Efficient Numerical Method Research Articles

Simulation of transient heat transfer in multilayer walls with photovoltaic cell and air by using efficient numerical methods

This paper focuses on simulating nonlinear transient heat transfer in multilayer walls under varying heat loads, using efficient numerical algorithms. The algorithms employed are completely explicit and unconditionally stable for the linear conduction-convection case. The study investigates heat transfer through building walls, considering different wall geometries and heat load scenarios, encompassing both cooling and heating. Our primary objective is to analyse how heat transfer depends on the wall materials and evaluate algorithm performance in cases involving heat transfer between solid surfaces and fluid (convection) on the outdoor surface, particularly across an air gap between insulation and Photovoltaic Cells (PVC). The results reveal that insulation prevents heat from entering the building, maintaining a comfortable indoor environment. Forced convection significantly enhances heat dissipation, especially during cooling operations to protect PVC with limited working temperature. Furthermore, the simulations highlight the air gap's efficiency in cooling PVC and reducing maximum temperatures on the insulation's outer surface, especially under forced convection conditions. The test results show that the leapfrog hopscotch algorithm offers the best solution for this highly stiff system, followed by the asymmetric and shifted hopscotch algorithms. This study offers insights into optimizing heat transfer within multilayer walls, with implications for building energy efficiency and thermal comfort.

Nanoparticle aggregation effect over MHD oblique stagnation point flow of the convective surface of nanoliquids

This research could be useful in the construction of heavy power engines and the cooling of nuclear reactors, as well as in the insulation of buildings, the management of petroleum reservoirs, and the production of solar energy collectors. In this study, a numerical model is used to look at the steady, laminar, and incompressible MHD oblique stagnation point flow of nanofluids over a stretched convective surface. It is possible to evaluate nanoparticle aggregation with modified versions of the Krieger-Dougherty and Maxwell-Bruggeman models. Transforming the governing nonlinear PDEs into connected nonlinear ODEs through similarity transformations is necessary. We use a very efficient and precise numerical method to solve these equations. Mathematical implementation of the shooting technique using the Runge-Kutta-IV algorithm. Annotations on graphs illustrate how major physical factors affect the velocity, Nusselt number, streamlines, and temperature profiles. The local skin friction coefficient is tabulated. Velocity and temperature profiles increase with increasing values of stagnation parameters. The rate of heat transfer increases noticeably after incorporating the parameters [Formula: see text] and [Formula: see text]. When using the aggregation model, the heat transfer rate is increased by around 0.89%, compared to an increase of about 0.59% when not using the model. These enhancements manifest themselves at a 1% concentration of [Formula: see text] when the parameter [Formula: see text] is set to 0.5. With aggregation effects, ethylene glycol-based fluids performed better than without aggregation in terms of heat transfer rate. Consequently, we are confident in our numerical findings, which match earlier material for the limiting condition.

Time warping between main epidemic time series in epidemiological surveillance.

The most common reported epidemic time series in epidemiological surveillance are the daily or weekly incidence of new cases, the hospital admission count, the ICU admission count, and the death toll, which played such a prominent role in the struggle to monitor the Covid-19 pandemic. We show that pairs of such curves are related to each other by a generalized renewal equation depending on a smooth time varying delay and a smooth ratio generalizing the reproduction number. Such a functional relation is also explored for pairs of simultaneous curves measuring the same indicator in two neighboring countries. Given two such simultaneous time series, we develop, based on a signal processing approach, an efficient numerical method for computing their time varying delay and ratio curves, and we verify that its results are consistent. Indeed, they experimentally verify symmetry and transitivity requirements and we also show, using realistic simulated data, that the method faithfully recovers time delays and ratios. We discuss several real examples where the method seems to display interpretable time delays and ratios. The proposed method generalizes and unifies many recent related attempts to take advantage of the plurality of these health data across regions or countries and time, providing a better understanding of the relationship between them. An implementation of the method is publicly available at the EpiInvert CRAN package.

Lower-Dimensional Model of the Flow and Transport Processes in Thin Domains by Numerical Averaging Technique

In this work, we present a lower-dimensional model for flow and transport problems in thin domains with rough walls. The full-order model is given for a fully resolved geometry, wherein we consider Stokes flow and a time-dependent diffusion–convection equation with inlet and outlet boundary conditions and zero-flux boundary conditions for both the flow and transport problems on domain walls. Generally, discretizations of a full-order model by classical numerical schemes result in very large discrete problems, which are computationally expensive given that sufficiently fine grids are needed for the approximation. To construct a computationally efficient numerical method, we propose a model-order-reduction numerical technique to reduce the full-order model to a lower-dimensional model. The construction of the lower-dimensional model for the flow and the transport problem is based on the finite volume method and the concept of numerical averaging. Numerical results are presented for three test geometries with varying roughness of walls and thickness of the two-dimensional domain to show the accuracy and applicability of the proposed scheme. In our numerical simulations, we use solutions obtained from the finite element method on a fine grid that can resolve the complex geometry at the grid level as the reference solution to the problem.

