Weight Function Technique Research Articles

A Class of Efficient Sixth-Order Iterative Methods for Solving the Nonlinear Shear Model of a Reinforced Concrete Beam

In this paper, we present a three-step sixth-order class of iterative schemes to estimate the solutions of a nonlinear system of equations. This procedure is designed by means of a weight function technique. We apply this procedure for predicting the shear strength of a reinforced concrete beam. The values for the parameters of the nonlinear system describing this problem were randomly selected inside the prescribed ranges by technical standards for structural concrete. Moreover, some of these parameters were fixed taking into consideration the solvability region of the adopted steel constitutive model. The effectiveness of the new class is also compared with other current schemes in terms of the computational efficiency and numerical performance, with very good results. The advantages of this new class come from the low computational cost, due to the existence of an only inverse operator.

A General Family of Fifth-order Iterative Methods for Solving Nonlinear Equations

A family of fifth-order iterative methods is proposed for solving nonlinear equations using the weight function technique. This family offers flexibility through its structure and the choice of weight functions, resulting in a wide range of new specific schemes. It is demonstrated that this proposed family includes several well-known and recent methods as special cases. In addition, several new particular methods are designed to achieve better results than existing methods of the same type. Convergence analysis is conducted, and numerical examples in both real and complex domains are provided for several specific schemes within the proposed family. Comparisons between the existing methods within this family and the newly introduced methods generally indicate improved performance among the new members. Notably, the study of complex dynamics and basins of attraction reveals that our new specific schemes have broader sets of initial points that lead to convergence.

Convergence and dynamical study of a new sixth order convergence iterative scheme for solving nonlinear systems

<abstract><p>A novel family of iterative schemes to estimate the solutions of nonlinear systems is presented. It is based on the Ermakov-Kalitkin procedure, which widens the set of converging initial estimations. This class is designed by means of a weight function technique, obtaining 6th-order convergence. The qualitative properties of the proposed class are analyzed by means of vectorial real dynamics. Using these tools, the most stable members of the family are selected, and also the chaotical elements are avoided. Some test vectorial functions are used in order to illustrate the performance and efficiency of the designed schemes.</p></abstract>

Process optimization using a dynamic self-adaptive constraint handling technique coupled to a Differential Evolution algorithm

Nowadays, chemical processes are projected to obtain their best performance in energy and water consumption, pollutant emissions and total annual costs, while still meeting quality of products and good operational performance. These goals are accomplished through adequate optimization of the fitness function by manipulating the operational variables (decision variables) of the process. However, a successful optimization process depends completely on the constraint handling established in the modeling of the process. The weighted summation of constraint violations (weighting function technique, WF) is one of the most common approaches for handling constraints in optimization problems. Nevertheless, in spite of this technique yielding good results, in this work we show a novel self-adaptive constraint handling technique (SA) based on a self-adaptation dynamic threshold and self-adaptation (weight) factors. This technique deals with real and discrete variables and converts equality constraints into inequality constraints through a dynamic threshold. Both penalization techniques (WF and SA) were, respectively, coupled to a Differential Evolution (DE) algorithm to optimize some benchmark functions and chemical engineering optimization problems. In addition, the rigorous model of a distillation train was optimized in Aspen One for the first time with a self-adaptive constraint handling technique in chemical engineering. Although both penalization techniques were coupled to the same DE algorithm and both cases were run under the same conditions, the results show that the dynamic self-adaptive constraint handling technique coupled to DE (DE-SA) achieves considerably better best-solutions than the best-solutions obtained by the weighting function technique coupled to DE (DE-WF). In addition, DE-SA led to substantial reductions of numerical effort in relation to DE-WF. These conclusions are supported by statistical analysis of the results of 30 runs of the optimization process for each constraint handling technique, for a distillation train.

