Application Of Nonlinear Analysis Research Articles

Editorial Conclusion for the Special Issue “New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization”

Nonlinear analysis has widespread and significant applications in many areas at the core of many branches of pure and applied mathematics and modern science, including nonlinear ordinary and partial differential equations, critical point theory, functional analysis, fixed point theory, nonlinear optimization, fractional calculus, variational analysis, convex analysis, dynamical system theory, mathematical economics, data mining, signal processing, control theory, and many more [...]

Unraveling the Complexity with Applications of Nonlinear Analysis in Signal Processing and Communication Engineering

Nonlinear analysis is an essential area of signal and communication engineering. Many different kinds of nonlinear analysis can be used in this study. Linear signal processing techniques had previously struggled to accurately capture complex data pattern and communication signals. However, the development of non-linear analysis is a game-changing advancement since it provides a solid framework for understanding and manipulating these complex and non-trivial dynamics. In order to understand the behaviour of a wide variety of signals, from financial market trends to biological and environmental data, non-linear techniques are essential in signal processing. Improved understanding of system behaviour and predictive models may result from the use of such techniques to extract complex data patterns that are not often used. Non-linear approaches also contribute to the development of image and video processing by easing the development of advanced compression, tracking, and object recognition algorithms. Non-linear analysis is crucial in communication engineering as it aids in comprehending and optimising the flow of information across various channels. A greater understanding of signal distortion is made possible by the nonlinear perspective, which makes it possible to build reliable communication systems that can withstand interference and noise. By facilitating the creation of sophisticated modulation systems, adaptive equalisation methods, and error control coding, it raises the communication networks' dependability and effectiveness. Designing complex systems that can handle the complexities of today's data and communication demands is made possible by the paradigm-shifting synergy of nonlinear analysis, signal processing, and communication engineering. This abstract highlights the role that nonlinear analysis plays in transforming various fields and highlights how it can be used to solve changing problems related to information comprehension, processing, and transmission in a technologically sophisticated world.

A mixed micromechanical homogenisation scheme for the prediction of the 3D orthotropic elastic properties of different masonry bond typologies

Masonry structures composed of two distinct material phases (units and mortar) with potentially vastly different elastic properties are characterised by strong orthotropy. The variety of commonly employed bond typologies for masonry walls, featuring unit groups and mortar joints with different orientation with respect to the plane of the wall, result in complex stress and strain interactions between the material phases. This interaction is further complicated by the high volume ratio of the units within the masonry composite. Therefore, classical homogenisation schemes for matrix-inclusion composite materials, mortar being the matrix and units the inclusions, are suitable for only a narrow range of relative unit and mortar stiffness, expressed as the ratio of the Young’s moduli of the two phases. This feature makes their application in nonlinear analysis of masonry structures, which have a strong tendency to crack under mechanical loading, problematic.The paper presents a novel homogenisation scheme based on different mean-field homogenisation schemes combined through a process derived from the discretisation of the masonry cells using a method-of-cells approach. This approach allows the intuitive interpretation of the stress and strain interaction of the material phases and yields good predictions of the elastic properties of a variety of masonry bonds over an extensive range of relative unit and mortar stiffness.

Preface to the Special Issue “Fixed Point Theory and Dynamical Systems with Applications”

Since the celebrated Brouwer’s fixed point theorem and Banach contraction principle were established, the rapid growth of fixed point theory and its applications have led to a number of scholarly essays studying the importance of its promotion and application in nonlinear analysis, applied mathematical analysis, economics, game theory, integral and differential equations and inclusions, dynamic systems theory, signal and image processing, etc [...]

Applications of Nonlinear Analysis in Engineering Design

Applications of Nonlinear Analysis in Engineering Design

Fragility and Vulnerability Analysis of an RC Building with the Application of Nonlinear Analysis

In this paper, the seismic response of a five-story reinforced concrete (RC) frame system building is analysed through fragility analysis. The structure is designed in accordance with structural Eurocodes EN1990, EN1991, EN1992 and EN1998, as a high-ductility (DCH) system. For the analysis of the response of a structural system to earthquake actions, the methods of nonlinear static (NSA) and nonlinear dynamic analyses (NDA) are applied and, based on the obtained results, fragility curves are constructed using statistical methods. A relationship between the intensity measure (IM) and engineering demand parameters (EDPs) is needed in order to estimate a fragility curve. Fragility functions represent a possibility for different states of damage to occur in a certain structural systems at the observed value of the specified IM. Ten accelerograms, used in NDA, are selected and scaled, according to EN1998 provisions, for the chosen elastic response spectrum and referent PGA. Obtained results are used for the statistical analysis and construction of fragility curves. Structural damage state threshold parameters are determined based on the Park and Ang modified damage index methodology and provisions given in FEMA, HAZUS, VISION 2000 and EN codes. Comparative analysis of the structural damage probability for the analysed RC building, calculated using different methodologies to determine damage states, is applied. The fragility analysis results showed the difference between the probabilities of the damage states to occur when different calculation methods are used. This reflects on the assessment of vulnerability curves as well. The obtained results, calculated using different methods, are analysed using comparative analysis.

