Simple Numerical Method Research Articles

Disconnected Stationary Solutions in 3D Kolmogorov Flow and Their Relation to Chaotic Dynamics

This paper aims to investigate the nonlinear transition to turbulence in generalized 3D Kolmogorov flow. The difference between this and classical Kolmogorov flow is that the forcing term in the x direction sin(y) is replaced with sin(y)cos(z). This drastically complicates the problem. First, a stability analysis is performed by deriving the analog of the Orr–Sommerfeld equation. It is shown that for infinite stretching, the flow is stable, contrary to classical forcing. Next, a neutral curve is constructed, and the stability of the main solution is analyzed. It is shown that for the cubic domain, the main solution is linearly stable, at least for 0<R≤100. Next, we turn our attention to the numerical investigation of the solutions in the cubic domain. The main feature of this problem is that it is spatially periodic, allowing one to apply a relatively simple pseudo-spectral numerical method for its investigation. We apply the method of deflation to find distinct solutions in the discrete system and the method of arc length continuation to trace the bifurcation solution branches. Such solutions are called disconnected solutions if these are solutions not connected to the branch of the main solution. We investigate the influence of disconnected solutions on the dynamics of the system. It is demonstrated that when disconnected solutions are formed, the nonlinear transition to turbulence is possible, and dangerous initial conditions are these disconnected solutions.

Simplified Procedure for the Design and Optimisation of Oval Tensile Spoke Wheels for Roof Structures

This paper presents a simplified procedure for the design and shape optimisation of oval tensile spoke wheels to be used in roof structures for grandstands in sports stadiums. This procedure combines graphical methods with simple numerical methods to obtain the geometrical definition of an oval wheel and to estimate the prestressing forces. It also allows for the compression forces in the outer ring to be estimated and compared with those in a circular wheel. This makes it easy to make small geometrical adjustments for optimisation. Application examples are given for each step of the procedure. Several wheel designs are defined, showing how the shape of the inscribing oval determines the magnitude of the forces in the outer ring.

A volumetric approach to Monge's optimal transport on surfaces

We propose a volumetric formulation for computing the Optimal Transport problem defined on surfaces in R3, found in disciplines like optics, computer graphics, and computational methodologies. Instead of directly tackling the original problem on the surface, we define a new Optimal Transport problem on a thin tubular region, Tϵ, adjacent to the surface. This extension offers enhanced flexibility and simplicity for numerical discretization on Cartesian grids. The Optimal Transport mapping and potential function computed on Tϵ are consistent with the original problem on surfaces. We demonstrate that, with the proposed volumetric approach, it is possible to use simple and straightforward numerical methods to solve Optimal Transport for Γ=S2 and the 2-torus.

Analysis of Two Hybrid Schemes to Solve the Benjamin-Bona-Mahony (BBM) Equation

In this research, we compare and contrast two numerical methods, the Laplace Adomian Decomposition Method (LADM) and the Elzaki Projected Differential Transform Method (EPDTM), for solving the Benjamin-Bona-Mahony (BBM) equation, and this helps in evaluating their performance, assessing robustness, gaining insights into solution behavior, validating results, generalizing their applicability, and advancing the field. This rigorous comparisons validates the efficacy of the methods as we compare the exact results and the results of LADM and EPDTM, plotting the convergent of the two methods to identify the method that is highly convergent and accurate. Results identify the Elzaki Projected Differential Transform Method (EPDTM) as having a simple and accurate numerical method for solving the BBM equation. EPDTM enhances accessibility, computational efficiency, accuracy, robustness, and validation of the solutions, supporting the understanding and analysis of wave phenomena described by the BBM equation. This study contributes to the advancement of the field of numerical analysis and computational science as practical applications of BBM equation in various fields, including coastal engineering, wave modeling, and wave stability analysis cannot be over emphasize.

