Semi-analytical Solution Research Articles

A simplified theoretical model to predict the shear behavior of concrete/rock joints with Gaussian irregularities

The inherent randomness in the roughness of concrete-rock joints poses challenges for analyzing their shear behaviour. Traditionally, theoretical models simplify these complex geometries by approximating them as regular triangular asperities. This study presents a novel micro-mechanical model developed to simulate the shear behaviour. The model accounts for the complex and random roughness of joint profiles by considering varying asperity heights as Gaussian distribution. Local stress distribution among individual asperities was determined. A failure envelope, mapping joint dilation against asperity height, was obtained, allowing us to identify the critical joint dilation for individual asperities at which individual asperity failure results. The parameter of shear area ratio, which captures the portion of failed asperities relative to all asperities, is introduced, and its semi-analytical solution was derived. Additionally, a wedge-shaped planar failure surface is determined based on laboratory observations and equations to estimate post-failure residual resistance is obtained. A series of direct shear tests on rock/concrete joints with a range of geometries was conducted to validate the proposed model and reasonably close agreement was obtained between predicted and measured performance. An additional case study analyzing the results of field tests on a drilled pile socketed in rock has been conducted using the proposed method. The results showed good agreement with the measured data, demonstrating the validity and practicality of the proposed approach.

A numerical study on the oscillatory dynamics of tip vortex cavitation

In this paper, we numerically study the mechanism of the oscillatory flow dynamics associated with the tip vortex cavitation (TVC) over an elliptical hydrofoil section. Using our recently developed three-dimensional variational multiphase flow solver, we investigate the TVC phenomenon at Reynolds number $Re = 8.95 \times 10^5$ via dynamic subgrid-scale modelling and the homogeneous mixture theory. To begin, we examine the grid resolution requirements and introduce a length scale considering both the tip vortex strength and the core radius. This length scale is then employed to non-dimensionalize the spatial resolution in the tip vortex region, the results of which serve as a basis for estimation of the required mesh resolution in large eddy simulations of TVC. We next perform simulations to analyse the dynamical modes of tip vortex cavity oscillation at different cavitation numbers, and compare them with the semi-analytical solution. The breathing mode of cavity surface oscillation is extracted from the spatial-temporal evolution of the cavity's effective radius. The temporally averaged effective radius demonstrates that the columnar cavity experiences a growth region followed by decay as it progresses away from the tip. Further examination of the characteristics of local breathing mode oscillations in the growth and decay regions indicates the alteration of the cavity's oscillatory behaviour as it travels from the growth region to the decay region, with the oscillations within the growth region being characterized by lower frequencies. For representative cavitation numbers $\sigma \in [1.2,2.6]$ , we find that pressure fluctuations exhibit a shift of the spectrum towards lower frequencies as the cavitation number decreases, similar to its influence on breathing mode oscillations. The results indicate the existence of correlations between the breathing mode oscillations and the pressure fluctuations. While the low-frequency pressure fluctuations are found to be correlated with the growth region, the breathing mode oscillations within the decay region are related to higher-frequency pressure fluctuations. The proposed mechanism can play an important role in developing mitigation strategies for TVC, which can reduce the underwater radiated noise by marine propellers.

Generalised hydrogen interactions with CINCO: a window to new physics

We present semi-analytic solutions for atomic transition rates in hydrogenic atoms induced by scalar, pseudoscalar, vector, axial-vector, and tensor interactions. Our results agree with quantum electrodynamics predictions to ~ 0.005 % precision, and further allow us to calculate absorption and emission rates for axions, hidden photons, light scalars or other dark matter candidates for hydrogen and hydrogenic ions. These results can be used to inform searches for light new physics as well as in calculations relevant to searches for fifth forces or varying fundamental constants, with applications from astrophysics to laboratory spectroscopy experiments. We also provide a dedicated tool for the construction of hydrogenic transition amplitudes: “Computation of hydrogen radial INtegrals and COefficients” (CINCO).

