Numerical Integration Approach Research Articles

Finite element approach for numerical integration over family of eight node linear quadrilateral element for solving Laplace equation

This paper describes the finite element solution of the boundary value problem or Laplace equation in curved and square region, the boundary region is divided into arbitrary triangles, and every triangle is divided in to three eight node quadrilateral elements, to generate a mesh over rectangular and curved region, performance of the new approach are demonstrated with numerical examples.

Generalised Modified Taylor Series Approach of Developing k-step Block Methods for Solving Second Order Ordinary Differential Equations

Various algorithms have been proposed for developing block methods where the most adopted approach is the numerical integration and collocation approaches. However, there is another conventional approach known as the Taylor series approach, although it was utilised at inception for the development of linear multistep methods for first order differential equations. Thus, this article explores the adoption of this approach through the modification of the aforementioned conventional Taylor series approach. A new methodology is then presented for developing block methods, which is a more accurate method for solving second order ordinary differential equations, coined as the Modified Taylor Series (MTS) Approach. A further step is taken by presenting a generalised form of the MTS Approach that produces any k-step block method for solving second order ordinary differential equations. The computational complexity of this approach after being generalised to develop k-step block method for second order ordinary differential equations is calculated and the result shows that the generalised algorithm involves less computational burden, and hence is suitable for adoption when developing block methods for solving second order ordinary differential equations. Specifically, an alternate and easy-to-adopt approach to developing k-step block methods for solving second order ODEs with fewer computations has been introduced in this article with the developed block methods being suitable for solving second order differential equations directly.

Integrated Sustainability Assessment of Divergent Mediterranean Farming Systems: Cyprus as a Case Study

A variety of indicator-based methods have been developed for the sustainability assessment of farming systems (FSs). However, many of them lack holisticity, focus on a specific agricultural sector/product, and do not provide aggregated results to better support decision-making process. The goal of this study was, for the first time, to assess, in a holistic manner, the sustainability performance of different FSs in southeastern Cyprus. The methodological framework involved three major steps. First, the sustainability context was set, and a list of 41 environmental, social, and economic indicators was created. The indicators were then calculated using data from 324 farms. Second, six FSs were identified using multivariate analysis. Finally, the sustainability of FSs was assessed by combining numerical (construction of four composite sustainability indices) and visual (presentation of indicator scores and values with graphs and tables) integration approaches. While the indices provided the “big picture”, visual integration revealed the areas where policy interventions are needed. The analysis showed that sustainable agricultural practices are already used by some farmers in the area. The results could be used for benchmarking purposes and to aid decision-making process in Cyprus but might also be useful for other Mediterranean regions with similar agro-ecological conditions.

A Boundary Integral Method Based on the Symplectic System

In this paper, we introduce a numerical boundary integral approach to study mechanical problems of curvilinear boundaries, multi-materials and fractures of solids and structures. Based on the equations of the eigensolution expansion and the weight functions of the basic solutions, the integral weighted equations of the undetermined coefficients can be set up by using the boundary conditions and the continuity conditions between different materials. Due to the completeness of the solution space and the independent property between the eigensolutions, the existence and the uniqueness of the solution of the boundary integral equations are seriously guaranteed. Therefore, the problems can be solved numerically by this approach.

Option Pricing under Double Stochastic Volatility Model with Stochastic Interest Rates and Double Exponential Jumps with Stochastic Intensity

We present option pricing under the double stochastic volatility model with stochastic interest rates and double exponential jumps with stochastic intensity in this article. We make two contributions based on the existing literature. First, we add double stochastic volatility to the option pricing model combining stochastic interest rates and jumps with stochastic intensity, and we are the first to fill this gap. Second, the stochastic interest rate process is presented in the Hull–White model. Some authors have concentrated on hybrid models based on various asset classes in recent years. Therefore, we build a multifactor model with the term structure of stochastic interest rates. We also approximated the pricing formula for European call options by applying the COS method and fast Fourier transform (FFT). Numerical results display that FFT and the COS method are much faster than the numerical integration approach used for obtaining the semi-closed form prices. The COS method shows higher accuracy, efficiency, and stability than FFT. Therefore, we use the COS method to investigate the impact of the parameters in the stochastic jump intensity process and the existence of the process on the call option prices. We also use it to examine the impact of the parameters in the interest rate process on the call option prices.

Multi-Symplectic Method for the Logarithmic-KdV Equation

The multi-symplectic integrator, as a numerical integration approach with symmetry, is known to have the characteristic of preserving the qualitative features and geometric properties of certain systems. Using the multi-symplectic integrator, the numerical simulation of the Gaussian solitary wave propagation of the logarithmic Korteweg–de Vries (logarithmic-KdV) equation was investigated. The multi-symplectic formulation of the logarithmic-KdV equation was explored by introducing some intermediate variables. A fully implicit version of the centered box scheme was used to discretize the multi-symplectic equations. In addition, numerical experiments were carried out to show the conservative properties of the proposed scheme.