Analysis of two hybrid schemes to solve the Benjamin-Bona-Mahony (BBM) equation

In this research, we compare and contrast two numerical methods, the Laplace Adomian Decomposition Method (LADM) and the Elzaki Projected Differential Transform Method (EPDTM), for solving the Benjamin-Bona-Mahony (BBM) equation, and this helps in evaluating their performance, assessing robustness, gaining insights into solution behavior, validating results, generalizing their applicability, and advancing the field. This rigorous comparisons validates the efficacy of the methods as we compare the exact results and the results of LADM and EPDTM, plotting the convergent of the two methods to identify the method that is highly convergent and accurate. Results identify the Elzaki Projected Differential Transform Method (EPDTM) as having a simple and accurate numerical method for solving the BBM equation. EPDTM enhances accessibility, computational efficiency, accuracy, robustness, and validation of the solutions, supporting the understanding and analysis of wave phenomena described by the BBM equation. This study contributes to the advancement of the field of numerical analysis and computational science as practical applications of BBM equation in various fields, including coastal engineering, wave modeling, and wave stability analysis cannot be over emphasize.

Modeling oxygen transport in the brain: An efficient coarse-grid approach to capture perivascular gradients in the parenchyma.

Recent progresses in intravital imaging have enabled highly-resolved measurements of periarteriolar oxygen gradients (POGs) within the brain parenchyma. POGs are increasingly used as proxies to estimate the local baseline oxygen consumption, which is a hallmark of cell activity. However, the oxygen profile around a given arteriole arises from an interplay between oxygen consumption and delivery, not only by this arteriole but also by distant capillaries. Integrating such interactions across scales while accounting for the complex architecture of the microvascular network remains a challenge from a modelling perspective. This limits our ability to interpret the experimental oxygen maps and constitutes a key bottleneck toward the inverse determination of metabolic rates of oxygen. We revisit the problem of parenchymal oxygen transport and metabolism and introduce a simple, conservative, accurate and scalable direct numerical method going beyond canonical Krogh-type models and their associated geometrical simplifications. We focus on a two-dimensional formulation, and introduce the concepts needed to combine an operator-splitting and a Green's function approach. Oxygen concentration is decomposed into a slowly-varying contribution, discretized by Finite Volumes over a coarse cartesian grid, and a rapidly-varying contribution, approximated analytically in grid-cells surrounding each vessel. Starting with simple test cases, we thoroughly analyze the resulting errors by comparison with highly-resolved simulations of the original transport problem, showing considerable improvement of the computational-cost/accuracy balance compared to previous work. We then demonstrate the model ability to flexibly generate synthetic data reproducing the spatial dynamics of oxygen in the brain parenchyma, with sub-grid resolution. Based on these synthetic data, we show that capillaries distant from the arteriole cannot be overlooked when interpreting POGs, thus reconciling recent measurements of POGs across cortical layers with the fundamental idea that variations of vascular density within the depth of the cortex may reveal underlying differences in neuronal organization and metabolic load.

The Way to Construct Innovative Methods for Solving Initial-Value Problem of the Volterra Integro-Differential Equation

Mathematical models for many problems in the natural sciences are often simplified to solving initial-value problems (IVPs) for the Volterra integro-differential equations (VIDE). Numerical methods of a multistep type are typically used to solve these problems. It is known that in some cases, the multi-step method (MSM) is applied to solving the IVPs of both ordinary differential equations (ODEs) and VIDE encountered in solving some problems in mathematical biology. Here, to solve such problems by combining different methods, some modifications of established methods were developed, and it was demonstrated that these methods outperform the existing ones. As is known, one of the main issues in solving the aforementioned problems is determining the reliability of calculating values using the known mathematicalstatistical models (MSMs). In this regard, some experts utilize the predictor-corrector method. Having highlighted the disadvantages of this method, the proposal is to develop an innovative approach and assess the errors that may arise when applying this method to solve various problems. Here, the IVPs for the VIDE of the first order are primarily investigated. To illustrate the benefits of the innovative methods proposed here, we discuss the use of simple numerical methods to solve some common examples.

