Efficient Numerical Approach Research Articles

Stability analysis study for the time-fractional Galilei invariant advection-diffusion model of distributive order using an efficient hybrid approach

Abstract In this manuscript, a new model of the time-fractional Galilei-invariant advection-diffusion model of distributed order is studied. An efficient hybrid numerical approach with high accuracy is used to estimate this equation. The finite difference numerical method is used to approximate the fractional operator in terms of the time variable and to approximate the integral term of distributed order, the Gaussian–Legendre integration is applied. To obtain a fully discrete numerical approach, we used a spectral element numerical approach, in which Legendre polynomials are used as the basis function. For the proposed numerical approach, the error and stability analysis are studied. For the efficiency of the numerical approach, some numerical examples are presented with graphs and tables.

An efficient numerical approach for solving three‐dimensional Black‐Scholes equation with stochastic volatility

An efficient numerical approach for solving three‐dimensional Black‐Scholes equation with stochastic volatility

Grassmann time-evolving matrix product operators: An efficient numerical approach for fermionic path integral simulations.

Developing numerical exact solvers for open quantum systems is a challenging task due to the non-perturbative and non-Markovian nature when coupling to structured environments. The Feynman-Vernon influence functional approach is a powerful analytical tool to study the dynamics of open quantum systems. Numerical treatments of the influence functional including the quasi-adiabatic propagator technique and the tensor-network-based time-evolving matrix product operator method have proven to be efficient in studying open quantum systems with bosonic environments. However, the numerical implementation of the fermionic path integral suffers from the Grassmann algebra involved. In this work, we present a detailed introduction to the Grassmann time-evolving matrix product operator method for fermionic open quantum systems. In particular, we introduce the concepts of Grassmann tensor, signed matrix product operator, and Grassmann matrix product state to handle the Grassmann path integral. Using the single-orbital Anderson impurity model as an example, we review the numerical benchmarks for structured fermionic environments for real-time nonequilibrium dynamics, real-time and imaginary-time equilibration dynamics, and its application as an impurity solver. These benchmarks show that our method is a robust and promising numerical approach to study strong coupling physics and non-Markovian dynamics. It can also serve as an alternative impurity solver to study strongly correlated quantum matter with dynamical mean-field theory.

An efficient moving-mesh strategy for predicting crack propagation in unidirectional composites: Application to materials reinforced with aligned CNTs

This paper presents an efficient numerical approach for reproducing the process of crack propagation inside unidirectional composites subjected to general loading conditions, with special reference to epoxy materials enhanced with embedded aligned CNTs. This approach involves a traditional FE framework improved by the Moving Mesh (MM) technique based on the Arbitrary Lagrangian-Eulerian (ALE) formulation and the Interaction Integral Method (M−integral). The MM serves as a powerful numerical tool to simulate the discrete crack advance with minimal remeshing, thus reducing the computational complexities. Instead, the M−Integral method, formulated for generally anisotropic materials, has been employed to extract the mixed-mode Stress Intensity Factors (SIFs), which are necessary to define the crack onset condition and propagation direction on the basis of the modified Maximum Hoop Stress Criterion. The proposed strategy includes the extended rule of mixtures to evaluate the homogenized elastic properties of nano-reinforced composites. The validity of the proposed methodology has been assessed through comparisons with experimental data and numerical results available in the literature.

An Efficient Numerical Approach to Solve Fractional Coupled Boussinesq Equations

An Efficient Numerical Approach to Solve Fractional Coupled Boussinesq Equations

Simulation of the SIR dengue fever nonlinear model: A numerical approach

This paper introduces a new numerical technique that utilizes the Chebyshev collocation method to solve the nonlinear SIR mathematical model for dengue disease dynamics. The main goal of this study is the development of a highly accurate and efficient numerical approach that allows for the simulation of dengue fever infection and transmission mechanisms. The suggested methodology provides a strong and adaptable tool for studying the intricate dynamics of dengue fever, which is essential for improving our comprehension and control of this significant public health problem. The numerical approach is totally programmable and can be customized to incorporate a user-friendly interface for data input, hence enhancing its accessibility and practicality for researchers and public health practitioners. The conducted numerical simulations provide evidence of the accuracy and computational efficiency of the suggested technique. This unique numerical framework is positioned as a standard for managing and regulating a diverse array of biological problems. The creation of this sophisticated numerical tool signifies significant progress in the realm of dengue fever modeling. It offers a versatile and dependable platform to investigate different epidemiological scenarios and assess the possible effects of intervention efforts. The results of this study have significant consequences for enhancing dengue monitoring, safeguarding, and control initiatives in areas where the disease is prevalent.

