A multilevel algorithm for force-directed graph drawing

We describe a heuristic method for drawing graphs which uses a multilevel technique combined with a force-directed placement algorithm. The multilevel process groups vertices to form clusters, uses the clusters to define a new graph and is repeated until the graph size falls below some threshold. The coarsest graph is then given an initial layout and the layout is successively refined on all the graphs starting with the coarsest and ending with the original. In this way the multilevel algorithm both accelerates and gives a more global quality to the force- directed placement. The algorithm can compute both 2 & 3 dimensional layouts and we demonstrate it on a number of examples ranging from 500 to 225,000 vertices. It is also very fast and can compute a 2D layout of a sparse graph in around 30 seconds for a 10,000 vertex graph to around 10 minutes for the largest graph. This is an order of magnitude faster than recent implementations of force-directed placement algorithms.

A multilevel algorithm for force-directed graph-drawing

We describe a heuristic method for drawing graphs which uses a multilevel framework combined with a force-directed placement algorithm. The multilevel technique matches and coalesces pairs of adjacent vertices to define a new graph and is repeated recursively to create a hierarchy of increasingly coarse graphs, G0, G1, …, GL. The coarsest graph, GL, is then given an initial layout and the layout is refined and extended to all the graphs starting with the coarsest and ending with the original. At each successive change of level, l, the initial layout for Gl is taken from its coarser and smaller child graph, Gl+1, and refined using force-directed placement. In this way the multilevel framework both accelerates and appears to give a more global quality to the drawing. The algorithm can compute both 2 & 3 dimensional layouts and we demonstrate it on examples ranging in size from 10 to 225,000 vertices. It is also very fast and can compute a 2D layout of a sparse graph in around 12 seconds for a 10,000 vertex graph to around 5-7 minutes for the largest graphs. This is an order of magnitude faster than recent implementations of force-directed placement algorithms.

Wetting and Reaction of Sn-2.8Ag-0.5Cu-1.0Bi Solder with Cu and Ni Substrates

The wettability of newly developed Sn-2.8Ag-0.5Cu-1.0Bi lead-free solder on Cu and Ni substrates was assessed through the wetting balance tests. The wettability assessment parameters such as contact angle (ϑc) and maximum wetting force (Fw) were documented for three solder bath temperatures with three commercial fluxes, namely, no-clean (NC), nonactivated (R), and water-soluble organic acid flux (WS). It was found that the lead-free Sn-2.8Ag-0.5Cu-1.0Bi solder exhibited less wetting force, i.e., poorer wettability, than the conventional Sn-37Pb solder for all flux types and solder bath temperatures. The wettability of Sn-2.8Ag-0.5Cu-1.0Bi lead-free solder on Cu substrate was much higher than that on Ni substrate. Nonwetting for Sn-2.8Ag-0.5Cu-1.0Bi and Sn-Pb solders on Ni substrate occurred when R-type flux was used. A model was built and simulations were performed for the wetting balance test. The simulation results were found very close to the experimental results. It was also observed that larger values of immersion depth resulted in a decrease of the wetting force and corresponding meniscus height, whereas the increase in substrate perimeter enhanced the wettability. The wetting reactions between the solder and Cu/Ni substrates were also investigated, and it was found that Cu atoms diffused into the solder through the intermetallic compounds (IMCs) much faster than did the Ni atoms. Rapid formation of IMCs inhibited the wettability of Sn-2.8Ag-0.5Cu-1.0Bi solder compared to the Sn-Pb solder.

Droplet formation with electromagnetic pulse force

Methods for serial generation of droplets from a liquid jet are shortly reviewed. A method of liquid metal droplet generation based on AC high frequency magnetic field is considered in detail. Numerical model for direct simulation of the time dependent droplet generation process is presented. Computed examples demonstrate the liquid silicon droplet formation for the cases of 500-1500 μm diameter.

Comparative wetting behavior of Sn-0.7Cu and Sn-0.7Cu-0.3Ni solders on Cu and Ni substrates

Electronic packaging industries are now in great challenge to find a suitable lead-free solder as an interconnection material to replace the conventional SnPb solders. Many solders such as SnCu, SnAg, SnAgCu, SnZn, SnBi have already been proposed as the replacement but none of them has reached the physical and metallurgical properties similar to the SnPb solder. However, wetting is one of the basic problems that make the lead-free solder inferior as compared to the SnPb solder. Therefore, alloying with the help of third, fourth or fifth element is the researchers' interest to improve the wetting behavior of lead-free solders. This paper describes the comparative wetting behavior of Sn-0.7Cu and Sn-0.7Cu-0.3Ni solders on Cu and Ni substrates. Wetting balance tests were performed to assess the wetting behaviors. Three different commercial fluxes namely no-clean (NC), non-activated (R) and water soluble organic acid (WS)fluxes were used to assess the wettability for three solder bath temperatures. It was found that Sn0.7Cu-03Ni solder exhibits better wettability on Cu substrate for NC and WS fluxes whereas reverse results were found for R-type flux. In the case of Ni substrate, Sn-0.7Cu-0.3Ni solder showed better wetting behavior compared to the well-known Sn-0.7Cu solder. Among the three fluxes, R-type flux showed the worst performance. Very large contact angles were documented for both solders with this flux. Higher solder bath temperature lowered the contact angles, increased the wetting forces and enhanced the wettability. Computer modeling of wetting balance test revealed that both the wetting force and meniscus height are inversely proportional to the contact angles. Modeling results also reveal that increase in solder bath depths and radiuses do not affect significantly on the wetting behavior.

