Solidification shrinkage has been recognized as an important factor for macrosegregation formation. An arbitrary Lagrangian-Eulerian (ALE) model is constructed to predict the macrosegregation caused by thermal-solutal convection and solidification shrinkage. A novel mesh update algorithm is developed to account for the domain change induced by solidification shrinkage. The velocity-pressure coupling between the non-homogenous mass conservation equation and momentum equation is addressed by a modified pressure correction method. The governing equations are solved by the streamline-upwind/ Petrov-Galerkin-stabilized finite element algorithm. The application of the model to the Pb-19.2 wt%Sn alloy solidification problem is considered. The inverse segregation is successfully predicted, and reasonable agreement with the literature results is obtained. Thus, the ALE model is established to be a highly effective tool for predicting the macrosegregation caused by solidification shrinkage and thermal-solutal convection. Finally, the effect of solidification shrinkage is analyzed. The results demonstrate that solidification shrinkage delays the advance of the solidification front and intensifies the segregation.

Purpose The purpose of this paper is to simulate two macrosegregation benchmarks with a newly developed stabilized finite element algorithm based on a semi-implicit pressure correction scheme. Design/methodology/approach A streamline-upwind/Petrov–Galerkin (SUPG) stabilized finite element algorithm is developed for the coupled conservation equations of mass, momentum, energy and species. A semi-implicit pressure correction method combined with SUPG stabilization technique is proposed to solve the convection flow during solidification. An analytically derived enthalpy method is adopted to solve the energy conservation equation. The nonlinearities of the energy and species equations are tackled by Newton–Raphson method. Two macrosegregation benchmarks considering the solidification of an Al-4.5 per cent Cu alloy and a Sn-10 per cent Pb alloy are simulated. Findings A very good agreement is achieved by comparison with the classical finite volume method and a novel meshless method for the Al-4.5 per cent Cu alloy solidification benchmark. Moreover, a unique reference numerical solution has been obtained. Besides, it is demonstrated that the stabilized finite element algorithm can capture the flow instability and channel segregation during solidification of the Sn-10 per cent Pb alloy. Originality/value A semi-implicit pressure correction method combined with SUPG stabilization technique is adopted to develop robust stabilized finite element algorithm for the macrosegregation model. A new enthalpy formulation for heat transfer problems with phase change is derived analytically.

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This article explores the internal and external motivations of company's repurchase decision. In terms of methods, the author uses a combination of theory and empirical research methods and logically uses an analysis process from general rules to special cases to make the conclusions more accurate and have practical meaning. In the research process, the article firstly makes a theoretical analysis, summarizing and expanding various hypotheses about corporate share repurchase motives put forward by predecessors. In addition to analyzing the common signal transmission hypothesis, management incentive hypothesis, and other internal factors, the author also added external factors such as share liquidity and volatility, making the theoretical analysis more comprehensive and innovative. At the same time, to make the research more specific, this article selects Starbucks, a representative company, to analyze its repurchase behavior. In the case analysis, the article at first gives concrete indicators and indexes of each hypothesis based on theoretical research, transforming the qualitative analysis into quantitative research, making the research process clearer and more specific. The article selects Starbucks' financial data from 2009 to 2020 and obtains the corresponding indicator values through integration and calculation. Finally, to increase the conclusions' accuracy, the article selected several main factors through the previous analysis to conduct an empirical test and used Eviews software to perform regression analysis on time series data such as the number of share repurchases and company share prices. The regression results show that a company's number of shares repurchases is negatively correlated with the share price growth rate. The significance level is the highest, proving that the main motivation for corporate share repurchases is to increase the share price. Finally, this article summarizes Starbucks' share repurchase behavior based on relevant analysis results and non-financial information. It provides suggestions for other companies' share repurchase decisions, which have practical value.