Homotopy Approach Research Articles

Study on a tangent hyperbolic thermal fluid flow over a porous stretching sheet with a magnetic field and the effect of suction/injection

This work examines a steady and incompressible flow of 2-dimensional MHD tangent hyperbolic fluid over a linearly stretchable surface with an injection/suction effect. It also investigates the impact of thermal radiation, heat source/sink, a porous medium and variable thermal conductivity. The slippery boundary conditions are employed to consider the presence of velocity slip near the surface. The Lie symmetric analysis with a novel homotopy approach is used to study the influence of non-dimensional characteristics, including the Weissenberg number, the Hartmann number, the power-law index, the Prandtl number, the porosity parameter and the heat absorption/generation parameter, on the fluid flow and temperature. Moreover, the numerical data include an examination of the influence of various parameters on both the surface drag force and the heat transmission rate, presented through graphs and tables. Ultimately, the acquired outcomes have been numerically verified with the pre-existing results. Finally, the heat transmission rate escalates with the augmentation of the Prandtl number and the variable thermal conductivity parameter.

Modified homotopy approach for diffractive production in the saturation region

In this paper we continue to develop the homotopy method for solving of the nonlinear evolution equation for the diffractive production in deep inelastic scattering. We introduce part of the nonlinear corrections as a first step of this approach. This simplified nonlinear evolution equation is solved analytically taking into account the initial and boundary conditions for the process. At the second step of our approach we demonstrated that the perturbative procedure can be used for the remaining parts of the nonlinear corrections. It turns out that these corrections are small and can be estimated in the regular iterative procedure. Published by the American Physical Society 2024

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.

Examining stagnation-point flow impinging on a deforming cylinder in Reiner-Rivlin fluid with integrated heat and mass transfer

Here the Reiner-Rivlin model is applied to explore non-Newtonian flow in stagnation-point region around a stretching cylinder. Under the influence of viscous heating and thermos-diffusion, the phenomena of combined heat and mass transport are investigated. To explore the intrinsic irreversible nature of transport phenomena, the current work incorporates an analysis of entropy generation. Additionally, prescribed wall temperature and concentration are taken into consideration when developing the computational framework, and it will be shown later that these assumptions are essential to achieving self-similar analysis. Numerical computations are developed by the MATLAB tool “bvp4c” while analytical solutions are found by the optimal homotopy approach embedded in the routine “BVPh 2.0” of MATHEMATICA. For a reasonable choice of embedded parameters, both techniques produce the same numerical results. For various regulating parameters, the importance of curvature effects on the boundary layers is explained. Flow visualization utilizing streamlines and isotherms reveals interesting patterns for both Newtonian and non-Newtonian flows. The force required by the stretching cylinder is analyzed by evaluating skin friction coefficient, in the case of Reiner-Rivlin fluid. Furthermore, surface cooling rate and mass flux are scrutinized graphically and in tabular form. The cooling rate of the cylindrical boundary is higher than that of the flat plate boundary. Furthermore, a noticeable reduction in surface cooling rate is found for increasing values of Reiner-Rivlin fluid parameter.

Topology optimization for magnetic circuits with continuous adjoint method in 3D

PurposeThe purpose of this paper is to present a new approach to optimizing the design of 3D magnetic circuits. This approach is based on topology optimization, where derivative calculations are performed using the continuous adjoint method. Thus, the continuous adjoint method for magnetostatics has to be developed in 3D and has to be combined with penalization, filtering and homotopy approaches to provide an efficient optimization code.Design/methodology/approachTo provide this new topology optimization code, this study starts from 2D magnetostatic results to perform the sensitivity analysis, and this approach is extended to 3D. From this sensitivity analysis, the continuous adjoint method is derived to compute the gradient of an objective function of a 3D topological optimization design problem. From this result, this design problem is discretized and can then be solved by finite element software. Thus, by adding the solid isotropic material with penalization (SIMP) penalization approach and developing a homotopy-based optimization algorithm, an interesting means for designing 3D magnetic circuits is provided.FindingsIn this paper, the 3D continuous adjoint method for magnetostatic problems involving an objective least-squares function is presented. Based on 2D results, new theoretical results for developing sensitivity analysis in 3D taking into account different parameters including the ferromagnetic material, the current density and the magnetization are provided. Then, by discretizing, filtering and penalizing using SIMP approaches, a topology optimization code has been derived to address only the ferromagnetic material parameters. Based on this efficient gradient computation method, a homotopy-based optimization algorithm for solving large-scale 3D design problems is developed.Originality/valueIn this paper, an approach based on topology optimization to solve 3D magnetostatic design problems when an objective least-squares function is involved is proposed. This approach is based on the continuous adjoint method derived for 3D magnetostatic design problems. The effectiveness of this topology optimization code is demonstrated by solving the design of a 3D magnetic circuit with up to 100,000 design variables.

