Phi-Functions for 2D Objects Formed by Line Segments and Circular Arcs

We study the cutting and packing (C&P) problems in two dimensions by using phi-functions. Our phi-functions describe the layout of given objects; they allow us to construct a mathematical model in which C&P problems become constrained optimization problems. Here we define (for the first time) a complete class of basic phi-functions which allow us to derive phi-functions for all 2D objects that are formed by linear segments and circular arcs. Our phi-functions support translations and rotations of objects. In order to deal with restrictions on minimal or maximal distances between objects, we also propose adjusted phi-functions. Our phi-functions are expressed by simple linear and quadratic formulas without radicals. The use of radical-free phi-functions allows us to increase efficiency of optimization algorithms. We include several model examples

Математическое моделирование отношений эллипсов в задачах оптимальной кластеризации объектов

В статье рассматриваются конструктивные средства математического и компьютерного моделирования отношений (включения, пересечения, касания, непересечения) эллиптических объектов. Определяется полный класс свободных от радикалов аппроксимаций phi-функций для эллипсов и их дополнений. Строится математическая модель задачи оптимальной кластеризации эллиптических объектов в виде последовательности задач нелинейной оптимизации. Приводятся результаты тестовых примеров.У статті розглянуті конструктивні засоби математичного та комп’ютерного моделювання відносин (включення, перетину, дотику, неперетину) еліптичних об’єктів. Визначено повний клас апроксимацій phi-функцій (що вільні від радикалів) для еліпсів та їх доповнень. Побудовано математичну модель задачі оптимальної кластеризації еліптичних об’єктів у вигляді послідовності задач нелінійної оптимізації. Наведено результати тестових прикладів.The article considers constructive tools of mathematical modeling and computer simulation technique of relations (inclusion, non-intersecting, touching, intersecting) between elliptic objects. A complete class of free radical approximations of phi-functions for ellipses and their complements are defined. We provide a mathematical model of the optimal clustering of elliptic objects in the form of constrained optimisation problem. A number of computational results are given

Deep learning-based initialization for object packing

One of the most important optimization tasks in the industry at the current time is the object packing problem. Although several methods have been developed for the purpose of solving it, they are usually only able to optimize placement locally and as such are heavily dependent on the choice of the initial setting -- hence the need for trying out multiple possible starting points, which impacts algorithm running time. In this paper we present a neural network-based model which provides sensible starting points in a linear time

Побудова та дослідження операторів ермітової інтерлінації функцій двох змінних на системі неперетинних ліній із збереженням класу диференційовності

Побудовано та досліджено оператори інтерлінації функцій двох змінних із збереженням класу диференційовності, якому належить наближувана функція за умови, що сліди цих операторів і сліди їх частинних похідних за однією із змінних до фіксованого порядку співпадають на заданій системі ліній з відповідними слідами наближуваної функції.Построены и исследованы операторы интерлинации функций двух переменных с сохранением класса дифференцируемости, которому' принадлежит приближаемая функция при условии, что следы этих операторов и следы частных производных по одной из переменных до фиксированного порядка совпадают на заданной системе линий с соответствующими следами приближаемой функции.Investigates methods for constructing Hermitian operators interlination recovery of differentiated functions of two variables between the smooth continuous curves that preserve the class of differentiability Cr(R2). To construct these operators are used traces of the interpolated function and its partial derivatives with respect to one variable to a given order. The method of constructing these operators are based on the method first proposed in O. N. Lytvyn and uses a linear combination of the integral operators, allowing to increase the relevant classes of differentiable functions that are built with the following, which are assumed not to have the required class of differentiability. Thus said linear combination belongs to a specific class of differentiability despite the value of the linear combination coefficients. These values are found from the condition that corresponding derivatives of the variable y have the same traces as the approximated function on all M nonintersecting curves. Thus constructed operators retain the same differentiability class r, which owns the approximated function f(x, y) and at the same has the same traces as the approximated function with partial derivatives y with respect to the order N inclusive. In this paper, accepted that the functions describing these curves have continuous derivatives to order r including and those curves do not intersect

Optimal clustering of a pair of irregular objects

Cutting and packing problems arise in many fields of applications and theory. When dealing with irregular objects, an important subproblem is the identification of the optimal clustering of two objects. Within this paper we consider a container (rectangle, circle, convex polygon) of variable sizes and two irregular objects bounded by circular arcs and/or line segments, that can be continuously translated and rotated. In addition minimal allowable distances between objects and between each object and the frontier of a container, may be imposed. The objects should be arranged within a container such that a given objective will reach its minimal value. We consider a polynomial function as the objective, which depends on the variable parameters associated with the objects and the container. The paper presents a universal mathematical model and a solution strategy which are based on the concept of phi-functions and provide new benchmark instances of finding the containing region that has either minimal area, perimeter or homothetic coefficient of a given container, as well as finding the convex polygonal hull (or its approximation) of a pair of objects

