Criss-cross methods: A fresh view on pivot algorithms

Criss-cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution. The first finite criss-cross method was invented by Chang, Terlaky and Wang independently. Unlike the simplex method that follows a monotonic edge path on the feasible region, the trace of a criss-cross method is neither monotonic (with respect to the objective function) nor feasibility preserving. The main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss-cross pivot methods. A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms. The origins and the history of criss-cross methods are also touched upo

Criss-cross methods: a fresh view on pivot algorithms

Criss-cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution. The first finite criss-cross method was invented by Chang, Terlaky and Wang independently. Unlike the simplex method that follows a monotonic edge path on the feasible region, the trace of a criss-cross method is neither monotonic (with respect to the objective function) nor feasibility preserving. The main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss-cross pivot methods. A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms. The origins and the history of criss-cross methods are also touched upon

Doctor of Philosophy

dissertationSmoothness-increasing accuracy-conserving (SIAC) filters were introduced as a class of postprocessing techniques to ameliorate the quality of numerical solutions of discontinuous Galerkin (DG) simulations. SIAC filtering works to eliminate the oscillations in the error by introducing smoothness back to the DG field and raises the accuracy in the L2-n o rm up to its natural superconvergent accuracy in the negative-order norm. The increased smoothness in the filtered DG solutions can then be exploited by simulation postprocessing tools such as streamline integrators where the absence of continuity in the data can lead to erroneous visualizations. However, lack of extension of this filtering technique, both theoretically and computationally, to nontrivial mesh structures along with the expensive core operators have been a hindrance towards the application of the SIAC filters to more realistic simulations. In this dissertation, we focus on the numerical solutions of linear hyperbolic equations solved with the discontinuous Galerkin scheme and provide a thorough analysis of SIAC filtering applied to such solution data. In particular, we investigate how the use of different quadrature techniques could mitigate the extensive processing required when filtering over the whole computational field. Moreover, we provide detailed and efficient algorithms that a numerical practitioner requires to know in order to implement this filtering technique effectively. In our first attempt to expand the application scope of this filtering technique, we demonstrate both mathematically and through numerical examples that it is indeed possible to observe SIAC filtering characteristics when applied to numerical solutions obtained over structured triangular meshes. We further provide a thorough investigation of the interplay between mesh geometry and filtering. Building upon these promising results, we present how SIAC filtering could be applied to gain higher accuracy and smoothness when dealing with totally unstructured triangular meshes. Lastly, we provide the extension of our filtering scheme to structured tetrahedral meshes. Guidelines and future work regarding the application of the SIAC filter in the visualization domain are also presented. We further note that throughout this document, the terms postprocessing and filtering will be used interchangeably

SciTech News Volume 71, No. 2 (2017)

Columns and Reports From the Editor 3 Division News Science-Technology Division 5 Chemistry Division 8 Engineering Division 9 Aerospace Section of the Engineering Division 12 Architecture, Building Engineering, Construction and Design Section of the Engineering Division 14 Reviews Sci-Tech Book News Reviews 16 Advertisements IEEE

The Design and Analysis of Electrically Large Custom-Shaped Reflector Antennas

Designing and analyzing electrically large reflectors poses numerically complex problems because the reflector must be sampled finely to obtain an accurate solution, causing an unwieldy number of samples. In addition to these complexities, a custom-shaped reflector poses a new analysis problem. Previously developed methods and theorems including Geometric Optics, Ray-Tracing, Surface Equivalence Theorems, Image Theory, and Physical Optics can be applied to these custom-shaped reflectors however. These methods all share in common their capability to provide accurate results in the analysis of electrically large structures. In this thesis, two custom-shaped reflector concepts are explored which include a rectangular shaped, spherically contoured reflector with largest dimension of 305 meters and a cross-shaped, parabolically contoured reflector with largest dimension of 150 meters. Each reflector is intended to operate in the Institute of Electrical and Electronics Engineers (IEEE) L-Band. The reflectors produced differing results, but the same methods apply to each. The motivation for pursuing these custom-shaped reflectors is for earth-based and space-based satellite communications respectively. In this thesis, the plane wave analysis and the ray tracing results are presented for each reflector, and the initial feed design results for the cross-shaped reflector are presented

Multi-objective design optimization of a mobile-bearing total disc arthroplasty considering spinal kinematics, facet joint loads, and metal-on-polyethylene contact mechanics

Total disc arthroplasty (TDA) is a motion-preserving surgical technique used to treat spinal disorders, when more conservative medical therapies fail. Unfortunately, a high incidence of revision surgery exists due to postoperative complications including abnormal kinematics, facet joint arthritis, and implant failures. However, TDA is still an attractive option, since an optimally designed artificial disc is expected to reproduce native segmental biomechanics. Correspondingly, it would mitigate the development of adjacent segment diseases (a major concern of spinal fusion) caused by altered segmental biomechanics. Design optimization is a process of finding the best design parameters for a component/system to satisfy one/multiple design requirements using optimization algorithms. The shape of a candidate design is parametrized using computer-aided design, such that design parameters are manipulated to minimize one/multiple objective functions subject to performance constraints and design space bounds. Optimization algorithms typically require the gradients of the objective/constraint functions with respect to each design variable. In the traditional design optimization, due to the high computational cost to calculate the gradients by performing finite element analysis in each optimization iteration, it often results in a slow process to seek the optimal solution. To address the problem, an artificial neural network (ANN) was implemented to derive the analytical expressions of the objective/constraint function and their gradients. By incorporating analytical gradients, we successfully developed a multiobjective optimization (MOO) framework considering three performance metrics simultaneously. Furthermore, a new mobile-bearing TDA design concept featuring a biconcave polyethylene (PE) core was proposed, to strengthen the PE rim, where a high risk of fracture exists. It was hypothesized that there is a trade-off relationship among postoperative performance metrics in terms of spinal kinematics, facet joint loading, and metal-on-polyethylene contact mechanics. We tested this hypothesis by refining the new TDA to match normal segmental biomechanics and alleviate PE core stress. After performing MOO, the best-trade-off TDA design was determined by the solved three-dimensional Pareto frontier. The novel MOO framework can be also used to improve existing TDA designs, as well as to push the cutting edge of surgical techniques for the treatment of spinal disorders