Semi-analytical Method Research Articles

Self-weight and thermal bifurcation analysis of functionally graded vertical annular plates

The present paper thoroughly investigates the influence of structural weight on the thermal buckling of standing circular and annular plates. The structure is composed of symmetrically functionally graded materials (FGM) in terms of its thickness. The first-order theory of shear deformation along with nonlinear strain–displacement equations has been employed to derive the governing equations of the FG plate. First, the asymmetric pre-buckling path of the structure under the influence of weight and heat is calculated by a semi-analytical method. The plate's stability relations are extracted using the adjacent equilibrium criterion and virtual work principle. Then, the stability equations are analyzed by the 2D generalized differential quadrature (GDQ) method. Parametric results are presented based on the analysis of the FGM power-law index, geometric ratios, and edge constraints on the buckling of the vertical plate. The study reveals the critical power-law index and thickness at which the structure becomes unstable, shedding light on the previously neglected calculation of weight-induced buckling in vertical structures. Also, the effect of weight on the buckling mode shapes of heavy plates and disturbing its symmetry is demonstrated. Additionally, this study presents the first investigation into the effect of weight on the thermal stability of annular FG plates.

A Semi-Analytical Method for the S-Parameter Calculations of an N × M Multimode Interference Coupler

A semi-analytical method for the S-parameter calculations of an N×M multimode interference coupler (MMI coupler) is presented. The proposed semi-analytical method is based on the mode decomposition and utilizes an effective index method to approximate the channel waveguide using an equivalent slab waveguide whose modes are described by exact analytic expressions. In comparison to the commonly used beam propagation method (BPM) and finite difference time domain method, which require significant time and computational resources, the proposed method accelerates the design process of photonic integrated circuits and basic building blocks such as an MMI coupler. The simulation results obtained using the developed method and the BPM were compared and showed very similar outcomes for different topologies of the MMI coupler. The key advantage of the proposed semi-analytical method over other analytical models is its ability to accurately simulate MMI couplers with an arbitrary position and number of input and output waveguides. In addition, this method can be extended using the theory of local coupled modes by taking into account the reflections from the end face of the MMI box.

Semi-analytical method for solving a model of the evolution of smoking habit using global rational approximants

Semi-analytical method for solving a model of the evolution of smoking habit using global rational approximants

Finding the polarizability of dielectric half-ellipse

This article studies the electrostatic response of a two-dimensional dielectric half-ellipse. A semianalytical method is presented for solving the polarizability of a half-ellipse as a function of its relative permittivity. The polarizability depends on the direction of incident field, and therefore, it is written as a dyadic with two perpendicular components. These are called series and parallel polarizability, and they are solved separately. The solution requires a series expansion of the potential function, and the coefficients must be obtained numerically, through an inversion of a finite matrix equation. Numerical results are presented for the polarizability components of the half-ellipse as functions of its geometry and permittivity.

Static Mechanics and Dynamic Analysis and Control of Bridge Structures Under Multi-Load Coupling Effects

With the rapid development and wide application of large-span bridges, the problem of dynamics and safety control of bridge structures under multiple loads is becoming more and more prominent. To realize the dynamic analysis and mechanical control of the girder structure, this paper designs a new type of magnetorheological (MR) mechanical damper based on the mechanistic analytical method and establishes a coupled dynamics model of the vehicle-rail-MR mechanical damper. The simulation and validation results show that the error between the semi-analytical method and the finite element theory calculation results is only 1.0%, while the kinetic simulation is consistent with the measured frequency domain trend. The analytical results show that: After applying MR mechanical damper for mechanical control, the moment live load and shear values of side spans were reduced by 27.68% and 10.79%, respectively; and the maximum moment and shear values generated at the center pivot were reduced by 28.19% and 10.81%, respectively. After applying the mechanical damper, the stress distribution of the cable-stayed bridge is more balanced, and the maximum diagonal stress of the overall structure is reduced from 3.8 MPa to 2.9 MPa. After safety control, the root-mean-square (RMS) value of the mid-span displacement amplitude was reduced by 59.32% and the maximum value was reduced by 11.46%, which improved the stability of the girder dynamics. After mechanical control, the dynamic acceleration response of the beam within the span decreased between 2 and 8 seconds and increased between 8 and 10 seconds. The overall response fluctuated around −5 m/s2 with a relatively smooth trend.

