Homotopy Series Research Articles

Dynamic characteristics of vertically irregular structures with random fields of different probability distributions based on stochastic homotopy method

Considering the uncertainty of elastic modulus in different materials as well as the strong nonlinearity between the structural frequency and elastic modulus, it is a great challenge for computing the dynamic characteristics of vertically irregular structures. In this research, a new stochastic homotopy approach is presented to compute the basic dynamic characteristics of the vertically irregular structures, where the elastic modulus of different materials are assumed as the random fields with different probability distributions. To obtain the dynamic characteristics of the vertically irregular structures, a stochastic eigenvalue equation is established firstly. Then the stochastic eigenvalue and eigenvector are represented by the homotopy series, and the stochastic residual error in relation to the stochastic eigenvalue equation is minimized to obtain the coefficients of the homotopy series. Afterwards, the basic dynamic characteristics, including stochastic natural frequencies and modal shapes, of the vertically irregular structures are obtained. In comparison to the sampling-based stochastic surrogate model methods like the Kriging and non-intrusive arbitrary polynomial chaos methods, the presented approach can efficiently yield more stable statistical moments of the natural frequencies. Meanwhile the proposed method can generate the statistical moments of the modal shapes, which is difficult to achieve with the stochastic surrogate model methods. Finally, the effectiveness of the suggested method is testified through two examples of a reinforced concrete-steel column and a realistic reinforced concrete-steel frame structure.

Static homotopy response analysis of structure with random variables of arbitrary distributions by minimizing stochastic residual error

The modelling of realistic engineering structures with uncertainties often involves various probabilistic distribution types, which bring forward higher requirements for the generality of stochastic analysis methods. This paper proposes a novel stochastic homotopy method to evaluate the static response of the structure with random variables of arbitrary distributions. In this method, a homotopy series expansion is used to approximate the stochastic static response, and the optimal form of the expansion can be determined by minimizing the residual error about the stochastic static equilibrium equation regardless of distribution types of random variables. The numerical results of a logarithmic function and a thin plate show that the new method exhibits excellent accuracy and stability compared to the homotopy stochastic finite element method depending on the sample selection. Compared with the arbitrary polynomial chaos method (aPC), the proposed method is more efficient under equivalent accuracy. On the other hand, as the expansion order increases, this new method shows better convergence than the aPC method and the perturbation stochastic finite element method in the case of non-Gaussian distributed random variables of large fluctuation. In addition, a cable-stayed bridge example illustrates the application of the proposed method on the large-scale structure.

Accurate solutions of a thin rectangular plate deflection under large uniform loading

The large deflection of a thin rectangular plate under uniform loading is a classic problem in solid mechanics. However, the equations are challenging to solve due to their strong nonlinearity. To the author’s best knowledge, existing studies have only reported solutions for rectangular plates with immovable clamped edges within the range of loadings a4q/(Dh)<500. In this study, we employ the homotopy analysis method (HAM) to address this problem and obtain accurate solutions even when the loading a4q/(Dh) reaches as high as 105. Additionally, we discover that the central deflection w(0,0)/h follows a scaling law of [a4q/(Dh)]1/3 for loadings a4q/(Dh)>104, which agrees with the scaling law for extremely large deflections. Consequently, we can utilize this scaling law to extrapolate our homotopy series solution from a4q/(Dh)=104 to even larger loadings. Our investigation also reveals that the bending and membrane stresses increase with the loading. The maximum bending stress is located at the edge, while the location of the maximum membrane stress shifts from the plate center towards the edge as the loading increases. Moreover, we find that even for large deflections, the bending stress remains significant at the edge, challenging the assumption of negligible bending stress adopted by the membrane equations. Thus, the membrane equation may over-predict the plate deflection even under large loadings. These findings advance our understanding of the large deflection of a thin rectangular plate with clamped edges, which has important applications in aerospace, ocean, chemical, and microelectronic engineering. The results presented in this paper can also be used to verify other algorithms designed for highly nonlinear problems.

