Semi-analytical Method Research Articles

In Situ Non-Destructive Stiffness Assessment of Fiber Reinforced Composite Plates Using Ultrasonic Guided Waves.

The multimodal and dispersive character of ultrasonic guided waves (UGW) offers the potential for non-destructive evaluation of fiber-reinforced composite (FRC) materials. In this study, a methodology for in situ stiffness assessment of FRCs using UGWs is introduced. The proposed methodology involves a comparison between measured wave speeds of the fundamental symmetric and antisymmetric guided wave modes with a pre-established dataset of UGW speeds and translation of them to corresponding stiffness properties, i.e., ABD-components, in an inverse manner. The dispersion relations of guided waves have been calculated using the semi-analytical finite element method. First, the performance of the proposed methodology has been assessed numerically. It has been demonstrated that each of the independent ABD-components of the considered laminate can be approximated with an error lower than 10.4% compared to its actual value. The extensional and bending stiffness properties can be approximated within an average error of 3.6% and 9.0%, respectively. Secondly, the performance of the proposed methodology has been assessed experimentally. This experimental assessment has been performed on a glass fiber-reinforced composite plate and the results were compared to mechanical tensile and four-point bending tests on coupons cut from the plate. Larger differences between the estimated ABD-components according to UGW and mechanical testing were observed. These differences were partly attributed to the variation in material properties across the test plate and the averaging of properties over the measurement area.

New semi-analytical method for fast transcranial ultrasonic field simulation

Objective. To optimize and ensure the safety of ultrasound brain therapy, personalized transcranial ultrasound simulations are very useful. They allow to predict the pressure field, depending on the patient skull and probe position. Most transcranial ultrasound simulations are based on numerical methods which have a long computation time and a high memory usage. The goal of this study is to develop a new semi-analytical field computation method that combines realism and computation speed. Approach. Instead of the classic ray tracing, the ultrasonic paths are computed by time of flight minimization. Then the pressure field is computed using the pencil method. This method requires a smooth and homogeneous skull model. The simulation algorithm, so-called SplineBeam, was numerically validated, by comparison with existing solvers, and experimentally validated by comparison with hydrophone measured pressure fields through an ex vivo human skull. Main results. SplineBeam simulated pressure fields were close to the experimentally measured ones, with a focus position difference of the order of the positioning error and a maximum pressure difference lower than 6.02%. In addition, for those configurations, SplineBeam computation time was lower than another simulation software, k-Wave’s, by two orders of magnitude, thanks to its capacity to compute the field only at the focal spot. Significance. These results show the potential of this new method to compute fast and realistic transcranial pressure fields. The combination of this two assets makes it a promising tool for real time transcranial pressure field prediction during ultrasound brain therapy interventions.

Buckling optimization of variable stiffness helicoidal composite laminates based on semi-analytical method

Variable stiffness laminates and helicoidal composites have gained significant attention in recent years. This paper introduces a novel structure called variable stiffness helicoidal composite laminates, which combines these two materials. The fibers in each layer are arranged in curved paths, and all the layers are stacked in a helicoidal pattern. To ensure uniform thickness throughout the laminate, an equidistant placement method is proposed for designing the curvilinear fibers. By establishing the energy equation and fitting the displacements with Legendre polynomials, a semi-analytical method is proposed to determine the critical buckling load. An integrated design framework is developed to obtain the optimal design. The optimized models show more than 10% enhancement in the critical buckling load compared to the quasi-isotropic laminate under uniaxial compression and in-plane shear.

Machine learning-based optimal design of an acoustic black hole metaplate for enhanced bandgap and load-bearing capacity

This paper introduces a novel machine learning-based optimization strategy for multi-functional acoustic black hole (ABH) metaplates. The primary objective is to achieve a multi-functional metaplate with excellent performance in elastic wave attenuation and load-bearing capacity simultaneously. The paper begins by describing the design of nanocomposite ABH metaplates, presenting a new pathway to realize multi-functional metaplates. Then, a semi-analytical method, based on plate theory and the Bloch–Floquet theorem, is introduced to consider the band structure of the nanocomposite metaplates. Through systematic analysis, the impacts of the ABH effect, nanocomposite reinforcements, and the viscoelastic damping layer on the bandgaps and strain energy compliance are highlighted. Meanwhile, two optimization objectives representing bandgap characteristics and in-plane stiffness are derived respectively. Subsequently, a deep learning surrogate model is employed to establish a relationship involving significant parameters with the optimization objectives. The performance evaluation confirms accuracy and computational speed of the surrogate model. Finally, an optimization strategy based on deep reinforcement learning is proposed to obtain multi-functional metaplates with superior bandgaps, enhanced in-plane stiffness, or both. The robustness and efficiency of the strategy are demonstrated under different tests. The results show that the proposed strategy can achieve identical results as the genetic algorithm and nondominated sorting genetic algorithm-II, while surpassing them in computational efficiency and balancing multiple objectives. The findings of this study serve as valuable references for the future development and application of multi-functional advanced metamaterials.

