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Under conventional solidification conditions, immiscible alloy melt would undergo large-scale composition segregation after liquid–liquid phase separation, resulting in the loss of properties and application value. In the present study, the ternary immiscible Al70Bi10Sn20 alloy was chosen to study the effect of cooling rate on its resultant microstructure by casting the melt under different cooling conditions. The results indicated that the Al–Bi–Sn alloy with a slow cooling rate exhibits a strong spatial phase separation trend during solidification. However, as the cooling rate increases, the decreasing volume fraction of the segregated Bi–Sn-rich regions indicates the efficient suppression of spatial phase separation. The relatively dispersed distribution of Bi–Sn phase in the Al-rich matrix can be obtained by quenching the melt into water. The influence mechanism of cooling rate on the microstructure of the alloy is also discussed. The present study is beneficial to further tailoring the microstructure of immiscible alloys.
This introduction to finite difference and finite element methods is aimed at graduate students who need to solve differential equations. The prerequisites are few (basic calculus, linear algebra, and ODEs) and so the book will be accessible and useful to readers from a range of disciplines across science and engineering. Part I begins with finite difference methods. Finite element methods are then introduced in Part II. In each part, the authors begin with a comprehensive discussion of one-dimensional problems, before proceeding to consider two or higher dimensions. An emphasis is placed on numerical algorithms, related mathematical theory, and essential details in the implementation, while some useful packages are also introduced. The authors also provide well-tested MATLAB® codes, all available online.
A nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.