The probability distribution of the nonlinear, up to the second order, crest height on a vertical wall is determined under the assumption of finite spectral bandwidth, finite water depth and long-crested waves. The distribution is derived by relying on the Quasi-Deterministic representation of the free surface elevation on the vertical wall. The theoretical results are compared against experimental data obtained by utilizing a compressive sensing algorithm for reconstructing the free surface elevation on the wall. The reconstruction is pursued by starting from recorded wave pressure time histories obtained by utilizing a sequence of pressure transducers located at various levels. The comparison demonstrates an excellent agreement between the proposed distribution and the experimental data, while, notably, the deviation of the crest height distribution from the Rayleigh one is considerable.

Estimating the space-time characteristics of a sea state is of crucial importance to a number of engineering applications, such as the ones involving three-dimensional waves interacting with marine structures. In this context, developing a technique that allows extrapolating information about the wave field utilizing only a relatively small number of records is highly impactful, as it allows minimizing the use of expensive and sophisticated measurement techniques. In this paper, a Compressive Sampling (CS) based technique is developed for extrapolating free surface displacement data. The technique relies on a directional spectrum compatible sparse representation in conjunction with formulating and solving an L-1-norm optimization problem. Further, the accuracy of the developed technique is significantly enhanced via the use of an adaptive basis re-weighting procedure. Pertinent numerical examples demonstrate that the technique is capable of reconstructing the time history of a free surface displacement record successfully, while capturing the main features of the target frequency spectrum and of the cross-correlation function satisfactorily.

A framework is developed for determining the stochastic response of linear multi-degree-of-freedom (MDOF) structural systems with singular matrices. This system modeling can arise when using more than the minimum number of coordinates, and can be advantageous, for instance, in cases of complex multibody systems whose dynamics satisfy a number of constraints. In such cases the explicit formulation of the equations of motion can be a nontrivial task, whereas the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labor-intensive manner. Relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, standard concepts, relationships, and equations of the linear random vibration theory are extended and generalized herein to account for systems with singular matrices. Adopting a state-variable formulation, equations governing the system response mean vector and covariance matrix are formed and solved. Further, it is shown that a complex modal analysis treatment, unlike the standard system modeling case, does not lead to decoupling of the equations of motion. However, relying on a singular value decomposition of the system transition matrix significantly facilitates the efficient computation of the system response statistics. A linear structural system with singular matrices is considered as a numerical example for demonstrating the applicability of the methodology and for elucidating certain related numerical aspects. (C) 2015 American Society of Civil Engineers.

An efficient stochastic incremental dynamic analysis (IDA) methodology for nonlinear/hysteretic oscillators is developed by resorting to nonlinear stochastic dynamics concepts and tools such as stochastic averaging and statistical linearization. Specifically, modeling the excitation as a nonstationary stochastic process possessing an evolutionary power spectrum (EPS), an approximate closed-form expression is derived for the parameterized oscillator response amplitude probability density function (PDF). In this regard, an IDA surface is determined providing the PDF of the engineering demand parameter (EDP) for a given intensity measure (IM) value. In contrast to a computationally expensive Monte Carlo simulation (MCS) based determination of the IDA surface, the methodology developed herein determines the EDP PDF at minimal computational cost. Further, an approximate closed-form expression is derived for the parameterized nonlinear oscillator response EPS as well; thus, a conceptually novel IDA surface is determined where the EDP relates to the nonlinear oscillator response EPS. The stochastic IDA framework can account for physically realistic excitation models possessing not only time-varying intensities but time-varying frequency contents as well. A bilinear/hysteretic single-degree-of-freedom oscillator is considered as a numerical example, whereas comparisons with pertinent MCS data demonstrate the accuracy and efficiency of the developed stochastic IDA methodology. (C) 2016 American Society of Civil Engineers.