Application of Nonlinear Finite Element Analysis on Shear-Critical Reinforced Concrete Beams

This paper presents the application of a smeared fixed crack approach for nonlinear finite element analysis of shear-critical reinforced concrete beams. The experimental data was adopted from tests undertaken on twelve reinforced concrete beams by Bresler and Scordelis in 1963, and from duplicate tests undertaken by Vecchio and Shim in 2004. To this end, all beams were modeled in 3D using the software package ATENA-GiD. In the modeling, the nonlinear behaviors of the concrete were represented by fracture-plastic constitutive models, which were formulated within the smeared crack and crack/crush band approaches. The applicability of nonlinear analysis was demonstrated through accurate simulations of the full load-deflection responses, underlying mechanisms, crack patterns, and failure modes of all 24 beams. Detailed documentation of the results is presented to demonstrate the potential and practical value of nonlinear finite element analysis in providing an informed assessment of the safety and performance of reinforced concrete structures.

Approximation Results for Equilibrium Problems Involving Strongly Pseudomonotone Bifunction in Real Hilbert Spaces

A plethora of applications in non-linear analysis, including minimax problems, mathematical programming, the fixed-point problems, saddle-point problems, penalization and complementary problems, may be framed as a problem of equilibrium. Most of the methods used to solve equilibrium problems involve iterative methods, which is why the aim of this article is to establish a new iterative method by incorporating an inertial term with a subgradient extragradient method to solve the problem of equilibrium, which includes a bifunction that is strongly pseudomonotone and meets the Lipschitz-type condition in a real Hilbert space. Under certain mild conditions, a strong convergence theorem is proved, and a required sequence is generated without the information of the Lipschitz-type cost bifunction constants. Thus, the method operates with the help of a slow-converging step size sequence. In numerical analysis, we consider various equilibrium test problems to validate our proposed results.

Deterministic chaos and forecasting in Amazon?s share prices

Research background: The application of non-linear analysis and chaos theory modelling on financial time series in the discipline of Econophysics.
 Purpose of the article: The main aim of the article is to identify the deterministic chaotic behavior of stock prices with reference to Amazon using daily data from Nasdaq-100.
 Methods: The paper uses nonlinear methods, in particular chaos theory modelling, in a case study exploring and forecasting the daily Amazon stock price.
 Findings & Value added: The results suggest that the Amazon stock price time series is a deterministic chaotic series with a lot of noise. We calculated the invariant parameters such as the maxi-mum Lyapunov exponent as well as the correlation dimension, managed a two-days-ahead forecast through phase space reconstruction and a grouped data handling method.

Abstract convexity in KKM theorems via set-valued maps

Different aspects of the KKM Lemma besides its vast applications in nonlinear analysis have been investigated over the years. Some researchers tried to present the KKM results such as generalized L-KKM type theorems in the absence of usual convexity. Here, first we present a counterexample to show that the main results in some generalized KKM type theorems and their consequences are not valid. Afterward, the uniform mapconvex space is introduced and a nonempty intersection result in this space is proved.

Optimization for the Sum of Finite Functions Over the Solution Set of Split Equality Optimization Problems with Applications

In this paper, we adopt an iterative approach to solve the class of optimization problem for the sum of finite functions over split equality optimization problems for the sum of two functions. This type of problem contains many optimization problems, and bilevel problems, as well as split equality problems, and split feasibility problems as special cases. Here, we are able to establish a strong convergence theorem for an iterative method for solving this problem. As consequences of this convergence theorem, we study the following problems: optimization for the sum of finite functions over the common solution set of optimization problems for the sum of two functions; optimization for the sum of finite functions; optimization for the sum of finite functions with split equality inconsistent feasibility constraints; optimization for the sum of finite functions over the solution set for split equality constrained quadratic signal recovery problem; optimization for the sum of finite functions over the solution set of generalized split equality multiple set feasibility problem, and optimization for the sum of finite functions over the solution set of split equality linear equations problem. We use simultaneous iteration to establish strong convergence theorems for these problems. Our results generalize and improve many existing theorems for these types of problems in the literature and will have applications in nonlinear analysis, optimization problems and signal processing problems.