Numerical Solution of the Newtonian Plane Couette Flow with Linear Dynamic Wall Slip

An efficient numerical approach based on weighted-average finite differences is used to solve the Newtonian plane Couette flow with wall slip, obeying a dynamic slip law that generalizes the Navier slip law with the inclusion of a relaxation term. Slip is exhibited only along the fixed lower plate, and the motion is triggered by the motion of the upper plate. Three different cases are considered for the motion of the moving plate, i.e., constant speed, oscillating speed, and a single-period sinusoidal speed. The velocity and the volumetric flow rate are calculated in all cases and comparisons are made with the results of other methods and available results in the literature. The numerical outcomes confirm the damping with time and the lagging effects arising from the Navier and dynamic wall slip conditions and demonstrate the hysteretic behavior of the slip velocity in following the harmonic boundary motion.

Enhancing the stability and efficiency of the advanced concrete structures surrounded by an auxetic foundation

This study delves into the vibrational characteristics of concrete annular plates reinforced with graphene oxide powders (GOP), underpinned by an innovative auxetic-elastic foundation. By applying Hamilton’s principle, the governing dynamic equations are meticulously derived, incorporating the effects of internal forces and GOP geometrical properties. The Differential Quadrature Method (DQM) is utilized to solve these equations, providing a robust and efficient numerical approach for analyzing the system’s complex dynamic behavior. GOP reinforcement significantly enhances the mechanical properties of the concrete plates, yielding improved stiffness and superior damping capabilities. The auxetic-elastic foundation, known for its unique property of becoming thicker perpendicular to the applied force, plays a pivotal role in altering the vibrational response of the system. The study meticulously examines the influence of this foundation on the system’s vibrational behavior, highlighting its potential to enhance performance under varying operational conditions. A comprehensive analysis is conducted to understand the effects of different boundary conditions on the vibrational characteristics of the reinforced concrete annular plates. The results underscore the importance of integrating advanced materials like GOP and innovative foundation designs in optimizing structural components. This study provides valuable insights into the design and application of reinforced concrete structures, emphasizing the effectiveness of GOP reinforcement and auxetic-elastic foundations in enhancing vibrational performance. These findings offer a significant contribution to the field of structural engineering, paving the way for future innovations in the design of resilient and efficient structural systems.

An efficient numerical approach for singularly perturbed time delayed parabolic problems with two-parameters

ObjectivesThe main objective of this work is to design an efficient numerical scheme is proposed for solving singularly perturbed time delayed parabolic problems with two parameters.ResultsThe scheme is constructed via the implicit Euler and non-standard finite difference method to approximate the time and space derivatives, respectively. Besides, to enhance the accuracy and order of convergence of the method Richardson extrapolation technique is employed. Intensive numerical experimentation has been done on some model examples. Further, the layer behavior of the solutions is presented using graphs and observed to agree with the existing theories. Finally, error analysis of the scheme is done and observed that the proposed method is parameter uniform convergent with the order of convergence (Δt)2+h2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( (\\Delta t )^2+h^2\\right) $$\\end{document}

Stress-driven design of incompressible multi-materials under frequency constraints

This paper presents an efficient and comprehensive numerical approach to address topology optimization challenges within the multi-material framework. The focus encompasses several key aspects: solving various stress-related problems from compressible to incompressible materials, broadening the scope to encompass multi-material systems, adopting a flexible methodology suitable for various elements (triangular, quadrilateral, and polygonal), and expanding to incorporate frequency constraints within the stress-driven system. The core idea to accommodate a broad range of materials, from compressible to nearly incompressible, is to implement a specialized polytopal composite finite element (PCE) technique to mitigate volumetric locking issues often prevalent in nearly incompressible materials. Then, to effectively solve the stress-related multi-material system, a well-known P-norm function integrated with the Moved and Regularized Heaviside function (MRHF) is employed, capitalizing on the correlation between topological phases and material allocation to handle multi-material problems effectively. Additionally, to deal with frequency-related multi-material problems, an extended technique addressing localized mode issues based on the relationship between representative solid and void is also employed. Several numerical examples are tested to validate the method’s efficiency and reliability within a multi-material framework, considering a diverse range of materials from compressible to nearly incompressible.