Layout problems for arc objects in convex domains

We introduce a new methodology for solving layout problems. Our objects and containers are bounded by circular arcs and line segments. We allow continuous object translations and rotations as well as minimal allowable distances between objects. For describing non-overlapping, containment and distance constraints the phi-function technique is used. We provide a general mathematical model as nonlinear programming problem with nonsmooth functions. We propose here the automatic feasible region generator, using phi-trees. The generator allows us to form ready-to-use systems of inequalities with smooth functions in order to apply efficient nonlinear optimisation procedures. We develop an efficient solution algorithm and original solver for layout problems which uses the core representation of inequlities in a sybmol form and provides exact calculation of Jacobian and Hessian matrixes. The search for local minima of NLP-problems is performed by IPOPT algorithm. An essential part of our local optimisation scheme is LORA algorithm that simplifies description of feasible region of the problem and reduces the runtime of local optimisation. It is due to this reduction our strategy can work efficiently with collections of composed objects and search for “good” local-optimal solutions for layout problems in reasonable time.Розглянуто отпимізаційну задачу упаковки довільних об'єктів, обмежених дугами кіл та відрізками прямих в опукіі області. Побудовано математичну модель у вигляді задачі недиференційованої оптимізації, множина реалізацій яко? покриває широкий клас наукових і прикладних задач геометричного проектування. Розроблено методологію розв'язання задач упаковки з урахуванням технологічних обмежень (мінімально допустимі відстані, зони заборони, можливість неперервних трансляцій та обертань об'єктів). Запропоновано генератор простору розв'язків та вирішувач (solver) для автоматичного розв'язання NLP-задач розглянутого класу.Предлагается новая методология решения оптимизационных задач компоновки произвольных объектов в контейнерах, ограниченных дугами окружностей и отрезками прямых. Строится математическая модель в виде задачи нелинейного программирования. Описывается процедура генерации области допустимых решений с применением phi-деревьев, которая позволяет формировать системы неравенств с гладкими функциями. Предлагается эффективный алгоритм поиска локально оптимальных решений.Разработан оригинальный решатель для задач негладкой оптимизации, который использует символьное представление неравенств и обеспечивает точное вычисление элементов матриц Якобиана и Гессиана. Предлагаемая методология эффективна для решения задач компоновки произвольных объектов и позволяет получать «хорошие» локально оптимальные решения за приемлемое время

Utilizing RepRap Style 3D Printers for the Manufacturing of Composite Heat Exchangers

The low cost 3D printing market is currently dominated by the application of RepRap (self-replicating rapid-prototyper) variants. Presented in this document are practical utilizations of RepRap technology. Developed are innovative processes to manufacture composite materials systems for thermal management solutions. First, a laser polymer welder system is validated by quantifying maximum peak load and weld width of linear low density polyethylene (LLDPE) lap welds as a function of linear energy density. The development of practical engineering data, in this application, is critical to producing mechanically durable welds. Developed laser and printer parameter sets allow for manufacturing of LLDPE multi-layered heat exchangers Second, newly introduced metal-polymer composite materials (e.g. copper-PLA, bronze-PLA, iron-PLA and stainless steel-PLA) were shown to influence the thermal conductivity (W/m·K) of the composite matrix. Increased volume percentage of metallic constituent was shown to increase thermal conductivity. Air void fraction, a resultant of the manufacturing process, reduced the bulk composite 3D printed component. No significant effects were realized dependent upon the metallic constituent morphology (i.e. flake-like vs. spherical). Third, development and fabrication of a large format multi-head RepRap 3D printer displays the ability of large-scale manufacturing potential. Energy efficiencies are realized upon utilization of all hot-ends (i.e. the embodied energy of each printer movement (X, Y and Z)) and are simultaneously shown at each hot-end. Furthermore, multi-head format printers are proven to develop composite components. Utilizing a novel weaving and layering method 1000-series aluminum wire is embedded into a polyethylene terephthalate glycol modified (PETG) matrix. Parametric customized gcode commands allow for innovative manufacturing. In total, laser parameter development, material characterization, custom machine fabrication and printing process development are quantified. The three presented projects demonstrate the engineering advancement of RepRap technology in application to thermal management solutions and composite material development

Additive Manufacturing Technologies and Applications

The present Special Issue proposes articles in the area of Additive Manufacturing with particular attention to the different employed technologies and the several possible applications. The main investigated technologies are the Selective Laser Sintering (SLS) and the Fused Deposition Modelling (FDM). These methodologies, combined with the Computer Aided Design (CAD), provide important advantages. Numerical, analytical and experimental knowledge and models are proposed to exploit the potential advantages given by 3D printing for the production of modern systems and structures in aerospace, mechanical, civil and biomedical engineering fields. The 11 selected papers propose different additive manufacturing methodologies and related applications and studies