## 9. SPARSE REPRESENTATIONS AND COMPRESSIVE SAMPLING APPROACHES IN ENGINEERING MECHANICS: A REVIEW OF THEORETICAL CONCEPTS AND DIVERSE APPLICATIONS

Under review - Probabilistic Engineering Mechanics

Kougioumtzoglou I.A., Petromichelakis I., Psaros A.F.

A review of theoretical concepts and diverse applications of sparse representations and compressive sampling (CS) approaches in engineering mechanics problems is provided from a broad perspective. First, following a presentation of well-established CS concepts and optimization algorithms, attention is directed to currently emerging tools and techniques for enhancing solution sparsity and for exploiting additional information in the data. These include alternative to l1-norm minimization formulations and iterative re-weighting solution schemes, Bayesian approaches, as well as structured sparsity and dictionary learning strategies. Next, CS-based research work of relevance to engineering mechanics problems is categorized and discussed under three distinct application areas: a) inverse problems in structural health monitoring, b) uncertainty modeling and simulation, and c) computationally efficient uncertainty propagation.

## 8. FUNCTIONAL SERIES EXPANSIONS AND QUADRATIC APPROXIMATIONS FOR ENHANCING THE ACCURACY OF THE WIENER PATH INTEGRAL TECHNIQUE

Accepted - Journal of Engineering Mechanics

Psaros A.F., Kougioumtzoglou I.A.

A novel Wiener path integral (WPI) technique is developed for determining the response of stochastically excited nonlinear oscillators. This is done by resorting to functional series expansions in conjunction with quadratic approximations. The technique can be construed as an extension and enhancement in terms of accuracy of the standard (semi-classical) WPI solution approach where only the most probable path connecting initial and final states is considered for determining the joint response probability density function (PDF). In contrast, the herein developed technique accounts also for fluctuations around the most probable path; thus, yielding an increased accuracy degree. An additional significant advantage of the proposed enhancement as compared to the most probable path approach relates to the fact that low probability events (e.g., failure probabilities) can be estimated directly in a computationally efficient manner by determining a few only points of the joint response PDF.

## 7. AN EXACT CLOSED-FORM SOLUTION FOR LINEAR MULTI-DEGREE-OF-FREEDOM SYSTEMS UNDER GAUSSIAN WHITE NOISE VIA THE WIENER PATH INTEGRAL TECHNIQUE

2020 - Probabilistic Engineering Mechanics - Draft

Psaros A.F., Zhao Y., Kougioumtzoglou I.A.

The exact joint response transition probability density function (PDF) of linear multi-degree-of-freedom oscillators under Gaussian white noise is derived in closed-form based on the Wiener path integral (WPI) technique. Specifically, in the majority of practical implementations of the WPI technique, only the first couple of terms are retained in the functional expansion of the path integral related stochastic action. The remaining terms are typically omitted since their evaluation exhibits considerable analytical and computational challenges. Obviously, this approximation affects, unavoidably, the accuracy degree of the technique. However, it is shown herein that, for the special case of linear systems, higher than second order variations in the path integral functional expansion vanish, and thus, retaining only the first term (most probable path approximation) yields the exact joint response transition Gaussian PDF. Both single- and multi degree-of-freedom linear systems are considered as illustrative examples for demonstrating the exact nature of the derived solutions.

## 6. STOCHASTIC RESPONSE DETERMINATION OF NONLINEAR SYSTEMS WITH SINGULAR DIFFUSION MATRICES: A WIENER PATH INTEGRAL VARIATIONAL FORMULATION WITH CONSTRAINTS

2020 - Probabilistic Engineering Mechanics - Draft

Petromichelakis I., Psaros A.F., Kougioumtzoglou I.A.

The Wiener path integral (WPI) approximate semi-analytical technique for determining the joint response probability density function (PDF) of stochastically excited nonlinear oscillators is generalized herein to account for systems with singular diffusion matrices. Indicative examples include (but are not limited to) systems with only some of their degrees-of-freedom excited, hysteresis modeling via additional auxiliary state equations, and energy harvesters with coupled electro-mechanical equations. Several numerical examples pertaining to both linear and nonlinear constraint equations are considered, including various multi-degree-of-freedom systems, a linear oscillator under earthquake excitation and a nonlinear oscillator exhibiting hysteresis following the Bouc-Wen formalism.

## 5. WIENER PATH INTEGRALS AND MULTI-DIMENSIONAL GLOBAL BASES FOR NON-STATIONARY STOCHASTIC RESPONSE DETERMINATION OF STRUCTURAL SYSTEMS

2019 - Mechanical Systems and Signal Processing - Draft

Psaros A.F., Petromichelakis I., Kougioumtzoglou I.A.

