。基于复杂模型的特点和难点,结合辅助变量方法等多种方法,我们提出并分析了多类高效无条件能量稳定的数值格式。论文主要内容包含在如下几个章节中:第一章,从相场模型背景出发,概述若干多相问题相场模型的数值方法研究现状,阐述本文的研究动机和研究的主要内容,并给出一些相关的预备知识。第二章,研究了Navier-Stokes-Cahn-Hilliard方程的数值解法,构造和分析了一类高效的时间空间离散方法。新算法的基本思想是,针对数值求解Navier-Stokes-Cahn-Hilliard方程的难点,首先通过引入辅助变量改写相场方程,然后在对扩展的相场方程进行差分离散时添加适当的附加项用于增加格式的稳定性以及减小计算复杂度。结合Navier-Stokes的压力投影方法,我们设计了 一个质量守恒且无条件能量稳定的一阶和二阶时间半离散格式,严格证明了质量守恒性和无条件稳定性。分析显示,所构造的格式的整体计算复杂度相当于求解六个常系数的二阶椭圆型方程和一个单变量代数方程。据我们所知,这是已有文献中最低成本的方法。空间离散格式基于标准的谱方法,可以证明全离散格式同样具有无条件稳定性。此外,给出了两个改进精度的数值格式。我们借助数个算例评估了算法的有效性,特别是验证了上述分析结果。第三章,考虑Navier-Stokes-Cahn-Hilliard方程一类唯一可解的高效数值解法。借助辅助变量方法及压力投影方法,我们首先通过引入辅助变量对整个方程组作等价变换,结合耦合项及稳定项的新处理技巧,提出了一阶和二阶全解耦的线性时间半离散格式。该格式同样具有质量守恒性和无条件稳定性。我们通过能量分析方法严格证明了任意步长下数值解的存在唯一性。由分析可知,在使用合适分裂项时,数值格式的计算复杂度仅比上一章多求解一个常系数的二阶椭圆型方程。通过与其他文献对比,该算法计算复杂度仍具有较大优势。多个数值实例验证了格式的有效性。第四章,考虑变系数各向异性枝晶生长相场模型的高效数值算法。我们基于辅助变量的方法,导出等价的多方程耦合模型。结合新模型的特点和难点,通过引进多个合适的稳定项,构造了一阶和二阶无条件能量稳定且全解耦的数值格式。该算法在计算实现上等价于求解四个线性椭圆型方程及一个存在唯一解的代数方程。最后通过一些数值例子验证了我们的理论结果,探讨了多类边界条件下枝晶生长过程。据我们了解,这是首次给出变系数各向异性枝晶生长相场模型的二阶线性全解耦的无条件能量稳定格式。更多还原

【Abstract】 The phase field is considered as an order parameter which is introduced to describe the moving interfacial boundary between unstable and stable phases during phase transformation processes.By asymptotic expansions,it can be shown that the phase-field methods relate to classical sharp interface models such as Hele-Shaw type models and Stefan problems in the limit of zero interfacial thickness,see,e.g.[11].A fundamental advantage of the phase-field approach is that the governing equations in the model can be naturally derived from an energy-based variational principle.The energy-based variational framework of phase-field formulations makes them a thermodynamically-consistent and physically attractive in modelling some complex models.The main focus of this thesis is to develop some highly efficient methods to solve some complex phase-field models of some multiphase systems.The outline of the thesis is as follows:In Chapter 1,we give a brief review about the recent progress on numerical investigation of some complex phase-field models,present the motivations and main contents of the thesis,and list some relevant preliminaries.In Chapter 2,we construct and analyze a class of efficient discretization schemes in time and space for the Navier-Stokes-Cahn-Hilliard equations stemming from phasefield modeling of two-phase incompressible flows.The proposed schemes are based on an auxiliary variable approach for the Cahn-Hilliard equation and delicate treatment of the terms coupling the Navier-Stokes equation and the Cahn-Hilliard system.In the theoretical aspect,we rigorously prove that the designed schemes are mass conserving and unconditionally stable in the sense that some kind of energy remains bounded during the time stepping.In the implementation,we show how the schemes can be reformulated into six linear second-order elliptic equations with constant coefficients.To the best of our knowledge,this method only requires half of the cost of the best method in the literature.Furthermore,we give two new schemes to improve the numerical accuracy.The efficiency of the proposed schemes is verified through several numerical examples.In Chapter 3,we consider a class of uniquely solvable numerical methods for the Navier-Stokes-Cahn-Hilliard equations.We construct and analyze two fully decoupled,unconditionally stable,uniquely solvable schemes.In the implementation,the proposed schemes need to solve one more second-order elliptic equation with constant coefficients than the ones developed in the previous chapter.Some numerical experiments are carried out to confirm the efficiency of the new method.In Chapter 4,we propose and analyze two time-stepping schemes for the anisotropic phase-field dendritic crystal growth model.The proposed methods are based on an auxiliary variable approach for the Allen-Cahn equation and delicate treatment of the terms coupling the Allen-Cahn equation and temperature equation.A new technique is proposed to treat the coupling terms involved in the crystal growth model,together with additional terms to stabilize the schemes.We show that the proposed schemes can be realized by solving four linear elliptic equations and a simple algebraic equation at each time step.A detailed comparison with existing schemes is given,and the advantage of the new schemes is emphasized.Finally,some numerical experiments are carried out to confirm the efficiency of the proposed schemes.As far as we know,this is the first secondorder scheme that is totally decoupled,linear,unconditionally stable for the dendritic crystal growth model with variable mobility parameters.更多还原