Third-order Theory Research Articles

Spectral Proper Orthogonal Decomposition Reduced-Order Model for Analysis of Aerothermoelasticity

Analysis and simulation of fluid–thermal–structure coupling play an essential role in hypersonic flight vehicle design. In this study, the spectral proper orthogonal decomposition (SPOD) method is employed to establish the reduced-order model (ROM) for a hypersonic panel with a thermal protection system. First, the subsystem of the aerothermoelastic (AETI) system is modeled. Specifically, the aerodynamic heating is calculated with Eckert’s reference enthalpy method. The heat transfer is carried out with the finite difference method, and the aeroelastic equation is built by integrating the von Kármán plate theory and the third-order piston theory. Additionally, the two-way coupling between aerothermal and aeroelastic subsystems is used to construct the AETI system. Second, the optimal snapshots with the resultant local SPOD modes (SPOMs) are evaluated in terms of accuracy. Third, the global SPOMs for variable flight parameters are promoted to develop the robustness of the proposed SPOD/ROM. The findings indicate that taking the transient chaotic response as snapshots can produce global SPOMs. SPOD/ROM for the present AETI system improves the efficiency by 10–50 times compared to the classic Galerkin method. Moreover, the global SPOMs to variable Mach number and flight altitude are essentially the natural modes of the AETI panel, and thus are dependent on the temperature of the fluttering panel, which is our new finding in the present study. Overall, the current research will provide methods for accurately and efficiently predicting thermal loads, flutter boundary, and dynamic response for hypersonic flight vehicles.

Investigate the bending and free vibration responses of multi-directional functionally graded plates with variable thickness based on isogeometric analysis

This study investigates the static bending and free vibration behaviour of Multi-directional Functionally Graded (MFGM) plates having a variable thickness. With this regard, the third-order shear deformation plate theory is used to describe the kinematic relation. Material properties of MFGM are assumed to be graded in spatial direction within the plate’s volume and the effective material properties are calculated based on the power-law approach. The governing equations are derived based on Hamilton’s principle and the Isogeometric Analysis (IGA) is employed as a discretization tool to develop system equations. Several numerical examples are conducted to verify the accuracy of the proposed approach and investigate the influence of different geometrical and material parameters on the behaviour of MFGM plates

Reddy’s third-order shear deformation shell theory for free vibration analysis of rotating stiffened advanced nanocomposite toroidal shell segments in thermal environments

Reddy’s third-order shear deformation shell theory for free vibration analysis of rotating stiffened advanced nanocomposite toroidal shell segments in thermal environments

Use of an achromatic meniscus to compensate for astigmatism of the optical system

In this paper, we present a procedure for calculating the initial configuration of one type of simple optical system consisting of an achromatic meniscus followed by an achromatic member in close proximity. The initial parameters for further optimisation are calculated using third-order aberration theory. The procedure is demonstrated by an example.

Analysis of FGM plates based on physical neutral surface using general third-order plate theory

We present a nonlinear finite element analysis to investigate the nonlinear behaviour of functionally graded materials (FGM) plates. The material properties are assumed to vary gradually across the thickness according to a power law distribution. The starting point of the investigation is the generalized third-order plate theory which is modified in the present analysis to include the position of the neutral surface and enhanced with additional terms to represent the distribution stresses better. The plates are subjected to bending and compression loads including buckling and post-buckling analysis.

Aberration analysis of decentered lenses for the compensation of vergence–accommodation conflict in virtual reality systems

Subject of study. This study considered optical virtual reality systems in which the use of decentered lenses was proposed to remove or reduce the vergence–accommodation conflict, which results in discomfort or quick fatigue of the user. Aim of study. The study aimed to investigate the possibility of using decentered lenses to compensate the vergence–accommodation conflict in virtual reality systems and to determine the optimal configuration of a two-component optical system. Methods. The required decentering values in the case of a shift of one or both components and aberration characteristics were analyzed. Main results. More exact equations for aberrations introduced into an optical system by decentered lenses upon a relatively large shift of lenses were obtained based on the relations of the third-order aberration theory. The minimum decentering and, consequently, minimum aberrations are obtained in the scheme when both components are decentered. An example of an optical system with decentered lenses designed considering the requirements on size and high image quality is presented. Practical significance. The obtained equations for the aberrations of decentered lenses enable the assessment of the effect of a transverse shift of lenses on image quality in virtual reality systems and can also be applied to analyze decentering aberrations in other schemes. The proposed designs of optical schemes can be utilized in virtual reality systems with reduced vergence–accommodation conflict.