A simplified model of the motion of a grounding iceberg for determining the gouge depth into the seabed is proposed. Specifically, taking into account uncertainties relating to the soil strength, a nonlinear stochastic differential equation governing the evolution of the gouge length/depth in time is derived. Further, a recently developed Wiener path integral (WPI) based approach for solving approximately the nonlinear stochastic differential equation is employed; thus, circumventing computationally demanding Monte Carlo based simulations and rendering the approach potentially useful for preliminary design applications. The accuracy/reliability of the approach is demonstrated via comparisons with pertinent Monte Carlo simulation (MCS) data.

A generalized statistical linearization technique is developed for determining approximately the stochastic response of nonlinear dynamic systems with singular matrices. This system modeling can arise when a greater than the minimum number of coordinates is utilized, and can be advantageous, for instance, in cases of complex multibody systems where the explicit formulation of the equations of motion can be a nontrivial task. In such cases, the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labor-intensive manner. Specifically, relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, a family of optimal and response-dependent equivalent linear matrices is derived. This set of equations in conjunction with a generalized excitation-response relationship for linear systems leads to an iterative determination of the system response mean vector and covariance matrix. Further, it is proved that setting the arbitrary element in the M-P solution for the equivalent linear matrices equal to zero yields a mean square error at least as low as the error corresponding to any nonzero value of the arbitrary element. This proof greatly facilitates the practical implementation of the technique because it promotes the utilization of the intuitively simplest solution among a family of possible solutions. A pertinent numerical example demonstrates the validity of the generalized technique.

Measuring the free-surface displacement on a vertical wall of a marine structure is not a trivial problem. In this context, the efficacy of ultrasonic probes is affected by the interaction between the signal emitted by the sensor and the vertical wall, whereas image-based techniques are computationally demanding, especially if long-time series are utilized. Considering these difficulties, this paper proposes a novel approach for measuring the sea surface elevation on vertical breakwaters. The proposed methodology involves the use of pressure measurements and a reconstruction algorithm based on a compressive sensing (CS) technique in conjunction with a generalized harmonic wavelet (GHW) basis. In particular, a constrained CS optimization approach is proposed by utilizing the known values of the free-surface data to reconstruct all other missing data while adhering at the same time to prescribed upper and lower bounds at all time instants. The reliability of the methodology was assessed against field data pertaining to a vertical wall equipped with pressure transducers recorded at the Natural Ocean Engineering Laboratory of Reggio Calabria. It was shown that direct application of an unconstrained GHW-based CS optimization approach yielded physically inconsistent minima and maxima values; thus, it was inadequate for reliably reconstructing the free surface. These drawbacks were removed by the constrained GHW-based CS. Furthermore, examination of the reconstructed sea surface profiles in the vicinity of extremely high wave crests or wave troughs showed that they are in agreement with pertinent theoretical data obtained by using the nonlinear quasi-determinism theory.

An approximate analytical dimension reduction approach is developed for determining the response of a multi-degree-of-freedom (MDOF) nonlinear/hysteretic system subject to a non-stationary stochastic excitation vector. The approach is based on the concepts of statistical linearization and of stochastic averaging. It is readily applicable even for excitations possessing evolutionary in time power spectra. Further, it can be potentially used in conjunction with design spectrum based analyses to obtain peak system response estimates. Numerical examples include MDOF systems comprising the versatile Bouc-Wen (hysteretic) model. The reliability of the approach is demonstrated by pertinent Monte Carlo simulations. (C) 2012 Elsevier Ltd. All rights reserved.

An approximate analytical technique for assessing the reliability of a softening Duffing oscillator subject to evolutionary stochastic excitation is developed. Specifically, relying on a stochastic averaging treatment of the problem, the oscillator time-dependent survival probability is determined in a computationally efficient manner. In comparison with previous techniques that neglect the potential unbounded response behavior of the oscillator when the stiffness element acquires negative values, the technique developed in this paper readily takes this aspect into account by introducing a special form for the oscillator nonstationary response amplitude probability density function (PDF). A significant advantage of the technique relates to the fact that it can readily handle cases of stochastic excitations that exhibit variability in both the intensity and the frequency content. Numerical examples include a softening Duffing oscillator under evolutionary earthquake excitation and a softening Duffing oscillator with nonlinear damping modeling the nonlinear ship roll motion in beam seas. Comparisons with pertinent Monte Carlo simulation data demonstrate the efficiency of the technique. (C) 2015 American Society of Civil Engineers.

