Purpose

This paper aims to propose a semi-analytical benchmarking framework for enthalpy-based methods used in problems involving phase change with latent heat. The benchmark is based on a class of semi-analytical solutions of spatially symmetric Stefan problems in an arbitrary spatial dimension. Via a public repository this study provides a finite element numerical code based on the FEniCS computational platform, which can be used to test and compare any method of choice with the (semi-)analytical solutions. As a particular demonstration, this paper uses the benchmark to test several standard temperature-based implementations of the enthalpy method and assesses their accuracy and stability with respect to the discretization parameters.

Design/methodology/approach

The class of spatially symmetric semi-analytical self-similar solutions to the Stefan problem is found for an arbitrary spatial dimension, connecting some of the known results in a unified manner, while providing the solutions’ existence and uniqueness. For two chosen standard semi-implicit temperature-based enthalpy methods, the numerical error assessment of the implementations is carried out in the finite element formulation of the problem. This paper compares the numerical approximations to the semi-analytical solutions and analyzes the influence of discretization parameters, as well as their interdependence. This study also compares accuracy of these methods with other traditional approach based on time-explicit treatment of the effective heat capacity with and without iterative correction.

Findings

This study shows that the quantitative comparison between the semi-analytical and numerical solutions of the symmetric Stefan problems can serve as a robust tool for identifying the optimal values of discretization parameters, both in terms of accuracy and stability. Moreover, this study concludes that, from the performance point of view, both of the semi-implicit implementations studied are equivalent, for optimal choice of discretization parameters, they outperform the effective heat capacity method with iterative correction in terms of accuracy, but, by contrast, they lose stability for subcritical thickness of the mushy region.

Practical implications

The proposed benchmark provides a versatile, accessible test bed for computational methods approximating multidimensional phase change problems. The supplemented numerical code can be directly used to test any method of choice against the semi-analytical solutions.

Originality/value

While the solutions of the symmetric Stefan problems for individual spatial dimensions can be found scattered across the literature, the unifying perspective on their derivation presented here has, to the best of the authors’ knowledge, been missing. The unified formulation in a general dimension can be used for the systematic construction of well-posed, reliable and genuinely multidimensional benchmark experiments.

中文翻译：

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任意维度的 Stefan 问题的半解析基准——评估基于焓的方法的准确性

目的

本文旨在提出一种基于焓的方法的半分析基准框架，该方法用于涉及潜热相变的问题。该基准基于任意空间维度中空间对称 Stefan 问题的一类半解析解。通过公共存储库，本研究提供了基于 FEniCS 计算平台的有限元数字代码，可用于测试和比较任何选择的方法与（半）分析解决方案。作为一个具体的演示，本文使用基准测试了几种标准的基于温度的焓法实现，并评估了它们在离散化参数方面的准确性和稳定性。

设计/方法/方法

Stefan 问题的空间对称半解析自相似解类是针对任意空间维度找到的，以统一的方式连接一些已知结果，同时提供解的存在性和唯一性。对于两种选择的标准半隐式基于温度的焓法，在问题的有限元公式中进行实施的数值误差评估。本文将数值近似与半解析解进行了比较，分析了离散化参数的影响以及它们的相互依赖性。本研究还将这些方法的准确性与其他基于时间显式处理有效热容的传统方法进行了比较，其中有和没有迭代校正。

发现

本研究表明，对称 Stefan 问题的半解析解和数值解之间的定量比较可以作为确定离散化参数最优值的稳健工具，无论是在准确性还是稳定性方面。此外，本研究得出结论，从性能的角度来看，所研究的两种半隐式实现是等效的，对于离散化参数的最佳选择，它们在精度方面优于具有迭代校正的有效热容方法，但是，相反，它们在糊状区域的亚临界厚度下失去稳定性。

实际影响

所提出的基准为近似多维相变问题的计算方法提供了一个通用的、可访问的测试平台。补充的数字代码可直接用于针对半解析解决方案测试任何选择的方法。

原创性/价值

虽然单个空间维度的对称 Stefan 问题的解决方案散布在文献中，但据作者所知，这里所提出的关于其推导的统一观点已经缺失。一般维度上的统一公式可用于系统构建适定、可靠和真正的多维基准实验。