Analysis of the stress-strain state of the rotary device fastening part by the semi-analytical finite element method

The possibilities of the numerical apparatus developed on the basis of the semi-analytical finite element method are shown on the example of solving a specific problem. In the areas of maximum values, the parameters of the stress-strain state of the re-device fastening part were compared. It is worth noting the relatively complex configuration of the cross-section of the object, which led to a thorough study of the principles of constructing a grid area. The surface of interaction between the axis and the hull was modeled by a fairly thin layer of elements in relation to their thickness, which absorb only normal stresses. Analysis of the stress-strain state of the rotary device fastening part should be carried out from the standpoint of a spatial problem. Moreover, according to the results of the elastic calculation, the transition section between the body and the base plate turned out to be more loaded compared to the zone of contact interaction between the body and the axle. The data of the calculation performed in the elastic-plastic formulation made it possible to clarify the idea of the operating conditions of individual sections of the object and to draw a conclusion about the uniformity of the hull. The carried out studies confidently demonstrated the wide possibilities of the developed approach in solving new, practically important problems of elastic deformation of prismatic bodies of complex shape in the spatial formulation of new ones. To substantiate the reliability of the results of the calculation of the elements under consideration, a sequential increase in the number of FE in the cross-section and the number of retained terms of the decomposition along the length of the body, as well as an increase in the accuracy of solving systems of equations, are given. In addition, an assessment of the satisfaction of natural boundary conditions on the surface of the body and the conditions of equilibrium in the integral sense is given according to the characteristic sections within the region, which showed their fairly good performance. New data on the regularities of behavior of critical structures in the process of loading, due to the consideration of their physical and geometric parameters, have been obtained.

Strength analysis and force distribution in a tensile metal joint

One of the methods of connecting several elements of mechanical structures to form an assembly is bolting. Unlike other types of connections, such as welding or riveting, this type of connection has the advantage of being disassembled, which is why bolted connections are quite common. At the same time, like other types of connections, it is an important structural element that requires additional attention in calculations. The most common calculation of a bolted connection is using analytical formulas. Every day, the requirements for accuracy, speed and versatility of approaches to the calculation of building structures are increasing. In addition, special attention should be paid to the calculation of nodes where structures abut each other, since depending on the operating conditions of the node, different distribution of the NDP is possible, which in turn affects the geometr2ic characteristics and operation of structures. In this regard, we compared the results of the distribution of stresses and forces in the bolt using the finite element method with the use of universal quadrilateral shell FEs, universal spatial eight-node isoparametric FEs, and the semi-analytical finite element method. The obtained results of solving the test case using universal quadrilateral shell FEs and universal spatial eight-node isoparametric FEs allow us to conclude that there is a slight difference in the stress distribution in the spatial formulation of the FEM and SFEM. In addition, based on the FEM and NEM calculations and, accordingly, the calculation according to the DBN, it can be concluded that when calculating according to the DBN standards, an analytical calculation can be used to determine the safety margin of a bolt for simple tasks.

Semi-Analytical Assessment of Multiple Gravity-Assist Opportunities Based on Feasibility Domains

This paper proposes a semi-analytical method for multiple gravity-assist (MGA) opportunity assessment, which can judge the feasibility of MGA sequences and predict corresponding opportunities in terms of the relative phases between flyby planets. The proposed method features the feasible domains of gravity-assist states and the phase differences in MGA rather than focusing on a limited number of optimal solutions. First, the planet flyby departure and arrival conditions for successive gravity assist are analytically derived based on the planar circular model. Second, the feasible domain of gravity assist described by flyby parameters is established and pruned based on the conditions, and the corresponding relative phases between planets are calculated. Third, a MGA opportunity search algorithm via feasible domain is formulated to match the phase differences in the ephemeris. Lastly, the method is examined in two types of MGA missions: the rendezvous and flyby missions. The results from the semi-analytical approach can cover all optimized opportunities in the rendezvous mission and 94.44% optimized opportunities in the flyby mission under the accurate three-dimensional model. The results of the proposed semi-analytical method can serve as a good initial guess for further accurate trajectory optimization and provide reference for future MGA mission design.

