Chebyshev Pseudospectral Method Research Articles

Stability examination of non-linear convection flow with partial slip phenomenon in a Riga plate channel

Abstract The present work examines the linear stability of non-linear convected flow inside a Riga plate channel. The channel is filled with hybrid nanoliquid and is under the novel influence of the partial slip phenomenon in the present scenario. The left domain of the channel is supported by the Riga sheet whereas the right part is bounded by a sheet of slippery nature. The stability model for this partial slip mechanism is developed in the form of an eigenvalue problem which is explored via the Chebyshev pseudospectral method
in combination with the QZ-algorithm. It is reported that the convection forces in hybrid nanofluid are amplified with Riga magnetic number (Hr) under slip/no-slip assumptions. It is interestingly noted that the flow is destabilized by 11.47% with non-linear convection (Nc) when considering no-slip at the right-hand sheet. However, the stability region is enlarged with Nc by 9.53% in the presence of slip at the right-hand sheet. The partial-slip (γ) assumption in the channel decelerates the growth rate of disturbances. The increment in Fe3O4- nanoparticles over the fixed volume of CoFe2O4-nanoparticles hampers the instability of the hybrid nanofluid mixture.

Chebyshev Pseudospectral Method for Fractional Differential Equations in Non-Overlapping Partitioned Domains

Chebyshev Pseudospectral Method for Fractional Differential Equations in Non-Overlapping Partitioned Domains

An approximate solution of the Blasius problem using spectral method

This paper aims at finding the numerical approximation of a classical Blasius flat plate problem using spectral collocation method. This technique is based on Chebyshev pseudospectral approach that involves the solution is approximated using Chebyshev polynomials, which are orthogonal polynomials defined on the interval [−1, 1]. The Chebyshev pseudospectral method employs Chebyshev- Gauss- Lobatto points, the extrema of the Chebyshev polynomials. The differential equation is approximated as a sum of Chebyshev polynomials. A differentiation matrix, based on these polynomials and their derivatives at the collocation points, transforms the differential equation into a system of algebraic equations. By evaluating the differential equation at these points and applying boundary conditions, the original boundary value problem reduced the solution to the solution of a system of algebraic equations. Solving for the coefficients of the polynomials yields the numerical approximation of the solution. The implementation of this method is carried out in Mathematica and its validity is ensured by comparing it with a built in MATLAB numerical routine called bvp4c.

Iterative Chebyshev approximation method for optimal control problems

We present a novel numerical approach for solving nonlinear constrained optimal control problems (NCOCPs). Instead of directly solving the NCOCPs, we start by linearizing the constraints and dynamic system, which results in a sequence of sub-problems. For each sub-problem, we use finite number of Chebyshev polynomials to estimate the control and state vectors. To eliminate the errors at non-collocation points caused by conventional collocation methods, we additionally estimate the coefficient functions involved in the linear constraints and dynamic system by Chebyshev polynomials. By leveraging the characteristics of Chebyshev polynomials, the approximate sub-problem is changed into an equivalent nonlinear optimization problem with linear equality constraints. Consequently, any feasible point of the approximate sub-problem will satisfy the constraints and dynamic system throughout the entire time scale. To validate the efficacy of the new method, we solve three examples and assess the accuracy of the method through the computation of its approximation error. Numerical results obtained show that our approach achieves lower approximation error when compared to the Chebyshev pseudo-spectral method. The proposed method is particularly suitable for scenarios that require high-precision approximation, such as aerospace and precision instrument production.

