Linear Interpolation Functions Research Articles

Methodology for the Differential Classification of Dengue and Chikungunya According to the PAHO 2022 Diagnostic Guide.

Arboviruses such as dengue, Zika, and chikungunya present similar symptoms in the early stages, which complicates their differential and timely diagnosis. In 2022, the PAHO published a guide to address this challenge. This study proposes a methodological framework that transforms qualitative information into quantitative information, establishing differential weights in relation to symptoms according to the medical evidence and the GRADE scale based on recommendation 1 of the said guide. To achieve this, common variables from the dataset were identified using the PAHO guide, and quality rules were established. A linear interpolation function was then parameterised to assign weights to the symptoms according to the evidence. Machine learning was used to compare the different models, achieving 99% accuracy compared with 79% without the methodology. This proposal represents a significant advancement, allowing the direct application of the PAHO recommendations to the dataset and improving the differential classification of arboviruses.

A comprehensive methodology for residual capacity evaluation of damaged steel tubular columns

Steel tubular sections are extensively used in modern-day construction due to their superior structural efficiency and aesthetic appearance. The thin-walled tubular sections are often subjected to accidental loading due to heavy object impact, collision, or improper handling during transportation. Due to induced geometric defects by unanticipated loads, the load-carrying capacity of the section decreases. This study proposes a methodology to estimate the residual capacity of damaged circular and square tubular specimens in axial compression. Initially, the specimen's surface coordinates were obtained in the form of a dense point cloud using a 3D digital image correlation (DIC) setup. Using a linear interpolation function based on Delaunay triangulation, the dense point cloud was regularized into a set of sparsely spaced nodes that would yield computationally efficient finite element (FE) analysis. The interpolation introduced errors at the locations of sharp curvature on the surface. A systematic smoothing (filtering) approach was proposed to minimize the effect of the interpolation error. The proposed methodology was validated against compression tests on five damaged and three as-built specimens, which consisted of circular and square tubular sections. The average residual strength estimated was found to be 93% of the experimental observations.

A visco-hyperelastic model for hydrogels with different water content and its finite element implementation

The visco-hyperelastic properties of hydrogels, which are greatly influenced by water content, play a crucial role in the potential applications in cutting-edge fields. Therefore, quantifying the effect of water content on the visco-hyperelastic behavior of hydrogels is necessary. In this work, we have proposed a visco-hyperelastic constitutive model which considers the water content for large deformation of hydrogels by introducing internal variables that can characterize the viscous dissipation. In the model, the evolution equation of the internal variable is established to achieve the coupling of viscoelastic and hyperelastic responses. To further understand the influence of water content on the visco-hyperelastic behavior of hydrogels, we modified the power-law relationship between water content and initial modulus through experiments. The accuracy of our visco-hyperelastic model is validated by uniaxial tension and relaxation experiments on hydrogels with varying water content. To facilitate the application of the constitutive model, we implement the constitutive model into ABAQUS user subroutine UMAT through constructing linear interpolation function. The results of the finite element analysis further confirm the capability of proposed model to describe the viscoelastic response of hydrogels under complex loading conditions.

A constitutive model of nanoporous NiTi shape memory alloys considering tensile-compressive asymmetry, grain size and porosity

A constitutive model of the macroscopic behaviors of nanoporous Shape Memory Alloys (SMAs) is developed, and the effects of tensile-compressive asymmetry, grain size, and porosity are all considered because the influences of these factors cannot be ignored. A finite-volume, three-phase micromechanical model that contains the grain boundary phase, grain-core phase, and uniformly distributed pores is first given to represent the grains of nanoporous NiTi SMAs. The secant equivalent bulk modulus and shear modulus of the nanoporous NiTi SMAs were calculated by using the extended Mori-Tanaka method. Due to the high plastic yield limit of the grain boundary phase, the grain boundary phase is regarded as the region where no phase change occurs. And the overall phase transformation can be caused by the grain-core phase. The phase transformation is established according to the J2, J3 theory and the von Mises equivalence effect. The martensite volume fraction is represented in this article as a linear interpolation function of equal effectiveness. Based on the strain-hardening behavior, the equivalent stress of nanoporous NiTi SMAs is represented by a power function consisting of transformation onset stress, hardening modulus, equivalent phase change strain, and hardening index. Then, the constitutive model describing the superelastic behavior can be obtained. By comparing the numerical simulation results with the experimental results, it is verified that the constitutive model proposed in this paper can better describe the superelastic behavior of the nanoporous NiTi SMAs. Finally, through numerical simulations, it was found that tensile-compressive asymmetry, grain size, and porosity have significant effects on the superelastic behavior of nanoporous NiTi SMAs. This study will provide a theoretical basis for the application of nanoporous NiTi SMAs.