Azimuthally-driven subharmonic thermoacoustic instabilities in a swirl-stabilised combustor

A joint experimental and computational study of thermoacoustic instabilities in a model swirl-stabilised combustor is presented. This paper aims to deliver a better characterisation of such instabilities through the examination of measurements in conjunction with the numerical results obtained via Large Eddy Simulation (LES). The experimental configuration features a cylindrical combustion chamber where the lean premixed methane/air flame experiences self-sustained thermoacoustic oscillations. The nonlinear behaviour of the acoustically excited flame is experimentally investigated by broadband chemiluminescence and dynamic pressure measurements. In LES, the Eulerian stochastic field method is employed for the unknown turbulence/chemistry interaction of the gas-phase. Comparisons of the predicted frequency spectrum and phase-resolved flame structure with measurements are found to be in good agreement, confirming the predictive capabilities of the LES methodology. The presence of Period 2 thermoacoustic oscillations, as a consequence of period-doubling bifurcation, is also confirmed in LES. Through the application of nonlinear analysis both to the experimental and numerical acoustic fluctuations, it is highlighted that the nonlinear behaviour of combustion instabilities in the burner under investigation follows a pattern typical of Period 2 oscillations. Furthermore, the current work demonstrates a useful approach, through the use of dynamic mode decomposition (DMD), for the investigation of unstable flame modes at a specific frequency of interest. The experimental and numerical DMD reconstructions suggest that hot combustion products are convected azimuthally at a rate dictated by the subharmonic frequency.

A novel approach to enhance the accuracy of vibration control of Frames

All structures built within known seismically active regions are typically designed to endure earthquake forces. Despite advances in earthquake resistant structures, it can be inferred from hindsight that no structure is entirely immune to damage from earthquakes. Active vibration control systems, unlike the traditional methods which enlarge beams and columns, are highly effective countermeasures to reduce the effects of earthquake loading on a structure. It requires fast computation of nonlinear structural analysis in near time and has historically demanded advanced programming hosted on powerful computers. This research aims to develop a new approach for active vibration control of frames, which is applicable over both elastic and plastic material behavior. In this study, the Force Analogy Method (FAM), which is based on Hook’s Law is further extended using the Timoshenko element which considers shear deformations to increase the reliability and accuracy of the controller. The proposed algorithm is applied to a 2D portal frame equipped with linear actuator, which is designed based on full state Linear Quadratic Regulator (LQR). For comparison purposes, the portal frame is analysed by both the Euler Bernoulli and Timoshenko element respectively. The results clearly demonstrate the superiority of the Timoshenko element over Euler Bernoulli for application in nonlinear analysis.

04.05: Analysis type influence on the durability prognosis for a steel tank corroded shell

ABSTRACTAn original approach to the evaluation of remaining service time of a corroded on the ground steel shell of a tank used to store liquid petroleum products is presented and discussed. A typical steel tank equipped with floating roof is analysed. The limit state for such structure is determined by the tensile hoop stress at the completely filled tank reaching the bearing capacity of steel plating. It is assumed, that in the foreseeable future the tank will not be modernized nor strengthened in any manner. The service mode of the tank will not change as well. The future progress of corrosion is forecast by the extrapolation of trends, which have appeared until now, and identified during the cyclical obligatory examinations of the tank technical condition. Detailed map of the corrosion reach and intensity determined in this way is superimposed over the numerical tank model, replicating its real imperfect geometry measured by geodetic methods. In this paper two alternative methods of analysis are compared. The first one is geometrically nonlinear (GNA), while the second one is geometrically linear (LA). The obtained results are compared with the analogous results obtained under the assumption of perfect tank geometry. It turns out that the quantitative differences with respect to the forecast tank shell life, due only to the numerical approach recommended for analysis, are substantially bigger than those generated by monotonically progressing, but locally varying corrosion progress. In the case considered in this paper, the linear analysis (LA) yields the most restrictive estimate of authoritative hoop stresses. The application of nonlinear analysis (GNA) proves to be the less restrictive approach in this field, as it takes into the account the smoothing of possible local folds in the deformed shell, and thus as a consequence making it closer to the shell with perfect geometry.

A primal-dual method of partial inverses for composite inclusions

Spingarn’s method of partial inverses has found many applications in nonlinear analysis and in optimization. We show that it can be employed to solve composite monotone inclusions in duality, thus opening a new range of applications for the partial inverse formalism. The versatility of the resulting primal-dual splitting algorithm is illustrated through applications to structured monotone inclusions and optimization.