Discrete Element Simulation of Macro and Micro Mechanical Properties of Round Gravel Material under Triaxial Stress

In this study, the effects of relative density and confining pressure on the shear characteristics of round gravel are investigated using a large-scale triaxial apparatus and the discrete element method. A simple and efficient numerical method for simulating flexible membranes is introduced. The results show that the stress-strain curves develop from hardened to softened type with increasing relative density, while the stress-strain curves develop from softened to hardened type with increasing confining pressure. As the axial strain increases, the strong contact force chains are vertically distributed, and the larger the relative density and confining pressure, the greater the number and thickness of the strong contact force chains. In the shear process, the distribution of average normal and tangential contact forces show “peanut-shaped” and “petal-shaped”, respectively. The increase in relative density increases the anisotropy of the specimen, while the increase in confining pressure results in a decrease. A linear relationship exists between the macroscopic stress ratio and the anisotropy coefficient. The anisotropy coefficient of the normal contact force provides the greatest contribution to the macroscopic shear strength (about 55%), followed by the anisotropy coefficient of the contact normal (about 26%) and that of the tangential contact force (about 19%).

Optimizing numerical k-sampling for swept-source optical coherence tomography angiography.

High-quality swept-source optical coherence tomography (SS-OCT) requires accurate k-sampling, which is equally vital for optical coherence tomography angiography (OCTA). Most SS-OCT systems are equipped with hardware-driven k-sampling. However, this conventional approach raises concerns over system cost, optical alignment, imaging depth, and stability in the clocking circuit. This work introduces an optimized numerical k-sampling method to replace the additional k-clock hardware. Using this method, we can realize high axial resolution (4.9-µm full-width-half-maximum, in air) and low roll-off (2.3 dB loss) over a 4-mm imaging depth. The high axial resolution and sensitivity achieved by this simple numerical method can reveal anatomic and microvascular structures with structural OCT and OCTA in both macular and deeper tissues, including the lamina cribrosa, suggesting its usefulness in imaging retinopathy and optic neuropathy.

Realization of collimated specific profiles in rotation-symmetrical beam shaping system with divergent light source

A simple numerical method is proposed for the design of two aspherical surfaces, each comprising multiple segmented refractive planes, for generating a collimated beam with a specific irradiance profile in a beam shaping system with a divergent light source. However, in real-world manufacturing, this performance improvement is obtained at the expense of a greater cost and complexity. Accordingly, a second algorithm is proposed which maximizes the number of rays passing through the central regions of the refractive planes in the second aspherical surface and hence minimizes the total number of segments required to achieve the same beam shaping performance. The feasibility of the proposed method is demonstrated through the design of two aspherical lenses for generating collimated output beams with ring- and triangle-like irradiance profiles, respectively. The experimental results show that the beam profiles are in close agreement with the desired irradiance distributions. In general, the results indicate that the proposed method provides a versatile and efficient approach for achieving the desired collimated profile in beam forming systems with a divergent light source.

The M3 pressure-differential wave energy converter in Persian shores; numerical modeling and design

Following a double objective, this paper develops a numerical model for the M3 fixed-bottom pressure-differential wave energy converter (WEC) and then employs the model in the design of the WEC with available wave data for three Persian shores. To reach the first objective, using the potential theory, a simple numerical method, based on the conventional finite element method (FEM), has been developed for calculating the added mass and damping coefficients of the WEC. This paper shows that a complex engineering problem can be solved easily by making adjustments and making simple assumptions. These assumptions, though some of which may not be applicable to other types of WECs, are shown to be very effective in the modeling of the WEC considered in this paper. By comparing the results of the proposed method with those previously reported in the literature, the obtained results are validated. With a simple and practical numerical tool in hand, on the sequel and as the second objective, the energy absorption of the WEC is investigated based on the wave data in the Persian Gulf and the Gulf of Oman. For this purpose, several cases are considered for the dimensions of the WEC. Other effective parameters on the energy absorption of the WEC, i.e. the stiffness and damping coefficient of the power take-off (PTO) system, are optimized for each dimensional case to achieve the maximum absorbed power. A comparison of the absorbed power in different dimensional cases suggests appropriate WEC dimensions and the corresponding stiffness and damping coefficient of the PTO system for different application locations.