A semi-analytical method for vibration localization of plates integrated with low-frequency plate-type resonators

A plate-type local resonator with varying free boundaries is integrated within the plate to transform the initial low-order global vibration modes into localized vibration modes. A novel semi-analytical method is proposed to analyze the free vibration of the plate with discontinuities in thickness and displacement. The host plate and the resonator are modeled separately and coupled by the condition of displacement compatibility, based on the geometry configuration. A set of local admissible functions, consisting of global and local parts in the resonator domain, is proposed to describe the vibration localization and displacement discontinuity. The Ritz method with the proposed admissible functions is employed to investigate the effect of geometry parameters and boundary conditions on the vibration characteristics of the plates. The lack of orthogonality between the global and localized modes is determined using the analytic mode functions obtained by the proposed method and can be altered by the free boundary conditions of the resonator. The results demonstrate that by applying free boundary conditions to a resonator, the low-order localized vibration frequencies can be significantly reduced by up to 90 %, with negligible effect on low-order global frequencies. An original and exciting finding is that the global modes can be assimilated by the corresponding localized vibration modes with close frequencies.

A Novel Differential Quadrature Galerkin Method for Dynamic and Stability Behaviour of Bi-directional Functionally Graded Porous Micro Beams

The free vibration and buckling behaviours of 2D-FG porous microbeams are explored in this paper utilizing the Quasi-3D beam deformation theory based on the modified couple stress theory and a Differential Quadrature Galerkin Method (DQGM) systematically, as a combination of the Differential Quadrature Method (DQM) and the semi-analytical Galerkin method, which has used to reduce computational cost for problems in dynamics. The governing equations are obtained using the Lagrange’s principle. The mass and stiffness matrices are calculated using the weighting coefficient matrices given by the differential quadrature (DQ) and Gauss-Lobatto quadrature rules. The matrices are expressed in a similar form to that of the Differential Quadrature Method by introducing an interpolation basis on the element boundary of the Galerkin method. The sampling points are determined by the Gauss-Lobatto node method. The influence of the thickness-to-material length scale parameter (MLSP) on the nondimensional natural frequencies and nondimensional critical buckling loads of 2D-FG porous microbeams are investigated, along with the effects of the boundary condition, aspect ratio and gradient index. The results are validated with literature to establish the accuracy of the procedure described. This work will provide a numerical basis for the design of FG microstructures in the field of micromechanics. These results can be applied to the engineering design of porous FG microstructures.

Analysis of Simply Supported Laminated Composite Plate by Semi Analytical and Finite Element Method for Different Orientation Angles

Abstract Deflection of simply supported laminated composite plate under transverse load is found out by semi-analytical method (Classical Lamination Theory) and the results are compared with Finite element method using FEA software ANSYS. The equations developed by classical lamination theory for symmetric angle ply and cross ply laminated plate are solved by Navier and Rayleigh-Ritz method. The efficiency of Navier and Rayleigh-Ritz method in solving the problems of symmetric angle ply and cross ply laminated plate are investigated by comparing their results with ANSYS. Percentage error of the results from finite element method with Navier and Rayleigh-Ritz methods are studied and compared. The model is extended to study the effect of orientation angle, number of lamina and length to thickness ratio on the deflection of simply supported composite plates for symmetric angle ply plates of stacking sequence (orientation angle) of [+θ/− θ]s . From the discussion, it is found that the percentage error depends on the variables: number of lamina, thickness of each lamina, orientation angle and mesh size. The significance of these variables in reducing the percentage error is studied by statistical tool: ANOVA and the relationship between these variables are established by regression equation. Statistical analysis is conducted by testing their significance at the 5% significance level. The comparative study and analysis of the percentage errors of the semi-analytical methods with finite element methods in solving the problem of laminated composite plates is essential to study the degree of approximation in the results.

Wave interactions with a cylinder surrounded by an arc-shaped breakwater

Inspired by the energy island used for energy storage, conversion, and transmission, this study explores the water wave interactions with a bottom-standing cylinder surrounded by an arc-shaped breakwater of negligible thickness. The arc-shaped breakwater can be either impermeable or porous. To analyze the problem, a semi-analytical matched eigenfunction expansion method is developed within the framework of linear potential flow theory. The fluid domain is divided into subdomains, with the velocity potentials in each subdomain expressed as eigenfunction expansions. Matching conditions, ensuring continuity in pressure and normal velocity, are imposed at the juncture boundary. The accuracy of the developed semi-analytical method is verified through comparison with results obtained using the numerically-based boundary element method. Extensive discussions are made on hydrodynamic responses, including wave exciting force and free surface elevations. When the breakwater is impermeable, fluid resonance occurs in the gap between the cylinder and the arc-shaped breakwater, leading to significantly large responses in both free surface elevation and hydrodynamic forces. A simple method is developed to estimate the resonant frequencies and mode shapes associated with these resonances. On the other hand, it is observed that the presence of the porous breakwater considerably reduces the free surface responses in the gap as well as the hydrodynamic loads on the cylinder. The porous breakwater effectively mitigates the occurrence of fluid resonance and offers an efficient means to dampen wave energy, providing valuable insights for wave energy dissipation and coastal engineering applications.