Efficient iterative scheme for solving non-linear equations with engineering applications

ABSTRACT A family of three-step optimal eighth-order iterative algorithm is developed in this paper in order to find single roots of nonlinear equations using the weight function technique. The newly proposed iterative methods of eight order convergence need three function evaluations and one first derivative evaluation that satisfies the Kung–Traub optimality conjecture in terms of computational cost per iteration (i.e.). Furthermore, using the primary theorem that establishes the convergence order, the theoretical convergence properties of our schemes are thoroughly investigated. On several engineering applications, the performance and efficiency of our optimal iteration algorithms are examined to those of existing competitors. The new iterative schemes are more efficient than the existing methods in the literature, as illustrated by the basins of attraction, dynamical planes, efficiency, log of residual, and numerical test examples.

Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction

Research interest in iterative multipoint schemes to solve nonlinear problems has increased recently because of the drawbacks of point-to-point methods, which need high-order derivatives to increase the order of convergence. However, this order is not the only key element to classify the iterative schemes. We aim to design new multipoint fixed point classes without memory, that improve or bring together the existing ones in different areas such as computational efficiency, stability and also convergence order. In this manuscript, we present a family of parametric iterative methods, whose order of convergence is four, that has been designed by using composition and weight function techniques. A qualitative analysis is made, based on complex discrete dynamics, to select those elements of the class with best stability properties on low-degree polynomials. This stable behavior is directly related with the simplicity of the fractals defined by the basins of attraction. In the opposite, particular methods with unstable performance present high-complexity in the fractals of their basins. The stable members are demonstrated also be the best ones in terms of numerical performance of non-polynomial functions, with special emphasis on Colebrook-White equation, with wide applications in Engineering.

Weight function procedure for reduced order fracture analysis of arbitrary flaws in cylindrical pressure vessels

Fracture mechanics calculations using the finite element method can be computationally expensive, which makes them challenging to use in engineering evaluations of flaws in pressure vessels. The weight function (WF) technique is a reduced order fracture modeling method that is widely used to greatly reduce these computational costs. Its computational efficiency is essential for use in probabilistic fracture mechanics evaluations of embrittled nuclear reactor pressure vessels (RPVs), due to the large number of sampled flaws that must be evaluated. Although the WF technique is general, it is typically used only for axis-aligned flaws. Recent discoveries of off-axis flaws in operating nuclear reactors have necessitated detailed simulations of such flaws and the interactions between neighboring flaws. This paper presents a generalized WF approach applicable for analyzing arbitrary flaw geometries in thick-walled cylindrical vessels, including surface-breaking and subsurface flaws, which can either axis-aligned or off-axis, and can account for interactions with other flaws. This approach is demonstrated on representative simulations of multiple flaw geometries in a RPV subjected to transient loading conditions. In all cases, the WF technique gives good comparison with benchmark results from direct simulation with greatly reduced computational effort.

A new fifth-order iterative method for solving non-linear equations using weight function technique and the basins of attraction

In this paper, new iterative method is presented of fifth-order for solving non-linear equations \(f\left(x\right)=0\) a devoid of the second derivative which requires two derivative functions and evaluations for each step, using both weight functions and synthesis techniques together. This method improves Newton's method and thus the efficiency index has been improved from \(1.414\) to \(1.495\). The convergence analysis for the new method is discussed. We provide some numerical examples that illustrate the performance of our proposed method by comparing them with numerical methods of fifth-order also the complex dynamics and basins of attraction is discussed, comparing it with several methods of the same order, thus comparisons show that new method gives the best results.

A Derivative Free Fourth-Order Optimal Scheme for Applied Science Problems

We suggest a new and cost-effective iterative scheme for nonlinear equations. The main features of the presented scheme are that it does not involve any derivative in the structure, achieves an optimal convergence of fourth-order factors, has more flexibility for obtaining new members, and is two-point, cost-effective, more stable and yields better numerical results. The derivation of our scheme is based on the weight function technique. The convergence order is studied in three main theorems. We have demonstrated the applicability of our methods on four numerical problems. Out of them, two are real-life cases, while the third one is a root clustering problem and the fourth one is an academic problem. The obtained numerical results illustrate preferable outcomes as compared to the existing ones in terms of absolute residual errors, CPU timing, approximated zeros and absolute error difference between two consecutive iterations.