Design of a free-form off-axis three-mirror optical system with a low f-number based on the same substrate.

In this paper, we design a free-form off-axis three-mirror optical system with a low f-number and compact structure, which can be used as an infrared reflection imager. The initial structure is calculated from the near-axis optical transfer matrix based on third-order aberration theory. Particular constraints are designed to install all mirrors on the same substrate for simultaneous milling, which reduces the processing difficulty and effectively avoids errors caused by component assembly. Zernike free-form surfaces are introduced to correct aberrations. This optical system has a field of view of 5∘×5∘ and an f-number of 1.82; the modulation transfer function of the system is higher than 0.6 at 30 lp/mm. The results of the tolerance assignment of the system were verified by the Monte Carlo method, and the machining tolerance is reasonable and easy to achieve. This design not only improves the optical performance of the system but also enhances the feasibility of manufacturing.

Efficient time-domain spectral element with zigzag kinematics for multilayered strips

Efficient structural elements are widely sought for fast computation of wave propagation response of anisotropic multilayered structures for their design, non-destructive testing, and structural health monitoring purposes. Although zigzag theories are known to combine excellent accuracy with high computational efficiency, no spectral element has been developed so far based on these theories, which would enable fast and accurate simulation of guided waves in such structures. In this article, we develop an efficient time-domain spectral strip element for analyzing ultrasonic wave propagation in multi-material laminated beams and panels while accounting for the zigzag profile of the in-plane displacement across the thickness. The theory superimposes a layerwise linear (zigzag) variation of the in-plane displacement of a layerwise theory onto the global third-order variation of an equivalent single layer (ESL) third-order theory while retaining the computational advantage of the latter. The high-order element has an arbitrary number of unequally spaced nodes with only four kinematic variables at each node, regardless of the number of laminas. The recently proposed generalized Hermite-type C1-continuous Lobatto shape functions interpolate the deflection, while the axial displacement and shear rotation are interpolated using the C0-continuous Lobatto shape functions. A comprehensive numerical study establishes high efficiency, excellent accuracy, and fast convergence of the new element in analyzing free vibration and guided wave propagation in single-material composite and multi-material fiber-metal laminate structures. Its combined performance in terms of these parameters is much superior to its standard FE counterpart, the ESL theory-based elements, and the continuum elements in the considered frequency range.

Biaxial buckling analysis of an innovative active sandwich plate

Fragility and the presence of a poisonous substance (lead) are the two major issues with traditional piezoceramics and structures that include them. This work suggests an active structure that is lightweight, flexible, and eco-friendly such that a polymeric foam plate is sandwiched between two piezoelectrically activated nanocomposite layers. A combination of a polymer and piezoelectric nanowires (NWs) of zinc oxide (ZnO) is used in the nanocomposite layers. For the proposed active structure, two types of void distributions and functionally graded (FG) distributions of the NWs have been assumed in each corresponding layer. The buckling characterization of the proposed eco-friendly active sandwich plate (EFASP) under biaxial mechanical loads has been conducted using an extended meshless solution based on a third-order theory. An in-depth parametric study on the EFASPs’ biaxial buckling resistance revealed that increasing NW content considerably enhances buckling resistance, with this enhancement strongly reliant on the distribution profile of the NWs. Moreover, it was observed that embedding voids in the substrate layer can lead to a large structural weight reduction while only marginally lowering the buckling performance of suggested EFASPs.

Polarization effects in the total rate of biharmonic ω+3ω ionization of atoms

The total ionization rate of biharmonic ($\ensuremath{\omega}+3\ensuremath{\omega}$) ionization is studied within the independent particle approximation and the third-order perturbation theory. Particular attention is paid to how the polarization of the biharmonic light field affects the total rate. The ratios of the biharmonic ionization rates for linearly and circularly polarized beams as well as for corotating and counter-rotating elliptically polarized beams are analyzed, and how they depend on the beam parameters, such as photon frequency or phase between $\ensuremath{\omega}$ and $3\ensuremath{\omega}$ light beams. We show that the interference of the biharmonic ionization amplitudes determines the dominance of a particular beam polarization over another and that it can be controlled by an appropriate choice of beam parameters. Furthermore, we demonstrate our findings for the ionization of neon $L$ shell electrons.