A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

AbstractThe large fluctuation of uncertain parameters introduces a great challenge in the stability analysis of structures. To address this problem, a novel stochastic residual error based homotopy method is proposed in this article. This new method used the concept of homotopy to reconstruct a new governing equation for stochastic elastic buckling analysis, and the closed‐form solutions of the isolated buckling eigenvalues and eigenvectors are obtained by the stochastic homotopy analysis method. On this basis, apth order origin moment of the stochastic residual error with respect to the elastic buckling equation is defined. Then, the optimal form of the homotopy series can be determined automatically by minimizing thepth order origin moment, which overcomes the disadvantage of highly relying on sample values of the existing homotopy stochastic finite element method. Moreover, the proposed method is developed to deal with the stochastic closely spaced buckling eigenvalue problem. Three mathematical examples and three buckling eigenvalue examples, including a variable cross‐section column, a 7‐story frame, and a Kiewitt single‐layer latticed spherical shell, are performed to illustrate the accuracy and effectiveness of the proposed method by comparing with the existing methods when dealing with large fluctuation of random parameters.

A linearization-based computational algorithm of homotopy analysis method for nonlinear reaction–diffusion systems

In this study, an optimal homotopy analysis algorithm is outlined by means of the nonlinear reaction–diffusion systems. This algorithm, the linearization-based algorithm, employs Taylor series approximations of the nonlinear equations to construct an optimal decomposition of the homotopy series solutions. Numerical comparisons between the proposed algorithm and the standard homotopy approach, as tools for analytically solving reaction–diffusion systems, are performed to test the computational efficiency and the pertinent features of the suggested algorithm. The illustrated numerical results demonstrate that the linearization-based algorithm improves the accuracy and the convergence of the homotopy series solutions. The suggested algorithm can be further used to get rapid convergent series solutions for different types of systems of partial differential equations.

Analytical solution for arbitrary large deflection of geometrically exact beams using the homotopy analysis method

Beam-like compliant elements have found wide-ranging application in many fields of engineering and science often where 3D large deflections can be of concern such as soft robotics, DNA mechanics and helicopter/wind turbine rotor blades. The homotopy analysis method (HAM) is used for the first time to obtain a novel analytical solution in converged series form for the arbitrary large deflection of geometrically exact beams subject to both conservative and follower loading scenarios. The homotopy analysis method, which offers desirable characteristics such as being free from small or large parameters, coupled with auxiliary parameters controlling convergence, is applied directly to the intrinsic governing equations of a geometrically exact beam theory. The system of first-order differential governing equations of geometrically exact beams with intrinsic formulation is free from rotation and displacement variables, and offers a low degree of nonlinearity (quadratic at most) and compact mathematical form, making it suitable for analytical solutions. Due to the relatively poor convergence of the original HAM algorithm, the iterative HAM technique is employed which is known to accelerate convergence and to improve the computational efficiency of the homotopy analysis method. The obtained homotopy series offers a number of novel features in the context of the analytical solutions for the large deflection of beams, including (a) the direct calculation of internal forces and moments which is significant for engineering design purposes, (b) being able to capture 3D deflections, (c) considering transverse shear effects which can be important for thicker beams or when the Young’s modulus to shear modulus ratio is significant (such as composite materials) and (d) considering conservative and follower tip and distributed loads, in a unified framework. In order to investigate the efficacy, applicability and accuracy of the proposed method, a number of numerical examples are considered in which a cantilever beam subject to tip or distributed loads undergoes large deflection. Large deflection results for both conservative and follower loads are compared against those of less comprehensive analytical solutions as well as against numerical methods including finite element and Chebyshev collocation methods where good agreement is observed. These results demonstrate the applicability and effectiveness of HAM for the large deflection analysis of geometrically exact beams.

A new homotopy approach for stochastic static model updating with large uncertain measurement errors

Large measurement errors are a major challenge in structural model updating. Based on the concept of homotopy, a new stochastic static model updating method is proposed to update structural models using uncertain static data with large measurement errors. First, considering the uncertainty of the static measurement, a stochastic model updating equation for element update factors is set up. To solve the stochastic model updating equation, a series of homotopy deformation equations are presented to establish the relationship between the deterministic update factors and the random update factors. Furthermore, the homotopy series expansions of the random update factors can be determined by solving the homotopy deformation equations. Since the measured degrees of freedom of updated structures are usually limited or unavailable, a static condensation technique is used for stochastic model updating. To address the ill-posed problems caused by incomplete measurement information and static measurement errors, the Tikhonov regularization method is used in the process of solving the homotopy deformation equations. Three numerical examples are given to demonstrate the validity of the proposed stochastic model updating method. The numerical results clearly show that unlike the second-order perturbation method, this new method can produce better accuracy in cases with large measurement errors. Compared with the Bayesian method with the Delayed Rejection Adaptive Metropolis sampling technique and even a fast Bayesian method, the proposed method utilizes much less computational time and provides an equivalent accuracy. Finally, static loading experiments on a two-span continuous concrete beam are implemented to validate the proposed model updating method.