Analysis of the co-rotating vortex pair as a test case for computational aeroacoustics

The co-rotating vortex pair arrangement is a frequently applied test case in computational aeroacoustics. Its flow quantities and far-field radiated sound pressure can be computed analytically, assuming delta-distribution potential vortices. When this configuration is used for verification in a numerical computation, the singular vortex cores cannot be modeled directly; they have to be desingularized, which affects the associated flow and acoustical fields. Closed-form expressions for the aeroacoustic source terms are derived for the Scully vortex model and used for detailed numerical computations of the radiated sound pressure. The effects of desingularization and finite source region size are analyzed in the frequency domain for Lighthill’s equation and the perturbed convective wave equation by means of an efficient numerical approach that computes the convolution of the aeroacoustic source terms and the 2D free-field Green’s function. It is shown that the deviations of the far-field radiated sound pressure from the idealized analytical solution depend on the Mach number, vortex core radius, and truncation limit of the source region. A significant increase in the effective size of the source region due to the desingularization is demonstrated. Minimal error bounds compared to the analytic far-field pressure solution are established. The presented converged numerical results serve as reference solutions for validating implementations of hybrid computational aeroacoustic workflows.

Multichannel multimodal piezoelectric middle ear implant concept based on MEMS technology for next-generation fully implantable cochlear implant applications

This paper introduces a unique multimode, multichannel piezoelectric vibration sensor for the next-generation fully implantable cochlear implant (FICI) systems. The sensor, which can be implanted on the middle ear chain to collect and filter the ambient sound in eight frequency bands, comprises an array of 4 M-shape multimode and 11 single cantilevers. Finite element (FE) analysis indicates a 2.05-fold improvement in capturing frequency information for the multimodal sensor compared to its single-mode counterpart. Under an acoustic excitation at 100 dB SPL, the sensor, mounted on an artificial tympanic membrane, yielded a peak output voltage of 546.16 mVpp and a peak sensitivity of 285.28 mVpp/Pa at 1613 Hz. The extrapolated acoustic results indicated a dynamic frequency range between 300 Hz and 6 kHz, even at 30 dB SPL. Furthermore, a lightweight titanium coupler, employing a two-sided clipping structure with a maximum wall thickness of 70 μm, is micromachined for surgical attachment of the transducer to the middle ear chain. A commercial accelerometer, implanted on the incus short process (SP) of a cadaver using the titanium coupler, successfully recorded 0.1 g for 100 dB SPL at 500 Hz, revealing the potential feasibility of the coupler for vibration sensor implantation. Moreover, the presented anatomically accurate FE model of the middle ear, exhibiting a high correlation coefficient (R2) of 0.97 with the cadaveric experiment, suggests an efficient numerical approach for evaluating the implantation of middle ear prostheses. In this regard, the study holds great promise for clinical application in the field of implantable hearing aids.

Wave finite element method based on isogeometric analysis for periodic structures

In this paper, an efficient numerical approach is proposed to calculate the band structure of periodic structures. It combines the advantages of isogeometric analysis (IGA) and the wave finite element method (WFEM). By utilizing NURBS basis functions, the new method achieves high−order cross−element continuity, resulting in a more accurate evaluation of the band structure. The high order element exhibits improved stability and faster convergence. It also has small dispersion errors, thereby partially overcoming the limitations imposed by the thumb rule. However, in scenarios involving material sudden change, IGA−WFEM necessitates the insertion of repeated knots to prevent over continuity in the analysis. The aforementioned characteristics of the method are demonstrated through numerical evaluations of one− and two−dimensional periodic structures.