Quantum third-order nonlinear Hall effect of a four-terminal device with time-reversal symmetry

The third-order nonlinear Hall effect induced by Berry-connection polarizability tensor has been observed in Weyl semimetals T$_d$-MoTe$_2$ as well as T$_d$-TaIrTe$_4$. The experiments were performed on bulk samples, and the results were interpreted with the semiclassical Boltzmann approach. Beyond the bulk limit, we develop a quantum nonlinear transport theory to investigate the third-order Hall response of a four-terminal setup with time-reversal symmetry in quantum regime. The quantum nonlinear theory is verified on a model system of monolayer MoTe$_2$, and numerical results on the angle-resolved Hall currents are qualitatively consistent with the experiment. More importantly, quantum signatures of the third-order Hall effect are revealed, which are independent of the system symmetry. The first quantum signature is quantum enhancement of the third-order Hall current, which is characterized by sharp current peaks whose magnitudes are three orders larger than the first-order Hall current. Such quantum enhancement originates from quantum interference in coherent transport, and it can be easily destroyed by dephasing effect. The second quantum signature is disorder-induced enhancement of the third-order Hall current for weak disorders. Our findings reveal quantum characteristics of the third-order Hall effect, and we propose feasible ways to enhance it in nanoscale systems. The quantum third-order theory developed in this work provides a general formalism for describing nonlinear coherent transport properties in multi-terminal devices, regardless of the system symmetry.

Benchmarking CASPT3 vertical excitation energies.

Based on 280 reference vertical transition energies of various excited states (singlet, triplet, valence, Rydberg, n → π*, π → π*, and double excitations) extracted from the QUEST database, we assess the accuracy of complete-active-space third-order perturbation theory (CASPT3), in the context of molecular excited states. When one applies the disputable ionization-potential-electron-affinity (IPEA) shift, we show that CASPT3 provides a similar accuracy as its second-order counterpart, CASPT2, with the same mean absolute error of 0.11eV. However, as already reported, we also observe that the accuracy of CASPT3 is almost insensitive to the IPEA shift, irrespective of the transition type and system size, with a small reduction in the mean absolute error to 0.09eV when the IPEA shift is switched off.

A semi-analytical approach for instability analysis of composite cylindrical shells subjected to harmonic axial loading

In this article, nonlinear vibration and dynamic stability analyses of simply supported laminated composite circular cylindrical shells subjected to periodic edge loading are carried out. A third-order shear deformation shell theory that considers all the nonlinear terms in all five kinematic parameters and rotary inertia is used to develop the present mathematical model so that the model is also valid for thick cylindrical shells. Hamilton’s principle, an energy-based approach, is used to obtain the governing partial differential equations (PDEs) of motion of the cylindrical shell. Further, these equations are reduced into ordinary differential equations by employing Galerkin’s method. The incremental harmonic balance (IHB) method in conjunction with the pseudo-arc-length method is used to obtain the frequency-amplitude response of the system. For obtaining the zone of instability regions, Bolotin’s method is adopted. For more practical significance, analysis of results is also extended by considering damping into account for the composite cylindrical shells. Time history response and phase portrait are plotted by adopting Newmark-beta method. The effects of the static load factor, dynamic load factor, modal damping coefficient, and stacking sequence on nonlinear vibration, instability regions and time history responses are also examined.

Nonlinear bending analysis of variable thickness multi-directional functionally graded plates based on isogeometric analysis

In this study, the nonlinear bending behavior of multi-directional functionally graded plates with variable thickness is investigated. To this regard, the governing equation of the plate problem is developed based on the principle of virtual work, while the third-order shear deformation plate theory proposed by Reddy is employed to describe the kinematic relations. The geometrical nonlinearity is taken into account by adopting the von Kármán’s assumptions for nonlinear problems with small strains and moderate rotations. The plates investigated in this study are assumed to be made from two different material constituents, whose volume fractions vary spatially within the plate domain. The effective material properties are calculated based on the mixture approach. The Isogeometric Analysis approach is used as a numerical method to solve the governing equations. NURBS basis functions are used as interpolation functions for geometrical modeling and discretization procedure, where the C 1-continuity requirement is meet efficiently. Various numerical examples are conducted to validate the accuracy of the proposed approach and investigate the influences of geometrical and material factors on the nonlinear bending responses of the plates.

