Tuning Monotonic Basin Hopping: Improving the Efficiency of Stochastic Search as Applied to Low-Thrust Trajectory Optimization

Trajectory optimization methods using monotonic basin hopping (MBH) have become well developed during the past decade [1, 2, 3, 4, 5, 6]. An essential component of MBH is a controlled random search through the multi-dimensional space of possible solutions. Historically, the randomness has been generated by drawing random variable (RV)s from a uniform probability distribution. Here, we investigate the generating the randomness by drawing the RVs from Cauchy and Pareto distributions, chosen because of their characteristic long tails. We demonstrate that using Cauchy distributions (as first suggested by J. Englander [3, 6]) significantly improves monotonic basin hopping (MBH) performance, and that Pareto distributions provide even greater improvements. Improved performance is defined in terms of efficiency and robustness. Efficiency is finding better solutions in less time. Robustness is efficiency that is undiminished by (a) the boundary conditions and internal constraints of the optimization problem being solved, and (b) by variations in the parameters of the probability distribution. Robustness is important for achieving performance improvements that are not problem specific. In this work we show that the performance improvements are the result of how these long-tailed distributions enable MBH to search the solution space faster and more thoroughly. In developing this explanation, we use the concepts of sub-diffusive, normally-diffusive, and super-diffusive random walks (RWs) originally developed in the field of statistical physics

Walking the Filament of Feasibility: Global Optimization of Highly-Constrained, Multi-Modal Interplanetary Trajectories Using a Novel Stochastic Search Technique

Interplanetary trajectory optimization problems are highly complex and are characterized by a large number of decision variables and equality and inequality constraints as well as many locally optimal solutions. Stochastic global search techniques, coupled with a large-scale NLP solver, have been shown to solve such problems but are inadequately robust when the problem constraints become very complex. In this work, we present a novel search algorithm that takes advantage of the fact that equality constraints effectively collapse the solution space to lower dimensionality. This new approach walks the filament'' of feasibility to efficiently find the global optimal solution

The Equivalence of the Radial Return and Mendelson Methods for Integrating the Classical Plasticity Equations

The radial return and Mendelson methods for integrating the equations of classical plasticity, which appear independently in the literature, are shown to be identical. Both methods are presented in detail as are the specifics of their algorithmic implementation. Results illustrate the methods' equivalence across a range of conditions and address the question of when the methods require iteration in order for the plastic state to remain on the yield surface. FORTRAN code implementations of the radial return and Mendelson methods are provided in the appendix

FORTRAN Versions of Reformulated HFGMC Codes

Several FORTRAN codes have been written to implement the reformulated version of the high-fidelity generalized method of cells (HFGMC). Various aspects of the HFGMC and its predecessors were described in several prior NASA Tech Briefs articles, the most recent being HFGMC Enhancement of MAC/GMC (LEW-17818-1), NASA Tech Briefs, Vol. 30, No. 3 (March 2006), page 34. The HFGMC is a mathematical model of micromechanics for simulating stress and strain responses of fiber/matrix and other composite materials. The HFGMC overcomes a major limitation of a prior version of the GMC by accounting for coupling of shear and normal stresses and thereby affords greater accuracy, albeit at a large computational cost. In the reformulation of the HFGMC, the issue of computational efficiency was addressed: as a result, codes that implement the reformulated HFGMC complete their calculations about 10 times as fast as do those that implement the HFGMC. The present FORTRAN implementations of the reformulated HFGMC were written to satisfy a need for compatibility with other FORTRAN programs used to analyze structures and composite materials. The FORTRAN implementations also afford capabilities, beyond those of the basic HFGMC, for modeling inelasticity, fiber/matrix debonding, and coupled thermal, mechanical, piezo, and electromagnetic effects

Analysis of Fiber Clustering in Composite Materials Using High-Fidelity Multiscale Micromechanics

A new multiscale micromechanical approach is developed for the prediction of the behavior of fiber reinforced composites in presence of fiber clustering. The developed method is based on a coupled two-scale implementation of the High-Fidelity Generalized Method of Cells theory, wherein both the local and global scales are represented using this micromechanical method. Concentration tensors and effective constitutive equations are established on both scales and linked to establish the required coupling, thus providing the local fields throughout the composite as well as the global properties and effective nonlinear response. Two nondimensional parameters, in conjunction with actual composite micrographs, are used to characterize the clustering of fibers in the composite. Based on the predicted local fields, initial yield and damage envelopes are generated for various clustering parameters for a polymer matrix composite with both carbon and glass fibers. Nonlinear epoxy matrix behavior is also considered, with results in the form of effective nonlinear response curves, with varying fiber clustering and for two sets of nonlinear matrix parameters

MAC/GMC Code Enhanced for Coupled Electromagnetothermoelastic Analysis of Smart Composites