Linear Time Electromigration Analysis Based on Physics-Informed Sparse Regression

In this work, we propose a novel physics-informed sparse regression (PISR) framework to solve stress evolution (described by Korhonen’s equations) in general multi-segment wires using an unsupervised learning scheme. Unlike the existing physics-informed neural network framework (PINN), the PISR method trains the trainable weights through the Moore-Penrose generalized inverse algorithm used in extreme learning machine (ELM), which is extremely faster than the back-propagation algorithm. To improve the accuracy of PISR for complex multi-segment interconnects, we employ domain decomposition schemes in both space and time. For each subdomain, we use different trainable weights but the same shared neural network to represent each subsolution, which leads to more efficient memory usage. Furthermore, we propose to use sparse matrix techniques to accelerate the training speed of the PISR method and prove that the resulting PISR has linear time complexity for analyzing tree-structured interconnects. Finally, we divide the time into many time internals and apply an autoregressive model to simulate each time interval in sequence to further improve scalability and reduce memory cost so that the PISR method can perform EM analysis for large-scale multi-segment interconnects. Experimental results on different kinds of interconnect structures show that the proposed PISR method has the same accuracy level as the numerical methods. The results on NT T-junctions interconnect trees show that the proposed PISR method indeed demonstrates true linear time complexity. Furthermore, PISR can deliver 8.9×, 20.6×, and 1284× speedups over the recently proposed semi-analytic method (ASOV), finite difference method accelerated with model order reduction (FDM-MOR), FDM for the interconnect with NT= 5000, respectively. Furthermore, we show that PISR also achieves an 818× speedup in training over the plain PINN method based on the traditional back-propagation algorithm.

Sound insulation prediction and band gap characteristics of four vibrators acoustic metamaterial with composite phononic crystal structure

Acoustic Metamaterial have been widely concerned because of their strong sound insulation characteristics for specific low-frequency bands. However, further optimization is still needed in terms of broadband sound insulation characteristics, and phononic crystals can generate multiple sound insulation bandwidths due to their unique bandgap characteristics. Therefore, a four vibrators acoustic Metamaterial structure with phononic crystals is proposed to broaden the sound insulation performance at medium and high frequencies while retaining the advantages of sound insulation at low frequencies. Firstly, a semi analytical method combining finite element analysis was proposed for the sound insulation performance of the structure. Afterwards, a bandgap theory based on the equivalent medium method was proposed to reveal the reasons for its characteristics. Finally, a cylindrical sleeve structure was designed for a certain rolling rotor compressor, and its sound insulation performance in the cylindrical state was verified through finite element simulation.

Mathematical solutions in internal dose assessment: A comparison of Python-based differential equation solvers in biokinetic modeling

In biokinetic modeling systems employed for radiation protection, biological retention and excretion have been modeled as a series of discretized compartments representing the organs and tissues of the human body. Fractional retention and excretion in these organ and tissue systems have been mathematically governed by a series of coupled first-order ordinary differential equations (ODEs). The coupled ODE systems comprising the biokinetic models are usually stiff due to the severe difference between rapid and slow transfers between compartments. In this study, the capabilities of solving a complex coupled system of ODEs for biokinetic modeling were evaluated by comparing different Python programming language solvers and solving methods with the motivation of establishing a framework that enables multi-level analysis. The stability of the solvers was analyzed to select the best performers for solving the biokinetic problems. A Python-based linear algebraic method was also explored to examine how the numerical methods deviated from an analytical or semi-analytical method. Results demonstrated that customized implicit methods resulted in an enhanced stable solution for the inhaled 60Co (Type M) and 131I (Type F) exposure scenarios for the inhalation pathway of the International Commission on Radiological Protection (ICRP) Publication 130 Human Respiratory Tract Model (HRTM). The customized implementation of the Python-based implicit solvers resulted in approximately consistent solutions with the Python-based matrix exponential method (expm). The differences generally observed between the implicit solvers and expm are attributable to numerical precision and the order of numerical approximation of the numerical solvers. This study provides the first analysis of a list of Python ODE solvers and methods by comparing their usage for solving biokinetic models using the ICRP Publication 130 HRTM and provides a framework for the selection of the most appropriate ODE solvers and methods in Python language to implement for modeling the distribution of internal radioactivity.