A novel approximate technique based on Wiener path integrals (WPIs) is developed for determining, in a computationally efficient manner, the non-stationary joint response probability density function (PDF) of nonlinear multi-degree-of-freedom dynamical systems. Specifically, appropriate multi-dimensional (time-dependent) global bases are constructed for approximating the non-stationary joint response PDF. Several numerical examples pertaining to diverse structural systems are considered, including both single-and multi-degree-of-freedom nonlinear dynamical systems subject to non-stationary excitations, as well as a bending beam with a non-Gaussian and non-homogeneous Young's modulus.

## 4. WIENER PATH INTEGRAL BASED RESPONSE DETERMINATION OF NONLINEAR SYSTEMS SUBJECT TO NON-WHITE, NON-GAUSSIAN, AND NON-STATIONARY STOCHASTIC EXCITATION

2018 - Journal of Sound and Vibration - Draft

Psaros A.F., Brudastova O., Malara G., Kougioumtzoglou I.A.

The recently developed Wiener Path Integral (WPI) technique for determining the joint response probability density function of nonlinear systems subject to Gaussian white noise excitation is generalized herein to account for non-white, non-Gaussian, and non-stationary excitation processes. Specifically, modeling the excitation process as the output of a filter equation with Gaussian white noise as its input, it is possible to define an augmented response vector process to be considered in the WPI solution technique. Several numerical examples pertaining to both single-and multi-degree-of-freedom systems are considered, including a marine structural system exposed to flow-induced non-white excitation, as well as a bending beam with a non-Gaussian and non-homogeneous Young's modulus.

## 3. SPARSE REPRESENTATIONS AND COMPRESSIVE SAMPLING FOR ENHANCING THE COMPUTATIONAL EFFICIENCY OF THE WIENER PATH INTEGRAL TECHNIQUE

2018 - Mechanical Systems and Signal Processing - Draft

Psaros A.F., Kougioumtzoglou I.A., Petromichelakis I.

The computational efficiency of the Wiener path integral (WPI) technique for determining the stochastic response of diverse dynamical systems is enhanced by exploiting recent developments in the area of sparse representations. Specifically, an appropriate basis for expanding the system joint response probability density function (PDF) is utilized. Next, only very few PDF points are determined based on the localization capabilities of the WPI technique. Further, compressive sampling procedures in conjunction with group sparsity concepts and appropriate optimization algorithms are employed for efficiently determining the coefficients of the system response PDF expansion. It is shown that the herein developed enhancement renders the technique capable of treating readily relatively high-dimensional stochastic systems. Two illustrative numerical examples are considered.

## 2. STOCHASTIC RESPONSE DETERMINATION AND OPTIMIZATION OF A CLASS OF NONLINEAR ELECTROMECHANICAL ENERGY HARVESTERS: A WIENER PATH INTEGRAL APPROACH

2018 - Probabilistic Engineering Mechanics - Draft

Petromichelakis I., Psaros A.F., Kougioumtzoglou I.A.

A methodology based on the Wiener path integral technique (WPI) is developed for stochastic response determination and optimization of a class of nonlinear electromechanical energy harvesters. To this aim, first, the WPI technique is extended to address the particular form of the coupled electromechanical governing equations, which possess a singular diffusion matrix. Specifically, a constrained variational problem is formulated and solved for determining the joint response probability density function (PDF) of the nonlinear energy harvesters. This is done either by resorting to a Lagrange multipliers approach, or by utilizing the nullspace of the constraint equation. Next, the herein extended WPI technique is coupled with an appropriate optimization algorithm for determining optimal energy harvester parameters. It is shown that due to the relatively high accuracy exhibited in determining the joint response PDF, the WPI technique is particularly well-suited for constrained optimization problems , where the constraint refers to low probability events (e.g. probabilities of failure).

## 1. STRENGTH, STIFFNESS AND CYCLIC DEFORMATION CAPACITY OF RC FRAMES CONVERTED INTO WALLS BY INFILLING WITH RC

2016 - Bulletin of Earthquake Engineering

Biskinis, D., Fardis, M.N., Psaros A.F.

In seismic retrofitting of concrete buildings, frame bays are converted into reinforced concrete (RC) walls by infilling the space between the frame members with RC of a thickness of not more than their width. The cyclic behavior of the resulting wall depends on the connection between the RC infill and the surrounding RC members. The paper uses the results from 56 cyclic tests on such composite walls to express their properties in terms of the geometry, the reinforcement and the connection.