Static analysis of functionally graded porous beam-column frames by the complementary functions method

In this paper, an efficient and accurate numerical approach is implemented to examine the static behavior of Functionally Graded Porous (FGP) planar frames for the first time. The symmetric material constitutive relationship (SMCR), monotonic material constitutive relationships (MMCR), and uniform material constitutive relationship (UMCR) models are used to define the gradation of porosity along the thickness direction. The effects of porosity coefficient and material porosity distribution type on the static behavior of the FGP frames are investigated in detail. A direct relation between porosity coefficient and nodal deformations is observed.The effect of shear deformation is considered in the governing equations based on the Timoshenko Beam Theory (TBT). Within the scope of the presented procedure, fifth-order Runge-Kutta (RK5) algorithm is applied in the solution of the initial value problems. Imposing the rigidity matrix-based Complementary Functions Method (CFM) on the static analysis of the FGP planar frames is the novelty of this research. The results of the suggested approach are verified with the those of the finite element method (FEM), and those of the available literature. It has been carried out that type of the porosity distribution and porosity gradient index have an important impact on the response of FGP planar frames. By increasing the porosity coefficient, the displacement values of the frames made of UMCR and MMCR porosity distribution are affected more than those of the UMCR distributions. It can be noted that the SMCR can be preferred for designing the FGP beam-column structure as the minimum deformation values are obtained.

A Robust Numerical Method for a Singularly Perturbed Semilinear Problem with Integral Boundary Conditions

In the present study, we provide an efficient numerical approach for solving singularly perturbed nonlinear ordinary differential equations with two integral boundary conditions. We specifically propose a numerical approach for the solution of a nonlinear singular perturbed problem with integral boundary conditions. To solve the nonlinear singularly perturbed issue, we also apply finite difference methods. It explores how a specific derivative and a problemsolving approach behave. Finally, a numerical technique that employs a finite difference scheme is built using a nonuniform mesh.

Developing Mathematical Models to Study Changes in Groundwater Levels and Salt Concentration

Understanding how groundwater and salt concentrations change when freshwater supplies are monitored is crucial. In this work, groundwater fluxes and salt migration were considered when studying groundwater level changes and mineralisation processes. To study salt and water level changes, a more realistic mathematical model was developed to reflect the object’s basic features. To monitor geofiltration and geomigration using mathematical and numerical modelling to provide recommendations, nonlinear differential equations were applied. Scientific research on mathematical and numerical modelling and computer experimentation of the issue is briefly reviewed. Also developed were a mathematical model with two integer-valued boundary conditions and an efficient finite difference numerical approach. Changing groundwater and pressure water levels, filtration permeability, water loss coefficient, and filtration rate connected to water level all affect the environment when building a mathematical model for salt migration filtration. The article predicts groundwater, pore water, and salt concentrations in high- and low- permeability layers. Scientific research on mathematical and numerical modelling and computer experimentation of the issue is briefly reviewed. Also developed were a mathematical model with two integer-valued boundary conditions and an efficient finite difference numerical approach. Changing groundwater and pressure water levels, filtration permeability, water loss coefficient, and filtration rate connected to water level all affect the environment when building a mathematical model for salt migration filtration. This study considered soil density, active porosity, and third-order boundary conditions Open, unlike earlier studies. Since the problem is represented by nonlinear differential equations with free variables, analytical solutions are unattainable. An effective numerical method and a stable implicit system with high-accuracy approximation, finite difference approach, and forward and backward sweep approaches were given to overcome these challenges.

