Gaussian,Lobatto and Radau positive quadrature rules with a prescribed abscissa

For a given $\theta \in (a,b)$ , we investigate the question whether there exists a positive quadrature formula with maximal degree of precision which has the prescribed abscissa $\theta $ plus possibly $a$ and/or $b$ , the endpoints of the interval of integration. This study relies on recent results on the location of roots of quasi-orthogonal polynomials. The above positive quadrature formulae are useful in studying problems in one-sided polynomial $L_1$ approximation.     

Prediction of reoxidation inclusion composition in casting of steel

A model is developed to calculate the composition of reoxidation inclusions that form during pouring of steel castings. The software package Thermo-Calc is used to obtain the inclusion phase fractions and compositions as a function of the temperature and oxygen content of the steel. Oxygen is assumed to be continually absorbed by the steel until the liquidus temperature is reached. Both lever rule and Scheil-type analyses are performed. The model is applied to reoxidation of two carbon steels, one low-alloy steel and one high-alloy steel. The effects of variations in the steel composition and the oxygen absorption rate on the inclusion composition are investigated in a parametric study. The mass fraction of absorbed oxygen is determined by matching predicted with previously measured reoxidation inclusion compositions for the various steels. Good agreement is obtained for most phases present in the inclusions. Interestingly, the agreement in the inclusion compositions occurs for all steel grades when the percentage of absorbed oxygen is equal to 0.9 wt pct. This value is explained using a separate model for the rate of oxygen absorption at the steel-atmosphere interface. Various scenarios are outlined that allow for the 0.9 wt pct of absorbed oxygen to be achieved. The model is then used to calculate the amount of alloy elements consumed and inclusions formed as a function of the oxygen boundary layer thickness in the atmosphere and the integrated free surface area of the liquid steel during pouring. It is found that for unprotected liquid steel transfer operations, such as tapping and ladle filling, the integrated free surface area and exposure time product can reach values of the order of 100 m²s per ton of steel, and that the air-to-steel volume ratio during pouring can be as large as 40. It is concluded that, in order to create a comprehensive tool for simulating reoxidation formation, more detailed models are needed for the external oxygen transfer in the atmosphere, the flow of the liquid steel during pouring, and the internal transport and reactions of chemical species in the steel.     

Equiaxed dendritic solidification with convection: Part II. Numerical simulations for an Al-4 Wt pct Cu alloy

The multiphase model developed in part I for equiaxed dendritic solidification with melt convection and solid-phase transport is applied to numerically predict structural and compositional development in an Al-4 wt pct Cu alloy solidifying in a rectangular cavity. A numerical technique combining a fully implicit control-volume-based finite difference method with a multiple time-step scheme is developed for accurate and efficient simulations of both micro- and macroscale phenomena. Quantitative results for the dendritic microstructure evolution in the presence of melt convection and solid movement are obtained. The remarkable effects of the solid-liquid multiphase flow pattern on macrosegregation as well as the grain size distribution are illustrated.     

Effective Computation of Rational Approximants and Interpolants

This paper considers the problem of effective algorithms for some problems having structured coefficient matrices. Examples of such problems include rational approximation and rational interpolation. The corresponding coefficient matrices include Hankel, Toeplitz and Vandermonde-like matrices. Effective implies that the algorithms studied are suitable for implementation in either a numeric environment or else a symbolic environment.The paper includes two algorithms for the computation of rational interpolants which are both effective in symbolic environments. The algorithms use arithmetic that is free of fractions but at the same time control the growth of coefficients during intermediate computations. One algorithm is a look-around procedure which computes along a path of closest normal points to an offdiagonal path while the second computes along an arbitrary path using a look-ahead strategy. Along an antidiagonal path the look-ahead recurrence is closely related to the Subresultant PRS algorithm for polynomial GCD computation. Both algorithms are an order of magnitude faster than alternative methods which are effective in symbolic environments.     