Implementation of the Hybrid ADI-FDTD Scheme to Maxwell Equation for Mathematical Modeling of Breast Tumor

Breast cancer is the most common cancer in women, and non-destructive detection of the tumor is vital. The interaction of electromagnetic waves with breast tissue and the behavior of waves after interaction are used to model tumor detection mathematically. The behavior of electromagnetic waves in a medium is described using Maxwell's equations. Electromagnetic waves propagate according to the electrical properties of a medium. Since the electrical properties of tumor tissue are different from those of normal breast tissue, it is assumed that the tumor is a lossy dielectric sphere, and the breast is a lossy dielectric medium. Under this assumption, Maxwell's equations are used to calculate the scattered field from the tumor. The field scattered by the tumor is different from other tissues because their dielectric properties are different. The location and size of the tumor can be determined by utilizing the difference in scattering from the tissues. While the scattering field from the tumor in spherical geometric form is analytically calculated, it is not analytically possible to calculate the scattering field from the tumor in different geometric shapes. In addition to non-destructive detection of the tumor, an efficient numerical method, the finite difference time domain method (FDTD), is used to simulate the field distribution. After the location of the tumor is determined, the Alternating Direction Implicit (ADI) FDTD method, which gives simulation results by dividing the computation domain into smaller sub-intervals, can be used. Scattered fields are calculated analytically in the geometry where the tumor is in the form of a smooth sphere, and in more complex geometry, the field distributions are successfully obtained with the help of MATLAB using FDTD and ADI-FDTD algorithms.

An efficient multi-derivative numerical method for chemical boundary value problems

An efficient multi-derivative numerical method for chemical boundary value problems

Simulation of a Composite with a Polyhydroxybutyrate (PHB) Matrix Reinforced with Cylindrical Inclusions: Prediction of Mechanical Properties.

Biocomposite development, as a sustainable alternative to fossil-derived materials with diverse industrial applications, requires expediting the design process and reducing production costs. Simulation methods offer a solution to these challenges. The main aspects to consider in simulating composite materials successfully include accurately representing microstructure geometry, carefully selecting mesh elements, establishing appropriate boundary conditions representing system forces, utilizing an efficient numerical method to accelerate simulations, and incorporating statistical tools like experimental designs and re-regression models. This study proposes a comprehensive methodology encompassing these aspects. We present the simulation using a numerical homogenization technique based on FEM to analyze the mechanical behavior of a composite material of a polyhydroxybutyrate (PHB) biodegradable matrix reinforced with cylindrical inclusions of flax and kenab. Here, the representative volume element (RVE) considered the geometry, and the numerical homogenization method (NHM) calculated the macro-mechanical behavior of composites. The results were validated using the asymptotic homogenization method (AHM) and experimental data, with error estimations of 0.0019% and 7%, respectively. This model is valuable for predicting longitudinal and transverse elastic moduli, shear modulus, and Poisson's coefficient, emphasizing its significance in composite materials research.

On Highly Efficient Fractional Numerical Method for Solving Nonlinear Engineering Models

We proposed and analyzed the fractional simultaneous technique for approximating all the roots of nonlinear equations in this research study. The newly developed fractional Caputo-type simultaneous scheme’s order of convergence is 3ς+5, according to convergence analysis. Engineering-related numerical test problems are taken into consideration to demonstrate the efficiency and stability of fractional numerical schemes when compared to previously published numerical iterative methods. The newly developed fractional simultaneous approach converges on random starting guess values at random times, demonstrating its global convergence behavior. Although the newly developed method shows global convergent behavior when all starting guess values are distinct, the method diverges otherwise. The total computational time, number of iterations, error graphs and maximum residual error all clearly illustrate the stability and consistency of the developed scheme. The rate of convergence increases as the fractional parameter’s value rises from 0.1 to 1.0.

A kernel-based method for fractional integro-differential equations with a weakly singular kernel in multi-dimensional complex domains

We present an efficient and practical numerical method for solving fractional integro-differential equations with a weakly singular kernel in complex two and three-dimensional domains. The reproducing kernel pseudo-spectral method is used to discretize the multidimensional regular and irregular spatial domains, while time discretization is done using a third-order backward differentiation formula. We have investigated the unconditional stability of the utilized temporal discretization. Finally, we have used the proposed method to solve the examples presented in the articles and compared the results. The obtained results show the accuracy and ability of the presented method.