Novel Numerical Investigations of Some Problems Based on the Darcy–Forchheimer Model and Heat Transfer

In this study, we introduced a new type of basis function and subsequently a Chebyshev delta shaped collocation method (CDSC). We then use this method to numerically investigate both the natural convective flow and heat transfer of nanofluids in a vertical rectangular duct on the basis of a Darcy–Brinkman–Forchheimer model and the electroosmosis-modulated Darcy–Forchheimer flow of Casson nanofluid over stretching sheets with Newtonian heating problems. The approximate solution is represented in terms of Chebyshev delta shaped basis functions. Novel error estimates for interpolating polynomials are derived. Computational experiments were carried out to corroborate the theoretical results and to compare the present method with the existing Chebyshev pseudospectral method. To demonstrate our proposed approach, we also compared the numerical solutions with analytic solutions of the Poisson equation. Computer simulations show that the proposed method is computationally cheap, fast, and spectrally accurate and more importantly the obtained approximate solution can easily be used by researchers in this field.

A Reentry Trajectory Planning Algorithm via Pseudo-Spectral Convexification and Method of Multipliers

The reentry trajectory planning problem of hypersonic vehicles is generally a continuous and nonconvex optimization problem, and it constitutes a critical challenge within the field of aerospace engineering. In this paper, an improved sequential convexification algorithm is proposed to solve it and achieve online trajectory planning. In the proposed algorithm, the Chebyshev pseudo-spectral method with high-accuracy approximation performance is first employed to discretize the continuous dynamic equations. Subsequently, based on the multipliers and linearization methods, the original nonconvex trajectory planning problem is transformed into a series of relaxed convex subproblems in the form of an augmented Lagrange function. Then, the interior point method is utilized to iteratively solve the relaxed convex subproblem until the expected convergence precision is achieved. The convex-optimization-based and multipliers methods guarantee the promotion of fast convergence precision, making it suitable for online trajectory planning applications. Finally, numerical simulations are conducted to verify the performance of the proposed algorithm. The simulation results show that the algorithm possesses better convergence performance, and the solution time can reach the level of seconds, which is more than 97% less than nonlinear programming algorithms, such as the sequential quadratic programming algorithm.

Reentry trajectory planning for hypersonic vehicles via an improved sequential convex programming method

The reentry trajectory planning problem of hypersonic vehicles is generally a continuous and nonconvex optimization problem. The solution efficiency and convergence properties of the planning algorithm become an essential point when considering the practical application scenarios. In this paper, an augmented-Lagrange-based improved sequential convex programming algorithm is proposed to solve it and achieve superior solutions with higher accuracy and efficiency. Firstly, the mapped Chebyshev pseudo-spectral method is utilized to convert the continuous optimal control problem into a highly approximately discretized one. In this framework, the discretization technique is employed to discretize the states and control variables at the time interval, which leads to the improvement of ill-conditioned differential matrix and better approximation properties of the variables. Moreover, by introducing slack variables, automatically updated penalty parameters and Lagrange multipliers during the convexification processing, the original nonconvex problem is transformed into an augmented-Lagrange-function-formed relaxed convex one. Then, the arrived convex subproblem can be solved iteratively by the matured interior-point method until the predetermined precision is satisfied. Based on this transformation, it can effectively alleviate the numerical difficulty and improve the convergence property of the proposed improved sequential convex programming algorithm. Finally, Rigorous theoretical analysis is conducted to prove that the solution sequence generated by the proposed algorithm could converge to an approximate global solution of the original problem. Numerical simulations are also presented, and it is demonstrated that the solution time of the proposed algorithm is more than 95% less than that of the traditional nonlinear programming methods without losing the merits of high precision.

Novel Numerical Investigation of Reaction Diffusion Equation Arising in Oil Price Modeling

Consideration is given to a reaction–diffusion free boundary value problem with one or two turning points arising in oil price modeling. First, an exact (analytical) solution to the reduced problem (i.e., no diffusion term) was obtained for some given parameters. The space–time Chebyshev pseudospectral and superconsistent Chebyshev collocation method is proposed for both reaction diffusion (RDFBP) and reduced free boundary value problem. Error bounds on the discrete L2–norm and Sobolev norm (Hp) are presented. Adaptively graded intervals were introduced and used according to the value of turning points to avoid the twin boundary layers phenomena. Excellent convergent (spectrally) and stable results for some special turning points were obtained for both reduced and RDFBP equations on an adaptively graded interval and this has been documented for the first time.

