Analytical Numerical Solutions Research Articles

Modeling of the Combined Effect of the Surface Roughness and Coatings in Contact Interaction

The model of indentation of a spherical punch with a rough surface into a thin elastic layer lying on an elastic half-space has been developed. A numerical-analytical solution is suggested based on the two-scales approach. At macroscale, the integral equation of the second kind is reduced to calculate the nominal contact characteristics, taking into account the surface layer thickness and its mechanical characteristics, as well as additional compliance due to surface roughness calculated from the microscale analysis. The influence of the punch roughness and the surface layer mechanical and geometrical characteristics on the nominal contact pressure distribution, on the nominal contact area and the indentation depth, as well as on the real contact area and pressures at the individual contact spots, is analyzed. The developed contact model can be used to analyze the indentation of the punch into the layered elastic base, taking into account the roughness of the punch surface, and can also be used to give a complete analysis of the roughness effect on the contact process at both scale levels. The results can be used to control the indentation of the rough-coated bodies.

Salt intrusion dynamics in a well-mixed sub-estuary connected to a partially to well-mixed main estuary

Abstract. Salt intrusion in estuaries has been exacerbated by climate change and human activities. Previous studies have primarily focused on salt intrusion in the mainstem of estuaries, whereas those in sub-estuaries (those that branch off their main estuaries) have received less attention. During an extended La Niña event from 2021 to 2022, a sub-estuary (the East River estuary) alongside the Pearl River estuary, China, experienced severe salt intrusions, posing a threat to the freshwater supply in the surrounding area. Observations revealed that maximum salinities in the main estuary typically preceded spring tides, exhibiting significant asymmetry in salinity rise and fall over a fortnightly timescale. In contrast, in the upstream region of the sub-estuary, the variation in salinity was in phase with that of the tidal range, and the rise and fall of the salinity were more symmetrical. Inspired by these observations, we employed idealized numerical models and analytical solutions to investigate the underlying physics behind these behaviors. It was discovered that under normal dry conditions (with a river discharge of 1500 m3 s−1 at the head of the main estuary), the river–tide interaction and change in horizontal dispersion accounted for the in-phase relationship between the salinity and tidal range in the upstream region of the sub-estuary. Under extremely dry conditions (i.e., a river discharge of 500 m3 s−1 at the head of the main estuary), salinity variations were in phase with those of the tidal range in the middle as well as the upstream region of the sub-estuary. The variation in salinity in the main estuary along with those in salt dispersion and freshwater influx inside the sub-estuary collectively influenced salinity variation in the well-mixed sub-estuary. These findings have important implications for water resource management and salt intrusion prevention in the catchment area.

A numerical and closed-form analytical solution for the global buckling critical load of tall buildings considering soil flexibility: A coupled shear-flexural model

The critical global buckling load is a fundamental parameter governing the performance of tall buildings, however, the existing literature lacks an efficient solution that comprehensively captures the behavior of tall structures with uniform or variable properties. In response to this gap, this study introduces a novel and generalized solution, employing the coupled shear-flexural model. This solution is designed for determining the critical global buckling load in tall buildings subjected to vertical loads with diverse profiles, while also incorporating soil flexibility. The analytical solution, designed for the uniform continuous model, offers a breakthrough by providing graphical representations facilitating the direct determination of eigenvalues based on a single dimensionless parameter. Addressing the unique challenge posed by tall buildings with variable properties, a numerical transfer matrix method based on a Laplacian approach is proposed. This method enables the direct calculation of the transfer matrix for each level, displaying exceptional convergence in just three iterations. The study sheds light on a crucial finding that emphasizes the adverse impact of rotational flexibility on soil, leading to a decrease in eigenvalues as soil flexibility increases. Importantly, the proposed solution methods not only show excellent accuracy but also serve as invaluable tools for performing parametric analyses. Furthermore, they present cost-effective alternatives to exact methods, providing a solid foundation for studying the overall stability of tall buildings. This research contributes significantly to the existing literature by introducing efficient methodologies that improve our understanding and analysis of the behavior of tall buildings under vertical loads.