An analytic modeling strategy for memristor cell applicable to large-scale memristive networks

Memristive networks are large-scale non-linear circuits based on memristor cells, playing a crucial role in developing the emerging researches such as next-generation artificial intelligence, bioelectronics, and high-performance memory. The performance of memristive networks is greatly affected by the memristor model describing physical and electrical characteristics of a memristor cell. However, existing models are mainly non-analytic and, accordingly, may have convergence issues in their applications in memristive networks’ analyses. Therefore, aiming at improving convergence of memristive networks, we propose an analytic modeling strategy for memristor based on homotopy analysis method (HAM). In this strategy, the HAM is used to obtain an analytic memristor model through solving the state equations of memristors in original physical model. Specifically, the HAM is used to solve the analytic approximate solution of the core parameter of memristor—state variable, from the state equations, in the form of analytic homotopy series. Then the analytic approximate model of memristor is obtained by using the solved state variables. The characteristics of the proposed strategy are as follows. 1) Its solution has a closed-form expression, i.e. an explicit function, 2) its approximation error is optimized, thereby realizing the convergence optimization. Moreover, according to the characteristics of memristive networks, we introduce an analysis criterion for memristor model applicable to memristive networks. Through the long-time evolution experiments of a memristor cell and a benchmark memristive matrix network with different inputs, and the comparisons with the traditional non-analytic (numeric) method, we verify the analyticity and convergence superiority of the modeling strategy. Besides, based on this strategy and the comparison experiments, we reveal that one of the underlying reasons for non-convergence in the large-scale memristive network simulation possesses the non-analyticity of the used memristor model. The strategy can be further used for analyzing the performances of a memristor cell and memristive networks in long-time. It also has potential applications in emerging technologies.

Non-similar Solution of Eyring–Powell Fluid Flow and Heat Transfer with Convective Boundary Condition: Homotopy Analysis Method

The study presents the mixed convective boundary layer flow of Eyring–Powell fluid over a vertical plate with variable velocity and temperature distribution taking convective boundary condition into account. The transformed non-dimensional governing equations in non-similar nature are solved by employing hybrid technique: local non-similarity method in conjunction with homotopy analysis method. The convergence of homotopy series solution is obtained and presented for various order of approximations. The series solutions have been validated by comparing the results available in the literature and found good agreement. The obtained results are shown properly by graphs and discussed for various values of thermo-physical parameters. It is found that the Eyring–Powell fluid shows higher velocity than Newtonian fluid whereas lower temperature than Newtonian fluid. Furthermore as increase in fluid parameter, ratio of relaxation and retardation parameter, skin friction coefficient decreased and heat transfer coefficient increased. The study finds wide applications in the field of design of heat exchangers, process of cooling of metallic plate, extrusion of plastic sheets, in polymer and glass industries etc.

Hydromagnetic Falkner-Skan fluid rheology with heat transfer properties

This article addresses the effects of heat transfer on magnetohydrodynamic Falkner-Skan wedge flow of a Jeffery fluid. The continuity, momentum and energy balance equations yield the relevant PDE which are transforms to ODE by exploitation of similarity variables. Strength of optimal homotopy series solutions is practiced to solved analytically the transformed ODE model of hydromagnetic Falkner-Skan fluid rheology with heat transfer scenarios. The graphical and numerical illustrations of the result are presented for different interesting flow parameters. Numerical values of Nusselt number are tabulated. It is observed that for the Falkner-Skan rheology, the applied magnetic field acts as a controlling agnet which controls the fluids velocity up to the desired value whereas Debrorah number enhances the fluid velocity.

Influence of chemically radiative nanoparticles on flow of Maxwell electrically conducting fluid over a convectively heated exponential stretching sheet