Variational Analysis, Optimization, and Fixed Point Theory

In the last two decades, the theory of variational analysis including variational inequalities (VI) emerged as a rapidly growing area of research because of its applications in nonlinear analysis, optimization, economics, game theory, and so forth; see, for example, [1] and the references therein. In the recent past, many authors devoted their attention to study the VI defined on the set of fixed points of a mapping, called hierarchical variational inequalities. Very recently, several iterative methods have been investigated to solve VI, hierarchical variational inequalities, and triple hierarchical variational inequalities. Since the origin of the VI, it has been used as a tool to study optimization problems. Hierarchical variational inequalities are used to study the bilevel mathematical programming problems. A triple level mathematical programming problem can be studied by using triple hierarchical variational inequalities. The Ekeland’s variational principle provides the existence of an approximate minimizer of a bounded below and lower semicontinuous function. It is one of the most important results from nonlinear analysis and it has applications in different areas of mathematics and mathematical sciences, namely, fixed point theory, optimization, optimal control theory, game theory, nonlinear equations, dynamical systems, and so forth, for example, [2–9] and the references therein. During the last decade, it has been used to study the existence of solutions of equilibrium problems in the setting of metric spaces, for example, [2, 3] and the references therein. Banach’s contraction principle is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theory in all of analysis. This is because the contractive condition on the mapping is simple and easy to verify, and because it requires only completeness of the metric space. Although, the basic idea was known to others earlier, the principle first appeared in explicit form inBanach’s 1922 thesis where it was used to establish the existence of a solution to an integral equation. Caristi’s fixed point theorem [10, 11] has found many applications in nonlinear analysis. It is shown, for example, that this theorem yields essentially all the known inwardness results of geometric fixed point theory in Banach spaces. Recall that inwardness conditions are the ones which assert that, in some sense, points from the domain are mapped toward the domain. This theorem is amazing equivalent to Ekeland’s variational principle.

Computational Methods in the Theory of Synthesis of Radio and Acoustic Radiating Systems

A brief review of the works of the author and his co-authors on the application of nonlinear analysis, numerical and analytical methods for solving the nonlinear inverse problems (synthesis problems) for optimizing the different types of radiating systems, is presented in the paper. The synthesis problems are formulated in variational statements and further they are reduced to research and numerical solution of nonlinear integral equations of Hammerstein type. The existence theorems are proof, the investigation methods of nonuniqueness problem of solutions and numerical algorithms of finding the optimal solutions are proved.

Ekeland’s variational principle for vectorial multivalued mappings in a uniform space

In this paper, we establish Ekeland’s variational principle and an equilibrium version of Ekeland’s variational principle for vectorial multivalued mappings in the setting of separated, sequentially complete uniform spaces. Our approaches and results are different from those in Chen et al. (2008), Hamel (2005), and Lin and Chuang (2010) [13–15]. As applications of our results, we study vectorial Caristi’s fixed point theorems and Takahashi’s nonconvex minimization theorems for multivalued mappings and their equivalent forms in a separated, sequentially complete uniform space. We also apply our results to study maximal element theorems, which are unified methods of several variational inclusion problems. Our results contain many known results in the literature Fang (1996) [21], and will have many applications in nonlinear analysis.

Variational relation problems and equivalent forms of generalized Fan-Browder fixed point theorem with applications to Stampacchia equilibrium problems

In this paper, we study the existence theorems of solution for variational relation problems. From the existence theorems of solution for variational relation problems, we study equivalent forms of generalized Fan-Browder fixed point theorem, existence theorems of solutions for Stampacchia vector equilibrium problems and generalized Stampacchia vector equilibrium problems. Our results contains many orginal results and have many applications in Nonlinear Analysis.

Common Fixed Point Theorems for a Finite Family of Discontinuous and Noncommutative Maps

AbstractWe study common fixed point theorems for a finite family of discontinuous and noncommutative single-valued functions defined in complete metric spaces. We also study a common fixed point theorem for two multivalued self-mappings and a stationary point theorem in complete metric spaces. Throughout this paper, we establish common fixed point theorems without commuting and continuity assumptions. In contrast, commuting or continuity assumptions are often assumed in common fixed point theorems. We also give examples to show our results. Results in this paper except those that generalized Banach contraction principle and those improve and generalize recent results in fixed point theorem are original and different from any existence result in the literature. The results in this paper will have some applications in nonlinear analysis and fixed point theory.