Deformations of Single-Crystal Silicon Circular Plate: Theory and Experiment

In this paper, the experimental methodology for the single-crystal circular plate deformation measurement and subsequent procedure for the quantitation of its mechanical properties are developed. The procedure is based on a new numerical-analytical solution of non-linear boundary-value problem for finite deformations of a circular anisotropic plate. Using the developed method, a study of the deformation of single-crystal circular plates formed on the basis of a silicon-on-insulator structure was carried out. The values of residual stresses are determined and it is shown that the presence of these stresses increases the flexural rigidity of the plate by several times.

Axisymmetric Stokes flow of a spherical droplet or slip particle in the presence of a circular disk

This article deals with the axisymmetric flow of a Newtonian fluid droplet moving along the axis of a thin circular disk under the conditions of a low Reynolds number in the quasi-steady limit. The system is immersed in a second Newtonian fluid. With the same geometry, the motion of a slip spherical particle is treated also. Basic solutions are constructed in cylindrical and spherical coordinates according to the appropriate flow region. A combined analytic-numerical solution procedure in conjunction with a dual integral equation is used. Numerical results show that the normalized drag force coefficient acting on the droplet/ slip particle is obtained with good convergence for various values of viscosity ratio, slip parameter, and clearance parameter. The findings demonstrate that the results of the drag coefficient are in good agreement with the limiting cases available in the literature.

Laboratory experiments and theoretical study of pressure and fluid influences on acoustic response in tight rocks with pore microstructure

AbstractWave‐induced fluid flow is considered to be a major source of seismic attenuation and dispersion in porous rocks. From the physical description of partially saturated reservoirs, numerous analytical solutions based on upscaling homogenization theories have been employed to calculate equivalent frequency‐dependent poroelastic media. Nevertheless, dispersion and attenuation predictions are often not reasonably consistent with laboratory and field measurements in a broad frequency range, particularly due to influences of biphasic fluids and their distribution, presence of heterogeneities on various length scales, and pore microstructure. We investigate the role of pore microstructure on pressure and fluid saturation dependence of elastic velocities in tight sandstones. Previous work points out that differentiating the impacts of heterogeneities at various scales on dispersion within seismic exploration and sonic frequencies can be very difficult. In practice, this is because fluid‐related dispersion mechanisms are impossible to be independent. Thus, it is important for a theoretical and more quantitative analysis of the relative contribution of interrelated energy dissipation processes through a better understanding of combined influences due to the presence of microscopic and mesoscopic heterogeneities. Based on microscopic squirt flow and mesoscopic flow in a partially saturated medium, we develop a poroelastic model that allows evaluating the overall frequency‐dependent dispersion via considering a random distribution of fluid heterogeneities as well as the broadly distributed aspect ratio of compliant pores. Experimental validation of the model is accomplished via a comprehensive comparison of predictions with measurements of partially saturated velocities versus pressure and fluid for sandstones with specific pore microstructures.

A kinetic non-steady state analysis of immobilized enzyme systems with external mass transfer resistance

<abstract><p>The goal of this paper is to utilize the homotopy perturbation method (HPM) and Laplace transform to provide an approximate analytical expression to the non-linear time-dependent reaction diffusion equation arising in a mathematical model of an immobilized enzyme system with external mass transfer resistance. This mathematical model is a non-steady, non-linear reaction diffusion equation based on Michaelis–Menten kinetics. Approximate analytical expressions are also provided for various geometries of the enzyme catalytic pellets, namely, planar, cylindrical, and spherical. Obtained semi-analytical expressions are proven to fit for all the parameters appearing in the system and for all the geometries of enzyme catalytic pellets. When comparing the numerical and approximate analytical solutions, satisfactory results are obtained. Also, approximate analytical expressions of the effectiveness factor (EF) of the immobilized system are presented, and the effect of parameters on the EF is also analyzed.</p></abstract>

Influence of thermophysical and acoustic properties on temperature elevation at muscle/bone interface induced by ultrasound hyperthermia

Therapeutic ultrasound heating is a very common technique in the treatment of muscular and joints injuries. However, insufficient or excessive heating delivered to living tissues can be either innocuous or hazardous to the patient. The present work deals with the heat transfer process in living tissues, with particular concern on the behavior at the muscle-bone interface, when ultrasound irradiation is applied. Based on the Generalized Integral Transform Technique (GITT), a hybrid numerical-analytical solution of the Pennes’ equation using the open-source UNIT code (“UNified Integral Transforms”) is proposed, considering different boundary conditions. The influence of physical properties and parameters, such as beam incidence angle, acoustic attenuation coefficients for refracted longitudinal and shear waves in the bone, and blood perfusion in the muscle, were more closely analyzed. Numerical results of temperature profiles showing variations of up to 2 °C for different situations are presented, illustrating a marked dependence of heating patterns on the perfusion rates as well as on the effect of shear waves. The conclusions are that such type of more detailed analysis is essential for the accurate identification of physical properties in living tissues and phantoms and can contribute to the treatment planning for therapeutic ultrasound.

