An explicit scheme for coupling temperature and concentration fields in solidification models

A numerical scheme for coupling temperature and concentration fields in a general solidification model is presented. A key feature of this scheme is an explicit time stepping used in solving the governing thermal and solute conservation equations. This explicit approach results in a local point-by-point coupling scheme for the temperature and concentration and avoids the multi-level iteration required by implicit time stepping schemes. The proposed scheme is validated by predicting the concentration field in a benchmark solidification problem. Results compare well with an available similarity solution. The simplicity of the proposed explicit scheme allows for the incorporation of complex microscale models into a general solidification model. This is demonstrated by investigating the role of dendrite coarsening on the concentration field in the solidification benchmark problem.

Multilevel refinement for combinatorial optimisation problems

We consider the multilevel paradigm and its potential to aid the solution of combinatorial optimisation problems. The multilevel paradigm is a simple one, which involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found (sometimes for the original problem, sometimes the coarsest) and then iteratively refined at each level. As a general solution strategy, the multilevel paradigm has been in use for many years and has been applied to many problem areas (most notably in the form of multigrid techniques). However, with the exception of the graph partitioning problem, multilevel techniques have not been widely applied to combinatorial optimisation problems. In this paper we address the issue of multilevel refinement for such problems and, with the aid of examples and results in graph partitioning, graph colouring and the travelling salesman problem, make a case for its use as a metaheuristic. The results provide compelling evidence that, although the multilevel framework cannot be considered as a panacea for combinatorial problems, it can provide an extremely useful addition to the combinatorial optimisation toolkit. We also give a possible explanation for the underlying process and extract some generic guidelines for its future use on other combinatorial problems.

Multilevel refinement for the vehicle routing problem

Multilevel approaches to computational problems are pervasive across many areas of applied mathematics and scientific computing. The multilevel paradigm uses recursive coarsening to create a hierarchy of approximations to the original problem, then an initial solution is found for the coarsest problem and iteratively refined and improved at each level, coarsest to finest. The solution process is aided by the global perspective (or `global view') imparted to the optimisation by the coarsening. This paper looks at their application to the Vehicle Routing Problem.

Effect of 0.5 wt % Cu addition in Sn–3.5% Ag solder on the dissolution rate of Cu metallization

The dissolution of thin film under-bump-metallization (UBM) by molten solder has been one of the most serious processing problems in electronic packaging technology. Due to a higher melting temperature and a greater Sn content, a molten lead-free solder such as eutectic SnAg has a faster dissolution rate of thin film UBM than the eutectic SnPb. The work presented in this paper focuses on the role of 0.5 wt % Cu in the base Sn–3.5%Ag solder to reduce the dissolution of the Cu bond pad in ball grid array applications. We found that after 0.5 wt % Cu addition, the rate of dissolution of Cu in the molten Sn–3.5%Ag solder slows down dramatically. Systematic experimental work was carried out to understand the dissolution behavior of Cu by the molten Sn–3.5%Ag and Sn–3.5%Ag–0.5%Cu solders at 230–250 °C, for different time periods ranging from 1 to 10 min. From the curves of consumed Cu thickness, it was concluded that 0.5 wt % Cu addition actually reduces the concentration gradient at the Cu metallization/molten solder interface which reduces the driving force of dissolution. During the dissolution, excess Cu was found to precipitate out due to heterogeneous nucleation and growth of Cu6Sn5 at the solder melt/oxide interface. In turn, more Cu can be dissolved again. This process continues with time and leads to more dissolution of Cu from the bond pad than the amount expected from the solubility limit, but it occurs at a slower rate for the molten Sn–3.5%Ag–0.5%Cu solder. © 2003 American Institute of Physics.

Multiple reflow study of ball grid array (BGA) solder joints on Au/Ni metallization

This paper evaluates the shearing behavior of ball grid array (BGA) solder joints on Au/Ni/Cu pads of FR4 substrates after multiple reflow soldering. A new Pb-free solder, Sn–3Ag–0.5Cu–8In (SACI), has been compared with Sn–3Ag–0.5Cu (SAC) and Sn–37Pb (SP) solders, in terms of fracture surfaces, shearing forces and microstructures. Three failure modes, ball cut, a combination of solder shear and solder/pad bond separation, and pad lift, are assessed for the different solders and reflow cycles. It is found that the shearing forces of the SP and SAC solder joints tend to increase slightly with an increase in the number of reflow cycles due to diffusion-induced solid solution strengthening of the bulk solder and augmentation of the shearing area. However, the shearing forces of the SACI solder joints decrease slightly after four cycles of reflow, which is ascribed to the thermal degradation of both the solder/intermetallic compound (IMC) and IMC/Ni interfaces. The SACI solder joints yield the highest strengths, whereas the SP solder joints give the smallest values, irrespective of the number of reflow cycles. Thickening of the interfacial IMC layer and coarsening of the dispersing IMC particles within the bulk solders were also observed. Nevertheless, the variation of shearing forces and IMC thickness with different numbers of reflow cycles was not so significant since the Ni under layer acted as an effective diffusion barrier. In addition, the initially-formed IMC layer retarded the further extensive dissolution of the pad material and its interaction with the solder

Multilevel refinement for combinatorial optimisation: boosting metaheuristic performance

The multilevel paradigm as applied to combinatorial optimisation problems is a simple one, which at its most basic involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found, usually at the coarsest level, and then iteratively refined at each level, coarsest to finest, typically by using some kind of heuristic optimisation algorithm (either a problem-specific local search scheme or a metaheuristic). Solution extension (or projection) operators can transfer the solution from one level to another. As a general solution strategy, the multilevel paradigm has been in use for many years and has been applied to many problem areas (for example multigrid techniques can be viewed as a prime example of the paradigm). Overview papers such as [] attest to its efficacy. However, with the exception of the graph partitioning problem, multilevel techniques have not been widely applied to combinatorial problems and in this chapter we discuss recent developments. In this chapter we survey the use of multilevel combinatorial techniques and consider their ability to boost the performance of (meta)heuristic optimisation algorithms.

Precipitation of heavy metals from wastewater using simulated flue gas: sequent additions of fly ash, lime and carbon dioxide

Lime is a preferred precipitant for the removal of heavy metals from industrial wastewater due to its relatively low cost. To reduce heavy metal concentration to an acceptable level for discharge, in this work, fly ash was added as a seed material to enhance lime precipitation and the suspension was exposed to CO2 gas. The fly ash-lime-carbonation treatment increased the particle size of the precipitate and significantly improved sedimentation of sludge and the efficiency of heavy metal removal. The residual concentrations of chromium, copper, lead and zinc in effluents can be reduced to (mg L-1) 0.08, 0.14, 0.03 and 0.45, respectively. Examination of the precipitates by XRD and thermal analysis techniques showed that calcium-heavy metal double hydroxides and carbonates were present. The precipitate agglomerated and hardened naturally, facilitating disposal without the need for additional solidification/stabilization measures prior to landfill. It is suggested that fly ash, lime and CO2, captured directly from flue gas, may have potential as a method for wastewater treatment. This method could allow the ex-situ sequestration of CO2, particularly where flue-gas derived CO2 is available near wastewater treatment facilities. (C) 2009 Elsevier Ltd. All rights reserved.