Local randomized neural networks with discontinuous Galerkin methods for diffusive-viscous wave equation

The diffusive-viscous wave equation is an advancement in wave equation theory, as it accounts for both diffusion and viscosity effects. This has a wide range of applications in geophysics, such as the attenuation of seismic waves in fluid-saturated solids and frequency-dependent phenomena in porous media. Therefore, the development of an efficient numerical method for the equation is of both theoretical and practical importance. Recently, local randomized neural networks with discontinuous Galerkin (LRNN-DG) methods have been introduced in [21] to solve elliptic and parabolic equations. Numerical examples suggest that LRNN-DG can achieve high accuracy, and can handle time-dependent problems naturally and efficiently by using a space-time framework. In this paper, we develop LRNN-DG methods for solving the diffusive-viscous wave equation and present numerical experiments with several cases. The numerical results show that the proposed methods can solve the diffusive-viscous wave equation more accurately with less computing costs than traditional methods.

Interaction of flutter and forced response in a low pressure turbine rotor with friction damping and mistuning effects

In modern LPT designs, which typically can assume a tolerable level of flutter, the simultaneous presence of forced response and flutter in different operation regimes becomes unavoidable. Current design rules rely on the linear superposition of vibration levels from the isolated problems. However, recent evidence from reduced order model calculations and from experiments suggests that this may be an over conservative approach. In order to check this finding, this study examines the flutter and forced response interaction in a realistic low pressure turbine rotor. A high fidelity model is used, and an efficient numerical method is implemented for the computation of the aeroelastic vibrations of the rotor with nonlinear friction effects at the blade-disk contact interfaces. A hybrid approach is used that combines a traveling wave description for the linear modes with a physical space representation for the displacement of the nonlinear contacts. The resulting equations are integrated in the time domain in order to account for the unknown flutter frequencies present in the response. Both tuned and mistuned configurations are considered, and it is confirmed that the actual response of the system is much smaller than that coming from the linear superposition of the separated effects.

An Efficient Numerical Method Based on Exponential B-splines for a Time-Fractional Black–Scholes Equation Governing European Options

An Efficient Numerical Method Based on Exponential B-splines for a Time-Fractional Black–Scholes Equation Governing European Options

Accurate and efficient numerical methods for the nonlinear Schrödinger equation with Dirac delta potential

Accurate and efficient numerical methods for the nonlinear Schrödinger equation with Dirac delta potential

An efficient approach for free vibration analysis of functionally graded sandwich beams of variable cross-section

In this research paper, a powerful numerical method is implemented to examine the free vibration analysis of functionally graded (FG) sandwich beams of variable cross-section. The core of the beam is considered to be made from homogenous isotropic material and the faces of the sandwich beam are considered to be made of FG materials. The effect of the shear strain is considered in the formulations based on the first-order shear deformation theory (FSDT). The governing canonical equations for the dynamic response of FG sandwich beams are derived with the aid of the canonically conjugate momentums for the first time. To solve these equations numerically the Complementary Functions Method (CFM) is implemented. The free vibration behavior of the symmetric and asymmetric FG sandwich beams is investigated for different boundary conditions. The presented efficient numerical method is validated with the available literature. Moreover, detailed parametric studies are conducted to demonstrate the influence of layers, geometric constants, material volume fraction indexes, and beam slenderness on the natural frequencies of FG sandwich beams.

Numerical solution of magnetohydrodynamic flow through duct with perturbated boundary using RBF-FD method

This paper explores the magnetic field and velocity profile within a square and rectangular duct that experiences a perturbation in the steady magnetohydrodynamic flow. The magnetic field applied forms an angle with the x-axis. To solve for the solutions, a radial basis function-based finite difference approach is employed, known as one of the most efficient numerical methods. The Hartmann number and the perturbation magnitude in the region with insulated walls are adjusted to compute the findings. Numerical solutions are obtained by varying the angles of the applied magnetic field. The results are computed by varying the width and magnitude of the perturbation on the wall. The computed results of the induced magnetic field and velocity are represented through contours.

Stability and optimal control of a cytokine-enhanced general HIV infection model with antibody immune response and CTLs immune response

In this article, a cytokine-enhanced viral infection model with cytotoxic T lymphocytes (CTLs) immune response and antibody immune response is proposed and analyzed. The model contains six compartments: uninfected CD4+T cells, infected CD4+T cells, inflammatory cytokines, viruses, CTLs and antibodies. Different from the previous works, this model not only considers virus-to-cell transmission and cell-to-cell transmission, but also includes a new infection mode, namely cytokine-enhanced viral infection. The incidence rates of the healthy CD4+T cells with viruses, infected cells and inflammatory cytokines are given by general functions. Moreover, the production/proliferation and removal/death rates of all compartments are represented by general functions. Firstly, we prove that all the solutions of the model are nonnegative and uniformly bounded. Then, five key parameters with strong biological significance, namely the virus basic reproduction number R 0, CTLs immune response reproduction number R 1, antibody immune response reproductive number R 2, CTLs immune competitive reproductive number R 3 and antibody immune competitive reproductive number R 4 are derived. Then, by using Lyapunov’s method and LaSalle’s invariance principle, we have shown the global stability of each equilibrium. In addition, the numerical simulation results also show that the theoretical results are correct. Finally, we formulate an optimal control problem and solve it using Pontryagins Maximum Principle and an efficient iterative numerical methods. The results of our numerical simulation show that it is very important to control the infection between viruses and cells and between cells and inflammatory cytokines for controlling HIV.