A modified numerical integration method to calculate the view factor between finite and infinite cylinders in arbitrary array

The view factor has a critical impact on the estimation of the radiation heat transfer between different surfaces of energy systems. This paper presents an alternate and simple method for the blocking judgment and a modified numerical integration approach to calculate the view factor between finite and infinite length cylinders in an arbitrary array. The proposed method has been successfully validated through a comparison with the state-of-the-art published methods. Two cases are studied: (i) cylinders arranged in a staggered array for different spacing and lengths; (ii) cylinders arranged in an arbitrary array. The results show that the proposed method provides a fast and accurate way to evaluate the view factor for cylinder arrays with similar length. The analysis reveals that the length and arrangements have a strong influence on the view factor. The proposed method might be very useful to design an improved compact heat exchanger using a nonstandard arrangements.

Potential field continuation in spatial domain: A new kernel function and its numerical scheme

As a significant data processing method, the potential field continuation is usually implemented in frequency domain, but rarely in spatial domain. However, the space-domain method has the advantages of being more accurate and flexible. In this study, we propose a new numerical integral approach to solve the integration equation. The algorithm can be applied to upward continuation as well as stable downward continuation, and can be used to obtain continued potential field data from a plane upward to a surface and from a surface downward to a plane. A smooth fitting curve is proposed to determine optimal iterative number for the iteration procedure of the algorithm. Both synthetic and field data are used to test the proposed approach and compare with previous studies. The numerical experiment results indicate the proposed method has a stable numerical performance and a wide range of application.

Nonlinear Dynamic Characteristic Analysis of a Coated Gear Transmission System

Coatings can significantly improve the load-carrying performance of a gear surface, but how they affect the vibration characteristic of the system is an urgent issue to be solved. Taking into account the nonlinear factors like the variable mesh stiffness, friction, backlash, and transmission error, a six-degree-of-freedom spur gear transmission system with coatings is presented. Meanwhile, the finite element method is applied to acquire the time-varying mesh stiffness of the coated gear pair in the engagement process. With the support of the time-history curve, phase curve, Poincare map, and fast Fourier transform spectrum, the dynamic characteristics and the effects of the coating elastic modulus on vibration behaviors of a gear transmission system are minutely dissected by using a numerical integration approach. Numerical cases illustrate that the dynamic characteristic of a gear transmission system tends toward a one-period state under the given operating condition. They also indicate that, compared with softer coatings, stiffer ones can properly enhance the transmission performance of the coated gear pair. Numerical results are also compared with previous studies, and can establish a theoretical basis for dynamic design and vibration control of the coated gear transmission system.

A numerical integration approach for fractional‐order viscoelastic analysis of hereditary‐aging structures

SummaryIn this paper, a novel numerical integration scheme is proposed for fractional‐order viscoelastic analysis of hereditary‐aging structures. More precisely, the idea of aging is first introduced through a new phenomenological viscoelastic model characterized by variable‐order fractional operators. Then, the presented fractional‐order viscoelastic model is included in a variational formulation, conceived for any viscous kernel and discretized in time by employing a discontinuous Galerkin method. The accuracy of the resulting finite element (FE) scheme is analyzed through a model problem, whose exact solution is known; and the most significant variables affecting the solution quality, such as the number of Gaussian quadrature points and time subintervals, are then investigated in terms of error and computational cost. Moreover, the proposed FE integration scheme is applied to study the short‐ and long‐term behavior of concrete structures, which, due to the severe aging exhibited during their service life, represents one of the most challenging time‐dependent behavior to be investigated. Eventually, also the Euler implicit method, commonly used in commercial software, is compared.

Influence of surface roughness on cyclists’ velocity choices

The purpose of this paper is to examine the influence of different roadway surfaces’ roughness on cyclists’ velocity choices. Could surface roughness be efficiently used for designing velocity choices according to needs in different locations? First, a survey to identify environmental conditions’ impact on cyclist velocity choices is employed. The main part analyses the speeds of 3750 cyclists, measured with a radar gun on seven different surfaces: smooth and rough asphalt, painted cycle lanes, concrete, large and small cobblestones and gravel. By utilising a smartphone's built-in accelerometer affixed to a bicycle, the three-dimensional acceleration-based roughness of the surfaces is measured over a fixed distance. The analysis of scatter plots reveals that the vertical acceleration component approximates the total acceleration best. Ultimately, a numerical integration approach is used to derive comparable vertical distance values per surface type and fork stiffness. The collected data show that the 50th and 85th percentiles of cyclist speeds are similar on all surface types: 19·4–20·7 and 23–25·5 km/h. Furthermore, the data show only a weak correlation between measured velocity and surface roughness, thus suggesting that other, unconsidered factors outweigh the influence of the road surface and point the way to future research needs.

