Chebyshev Method Research Articles

Localized Stochastic Dynamic Solution for Laminated Coupled Open Conical-Cylindrical Cabin System Based on the Condensation Method

This paper proposed a technical solution for quickly and accurately solving the stochastic dynamic characteristics of laminated coupled open conical-cylindrical cabin system. On the basic of first-order shear deformation theory (FSDT), the two-dimensional spectral Chebyshev method was introduced to obtain a unified dynamic matrix that included laminated conical and cylindrical panels. The middle part of the cabin was selected as the target substructure, while the precise dynamic condensation theory was used as a tool to establish a local dynamic analysis model of the cabin system based on comprehensive consideration of complex boundary conditions, coupling conditions, random load excitation and other factors. From a numerical analysis perspective, the local-level dynamic responses obtained from the model were validated to match well with the global-level results from the finite element method. Based on this numerical validation, a dynamic parametric analysis scheme was proposed, focusing on the local dynamic characteristics of the target substructure. This scheme analyzed the impact of the structural parameters of the middle part of cabin on the dynamic behaviors of the cabin system, providing technical guidance for the design optimization of the cabin system in engineering applications.

Solving for the Thermal Vibration Behaviors of Functionally Graded Stepped Conical Curved Plates with Point Connection Conditions

This paper proposed a degree-of-freedom reduction model for the rapid and unified solution of the modal and steady-state response behavior of functionally graded stepped conical curved plates with point connections under thermal environments. Within the framework of the first-order shear deformation theory, the two-dimensional spectral Chebyshev method and the fixed interface modal synthesis method were combined to derive the degree-of-freedom reduction model for functionally graded conical curved plates under thermal environments. Each reduction model was assembled considering external load excitations. Virtual springs were used to equivalently handle the coupling interface forces and point connection interface forces, further establishing a thermal vibration response analysis model for functionally graded stepped conical curved plates with point connections. In numerical examples, this degree-of-freedom reduction model was compared with experimental results and overall finite element results, validating the accuracy of the model and the advantages of rapid solution. Additionally, this degree-of-freedom reduction model was applied to quickly analyze the effects of material parameters and point connection parameters on the thermal steady-state response of the structure, providing a theoretical basis for the environmental adaptability design, verification, and evaluation of such structures.

Stability of thermal convection in two superimposed miscible viscous fluids under gravitational modulation

Abstract The thermal stability of a system of two superimposed miscible viscous fluid layers, under gravitational modulation, is investigated using linear stability analysis. It is well known in the literature that in the absence of gravitational modulation, there exist two convection regimes depending on the buoyancy number: the single-cell regime, where convection is oscillating and occupies the entire volume of the two fluid layers, and the stratified regime where stationary convection occurs in each fluid layer. Using Chebyshev's spectral collocation method and Floquet's theory, the numerical results show that under gravitational modulation, single-cell convection oscillates at half the frequency of the external oscillation (subharmonic threshold), while the stratified regime oscillates at a frequency equal to that of the forcing (harmonic threshold). The critical buoyancy number corresponding to the transition between the two convection regimes depends, in addition to the viscosity ratio and the depth of the two layers, on certain dimensionless frequencies permeating this transition and also on the amplitude of oscillation via the Froude number. In the plane of the critical Rayleigh number versus dimensionless modulation frequency, the branch associated with stratified convection does not depend on the buoyancy number for equal viscosities whereas this is the case for the whole-convection regime. For different viscosities, both convection regimes depend on the buoyancy number. At last, in the stratified regime (harmonic threshold), the critical value of the viscosity contrast, corresponding to the passage from an active layer to a passive layer and vice versa, increases with the modulation frequency.