Two Simple Numerical Implementation Methods for Damage-Coupled Viscoplastic Constitutive Model

Abstract This paper is concerned with the two simple numerical implementation methods for a damage-coupled Chaboche-type viscoplastic constitutive model. By considering the damage variable as a constant in each incremental step, the return-mapping procedure is reduced to the solution of only one nonlinear scalar equation. Depending on the use of damage value in the current or prior incremental state, the two methods are named the backward difference implicit integration scheme and the two-step explicit integration scheme respectively. These two numerical algorithms are implemented into the ansys software by developing the usermat subroutine and verified by comparing them with available experimental data. Several numerical examples on the Gauss point level are studied in terms of stability, accuracy, computational efficiency, and applicability for further numerical observation. In addition to higher computational efficiency and lower memory requirements, the two methods can be easily extended to other damage models due to their simplicity.

On equilibrium states of fluid membranes

The paper studies the equilibrium configurations of inextensible elastic membranes exhibiting lateral fluidity. Using a continuum description of the membrane's motions based on the surface Navier–Stokes equations with bending forces, the paper derives differential equations governing the mechanical equilibrium. The equilibrium conditions are found to be independent of lateral viscosity and relate tension, pressure, and tangential velocity of the fluid. These conditions suggest that either the lateral fluid motion ceases or non-decaying stationary flow of mass can only be supported by surfaces with Killing vector fields, such as axisymmetric shapes. A shape equation is derived that extends the classical Helfrich model with an area constraint to membranes of non-negligible mass. Furthermore, the paper suggests a simple numerical method to compute solutions of the shape equation. Numerical experiments conducted reveal a diverse family of equilibrium configurations. The stability of equilibrium states involving lateral flow of mass remains an unresolved question.

Cropland and rooftops: the global undertapped potential for solar photovoltaics

The utilization of cropland and rooftops for solar photovoltaics (PVs) installation holds significant potential for enhancing global renewable energy capacity with the advantage of dual land-use. This study focuses on estimating the global area suitable for agrivoltaics (PV over crops) and rooftop PVs by employing open-access data, existing literature and simple numerical methods in a high spatial resolution of 10 km × 10 km. For agrivoltaics, the suitability is assessed with a systematic literature review on crop-dependent feasibility and profitability, especially for 18 major crops of the world. For rooftop PV, a non-linear curve-fitting method is developed, using the urban land cover to calculate the PV-suitable built-up areas. This method is then verified by comparing the results with open-access building footprints. The spatially resolved suitability assessment unveils 4.64 million km2 of global PV-usable cropland corresponding to a geographic potential of about 217 Terawatts (TW) in an optimistic scenario and 0.21 million km2 of rooftop-PV suitable area accounting for about 30.5 TW maximum installable power capacity. The estimated suitable area offers a vast playground for energy system analysts to undertake techno-economic assessments, and for technology modellers and policy makers to promote PV implementation globally with the vision of net-zero emissions in the future.

Numerical simulation of buoyant plumes using a fixed point iterative method

A numerical simulation of unsteady plumes for driven thermal high convection dominated flows is presented in this work. The problem is of practical interest for example in the cooling of electronic devices and heat transfer from pipes in heat exchange systems. This kind of thermal flows may be modeled using the unsteady Boussinesq approximation in the Stream function-vorticity formulation. Results are obtained with a simple numerical method previously reported for isothermal/natural and mixed convection flows. The numerical method is based on a fixed point iterative process to solve the non-linear elliptic system that results after an appropriate discretization in time. The iterative process leads us to the solution of uncoupled, well-conditioned and symmetric linear elliptic problems for which very efficient solvers are known to exist regardless of the space discretization.

Lucas Polynomial Solution of the Single Degree of Freedom System

Free vibration of a single degree of freedom system is a fundamental topic in mechanical vibrations. The present study introduces a novel and simple numerical method for the solution of this system in terms of Lucas polynomials in the matrix form. Particular and general solutions of the differential equation can be determined by this method. The method is illustrated by a numerical application and the results obtained are compared with those of the exact solution.