PurposeThis paper aims to examine the influence of radiative nanoparticles on incompressible electrically conducting upper convected Maxwell fluid (rate type fluid) flow over a convectively heated exponential stretching sheet with suction/injection in the presence of heat source taking chemical reaction into account. Also, a comparison of the flow behavior of Newtonian and Maxwell fluid containing nanoparticles under the effect of different thermophysical parameters is elaborated. Velocity, temperature and nanoparticle volume fractions are assumed to have exponential distribution at boundary. Buongiorno model is considered for nanofluid transport.Design/methodology/approachThe equations, which govern the flow, are reduced to ordinary differential equations using suitable transformation. The transformed equations are solved using a robust homotopy analysis method. The convergence of the homotopy series solution is explicitly discussed. The present results are compared with the results reported in the literature and are found to be in good agreement.FindingsIt is observed from the present study that larger relaxation time leads to slower recovery, which results in a decrease in velocity, whereas temperature and nanoparticle volume fraction is increased. Maxwell nanofluid has lower velocity with higher temperature and nanoparticle volume fraction when compared with Newtonian counterpart. Also, the presence of magnetic field leads to decrease the velocity of the nanofluid and enhances the skin coefficient friction. The existence of thermal radiation and heat source enhance the temperature. Further, the presence of chemical reaction leads to decrease in nanoparticle volume fraction. Higher value of Deborah number results in lower the rate of heat and mass transfer.Originality/valueThe novelty of present work lies in understanding the impact of fluid elasticity and radiative nanoparticles on the flow over convectively heated exponentially boundary surface in the presence of a magnetic field using homotopy analysis method. The current results may help in designing electronic and industrial applicants. The present outputs have not been considered elsewhere.

Estimating the approximate analytical solution of HIV viral dynamic model by using homotopy analysis method

Abstract Viruses have different mechanisms in causing a disease in an organism, which largely depend on the viral species. The recent advancement, through coupling data analysis and mathematical modeling, has allowed the identification and characterization of the nature of the virus. In the present paper, the homotopy analysis method is applied to provide an approximate solution of the basic HIV viral dynamic model describing the viral dynamics in a susceptible population. The proposed method allows for the solution of the governing system of differential equations to be calculated in the form of an infinite series with components which can be easily calculated. The homotopy analysis method utilizes a simple method to adjust and control the convergence region of the infinite series solution by using an auxiliary parameter. By using the homotopy series solutions, firstly, several β − curves using an appropriate ratio are plotted to demonstrate the regions of convergence and the optimum value of ℏ, then the residual and absolute errors are obtained for different values of these regions. Secondly, the residual error functions are applied to show the accuracy of the applied homotopy analysis method. Also, the convergence theorem of homotopy analysis method for the HIV viral dynamic model is proved. Mathematica software is used for the calculations and numerical results. The results obtained show the effectiveness and strength of the homotopy analysis method.

Approximate solution of imbibition phenomenon arising in heterogeneous porous media by optimal homotopy analysis method

This paper deals with approximate homotopy series solution of imbibition phenomenon occurring in multiphase flow during the secondary oil recovery process. In heterogeneous porous media, the geometry of pores is irregular while in homogeneous porous media, the geometry of pores is uniformly same. The comparative study of counter-current imbibition phenomenon in heterogeneous and homogeneous porous medium has been also discussed. The governing partial differential equation obtained by mathematical formation of imbibition phenomenon has been solved by the optimal homotopy analysis method. The numerical as well as graphical interpretation of the solution have been given.

Surprising solutions for some challenging problems arising from boundary layer theory by a new technique: the homotopy contraction mapping technique (HCMT)

In this work, the homotopy structure $$ (1 - p)L\,[f] = - p\,N\,[f] $$ is considered to solve some newly developed nonlinear problems in fluid dynamics. In summary, deformation in homotopy series solutions occurs from the initial trial function to the actual solution. The main challenging part in many similarity equations associated with the boundary layer theory is to identify a certain quantity of the exact solution such as $$ f^{{\prime \prime }} (0) $$ . This quantity is initially unknown and is automatically guessed through the zeroth function. Therefore, it makes sense that in the previous homotopy series solutions one would need at least another parameter i.e. the controlling parameter ( $$ \hbar $$ ) to handle the convergence through the so-called $$ \hbar $$ -curves analysis and eventually get an approximate value for $$ f^{{\prime \prime }} (0) $$ . Here it is accounted the basic homotopy structure with No controlling parameter ( $$ \hbar $$ ) and it is shown for the 1st time that the zeroth order solution in homotopy series may be potential to contain some certain quantities of the exact solution, here is to be $$ f^{{\prime \prime }} (0) $$ ; i.e. $$ f_{0}^{{\prime \prime }} (0) $$ could be $$ f^{{\prime \prime }} (0) $$ . This hypothesis is checked through a theorem, being totally linked to the Fixed Point Property (FPP), a topological invariant; i.e. being preserved by any homeomorphism. The theorem enables us Not only to check the validity of the hypothesis, but also to extract the value of the quantity as the unique description of the homotopy series solutions truncated at any order of approximation. The technique is initially introduced in a simple and straightforward manner and then it is applied to some fluid mechanical problems to be further compared with traditional homotopy analysis method, homotopy perturbation method and Numerical solutions. The outcomes indicated an excellent improvement by the present approach as only a few series terms were accounted to obtain highly accurate solutions.