Dynamic Behaviors of Composite-Laminated Concentrically Stiffened Rectangular Plates by Substructure Modal Synthesis

Existing numerical analysis methods often incur high computational costs when directly solving multi-degree-of-freedom systems due to computer capacity limitations. To address this issue, this paper took the composite laminated concentrically stiffened rectangular plate as the research object. The modal synthesis method was employed to reduce the degrees of freedom of the model, enabling the solution of complex dynamic characteristics. In the research process, the domain decomposition theory was used to divide the research object into plate and rib substructures. Based on the spectral Chebyshev method, the substructure dynamic model of the composite laminated rectangular plate was constructed. Subsequently, the function expressions of the complex load excitation on each substructure were derived, and the fixed modal synthesis method was introduced to construct the spectral Chebyshev reduced model of the substructures. According to the artificial spring theory equivalent, the interface force, and the whole reduction model of the composite laminated concentrically stiffened rectangular plate were obtained by assembling the substructure models. The time–domain and frequency–domain dynamic behaviors of composite laminated concentrically stiffened rectangular plates are yielded by solving the whole reduced model. The novelty of this work lies in combining the spectral Chebyshev method with the modal synthesis method. The established reduced model no longer depended on mesh quality and required fewer matrix dimensions and computation time compared to the whole model, while still achieving sufficiently accurate results. Additionally, based on the reduced model, an effective parametric study scheme was proposed to analyze the impact of dimensions and material properties of rib substructures on the dynamic behaviors of the composite laminated concentrically stiffened rectangular plate. These findings can be used to guide the optimal design of laminated concentrically stiffened rectangular plates in engineering practice.

Third order two-step Runge–Kutta–Chebyshev methods

The well-known high order stabilized codes (such as DUMKA and ROCK) have several drawbacks: numerically obtained stability polynomials (which do not have a closed analytic form), poor internal stability and convergence. RKC-type methods have much better computational properties. However, these types of methods currently have a second order maximum. In this paper, a family of third order stabilized methods with an explicit analytical solution of stability polynomials is presented. This was made possible by usage of two-step Runge–Kutta methods. A new code TSRKC3 is proposed, illustrated by several examples, and compared to existing programs.

Theoretical design and implementation of equal and unequal split ultra-wide band Wilkinson power divider with Chebyshev impedance transform

Abstract In this study, we designed multi-section UWB Wilkinson power dividers for both equal and unequal power divider. The key innovation is successfully matching these power dividers using the Chebyshev method. The proposed method achieves very low insertion losses and high transmission coefficients across the ports. For the three-section equal power divider, Chebyshev polynomials were used to model the reflection coefficient, and the characteristic impedance for each section was calculated symmetrically. The design achieved a fractional bandwidth of 106% for a 20 dB return loss, with an insertion loss of 0.3 dB. For unequal multi-section dividers, power division ratios of 1.5:1 and 2:1 were chosen. Theoretical microwave calculations were used to determine the input and output transmission line impedances. Chebyshev approximation was then applied to design the three-section unequal power dividers. The return losses for these unequal dividers were measured at 17 dB and 15 dB, with fractional bandwidths of 120% and 126%. Both circuits had insertion losses of 0.25 dB. The consistent results from analytical calculations, simulations, and measurements confirm the effectiveness of the Chebyshev method for both equal and unequal power division.

Early-time resonances in the three-dimensional wall-bounded axisymmetric Euler and related equations

We investigate the complex-time analytic structure of solutions of the three-dimensional (3D)-axisymmetric, wall-bounded, incompressible Euler equations, by starting with the initial data proposed in Luo and Hou [“Potentially singular solutions of the 3D axisymmetric Euler equations,” Proc. Natl. Acad. Sci. U. S. A. 111(36), 12968–12973 (2014)], to study a possible finite-time singularity. We use our pseudospectral Fourier–Chebyshev method [Kolluru et al., “Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation,” Phys. Rev. E 105(6), 065107 (2022)], with quadruple-precision arithmetic, to compute the time-Taylor-series coefficients of the flow fields, up to a high order. We show that the resulting approximations display early-time resonances; the initial spatial location of these structures is different from that for the tygers, which we have obtained in Kolluru et al. [“Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation,” Phys. Rev. E 105(6), 065107 (2022)]. We then perform asymptotic analysis of the Taylor-series coefficients, by using generalized ratio methods, to extract the location and nature of the convergence-limiting singularities and demonstrate that these singularities are distributed around the origin, in the complex-t2 plane, along two curves that resemble the shape of an eye. We obtain similar results for the one-dimensional (1D) wall-approximation (of the full 3D-axisymmetric Euler equation) called the 1D Hou–Luo model, for which we use Fourier-pseudospectral methods to compute the time-Taylor-series coefficients of the flow fields. Our work examines the link between tygers, in Galerkin-truncated pseudospectral studies, and early-time resonances, in truncated time-Taylor expansions of solutions of partial differential equations, such as those we consider.