Numerical Solution of Time-fractional BBMB Equation via LRPSM

The time-fractional Benjamin-Bona-Mahony-Burger (BBMB) differential equation plays an important role in explaining the unidirectional propagation of long waves in definite nonlinear differential systems. This work presents an analytical numerical solution of considered equation by Laplace transform with residual power series method (LRPSM) which is a generalized Taylor series together with Laplace transform and the residual error function. Using the proposed approach, the series solution of this equation is obtained. The analytical numerical solution shows that the LRPSM is a reliable and powerful method for solving the time fractional differential equations (FDEs) with less number of terms. The obtained results of numerical solutions are also compared with the exact solution and presented as absolute errors at different time levels.

An improved Coupled Level Set and Volume of Fluid (i-CLSVoF) framework for sessile droplet evaporation

Surface-tension-dominant droplet evaporation is ubiquitous and of importance to many applications. We present an improved Coupled Level Set and Volume of Fluid (i-CLSVoF) framework without explicit interface reconstruction for modelling micro-sized droplets with and without evaporation. In the i-CLSVoF framework, an improved surface tension force model with additional filtering steps to filter un-physical spurious velocities is developed and implemented. A simple, yet efficient, velocity-potential based approach is proposed to reconstruct a divergence-free velocity field for the advection of the free surface during droplet evaporation. This approach fixes the numerical issues resulting from the evaporation-induced velocity jump at the interface. The smeared mass source term approach incorporated in this work guarantees greater numerical stability than the non-smeared approach. Three different evaporation models (constant mass flux, thermally driven evaporation and droplet evaporation at room temperature) are implemented in the i-CLSVoF. Corresponding numerical benchmark cases (dam break, droplet relaxation and droplet evaporation subjected to different evaporation models) are conducted to validate the surface tension and the evaporation models. Good agreement between the numerical and corresponding analytical solutions is found. The model developed in this work shows convincing performance in modelling surface-tension-dominant flow with and without evaporation.

Fractional-order numerical modeling to study chloride ion transport in concrete with fly ash or slag additions.

To better study the chloride ion migration in concrete with fly ash or ground granulated blast furnace slag under low fatigue load, a Caputo time fractional-order chloride diffusion model is developed in this paper. The model, grounded in Fick's second law with a fractional-order derivative, employs an implicit numerical method for discretization, resulting in a fractional-order numerical scheme. The stability and convergence of the scheme are rigorously proven within the paper. The model's unknown parameters are estimated using genetic algorithm with a grid method. To validate the model's effectiveness, its numerical solution is juxtaposed with experimental results from chloride erosion studies. Furthermore, the fitting efficacy of the Caputo time fractional-order numerical scheme is compared with that of the classical Fick's second law numerical scheme and analytical solution. The research findings demonstrate that the fractional-order numerical scheme can more accurately simulate the chloride concentration in concrete containing fly ash or slag. Additionally, the model shows promise in predicting the service life of fly ash or slag concrete.

