Formulation space search for two-dimensional packing problems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The two-dimension packing problem is concerned with the arrangement of items without overlaps inside a container. In particular we have considered the case when the items are circular objects, some of the general examples that can be found in the industry are related with packing, storing and transportation of circular objects. Although there are several approaches we want to investigate the use of formulation space search. Formulation space search is a fairly recent method that provides an easy way to escape from local optima for non-linear problems allowing to achieve better results. Despite the fact that it has been implemented to solve the packing problem with identical circles, we present an improved implementation of the formulation space search that gives better results for the case of identical and non-identical circles, also considering that they are packed inside different shaped containers, for which we provide the needed modifications for an appropriate implementation. The containers considered are: the unit circle, the unit square, two rectangles with different dimension (length 5, width 1 and length 10 width 1), a right-isosceles triangle, a semicircle and a right-circular quadrant. Results from the tests conducted shown several improvements over the best previously known for the case of identical circles inside three different containers: a right-isosceles triangle, a semicircle and a circular quadrant. In order to extend the scope of the formulation space search approach we used it to solve mixed-integer non-linear problems, in particular those with zero-one variables. Our findings suggest that our implementation provides a competitive way to solve these kind of problems.This study was funded by the Mexican National Council for Science and Technology (CONACyT)

Packing equal circles in a damaged square using simulated annealing and greedy vacancy search.

This thesis defines and investigates a generalized circle packing problem, called Packing Equal Circles into a Damaged Square (PECDS). We introduce a new heuristic algorithm that enhances and combines the Greedy Vacancy Search (GVS) and Stimulated Annealing (SA), and demonstrate, through a series of experiments, its ability to find better solutions than either GVS or SA alone. The synergy between the enhanced GVS and SA, along with explicit convergence detection, makes the algorithm robust in escaping the points of local optimum. --Leaf ii.The original print copy of this thesis may be available here: http://wizard.unbc.ca/record=b200686

A New Approach to CNC Programming of Plunge Milling

ABSTRACT A New Approach to CNC Programming of Plunge Milling Sherif Abdelkhalek, PhD. Concordia University, 2013. In current industrial applications many engineering parts are made of hard materials including dies, mold cavities and aerospace parts. Manufacturing these types of parts is classified as pocket milling. By using the regular machining methods, pocket milling takes a long time accompanied by high cost. Plunge milling, is a new machining strategy that has proven to have an excellent performance in the rough machining of hard materials. In plunge milling, the cutter is fed in the direction of the spindle axis, with the highest structural rigidity which showed a very interesting performance in removing the excess material rapidly in the rough operations. Mainly, according to the previous researchers, two directions are adopted to improve the efficiency of the plunge milling process. First, to reduce the cutting forces and increase chatter stability which attracts the majority of the researchers. Second, to optimize the tool path planning which has less attention. Therefore, in the first part of the research, a new practical approach is established in optimized procedures to generate the tool paths for plunge milling of pockets, even for these with free-form boundaries and islands. This innovative approach is proposed as follows: (1) fill a pocket with minimum number of specified radii circles which are tangent to each other and/or the pocket boundary without overlapping by building an algorithm using the maximum hole degree (MHD) theory for solving the circle packing problem. (2) cover the areas left between the non-overlapped circles by the same used specified radii. Finally, solve the travelling sales man problem (TSP) for the circles with the same radii by using the simulated annealing algorithm. According to the results, this approach significantly advances the tool path planning technique for pockets plunge milling. In the second part of the research, a new algorithm is proposed to calculate the global solution for constraint polynomial functions by using subtractive clustering which makes the results more accurate and faster to be obtained. This part is extremely useful to calculate the depth of cut for each plunging place in case of having a polynomial surface as a bottom of the machined pocket with high accuracy, and less calculation time to avoid gauging between the tool and the bottom surface. The polynomial function can be classified according to the number of variables. In the proposed research, the functions with one and two variables have more importance because they graphically represent curves and surfaces which are the cases under study. Since the polynomial function under study can be represented graphically according to the number of the variables, the change in the function’s shape can be detected by the feature recognition. The feature recognition is done for the function’s shape by calculating the surface or curve curvature at the data points. The main procedure is; (1) identifying the entire features of the objective function which are classified according to the curvature as convex, concave, plane, and hyperbolic, (2) applying the sub-clustering technique for convex and concave regions to find the approximated centers of these regions, and eventually, (3) the clusters’ centers are calculated and used as initial points for local optimization technique which gives the local critical point for each region. The local minima are calculated, the global minimum is the minimum of the local minima