Study on uncertain impact of thermal radiation on micropolar hybrid nanofluid flow between porous channel walls with distinct nanoparticles

ABSTRACT This work considers the micropolar hybrid nanofluid flows between porous channel walls with distinct nanoparticles such as Copper (Cu) and Alumina ( A l 2 O 3 ) with fuzzy extension. The volume fraction of hybrid nanoparticles is regarded as an uncertain parameter and represented using a triangular fuzzy number varying between 0 % and 2 % to enhance the uncertainty in the current study. Thermal radiation is a kind of heat transmission and one of several strategies to improve thermal efficiency in various systems. The proposed micropolar hybrid nanofluid flow examines the thermal radiation effect that enhances the thermal efficiency of the flow between the porous channels. A novel generalized double parameter-based homotopy approach is used to study and analyse the uncertain impact of different parameters on the thermal and velocity profiles of the micropolar hybrid nanofluid flow. Finally, the results obtained are validated in a crisp case to test the accuracy of the proposed technique.

Study on MHD hybrid nanofluid flow with heat transfer in a stretchable converging/diverging channels embedded in a porous medium with uncertain volume fraction

This study considers a hybrid nanofluid flow in a non-parallel plate embedded in permeable stretchable convergent and divergent channels. It investigates the impact of the magnetic field and heat transmission on the velocity and temperature profiles of the hybrid nanofluid flow. Here, the walls of the stationary channels are considered to be allowed to stretch or shrink, and the hybrid nanofluids are Cu-Ag, with water as the base fluid. The nanoparticle volume fraction of hybrid nanofluid is considered an uncertain parameter, represented with a triangular fuzzy number ranging from 0.0 − 0.04 . A novel double parametric homotopy approach is used to study the impacts of the relevant parameters on the velocity and temperature profiles and validate the obtained result by comparing it with previously published literature in the crisp case without the porous medium. Finally, the fuzzy velocity and temperature profiles at distinct channel locations are graphically illustrated, accompanied by numerical representations of velocity and temperature bounds at various positions along the stretchable convergent/divergent channels.

Numerical simulation of chemical reactive flow of Boger fluid over a sheet with heat source and local thermal non-equilibrium conditions

In this article, the effect of nonlinear heat source on chemical reactive flow of magnetized Boger fluid over a sheet with Marangoni convection and porous medium is explored. In the modelling, local thermal non-equilibrium (LTNE) conditions and heat sources are taken into account. The unique heat transport for both liquid and solid phases is provided by the LTNE model, which is based on thermal equations. As a result, various temperature profiles are used in this work for both the solid and fluid phases. It is essential for streamlining industrial procedures, improving material processing methods, comprehending biological systems like blood flow for medicine delivery, determining the environmental impact of pollutant dispersion, and increasing the effectiveness of oil and gas transportation. This model supports crucial decision-making processes in numerous disciplines by offering insights into the intricate interplay between reactive fluids, heat transmission, and chemical reactions. The appropriate similarity variables are used to compress the model equation system, and the shooting method and the conventional bvp4c solver are then used to solve the system numerically. Optimal homotopy approach is used to achieve residual errors. There are suggested best estimates for auxiliary variables. Results show that for larger values of the solvent fraction parameter, Boger fluid exhibits an enhanced velocity profile and a declining thermal profile of the liquid phase. The concentration and thermal profiles of both the solid and liquid phases are declined and enhanced the velocity of the Boger fluid for higher values of the Marangoni convection parameter.

Mixed convection flow in a square lid-driven cavity subject to inclined magnetic field with highly accurate wavelet-homotopy solutions

Numerical studies of flow in square enclosures attain great interest due to their wide applications in engineering, such as in the meteorology, navigation, mechanology and mining, etc. In this paper, the solutions are obtained to the square lid driven cavity describing classical mixed convection in the presence of an inclined magnetic field, chemical radiation and heat absorption/generation. The problem is mathematically reduced into Navier-Stokes equations. The Coiflet wavelet homotopy approach, a newly suggested computational technique, is then used to solve these nonlinear partial differential equations with their non-homogeneous boundary conditions. The numerical solutions for streamlines, isotherms, iso-concentrations contours, Nusselt number are carefully determined. The solutions so obtained are analyzed through various plots to demonstrate the effects of several physical parameters such as Reynolds numbers (Re), Richardson number (Ri), Grashof number (Gr), Hartman number (Ha), thermal radiation parameter (NR), chemical reaction coefficient (K), heat generation/absorption coefficient (Qh), the angle of inclination of magnetic field (γ), slipping parameter (λ) and Schmidt number (Sc) and their physical insights are also reported. Rigid comparison of with previous studies shows excellent computational effectiveness of newly proposed method. Additionally, according to the results, our recently proposed approach has superior characteristics and strong nonlinear processing capabilities, which render it more appropriate than the Coiflet wavelet analysis method and the traditional homotopy analysis method for tackling complex nonlinear problems.

