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Applied nonlinear functional analysis : an introduction
The aim of this book is to provide a concise but complete introduction to the main mathematical tools of nonlinear functional analysis, which are also used in the study of concrete problems in economics, engineering, and physics. This volume gathers the mathematical background needed in order to conduct research or to deal with theoretical problems and applications using the tools of nonlinear functional analysis.
Contents
Basic Topology
Measure Theory
Basic Functional Analysis
Banach Spaces of Functions and Measures
Convex Functions – Nonsmooth Analysis
Nonlinear Analysis
eBook
A nonlinear numerical scheme to investigate the influence of geometric nonlinearity on post-flutter responses of bridges
The present study aims to investigate the influence of geometric nonlinearity on post-flutter responses by developing a full-mode coupled nonlinear flutter analysis method (frequency-domain method) and a time-dependent nonlinear analysis scheme (time-domain method). This approach integrates the three-dimensional (3D) nonlinear finite element model and nonlinear self-excited force described by amplitude-dependent rational functions (RFs). By comparing post-flutter responses obtained from frequency-domain and time-domain methods, not only the influence of geometric nonlinearity on post-flutter responses is quantified, but also the underlying physical mechanism is revealed. The results show that the geometric nonlinear effect will become more significant with the increase of the amplitude and thus will induce a super-harmonic resonance behavior. The behavior is mainly characterized by the higher harmonic frequencies vibrations with higher-order mode shapes involved in the vertical and torsional displacement responses. Meanwhile, the larger the vibration amplitude, the more significant the super-harmonic resonance behavior. Besides, the geometric nonlinear effect will also cause a significant uplifting of the bridge deck in the vertical direction during 3D nonlinear flutter process. The main physical mechanism for the reduction in the amplitude of post-flutter response (dominated by the vibration with fundamental harmonic frequency) after considering the geometric nonlinear behavior is that the vibrations with higher harmonic frequencies play a role of absorbing energy and reducing vibration (similar to tuned mass damper effect) for the vibration with fundamental harmonic frequency. For the long-span suspension bridge with a main span of 1650 m studied in this study, the geometric nonlinear effect may need to be considered when the torsional amplitude at mid-span is only greater than 1.5°.
Journal Article
Spatiotemporal dynamics of a prey–predator model with Allee effect in prey and hunting cooperation in a Holling type III functional response
In this work, we have studied a spatiotemporal prey–predator model with Allee effect in prey and hunting cooperation in predator. In available literature, a prey-dependent functional response is mostly considered to model the prey–predator interaction. But empirical data show that functional response can depend on both prey and predator populations. Here, we have introduced the cooperative hunting in a Holling type III functional response for the predator population and extended the model spatially. Both Turing and non-Turing patterns produced by the diffusion added prey–predator model have been studied in detail. Emphasis is given to the analytical study of the spiral and target patterns applying the amplitude equation through weakly nonlinear analysis. The analytical results are verified with extensive numerical simulations.
Journal Article
Advanced numerical methods with Matlab 2 : resolution of nonlinear, differential and partial differential equations
The purpose of this book is to introduce and study numerical methods basic and advanced ones for scientific computing. This last refers to the implementation of appropriate approaches to the treatment of a scientific problem arising from physics (meteorology, pollution, etc.) or of engineering (mechanics of structures, mechanics of fluids, treatment signal, etc.). Each chapter of this book recalls the essence of the different methods resolution and presents several applications in the field of engineering as well as programs developed under Matlab software.
Book
Bifurcations analysis and pattern formation in a plant-water model with nonlocal grazing
To investigate the formation mechanism of vegetation patterns in dry-land ecosystems, this paper delves into the impact of non-local grazing on the stability and spatiotemporal dynamics of a plant-water model. We first establish the conditions for the occurrence of codimension-1 bifurcations: Turing bifurcations, Hopf bifurcations, as well as codimension-2 bifurcations: Turing–Turing bifurcations, Turing-Hopf bifurcations, and determine the stable and unstable regions of the positive equilibrium. Regarding Turing bifurcations, utilizing weakly nonlinear analysis methods to derive amplitude equations, we conclude that under the influence of non-local grazing, the system exhibits complex patch patterns, including spot, mixed, and stripe patterns. The main analytical challenges arise from non-local interactions, which increase the difficulty of deriving the amplitude equations. From a biological perspective, besides water and nutrients, herbivores also play a significant role in the self-organization of patch patterns in dry-land ecosystems.
Journal Article
Numerical Design Calculation According to EN 1993-1-14 of Innovative Thin-Walled Columns with Sectional Transverse Strengthening
This paper presents a numerical analysis of cold-formed thin-walled columns reinforced with sectional transverse stiffeners (STSs) based on the recent part of EC3 concerning the finite element analysis. Columns that are 1 m tall with various arrangements of STSs were modeled in the AxisVM environment. Numerical design calculations were completed using an analysis requiring a subsequent design check. This included a geometrically nonlinear analysis considering imperfections (GNIA) along with linear analysis (LBA) to assess the columns’ susceptibility to second-order effects. Reinforcing columns with STSs did not show a significant effect on the local buckling behavior of the elements. However, the results indicated that increasing the number of STSs positively influenced the columns’ resistance. This modification reduced the magnitudes of distortional, global flexural, and torsional buckling. Additionally, adding more than three STSs increased the critical loads related to distortional, flexural, and torsional buckling by 58–90%, 52–119%, and 19–154%, respectively. For the GNIA, two combinations of imperfections were analyzed: global flexural imperfection paired with either local or distortional imperfection. LBA was used to apply the imperfect geometry of the columns with the appropriate magnitudes of imperfections. The results between LBA and GNIA for the single-branched columns varied by 8–24%, while for the double-branched columns, the differences were less than 3%.
Journal Article
Anisotropic mesoscale turbulence and pattern formation in microswimmer suspensions induced by orienting external fields
This paper studies the influence of orienting external fields on pattern formation, particularly mesoscale turbulence, in microswimmer suspensions. To this end, we apply a hydrodynamic theory that can be derived from a microscopic microswimmer model (Reinken et al 2018 Phys. Rev. E 97, 022613). The theory combines a dynamic equation for the polar order parameter with a modified Stokes equation for the solvent flow. Here, we extend the model by including an external field that exerts an aligning torque on the swimmers (mimicking the situation in chemo-, photo-, magneto- or gravitaxis). Compared to the field-free case, the external field breaks the rotational symmetry of the vortex dynamics and leads instead to strongly asymmetric, traveling stripe patterns, as demonstrated by numerical solution and linear stability analysis. We further analyze the emerging structures using a reduced model which involves only an (effective) microswimmer velocity field. This model is significantly easier to handle analytically, but still preserves the main features of the anisotropic pattern formation. We observe an underlying transition between a square vortex lattice and a traveling stripe pattern. These structures can be well described in the framework of weakly nonlinear analysis, provided the strength of nonlinear advection is sufficiently weak.
Journal Article