Boundary element analysis for finite-size quasicrystal plates with a hole

The quasicrystal thermal barrier coating technology for engines is one effective means to address the energy-saving and emission-reduction issues in fuel-powered vehicles. It is noted that under loads, quasicrystal components with holes will experience stress concentration at hole edges, which can induce cracks and lead to material failure. Therefore, studying the mechanical behavior of quasicrystal defects is of great significance for improving the reliability of components. However, the boundary value problem of finite-sized icosahedral quasicrystal plates with holes is very complex and difficult to solve analytically. In the article, based on the extended Stroh method, a boundary element analysis is conducted for finite-size quasicrystal plates with a hole. Mainly, Green’s function is derived to obtain the fundamental solutions of an infinite-sized quasicrystal specimen with a hole, and linear interpolation functions and the Gauss integral formula are used to discretize integral equations. Finally, the effects of the size of the medium and the coupling elastic constant on the stress around the hole are discussed. The numerical results show that the boundary element method can be used to analyze both the finite-sized problem and the infinite-sized problem.

Data-driven modelling for assessing trophic status in marine ecosystems using machine learning approaches

Assessing eutrophication in coastal and transitional waters is of utmost importance, yet existing Trophic Status Index (TSI) models face challenges like multicollinearity, data redundancy, inappropriate aggregation methods, and complex classification schemes. To tackle these issues, we developed a novel tool that harnesses machine learning (ML) and artificial intelligence (AI), enhancing the reliability and accuracy of trophic status assessments. Our research introduces an improved data-driven methodology specifically tailored for transitional and coastal (TrC) waters, with a focus on Cork Harbour, Ireland, as a case study.Our innovative approach, named the Assessment Trophic Status Index (ATSI) model, comprises three main components: the selection of pertinent water quality indicators, the computation of ATSI scores, and the implementation of a new classification scheme. To optimize input data and minimize redundancy, we employed ML techniques, including advanced deep learning methods. Specifically, we developed a CHL prediction model utilizing ten algorithms, among which XGBoost demonstrated exceptional performance, showcasing minimal errors during both training (RMSE = 0.0, MSE = 0.0, MAE = 0.01) and testing (RMSE = 0.0, MSE = 0.0, MAE = 0.01) phases.Utilizing a novel linear rescaling interpolation function, we calculated ATSI scores and evaluated the model's sensitivity and efficiency across diverse application domains, employing metrics such as R2, the Nash-Sutcliffe efficiency (NSE), and the model efficiency factor (MEF). The results consistently revealed heightened sensitivity and efficiency across all application domains. Additionally, we introduced a brand new classification scheme for ranking the trophic status of transitional and coastal waters. To assess spatial sensitivity, we applied the ATSI model to four distinct waterbodies in Ireland, comparing trophic assessment outcomes with the Assessment of Trophic Status of Estuaries and Bays in Ireland (ATSEBI) System. Remarkably, significant disparities between the ATSI and ATSEBI System were evident in all domains, except for Mulroy Bay.Overall, our research significantly enhances the accuracy of trophic status assessments in marine ecosystems. The ATSI model, combined with cutting-edge ML techniques and our new classification scheme, represents a promising avenue for evaluating and monitoring trophic conditions in TrC waters. The study also demonstrated the effectiveness of ATSI in assessing trophic status across various waterbodies, including lakes, rivers, and more. These findings make substantial contributions to the field of marine ecosystem management and conservation.