Thermal Analysis of MHD Hybrid Nanofluid on Stretching/Shrinking Non-Parallel Walls with Uncertain Volume Fractions

This study examines the heat transmission of magneto-hydrodynamic (MHD) hybrid nanofluid (Ag–[Formula: see text]O) flows between two stretchable or shrinkable walls. The nanoparticle volume fraction is considered an uncertain parameter and is represented using triangular and Gaussian fuzzy numbers. The governing radial momentum and energy equations are transformed into nonlinear ordinary differential equations using a similarity transformation and with the required boundary conditions. A generalized homotopy approach is used to analyze the impacts of embedding parameters, including the Reynolds number, stretching/shrinking parameter, opening angle, radiation parameter, Magnetic parameter, and solid volume fraction on velocity and thermal profiles in crisp and uncertain cases. The findings demonstrate that fluid velocity and temperature distribution increase as the stretching parameter increases. The correctness of the suggested technique is validated by comparing the results with the available numerical solution. It has been observed that there is a great agreement between the present results and the numerical ones.

Differential contracting homotopy in higher-spin theory

A new efficient approach to the analysis of nonlinear higher-spin equations, that treats democratically auxiliary spinor variables ZA and integration homotopy parameters in the non-linear vertices of the higher-spin theory, is developed. Being most general, the proposed approach is the same time far simpler than those available so far. In particular, it is free from the necessity to use the Schouten identity. Remarkably, the problem of reconstruction of higher-spin vertices is mapped to certain polyhedra cohomology in terms of homotopy parameters themselves. The new scheme provides a powerful tool for the study of higher-order corrections in higher-spin theory and, in particular, its spin-locality. It is illustrated by the analysis of the lower order vertices, reproducing not only the results obtained previously by the shifted homotopy approach but also projectively-compact vertices with the minimal number of derivatives, that were so far unreachable within that scheme.

Dynamical homotopy transient-based technique to improve the convergence of ill-posed power flow problem

This paper proposes a hybrid technique to solve the ill-posed Power Flow Problem (PFP), considering a homotopy approach. The primary proposal is to solve large-scale problems where the traditional Newton–Raphson (NR) fails to converge, as in the case of ill-posed systems. The method explores a dynamical homotopy transient-based technique to improve the convergence of the ill-conditioned problem instead of using the classical static method. Depending on the integration selected scheme and the integration step, the result furnished by the dynamical homotopy method has low accuracy. Then, the NR method is employed to refine the low-accuracy result and accurately determine the ill-posed PFP solution. The proposed approach can be implemented efficiently using only one Jacobian matrix computation and LU factorization per point of the homotopy path. In the static homotopy problem, a PFP using previous results must be solved per path point. In this case, some LU factorizations are necessary for each path point. The technique’s performance was evaluated through experiments, including a 70,000-bus large-scale system. The approximate dynamical homotopy result used as an initial estimate provided appropriate convergence quality for the NR method to determine a high-precision solution to the PFP.

Disentanglement of topological and dynamical fields in 3d higher-spin theory within shifted homotopy approach

The first-order correction to the one-form sector of equations of the 3d higher-spin theory is derived from the generating nonlinear HS system by virtue of the shifted homotopy approach. The family of solutions to the generating system that disentangles equations for dynamical and topological fields in the first order of perturbation theory is found. This family is shown to belong to the different cohomology class compared to the solution found earlier by the direct methods. The related cohomology is shown to be the same as that underlying the mass deformation in the matter sector of 3d higher-spin equations.

Safety of a quarter-vehicle car through negative derivative feedback controller

The aim of this paper is presenting the acts of a resonant control technique called Negative Derivative Feedback (NDF) controller on a quarter-vehicle car subjected to principal parametric excitation. This control technique is useful and effective as it helps to avoid any dangerous vibrations caused during its operation. Therefore, we considered the NDF controller that plays an important role in it. Moreover, robustness to natural frequency variation has been investigated for the first time on a quarter-vehicle car system. The multiple scale homotopy (MSH) approach has been used for solving the equations of the controlled system mathematically. At the studied resonance case Ω=2,ω=1, the acts of actual various parameters and system conduct are calculated. Frequency-response equations (FREs) governing the controlled system vibration amplitude have been derived. The simulation outcomes were completed with the help of MATLAB 7.0 programs. A comparison between PPF and NDF controllers has been made in the vibration system to show the best controller. Numerical outcomes reveal that the best one in removing the oscillations and decreasing the value of amplitudes is the NDF controller. Furthermore, after adding this controller, its gains lessened the steady-state amplitudes of the model in a minor time. In addition, this controller made the structure more stable than those of other controllers. The impacts of the effective parameters on the controlled system have been discovered. The different response curves for the framework when the controller is off and on action have been plotted by the obtained frequency-response equation displaying the stable and unstable zones. Finally, the obtained numerical results have shown an outstanding agreement per the analytical ones.