Frequency Design of a Vertical Cantilever Beam Carrying Tip Mass

Previous research has revealed that vertical cantilever beam structures, after considering the effects of gravity, can exhibit a 1:3 internal resonance between two transverse modes. However, whether a similar phenomenon would still occur in the system when a proof body is attached to the free end of cantilever beams, and how to perform parameter design to deliberately induce internal resonance are still open issues. The present study conducts research on the natural frequencies of a vertical cantilever beam with a tip mass. Within the framework of the Euler–Bernoulli beam theory, Hamilton’s principle is utilized to derive the integral-partial differential equation of free vibration of the system. The corresponding frequency equation is then solved using the Chebyshev pseudospectral method, with an algebraic equation of the natural frequency obtained. Moreover, an explicit analytical expression for the fundamental frequency is obtained using parameter fitting, which is then used to reveal the relationship among parameters under the critical state of buckling and evaluate the extent of gravity’s impact on the fundamental frequency. The study finally investigates the ratio of the first two natural frequencies. A parameter design method is proposed, which derives a parametric curve in terms of the fundamental frequency, ensuring a 1:3 ratio between the first two natural frequencies. The free vibration time history at the midpoint of the beam is acquired by directly solving the system’s partial differential equation through the finite element method. Fourier transformation of this time history verifies that the fundamental frequency aligns with a predetermined value, and the ratio between the first two natural frequencies is 1:3, thus confirming the efficacy of the proposed method. It is found that the addition of a tip mass provides more flexibility in adjusting parameters to achieve internal resonance, which facilitates the miniaturization of actual structures.

Global nonlinear stability of bidispersive porous convection with throughflow and depth-dependent viscosity

The linear instability and the nonlinear stability analyses have been performed to examine the combined impact of a uniform vertical throughflow and a depth-dependent viscosity on bidispersive porous convection using the Darcy theory with a single temperature field. The validity of the principle of exchange of stability is proved. The eigenvalue problems resulting from both linear instability and nonlinear stability analyses with variable coefficients are numerically solved using the Chebyshev pseudo-spectral method. The equivalence of linear instability and nonlinear stability boundaries is established in the absence of throughflow, while in its presence, the subcritical instability is shown to be evident. The stability of the system is independent of the direction of throughflow in the case of constant viscosity, whereas upflow is found to be more stabilizing than downflow when the viscosity is varying with depth. While the viscosity parameter offers a destabilizing influence on the onset of convection in the absence of throughflow, it imparts both stabilizing and destabilizing effects on the same in its presence. The influence of the ratio of permeabilities and the interphase momentum transfer parameter is to make the system more stable. The findings of a mono-disperse porous medium are presented as a specific case within the broader context of this investigation.

Study of conduction heat transfer in semi-elliptic cross sections using analytical and bivariate Chebyshev pseudospectral methods

Study of conduction heat transfer in semi-elliptic cross sections using analytical and bivariate Chebyshev pseudospectral methods

The onset of anisotropic Darcy–Brinkman penetrative convection with local thermal nonequilibrium due to heat generation and extraction

AbstractAn anisotropic, fluid‐saturated porous medium with local thermal nonequilibrium is the subject of this paper's analysis of the stability of the commencement of convection caused by internal heat generation. The classical Darcy's momentum equation is applied with the Brinkman model. To observe the thermal instabilities in the fluid layer, it is assumed that there is steady heat extraction through the bottom boundary and constant heat generation inside the fluid layer. The critical internal Darcy Rayleigh and associated wave numbers are computed using linear instability and energy approaches using the Chebyshev pseudospectral method to determine the existence of instability or otherwise. The fluid layer becomes unstratified as a result of heat generation inside the fluid, which encourages the occurrence of any subcritical instability. The critical Darcy–Rayleigh number has been found to be significantly influenced by a number of parameters, including the anisotropy parameter, internal heat parameter, porosity‐modified conductivity ratio, inter‐phase heat transfer parameter, and Darcy–Brinkman number.

