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The basin of attraction of an equilibrium of an ordinary differential equation can be determined using a Lyapunov function. A new method to construct such a Lyapunov function using radial basis functions is presented in this volume intended for researchers and advanced students from both dynamical systems and radial basis functions. Besides an introduction to both areas and a detailed description of the method, it contains error estimates and many examples.

Lyapunov functions. --- Radial basis functions. --- Electronic books. -- local. --- Lyapunov functions --- Radial basis functions --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Applied Mathematics --- Calculus --- Mathematical Theory --- Basis functions, Radial --- Functions, Radial basis --- Radial basis function method --- Functions, Liapunov --- Liapunov functions --- Mathematics. --- Approximation theory. --- Dynamics. --- Ergodic theory. --- Differential equations. --- Dynamical Systems and Ergodic Theory. --- Approximations and Expansions. --- Ordinary Differential Equations. --- 517.91 Differential equations --- Differential equations --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Math --- Science --- Approximation theory --- Differentiable dynamical systems. --- Differential Equations. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics

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This book presents the first “How To” guide to the use of radial basis functions (RBF). It provides a clear vision of their potential, an overview of ready-for-use computational tools and precise guidelines to implement new engineering applications of RBF. Radial basis functions (RBF) are a mathematical tool mature enough for useful engineering applications. Their mathematical foundation is well established and the tool has proven to be effective in many fields, as the mathematical framework can be adapted in several ways. A candidate application can be faced considering the features of RBF: multidimensional space (including 2D and 3D), numerous radial functions available, global and compact support, interpolation/regression. This great flexibility makes RBF attractive – and their great potential has only been partially discovered. This is because of the difficulty in taking a first step toward RBF as they are not commonly part of engineers’ cultural background, but also due to the numerical complexity of RBF problems that scales up very quickly with the number of RBF centers. Fast RBF algorithms are available to alleviate this and high-performance computing (HPC) can provide further aid. Nevertheless, a consolidated tradition in using RBF in engineering applications is still missing and the beginner can be confused by the literature, which in many cases is presented with language and symbolisms familiar to mathematicians but which can be cryptic for engineers. The book is divided in two main sections. The first covers the foundations of RBF, the tools available for their quick implementation and guidelines for facing new challenges; the second part is a collection of practical RBF applications in engineering, covering several topics, including response surface interpolation in n-dimensional spaces, mapping of magnetic loads, mapping of pressure loads, up-scaling of flow fields, stress/strain analysis by experimental displacement fields, implicit surfaces, mesh to cad deformation, mesh morphing for crack propagation in 3D, ice and snow accretion using computational fluid dynamics (CFD) data, shape optimization for external aerodynamics, and use of adjoint data for surface sculpting. For each application, the complete path is clearly and consistently exposed using the systematic approach defined in the first section.

Radial basis functions. --- Mathematics. --- Algorithms. --- Computer mathematics. --- Structural mechanics. --- Computational Science and Engineering. --- Algorithm Analysis and Problem Complexity. --- Structural Mechanics. --- Basis functions, Radial --- Functions, Radial basis --- Radial basis function method --- Approximation theory --- Computer science. --- Computer software. --- Mechanics. --- Mechanics, Applied. --- Solid Mechanics. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Software, Computer --- Computer systems --- Informatics --- Science --- Algorism --- Algebra --- Arithmetic --- Computer mathematics --- Electronic data processing --- Mathematics --- Foundations

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“Computational Mathematics, Algorithms, and Data Processing” of MDPI consists of articles on new mathematical tools and numerical methods for computational problems. Topics covered include: numerical stability, interpolation, approximation, complexity, numerical linear algebra, differential equations (ordinary, partial), optimization, integral equations, systems of nonlinear equations, compression or distillation, and active learning.

Research & information: general --- Mathematics & science --- interpolation --- constraints --- embedded constraints --- generalized multiscale finite element method --- multiscale model reduction --- deep learning --- Deep Neural Nets --- ReLU Networks --- Approximation Theory --- radial basis functions --- native spaces --- truncated function --- approximation --- surface modeling --- second order initial value problems --- linear multistep methods --- Obrechkoff schemes --- trigonometrically fitted --- Darcy-Forchheimer model --- flow in porous media --- nonlinear equation --- heterogeneous media --- finite element method --- multiscale method --- mixed generalized multiscale finite element method --- multiscale basis functions --- two-dimensional domain --- Thiele-like rational interpolation continued fractions with parameters --- unattainable point --- inverse difference --- virtual point --- polynomial chaos --- Szegő polynomials --- directional statistics --- Rogers-Szegő --- state estimation --- clustering --- interpolation --- constraints --- embedded constraints --- generalized multiscale finite element method --- multiscale model reduction --- deep learning --- Deep Neural Nets --- ReLU Networks --- Approximation Theory --- radial basis functions --- native spaces --- truncated function --- approximation --- surface modeling --- second order initial value problems --- linear multistep methods --- Obrechkoff schemes --- trigonometrically fitted --- Darcy-Forchheimer model --- flow in porous media --- nonlinear equation --- heterogeneous media --- finite element method --- multiscale method --- mixed generalized multiscale finite element method --- multiscale basis functions --- two-dimensional domain --- Thiele-like rational interpolation continued fractions with parameters --- unattainable point --- inverse difference --- virtual point --- polynomial chaos --- Szegő polynomials --- directional statistics --- Rogers-Szegő --- state estimation --- clustering