Terpolymerization of Triisopropylsilyl Acrylate, Methyl Methacrylate, and Butyl Acrylate: Reactivity Ratio Estimation

AbstractTernary monomer reactivity ratios of triisopropylsilyl acrylate (SiA), methyl methacrylate (MMA), and n‐butyl acrylate (BA), as common monomers in self‐polishing coatings (SPCs) binders are obtained using experimental data collected from free radical bulk polymerization at 70 °C. Different terpolymerizations at low and medium‐high conversions are performed at optimized feed compositions. Estimations are made using the error‐in‐variables model (EVM) framework, applying the recast form of the Alfrey–Goldfinger (AG) model and a direct numerical integration (DNI) approach to the collected data. Estimations from individual low and medium‐high conversion data are compared to those found with the combined data (full conversion range data). The highest certainty in point estimates are obtained with analysis of the full conversion range data. Furthermore, the reactivity ratios determined from the combined data fall between those found with analysis of individual low and medium‐high conversion data, another corroboration of reliable data collection. Reactivity ratios determined from analysis of the combined data (rSiA/MMA = 0.4185, rMMA/SiA = 1.3754, rSiA/BA = 0.8739, rBA/SiA = 0.5736, rBA/MMA = 0.3692, rMMA/BA = 1.7919) are used in the recast AG model to predict cumulative terpolymer composition as a function of conversion. The experimental data and model prediction show satisfactory agreement.

One-step validation method for surrogate endpoints using data from multiple randomized cancer clinical trials with failure-time endpoints.

A surrogate endpoint can be used instead of the most relevant clinical endpoint to assess the efficiency of a new treatment. Before being used, a surrogate endpoint must be validated based on appropriate methods. Numerous validation approaches have been proposed with the most popular used in a context of meta-analysis, based on a two-step analysis strategy. For two failure-time endpoints, two association measurements are usually used, Kendall's τ at the individual level and the adjusted coefficient of determination ( ) at the trial level. However, is not always available due to model estimation constraints. We propose a one-step validation approach based on a joint frailty model, including both individual-level and trial-level random effects. Parameters have been estimated using a semiparametric penalized marginal log-likelihood method, and various numerical integration approaches were considered. Both individual- and trial-level surrogacy were evaluated using a new definition of Kendall's τ and the coefficient of determination. Estimators' performances were evaluated using simulation studies and satisfactory results were found. The model was applied to individual patient data meta-analyses in gastric cancer to assess disease-free survival as a surrogate for overall survival, as part of the evaluation of adjuvant therapy.

Boundary modeling with moving least squares

Modeling geological boundaries is an important step in geological modeling workflows that aim to quantify the location, volume, and uncertainty of a resource. Several well established explicit and implicit techniques can be used to identify the limits of an ore zone; however, most techniques are unable to account for uncertainty in a straightforward manner, where uncertainty originates from the sparseness of sample data relative to the volume of the resource in question. An implicit technique that benefits from simplicity and has the capabilities to incorporate and account for uncertainty in boundary geometry is signed distance function (SDF) modeling. Sample data that is in the form of line segments from drill holes presents challenges for interpolators such as kriging and radial basis functions. Utilizing linear regression as an interpolator several advantages to the problem are realized. The particular variant of regression that is used is moving least squares (MLS) that permits the exact integration of line data from drill holes and human interpretations without the need for discretization or numerical integration approaches. Integrating line data is critical for including explicit interpretations into implicit boundary models because geologic interpretations commonly involve polygons digitized on two dimensional sections through a field of drill holes.

Computable form of the Born-Markov master equation for open multilevel quantum systems

The Redfield and Lindblad forms of the master equations are obtained under not only the Born and Markov approximations but also the secular approximation. In this paper, we propose a computable form of the master equation for open multilevel quantum systems beyond the secular approximation. The proposed form can retain some dynamical effects that are lost due to use of the secular approximation. Two kinds of open multilevel quantum systems are investigated by using this method, and the results are compared to the ones obtained from the Redfield equation and an exactly numerical path integral approach. It is shown that the results obtained from this computable form match those from the exact numerical path integral approach better than results from the Redfield equation. The normal-diffusion coefficient, momentum-damping coefficient, anomalous-diffusion coefficient, etc. in the master equation are also reevaluated.

