Stochastic Vibration Analysis of Functionally Graded Plates with Material Randomness Using Cell-Based Smoothed Discrete Shear Gap Method

Multi-scale stochastic dynamic response analysis of offshore risers with lognormal uncertainties

In this study, we present an efficient approach, called the probabilistic collocation method (PCM), for uncertainty analysis of flow in unsaturated zones, in which the constitutive relationship between the pressure head and the unsaturated conductivity is assumed to follow the van Genuchten‐Mualem model. Spatial variability of soil parameters leads to uncertainty in predicting flow behaviors. The aim is to quantify the uncertainty associated with flow quantities such as the pressure head and the effective saturation. In the proposed approach, input random fields, i.e., the soil parameters, are represented via the Karhunen‐Loeve expansion, and the flow quantities are expressed by polynomial chaos expansions (PCEs). The coefficients in the PCEs are determined by solving the equations for a set of carefully selected collocation points in the probability space. To illustrate this approach, we use two‐dimensional examples with different input variances and correlation scales and under steady state and transient conditions. We also demonstrate how to deal with multiple‐input random parameters. To validate the PCM, we compare the resulting mean and variance of the flow quantities with those from Monte Carlo (MC) simulations. The comparison reveals that the PCM can accurately estimate the flow statistics with a much smaller computational effort than the MC.

Robust topology optimization for multi-material structures considering material uncertainties

Free vibration analysis of functionally graded carbon nanotube-reinforced composite triangular plates using the FSDT and element-free IMLS-Ritz method

<p>The primary objective of this study is to develop new computational methods for robust design optimization (RDO) and reliability-based design optimization (RBDO) of high-dimensional, complex engineering systems. Four major research directions, all anchored in polynomial dimensional decomposition (PDD), have been defined to meet the objective. They involve: (1) development of new sensitivity analysis methods for RDO and RBDO; (2) development of novel optimization methods for solving RDO problems; (3) development of novel optimization methods for solving RBDO problems; and (4) development of a novel scheme and formulation to solve stochastic design optimization problems with both distributional and structural design parameters.</p> <p>The major achievements are as follows. Firstly, three new computational methods were developed for calculating design sensitivities of statistical moments and reliability of high-dimensional complex systems subject to random inputs. The first method represents a novel integration of PDD of a multivariate stochastic response function and score functions, leading to analytical expressions of design sensitivities of the first two moments. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For all three methods developed, both the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Secondly, four new methods were developed for RDO of complex engineering systems. The methods involve PDD of a high-dimensional stochastic response for statistical moment analysis, a novel integration of PDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms. The methods, depending on how statistical moment and sensitivity analyses are dovetailed with an optimization algorithm, encompass direct, single-step, sequential, and multi-point single-step design processes. Thirdly, two new methods were developed for RBDO of complex engineering systems. The methods involve an adaptive-sparse polynomial dimensional decomposition (AS-PDD) of a high-dimensional stochastic response for reliability analysis, a novel integration of AS-PDD and score functions for calculating the sensitivities of the failure probability with respect to design variables, and standard gradient-based optimization algorithms, resulting in a multi-point, single-step design process. The two methods, depending on how the failure probability and its design sensitivities are evaluated, exploit two distinct combinations built on AS-PDD: the AS-PDD-SPA method, entailing SPA and score functions; and the AS-PDD-MCS method, utilizing the embedded MCS of the AS-PDD approximation and score functions. In addition, a new method, named as the augmented PDD method, was developed for RDO and RBDO subject to mixed design variables, comprising both distributional and structural design variables. The method comprises a new augmented PDD of a high-dimensional stochastic response for statistical moment and reliability analyses; an integration of the augmented PDD, score functions, and finite-difference approximation for calculating the sensitivities of the first two moments and the failure probability with respect to distributional and structural design variables; and standard gradient-based optimization algorithms, leading to a multi-point, single-step design process.</p> <p>The innovative formulations of statistical moment and reliability analysis, design sensitivity analysis, and optimization algorithms have achieved not only highly accurate but also computationally efficient design solutions. Therefore, these new methods are capable of performing industrial-scale design optimization with numerous design variables.</p>

Combination of Karhunen-Loève and intrusive polynomial chaos for uncertainty quantification of thermomagnetic convection problem with stochastic boundary condition

In this paper, a parametric study is presented for free vibration analysis of rotating truncated conical shells reinforced with graphene nanoplatelets (GNPs). The composite shell is considered to be composed of epoxy as the matrix and the GNPs which are distributed along the thickness direction based on the various distribution patterns. The shell is modeled based on the first-order shear deformation theory (FSDT), and effective material properties are calculated based on the Halpin–Tsai model and the rule of mixture. Incorporating centrifugal and Coriolis accelerations along with initial hoop tension, the set of the governing equations and boundary conditions are derived using Hamilton’s principle and are solved numerically using generalized differential quadrature method. Convergence and accuracy of the presented solution are confirmed, and influences of various parameters on the forward and backward frequencies are investigated including circumferential mode number, boundary conditions, rotational speed, semi-vertex angle and also mass fraction, distribution pattern, width and thickness of the GNPs. It is noteworthy that for the first time, the initial hoop tension is incorporated for a rotating conical shell modeled based on the FSDT.