Conical-shaped variable stiffness composite laminates: Design and fiber path planning

This study presents an innovative design methodology for conical-shaped variable stiffness composite laminates. It employs the spectral Chebyshev method to solve dynamic equations, optimizing lamination parameters to maximize fundamental frequency. Subsequently, the fiber angles at the sampling points across the domain are retrieved. In the final step, a custom normalized cut segmentation approach is proposed to partition the domain into clusters and delineate fiber paths based on discrete fiber angles, while satisfying the manufacturing constraints such as curvature, gaps, and overlaps. The effectiveness of the proposed design methodology is demonstrated through several case studies, where maximum enhancement in fundamental frequency for the same part weight is compared to optimized constant stiffness composite laminates. The results show that an enhancement of 16% is achievable for a variable stiffness laminate disregarding manufacturing constraints for the fully clamped case study. However, the improvements reduce to 12.6% and 14% (deviating from the ideal optimal fundamental frequency by 4.5% and 3.3%), respectively, once the manufacturable constant and curvilinear fibers in each cluster are computed. This work can lead significant advances of engineering applications by enabling the design of variable stiffness laminates with complex geometries improving mechanical performance and/or reducing structural weight compared to conventional laminates.

Photocatalysis case study of wastewater treatment using magneto-radiative Williamson tri-hybrid nanofluid

As a result of the tremendous development in all aspects of life, the increase in population day after day, and the scarcity of freshwater resources sufficient to meet the needs of living organisms on the surface of the earth, all these reasons called for an attempt to treat wastewater supported by solar energy to raise the temperature and facilitate the operation of wastewater treatment systems. Health in areas rich in solar resources. This approach reduces reliance on traditional energy sources in the wastewater treatment process. This theoretical study aims to create a mathematical model for the problem of water turbidity due to impurities (wastewater). Non-Newtonian fluid flow equations are established in the case of using renewable energy in various forms of nano-photocatalysts, as well as magnetic force which has a clear effect on the process of removing impurities from the aqueous fluid. Chebyshev's method was used to obtain numerical solutions to the problem, which were represented by a set of drawings and tables, that highlighted the effective role of both the magnetic field and sunlight in eliminating impurities. Increasing the Lorentz force and thermal radiation increases the strength of hydrogen bonding of the aqueous liquid mixed with particles of tri-hybrid nanomaterials larger than the aqueous liquid, which in turn works to raise the thickness of the thermal layer in the blade shape higher than the rest of the shapes. This study gives future insights for further research endeavors.

Polynomial solution of Cauchy-type singular integro-differential equations with bivariate kernels

In this paper, we present Chebyshev and ultraspherical polynomials method to obtain numerical solutions of Cauchy-type singular integro-differential equations. We prove that the singular integro-differential equations with smooth/non-smooth bivariate kernels and general boundary conditions can be transformed into a constrained least square problem and solved via generalized QR factorization. By using open-source Chebfun code, low-rank approximate solutions of singular integro-differential equations can be obtained efficiently in the spectral accuracy range. The method can be easily extended to solve those singular integro-differential equations with smooth bivariate kernels, or convolution kernels with absolute value symbols (non-smooth cases), or when the solutions have (inverse) square root singularities.

Massive scalar field perturbations of black holes immersed in Chaplygin-like dark fluid

We consider massive scalar field perturbations in the background of black holes immersed in Chaplygin-like dark fluid (CDF), and we analyze the photon sphere modes, the de Sitter modes as well as the near extremal modes and discuss their dominance, by using the pseudospectral Chebyshev method and the third order Wentzel-Kramers-Brillouin approximation. We also discuss the impact of the parameter representing the intensity of the CDF on the families of quasinormal modes. Mainly, we find that the propagation of a massive scalar field is stable in this background, and it is characterized by quasinormal frequencies with a smaller oscillation frequency and a longer decay time compared to the propagation of the same massive scalar field within the Schwarzschild-de Sitter background.