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With rapid developments being made in the exploration of marine resources, coastal geohazard and offshore geotechnics have attracted a great deal of attention from coastal geotechnical engineers, with significant progress being made in recent years. Due to the complicated nature of marine environmnets, there are numerous natural marine geohazard preset throughout the world’s marine areas, e.g., the South China Sea. In addition, damage to offshore infrastructure (e.g., monopiles, bridge piers, etc.) and their supporting installations (pipelines, power transmission cables, etc.) has occurred in the last decades. A better understanding of the fundamental mechanisms and soil behavior of the seabed in marine environments will help engineers in the design and planning processes of coastal geotechnical engineering projects. The purpose of this book is to present the recent advances made in the field of coastal geohazards and offshore geotechnics. The book will provide researchers with information reagrding the recent developments in the field, and possible future developments. The book is composed of eighteen papers, covering three main themes: (1) the mechanisms of fluid–seabed interactions and the instability associated with seabeds when they are under dynamic loading (papers 1–5); (2) evaluation of the stability of marine infrastructure, including pipelines (papers 6–8), piled foundation and bridge piers (papers 9–12), submarine tunnels (paper 13), and other supported foundations (paper 14); and (3) coastal geohazards, including submarine landslides and slope stability (papers 15–16) and other geohazard issues (papers 17–18). The editors hope that this book will functoin as a guide for researchers, scientists, and scholars, as well as practitioners of coastal and offshore engineering.

Technology: general issues --- wave motion --- offshore deposits --- seabed response --- FEM --- pore pressure --- wave-current-seabed interaction --- RANS equations --- k-ε model --- current velocity --- seabed liquefaction --- liquefaction --- lateral displacement --- response surface method (RSM) --- artificial neural network (ANN) --- wave action --- silty sand --- seepage flow --- soil erosion --- pore-pressure accumulation --- three-phase soil model --- immersed tunnel --- trench --- numerical study --- porous seabed --- pumping well test --- groundwater fluctuation --- stratum deformation --- micro-confined aquifer --- wave-current-seabed interaction --- Reynolds-Averaged Navier-Stokesequations --- buried pipeline --- k-ε turbulence model --- oscillatory liquefaction --- wave-soil-pipeline interactions --- meshfree model --- local radial basis functions collocation method --- hydrate-bearing sediments --- damage statistical constitutive model --- multi-field coupling --- wellbore stability --- bridge scour --- identification --- ambient vibration --- field application --- natural frequency --- mode shape --- superstructure --- cable-stayed bridge --- Principal stress rotation --- dynamic loading --- wave (current)-induced soil response --- open-ended pile --- soil plug --- offshore wind turbines --- lateral cyclic loading --- model test --- discrete element simulation --- rock-socketed piles --- monopiles --- impedances --- dynamic responses --- buoyancy --- bottom-supported foundation --- field test --- numerical analysis --- giant submarine landslides --- shelf break --- South China Sea --- Himalayan orogeny --- repeated submarine landslides --- coastal-embankment slope --- stability --- unsaturated soil --- multilayered --- matric suction --- random searching algorithm --- rainfall infiltration --- scour --- soft clay --- monopile --- stress history --- hypoplastic model --- submarine pipeline --- dense seabed foundation --- seismic dynamics --- resonance of submarine pipeline --- FSSI-CAS 2D --- wave motion --- offshore deposits --- seabed response --- FEM --- pore pressure --- wave-current-seabed interaction --- RANS equations --- k-ε model --- current velocity --- seabed liquefaction --- liquefaction --- lateral displacement --- response surface method (RSM) --- artificial neural network (ANN) --- wave action --- silty sand --- seepage flow --- soil erosion --- pore-pressure accumulation --- three-phase soil model --- immersed tunnel --- trench --- numerical study --- porous seabed --- pumping well test --- groundwater fluctuation --- stratum deformation --- micro-confined aquifer --- wave-current-seabed interaction --- Reynolds-Averaged Navier-Stokesequations --- buried pipeline --- k-ε turbulence model --- oscillatory liquefaction --- wave-soil-pipeline interactions --- meshfree model --- local radial basis functions collocation method --- hydrate-bearing sediments --- damage statistical constitutive model --- multi-field coupling --- wellbore stability --- bridge scour --- identification --- ambient vibration --- field application --- natural frequency --- mode shape --- superstructure --- cable-stayed bridge --- Principal stress rotation --- dynamic loading --- wave (current)-induced soil response --- open-ended pile --- soil plug --- offshore wind turbines --- lateral cyclic loading --- model test --- discrete element simulation --- rock-socketed piles --- monopiles --- impedances --- dynamic responses --- buoyancy --- bottom-supported foundation --- field test --- numerical analysis --- giant submarine landslides --- shelf break --- South China Sea --- Himalayan orogeny --- repeated submarine landslides --- coastal-embankment slope --- stability --- unsaturated soil --- multilayered --- matric suction --- random searching algorithm --- rainfall infiltration --- scour --- soft clay --- monopile --- stress history --- hypoplastic model --- submarine pipeline --- dense seabed foundation --- seismic dynamics --- resonance of submarine pipeline --- FSSI-CAS 2D