Optimal Projected Circular Orbit for Spacecraft Formation Flying Near a Slowly Rotating Asteroid

In spacecraft formation flying missions, the projected circular (relative) orbit (PCO), which appears circular when viewed along the zenith-nadir direction of the nominal chief spacecraft, is suitable for constructing a synthetic aperture radar. In this paper, the analysis of a PCO for spacecraft formation flying near a slowly rotating asteroid is conducted. By matching the in-plane and out-of-plane relative motion frequency, the critical values of the initial phase angle that characterizes the PCO are identified, resulting in the minimum precession of relative orbit. The analysis is then utilized to develop the fuel-optimal control law for establishing a target PCO. In particular, the fuel-optimal problem is addressed with an indirect method, where a homotopy approach is adopted, and the initial (unknown) costate vector is calculated with a scaling technique, so that the sensitivity to the initial guess problem is mitigated.

Homotopy Optimization and ANN Modeling of Millimeter-Wave SIW Cruciform Coupler

The development of millimeter-wave and terahertz (THz) passive components such as couplers and filters is an intimidating task because of underlying ultrasensitivity of electrical performances to geometric dimensions and processing tolerances. It is a common practice for us to use an integrated optimizer of commercial electromagnetic (EM) software packages for the design and optimization of such geometric parameters. However, those optimizers may fail to achieve a desired performance if initial variables are not in a range close enough to the optimal solution. In this article, we introduce an homotopy approach to optimizing the geometric parameters of a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> -band cruciform coupler based on substrate integrated waveguide (SIW) technique in conjunction with an artificial neural network (ANN) model. Starting from a set of initial variables, a homotopy optimization is set to search for an optimum solution. The ANN technique is adopted as the surrogate in place of a usual time-consuming EM model to accelerate the homotopy optimization process of the cruciform coupler. We propose a feed-forward computational formulation inspired by the fundamental transmission line impedance equation. Such a transmission line knowledge-based feedforward network results in a faster convergence with better accuracy than its conventional counterpart. To demonstrate the homotopy optimization method based on the ANN model, an example of multiparameterized cruciform coupler design is detailed. This cruciform coupler with optimized geometric dimensions is simulated, fabricated, and measured. Measured and simulated results validate the combined ANN model and homotopy method. An equivalent lumped-element circuit model of the cruciform coupler is also proposed in this work. An ANN model development technique is described how to extract the equivalent circuit parameters for given coupler specifications. Extracted circuit parameters in connection with the desired coupler performance are then compared with published results which verify the ANN model development algorithm.

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.

Network Topology Invariant Stability Certificates for DC Microgrids With Arbitrary Load Dynamics

DC microgrids are prone to small-signal instabilities due to the presence of tightly regulated loads. This paper develops a decentralized stability certificate which is capable of certifying the small-signal stability of an islanded DC network containing such loads. Utilizing a novel homotopy approach, the proposed standards ensure that no system eigenmodes are able to cross into the unstable right half plane for a continuous range of controller gain levels. The resulting “standards” can be applied to variety of grid components which meet the specified, but non-unique, criteria. These standards thus take a step towards offering <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">plug-and-play</i> operability of DC microgrids. The proposed theorems are explicitly illustrated and numerically validated on multiple DC microgrid test-cases containing both buck and boost converter dynamics.

Affine-Invariant Ensemble Transform Methods for Logistic Regression

We investigate the application of ensemble transform approaches to Bayesian inference of logistic regression problems. Our approach relies on appropriate extensions of the popular ensemble Kalman filter and the feedback particle filter to the cross entropy loss function and is based on a well-established homotopy approach to Bayesian inference. The arising finite particle evolution equations as well as their mean-field limits are affine-invariant. Furthermore, the proposed methods can be implemented in a gradient-free manner in case of nonlinear logistic regression and the data can be randomly subsampled similar to mini-batching of stochastic gradient descent. We also propose a closely related SDE-based sampling method which again is affine-invariant and can easily be made gradient-free. Numerical examples demonstrate the appropriateness of the proposed methodologies.

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.