In this paper, the challenge of quantifying the uncertainty in stochastic process spectral estimates based on realizations with missing data is addressed. Specifically, relying on relatively relaxed assumptions for the missing data and on a kriging modeling scheme, utilizing fundamental concepts from probability theory, and resorting to a Fourier-based representation of stationary stochastic processes, a closed-form expression for the probability density function (PDF) of the power spectrum value corresponding to a specific frequency is derived. Next, the approach is extended for also determining the PDF of spectral moments estimates. Clearly, this is of significant importance to various reliability assessment methodologies that rely on knowledge of the system response spectral moments for evaluating its survival probability. Further, utilizing a Cholesky decomposition for the PDF-related integrals kept the computational cost at a minimal level. Several numerical examples are included and compared against pertinent Monte Carlo simulations for demonstrating the validity of the approach. (c) 2017 American Society of Civil Engineers.

A novel approximate analytical technique is developed for determining the nonstationary response probability density function (PDF) of randomly excited nonlinear multidegree-of-freedom (MDOF) systems. Specifically, the concept of the Wiener path integral (WPI) is used in conjunction with a variational formulation to derive an approximate closed-form solution for the system response PDF. Notably, determining the nonstationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by existing alternative numerical path integral solution schemes, which rely on a discrete version of the Chapman-Kolmogorov (C-K) equation. In this manner, the analytical WPI-based technique developed by the authors is extended and generalized herein to account for hysteretic nonlinearities and MDOF systems. This enhancement of the technique affords circumventing approximations associated with the stochastic averaging treatment of the previously developed technique. Hopefully the technique can be used as a convenient tool for assessing the accuracy of alternative, more computationally intensive stochastic dynamics solution methods. The accuracy of the technique is demonstrated by pertinent Monte Carlo simulations. (C) 2014 American Society of Civil Engineers.

A numerical path integral solution approach is developed for determining the response and first-passage probability density functions (PDFs) of nonlinear oscillators subject to evolutionary broad-band stochastic excitations. Specifically, based on the concepts of statistical linearization and of stochastic averaging, the system response amplitude is modeled as a one-dimensional Markov diffusion process. Further, using a discrete version of the Chapman-Kolmogorov equation and the associated first-order stochastic differential equation, the response amplitude and first-passage PDFs are derived. The main concept of the approach relates to the evolution of the response PDF in short time steps, assuming a Gaussian form for the conditional response PDF. A number of nonlinear oscillators are considered in the numerical examples section including the versatile Preisach hysteretic oscillator. For this oscillator, first-passage PDFs are derived for the first time to the authors' best knowledge. Comparisons with pertinent Monte Carlo data demonstrate the reliability of the approach.

A novel identification approach for linear and nonlinear time-variant systems subject to non-stationary excitations based on the localization properties of the harmonic wavelet transform is developed. Specifically, a single-input/single-output (SISO) structural system model is transformed into an equivalent multiple-input/single-output (MISO) system in the wavelet domain. Next, time and frequency dependent generalized harmonic wavelet based frequency response functions (GHW-FRFs) are appropriately defined. Finally, measured (non-stationary) input-output (excitation-response) data are utilized to identify the unknown GHW-FRFs and related system parameters. The developed approach can be viewed as a generalization of the well established reverse MISO spectral identification approach to account for non-stationary inputs and time-varying system parameters. Several linear and nonlinear time-variant systems are used to demonstrate the reliability of the approach. The approach is found to perform satisfactorily even in the case of noise-corrupted data. [All rights reserved Elsevier].