Renting Servers in the Cloud: The Case of Equal Duration Jobs

Renting servers in the cloud is a generalization of the bin packing problem, motivated by job allocation to servers in cloud computing applications. Jobs arrive in an online manner, and need to be assigned to servers; their duration and size are known at the time of arrival. There is an infinite supply of identical servers, each having one unit of computational capacity per unit of time. A server can be rented at any time and continues to be rented until all jobs assigned to it finish. The cost of an assignment is the sum of durations of rental periods of all servers. The goal is to assign jobs to servers to minimize the overall cost while satisfying server capacity constraints. We focus on analyzing two natural algorithms, NextFit and FirstFit, for the case of jobs of equal duration. It is known that the competitive ratio of NextFit and FirstFit are at most 3 and 4 respectively for this case. We prove a tight bound of 2 on the competitive ratio of NextFit. For FirstFit, we establish a lower bound of 2.519 on the competitive ratio, even when jobs have only two distinct arrival times. For the case when jobs have arrival times 0 and 1 and duration 2, we show a lower bound of 1.89 and an upper bound of 2 on the strict competitive ratio of FirstFit. Finally, using the weight function technique, we obtain stronger results for the case of uniform servers.

A two-point eighth-order method based on the weight function for solving nonlinear equations

In this work, we have designed a family of with-memory methods with eighth-order convergence. We have used the weight function technique. The proposed methods have three parameters. Three self-accelerating parameters are calculated in each iterative step employing only information from the current and all previous iteration. Numerical experiments are carried out to demonstrate the convergence and the e?ciency of our iterative method.

Family of optimal two-step fourth order iterative method and its extension for solving nonlinear equations

In this paper, the weight function technique is utilized to develop new family of two-step fourth order convergence iterative methods for approximating the solution of nonlinear equations. The methods require the evaluation of three distinct functions evaluation per iteration circle and as such, are optimal in agreement with the Kung-Traub’s conjecture. This developed family of methods is further extended to design another new family of four-step ninth order convergence with efficiency index EI = 1.5518. We carried out the convergence analysis of the two families of iterative methods. This analysis provided us with information about the flexibility of the weight function in the method used in constructing other new families of iterative methods. The methods are applied to solve some nonlinear equations and real life problems that are modeled into nonlinear equations. The results obtained from computation experience are compared with some of its existing contemporary methods.

A New Family of Fourth-Order Optimal Iterative Schemes and Remark on Kung and Traub’s Conjecture

Kung and Traub conjectured that a multipoint iterative scheme without memory based on m evaluations of functions has an optimal convergence order p = 2 m − 1 . In the paper, we first prove that the two-step fourth-order optimal iterative schemes of the same class have a common feature including a same term in the error equations, resorting on the conjecture of Kung and Traub. Based on the error equations, we derive a constantly weighting algorithm obtained from the combination of two iterative schemes, which converges faster than the departed ones. Then, a new family of fourth-order optimal iterative schemes is developed by using a new weight function technique, which needs three evaluations of functions and whose convergence order is proved to be p = 2 3 − 1 = 4 .

A stable class of modified Newton-like methods for multiple roots and their dynamics

Abstract There have appeared in the literature a lot of optimal eighth-order iterative methods for approximating simple zeros of nonlinear functions. Although, the similar ideas can be extended for the case of multiple zeros but the main drawback is that the order of convergence and computational efficiency reduce dramatically. Therefore, in order to retain the accuracy and convergence order, several optimal and non-optimal modifications have been proposed in the literature. But, as far as we know, there are limited number of optimal eighth-order methods that can handle the case of multiple zeros. With this aim, a wide general class of optimal eighth-order methods for multiple zeros with known multiplicity is brought forward, which is based on weight function technique involving function-to-function ratio. An extensive convergence analysis is demonstrated to establish the eighth-order of the developed methods. The numerical experiments considered the superiority of the new methods for solving concrete variety of real life problems coming from different disciplines such as trajectory of an electron in the air gap between two parallel plates, the fractional conversion in a chemical reactor, continuous stirred tank reactor problem, Planck’s radiation law problem, which calculates the energy density within an isothermal blackbody and the problem arising from global carbon dioxide model in ocean chemistry, in comparison with methods of similar characteristics appeared in the literature.

