Global Optimization for a Class of Nonlinear Sum of Ratios Problem

We present a branch and bound algorithm for globally solving the sum of ratios problem. In this problem, each term in the objective function is a ratio of two functions which are the sums of the absolute values of affine functions with coefficients. This problem has an important application in financial optimization, but the global optimization algorithm for this problem is still rare in the literature so far. In the algorithm we presented, the branch and bound search undertaken by the algorithm uses rectangular partitioning and takes place in a space which typically has a much smaller dimension than the space to which the decision variables of this problem belong. Convergence of the algorithm is shown. At last, some numerical examples are given to vindicate our conclusions

On some geometric optimization problems.

An optimization problem is a computational problem in which the objective is to find the best of all possible solutions. A geometric optimization problem is an optimization problem induced by a collection of geometric objects. In this thesis we study two interesting geometric optimization problems. One is the all-farthest-segments problem in which given n points in the plane, we have to report for each point the segment determined by two other points that is farthest from it. The principal motive for studying this problem was to investigate if this problem could be solved with a worst-case time-complexity that is of lower order than O(n 2), which is the time taken by the solution of Duffy et al. (13) for the all-closest version of the same problem. If h be the number of points on the convex hull of the point set, we show how to do this in O(nh + n log n) time. Our solution to this problem has also triggered off research into the hitherto unexplored problem of determining the farthest-segment Voronoi Diagram of a given set of n line segments in the plane, leading to an O(n log n) time solution for the all-farthest-segments problem (12). For the second problem, we have revisited the problem of computing an area-optimal convex polygon stabbing a set of parallel line segments studied earlier by Kumar et al. (30). The primary motive behind this was to inquire if the line of attack used for the parallel-segments version can be extended to the case where the line segments are of arbitrary orientation. We have provided a correctness proof of the algorithm, which was lacking in the above-cited version. Implementation of geometric algorithms are of great help in visualizing the algorithms, we have implemented both the algorithms and trial versions are available at www.davinci.newcs.uwindsor.ca/ ∼asishm.Dept. of Computer Science. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2006 .C438. Source: Masters Abstracts International, Volume: 45-01, page: 0349. Thesis (M.Sc.)--University of Windsor (Canada), 2006

Optimal Placement of Metal Foils in Ultrasonic Consolidation Process

Ultrasonic Consolidation is a combination of additive and subtractive manufacturing processes resulting in considerable material waste. This waste is a function of the geometry of the part being manufactured and of the relative placement of the layer with respect to the metal bands. Thus the waste may be minimized by careful choice of the layer angle and offset from the original position. Previous work done in this field had developed an automated algorithm which optimally places and orients the individual slices of the STL file of the artifact being manufactured. However, the problem was solved on a 2-D scale and the 3- D nature of the part was not considered for the development of the algorithm. The earlier algorithm employed approximation on the input data to minimize the computational expense. This resulted in convergence of the optimizer to suboptimal solutions. Further, as the final part is made of anisotropic material the relative angles and overlap between subsequent layers also plays an important role in the final part strength. Finally, it is noted that the build time required for the ultrasonic consolidation process is a function of the number of bands required to form each slice. Considering these limitations and opportunities, this thesis presents an algorithm which optimally orients and places the part layers with respect to aluminum bands in order to minimize the waste formed and the build time required. The algorithm has the capability of increasing the part strength by forming crisscross and brick structures using the metal foils. This research work also improves on the previous algorithm by extending the functionality of the algorithm by building in capability to handle multiple loops within the same slice and non convex slice data. Further, the research studies the choice of optimizer that needs to be employed for different types of input data