Stabilized mixed material point method for incompressible fluid flow analysis

This paper proposes novel and robust stabilization strategies for accurately modeling incompressible fluid flow problems in the material point method (MPM). To address the modeling of Newtonian fluids with incompressibility constraints, a new mixed implicit MPM formulation is proposed. Here, instead of solving the velocity and pressure fields as the unknown variables like the typical Eulerian computational fluid dynamics (CFD) solver, the proposed approach adopts a monolithic displacement–pressure formulation inspired by the mixed-form updated-Lagrangian Finite Element Method (FEM). To satisfy the inf–sup stability condition, two stabilization strategies are integrated into the formulation: the variational multiscale method (VMS) and the pressure-stabilization Petrov–Galerkin method (PSPG). By concurrently solving the displacement and pressure fields, the developed monolithic solver obviates the need for free-surface detection as well as Dirichlet and Neumann pressure imposition, in contrast to the fractional-step method. This attribute mitigates spurious pressure and velocity oscillations in simulating dynamic and transient flow problems. This study also addresses other prevalent challenges in MPM simulations, such as the pressure oscillations triggered by cell-crossing errors, particle-distribution-induced quadrature errors, and particle-grid information transfer. To resolve these issues, the quadratic B-Spline basis function, the delta-correction method, and the Taylor particle-in-cell method are incorporated into the proposed mixed MPM formulation, thereby enhancing numerical stability. The efficacy of the proposed stabilized incompressible MPM framework is validated through several benchmark cases, comparing the obtained results with other numerical methods and analytical solutions. Furthermore, the method’s capability in simulating real-world problems involving violent free-surface fluid motion is demonstrated through comparisons with experimental results of water sloshing and dam break scenarios.

The One-Dimensional (1D) Numerical Model: An Application to Oxygen Diffusion in Mitochondria Cell

The first model of oxygen transport was formulated by August Krogh. However, the investigations conducted have yet to yield a complete analytical model and a widely applicable solution for One-Dimensional (1D) network construction. The research sought to provide numerical and analytical solutions for the oxygen transfer model in mitochondrial cells to enable researchers to estimate the molecular dynamics and diffusion characteristics in mitochondrial cells. The oxygen diffusion process in mitochondria was modeled with ID numerical models. The numerical models used to solve the equations were explicit and implicit. The explicit model consisted of Forward Time Center Space (FTCS) and DuFort-Frankel. Meanwhile, the implicit model had Crank-Nicholson and Laasonen. The numerical solutions of the explicit and implicit were divided into four scenarios with a variation of Δt and compared with the analytical solutions. The results show that the Laasonen method is the best in describing the diffusion process. The best scenario with the lowest slope value and small Root Mean Square Error (RMSE) value is scenario 2 (Δt = 3,33E-4 s and Δx = 2,00E-5 cm). The numerical model and analytical solution show that the time required to reach a steady state is 0,7 s. It indicates oxygen exchange in two sides of the mitochondrial cell after 0,7 s.

Self-Diffusiophoresis and Symmetry-Breaking of a Janus Dimer: Analytic Solution

A self-diffusiophoretic problem is considered for a chemically active dimer consisting of two equal touching spherical colloids that are exposed to different fixed-flux and fixed-rate surface reactions. A new analytic solution for the autophoretic mobility of such a catalytic Janus dimer is presented in the limit of a small Péclet number and linearization of the resulting Robin-type boundary value problem for the harmonic solute concentration. Explicit solutions in terms of the physical parameters are first obtained for the uncoupled electrostatic and hydrodynamic problems. The dimer mobility is then found by employing the reciprocal theorem depending on the surface slip velocity and on the normal component of the shear stress acting on the inert dimer. Special attention is given to the limiting case of a Janus dimer composed of an inert sphere and a chemically active sphere where the fixed-rate reaction (Damköhler number) is infinitely large. Examples are given, comparing the numerical and approximate analytic solutions of the newly developed theory. Singular points arising in the model are discussed for a dimer with a fixed-rate reaction, and the flow field around the dimer is also analysed. The new developed theory introduces a fast way to compute the mobility of a freely suspended dimer and the induced flow field around it, and thus can also serve as a sub grid scale model for a multi-scale flow simulation.

A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation

One of the issues in numerical solution analysis is the non-linear distributed-order fractional Bagley–Torvik differential equation (DO-FBTE) with boundary and initial conditions. We solve the problem by proposing a numerical solution based on the shifted Legendre Gauss–Lobatto (SL-GL) collocation technique. The solution of the DO-FBTE is approximated by a truncated series of shifted Legendre polynomials, and the SL-GL collocation points are employed as interpolation nodes. At the SL-GL quadrature points, the residuals are computed. The DO-FBTE is transformed into a system of algebraic equations that can be solved using any conventional method. A set of numerical examples is used to verify the proposed scheme’s accuracy and compare it to existing findings.