Nonlinear modeling and analysis of spatial pipeline with arbitrary shapes supported by clamps considering displacement-dependent characteristic

Considering the displacement-dependent characteristics of the clamp, a general nonlinear semi-analytical modeling method for the spatial pipeline with arbitrary shapes supported by multiple clamps is proposed to study the vibration characteristics of the pipeline-clamp system. The coupling of the complex spatial pipeline is realized by introducing the connecting springs at the connection region of the straight pipe and curved pipe. The equation of motion for spatial pipeline structure is established based on the Lagrange energy equation and Timoshenko beam theory. High-order polynomials are used to characterize the stiffness and damping characteristics of the clamp, and the simple straight pipe is used as the basic model to reverse the clamp parameters. The influence of clamps on pipeline vibration is skillfully introduced into the nonlinear dynamic model of the spatial pipeline. According to the Newton-Raphson method, the nonlinear equation of motion is solved. Taking the spatial pipeline supported by double clamps as an example, the related experiment tests and ANSYS simulation are carried out, and the effectiveness of the established model is verified by comparing the results. Finally, the nonlinear vibration mechanism of the pipeline-clamp system is analyzed. The modeling method proposed in this paper can provide positive guidance for modeling and vibration analysis of the aero-engine pipeline with arbitrary shapes.

Performance analysis and material distribution optimization for sound barriers using a semianalytical meshless method

AbstractWith the increase in car ownership, traffic noise pollution has increased considerably and is one of the most severe types of noise pollution that affects living standards. Noise reduction by sound barriers is a common protective measure used in this country and abroad. The acoustic performance of a sound barrier is highly dependent on its shape and material. In this paper, a semianalytical meshless Burton–Miller‐type singular boundary method is proposed to analyze the acoustic performance of various shapes of sound barriers, and the distribution of sound‐absorbing materials on the surface of sound barriers is optimized by combining a solid isotropic material with a penalization method. The acoustic effect of the sound‐absorbing material is simplified as the acoustical impedance boundary condition. The objective of optimization is to minimize the sound pressure in a given reference plane. The volume of the sound‐absorbing material is used as a constraint. The density of the nodes covered with the sound‐absorbing material is used as the design variable. The method of moving asymptotes was used to update the design variables. This model completely avoids the mesh discretization process in the finite element method and requires only boundary nodes. In addition, the approach also does not require the singular integral calculation in the boundary element method. The method is illustrated and validated using numerical examples to demonstrate its accuracy and efficiency.

Time discretization for modeling migration of groundwater contaminant in the presence of micro-organisms via a semi-analytic method

The parabolic systems for migration of groundwater contaminants are studied through the semi-analytic method (SAM). For systems, a modification of the classic trigonometric basis function (TBF) is proposed that successfully obtains the solution on the available boundary data and improves accuracy. The method is based on a theory that removes instability and keeps the same accuracy. However, especially when using SAM schemes with acceptable stability properties, one is still faced with the considerable task of linearizing the systems, consequently, the linearization of the systems which approximate the nonlinear term with the first order derivative is introduced. The final approximation is given by the summation of the primary approximation, radial basis functions (RBFs), and the related TBFs which are determined by the homogeneous boundary conditions. Then the approximation is substituted back to the governing equations where the unknown coefficients can be determined. The efficiency of the algorithm is highlighted by numerical simulations relating to a model on test cases with analytical solutions and without an analytical solution.

Theory and optimisation of radiative recombination in broken-gap InAs/GaSb superlattices

We present a theoretical analysis of mid-infrared radiative recombination in InAs/GaSb superlattices (SLs). We employ a semi-analytical plane wave expansion method in conjunction with an 8-band Hamiltonian to compute the SL electronic structure, paying careful attention to the identification and mitigation of spurious solutions. The calculated SL eigenstates are used directly to compute spontaneous emission spectra and the radiative recombination coefficient . We elucidate the origin of the relatively large coefficients in InAs/GaSb SLs which, despite the presence of spatially indirect (type-II-like) carrier confinement, are close to that of bulk InAs and compare favourably to those calculated for mid-infrared type-I pseudomorphic and metamorphic quantum well structures having comparable emission wavelengths. Our analysis explicitly quantifies the roles played by carrier localisation (specifically, partial delocalisation of bound electron states) and miniband formation (specifically, miniband occupation and optical selection rules) in determining the magnitude of and its temperature dependence. We perform a high-throughput optimisation of the room temperature coefficient in InAs/GaSb SLs across the 3.5–7 m wavelength range, quantifying the dependence of on the relative thickness of the electron-confining InAs and hole-confining GaSb layers. This analysis provides guidance for the growth of optimised SLs for mid-infrared light emitters. Our results, combined with the expected low non-radiative Auger recombination rates in structures having spatially indirect electron and hole confinement, corroborate recently observed high output power in prototype InAs/GaSb SL inter-band cascade light-emitting diodes.