Density of States for the Unitary Fermi Gas and the Schwarzschild Black Hole

The density of states of a quantum system can be calculated from its definition, but, in some cases, this approach is quite cumbersome. Alternatively, the density of states can be deduced from the microcanonical entropy or from the canonical partition function. After discussing the relationship among these procedures, we suggest a simple numerical method, which is equivalent in the thermodynamic limit to perform a Legendre transformation, to obtain the density of states from the Helmholtz free energy. We apply this method to determine the many-body density of states of the unitary Fermi gas, a very dilute system of identical fermions interacting with a divergent scattering length. The unitary Fermi gas is highly symmetric due to the absence of any internal scale except for the average distance between two particles and, for this reason, its equation of state is called universal. In the last part of the paper, by using the same thermodynamical techniques, we review some properties of the density of states of a Schwarzschild black hole, which shares the problem of finding the density of states directly from its definition with the unitary Fermi gas.

Peak deconvolution with significant noise suppression and stability using a facile numerical approach in Fourier space

The deconvolution problem for resolution and feature enhancement of analytical signals is known to be a noise-enhancing procedure to the extent that some authors have called it “violently unstable.” The sharpened result is frequently buried in high-amplitude noise in direct deconvolution using the Fourier division method. With a Gaussian deconvolving function, one can get a “Not-a-Number (NaN) error” due to division by near zero (complex) numbers in Fourier space. Herein, a simple numerical method that stabilizes the calculation of direct Fourier deconvolution and ensures that the noise is controlled is proposed. In this calculation, a small positive constant or symmetric positive peak function is added to the denominator during division in Fourier space, resulting in stability and high-frequency noise suppression. The key advantages of this non-iterative method are speed, numerical stability during division, noise suppression, peak area and location invariance, and minimization of the edge effects after deconvolution. Also, ringing effects in the baseline and negative overshoots are significantly mitigated using the modified denominator method. The numerical approach can be applied to enhance the analytical signals consisting of peaks, e.g., in NMR, IR, Raman spectroscopy, or separation sciences. Both symmetric and one-sided (asymmetric) broadening functions can be used for deconvolution in this method. Applications are also demonstrated for various test functions (square pulse, Voigt, or highly overlapped Gaussian peaks) with clear advantages with respect to the standard Fourier division method. The new denominator addition method is compared with two constrained deconvolution procedures in matrix formalism and Fourier domain.

A simple and efficient numerical method for the Allen–Cahn equation on effective symmetric triangular meshes

<abstract><p>In this paper, we propose a novel, simple, efficient, and explicit numerical method for the Allen–Cahn (AC) equation on effective symmetric triangular meshes. First, we compute the net vector of all vectors starting from each node point to its one-ring neighbor vertices and virtually adjust the neighbor vertices so that the net vector is zero. Then, we define the values at the virtually adjusted nodes using linear and quadratic interpolations. Finally, we define a discrete Laplace operator on triangular meshes. We perform several computational experiments to demonstrate the performance of the proposed numerical method for the Laplace operator, the diffusion equation, and the AC equation on triangular meshes.</p></abstract>

A simple numerical simulation method for unsaturated stratum under the water environmental load

A simple numerical simulation method is proposed to investigate the mechanical response of the unsaturated stratum under the water environmental load. The proposed numerical simulation method applies the water environmental load to the unsaturated stratum as the load condition, and adopts the developed non-orthogonal elastoplastic constitutive model for unsaturated soil to describe the mechanical behaviors at the material scale. The mechanical response of unsaturated stratum is obtained by solving the discretized governing equation of the mechanical equilibrium. The proposed numerical simulation method is implemented based on the commercial finite element software ABAQUS. The effectiveness of the proposed method is verified through the element tests on the Botkin silt, the element tests on the Pearl clay, and a model test of the rising water level on a sand column. The numerical simulations of the rising and decreasing groundwater level in the unsaturated stratum are conducted. During the rising groundwater level, the non-collapsible stratum produces rebound deformation, the collapsible stratum below the groundwater level occurs rebound deformation, and the collapsible stratum above the groundwater level mainly produces settlement deformation. During the decreasing groundwater level, both the collapsible stratum and the non-collapsible stratum produce the settlement deformation.