A new semi-analytical approach for nonlinear dynamic responses of functionally graded porous graphene platelet-reinforced circular plates and spherical shells

This paper presents the semi-analytical approach for nonlinear dynamic responses of porous graphene platelet-reinforced circular plates (CiPs) and spherical shells (SpSs) resting on the Pasternak foundation in thermal environment. The graphene platelets (GPLs) are distributed in the functionally graded (FG) distribution patterns and uniform distribution pattern through the CiP and SpS thickness direction. The governing equations of the GPL-reinforced structures subjected to time-dependent external pressure respected to the harmonic and linear functions of time are established based on the geometrically nonlinear higher-order shear deformation theory. A simple and effective semi-analytical solution for the considered problem is established using the Euler-Lagrange equations, and the damping viscosity of the foundation can be counted using the Rayleigh dispassion function. The Runge-Kutta method is employed to determine the dynamic responses of the GPL-reinforced structures, while the fundamental frequencies of free and linear vibration, and frequency-amplitude curves are obtained in explicit form. The numerical results obtained reveal that the GPL and porous distributions, geometrical and material properties can significantly affect the frequency-amplitude curves, and dynamic responses of harmonically loaded and linearly loaded structures, specially, the dynamic buckling phenomenon is clearly observable with the SpSs but not with the CiPs.

Semi-Analytical Solution of Brent Equations

Semi-Analytical Solution of Brent Equations

Galerkin’s Method and Multivariate Newton’s Method for the Nonlinear Deformation of the Magneto-Electro-Elastic Bi-Layered Laminates

In this paper, mathematical modelling for the large deformation of a magneto-electro-elastic rectangular bi-layered laminate with general boundary conditions is presented. Constitutive equations involving the magneto-electro-elastic (MEE) material properties are introduced, Maxwell equations accounts for the electric and magnetic effects are also utilized. First-order shear deformation theory (FSDT) considering the von Karman nonlinear strain is adopted, and the plain strain/stress assumption applicable for thin plate analysis is used. A rather compact set of governing equations related to kinematical variables, electric/magnetic potentials and the Airy stress function is obtained as a consequence of the preliminary condensation for the electro-magnetic state to the plate kinematics. Semi-analytic solution for a bi-layered BaTiO3-CoFe2O4 laminate with specified boundary conditions subjected to various external applied loads is performed. By employing the Bubnov-Galerkin method, the set of nonlinear partial differential equations is transformed to a set of third-order nonlinear algebraic equations for the static deformation due to applied load. Numerical results are carried out by using the multivariate Newton's method with respect to various volume fractions indicating the volume ratio between piezoelectric BaTiO3 layer and piezomagnetic CoFe2O4. From the result, the nonlinearity of the von Karman strain appears to enhance system rigidity as smaller deformations will be detected when external load is applied. Also, some other interesting results are obtained which could be useful to future analysis and design of multiphase composite plates.

EMSA-IK: a real-time evolutionary multi-objective semi-analytical inverse kinematics algorithm for redundant manipulators

Purpose This paper aims to propose a real-time evolutionary multi-objective semi-analytical inverse kinematics (IK) algorithm (EMSA-IK) for solving the multi-objective IK of redundant manipulators. Design/methodology/approach Within EMSA-IK, the parameterization method is applied to reduce the number of optimization variables of the evolutionary algorithm and calculate semi-analytical solutions that meet high target pose accuracy. The original evolutionary algorithm is improved with the proposed adaptive search sub-space strategy so that the improved evolutionary algorithm can be used to efficiently perform global search within the parametric joint space to obtain the global optimal parametric joint angles that satisfy multi-objective constraints. Findings Ablation experiments show the effectiveness of the improved strategy used for evolutionary algorithms. Comparative experiments on different manipulators demonstrate the advantages of EMSA-IK in terms of generalizability and balancing multiple objectives, for example, motion continuity, joint limits and obstacle avoidance. Real-world experiments further validate the effectiveness of the proposed algorithm for real-time application. Originality/value The semi-analytical IK solution that simultaneously satisfies high target pose accuracy and multi-objective constraints can be obtained in real time. Compared to existing semi-analytical IK algorithms, the proposed algorithm achieves obstacle avoidance for the first time. The proposed algorithm demonstrates superior generalizability, applicable to not only redundant manipulators with revolute joints but also those with prismatic joints.

Nonlinear Crack-Wave Modulations in Shear Horizontal Wave Propagation for Fatigue Crack Detection

The nonlinear interaction of shear horizontal waves with fatigue cracks is investigated for structural damage detection. The focus is on the non-classical modulation effect in the presence of damage. The major principle of the method is explained theoretically and supported by a semi-analytical solution. The method is tested experimentally in aluminium beams with fatigue cracks. A combination of high-frequency ultrasonic and low-frequency vibration shear excitations are introduced to monitored beam components using low-profile piezoceramic shear actuators. Nonlinear vibro-acoustic responses are gathered using laser vibrometry. Vibration excitation frequencies are selected using experimental and numerical modal analysis. A series of vibro-acoustic experiments is performed for fatigue crack detection. The results show that the non-classical nonlinear vibro-acoustic effect – based on shear horizontal wave propagation – can be used for crack detection. In addition, the method also allows for damage localisation.