An efficient uncertainty propagation method for nonlinear dynamics with distribution-free P-box processes

The distribution-free P-box process serves as an effective quantification model for time-varying uncertainties in dynamical systems when only imprecise probabilistic information is available. However, its application to nonlinear systems remains limited due to excessive computation. This work develops an efficient method for propagating distribution-free P-box processes in nonlinear dynamics. First, using the Covariance Analysis Describing Equation Technique (CADET), the dynamic problems with P-box processes are transformed into interval Ordinary Differential Equations (ODEs). These equations provide the Mean-and-Covariance (MAC) bounds of the system responses in relation to the MAC bounds of P-box-process excitations. They also separate the previously coupled P-box analysis and nonlinear-dynamic simulations into two sequential steps, including the MAC bound analysis of excitations and the MAC bounds calculation of responses by solving the interval ODEs. Afterward, a Gaussian assumption of the CADET is extended to the P-box form, i.e., the responses are approximate parametric Gaussian P-box processes. As a result, the probability bounds of the responses are approximated by using the solutions of the interval ODEs. Moreover, the Chebyshev method is introduced and modified to efficiently solve the interval ODEs. The proposed method is validated based on test cases, including a duffing oscillator, a vehicle ride, and an engineering black-box problem of launch vehicle trajectory. Compared to the reference solutions based on the Monte Carlo method, with relative errors of less than 3%, the proposed method requires less than 0.2% calculation time. The proposed method also possesses the ability to handle complex black-box problems.

A family of two-step second order Runge–Kutta–Chebyshev methods

In this paper, a new family of second order two-step Runge–Kutta–Chebyshev methods is presented. These methods are a generalization of the one-step stabilized methods and have better computational properties compared to them. A new code TSRKC2 is developed and compared to existing solvers at sufficiently large set of examples.

A novel spectral element method with a higher-order coarse quad meshing approach to design laminated composite panels with arbitrarily shaped cutouts

Laminated panels are widely used in various industries due to their distinct advantages. The design of laminated composites requires efficient methodologies due to the vast design space. This study proposes a new spectral element modeling approach for accurate and computationally efficient analysis of laminated composites with arbitrarily shaped cutouts. The method employs a coarse quad meshing approach and a spectral Chebyshev method to determine the element matrices. The presented method enables obtaining high-fidelity models by applying h- and p-refinements following a non-dimensional approach, aiming to yield a model with the fewest degrees of freedom to achieve a desired convergence level. Benefiting from the advantages of both meshless methods (in terms of computational efficiency) and finite element methods (in terms of geometric capabilities), we performed several case studies for laminated panels with cutouts and it is shown that the presented spectral element method (SEM) enables calculating the natural frequencies and the mode shapes as accurate as FEM, yet decreases the analysis duration by 13 folds. Furthermore, the developed approach was employed with a gradient-based optimizer or genetic algorithm to demonstrate the design of (sandwich) laminated composites for obtaining optimal lamination parameters and 2D Pareto fronts.

Low sidelobe silicon optical phased array with Chebyshev amplitude distribution

Abstract We propose and demonstrate a silicon photonic optical phased array (OPA) with ultra-low sidelobe level. The arbitrary ratio power splitters (ARPSs) are introduced to manipulate the amplitude distribution between different channels and suppress the sidelobe level. A 32-channel OPA has been designed and demonstrated with the amplitude distribution determined by preferred Chebyshev method. The experimental results indicate that the sidelobe suppression ratio (SLSR) can be up to 25.3 dB. The measured field of view (FOV) is 84° × 13° with divergence of 2.8° × 1.7°. Furthermore, the frequency-modulated continuous-wave (FMCW) based ranging has been also demonstrated experimentally by utilizing the OPA as the transmitter.

THERMAL ANALYSIS OF BI-DIRECTIONAL FUNCTIONALLY GRADED PLATES SUBJECTED TO HEAT GENERATION

The two-dimensional steady-state heat transfer problem is analyzed under the most general boundary conditions in a heterogeneous environment. It is assumed that the internal heat generation varies arbitrarily in two directions, while the thermal conductivity coefficient of the material is bi-directly (thickness-length) graded with the Mori-Tanaka model. The linear inhomogeneous partial differential equations produced under these conditions may not be solved by conventional methods. The governing equation of the system is numerically solved with high accuracy using the 2D pseudospectral Chebyshev method. Benchmark problems are used to check the accuracy of the method. It has been observed that the effect of volumetric heat generation on the temperature distribution is more than the heat transmission coefficient. Besides, this method is simple and effective and can be easily applied to such problems.