Multi-Physics Analysis of Laser Solid Freeform Fabrication

The quality of parts fabricated using Laser Solid Freeform Fabrication (LSFF) is highly dependent on the physical phenomena and operating parameters which govern the process. For instance, the thermal stress patterns and intensity, induced throughout the process domain due to the layer-by-layer material deposition and the temperature distribution characteristics, contribute significantly to potential delamination and crack formation across the fabricated part. In this research, some of the main features as well as drawbacks of this technique are studied through a multi-physics analysis of the process. For this purpose, a coupled time-dependent 3D model is developed with which the geometry of the deposited material as well as temperature and thermal stress fields across the process domain can be predicted. In the proposed approach, coupled thermal and stress domains are numerically obtained assuming a decoupled interaction between the laser beam and powder stream. To predict the geometry of the deposited material, once the melt pool boundary is obtained, the process domain is discretized in a cross-sectional fashion based on the powder feed rate, elapsed time, and intersection of the melt pool and powder stream projected on the substrate. Layers of additive material are then added onto the non-planar domain. The main process parameters affected by a multilayer deposition due to the formation of non-planar surfaces, such as powder catchment, are incorporated into the modelling approach to enhance the accuracy of the results. To demonstrate the proposed algorithm and to study the main features of the process, a four-layer thin wall of AISI 304L steel on a substrate of the same material is numerically and experimentally fabricated. The numerical analyses along with the experimental results are then used to investigate the correlation between the temperature-thermal stress fields and crack formation across the fabricated parts. The trend of the results reveals that by preheating the substrate prior to the fabrication process, it is possible to substantially reduce the formed micro-cracks. To demonstrate the feasibility of preheating on the reduction of micro-cracks, several simulations and experiments are performed in which a crack-free result is obtained, with a 22 per cent reduction in thermal stresses when the substrate is preheated to 800 K. The numerical and experimental results are also used to study the circumstances of the microstructural formation during the fabrication process. To conclude this research, the developed modelling approach is further extended to briefly discuss the effects of the path patterns and the main operating parameters on the outcomes of the process. The effects of the material properties and their variations on the temperature distributions and thermal stress fields are studied by fabrication of a thin wall of two Stellite 6 layers and two Ti layers on a stainless steel substrate

The Development of Lightweight Cellular Structures for Metal Additive Manufacturing

Metal Additive Manufacturing (AM) technologies in particular powder bed fusion processes such as Selective Laser Melting (SLM) and Direct Metal Laser Sintering (DMLS) are capable of producing a fully-dense metal components directly from computer-aided design (CAD) model without the need of tooling. This unique capability offered by metal AM has allowed the manufacture of inter-connected lattice structures from metallic materials for different applications including, medical implants and aerospace lightweight components. Despite the many promising design freedoms, metal AM still faces some major technical and design barriers in building complex structures with overhang geometries. Any overhang geometry which exceeds the minimum allowable build angle must be supported. The function of support structure is to prevent the newly melted layer from curling due to thermal stresses by anchoring it in place. External support structures are usually removed from the part after the build; however, internal support structures are difficult or impossible to remove. These limitations are in contrast to what is perceived by designers as metal AM being able to generate all conceivable geometries. Because support structures consume expensive raw materials, use a considerable amount of laser consolidation energy, there is considerable interest in design optimisation of support structure to minimize the build time, energy, and material consumption. Similarly there is growing demand of developing more advanced and lightweight cellular structures which are self-supporting and manufacturable in wider range of cell sizes and volume fractions using metal AM. The main focuses of this research is to tackle the process limitation in metal AM and promote design freedom through advanced self-supporting and low-density Triply Periodic Minimal Surface (TPMS) cellular structures. Low density uniform, and graded, cellular structures have been developed for metal AM processes. This work presents comprehensive experimental test conducted in SLM and DMLS processes using different TPMS cell topologies and materials. This research has contributed to new knowledge in understanding the manufacturability and mechanical behaviour of TPMS cellular structures with varying cell sizes, orientations and volume fractions. The new support structure method will address the saving of material (via low volume cellular structures and easy removal of powder) and saving of energy (via reduced build-time)

Etats de surface de pièces métalliques obtenues en Fabrication Directe par Projection Laser (FDPL) : compréhension physique et voies d’amélioration

