Dynamic solid mechanics using finite volume methods

A procedure for evaluating the dynamic structural response of elastic solid domains is presented. A prerequisite for the analysis of dynamic fluid–structure interaction is the use of a consistent set of finite volume (FV) methods on a single unstructured mesh. This paper describes a three-dimensional (3D) FV, vertex-based method for dynamic solid mechanics. A novel Newmark predictor–corrector implicit scheme was developed to provide time accurate solutions and the scheme was evaluated on a 3D cantilever problem. By employing a small amount of viscous damping, very accurate predictions of the fundamental natural frequency were obtained with respect to both the amplitude and period of oscillation. This scheme has been implemented into the multi-physics modelling software framework, Physica, for later application to full dynamic fluid structure interaction.     

Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages

We introduce and develope a unifying multifractal framework. The framework developed in this paper is based on the notion of deformations of empirical measures. This approach leads to significant extensions of already know results. However, our approach not only leads to extensions of already know results, but also, by considering non-linear deformations, provides the basis for the study of several new and non-linear local characteristic. We also initiate a detailed study of the fractal structure of so-called divergence points. We define multifractal spectra that provides extremely precise quantitative information about the distribution of individual divergence points of arbitrary (possibly non-linear) deformations, thereby extending and unifying many diverse qualitative results on the behaviour of divergence points. The techniques used in proving the main results are taken from large deviation theory and are completely different from previous techniques in the literature.     

A fractal interpretation of size-scale effects on strength,friction and fracture energy of faults

Experimental results indicate that large faults involved in earthquakes possess low strength, low friction coefficient and high fracture energy, in comparison with data obtained according to small scale laboratory tests on the same material. The reasons for such an unexpected anomalous behaviour have been the subject of several researches in the past and are still under debate in the Scientific Community. In this note, we propose a unifying interpretation of these size-scale effects according to fractal geometry, which represents the proper mathematical framework for the analysis of the multi-scale properties of rough surfaces in contact. This contribution sheds a new light on the non-linear properties of friction and on the understanding the fundamental physics governing the scaling of the mechanical properties in geophysics from the laboratory to a planetary scale.     

Obtaining scattering kernels using invariant imbedding

The symplectic group structure associated with the Riccati equation is exploited to derive a unifying group-theoretical framework encompassing a large number of well-known results, such as partitioning, doubling, and Chandrasekhar algorithms, the algebraic Riccati equation, and the discrete-time case. It is also used to derive a new integration-free Chandrasekhar type algorithm, and to present a fairly complete characterization of the solutions of the Riccati equation in the periodic case.     

Hybrid pseudospectral–finite difference method for solving a 3D heat conduction equation in a submicroscale thin film

This research aims to develop a time‐dependent pseudospectral‐finite difference scheme for solving a 3D dual‐phase‐lagging heat transport equation in a submicroscale thin film. The scheme uses periodic pseudospectral discretization in space and a fully second‐order finite difference discretization in time. The three consecutive time steps model is then solved explicitly, by using a preconditioned conjugate gradient method. The scheme is illustrated by an example which is used to investigate the heat transfer in a gold submicroscale thin film. Comparisons are made with available literature. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A Uniform Approach to Fundamental Sequences and Hierarchies

In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one. Mathematics Subject Classification: 03D20, 03F15, 03E10.     

Convergence proof of the velocity field for a stokes flow immersed boundary method

The immersed boundary (IB) method is a computational framework for problems involving the interaction of a fluid and immersed elastic structures. It is characterized by the use of a uniform Cartesian mesh for the fluid, a Lagrangian curvilinear mesh on the elastic material, and discrete delta functions for communication between the two grids. We consider a simple IB problem in a two‐dimensional periodic fluid domain with a one‐dimensional force generator. We obtain error estimates for the velocity field of the IB solution for the stationary Stokes problem. We use this result to prove convergence of a simple small‐amplitude dynamic problem. We test our error estimates against computational experiments. © 2007 Wiley Periodicals, Inc.     