VOF method in two-stage fourth order time-stepping framework

The fraction volume of fluid (VOF) method is a successful approach to track the dynamics of free boundaries thanks to its flexibility, efficiency and simplicity. And various efficient numerical methods have been developed to solve the associated VOF equation. This paper applies the two-stage fourth order accurate time discretization in [21] for the evolution of the VOF equation, aiming to provide a compact diffuse-interface algorithm simple for implementation, in order to incorporate with CFD shock-capturing schemes for the simulation of multi-material/phase problems with sharp interfaces. As such, the algorithm is decoupled from those for fluid flows as long as the velocity of the interface is given. Numerical results demonstrate the high resolution and high-fidelity for the interface capturing.

A positivity-preserving numerical method for a thin liquid film on a vertical cylindrical fiber

When a thin liquid film flows down on a vertical fiber, one can observe the complex and captivating interfacial dynamics of an unsteady flow. Such dynamics are applicable in various fluid experiments due to their high surface area-to-volume ratio. Recent studies verified that when the flow undergoes regime transitions, the magnitude of the film thickness changes dramatically, making numerical simulations challenging. In this paper, we present a computationally efficient numerical method that can maintain the positivity of the film thickness as well as conserve the volume of the fluid under the coarse mesh setting. A series of comparisons to laboratory experiments and previously proposed numerical methods supports the validity of our numerical method. We also prove that our method is second-order consistent in space and satisfies the entropy estimate.

Cross-scale coupling analysis method for OTSG in SFR and its application

The once-through steam generators (OTSG) in sodium-cooled fast reactor (SFR) are characterized by full-regime flow boiling heat transfer, coupled heat transfer on both sides and complex structure with a large dimensional span. This makes the computational fluid dynamics (CFD) and 3D refinement modeling methods to analyze the 3D temperature field of OTSG containing large-scale heat transfer tube bundles generally face the problems of high computational difficulty, low solution efficiency and poor solution stability. In this paper, based on the full-regime flow boiling heat transfer model and the flow pattern discrimination model, a one-dimensional full-regime flow boiling heat transfer model from subcooled water to superheated steam on the water side (tube side) and its cross-scale coupling solution model with the three-dimensional refined CFD model on the sodium side (shell side) are established. The effectiveness of the coupling analysis model was verified by comparative analysis with experimental data of the Indian 19-tube Steam Generator Test Facility (SGTF). The independent coupling of the water side of each heat transfer tube in the OTSG with the sodium side 3D fluid domain is realized. The results show that the cross-scale coupling method using the one-dimensional full-regime flow boiling heat transfer model for the water side and the three-dimensional CFD model for the sodium side can accurately predict the temperature distribution, the main heat transfer characteristic points on the full height of each heat transfer tube, and can obtain the temperature difference between different heat transfer tubes and tube walls. The maximum error between the calculated results and experimental data for the distribution of sodium temperature along the axial direction of the tubes is less than 3%. The cross-scale coupled solution method avoids the direct adoption of the three-dimensional multiphase flow model, which significantly improves the efficiency and stability of the solution of the three-dimensional temperature field of OTSG. This study provides an efficient 3D refinement numerical analysis method for the design and safety verification of OTSG.

Implicit Runge‐Kutta with spectral Galerkin methods for the fractional diffusion equation with spectral fractional Laplacian

AbstractAn efficient numerical method with high accuracy both in time and in space is proposed for solving ‐dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an ‐stage implicit Runge‐Kutta method and approximating the space by a spectral Galerkin method with Fourier‐like basis functions. In view of the orthogonality, the mass matrix of the spectral Galerkin method is an identity and the stiffness matrix is diagonal, which makes the proposed method is efficient for high‐dimensional problems. The proposed method is showed to be stable and convergent with at least order in time, when the implicit Runge‐Kutta method with classical order () is algebraically stable. As another important contribution of this paper, we derive the spatial error estimate with optimal convergence order which depends on the regularity of the exact solution but not on the fractional parameter . This improves the previous result which depends on the fractional parameter . Numerical experiments verify and complement our theoretical results.