On rotavirus infectious disease model using piecewise modified $ ABC $ fractional order derivative

<abstract><p>The goal of this manuscript is to use a mathematical model with four compartments to examine the positive effects of rotavirus vaccinations. Susceptible, vaccinated, infected, and recovered (SVIR) classes are included in the suggested model. Some qualitative conclusions are established for the complicated pediatric disease epidemic model of rotavirus, which travels through a population at an inconsistent rate. The model has been fitted with piecewise equations of non-singular kernel-type derivatives in the modified Atangana-Balaneu-Caputo $ (mABC) $ sense. Using the Laplace transform and the notion of non-singular-type derivatives, we prove several basic conclusions regarding the solution's feasibility and positivity. We have used the matrix approach to compute the reproductive number further. Also, the sensitivity of the model has been computed. Additionally, we have used an efficient numerical approach to simulate the model by using some numerical values for the nomenclature of the model. Additionally, using the numerical approach, various graphical illustrations are given.</p></abstract>

A NURBS-based IGA using zig-zag plate theory for nonlinear passive/semi-active damping analysis of laminated FG-CNTRC plates

This study introduces an efficient and versatile numerical approach for simulating passive/semi-active damping treatments on laminated structures composed of functionally graded carbon nanotube-reinforced composites (FG-CNTRC) plates. This novel approach integrates an isogeometric finite element formulation that leverages basic functions generated from Non-Uniform Rational Basis Spline (NURBS) with the Murakami zig-zag theory, Von Kármán’s nonlinear strain–displacement model, and the electro-mechanical coupling properties of the constraining layers. The proposed isogeometric formulation can be transformed into the Laplace domain using the Golla–Hughes–McTavish model to successfully simulate the time-dependent passive/semi-active damping mechanism of viscoelastic materials. Subsequently, results obtained in the Laplace domain are reconverted into the time domain using inverse techniques. The validation of the proposed model through two numerical examples showcases its high reliability and demonstrates the feasibility of simulating and controlling FG-CNTRC plates with various geometries within a Computer-Aided Design-Computer-Aided Engineering (CAD-CAE) analysis framework.

An Efficient Numerical Approach to Modeling the Effects of Particle Shape on Rubble-pile Dynamics

We present an approach for the inclusion of nonspherical constituents in high-resolution N-body discrete element method (DEM) simulations. We use aggregates composed of bonded spheres to model nonspherical components. Though the method may be applied more generally, we detail our implementation in the existing N-body code pkdgrav. It has long been acknowledged that nonspherical grains confer additional shear strength and resistance to flow when compared with spheres. As a result, we expect that rubble-pile asteroids will also exhibit these properties and may behave differently than comparable rubble piles composed of idealized spheres. Since spherical particles avoid some significant technical challenges, most DEM gravity codes have used only spherical particles or have been confined to relatively low resolutions. We also discuss the work that has gone into improving performance with nonspherical grains, building on pkdgrav's existing leading-edge computational efficiency among DEM gravity codes. This allows for the addition of nonspherical shapes while maintaining the efficiencies afforded by pkdgrav's tree implementation and parallelization. As a test, we simulated the gravitational collapse of 25,000 nonspherical bodies in parallel. In this case, the efficiency improvements allowed for an increase in speed by nearly a factor of 3 when compared with the naive implementation. Without these enhancements, large runs with nonspherical components would remain prohibitively expensive. Finally, we present the results of several small-scale tests: spin-up due to the YORP effect, tidal encounters, and the Brazil nut effect. In all cases, we find that the inclusion of nonspherical constituents has a measurable impact on simulation outcomes.

Highly efficient and reliable numerical approach for predicting automotive hood displacement: consideration of composite materials and weatherstrip structures

The automotive hood, a closure system for the engine compartment, must be mounted in a suitable position to avoid irregular gaps and flushes when it is initially assembled. The hood panel is mounted with an elastomer part called weatherstrip that is generally made of ethylene propylene diene monomer foam material to seal and reduce vehicle vibration and whistling noise. Because the elastomer material exhibits non-linear mechanical properties and viscoelasticity, which result in unpredictable reaction forces and compression set, the finite element analysis (FEA) of the automotive hood panel displacement with weatherstrip model shows low precision and long computing time. To resolve this issue, in this study, a cost-effective methodology for FEA is introduced by applying compressive stress of a weatherstrip to the distributed surface load in the finite element model without weatherstrip modeling. In addition, a 2D FEA of the weatherstrip is performed to investigate the effects of structural parameters on the reaction force. This work demonstrates a numerical approach for predicting automotive hood displacement by considering the material properties and the structural design variables of the weatherstrip. Therefore, this approach can be applied to other closer parts where it requires highly efficient yet reliable predictive results for mounting position.