Dynamic stability of the three‐dimensional axisymmetric Navier‐Stokes equations with swirl

In this paper, we study the dynamic stability of the three‐dimensional axisymmetric Navier‐Stokes Equations with swirl. To this purpose, we propose a new one‐dimensional model that approximates the Navier‐Stokes equations along the symmetry axis. An important property of this one‐dimensional model is that one can construct from its solutions a family of exact solutions of the three‐dimensionaFinal Navier‐Stokes equations. The nonlinear structure of the one‐dimensional model has some very interesting properties. On one hand, it can lead to tremendous dynamic growth of the solution within a short time. On the other hand, it has a surprising dynamic depletion mechanism that prevents the solution from blowing up in finite time. By exploiting this special nonlinear structure, we prove the global regularity of the three‐dimensional Navier‐Stokes equations for a family of initial data, whose solutions can lead to large dynamic growth, but yet have global smooth solutions. © 2007 Wiley Periodicals, Inc.     

Fractal response of physiological signals to stress conditions,environmental changes,and neurodegenerative diseases

In the past two decades the biomedical community has witnessed several applications of nonlinear system theory to the analysis of biomedical time series and the development of nonlinear dynamic models. The development of this area of medicine can best be described as nonlinear and fractal physiology. These studies have been intended to develop more reliable methodologies for understanding how biological systems respond to peculiar altered conditions induced by internal stress, environment stress, and/or disease. Herein, we summarize the theory and some of our results showing the fractal dependency on different conditions of physiological signals such as inter‐breath intervals, heart inter‐beat intervals, and human stride intervals. © 2007 Wiley Periodicals, Inc. Complexity 12: 12–17, 2007     

耗散孤立波方程的吸引子

本文研究耗散孤立波方程的长期动力学行为：吸引子的存在性、吸引子的几何结构、耗散系统参数扰动下动力行为、吸引子的分形维估计．     

High order nonlocal symmetries and exact interaction solutions of the variable coefficient KdV equation

In this paper, the nonlocal symmetries and exact interaction solutions of the variable coefficient Korteweg–de Vries (KdV) equation are studied. With the help of pseudo-potential, we construct the high order nonlocal symmetries of the time-dependent coefficient KdV equation for the first time. In order to construct the new exact interaction solutions, two auxiliary variables are introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of the closed system, some exact interaction solutions are obtained. For some interesting solutions, such as the soliton–cnoidal wave solutions are discussed in detail, and the corresponding 2D and 3D figures are given to illustrate their dynamic behavior.     

Performance and numerical behavior of the second‐order scheme of precise time‐step integration for transient dynamic analysis

Spurious high‐frequency responses resulting from spatial discretization in time‐step algorithms for structural dynamic analysis have long been an issue of concern in the framework of traditional finite difference methods. Such algorithms should be not only numerically dissipative in a controllable manner, but also unconditionally stable so that the time‐step size can be governed solely by the accuracy requirement. In this article, the issue is considered in the framework of the second‐order scheme of the precise integration method (PIM). Taking the Newmark‐β method as a reference, the performance and numerical behavior of the second‐order PIM for elasto‐dynamic impact‐response problems are studied in detail. In this analysis, the differential quadrature method is used for spatial discretization. The effects of spatial discretization, numerical damping, and time step on solution accuracy are explored by analyzing longitudinal vibrations of a shock‐excited rod with rectangular, half‐triangular, and Heaviside step impact. Both the analysis and numerical tests show that under the framework of the PIM, the spatial discretization used here can provide a reasonable number of model types for any given error tolerance. In the analysis of dynamic response, an appropriate spatial discretization scheme for a given structure is usually required in order to obtain an accurate and meaningful numerical solution, especially for describing the fine details of traction responses with sharp changes. Under the framework of the PIM, the numerical damping that is often required in traditional integration schemes is found to be unnecessary, and there is no restriction on the size of time steps, because the PIM can usually produce results with machine‐like precision and is an unconditionally stable explicit method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Analysis of dynamic solutions of complex delay differential equation exemplified by the extended Mandelbrot’s equation

The paper deals with the mathematical–numerical analysis of the logistic Mandelbrot equation extended by the dynamic continuous term. The possibilities of generation of fractal patterns with the mathematical form, defined in such a manner, were investigated. The decay of fractal structure with time was demonstrated.     