Semi Analytic-Numerical Solution of Imbibition Phenomenon in Homogeneous Porous Medium Using Hybrid Differential Transform Finite Difference Method

Semi Analytic-Numerical Solution of Imbibition Phenomenon in Homogeneous Porous Medium Using Hybrid Differential Transform Finite Difference Method

Numerical and Analytical Investigation of Shock Wave Processes in Elastoplastic Media

The Wilkins model for an elastoplastic medium is considered. A theoretical analysis of discontinuous solutions under the assumption of one-dimensional uniaxial strain is performed. In this approximation, the material equations for the deviator stress tensor components are integrated exactly, and only the conservative part of the governing equations remains, which makes it possible to derive a class of exact analytical solutions for the model. To solve the full nonconservative system of equations (without assuming the uniaxial strain), a Godunov-type numerical method is developed, which uses an approximate Riemann solver based on integrating the system of equations along a path in the phase space. A special choice of path is proposed that reduces the two-wave HLL approximation to the solution of a linear equations. Numerical and exact analytical solutions are compared for a number of problems with various regimes of shockwave processes.

An efficient HLL-based scheme for capturing contact-discontinuity in scalar transport by shallow water flow

An efficient numerical scheme is developed for coupled hydrodynamic and scalar transport systems to guarantee the conservation and the positivity-preserving properties for water depth and scalar concentration. A second-order well-balanced positivity-preserving central-upwind scheme based on the finite volume method is adopted to discretize both the Saint-Venant system and the advective fluxes in the scalar transport system. In particular, an anti-diffusion modification is augmented in an ad hoc manner to the Harten–Lax–van Leer approximate Riemann solver for the scalar transport system, with the aim of significantly reducing the numerical diffusion near contact discontinuities. The proposed scheme is validated through seven numerical experiments, wherein the advection and diffusion processes of scalar transport are considered either separately or in combination. Accuracy is measured based on the difference between the numerical results and analytical solutions, and the performance of the proposed scheme is assessed by comparing it with that of the existing scheme. Convergence analysis confirms that the proposed scheme is accurate to the second order. The stationary steady state with wet-dry fronts is simulated, which verifies that the proposed scheme can preserve the exact C-property. The results for the pure diffusion case reveal that the original scheme has limitations in precisely predicting scalar diffusion when the diffusion coefficient is small. By contrast, the proposed scheme accurately approximates scalar diffusion, even at low diffusivity. The results under flow conditions with various Froude and Péclet numbers confirm that the proposed scheme is more accurate than the existing scheme in solving scalar transport problems over a wide range of shallow water flows.

Electrothermal actuation of NEMS resonators: Modeling and experimental validation

We study the electrothermal actuation of nanomechanical motion using a combination of numerical simulations and analytical solutions. The nanoelectrothermal actuator structure is a u-shaped gold nanoresistor that is patterned on the anchor of a doubly clamped nanomechanical beam or a microcantilever resonator. This design has been used in recent experiments successfully. In our finite-element analysis (FEA) based model, our input is an ac current; we first calculate the temperature oscillations due to Joule heating using Ohm’s law and the heat equation; we then determine the thermally induced bending moment and the displacement profile of the beam by coupling the temperature field to Euler–Bernoulli beam theory with tension. Our model efficiently combines transient and frequency-domain analyses: we compute the temperature field using a transient approach and then impose this temperature field as a harmonic perturbation for determining the mechanical response in the frequency domain. This unique modeling method offers lower computational complexity and improved accuracy and is faster than a fully transient FEA approach. Our dynamical model computes the temperature and displacement fields in the time domain over a broad range of actuation frequencies and amplitudes. We validate the numerical results by directly comparing them with experimentally measured displacement amplitudes of nano-electro-mechanical system beams around their eigenmodes in vacuum. Our model predicts a thermal time constant of 1.9 ns in vacuum for our particular structures, indicating that electrothermal actuation is efficient up to ∼80 MHz. We also investigate the thermal response of the actuator when immersed in a variety of fluids.

Emergence of chaotic hysteresis in a second-order non-autonomous chaotic circuit

The observation of hysteresis in the switching between the attractors of a third-order autonomous chaotic system namely, the Chua’s circuit has been reported recently (Gomes et al., 2023). In the present work, we make a detailed study on the emergence of chaotic hysteresis in the switching between the attractors of MLC circuit. The experimental realization of chaotic hysteresis is further validated by numerical simulation and analytical solutions. The presence of chaotic hysteresis in a second-order non-autonomous electronic circuit is reported for the first time. Multistable regions are observed in the dynamics of MLC with constant bias.