Natural frequencies and stability for Z-shaped fluid-conveying pipes with constrained-layer damping – Semi-analytic modeling with experimental validation

As the common fluid-conveying component, pipe structures are susceptible to excessive vibration due to internal fluid and external excitation. Constrained-layer damping (CLD) is commonly applied to suppress structural vibrations. However, effective modeling and vibration suppression analysis for Z-shaped fluid-conveying pipes are lacking to explore novel dynamic behaviors. This paper presents a dynamic model for analyzing the three-dimensional vibration of Z-shaped fluid-conveying pipes treated by partial CLD based on the semi-analytical method. Utilizing Timoshenko beam theory and calculating the centrifugal, Coriolis, and inertial force of fluids, the energy of straight and curved fluid-conveying pipes treated by CLD is derived. Based on the Lagrangian energy function, the equation of motion is obtained by using the Rayleigh-Ritz method. The improved artificial spring set is deployed to model the interaction of pipe sections and the non-uniform constraint distribution of clamps. Modal parameters are obtained by solving characteristic equations. The proposed method is verified by comparisons with references and modal experiment results. Frequency parameters, damping effects, and stability are analyzed under various fluid, geometric, and CLD parameters. In conclusion, a novel dynamic model is provided for analyzing the vibration and stability of the Z-shaped fluid-conveying pipe treated by CLD, which lays a foundation for vibration suppression research of fluid-conveying pipe systems.

Solutions of Surface-Wave Dispersion and Attenuation in Stratified Viscoelastic Media Using a Spectral-Element Approach

ABSTRACT Efficient and accurate calculation for the dispersion and attenuation of the surface waves in viscoelastic media is numerically challenging because the eigen wavenumbers are located in the complex domain. In this study, we propose a semianalytical spectral-element method (SASEM), which can determine the complex eigen wavenumbers by solving linear eigenvalue problems. By simplifying the structure of the eigenvalue problem, we significantly improve the calculation efficiency. The implementation of the frequency-dependent automatic discretization, semi-infinite element, and mode filter guarantees the correctness and accuracy of the modal solutions. Because no root-searching schemes are required, the root-skipping problem is naturally avoided. The numerical tests show that the SASEM can provide sufficiently accurate solutions with much less computation cost than traditional Muller’s method. Meanwhile, SASEM exhibits high flexibility when applied to media the parameters for which vary continuously with depth. To demonstrate the effectiveness of SASEM for complicated dispersion features, the dispersion curves and eigen wavefields of the viscoelastic media with a low-velocity layer are also analyzed. The results of numerical tests indicate the versatility, efficiency, and accuracy of our method. With further study, the proposed SASEM has the potential to become a promising tool for the investigation and retrieval of viscoelastic subsurface structures.

Response of Gaussian color noise excited oscillators with inertia nonlinearity based on the radial basis function neural network method

Given the widespread utilization of oscillators featuring inertial nonlinear terms across engineering disciplines and the omnipresence of noise excitations, comprehending their response to noise excitation holds paramount importance. In contrast to Gaussian white noise, colored noise stands out as a more adept mathematical model for simulating ambient noise scenarios. The widely used stochastic averaging method is somewhat ineffective in dealing with the response of oscillators with inertial nonlinearity under Gaussian colored noise excitation, as it cannot display the saddle shaped characteristics of the steady-state probability density function (PDF) caused by inertial nonlinearity. To overcome the shortcomings, this paper adopts a semi-analytical method called a radial basis function neural network method (RBFNN) to address this problem. This method represents the solution of the Fokker-Plank-Kolmogorov (FPK) equation of the system as the sum of a series of weighted radial basis functions, and obtains the optimal weight coefficients through the Lagrange multiplier method. This article investigates two examples, one of which has inertial nonlinearity and the other has multiple potential wells. The effects of inertial nonlinearity coefficients and the delay coefficients of colored noise on response are explored. The mean square errors between Monte Carlo (MC) and RBF are presented, which indicates the RBF results perfectly agree with the MC predictions.

A numerical assembly technique for free vibration of plane frames using a shifted and deflated Newton method

A semi-analytic method, namely the Numerical Assembly Technique (NAT) is extended to compute natural frequencies of arbitrary planar frame structures. The frame structure is systematically defined by a set of nodes, beams, bearings, springs, and external loads and the corresponding boundary and interface conditions are formulated. We consider the Timoshenko-Ehrenfest beam theory and take axial, bending and shear deformations as well as linear and rotational inertia into account. Since we use the exact solutions of the governing differential equations in the NAT, the accuracy of the results is independent from the spatial discretization. However, the resulting eigenvalue problem is non-linear and requires a numerical algorithm to solve. We consider three different representations of exact homogeneous solutions and study the numerical stability of the resulting system matrix. To solve for the eigenvalues we propose a Newton method enhanced with a shifted deflation algorithm. The stability and the efficiency of this approach is illustrated in numerical examples.