Semi-analytical Approach to Nonlinear Partial Differential Equations Using Homotopy Analysis Technique (HAM)

This work considers a novel semi-analytical method named the homotopy analysis method (HAM) to study the nonlinear gas dynamic equation. The obtained HAM solution is validated by comparing it with the exact available solution and compared with the (Adomian decomposition method) ADM solution and numerical solution to test the efficiency of the proposed method. The efficiency of the proposed approach can be demonstrated numerically and graphically, and it is found to be in excellent agreement with the current approach.

A semi-analytical and semi-numerical method for solving plasma instability of nonuniform two-dimensional electron gas

The plasma instability of two-dimensional electron gas (2DEG) is a crucial physical mechanism for generating terahertz radiation in field-effect transistors, especially in high electron mobility transistors (HEMTs). In this paper, we have proposed a new semi-analytical and semi-numerical method to deal with oscillation problems of any nonuniform 2DEG plasma, especially considering the steady-state distribution, which can be calculated and analyzed more quickly than only using numerical calculation. By constructing a wave equation, using the auxiliary function and Wentzel–Kramers–Brillouin approximation method, the wave vector of the plasma wave is obtained. On this basis, combined with the Dyakonov–Shur instability's boundary conditions, the oscillation frequency, the wave amplitude increment, and their correction caused by the nonuniformity can be obtained by numerical calculation. Furthermore, the analytical solution is obtained under reasonable approximate conditions for the linear distribution of electron concentration. It is proved that the electron concentration gradient in the channel will not only attenuate the wave increment but also decrease the plasmonic frequency in the case of linear distribution. Moreover, we get the reasons for the above conclusions through theoretical derivation. We also investigate the effects of various device parameters on attenuation, such as gate length, electron mobility, and voltage, which may explain the difference between the actual and theoretical values in HEMTs and provide new guidance for device design.

Semi-Analytical Prediction of Ground Surface Heave Induced by Shield Tunneling Considering Three-Dimensional Space Effect

The ground surface deformation induced by shield tunnels passing through enclosure structures of existing tunnels is a particular underground construction scenario that has been encountered in Wuhan Metro Line 12 engineering cases in China. Timely ground deformation prediction is important to keep shield tunneling safe. However, the classic ground deformation theory is difficult to accurately predict for this ground deformation. This paper develops a semi-analytical method to predict ground heave considering the space effect in this engineering condition. Based on the improved ground deformation theory, a novel deformation prediction method for the ground and enclosure structure is derived and combined with Kirchhoff plate theory. Comparing with field deformation measurements, the maximum difference between the measured and calculated deformation is 14.6%, which demonstrates that the proposed method can be used to predict the ground heave induced by shield tunnels passing through the enclosure structure of existing tunnels. The parameters of the underground diaphragm wall used in Wuhan Metro Line 12 are further studied in detail. The results show that the ground heaves have a positive correlation with the embedded ratio of the diaphragm wall, but a negative correlation with its elastic modulus and thickness. However, the thickness and embedded ratio have a limited effect on ground heaves. This study provides a technical reference for optimizing the setting of enclosure structures in order to protect existing buildings.

A new semi-analytic method for calculating the thermal–mechanical coupling stress intensity factor of interfacial crack

The interfacial crack thermal–mechanical coupling stress intensity factor (SIF) calculation plays an important role in anti-cracking design and safety assessment of compressed air energy storage (CAES) multi-layer cavern. In current SIF calculation method, there is no consideration of thermal–mechanical coupling effect caused by the difference between the singularity of stress field (λ) and temperature field (λT). In this study, a new semi-analytic method for calculating the thermal–mechanical coupling SIF of interfacial crack is established based on a new thermal–mechanical coupling Williams expansion for the cases of both insulated and constant temperature. The singularity index of temperature field at the crack-tip is dependent on the material parameters (heat conduction coefficient ki) for the crack perpendicular to and terminated at the interface, but independent on the material parameters (ki) for the crack located at the interface. The larger the difference of bi-material's heat conduction coefficient (ki), the larger the value of stress field when the crack is located on the side of larger Young's modulus (Ei) or on the side of lower ki. The semi-analytic method is degenerately verified by available literature results (without thermal–mechanical coupling term) and completely verified by finite element results (with thermal–mechanical coupling term), which can provide a theoretical basis for anti-cracking optimal design of CAES multi-layer caverns under thermal–mechanical loading condition.