A time-consistent stabilized finite element method for fluids with applications to hemodynamics

Several finite element methods for simulating incompressible flows rely on the streamline upwind Petrov–Galerkin stabilization (SUPG) term, which is weighted by tau _{text{SUPG}}. The conventional formulation of tau _{text{SUPG}} includes a constant that depends on the time step size, producing an overall method that becomes exceedingly less accurate as the time step size approaches zero. In practice, such method inconsistency introduces significant error in the solution, especially in cardiovascular simulations, where small time step sizes may be required to resolve multiple scales of the blood flow. To overcome this issue, we propose a consistent method that is based on a new definition of tau _{text{SUPG}}. This method, which can be easily implemented on top of an existing streamline upwind Petrov–Galerkin and pressure stabilizing Petrov–Galerkin method, involves the replacement of the time step size in tau _{text{SUPG}} with a physical time scale. This time scale is calculated in a simple operation once every time step for the entire computational domain from the ratio of the L2-norm of the acceleration and the velocity. The proposed method is compared against the conventional method using four cases: a steady pipe flow, a blood flow through vascular anatomy, an external flow over a square obstacle, and a fluid–structure interaction case involving an oscillatory flexible beam. These numerical experiments, which are performed using linear interpolation functions, show that the proposed formulation eliminates the inconsistency issue associated with the conventional formulation in all cases. While the proposed method is slightly more costly than the conventional method, it significantly reduces the error, particularly at small time step sizes. For the pipe flow where an exact solution is available, we show the conventional method can over-predict the pressure drop by a factor of three. This large error is almost completely eliminated by the proposed formulation, dropping to approximately 1% for all time step sizes and Reynolds numbers considered.

On the integral transform of fractal interpolation functions

This paper explores the integral transform of two distinct fractal interpolation functions, namely the linear fractal interpolation function and the hidden variable fractal interpolation function with variable scaling factors. Further, with a particular application of kernel functions, we investigate the integral transform of fractal functions, such as the Laplace transform and the Laplace Carson transform. Moreover, we show that the compositeness of two fractal interpolation functions, f1 in {tɛ,xɛ} and f2 in {xɛ,zɛ} remains a fractal interpolation function. It also generates iterated function system from given iterated function systems. In addition to this, the study is carried out on the composite linear fractal interpolation function of the integral transform, the Laplace transform, and the Laplace Carson transform.

Approximation by the linear fractal interpolation functions with the same fractal dimension

Approximation by the linear fractal interpolation functions with the same fractal dimension

An Element Decomposition Method for Three-Dimensional Solid Mechanics

This paper proposes an element decomposition method (EDM) for elastic-static, free vibration and forced vibration analyses of three-dimensional solid mechanics. The problem domain is first discretized using eight-node hexahedral elements. Then, each hexahedron is further subdivided into a set of sub-tetrahedral cells, and the local strains in each sub-tetrahedron are obtained using linear interpolation functions. For each hexahedron, the strain of the whole element is the weighted average value of the local strains, which means only one integration point is adopted to establish the stiffness matrix. To cure the numerical instability of one-point quadrature and improve the accuracy, a variation gradient item is complemented by variance of the local strains. Numerical examples, including both benchmark and practical engineering cases, demonstrate that the present method possesses the following interesting properties compared with the traditional finite element method using the same mesh discretization (1) super accuracy and faster convergence rate; (2) higher computational efficiency; (3) more immune to mesh distortion.

Transient analyses of wave propagations in nonhomogeneous media employing the novel finite element method with the appropriate enrichment function

An enriched finite element approach with appropriate enrichment function is proposed for the transient analyses of wave propagations in nonhomogeneous media. In present method, the traditional linear nodal interpolation functions for low order finite elements are enriched by the augmented enrichment function. For the purpose of improving the computation accuracy and reducing the numerical dispersion error, the polynomial bases and the trigonometric functions are utilized to form the enrichment functions. In addition, a simple and direct approach is introduced with the aim to tackle the possible linear dependence of the system equations. The numerical results show that the accuracy of the obtained numerical results can be markedly improved by the proposed method in comparison with the standard finite elements. What's more, it can be found that the calculated solutions from the proposed method with the appropriate temporal discretization scheme are able to monotonically converge to the exact solution in transient wave analysis. In consequence, it is possible to consistently increase the solution accuracy by directly decreasing temporal discretization step size. This valuable and important property enable the present approach to be particularly suitable to tackle the wave propagations in nonhomogeneous media.