Buckling and free vibration of grid-stiffened composite conical panels using Extended Kantorovich Method

This paper develops very accurate closed-form solutions for buckling and free vibration of grid-stiffened composite conical panels using a semi-analytical method. On the base of first-order shear deformation theory (FSDT), the governing formulations are derived while an appropriate unit cell is used to derive the material properties of the lattice from their bulk properties. Multiple systems of ordinary differential equations (ODEs) are obtained by applying the Extended Kantorovich Method (EKM) to solve the governing system of partial differential equations (PDEs). The resulting ODEs are then solved using semi-analytical closed-form solutions to study buckling and free vibration of grid-stiffened panels. It is interesting to note that due to the general formulation of the conical panel, it is very easy to obtain other useful geometries including cylindrical panels, and even rectangular plates with general unit cell configurations. Regarding stability, fast convergence, and very low computing cost, the effectiveness of the proposed technique is investigated. The accuracy of the critical buckling loads and natural frequencies for various cases is studied which shows very good agreement in comparison with results obtained from the commercial finite element code ANSYS.

Semi-analytic modeling and experimental verification of arbitrary aero-engine complex spatial pipeline

In-depth research is imperative for complex spatial pipelines with fluid-structure interaction. A comprehensive and universal three-dimensional semi-analytical modeling method is first established for complex fluid-conveying spatial pipelines under the base excitation with arbitrary connections and configurations under different boundary conditions. The virtual spring technology is used to connect the straight and curved pipeline segments with spatial angles and simulate boundary conditions. For a thorough analysis, the actual stiffness of the bracket-clamp system is experimentally measured for the first time. The multi-level coupling dynamic model of part, component, and system levels under actual boundary conditions is established. An efficient method for solving the fluid-structure coupling pipeline system dynamics considering the base excitation is proposed based on energy method. The mid-plane displacement function of the Fourier series is introduced and solved using the principle of modal superposition. Hammer impact modal experiments, ANSYS software and vibration response experiments are performed on spatial pipelines supported by flexible bracket-clamps; the numerical and experimental results support the semi-analytical results. The degenerate model is verified with the simple plane pipelines in other references, and the errors are less than 5 %. The proposed method exhibits broad applicability and generalizability, making it suitable for various configurations and boundary conditions.

Resonance response and chaotic analysis for an irrational pendulum system

We propose a novel rotating pendulum system with irrational nonlinearity, bi-stable phenomenon and quasi-zero stiffness characteristic, which bears a transition from standard double well dynamics to discontinuous double well dynamics by adjusting a geometric parameter. Nonlinear dynamical behaviors of the proposed irrational pendulum system subjected to viscus damping and periodic excitation are studied. To analytically demonstrate the primary resonance response for this perturbed irrational pendulum system, we introduce an approximate irrational system which is significant parallels with the original pendulum system from the perspective of qualitative analysis and quantitative calculation. Averaging method is applied to investigate the dynamic response of the perturbed irrational pendulum system with quasi-zero stiffness at origin. The effect of the internal and external parameters on the response curves are discussed. Numerical fitting technique and semi-analytical Melnikov method are used to detect the chaotic boundaries of the perturbed irrational pendulum system with two types of homoclinic orbits. Different types of periodic solutions and chaos are clarified in the perturbed irrational pendulum system by using numerical simulations. It is found that a class of chaos with oscillatory and rotational motions bears significant similarities to paroxysmal chaos due to sudden rotational motion and two chaotic motions formed by period-doubling bifurcation merge into one chaos.

Investigating the Response Variability of Statically Determined Sandwich Beams considering two Random Fields of Elastic Modulus

In this paper, the displacement variation in sandwich beams is determined by employing a semi-analytical approach. The classical displacement is calculated by integration using Mohr’s equation, although the integration is complicated due to the inclusion of random fields in the inertial moment term. Using the trapezoidal rule to compute these integrals, the random fields are discretized into random variables at the nodal point of the beam segments. Thus, the expected displacement, standard deviation, and coefficient of variation can be computed. To validate the results, the random fields are simulated using a previously described spectral method. The results of numerical examples were compared with the semi-analytical method and the Monte Carlo simulation demonstrating the high accuracy of the proposed method. The results also illustrate the influence of the parameters of the random fields of elastic modulus on the variability of displacement.