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As faster and more efficient numerical algorithms become available, the understanding of the physics and the mathematical foundation behind these new methods will play an increasingly important role. This Special Issue provides a platform for researchers from both academia and industry to present their novel computational methods that have engineering and physics applications.

Research & information: general --- Mathematics & science --- radial basis functions --- finite difference methods --- traveling waves --- non-uniform grids --- chaotic oscillator --- one-step method --- multi-step method --- computer arithmetic --- FPGA --- high strain rate impact --- modeling and simulation --- smoothed particle hydrodynamics --- finite element analysis --- hybrid nanofluid --- heat transfer --- non-isothermal --- shrinking surface --- MHD --- radiation --- multilayer perceptrons --- quaternion neural networks --- metaheuristic optimization --- genetic algorithms --- micropolar fluid --- constricted channel --- MHD pulsatile flow --- strouhal number --- flow pulsation parameter --- multiple integral finite volume method --- finite difference method --- Rosenau-KdV --- conservation --- solvability --- convergence --- transmission electron microscopy (TEM) --- convolutional neural networks (CNN) --- anomaly detection --- principal component analysis (PCA) --- machine learning --- deep learning --- neural networks --- Gallium-Arsenide (GaAs) --- radiation-based flowmeter --- two-phase flow --- feature extraction --- artificial intelligence --- time domain --- Boltzmann equation --- collision integral --- convolutional neural network --- annular regime --- scale layer-independent --- petroleum pipeline --- volume fraction --- dual energy technique --- prescribed heat flux --- similarity solutions --- dual solutions --- stability analysis --- RBF-FD --- node sampling --- lebesgue constant --- complex regions --- finite-difference methods --- data assimilation --- model order reduction --- finite elements analysis --- high dimensional data --- welding --- radial basis functions --- finite difference methods --- traveling waves --- non-uniform grids --- chaotic oscillator --- one-step method --- multi-step method --- computer arithmetic --- FPGA --- high strain rate impact --- modeling and simulation --- smoothed particle hydrodynamics --- finite element analysis --- hybrid nanofluid --- heat transfer --- non-isothermal --- shrinking surface --- MHD --- radiation --- multilayer perceptrons --- quaternion neural networks --- metaheuristic optimization --- genetic algorithms --- micropolar fluid --- constricted channel --- MHD pulsatile flow --- strouhal number --- flow pulsation parameter --- multiple integral finite volume method --- finite difference method --- Rosenau-KdV --- conservation --- solvability --- convergence --- transmission electron microscopy (TEM) --- convolutional neural networks (CNN) --- anomaly detection --- principal component analysis (PCA) --- machine learning --- deep learning --- neural networks --- Gallium-Arsenide (GaAs) --- radiation-based flowmeter --- two-phase flow --- feature extraction --- artificial intelligence --- time domain --- Boltzmann equation --- collision integral --- convolutional neural network --- annular regime --- scale layer-independent --- petroleum pipeline --- volume fraction --- dual energy technique --- prescribed heat flux --- similarity solutions --- dual solutions --- stability analysis --- RBF-FD --- node sampling --- lebesgue constant --- complex regions --- finite-difference methods --- data assimilation --- model order reduction --- finite elements analysis --- high dimensional data --- welding