Using self‐dissimilarity to quantify complexity

For many systems characterized as “complex” the patterns exhibited on different scales differ markedly from one another. For example, the biomass distribution in a human body “looks very different” depending on the scale at which one examines it. Conversely, the patterns at different scales in “simple” systems (e.g., gases, mountains, crystals) vary little from one scale to another. Accordingly, the degrees of self‐dissimilarity between the patterns of a system at various scales constitute a complexity “signature” of that system. Here we present a novel quantification of self‐dissimilarity. This signature can, if desired, incorporate a novel information‐theoretic measure of the distance between probability distributions that we derive here. Whatever distance measure is chosen, our quantification of self‐dissimilarity can be measured for many kinds of real‐world data. This allows comparisons of the complexity signatures of wholly different kinds of systems (e.g., systems involving information density in a digital computer vs. species densities in a rain forest vs. capital density in an economy, etc.). Moreover, in contrast to many other suggested complexity measures, evaluating the self‐dissimilarity of a system does not require one to already have a model of the system. These facts may allow self‐dissimilarity signatures to be used as the underlying observational variables of an eventual overarching theory relating all complex systems. To illustrate self‐dissimilarity, we present several numerical experiments. In particular, we show that the underlying structure of the logistic map is picked out by the self‐dissimilarity signature of time series produced by that map. © 2007 Wiley Periodicals, Inc. Complexity 12: 77–85, 2007     

Dynamic shortest paths and transitive closure: Algorithmic techniques and data structures

In this paper, we survey fully dynamic algorithms for path problems on general directed graphs. In particular, we consider two fundamental problems: dynamic transitive closure and dynamic shortest paths. Although research on these problems spans over more than three decades, in the last couple of years many novel algorithmic techniques have been proposed. In this survey, we will make a special effort to abstract some combinatorial and algebraic properties, and some common data-structural tools that are at the base of those techniques. This will help us try to present some of the newest results in a unifying framework so that they can be better understood and deployed also by non-specialists.     

Three dimensional elastodynamics of 2D quasicrystals: The derivation of the time-dependent fundamental solution

The time-dependent differential equations of elasticity for 2D quasicrystals with general structure of anisotropy (dodecagonal, octagonal, decagonal, pentagonal, hexagonal, triclinic) are considered in the paper. These equations are written in the form of a vector partial differential equation of the second order with symmetric matrix coefficients. The fundamental solution (matrix) is defined for this vector partial differential equation. A new method of the numerical computation of values of the fundamental solution is suggested. This method consists of the following: the Fourier transform with respect to space variables is applied to vector equation for the fundamental solution. The obtained vector ordinary differential equation has matrix coefficients depending on Fourier parameters. Using the matrix computations a solution of the vector ordinary differential equation is numerically computed. Finally, applying the inverse Fourier transform numerically we find the values of the fundamental solution. Computational examples confirm the robustness of the suggested method for 2D quasicrystals with arbitrary type of anisotropy.     

Nonextensive model of self‐organizing systems

The article concerns the new nonextensive model of self‐organizing systems and consists of two interrelated parts. The first one presents a new nonextensive model of interaction between elements of systems. The second one concerns the relationship between microscopic and macroscopic processes in complex systems. It is shown that nonextensivity and self‐organization of systems is a result of mismatch between its elements. © 2013 Wiley Periodicals, Inc. Complexity 18: 28–36, 2013     

Using a Functional Model to Develop a Mathematical Formula

The unifying theme of models was incorporated into a required Science Capstone course for pre‐service elementary teachers based on national standards in science and mathematics. A model of a teeter‐totter was selectedfor use as an example of a functional model for gathering data as well as a visual model of a mathematical equation for developing the mathematical relationship for a Class 1 lever, M1D1=M1D1. In this study, 20 student groups (n=72) collected data using the model in an inquiry‐based activity. All groups developed the qualitative relationship, 13 groups developed a correct mathematical formula, 6 groups developed one‐half of the relationship (X = mass × distance), and 1 group attempted to develop a procedural relationship. The pre‐service elementary teachers used a variety of model types in the activity including visual/pictorial, functional/physical and mathematical‐both graphs and formulas. The use of the teeter‐totter model as a visual and functional model of a mathematical formula was a factor in developing the mathematical relationship.     

SPDE in Hilbert space with locally monotone coefficients

The aim of this paper is to extend the usual framework of SPDE with monotone coefficients to include a large class of cases with merely locally monotone coefficients. This new framework is conceptually not more involved than the classical one, but includes many more fundamental examples not included previously. Thus our main result can be applied to various types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equation, stochastic 2-D Navier-Stokes equation, stochastic p-Laplace equation and stochastic porous media equation with non-monotone perturbations.