Dielectric equilateral triangle microresonators: integral equations and semiclassical physics approaches

We investigate in detail the resonant properties of a two-dimensional dielectric cavity with an equilateral triangle shape, using a numerical integral equations approach and a semiclassical approach. The homogeneous Muller boundary equations are used to calculate the resonant modes of a dielectric triangle in a wide range of frequencies. It is shown that the modes of dielectric triangles localized on families of periodic orbits can be well described in terms of a semiclassical superscar model. Special attention is given to the families of resonances based on the inscribed triangle periodic orbit.

Sequential ℓ1 Quadratic Programming for Nonlinear Model Predictive Control

Abstract In this paper, we present and describe a computationally efficient sequential l1 quadratic programming (Sl1QP) algorithm for Nonlinear Model Predictive Control (NMPC). We use a tailored trust region sequential quadratic programming for the solution of the optimal control problem (OCP) involved in the NMPC algorithm. We use a multiple shooting approach for numerical integration and sensitivity computation. A second order correction ensures a faster convergence of the SQP algorithm. We exploit the structure of the OCP by using an efficient primal-dual interior point algorithm based on Riccati factorizations and a block diagonal BFGS update of the Hessian matrix. The complexity scales linearly with the prediction horizon length. We numerically evaluate and compare the performance of our algorithm on a numerical example.

Improving the two-stage numerical integration in stability identification of oscillation with distributed delay

The vibration of the engineering systems with distributed delay is governed by delay integro-differential equations. Two-stage numerical integration approach was recently proposed for stability identification of such oscillators. This work improves the approach by handling the distributed delay—that is, the first-stage numerical integration—with tensor-based higher order numerical integration rules. The second-stage numerical integration of the arising methods remains the trapezoidal rule as in the original method. It is shown that local discretization error is of order [Formula: see text] irrespective of the order of the numerical integration rule used to handle the distributed delay. But [Formula: see text] is less weighted when higher order numerical integration rules are used to handle the distributed delay, suggesting higher accuracy. Results from theoretical error analyses, various numerical rate of convergence analyses, and stability computations were combined to conclude that—from application point of view—it is not necessary to increase the first-stage numerical integration rule beyond the first order (trapezoidal rule) though the best results are expected at the second order (Simpson’s 1/3 rule).

Effective Computational Approach for Prediction and Estimation of Space Object Breakup Dispersion during Uncontrolled Reentry

This paper provides an effective approach for the prediction and estimation of space debris due to a vehicle breakup during uncontrolled reentry. For an advanced analysis of the time evolution of space debris dispersion, new efficient computational approaches are proposed. A time evolution of the dispersion of space pieces from a breakup event to the ground impact time is represented in terms of covariance ellipsoids, and in this paper, two covariance propagation methods are introduced. First, a derivative-free statistical linear regression method using the unscented transformation is utilized for performing a covariance propagation. Second, a novel Gaussian moment-matching method is proposed to compute the estimation of the covariance of a debris dispersion by using a Gauss-Hermite cubature-based numerical integration approach. Compared to a linearized covariance propagation method such as the Lyapunov covariance equation, the newly proposed Gauss-Hermite cubature-based covariance computation approach could provide high flexibilities in terms of effectively representing an initial debris dispersion and also precisely computing the time evolution of the covariance matrices by utilizing a larger set of sigma points representing debris components. In addition, we also carry out a parametric study in order to analyze the effects on the accuracy of the covariance propagation due to modeling uncertainties. The effectiveness of the newly proposed statistical linear regression method and the Gauss-Hermite computational approach is demonstrated by carrying out various simulations.

Kinetic modeling of cellobiose by a β-glucosidase from Aspergillus fumigatus

The final step in lignocellulose enzymatic saccharification is the cellobiose conversion to glucose by β-glucosidases (BG). In this work, a valid kinetic model to describe cellobiose degradation for an industrial mixture of BG enzymes present in Aspergillus fumigatus is selected. Firstly, the enzyme mixture was characterised in terms of protein content and enzymatic activity on p-NPG (1326UmLpreparation−1), determining the molecular weight of the only BG activity band observed in zymograms by SDS-PAGE and MALDI-TOF: 94kDa. Subsequently, to select the correct kinetic model for the enzymatic hydrolysis of cellobiose, a combined strategy was performed: Firstly, non-linear regressions were applied to initial hydrolysis rate data for different enzyme concentrations and initial substrate and product concentrations, observing inhibition by cellobiose and glucose. Secondly, the optimal kinetic model was discriminated by a coupled non-linear regression-DOE numerical integration approach, by fitting several possible kinetic models involving different product inhibition mechanisms to progress curve data from runs at various initial substrate concentrations and temperatures. The best kinetic model involves non-competitive substrate inhibition and product competitive inhibition with two binding sites for glucose.