A Fully Explicit Integrator for Modeling Astrophysical Reactive Flows

Simulating complex astrophysical reacting flows is computationally expensive—reactions are stiff and typically require implicit integration methods. The reaction update is often the most expensive part of a simulation, which motivates the exploration of more economical methods. In this research note, we investigate how the explicit Runge–Kutta–Chebyshev (RKC) method performs compared to an implicit method when applied to astrophysical reactive flows. These integrators are applied to simulations of X-ray bursts arising from unstable thermonuclear burning of accreted fuel on the surface of neutron stars. We show that the RKC method performs with similar accuracy to our traditional implicit integrator, but is more computationally efficient when run on CPUs.

Design of manufacturable variable stiffness composite laminates using spectral Chebyshev and normalized cut segmentation methods

The in-plane fiber orientations of variable stiffness (VS) laminates can be tailored to achieve enhanced structural properties compared to conventional constant stiffness (CS) laminates. However, VS laminate manufacturing faces challenges such as wrinkles, gaps, and overlaps. To address these challenges, we present a novel three-step design methodology. First, the laminate is modeled using lamination parameters (LPs) and the spectral Chebyshev method, and the optimal LPs are determined to maximize the fundamental frequency. Then, the discrete fiber angles are retrieved using the optimal LP distribution. Lastly, a normalized-cut segmentation method is applied to divide the domain into clusters and to generate manufacturable curvilinear fiber paths. Case studies focusing on designing clusters containing both straight and curvilinear fiber paths demonstrate that the designed VS composites can significantly enhance the dynamic performance with up to 20% enhancement in the fundamental frequency compared to CS laminates, under fully clamped boundary conditions with manufacturing constraints.

Sixth-Kind Chebyshev and Bernoulli Polynomial Numerical Methods for Solving Nonlinear Mixed Partial Integrodifferential Equations with Continuous Kernels

In the present paper, a new efficient technique is described for solving nonlinear mixed partial integrodifferential equations with continuous kernels. Using the separation of variables, the nonlinear mixed partial integrodifferential equation is converted to a nonlinear Fredholm integral equation. Then, using different numerical methods, the Bernoulli polynomial method and the Chebyshev polynomials of the sixth kind, the nonlinear Fredholm integral equation has been reduced into a system of nonlinear algebraic equations. The Banach fixed-point theory is utilized in order to have a conversation about the nonlinear mixed integral equation’s solution, namely, its existence and uniqueness. In addition, we talk about the convergence and stability of the solution. Finally, a comparison between the two different methods and some other famous methods is presented through various examples. All the numerical results are calculated and obtained using the Maple software.

Investigation of stresses in a Kevlar (491PR-286) material disc by the Chebyshev pseudospectral method

In this study, a composite disc with Kevlar (491PR-286) material was modeled. Kevlar consists of very strong fibers of very light carbon origin. That is why they are used quite often in unmanned aerial vehicles and spacecraft. The disc has been subjected to thermal stress under a linearly increasing temperature distribution. The temperature limit conditions were applied as 25 °C, 50 °C, 75 °C, 100 °C, and 150 °C. The obtained findings were determined using a computer program, the psedudospectral Chebyshev method, and analytically in three different ways. The main difference between this study and other studies is that it investigates the thermal stresses occurring in circular discs using different methods. The results obtained are compared fairly among themselves and presented with graphs. It was determined that tangential stresses were higher than radial stresses at the studied temperature values. In the analytical study conducted, the radial stresses on the inner and outer surfaces of the disc were determined to be zero for the boundary conditions. Under the increasing temperature distribution from the inner surface to the outer surface, tangential stresses occurred as tensile stress on the inner part of the disc and compressive stress on the outer part. Under the decreasing temperature distribution from the inner surface to the outer surface, tangential stresses are pressed on the inner part of the disc, resulting in a tensile stress on the outer part. It is observed that with increasing temperature, there is an increase in radial and tangential stress values. At the end of the study, it was concluded that Kevlar (491PR-286) material discs can be used at high temperatures.