A linearization‐based approach of homotopy analysis method for non‐linear time‐fractional parabolic PDEs

In this paper, a novel approach, namely, the linearization‐based approach of homotopy analysis method, to analytically treat non‐linear time‐fractional PDEs is proposed. The presented approach suggests a new optimized structure of the homotopy series solution based on a linear approximation of the non‐linear problem. A comparative study between the proposed approach and standard homotopy analysis approach is illustrated by solving two examples involving non‐linear time‐fractional parabolic PDEs. The performed numerical simulations demonstrate that the linearization‐based approach reduces the computational complexity and improves the performance of the homotopy analysis method.

A new homotopy-based approach for structural stochastic analysis

This paper presents a homotopy-based method to calculate the stochastic responses of a structure under static loads involving uncertainties in materials, external loads, and structural geometries. The proposed method is based on the homotopy analysis method. In this method, the stochastic responses are represented by an infinite multivariate homotopy series of the involved random variables, and all the deterministic coefficients in the multivariate series are determined through solving a series of various order of deformation equations. This homotopy series solution obtained has a relatively large convergence domain due to the approach function compared with the Taylor series, which makes the series solution independent of random parameters with small fluctuation. Further, two dimension reduction approximations are proposed to improve the computational efficiency of the solution. Five mathematical functions and three solid-mechanics examples show that the proposed method can produce very accurate results with reduced or similar computational efforts compared with the existing methods.

Exact Analytical Nanofluid Flow and Heat Transfer Involving Asymmetric Wall Heat Fluxes with Nonlinear Velocity Slip

Nanofluid slip flow and heat transfer over the microchannel with asymmetric wall heat fluxes are investigated theoretically. The local thermal nonequilibrium model in which there is a difference between the solid temperature and nanofluid temperature is modified by considering the nonlinear velocity slip and temperature jump. Exact analytical solutions and homotopy series solutions for both velocity and temperature of nanofluid and solid are first obtained. Two groups of solutions agree well with existing references and residual errors are plotted. In addition, the effects of the physical factors on the heat transfer are graphically discussed. The results show that the first velocity slip enhances Nusselt numbers while the second slip coefficient has a reverse effect on them. As expected, the nanoparticle concentration enhances heat transfer on bottom wall.

Theoretical investigation of the doubly stratified flow of an Eyring-Powell nanomaterial via heat generation/absorption

.The mixed convective flow of an Eyring-Powell nanomaterial in a doubly stratified medium is addressed in this paper. The stretching surface has varying thickness. The nanofluid model given by Buongiorno is utilized in the formulation of energy and concentration expressions. Heat generation is also retained. Ordinary differential systems are obtained by utilizing the transformations procedure. Homotopy series solutions containing exponentially functions are developed. Significant characteristics of influential variables for velocity, temperature, nanoparticle concentration, skin friction coefficient and Nusselt and Sherwood numbers are reported through graphs and tables. It is found that stratification phenomenon leads to a decay in temperature and nanoparticle concentration.

The onset of modified Fourier and Fick’s theories in temperature-dependent conductivity flow of micropolar liquid

Nonlinear convective stagnation point flow of micropolar liquid is modeled. Stretching surface has variable thickness. Energy and concentration are modeled with modified Fourier and Fick’s relations. Temperature-dependent thermal conductivity is considered. Transformations are utilized in obtaining ordinary differential systems. Homotopy series solutions comprising exponentially declining functions are established. Salient features of various parameters for velocity, temperature and skin friction coefficient are addressed through graphs. It is found that both velocities are increasing function of material parameter. Moreover thermal and solutal relaxation time factors have effective contributions in adjusting the chilling process of the stretchable surface which is important in several industrial applications.

Homotopy Series Solutions to Time-Space Fractional Coupled Systems

We apply the homotopy perturbation Sumudu transform method (HPSTM) to the time-space fractional coupled systems in the sense of Riemann-Liouville fractional integral and Caputo derivative. The HPSTM is a combination of Sumudu transform and homotopy perturbation method, which can be easily handled with nonlinear coupled system. We apply the method to the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The simplicity and validity of the method can be shown by the applications and the numerical results.