Study of non-linear optimization techniques

Nonlinear optimization techniques in dynamic programming and solution of ordinary nonlinear differential equations by Runge-Kutta metho

Monotonically convergent optimization in quantum control using Krotov's method

The non-linear optimization method developed by Konnov and Krotov [Automation and Remote Control 60, 1427 (1999)] has been used previously to extend the capabilities of optimal control theory from the linear to the non-linear Schr\"odinger equation [Sklarz and Tannor, Phys. Rev. A 66, 053619 (2002)]. Here we show that based on the Konnov-Krotov method, monotonically convergent algorithms are obtained for a large class of quantum control problems. It includes, in addition to non-linear equations of motion, control problems that are characterized by non-unitary time evolution, non-linear dependencies of the Hamiltonian on the control, time-dependent targets and optimization functionals that depend to higher than second order on the time-evolving states. We furthermore show that the non-linear (second order) contribution can be estimated either analytically or numerically, yielding readily applicable optimization algorithms. We demonstrate monotonic convergence for an optimization functional that is an eighth-degree polynomial in the states. For the 'standard' quantum control problem of a convex final-time functional, linear equations of motion and linear dependency of the Hamiltonian on the field, the second-order contribution is not required for monotonic convergence but can be used to speed up convergence. We demonstrate this by comparing the performance of first and second order algorithms for two examples

Optimization of piezoelectric cantilever energy harvesters including non-linear effects

This paper proposes a versatile non-linear model for predicting piezoelectric energy harvester performance. The presented model includes (i) material non-linearity, for both substrate and piezoelectric layers, and (ii) geometric non-linearity incorporated by assuming inextensibility and accurately representing beam curvature. The addition of a sub-model, which utilizes the transfer matrix method to predict eigenfrequencies and eigenvectors for segmented beams, allows for accurate optimization of piezoelectric layer coverage. A validation of the overall theoretical model is performed through experimental testing on both uniform and non-uniform samples manufactured in-house. For the harvester composition used in this work, the magnitude of material non-linearity exhibited by the piezoelectric layer is 35 times greater than that of the substrate layer. It is also observed that material non-linearity, responsible for reductions in resonant frequency with increases in base acceleration, is dominant over geometric non-linearity for standard piezoelectric harvesting devices. Finally, over the tested range, energy loss due to damping is found to increase in a quasi-linear fashion with base acceleration. During an optimization study on piezoelectric layer coverage, results from the developed model were compared with those from a linear model. Unbiased comparisons between harvesters were realized by using devices with identical natural frequencies—created by adjusting the device substrate thickness. Results from three studies, each with a different assumption on mechanical damping variations, are presented. Findings showed that, depending on damping variation, a non-linear model is essential for such optimization studies with each model predicting vastly differing optimum configurations

Structural optimisation problem in support to building retrofitting decision

Various analysis methods, either linear elastic or non-linear, static or dynamic, are available for the performance analysis of existing buildings. Despite its advantages, it must be admitted that non-linear time history analysis can frequently become overly complex and impractical for general use as a first assessment. Simplified models, as the Capacity Spectrum Method, are frequently not able to accurately assess irregular structures. Considering these limitations, it is proposed and evaluated a simplified MDOF non-linear dynamic model, accounting for non-linear storey behaviour and storey damping. Based on the MDOF non-linear dynamic model, were developed optimization algorithms for the redesign of existing non-seismically designed structures. The optimization procedure searches for the optimum storey strengthening distribution (strength, stiffness or damping) in order to meet specific performance requirements, in terms of maximum inter-storey drift for a given seismic demand level. Numerical examples are presented in order to illustrate the capability of methodology