MHD Casson non-Newtonian fluid flow in a channel with expanding/contracting porous walls in the presence of thermal radiation

In this research, we explore the impact of changing viscosity in the context of the asymmetric laminar flow of Casson fluid with thermal radiation. The flow occurs within a channel that can either expand or contract, featuring porous walls. The flow equations are converted into ordinary differential equations (ODEs) through appropriate transformations to simplify the analysis. The velocity field and temperature profile expressions are obtained using Akbari-Ganji's method (AGM) and finite-element method (FEM). The graphs show the impact of different parameters on the flow characteristics, especially the viscosity-dependent parameter, Casson fluid parameter, and expansion ratio. The fluid velocity near the lower wall increases with the Casson fluid parameter but decreases after the mid-point. The fluid velocity is more affected by the temperature-dependent viscosity parameter for Newtonian fluid than non-Newtonian fluid. The results showed that as the Prandtl number increases, the temperature decreases, while the temperature increases with an increase in the Radiation parameter. Also, An increase in the wall expansion ratio leads to higher wall temperatures, whereas a decrease in the power-law index results in reduced temperatures. The results we've acquired were cross-referenced with prior studies, revealing that the employed methods exhibit high precision. The novelty of the present work is obtaining new results through the use of Akbari-Ganji's semi-analytical method.

Semianalytical Method for Controlling the Deformation of Retaining Structures Subjected to Asymmetrical Loads

Semianalytical Method for Controlling the Deformation of Retaining Structures Subjected to Asymmetrical Loads

The new analytical and numerical analysis of 2D stretching plates in the presence of a magnetic field and dependent viscosity

This study explores heat transfer in a system involving Jeffery fluid of MHD flow and a porous stretching sheet. The mathematical representation of this system is initially described using a partial differential equation (PDE), which is then converted into an ordinary differential equation (ODE) through numerical techniques such as Lie similarity and transformation methods, along with the shooting approach. The results indicate that altering the variables of Jeffery fluid, heat source, porosity on a stretching sheet, and the physical characteristics of the magnetic field within the system leads to an upward trend. Implementing this enhanced heat transfer system can yield benefits across various domains, including industrial machinery, mass data storage units, electronic device cooling, etc., thereby enhancing heating and cooling processes. Furthermore, the study also utilized Akbari-Ganji’s Method, a new semi-analytical method designed to solve nonlinear differential equations of heat and mass transfer. The results obtained from this method were compared with those from the finite element method for accuracy, efficiency, and simplicity. This research provides valuable insights into heat transfer dynamics in complex systems and offers potential applications in various industrial settings. It also contributes to developing more efficient and effective heat transfer techniques.

Mathematical modeling of droplet collisions in sprays under microgravity conditions

Two-phase flows under microgravity conditions play a crucial role in Space technology and science. Studying the behavior of dispersed droplets suspended in a gas flow and their influence on the characteristics of the carrier flow is the main task of computational modeling of turbulent two-phase flows. To understand the features and predict the behavior of such flows, along with experimental studies, it is necessary to develop mathematical models, which adequately describe all the basic physical and chemical processes in a gas-droplet environment at different scales.In this paper we obtain solutions for the problem of two droplets collision, using solutions for cumulative jets, and propose droplet collision outcome criterion. This research will focus on the process of two interacting droplets coalescence into one and coalescence with subsequent separation into two or more droplets. A semi-analytical method has been developed for application of the solution of the cumulative jet problem to the problem of droplet collision. A differential equation has been obtained that describes the following possible outcomes of droplet collision: coalescence and coalescence with subsequent separation. A criterion for determining the outcome of on-head droplet collision has been obtained. In accordance with the developed criterion, the number and size of formed droplets are obtained. The effect of collision on moving droplet evaporation in a heated atmosphere was regarded.

A homotopy-based computational scheme for two-dimensional fractional cable equation

In this paper, we examine the time-dependent two-dimensional cable equation of fractional order in terms of the Caputo fractional derivative. This cable equation plays a vital role in diverse areas of electrophysiology and modeling neuronal dynamics. This paper conveys a precise semi-analytical method called the q-homotopy analysis transform method to solve the fractional cable equation. The proposed method is based on the conjunction of the q-homotopy analysis method and Laplace transform. We explained the uniqueness of the solution produced by the suggested method with the help of Banach’s fixed-point theory. The results obtained through the considered method are in the form of a series solution, and they converge rapidly. The obtained outcomes were in good agreement with the exact solution and are discussed through the 3D plots and graphs that express the physical representation of the considered equation. It shows that the proposed technique used here is reliable, well-organized and effective in analyzing the considered non-homogeneous fractional differential equations arising in various branches of science and engineering.