Two-component compensators of field aberrations of optical systems

Subject of study. Two-component optical schemes of compensators of field aberrations of optical systems calculated based on the third-order aberration theory and modular method of optical system design [Izv. Vyssh. Uchebn. Zaved., Priborostr.57, 57 (2014)] are proposed. These compensators enable correction of two aberrations: Petzval field curvature and astigmatism or longitudinal chromatic aberration and lateral color or transverse chromatic aberration. Method. A modular method for the design of optical systems based on the third-order aberration theory was used in the calculation. This method consists of synthesis of the systems using optical elements (modules) with known correcting properties [Izv. Vyssh. Uchebn. Zaved., Priborostr.57, 57 (2014)]. Aplanatic menisci with different linear magnifications and a hyperchromatic lens were used as optical modules. The aplanatic menisci were used to calculate the compensator of Petzval field curvature and astigmatism, whereas the hyperchromatic lens was used to calculate the compensator of longitudinal and transverse chromatic aberrations. Main results. The use of two-component compensators of field aberrations enables significant enhancements in the aberrational properties of the objectives, improves the image quality of optical systems, and increases their optical parameters, particularly the angular field of doublet objectives (fivefold) using the correcting properties of optical modules (menisci and hyperchromatic lens). Practical significance. The introduction of a compensator composed of two aplanatic menisci positioned at a significant distance from each other into the optical system makes it possible to correct such aberrations as field curvature and astigmatism regardless of the number of lenses in the objective and increase the angular field. The introduction of a compensator of chromatic aberrations into an optical scheme of a high-aperture microscope objective with 100×1.25 oil immersion corrects the longitudinal and transverse chromatic aberrations. This makes it possible to cease the use of compensating eyepieces with a complex optical scheme and replace them with standard eyepieces as well as to use a microscope where the optical radiation is detected by a charge-coupled device array or complementary metal oxide semiconductor detector instead of the human eye.

Nonlocal integral elasticity for third-order small-scale beams

Small-scale beams are basic structural components of miniaturized electro-mechanical systems whose design requires accurate modeling of size effects. In this research, the size-dependent behavior of nonlocal elastic beams is investigated by adopting the stress-driven elasticity theory. Kinematics of beams is modeled by the Reddy variational third-order beam theory accounting for the effective distribution of shear stresses on cross sections without needing the evaluation of shear correction factors. Stress-driven integral elasticity is thus extended to third-order small-scale beams providing an equivalent constitutive formulation with boundary conditions. The relevant nonlocal elastic equilibrium problem is formulated and an analytical strategy is proposed to obtain closed-form solutions. The present approach is elucidated by solving some structural problems of current interest in Nanotechnology.

Size-dependent free vibration and buckling analysis of magneto-electro-thermo-elastic nanoplates based on the third-order shear deformable nonlocal plate model

In this paper, a size-dependent nanoplate model is developed to describe the free vibration and buckling behaviors of magneto-electro-thermo-elastic (METE) rectangular nanoplates. It is assumed that the METE nanoplate is subjected to external electric voltage, external magnetic potential, and uniform temperature rise. The nonlocal elasticity theory along with the third-order shear deformation plate theory is employed for the size-dependent mathematical modeling of nanoplates. The presented model has two advantages over available models: (1) the need for the correction factor is bypassed and (2) it can be successfully applied to thick nanoplates. The governing equations and the corresponding boundary conditions are derived using the Hamilton’s principle which are then discretized on the space domain based on the generalized differential quadrature (GDQ) method. Afterward, an efficient numerical Galerkin procedure is adopted to reduce the discretized equations into Duffing-type ordinary differential equations. Numerical results are presented to examine the influences of nonlocal parameter, length-to-thickness ratio, temperature rise, external electric potential, external magnetic potential and type of boundary condition on the free vibration, and buckling behaviors of METE nanoplates. It is revealed that frequency and critical buckling load of nanoplates are dependent on magneto-electro-mechanical loadings, whereas they are less dependent on thermal loading.