A posteriori error analysis of an HDG method for the Oseen problem

We propose a residual-type a posteriori error estimator for a hybridizable discontinuous Galerkin method applied to the Oseen problem in a gradient-velocity-pressure formulation. We state reliability and local efficiency results for our estimator with respect to an error measured in the natural norms, with constants depending explicitly on the physical parameters. A weighting function technique to control the L2-error of the velocity, and the approximation properties of the Oswald interpolation operator are the main ingredients in the analysis. Numerical experiments in three dimensions validate our theoretical results.

Fixed Point Root-Finding Methods of Fourth-Order of Convergence

In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of the derivative, is the unique difference between the two steps of each method, which is unusual. As it is proven that all the members of the class are optimal methods in the sense of Kung-Traub’s conjecture, the dynamical analysis is a good tool to determine the best elements of the family in terms of stability. Therefore, the dynamical behavior of this class on quadratic polynomials is studied in this work. We analyze the stability of the presented family from the multipliers of the fixed points and critical points, along with their associated parameter planes. In addition, this study enables us to select the members of the class with good stability properties.

Some New Higher Order Weighted Newton Methods for Solving Nonlinear Equation with Applications

Three new iterative methods for solving scalar nonlinear equations using weight function technique are presented. The first one is a two-step fifth order method with four function evaluations which is improved from a two-step Newton’s method having same number of function evaluations. By this, the efficiency index of the new method is improved from 1.414 to 1.495. The second one is a three step method with one additional function evaluation producing eighth order accuracy with efficiency index 1.516. The last one is a new fourth order optimal two-step method with efficiency index 1.587. All these three methods are better than Newton’s method and many other equivalent higher order methods. Convergence analyses are established so that these methods have fifth, eighth and fourth order respectively. Numerical examples ascertain that the proposed methods are efficient and demonstrate better performance when compared to some equivalent and optimal methods. Seven application problems are solved to illustrate the efficiency and performance of the proposed methods.

Local convergence of iterative methods for solving equations and system of equations using weight function techniques

This paper analyzes the local convergence of several iterative methods for approximating a locally unique solution of a nonlinear equation in a Banach space. It is shown that the local convergence of these methods depends of hypotheses requiring the first-order derivative and the Lipschitz condition. The new approach expands the applicability of previous methods and formulates their theoretical radius of convergence. Several numerical examples originated from real world problems illustrate the applicability of the technique in a wide range of nonlinear cases where previous methods can not be used.

Fatigue life evaluation of welded joints in steel bridge considering residual stress

Welded joints are widely used to connect structural components in steel truss bridges. Weld residual stresses (WRS) and weld residual stresses relaxation (WRSR) have notable influences on fatigue crack propagation in welded joints, and therefore affect the fatigue life of welded joints. Failing to properly consider the effect of WRS and WRSR in fatigue evaluation may lead to inaccurate results. This study presents a fatigue life prediction model based on the elastic fracture mechanics, with consideration of the WRS and WRSR. The solution for stress intensity factor caused by cyclic loading and WRS is derived. The WRS-induced stress intensity factor is calculated using a weight function technique. Fatigue tests of eight welded joint specimens are implemented, and the fatigue failure analysis of specimens is conducted. The proposed fatigue life prediction model is validated against fatigue test results of welded joints. By considering both WRS and WRSR, the model provides a prediction of fatigue life with a maximum error of 14%. Finally, the validated model is employed to investigate the fatigue life of a real bridge. The fatigue life is underestimated by 17% by considering WRS but not considering WRSR; the fatigue life is overestimated by 49% by neglecting WRS and WRSR.

On implementations of H 2 preview output feedback law with application to LFC with load demand prediction

ABSTRACT This paper addresses a continuous-time preview control problem in output feedback setting. By a state transformation approach, it is shown that the optimal output feedback law is implemented in two different ways. The first and second implementations have the structures similar to the Smith and observer-based predictors for input-delay systems, respectively. In order to illustrate the features of the optimal control law, we design an Load Frequency Control system, which utilises load demand prediction. Although the accompanying prediction error obstacles the performance improvement, it is demonstrated that the adverse effect is mitigated by the weighting function technique.