Intelligent materials are those that exhibit coupling between their electromagnetic response and their thermomechanical response. This coupling allows smart materials to react mechanically (e.g., an induced displacement) to applied electrical or magnetic fields (for instance). These materials find many important applications in sensors, actuators, and transducers. Recently interest has arisen in the development of smart composites that are formed via the combination of two or more phases, one or more of which is a smart material. To design with and utilize smart composites, designers need theories that predict the coupled smart behavior of these materials from the electromagnetothermoelastic properties of the individual phases. The micromechanics model known as the generalized method of cells (GMC) has recently been extended to provide this important capability. This coupled electromagnetothermoelastic theory has recently been incorporated within NASA Glenn Research Center's Micromechanics Analysis Code with Generalized Method of Cells (MAC/GMC). This software package is user friendly and has many additional features that render it useful as a design and analysis tool for composite materials in general, and with its new capabilities, for smart composites as well

Analyzing the Intensities of K‑Edge Transitions in X2 Molecules (X = F, Cl, Br) for Use in Ligand K‑Edge X‑ray Absorption Spectroscopy

Ligand K-edge X-ray absorption spectroscopy (XAS) is regularly used to determine the ligand contribution to metal-ligand bonds. For quantitative studies, the pre-edge transition intensities must be referenced to an intensity standard, and pre-edge intensities obtained from different ligand atoms cannot be compared without standardization due to different cross sections at each absorption edge. In this work, the intensities of the 1s → σ* transitions in F2, Cl2, and Br2 are analyzed for their use as references for ligand K-edge XAS. We show that the intensities of these transitions are equal to the intensities of the 1s → np transitions in the unbound halogens. This finding is supported by a comparison between the normalized experimental intensities for the molecules and the calculated oscillator strengths for the atoms. These results highlight the potential for these molecules to be used as intensity standards in F, Cl, and Br K-edge XAS experiments

HFGMC Enhancement of MAC/GMC

Additional information about a mathematical model denoted the high-fidelity generalized method of cells (HFGMC) and implementation of the HFGMC within version 4.0 of the MAC/GMC software has become available. MAC/GMC (Micromechanics Analysis Code With Generalized Method of Cells) was a topic of several prior NASA Tech Briefs articles, version 4.0 having been described in "Comprehensive Micromechanics-Analysis Code - Version 4.0" (LEW-17495-1), NASA Tech Briefs, Vol. 29, No. 9 (September 2005), page 54. MAC/GMC predicts elastic and inelastic thermomechanical responses of composite materials. MAC/GMC utilizes the generalized method of cells (GMC) - a model of micromechanics that predicts macroscopic responses of a composite material as functions of the properties, sizes, shapes, and responses of its constituents (e.g., matrix and fibers). The accuracy of the GMC is limited by neglect of coupling between normal and shear stresses. The HFGMC was developed by combining elements of the GMC and a related model, denoted the higher-order theory for functionally graded materials (HOTFGM), that can account for this coupling. Hence, the HFGMC enables simulation of stress and strain with greater accuracy. Some alterations of the MAC/GMC data structure were necessitated by the greater computational complexity of the HFGMC

Micromechanics of Spray-On Foam Insulation

Understanding the thermo-mechanical response of the Space Shuttle External Tank spray-on foam insulation (SOFI) material is critical, to NASA's Return to Flight effort. This closed-cell rigid polymeric foam is used to insulate the metallic Space Shuttle External Tank, which is at cryogenic temperatures immediately prior to and during lift off. The shedding of the SOFI during ascent led to the loss of the Columbia, and eliminating/minimizing foam lass from the tank has become a priority for NASA as it seeks to resume scheduled space shuttle missions. Determining the nature of the SOFI material behavior in response to both thermal and mechanical loading plays an important role as any structural modeling of the shedding phenomenon k predicated on knowledge of the constitutive behavior of the foam. In this paper, the SOFI material has been analyzed using the High-Fidelity Generalized Method of Cells (HFGMC) micromechanics model, which has recently been extended to admit a triply-periodic 3-D repeating unit cell (RUC). Additional theoretical extensions that mere made in order to enable modeling of the closed-cell-foam material include the ability to represent internal boundaries within the RUC (to simulated internal pores) and the ability to impose an internal pressure within the simulated pores. This latter extension is crucial as two sources contribute to significant internal pressure changes within the SOFI pores. First, gas trapped in the pores during the spray process will expand or contract due to temperature changes. Second, the pore pressure will increase due to outgassing of water and other species present in the foam skeleton polymer material. With HFGMC's new pore pressure modeling capabilities, a nonlinear pressure change within the simulated pore can be imposed that accounts for both of these sources, in addition to stmdar&-thermal and mechanical loading; The triply-periodic HFGMC micromechanics model described above was implemented within NASA GRC's MAC/GMC software package, giving the model access to a range of nonlinear constitutive models for the polymeric foam skeleton material. A repeating unit cell architecture was constructed that, while relatively simple, still accounts for the geometric anisotropy of the porous foam microstructure and its thin walls and thicker edges. With the lack of reliable polymeric foam skeleton materia1 properties, many simulations were executed aimed at backing out these material properties. Then, using these properties, predictions of the thermo-mechanical behavior of the foam, including calculated internal applied pressure profiles, were performed and compared with appropriate experimental data