Deep learning-based rapid prediction of temperature field and intelligent control of molten pool during directed energy deposition process

In this work, deep learning-based approaches are proposed to provide promising solutions to address the challenges in realizing intelligent manufacturing and digital twins for directed energy deposition (DED) process. Firstly, a rapid and accurate prediction of part temperature is realized by innovatively combining graph neural networks (GNNs) and recurrent neural networks (RNNs). Twenty parts with different structures are selected for demonstration. GPU parallel computing technique is adopted to accelerate the thermal finite element analysis, which is used to quickly construct the simulated graph dataset with sufficient samples. By embedding the memory optimization method into the GNN block, deeper GNNs with more trainable parameters are successfully trained with a 79.4% lower GPU memory footprint, which solves the difficulty of deeper GNNs are hard to train on large graph datasets, and the accuracy of temperature prediction on unseen DED parts is significantly improved. Secondly, for intelligent molten pool regulation, a semi-analytic temperature solution method is used to create an efficient DED environment in reinforcement learning (RL) workflows. The intelligent control of molten pool depth under complex deposition strategy is realized based on the environmental state represented by molten pool images. A tailored convolutional neural networks (CNNs) model is employed as the agent to output varying laser power and continuously interact with the dynamic environment. Compared with the traditional artificial neural network agent, the total reward scored by the CNN agent is improved by 9.7% in the zigzag deposition process, mitigating the fluctuations in the controlled molten pool depths. Moreover, CNNs are more compatible with in-situ thermal images. This work can provide theoretical and technical support for realizing real-time and even ahead-of-time temperature prediction and the corresponding feedback control during DED process.

Enhanced Heat Transfer in Ternary Tangent Hyperbolic Nanofluids through Non-Darcy Porous Media

The effectiveness of heat transfer fluids (HTFs) is pivotal in maximizing the efficiency and longevity of various devices, from small-scale applications to large industrial systems. A comprehensive understanding of the properties of innovative ternary heat transfer nanofluids (TNFs) is essential, particularly when utilized over a Darcy-Forchheimer porous medium. This study explores tangent hyperbolic thermal nanofluids (TNFs) made up of nanoparticles such as graphene, zirconium oxide and magnesium oxide and, suspended in an ethylene glycol base fluid, within a non-Darcy porous medium. Similarity variables are used to streamline the mathematical representation of fluid flow and heat transmission in TNFs. Then, semi-analytical solutions to the reduced governing equations are obtained using the homotopy analysis method. The influence of tri-nanoparticles, porosity, and the Forchheimer parameter on skin friction, fluid flow dynamics, heat transfer rates, and the Nusselt number is investigated. The Forchheimer parameter lowers the Nusselt number by 27.80% for TNFs, 21.27% for the hybrid nanofluid, and 21.08% for the nanofluid. As a result, the temperature within TNFs is more evenly distributed. TNFs can transfer more heat by raising the medium’s porosity and the tri-nanoparticle volume fraction. These results unveil groundbreaking insights into enhancing the efficiency of heat transfer fluids. The introduction of a porous medium emerges as an alternate strategy to boost the performance of TNFs.

Semi-analytical computation of bifurcation of orbits near collinear libration point in the restricted three-body problem

A unified semi-analytical solution is presented for constructing the phase space near collinear libration points in the Circular Restricted Three-body Problem (CRTBP), encompassing Lissajous orbits, quasihalo orbits, Axial orbits, and their invariant manifolds, as well as transit and non-transit orbits. Based on classical in-plane and out-of-plane frequency resonance mechanisms, the Lindstedt–Poincaré method could only derive separate analytical solutions for the invariant manifolds of Lissajous orbits and halo orbits, falling short for the invariant manifolds of quasihalo orbits. In this paper, by introducing a coupling coefficient η and a bifurcation equation, a unified series solution for these orbits is systematically developed using a coupling-induced bifurcation mechanism and Lindstedt–Poincaré method. Analyzing the bifurcation equation obtained from different coupling forms reveals bifurcation conditions for all kinds of orbits near collinear libration points. When η = 0, the series solution describes non-bifurcated orbits, while when η ≠ 0, the solution describes bifurcated orbits, including quasihalo orbits, Axial orbits, and their invariant manifolds, as well as newly bifurcated transit and non-transit orbits. This unified semi-analytical framework provides a more comprehensive understanding of the complex phase space structures near collinear libration points in the CRTBP.

