材料科学中复杂相场模型的高效稳定数值方法

The phase field model describes a microstructure by adopting a set of conserved and nonconserved field variables that are continuous across the interfacial regions. The phase field approach has recently emerged as an important and extremely versatile technique for modeling and predicting mesoscale morphological and microstructure evolution in materials. Due to the coupled nonlinear term of the complex multi-component multi-phase field model in materials, therefore simple fully-implicit or explicit type discretizations for the phase field model will induce unfavorable nonlinear and/or energy unstable schemes which are not efficient in practice. How to develop linear and/or decoupled schemes while preserving the energy stability at the discrete level for solving the nonlinear coupled system numerically has become one main topic for the algorithmic design. Based on the ideas of the scalar auxiliary variable and exponential time differencing approaches, we aim to design the high order, unconditionally energy stable and efficient numerical decoupled implicit schemes for the complex phase field model in material, the main idea is to reformulate and split the energy functional in the quadratic form by introducing extra variables and combine the stabilized integrating factor with Runge-Kutta method. From the point of view of the theoretical analysis, the rigorous stability and error analysis for the schemes are analyzed by applying the energy dissipation law of the gradient flow problems. From the numerical point of view, some fast algorithms and adaptive algorithms are developed for the energy stable schemes, and a series of numerical experiments is carried out to illustrate the efficiency of the proposed efficient unconditionally stable numerical schemes, and simulate and predict the microstructure evolution. The establishment of accurate, highly efficient numerical schemes and completed numerical analysis has important theoretical value and application aspects for deeply revealing the microstructure evolution and guiding science engineering practice.

相场模型用在界面区域上连续守恒和非守恒的场变量描述材料科学中涉及许多复杂过程的微结构，它的计算是近年来发展的一种强有力的模拟和预测材料中尺度形态和微结构演化的方法。由于材料科学中复杂相场模型的耦合非线性特点，显式或简单的全隐式离散格式将会导致能量不稳定格式或计算中出现效率较低且不易求解的非线性问题。本项目旨在利用标量辅助变量方法和指数时间差分方法的思想，分别通过引入辅助变量将能量泛函分裂为二次形式和结合龙格库塔方法，对材料科学中复杂相场模型设计时间上高阶、无条件能量稳定、易于计算的解耦线性隐式数值格式。在理论上利用梯度流问题的能量耗散定律，分析半离散数值格式的稳定性及全离散数值格式的误差和收敛性；在数值计算上，利用上述稳定数值格式进而设计快速算法，验证数值格式的准确性和稳定性，并模拟和预测材料微结构的演化过程，对深刻了解材料形变机理和及时更新材料具有重要的理论价值和应用前景。

材料科学中涉及许多复杂过程的微结构通常用相场模型中连续守恒和非守恒的场变量描述，它的计算是近年来发展的一种强有力的模拟和预测材料中尺度形态和微结构演化的方法。本项目主要研究了材料科学中代表性的金属腐蚀，它是易感金属与它所处环境间的破坏性电化学反应，导致材料的完整性和耐久性遭到破坏，大多数高强度的合金都会在其表面镀一层保护膜以防止被腐蚀。然而，合金表面的保护膜遭到局部破坏就很容易被腐蚀，加速穿孔处微观结构间的结构失效或形成裂缝。根据梯度流理论，用守恒的腐蚀液浓度和非守恒的相变量刻画金属腐蚀界面的演化过程，从而得到了由Cahn-Hilliard和Allen-Cahn耦合的非线性方程组。数值求解这个模型的主要困难是方程组的强刚性和耦合非线性，要求时间步长非常小。为了克服这些困难，在时间上采用近来发展起来的高阶隐显龙格库塔方法求解上述相场模型。我们为金属腐蚀模型设计了时间上的二阶和三阶数值格式。这个方法用隐式格式离散方程组的线性部分，用显式格式结合龙格库塔方法离散并解耦方程组的非线性部分，理论证明了本方法的无条件能量耗散规律；在空间上，采用自适应移动网格方法来提高计算效率。数值实验验证了方法的有效性和能量耗散规律，并模拟了金属腐蚀过程以及界面的演化过程。本方法对模拟和预测材料微结构的演化过程，对深刻了解材料形变机理和及时更新材料具有重要的理论价值和应用前景。

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