Effect of adding 0.3 wt% Ni into the Sn–0.7 wt%Cu solder: part I: wetting behavior on Cu and Ni substrates

Comparative wetting behavior of Sn-0.7Cu and Sn-0.7Cu-0.3Ni solders on Cu and Ni substrates were assessed through the wetting balance test. No-clean (NC), non-activated (R) and water soluble organic acid (WS) fluxes were used to assess the wetting behavior for three different solder bath temperatures of 255, 275 and 295 °C. Experimental results unveiled that adding of 0.3 wt% Ni into Sn-0.7Cu solder can improve the wetting on Cu substrate when NC and WS fluxes are used. However, such addition of Ni did not improve the wetting of Sn-0.7Cu solder for R-type flux. In the case of Ni substrate, addition of Ni helped to improve the wetting for all three types of fluxes as higher wetting forces were documented for Sn-0.7Cu-0.3Ni solder compared to the Sn-0.7Cu solder. Among the fluxes, worst performance was observed for R-type flux. Very large contact angles were recorded for both solders with this kind of flux. Experimental results also revealed that higher solder bath temperature played an important role to lower the contact angle, to increase the wetting force and to enhance the wetting. Computer modeling of wetting balance test also revealed that both the wetting force and meniscus height are inversely proportional to the contact angles. Besides, solder bath depth and radius do not affect significantly on the wetting behavior.

Determining surface tension using DC magnetic levitation

The values of material physical properties are vital for the successful use of numerical simulations for electromagnetic processing of materials. The surface tension of materials can be determined from the experimental measurement of the surface oscillation frequency of liquid droplets. In order for this technique to be used, a positioning field is required that results in a modification to the oscillation frequency. A number of previous analytical models have been developed that mainly focus on electrically conducting droplets positioned using an A.C. electromagnetic field, but due to the turbulent flow resulting from the high electromagnetic fields required to balance gravity, reliable measurements have largely been limited to microgravity. In this work axisymmetric analytical and numerical models are developed, which allow the surface tension of a diamagnetic droplet positioned in a high DC magnetic field to be determined from the surface oscillations. In the case of D.C. levitation there is no internal electric currents with resulting Joule heating, Marangoni flow and other effects that introduce additional physics that complicates the measurement process. The analytical solution uses the linearised Navier-Stokes equations in the inviscid case. The body force from a DC field is potential, in contrast to the AC case, and it can be derived from Maxwell equations giving a solution for the magnetic field in the form of a series expansion of Legendre polynomials. The first few terms in this expansion represent a constant and gradient magnetic field valid close to the origin, which can be used to position the droplet. Initially the mathematical model is verified in microgravity conditions using a numerical model developed to solve the transient electromagnetics, fluid flow and thermodynamic equations. In the numerical model (as in experiment) the magnetic field is obtained using electrical current carrying coils, which provides the confinement force for a liquid droplet. The model incorporates free surface deformation to accurately model the oscillations that result from the interaction between the droplet and the non-uniform external magnetic field. A comparison is made between the analytical perturbation theory and the numerical pseudo spectral approximation solutions for small amplitude oscillations.

Manufacturing of uniformly-sized silicon particles for solar cell from molten metal jet by electromagnetic pinch force

Spherical silicon solar cells are expected to serve as a technology to reduce silicon usage of photovoltaic (PV) power systems[1, 2, 3]. In order to establish the spherical silicon solar cell, a manufacturing method of uniformly sized silicon particles of 1mm in diameter is required. However, it is difficult to mass-produce the mono-sized silicon particles at low cost by existent processes now. We proposed a new method to generate liquid metal droplets uniformly by applying electromagnetic pinch force to a liquid metal jet[4]. The electromagnetic force was intermittently applied to the liquid metal jet issued from a nozzle in order to fluctuate the surface of the jet. As the fluctuation grew, the liquid jet was broken up into small droplets according to a frequency of the intermittent electromagnetic force. Firstly, a preliminary experiment was carried out. A single pulse current was applied instantaneously to a single turn coil around a molten gallium jet. It was confirmed that the jet could be split up by pinch force generated by the current. And then, electromagnetic pinch force was applied intermittently to the jet. It was found that the jet was broken up into mono-sized droplets in the case of a force frequency was equal to a critical frequency[5], which corresponds to a natural disturbance wave length of the jet. Numerical simulations of the droplet generation from the liquid jet were then carried out, which consisted of an electromagnetic analysis and a fluid flow calculation with a free surface of the jet. The simulation results were compared with the experiments and the agreement between the two was quite good.

Mode shape expansion using perturbed force vector approach

A new technique for mode shape expansion in structural dynamic applications is presented based on the perturbed force vector approach. The proposed technique can directly adopt the measured incomplete modal data and include the effect of the perturbation between the analytical and test models. The results show that the proposed technique can provide very accurate expanded mode shapes, especially in cases when significant modelling error exists in the analytical model and limited measurements are available.