CONSTRUCTION OF NEW AFFINE AND NON-AFFINE FRACTAL INTERPOLATION FUNCTIONS THROUGH THE WEYL–MARCHAUD DERIVATIVE

This paper investigates the Weyl–Marchaud fractional derivative of affine and non-affine fractal interpolation functions with function scaling factors. The dependence of fractal interpolation function on the scaling factor is mainly explored by choosing the scaling factor as a function instead of a constant. In addition, for some fixed order [Formula: see text], the Weyl–Marchaud fractional derivative of a linear fractal interpolation function is estimated by predefining the fractional derivative values at the end points. Similarly, the Weyl–Marchaud fractional derivative of a [Formula: see text]-fractal function is investigated for some fixed order [Formula: see text] with additional constraints on the derivative of prescribed continuous function and base function.

Approximation of Caputo‐Prabhakar derivative with application in solving time fractional advection‐diffusion equation

AbstractThis work aims to numerically approximate the Caputo‐Prabhakar derivative and use this approximation for solving the time‐fractional advection‐diffusion equation defined in Caputo‐Prabhakar sense which is widely used in fluid dynamics. In this approach, we approximate the time‐fractional derivative of the mentioned equation by two schemes, namely and , using linear and quadratic interpolation functions, respectively. The convergence order of the two schemes is , , respectively, for . The analytical error bounds for the two schemes are also discussed. Then, these schemes are applied to solve the time‐fractional advection‐diffusion equation defined in the Caputo‐Prabhakar sense numerically. We will prove the solvability and stability of the proposed methods. Numerical examples validate the analytical results. With the reference of an example, we have shown that the schemes work well for the fractional diffusion equation also.

Progressive debonding analysis of sandwich composite strips using a cohesive-layerwise spectral finite element model

This paper presents a novel numerical framework for the simulation of debonding growth in sandwich structures via a layerwise spectral finite element model. A variable kinematic layerwise theory is developed, which employs cubic Hermite splines for the composite facesheets and the core and linear interpolation functions for the thin resin-rich layers in connection with continuum damage mechanics for tracking of debonding propagation. The layerwise mechanics and continuum damage mechanics are integrated into a spectral finite element, termed thereafter as cohesive-layerwise spectral finite element, with integration points collocated to nodes and high-order spatial approximation in the plane of the structure. The developed computational model is applied to global mode I, mode II, and mixed-mode fracture problems and correlated with experimental results and reference high-fidelity finite element models. The evaluation cases of the present method reveal significant enhancements in computational speed and meshing process due to elimination of cohesive layer parameter calibration, alleviation of FE aspect ratio constraints and high-convergence rates.

An adaptive hybrid fractal model for short-term load forecasting in power systems

This paper develops a hybrid short-term load forecasting (STLF) modeling with adaptive parametrization, based on composite linear fractal interpolation function (CLFIF), iterative learning (IL) and chimp optimization algorithm (ChOA). More precisely, after selecting similar days in power load data, an amendatory CLFIF model is constructed for an-hour-ahead prediction in terms of hourly load curves. Then, iterative learning based on ChOA optimizes the parameters of the amendatory CLFIF model with higher accuracy. Moreover, to confirm effectiveness of the CLFIF-IL-ChOA model, numerical examples are tested on the historical power load data from PJM and ENTSOE. The numerical results show that the proposed method can obtain higher accuracy, compared to some common methods about time series analysis.