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.

Numerically efficient full-vectorial rational Chebyshev pseudo-spectral modal analysis for optical Waveguides

In this paper, an efficient full-vectorial modal analysis based on the rational Chebyshev pseudo-spectral method (V-RCPSM) is introduced to analyze 3 dimensional (3D) structures that are invariant along one spatial variable. Such structures are essential in silicon photonics and plasmonics applications where permittivity profiles with high-index contrast need precise treatment of the interface boundary conditions. Besides, such structures are open in general. Hence, good domain truncation is important. Our method handles these challenges via hybrid usage of the domain decomposition technique where the electromagnetic field is expanded in terms of Chebyshev functions in homogeneous regions, while the rational Chebyshev functions are used for semi-infinite homogeneous domains. The boundary conditions are rigorously imposed along the interfaces, a step that maintains the known exponential convergence rate of Chebyshev functions. Chebyshev functions have the ability to capture the correct rapid variation of the electromagnetic fields at the interfaces of the high-index-contrast waveguides using only a few basis functions; a critical feature for accurate mode computation. To show the accuracy and efficiency of our new approach, we studied rib and plasmonic waveguides and compared the results with those obtained using other full-vectorial approaches such as the finite elements method (FEM). Our developed approach has achieved a huge reduction in computational resources over the FEM.

Flow and heat transfer for MHD wall jet of hybrid nanofluids: theoretical and numerical solutions

PurposeThis paper aims to study the magnetohydrodynamic (MHD) wall jet of a hybrid nanofluid flow over a moving surface with a thermally convective surface, wall moving with suction/injection.Design/methodology/approachOn using appropriate similarity transformations, the governing equations that describe the model are converted into a system of nonlinear ordinary differential equations. These equations are solved both analytically and numerically using standard two-point boundary-value problem solvers and Chebyshev pseudospectral differentiation matrix method, respectively.FindingsThese results show that the HNF is heating/cooling with growth of the positive/negative values of the parameter measuring the velocity of the moving surface. The temperature distributions increase, where the thermal boundary layer gets thicker, as the magnetic field strengthens and with an increase in the absolute value of the Biot number.Originality/valueThe current findings for the HNFs are new and original. They generalize successfully the problems investigated previously by different researchers for the cases of fluids and also nanofluids.

Numerical scrutinization of micropolar and Walters-B non-Newtonian fluids motion under the influence of thermal radiation and chemical reaction

Purpose The purpose of this study is to explore numerical scrutinization of micropolar and Walters-B non-Newtonian fluids motion under the influence of thermal radiation and chemical reaction. Design/methodology/approach The two fluids micropolar and Walters-B liquid are considered to start flowing from the slot to the stretching sheet. A magnetic field of constant strength is imposed on their flow transversely. The problems on heat and mass transport are set up with thermal, chemical reaction, heat generation, etc. to form partial differential equations. These equations were simplified into a dimensionless form and solved using spectral homotopy analysis method (SHAM). SHAM uses the basic concept of both Chebyshev pseudospectral method and homotopy analysis method to obtain numerical computations of the problem. Findings The outcomes for encountered flow parameters for temperature, velocity and concentration are presented with the aid of figures. It is observed that both the velocity and angular velocity of micropolar and Walters-B and thermal boundary layers increase with increase in the thermal radiation parameter. The decrease in velocity and decrease in angular velocity occurred are a result of increase in chemical reaction. It is hoped that the present study will enhance the understanding of boundary layer flow of micropolar and Walters-B non-Newtonian fluid under the influences of thermal radiation, thermal conductivity and chemical reaction as applied in various engineering processes. Originality/value All results are presented graphically and all physical quantities are computed and tabulated.