Analytical, semi-analytical and numerical solutions for global stability analysis in buildings: Double-Beam Systems Timoshenko

In this article, two analytical solutions, one semi-analytical solution, and one numerical solution are proposed to calculate the global critical buckling load of buildings employing a continuum model parallel coupling two Timoshenko beams. The parallel coupling of two Timoshenko beams represents the four types of building behavior: global bending, global shear, local bending, and local shear, the latter of which has been widely ignored in the literature. The model is associated with one translational kinematic field and two rotational fields. The equilibrium equations, constitutive laws, and boundary conditions are derived using a variational approach by applying Hamilton’s principle to the relevant Lagrangian function. Specifically, to solve the classical problem of buildings with uniform properties along their height, the first analytical solution proposes exact closed-form equations for a vertical load applied at the top and a closed-form analytical solution based on the decomposition of two subsystems for a polynomial vertical load uniformly applied along the height of the beam. Meanwhile, the semi-analytical solution exactly solves the differential equation associated with a polynomial vertical load profile and is based on coefficient iteration, converging with only two or three iterations. To address the challenge of buildings with variable properties along their height, the combined application of the continuous method and the transfer matrix method allows us to propose an efficient numerical method leading to a 6 × 6 transfer matrix, constant regardless of the number of discretizations and derived analytically, which results in a computational advantage by drastically reducing computational costs. The numerical results are encouraging and show good agreement with the results of the finite element method, suggesting that the proposed method can be used for engineering purposes in both the preliminary and final phases of the structural analysis of tall buildings.

Unveiling the Dynamics of Nonlinear Landau-Ginzburg-Higgs (LGH) Equation: Wave Structures through Multiple Auxiliary Equation Methods

This comprehensive investigation delves deeply into the intricate dynamics governed by the nonlinear Landau-Ginzburg-Higgs equation. It uncovers a diversity of semi-analytical solutions by leveraging three auxiliary equation methods within the traveling wave framework. This article effectively utilizes the improved Kudryashov, Kudryashov's R, and Sardar's subequation methods. The methods discussed are advantageous because they are easy to implement and suitable for use with the Mathematica package program. Each method yields a distinct set of solutions, scrutinized across all cases. We elucidate the complex wave structures through 3D, 2D, and contour graphical representations, providing profound insights into their underlying characteristics. Furthermore, we scrutinize the influence of parameter variations on these wave structures, thereby offering a comprehensive understanding of their dynamic behavior.

One-dimensional consolidation analysis of layered unsaturated soils: An improved model integrating interfacial flow and air contact resistance effects

Layered unsaturated soils exhibit complex mechanical and physical properties. Owing to the roughness between unsaturated soil interfaces and the presence of irregularly distributed micro-pores, this study explores the laminar flow of pore water and counter-cyclonic flow of pore air through these channels at low velocities. In response to the complex consolidation behavior of unsaturated soils influenced by the flow and air contact resistance, an improved model is developed. The model incorporates the flow contact transfer coefficient (Rω), flow partition coefficient (ηω), air contact transfer coefficient (Ra) and air partition coefficient (ηa). Semi-analytical solutions for pore water pressure, pore air pressure and settlement in layered unsaturated soils are derived by employing the Laplace transform and its inverse transform. The rationality of the model is validated through comparative analysis with existing solutions. Analysis of the improved model yields critical insights: the presence of flow and air contact resistance leads to the development of relative pore pressure and air pressure gradients at interfaces, which diminishes the influence of the permeability coefficients of the water phase (kω) and air phase (ka) on the consolidation process. Moreover, neglecting the flow and air contact resistance effects may lead to an overestimation of settlement.

Cylindrical cavity expansion-contraction solutions in undrained MCC soils with the auxiliary variable approach

Cylindrical cavity exhibits non-self-similarity during contraction process following expansion. Previous studies solve this problem with total strain approach and simple constitutive models, but the approach is not applicable when using an advanced constitutive model. This paper presents a semi-analytical solution for a cylindrical cavity undergoing expansion-contraction in undrained soils with auxiliary variable approach, incorporating the Modified Cam-Clay (MCC) model. The stress states around the cavity are formed by the superposition of initial and superimposed stress states. By treating superimposed effective stresses as self-similar, a semi-analytical solution is derived for solving the cavity expansion-contraction problem. The elastoplastic stress-strain relationship is formulated as a set of first-order differential equations, which can be solved as an initial value problem though Runge-Kutta (RK) method. Then the stress distribution around the cavity during expansion-contraction process can be determined. To validate the proposed approach, a series of well-conduced self-boring pressuremeter (SBP) tests are used to verify the proposed approach, which shows good agreements. Additionally, a FEM simulation incorporating the MCC model is performed, and the simulation results are presented to carry out parametric studies on soil parameters. A significant influence on the range of the plastic and reverse plastic zones is shown for overconsolidation ratio, while the in-situ coefficient of the earth pressure only quantitatively affects the stress distribution.