Nodal integral methods in general 2D curvilinear coordinates - applied to convection–diffusion equation in domains discretized using quadrilateral elements

Nodal integral method (NIM) is derived in general, 2D, curvilinear coordinates and applied to solve the convection–diffusion equation in domains discretized using quadrilateral elements. The quadrilateral elements in the Cartesian system are transformed into rectangular elements in curvilinear coordinates using bi–linear Lagrangian interpolation functions. An approximation for the transverse-integration operator is developed in curvilinear coordinates. The convection–diffusion equation and the continuity conditions at the common edge between two adjacent nodes are derived in curvilinear coordinates. A new approximation of the cross–derivative terms resulting from transforming the governing equation to curvilinear coordinates is developed using the discrete unknowns of NIM (line–averaged variables). Three numerical test problems are solved to assess the accuracy and efficiency of the newly developed scheme. The results show that the scheme developed in this paper is second–order accurate in space and time, which is the same order of accuracy for traditional NIM for regularly-shaped elements. The results show that the accuracy of NIM for quadrilateral elements is maintained even for coarse mesh and highly distorted elements.

Dual-field measurement system for real-time material flow on conveyor belt

In this paper, an innovative dual-field measurement system is proposed for measuring the real-time material flow on the conveyor belt. The system consists of two light sources to illuminate the upper and the lower surface of the conveyor belt, respectively, and two binocular cameras to capture the dual-field contour images. The contour curves are extracted from the images by the contour acquisition algorithm and fitted with linear interpolation functions for the calculation of instantaneous cross-sectional area of material flow. Then the real-time volume of material flow is obtained according to the belt speed. Compared with conventional visual methods, the proposed method is no need to preliminarily acquire the data of the empty belt as well as hardly affected by belt deformation. Some measured objects are prepared for both sectional area and volume measurement. The results show that the accuracy of the proposed system can achieve up to 96.3% and 96.05% for the volume measurement of regular materials and coals, respectively, which is superior to the conventional visual method. The proposed measurement method has strong robustness and low construction cost, which is expected to generalize and apply in the bulk material transport field.

Isogeometric boundary element analysis of liquid nonlinear sloshing in two dimensional rectangular tanks

This paper presents a numerical model based on the Isogeometric Analysis (IGA), Boundary Element Method (BEM) and the second order Runge–Kutta (RK2) method for solving the nonlinear sloshing problem in a two-dimensional (2D) tank. In this study, the IGABEM is employed to solve the Laplace equation for the velocity potentials of the liquid, and the singular integral is calculated by the Radial Integral Method (RIM). The Semi-Lagrange (SL) approach is adopted to capture the movement of the free surface, then the RK2 method is used to update the velocity potential and elevation of the free surface at each time step. The numerical results agree very well with the reference results in different situations including free sloshing and forced sloshing under different excitations, which illustrates the validity of the present method. Moreover, in comparison with the traditional element-based method with linear polynomial interpolation function, the present RK2-IGABEM can evaluate the velocity potential and elevation of the free surface more smoothly with only a few degrees of freedom added, leading to a high accuracy analysis of nonlinear sloshing problems.

DISTRIBUTION OF LINEAR FRACTAL INTERPOLATION FUNCTION FOR RANDOM DATASET WITH STABLE NOISE

In this paper, we derive the probability distribution of linear fractal interpolation function (FIF) for a random dataset on [Formula: see text], where randomness in the data is induced, using [Formula: see text]-stable noise, in the second coordinate. In particular, we show that the distribution of linear FIF, at any point (on the first coordinate), is also stable, but with different parameters which can be estimated. Finally, we also provide statistical validity of our results.

Computable Interpolation Error Constants for the Geometric Simplex Finite Elements

In this paper, a novel high accuracy computation method for interpolation error constants is proposed over the geometric simplex finite elements. Firstly, the expansions of bounded linear operators are employed to derive the explicit estimate of interpolation error constants, which depend only on the shape of the geometric simplex finite elements and the definition of interpolation functions. Then, this method is applied to the linear interpolation function, and the results are consistent with our analysis. Finally, some numerical examples are given to validate our analysis. Such high accuracy computation method for interpolation error constants are beneficial attempts to accelerate the adaptive computation and verification of finite element solutions.