Constrained nonlinear high-efficiency model predictive technics for test mass capture

This paper presents a nonlinear high-efficiency model predictive control (NHMPC) with constraints designed for the test mass (TM) capture phase of the drag-free satellite about gravitational wave observatory. To avoid collisions between test mass and satellite cavity, TM capture is the essential technology for drag-free satellite. Test mass is located inside an electrostatic suspension and locked by a clamp mechanism initially. The test masses are released with high initial offsets and velocities when the mechanism is retracted. The purpose of this phase is to guarantee the TM to be positioned at the cage center and attitude aligned with the local cage frame. Due to the low actuation authority of electrostatic suspension along with critical initial offsets and velocities, it is a challenging task to design the attitude and translation control schemes with simultaneous consideration of system performance and energy consumption. For the problem given above, the TM capture can be reformulated into a nonlinear quadratic optimal control problem with the state and input constraints. A nonlinear model predictive control (NMPC) structure is also designed to handle the noises by forming a closed loop. This control framework can realize optimality and robustness in a compromise. To improve the speed of solving high-dimensional nonlinear optimal control online for MPC, the indirect Chebyshev pseudospectral method with constraints is employed. The convergence and stability of the control loop are analyzed. Simulation examples show that the feasibility and performance of the designed control loop are verified compared with the existing methods and computation times in the millisecond range can be achieved.

Thermorheological effect on Rayleigh–Bénard magnetoconvection in a biviscous Bingham fluid with rough boundary condition on velocity and Robin boundary condition on temperature

AbstractThe thermorheological effect on the onset of Rayleigh–Bénard convection in a biviscous Bingham fluid in the presence of a horizontal magnetic field is investigated considering rough boundary conditions on velocity and Robin boundary conditions on temperature. The viscosity of the electrically conducting fluid is assumed to be sensitive to temperature variation. Linear and global nonlinear stability analyses are performed using the Chebyshev pseudospectral method to determine the existence of instability or otherwise. A general interpretation is made from the results to show the effects of the magnetic field and the variable viscosity on the system's stability. The biviscous Bingham parameter and the Chandrasekhar number are shown to have a delay in the onset of convection, while the effect of temperature sensitivity is to advance the onset. It is found that the results of linear and global nonlinear stability are not in agreement, so the region of subcritical instability exists. Also, the results obtained for Rayleigh–Bénard convection agree pretty well with those of Platten and Legros and Siddheshwar et al. for the limiting cases.

Numerical Simulations of the Fractional Systems of Volterra Integral Equations within the Chebyshev Pseudo-Spectral Method

In this article, we find the solutions to fractional Volterra-type integral equation nonlinear systems through a Chebyshev pseudo-spectral method (CPM). The fractional derivative is described in the Caputo manner. The suggested method’s accuracy and reliability are confirmed by the results. The proposed method is implemented for solving various nonlinear systems; the results we obtained were compared with the exact solution and other method solutions. The graphical representation and tables show that our method’s error quickly converges as compared to other methods. By comparing the proposed method’s solution with the actual solution and other methods, we can confirm that CPM is more accurate and closer to the exact solution. We display the pointwise solution in the tables, which verifies the proposed method’s accuracy at each point and aids in a better comprehension of the suggested approach. Moreover, the results of using the suggested method at different fractional orders are examined, showing that when a value moves from a fractional order to an integer order, the result is closer to the precise solution. Furthermore, the proposed technique for handling fractional-order linear and non-linear physical problems in science and engineering is straightforward to implement.

Thermal Convection for an Oldroyd-B Fluid in an Anisotropic Porous Medium Underlying a Fluid Layer

Abstract We investigate the thermal convection in a fluid layer overlying an anisotropic porous medium with Oldroyd-B fluids in the paper. The linear stability analysis and the Chebyshev pseudo-spectral method are used to obtain the numerical results. We found that the neutral curves are bimodal at some depth ratios. The proper values of the permeability ratio and the thermal diffusivity ratio are found to stabilize the stationary convection and the oscillatory convection. Furthermore, the variation of permeability ratio does not affect the bimodal nature of the neutral curves. In addition, the increase in the stress relaxation time has an unstable effect on the oscillatory convection, but the strain retardation time has the opposite effect.