Phase-Field Software for the Quantitative Prediction of Coupled Thermal-Chemical Alloy Solidification

凝固微观结构的模拟是近年来人们研究的热点。大型铸造产品的性能受到微观和介观长度尺度上发生的过程的物理学的强烈影响。枝晶是凝固过程中最基本、最普遍的微观组织之一.这些枝晶微观结构的残余通常在随后的加工操作（例如轧制和锻造）中幸存，并且由枝晶建立的长度尺度不仅可以影响最终晶粒尺寸，而且可以影响微观偏析模式，从而影响宏观偏析模式。并且出现的用于模拟枝晶微观结构的最强大的技术之一是相场方法。相场方法的新奇在于，假设固相和液相之间的数学上的尖锐界面是扩散的，允许定义一个连续的、可微分的、代表材料相的序参数。这个相参数的演变是由一个自由能功能，可以解决使用偏微分方程的标准技术，而不明确跟踪固-液界面，从而允许模拟任意复杂的形态。然而，为了获得真实的凝固结构，扩散界面的宽度必须远小于要分辨的最小结构特征。因此，相场模拟通常需要非常精细的网格划分，但它们同样非常适合自适应网格细化。在这里，我们的目标是建立在我们之前的工作基础上，也是由EPSRC资助的，这使我们能够开发第一个完全隐式的，完全自适应（在空间和时间）数值工具的解决方案，以耦合方程组的演变相-场以及热和质量的传输。的自适应性允许非常精细的数值分辨率的相界面，而完全隐式的时间积分，使我们能够解决物理上现实的扩散参数的问题（这是无法实现的半隐式或显式计划，由于其刚度）。这些问题的高度非线性以前阻止了研究人员应用全隐式格式，因此，我们的主要贡献是通过使用鲁棒且有效的非线性多重网格方法求解由此产生的非线性代数方程组来克服这个问题。在本提案中，我们寻求采取最后步骤来开发一种软件工具，该软件工具将允许进行模拟，从而可以产生定量预测第一次在现实材料中的物理固化行为。这将需要两个额外的创新步骤：我们目前的方法推广到三个空间维度，以及高性能计算技术的应用。三个空间维度是必需的，以提供物理上逼真的接口和HPC是必需的，使这种模拟可行的内存和CPU的要求。我们将能够成功地进行这些步骤，建立在我们现有的2-D自适应，隐式相位场的工作，结合我们以前的经验与发展的3-D自适应和3-D HPC软件。

The modelling of solidification microstructures has become an area of intense interest in recent years. The properties of large-scale cast products are strongly influenced by the physics of processes occurring on the microscopic and mesoscopic lengths scales. One of the most fundamental and all pervasive microstructures produced duringsolidification is the dendrite. Remnants of these dendritic microstructures often survive subsequent processing operations such as rolling and forging and the length scales established by the dendrite can influence not only the final grain size but also the micro- and hence the macro-segregation patterns.Theoretical and experimental studies of the development of dendrites have been numerous in recent years, and one of the most powerful techniques to emerge for modelling dendritic microstructures is the phase-field method. The novelty of the phase-field method is that the mathematically sharp interface between the solid and liquid phases is assumed diffuse, allowing the definition of a continuous, differentiable, order parameter which represents the phase of the material. The evolution of this phase parameter is governed by a free energy functional which can be solved using standard techniques for partial differential equations without explicitly tracking the solid-liquid interface, thus allowing the simulation of arbitrarily complex morphologies. However, in order obtain realistic solidification structures the width of the diffuse interface must be much smaller than the smallest structural feature to be resolved. Consequently, phase-field simulations usually require very fine meshing, but equally they are ideally suited to adaptive mesh refinement.Here, our aim is to build upon our prior work, also funded by the EPSRC, which has allowed us to develop the first fully implicit, fully adaptive (in space and time) numerical tool for the solution to the coupled set of equations governing both the evolution of the phase-field and the transport of heat and mass. The adaptivity allows very fine numerical resolution of the phase interface, whilst the fully implict time integration allows us to tackle problems with physically realistic diffusion parameters (that are not attainable with semi-implicit or explicit schemes due to their stiffness). The high nonlinearity of these problems has previously prevented researchers from applying fully implicit schemes, and so our main contribution has been to overcome this problem by solving the resulting systems of nonlinear algebraic equations using a robust and efficient nonlinear multigrid method.In this proposal we seek to take the final steps towards developing a software tool that will permit simulations to be undertaken that can yield quantitative predictions of physical solidification behaviour in realistic materials for the first time. This will require two additional innovative steps: the generalisation of our current methodology to three space dimensions, and the application of high performance computing techniques. Three space dimensions are required in order to provide physically realistic interfaces and HPC is required to make such simulations feasible in terms of both memory and cpu requirements. We will be able to undertake these steps successfully by building upon our existing 2-d adaptive, implicit phase-field work, combined with our prior experience with the development of both 3-d adaptive and 3-d HPC software.