Thermal Backbone Curves of Nanocomposite Beams Reinforced with Graphene Platelet on Elastic Foundation

In this paper, an examination on the backbone curves of nanocomposite beams reinforced with graphene platelet (GPL) on elastic foundation exposed to a temperature increment is accomplished. By means of the Hamilton principle and in the framework of a third-order shear deformation beam theory, i.e. Reddy’s beam theory (RBT), along with the von Karman nonlinear strains, the nonlinear motion equations are disclosed. The Halpin–Tsai micromechanical model is exploited in order to release the effective modulus of elasticity of the nanocomposite beam while the thermal expansion coefficient, Poisson’s ratio and the mass density are estimated resorting to the rule of mixtures. Elastic foundation, and uniform temperature (UD) rising impacts are incorporated during the mechanical modeling of the nanocomposite beam. The Ritz scheme together with an iterative procedure reveals the nonlinear natural frequency associated to an assumed deflection in order to sketch the corresponding backbone curve. The outcomes are validated in comparison with the available results. Some case studies are established for the sake of clarifying the impressions of the distribution pattern of the GPL, the beam length to its thickness ratio, the weight fraction of the GPL, the elastic foundation, the boundary condition type, and the temperature changes on the first backbone curve of the nanocomposite beam. It is elucidated that the increment of the weight fraction of the GPL with X/O distribution pattern decreases/increases the hardening behavior of clamped–clamped (C-C) and simply-supported beam, while the softening behavior of a clamped-free (C-F) nanocomposite beam is independent of the division pattern, and the weight fraction of the GPL. Moreover, the temperature increment unlike the elastic foundation develops the hardening behavior of the backbone curves of simply supported and C-C nanocomposite beams. Although the backbone curve associated to a C-F nanocomposite beam is invariant with respect to the temperature, the elastic foundation develops the softening trend of C-F nanocomposite beams.

Nonlinear and chaotic vibrations of FG double curved sandwich shallow shells resting on visco-elastic nonlinear Hetenyi foundation under combined resonances

In this study, the nonlinear and chaotic instability of functionally graded (FG) double curved shallow sandwich shells resting on a viscoelastic Hetenyi foundation, under simultaneous effect of in-plane and transverse excitations is studied. Employing the third-order Reddy theory and von-Karman relations and the Hamilton principle, the partial differential equations of motion under movable SS boundary conditions are derived. Introducing the trigonometric Airy stress function and applying the Galerkin’s method, the equations are reduced to a set of nonlinear ODEs with time. Then, under forcing resonance conditions and using the perturbation method, the modulation equations at the stationary conditions are derived and solved numerically. Then stability of non-trivial solutions for the resonance amplitude corresponding to the presence of limit cycle oscillation is investigated. Then by extracting the characteristic resonance amplitude curves, the effect of various parameters, including: frequency detuning parameter, in-plane excitation amplitude, linear, shear and nonlinear stiffness and damping parameters of the foundation on nonlinear response are analyzed. Finally, by extracting two degrees of freedom system time response, bifurcation and chaotic characteristic curves of the problem, the conditions for occurrence the periodic, double periodic, multi-periodic and chaotic behaviors under the simultaneous effect of parametric and internal resonances are studied comprehensively.

Investigation of Nonlinear Thermo-Elastic Behavior of Fluid Conveying Piezoelectric Microtube Reinforced by Functionally Distributed Carbon Nanotubes on Viscoelastic-Hetenyi Foundation

In this paper, nonlinear and nonlocal thermo-elastic behavior of a microtube reinforced by Functionally Distributed Carbon Nanotubes, with internal and external piezoelectric layers, in the presence of nonlinear viscoelastic-Hetenyi foundation, and axial fluid flow inside the microtube is studied. Nonlinear partial differential equations governing the system are derived using Reddy’s third-order shear deformations theory along with the Von-Karman theory including the effect of fluid viscosity. Then, the equations are converted to time-dependent ordinary nonlinear equations using the Galerkin method. Afterward, the governing equations of the microtube’s lateral displacements are solved using the multiple scales method. The analysis of the piezoelectric’s parametric resonance is performed by obtaining trivial and nontrivial stationary solutions and plotting characteristic curves of the frequency response and voltage response. At the end, the effect of different parameters including the flow velocity, excitation voltage, parameters of the foundation, viscosity parameter, thermal loading and nanotubes’ volume fraction index on the nonlinear behavior of the system, under parametric resonance condition, is investigated.