Feeding of high-nickel alloy castings

Feeding of the nickel-based alloys CZ-100, M-35-1, and CW-12MW, as well as of the austenitic stainless steel CN-7M, is investigated, using a combination of casting experiments and simulation. Casting trials are performed at five foundries, to produce a total of 55 plates of varying lengths and radiographic soundness levels. In order to develop the property databases necessary to simulate the casting of these alloys, temperature data are recorded for each alloy during the casting trials. These measured data are used in conjunction with material property simulation to develop the necessary property data for each alloy, including the solidification path. These property data are used to simulate the casting trials. Good agreement between the simulation results and the radiographic testing (RT) results for the castings is obtained. A quantitative relation between the measured ASTM X-ray levels and the predicted minimum Niyama criterion value is established for all but the CZ-100 alloy, which does not appear to suffer from shrinkage defects. A large number of additional simulations are used to develop general feeding distance (FD) rules. The new rules are shown to provide accurate FDs for the casting trial plates. The FDs of the high-nickel alloys (except CZ-100) are found to be at least 25 pct shorter than those for typical low-alloy steels.     

An experimental study of deformation of a columnar dendritic mushy zone using a transparent succinonitrile-acetone alloy

The deformation of a directionally solidified columnar dendritic mushy zone in a transparent succinonitrile-acetone (SCN-ACE) alloy has been studied expermentally. In addition to solidifying dendritically like a metal alloy, this alloy also has mechanical properties that are similar to those of metals near the melting point. The experiments are relevant, for example, to the deformation of a partially solidified strand during continuous casting of steel slabs. A test cell was designed which allows for directional solidification of the alloy and controlled compression of the solid-liquid mush which forms. Measurements during solidification and deformation include temperatures, interface positions, local displacements of the solid skeleton in the mush, and liquid concentrations. Results are presented for a range of initial test-cell thicknesses, deformation amounts, and deformation start times. The measurements are suitable for validation of future models.     

Formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel

The formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel is simulated by simultaneously solving macroscopic mass, momentum, energy, and species conservation equations with full coupling of the temperature and concentrations through thermodynamic equilibrium at the solid/liquid interface. The flow field, solid fraction evolution, and macrosegregation patterns for four cases are presented. The results show both the formation of channel segregates and the formation of islands of mush surrounded by bulk melt. In examining the solidification of a ten-element steel, the global extent of macrosegregation of an element is found to be linearly dependent on its partition coefficient (more severe segregation for small partition coefficient), although such scaling is not possible locally. Results for the solidification of a binary Fe-C alloy (with the same carbon content as the ten-element alloy) are similar to those for the ten-element alloy due solely to the large contribution of carbon to buoyancy driven flow in the ten-element steel chosen for study. While including only those elements that make significant contributions to buoyancy driven flow reproduces the global extent of macrosegregation seen in the ten-element alloy, local differences in the predictions are visible. Finally, comparison of results for the solidification of the same ten-element steel using two different sets of data to describe the partition coefficients and change in liquidus temperature with concentration of the elements shows completely opposite behavior,i.e., upward flow through the mushy zone for one case and downward flow for the other. Thus, the need to have accurate phase-equilibrium data when modeling multicomponent macrosegregation is illustrated. Together, the results give an indication of what areas require more careful examination if accurate modeling of multicomponent solidification is to be accomplished.     

糊状区变形及浓度再分布的模拟实验

用丁二腈－丙酮合金实验研究试样变形开始时间、初始厚度及变形量对枝晶变形及溶质再分布的影响。结果表明，枝晶变形满足幂次大于1的能量函数关系；枝晶变形能力主要与变形时糊状区固相分数有关；从糊状区挤出的液相相对浓度与固相分数、变形比率成线性关系。实验结果对建立糊状区变形与宏观偏析数学模型及优化连铸液芯压下工艺有意义。     

Modeling segregation and grain structure development in equiaxed solidification with convection

Modeling the development of micro- and macrosegregation patterns and grain structures in the equiaxed solidification of metal alloys under the combined influences of melt convection and the motion of free solid has been the subject of intense recent research efforts. This article presents a summary of selected experimental and theoretical studies aimed at understanding the convective transport processes both for a single grain and at the scale of a casting. The need for much additional research is emphasized. C. Beckermann earned his Ph.D. in mechanical engineering at Purdue University in 1987. He is currently a professor at the University of Iowa. He is also a member of TMS.