An improved steady-state semianalytical solution for assessing the two-dimensional hydraulic head distribution induced by an underground dam in a sloping unconfined aquifer

An improved steady-state semianalytical solution for assessing the two-dimensional hydraulic head distribution induced by an underground dam in a sloping unconfined aquifer

Dynamical systems theory applied to short walking trials

Human walking is an extremely complex neuromuscular activity whose simplicity disappears when an attempt is made to provide a quantitative description of the process. The dynamical systems theory provides a framework for analyzing the stability and chaotic nature of dynamical systems, employing Floquet multipliers (FM) and long and short-term Lyapunov exponents (LE), respectively. This report compares FM and LE from three methods: method A (false nearest neighbors and numerical approximation), method B (false nearest neighbors and semi-analytical technique) and method C (singular value decomposition and semi-analytical technique). Data from 33 healthy older adults with no history of falls were used to explain the dynamic system. A surrogate center of mass trajectory was calculated for the analysis of sway in the transverse plane. Results revealed methodological differences in LE and FM calculations with semi-analytical solutions providing closer approximations to observed gait behavior. The long-term LE from Methods A and B were similar, but other LE pairings differed. Method A’s short-term LE indicated chaotic gaits for all subjects, while long-term LE from Methods A and B indicated chaos for half the subjects. Method C showed non-chaotic gait for most subjects. Method B’s FM indicated over 30% of subjects had unstable gait. Method C yielded values of LE and FM that most closely matched the subjects’ gait patterns. This study offers a methodological foundation for gait analysis using short time-series data, facilitating deeper insights into both stability and chaos within gait dynamics.

Semi-analytical solutions for dynamic analysis of fluid conveying pipes by a novel hybrid method

Semi-analytical solutions for dynamic analysis of fluid conveying pipes by a novel hybrid method

A semi-analytical method based on the Green's function for the laser melting process of the metal materials

Thermal responses of the metal materials subjected to intense laser irradiation are complicated due to the phase change in the melting process. In order to simplify this problem, previous studies usually ignored the liquid phase or took the usage of finite element method, which limited the achievement of theoretical breakthroughs. In the present study, a one-dimensional heat transfer model with consideration of phase change was established to describe the melting process of the metal materials caused by a constant intense laser beam. The governing equations with three domains were constructed in the piecewise form: the liquid phase region (melted already), the solid phase region (unmelted yet) and the melting region. The limited discretization method was employed to describe the moving of the soli-liquid interface and the energy exchange. The piecewise governing equations in liquid and solid region were solved separately and analytically by means of the Green's function method, and the calculation result was verified by the finite element simulation. The results of the analytical model were in good agreement with those of the finite element simulation, and the maximal prediction errors of the temperature field prediction between the theoretical model and FE simulation is limited to 2.8 %. In addition, the influences of heat source intensity, material heat conductivity, latent heat on the melting process were discussed in detail. The analytical and simulation results reveal that the semi-analytical solution in piecewise for 1-D laser melting process can be successfully used to reflect the dynamic temperature response of the metal materials and the movement of the solid-liquid interface. This work is of guiding significance for the analytical solution of the multi-dimensional melting process of the metal materials under the fixed and continuous laser beams.

Semi-analytical layerwise solutions for FG MEE complex shell structures

This paper presents a unique analytical layerwise solution for functionally graded magneto-electric-elastic shell with complex geometry. The middle surface of the shell is modeled by a parametric equation and the Lamé parameter and radii of curvature is modeled by Differential Geometry. The mechanical displacements, electrical and magnetic potentials are written in term of a simple cosine layerwise based on a unified formulation. The highly coupled differential equations are discretized by the Chebyshev-Gauss-Lobatto grid points and solved numerically via the Differential Quadrature Method (DQM). Lagrange interpolation polynomials are employed as the basis functions. The highly coupled differential equations are solved for shells subjected to different loads and boundary conditions. Since extremely few results on this topic is available in the literature, benchmarks complex shell problems and their solutions are introduced in this paper for the very first time.