Cylinder-plate combined structures (CPCS) are widely used in aerospace, marine engineering, and offshore platform systems. During service, they are frequently subjected to stochastic excitations induced by turbulent boundary layers, acoustic loads, hydrodynamic disturbances, and broadband operational vibrations. Excessive random vibration responses may significantly reduce structural reliability, accelerate fatigue damage, and compromise operational safety. To address these engineering challenges, a unified stochastic dynamic analysis and vibration suppression framework is established for functionally graded graphene platelet-reinforced composites (FG-GPLRC) CPCS equipped with distributed dynamic vibration absorbers (DVAs). Adopting the First-order Shear Deformation Theory (FSDT), a comprehensive energy functional for the CPCS is established, in which the penalty method is implemented to impose boundary conditions and ensure interface continuity. Subsequently, the Pseudo-excitation Method (PEM) is utilized to convert the stochastic vibration analysis into an equivalent deterministic harmonic problem, and the governing equations are spatially discretized by combining the spectral geometric method (SGM) with the Ritz variational procedure, enabling efficient evaluation of power spectral density (PSD) and root-mean-square (RMS) responses. The reliability of the proposed model is verified through a series of numerical validation comparisons. On this basis, comprehensive parametric investigations are conducted to assess how material properties, structural geometries, and critical DVA parameters influence system behavior. The results demonstrate that the incorporation of distributed DVAs can achieve superior vibration suppression performance. This study provides an efficient and reliable theoretical framework for stochastic vibration analysis and damping design of advanced composite plate-shell coupled structures operating in complex random environments, offering important theoretical support for dynamic optimization design in aerospace and marine engineering applications.

The application of polynomial chaos expansions (PCEs) to the propagation of uncertainties in stochastic dynamical models is well-known to face challenging issues. The accuracy of PCEs degenerates quickly in time. Thus maintaining a sufficient level of long-term accuracy requires the use of high-order polynomials. In numerous cases, it is even infeasible to obtain accurate metamodels with regular PCEs because PCEs cannot represent the dynamics. To overcome the problem, an original numerical approach was recently proposed that combines PCEs and non-linear autoregressive with exogenous input (NARX) models, which are a universal tool in the field of system identification. The approach relies on using NARX models to mimic the dynamical behavior of the system and dealing with the uncertainties using PCEs. The PC-NARX model was built by means of heuristic genetic algorithms. This paper aims at introducing the least angle regression (LARS) technique for computing PC-NARX models, which consists of solving two linear regression problems. The proposed approach is validated with structural mechanics case studies, in which uncertainties arising from both structures and excitations are taken into account. Comparison with Monte Carlo simulation and regular PCEs is also carried out to demonstrate the effectiveness of the proposed approach.

Response analysis of an accelerating unbalanced rotating system with both random and interval variables

Considering uncertainty is inarguably a crucial aspect of dynamic analysis, design, and control of a mechanical system. When it comes to multibody problems, the effect of uncertainty in the system’s parameters and excitations becomes even more significant due to the accumulation of inaccuracies. For this reason, this paper presents a detailed research on the use of the Polynomial Chaos Expansion (PCE) method for the control of nondeterministic multibody systems. PCE is essentially a way to compactly represent random variables. In this scheme, each stochastic response output and random input is projected onto the space of appropriate independent orthogonal polynomial basis functions. In the field of robotics, a required task is to force robotic arms to follow designated paths. Controlling such systems usually leads to difficulties since the dynamic equations of multibody problems are highly nonlinear. Computed Torque Control Law (CTCL) is able to overcome these difficulties by using feedback linearization to evaluate the required torque/force at any time to make the system follow a trajectory. In this paper, a mathematical framework is introduced to apply the Computed Torque Control Law to a multibody system with uncertainty. Surprisingly, it is shown that using this control scheme, uncertainty in geometry does not affect the closed-loop equations of controlled systems. Both the intrusive PCE method and the Monte Carlo approach are used to control a fully actuated two-link planar elbow arm where each link is required to follow a specified path. Lastly, a comparison of the time efficiency and accuracy between the traditionally used Monte Carlo method and the intrusive PCE is presented. The results indicate that the intrusive PCE approach can provide better accuracy with much less computation time than the Monte Carlo method.

Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies

Buckling analysis of CNT reinforced functionally graded laminated composite plates

Large deflection analysis of FG-CNT reinforced composite skew plates resting on Pasternak foundations using an element-free approach

Postbuckling of carbon nanotube reinforced functionally graded plates with edges elastically restrained against translation and rotation under axial compression