The process of direct manufacturing by projection laser ( FDPL), is a process of additive manufacturing which allows to make rooms(parts,plays) of complex shape directly from a file CAD, without tool and without mold(mussel). One of its major inconveniences is the bad quality of the states of surface obtained (Ra upper to 15 µm) which requires systematically stages of remanufacturing. In this context, and within the framework of the project ANR " aspect ", this thesis(theory) has for double objective a better understanding of the origin of the degraded states of surface, and the development of various innovative experimental solutions allowing to improve the states of surface. At first, by considering simple geometries (walls) in alloy of titanium Ti-6Al-4V, we studied the interactionLe procédé de fabrication directe par projection laser (FDPL), est un procédé de fabrication additive qui permet de fabriquer des pièces de forme complexe directement à partir d'un fichier CAO, sans outil et sans moule. L'un de ses inconvénients majeurs est la mauvaise qualité des états de surface obtenus (Ra supérieur à 15 μm) qui nécessite systématiquement des étapes de ré-usinage. Dans ce contexte, et dans le cadre du projet ANR « ASPECT », cette thèse a pour double objectif une meilleure compréhension de l'origine des états de surface dégradés, et le développement de différentes solutions expérimentales innovantes permettant d'améliorer les états de surface.Dans un premier temps, en considérant des géométries simples (murs) en alliage de titane Ti-6Al-4V, nous avons étudié l'interaction faisceau laser / jet de poudre / bain liquide métallique par différents diagnostics (caméra rapide, caméra thermique, pyrométrie…) pour comprendre l'évolution de la géométrie, la thermique et l'hydrodynamique de la zone fondue (ZF). Ces analyses nous ont permis de corréler les évolutions des ZF à celles des états de surface, et de mettre en évidence, sur le Ti-6Al-4V la prépondérance des effets de tension superficielle sur les effets de gravité, dans l'équilibre des ZF, et l'effet bénéfique de zones fondues larges et profondes combinées à de faibles hauteurs par couche, dans la réduction des micro et macro-rugosités. La réduction du débit massique local Dm* en vis-à-vis des parois latérales et l'augmentation du rayon de courbure des ZF avec l'élargissement des ZF (donc avec des rapports El= P/V (J/m) élevés) sont à l'origine des effets bénéfiques obtenus. Différents modèles analytiques et numériques ont également été utilisés ou développés, en complément des résultats expérimentaux, pour décrire le procédé (modèle d'interaction laser-poudre, modèle numérique thermique 3D du procédé, modèle de calcul des ondulations périodiques).En utilisant un large spectre de conditions expérimentales, et une caractérisation rigoureuse des conditions de fabrication (analyses de faisceau, de jet de poudre …) nous avons également apporté des améliorations notables à la qualité des états de surface obtenus. Ainsi, l'utilisation d'un éclairement laser uniforme plutôt que quasi-Gaussien, ou l'utilisation d'un régime quasi-pulsé plutôt que continu ont permis, sur le Ti-6Al-4V de réduire significativement les gradients thermiques en ZF et les mouvements de convection de Marangoni associés, et d'obtenir des qualités d'états de surface fortement améliorées (Ra< 3 µm) par rapport aux études antérieures sur le sujet.Pour finir, une partie de l'étude s'est concentrée sur l'utilisation d'un autre matériau: l'acier inoxydable 316L, afin d'analyser l'influence de la nature chimique et des propriétés thermo-physiques de la poudre projetée sur la qualité des états de surface. Les résultats ont montré que, contrairement à l'alliage de titane, les meilleures rugosités étaient obtenues pour les énergies linéiques El (J/m) les plus faibles, en raison de la formation, à El élevé, de macro-agglomérats de poudre sur les parois des murs. Ce résultat confirme la difficulté d'une approche prédictive globale des états de surface à partir des paramètres thermo-physiques des matériaux projetés

On some geometric optimization problems in layered manufacturing

Efficient geometric algorithms are given for optimization problems arising in layered manufacturing, where a 3D object is built by slicing its CAD model into layers and manufacturing the layers successively. The problems considered include minimizing the degree of stair-stepping on the surfaces of the manufactured object, minimizing the volume of the so-called support structures used, and minimizing the contact area between the supports and the manufactured object - all of which are factors that affect the speed and accuracy of the process. The stair-step minimization algorithm is valid for any polyhedron, while the support minimization algorithms are applicable to convex polyhedra only. Algorithms are also given for optimizing supports for non-convex, simple polygons. The techniques used to obtain these results include construction and searching of certain arrangements on the sphere, 3D convex hulls, halfplane range searching, ray-shooting, visibility and constrained optimization. (orig.)Available from TIB Hannover: RR 4485(1997,1) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

On some geometric optimization problems in layered manufacturing

Efficient goemetric algorithms are given for optimization problems arising in layered manufacturing, where a 3D object is built by slicing its CAD model into layers and manufacturing the layers successively. The problems considered include minimizing the stair-step error on the surfaces of the manufactured object under various formulations, minimizing the volume of the so-called support structures used, and minimizing the contact area between the supports and the manufactured object - all of which are factors that affect the speed and accuracy of the process. The stair-step minimization algorithm is valid for any polyhedron, while the support minimization algorithms are applicable only to convex polyhedra. The techniques used to obtain these results include construction and searching of certain arrangements on the sphere, 3D convex hulls, halfplane range searching, and constrained optimization. (orig.)Available from TIB Hannover: RR 4485(1998,2) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman