Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations

We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds. We developed the verification tool as a part of the Flyspeck project (a formal proof of the Kepler conjecture). The Flyspeck project includes about 1000 nonlinear inequalities. We successfully tested our method on more than 100 Flyspeck inequalities and estimated that the formal verification procedure is about 3000 times slower than an informal verification method implemented in C++. We also describe future work and prospective optimizations for our method.Comment: 15 page

Superplastic forming of Al-Li alloys for lightweight, low-cost structures

Superplastic forming of advanced aluminum alloys is being evaluated as an approach for fabricating low-cost, light-weight, cryogenic propellant tanks. Built-up structure concepts (with inherent reduced scrap rate) are under investigation to offset the additional raw material expenses incurred by using aluminum lithium alloys. This approach to fabrication offers the potential for significant improvements in both structural efficiency and overall manufacturing costs. Superplasticity is the ability of specially processed material to sustain very large forming strains without failure at elevated temperatures under controlled deformation conditions. It was demonstrated that superplastic forming technology can be used to fabricate complex structural components in a single operation and increase structural efficiency by as much as 60 percent compared to conventional configurations in skin-stiffened structures. Details involved in the application of this technology to commercial grade superplastic aluminum lithium material are presented. Included are identification of optimum forming parameters, development of forming procedures, and assessment of final part quality in terms of cavitation volume and thickness variation

Minimum-fuel Attitude Control of a Rigid Body in Orbit by an Extended Method of Steepest-descent

Minimum fuel control of spacecraft in orbit using extended method of steepest descen

Array-based iterative measurements of SmKS travel times and their constraints on outermost core structure

Vigorous convection in Earth's outer core led to the suggestion that it is chemically homogeneous. However, there is increasing seismic evidence for structural complexities close to the outer core's upper and lower boundaries. Both body waves and normal mode data have been used to estimate a P wave velocity, V_p, at the top of the outer core (the E’ layer), which is lower than that in the Preliminary Reference Earth Model. However, these low V_p models do not agree on the form of this velocity anomaly. One reason for this is the difficulty in retrieving and measuring SmKS arrival times. To address this issue, we propose a novel approach using data from seismic arrays to iteratively measure SmKS-SKKS-differential travel times. This approach extracts individual SmKS signal from mixed waveforms of the SmKS series, allowing us to reliably measure differential travel times. We successfully use this method to measure SmKS time delays from earthquakes in the Fiji‐Tonga and Vanuatu subduction zones. SmKS time delays are measured by waveform cross correlation between SmKS and SKKS, and the cross‐correlation coefficient allows us to access measurement quality. We also apply this iterative scheme to synthetic SmKS seismograms to investigate the 3‐D mantle structure's effects. The mantle structure corrections are not negligible for our data, and neglecting them could bias the V_p estimation of uppermost outer core. After mantle structure corrections, we can still see substantial time delays of S3KS, S4KS, and S5KS, supporting a low V_p at the top of Earth's outer core

Doctor of Philosophy

dissertationCorrosion in commonly used epoxy-coated steel reinforcing bars has raised concern with its use and has increased interest in the use of alternative reinforcement including fiber-reinforced polymer (FRP) bars. Glass FRP (GFRP) bars are a costcompetitive alternative in conditions where chloride-induced deterioration is an issue. There has been considerable research and code-writing effort towards using GFRP bars in construction of bridge deck and beam elements but these efforts are still somewhat limited with regards to the use of FRP bars in construction of column elements and particularly in the use of FRP-spiral confining hoops or spirals. Steel spiral reinforcement is commonly used in concrete columns for confinement and improvement of displacement ductility but little is known regarding the use of FRP-spiral reinforcement for confinement. The purpose of this research is to gain a better understanding of the behavior and performance of FRP reinforcement for short as well as slender concrete columns, particularly FRP-spiral confining reinforcement, to better determine its viability as a potential construction alternative. Tests were conducted on nine circular concrete columns reinforced with internal GFRP-spirals and either steel, GFRP or a combination of steel and GFRP longitudinal bars. Axial load tests were conducted on both slender and short columns with loads placed at two eccentricities to observe the mode of failure, load capacity, and general behavior associated with different geometric and loading conditions. The test data is analyzed and a confinement model based on plasticity theory is derived to describe the axial strength and stress-strain relationship for FRP-spiralconfined circular columns. The confinement model is used to predict the momentcurvature relationship for FRP-spiral-confined circular columns. An analytical buckling model is developed, based on a numerical integration method, to describe the loaddeflection behavior for slender FRP-spiral-confined circular columns. Axial load and bending moment interaction diagrams are produced for FRPspiral-confined slender columns. The analytical buckling model is also used in parametric studies of large-scale slender columns to compare steel with FRP spirals in addition to other comparisons between reinforcement configurations and concrete compressive strength