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

近年来，凝固微观组织的建模已成为一个备受关注的领域。大规模铸造产品的性能受到发生在微观和介观长度尺度上的物理过程的强烈影响。在凝固过程中产生的最基本和最普遍的微观组织之一是枝晶。这些枝晶组织的残余通常在随后的加工操作中幸存下来，如轧制和锻造，枝晶建立的长度尺度不仅会影响最终的晶粒尺寸，还会影响微观和宏观的偏析模式。近年来，对枝晶发展的理论和实验研究已经很多，其中最强大的枝晶微观结构建模技术之一是相场法。相场法的新颖之处在于，假定固体和液体之间的数学上清晰的界面是扩散的，从而允许定义一个连续的、可微的、有序的参数来表示材料的相。该相位参数的演变由自由能泛函控制，该泛函可以使用偏微分方程的标准技术求解，而无需显式跟踪固液界面，从而允许模拟任意复杂的形态。然而，为了获得真实的凝固组织，扩散界面的宽度必须远远小于要解决的最小结构特征。因此，相场模拟通常需要非常精细的网格划分，但同样地，它们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的基础上，也由EPSRC资助，这使我们能够开发出第一个完全隐式的，完全自适应的（在空间和时间上）数值工具，用于解决控制相场演化和热量和质量传输的耦合方程组。自适应性允许相位界面的非常精细的数值分辨率，而完全隐式时间积分允许我们处理物理上真实的扩散参数问题（由于其刚度，半隐式或显式方案无法实现）。这些问题的高度非线性先前阻碍了研究人员应用完全隐式格式，因此我们的主要贡献是通过使用鲁棒和高效的非线性多重网格方法求解非线性代数方程的结果系统来克服这一问题。在本提案中，我们试图采取最后步骤，开发一种软件工具，该软件工具将允许进行模拟，从而首次对现实材料的物理凝固行为进行定量预测。这将需要两个额外的创新步骤：将我们目前的方法推广到三维空间，以及高性能计算技术的应用。为了提供物理上真实的界面，需要三个空间维度，并且需要HPC使这种模拟在内存和cpu需求方面都可行。我们将能够在现有的二维自适应、隐式相场工作的基础上，结合我们之前在三维自适应和三维HPC软件开发方面的经验，成功地完成这些步骤。

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.