Epsilon‐stable quasi‐static brittle fracture evolution

We introduce a new definition of stability, ε‐stability, that implies local minimality and is robust enough for passing from discrete‐time to continuous‐time quasi‐static evolutions, even with very irregular energies. We use this to give the first existence result for quasi‐static crack evolutions that both predicts crack paths and produces states that are local minimizers at every time, but not necessarily global minimizers. The key ingredient in our model is the physically reasonable property, absent in global minimization models, that whenever there is a jump in time from one state to another, there must be a continuous path from the earlier state to the later along which the energy is almost decreasing. It follows that these evolutions are much closer to satisfying Griffith's criterion for crack growth than are solutions based on global minimization, and initiation is more physical than in global minimization models. © 2009 Wiley Periodicals, Inc.     

Invariant tracking for planar rigid body dynamics

A quasi‐static state feedback is proposed that achieves stable tracking for planar motions of rigid bodies and is invariant with respect to the choice of the inertial frame.     

Affine,quasi‐affine and co‐affine frames on local fields of positive characteristic

The concept of quasi‐affine frame in Euclidean spaces was introduced to obtain translation invariance of the discrete wavelet transform. We extend this concept to a local field K of positive characteristic. We show that the affine system generated by a finite number of functions is an affine frame if and only if the corresponding quasi‐affine system is a quasi‐affine frame. In such a case the exact frame bounds are equal. This result is obtained by using the properties of an operator associated with two such affine systems. We characterize the translation invariance of such an operator. A related concept is that of co‐affine system. We show that there do not exist any co‐affine frame in .     

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

A proposed discretized form of fractional‐order prey‐predator model is investigated. A sufficient condition for the solution of the discrete system to exist and to be unique is determined. Jury stability test is applied for studying stability of equilibrium points of the discretized system. Then, the effects of varying fractional order and other parameters of the systems on its dynamics are examined. The system undergoes Neimark‐Sacker and flip bifurcation under certain conditions. We observe that the model exhibits chaotic dynamics following stable states as the memory parameter α decreases and step size h increases. Theoretical results illustrate the rich dynamics and complexity of the model. Numerical simulation validates theoretical results and demonstrates the presence of rich dynamical behaviors include S‐asymptotically bounded periodic orbits, quasi‐periodicity, and chaos. The system exhibits a wide range of dynamical behaviors for fractional‐order α key parameter.     

Existence and convergence for quasi‐static evolution in brittle fracture

This paper investigates the mathematical well‐posedness of the variational model of quasi‐static growth for a brittle crack proposed by Francfort and Marigo in [15]. The starting point is a time discretized version of that evolution which results in a sequence of minimization problems of Mumford and Shah type functionals. The natural weak setting is that of special functions of bounded variation, and the main difficulty in showing existence of the time‐continuous quasi‐static growth is to pass to the limit as the time‐discretization step tends to 0. This is performed with the help of a jump transfer theorem which permits, under weak convergence assumptions for a sequence {u_n} of SBV‐functions to its BV‐limit u, to transfer the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence {u_n}. In particular, it is shown that the notion of minimizer of a Mumford and Shah type functional for its own jump set is stable under weak convergence assumptions. Furthermore, our analysis justifies numerical methods used for computing the time‐continuous quasi‐static evolution. © 2003 Wiley Periodicals, Inc.     

Stability of one‐dimensional boundary layers by using Green's functions

The aim of this paper is to investigate the stability of one‐dimensional boundary layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case and, more precisely, to prove that spectral stability implies linear and nonlinear stability of approximate solutions. In particular, we replace the smallness condition obtained by the energy method [10, 13] by a weaker spectral condition. © 2001 John Wiley & Sons, Inc.     

Numerical solutions of coupled systems of nonlinear elliptic equations

This article deals with numerical solutions of a general class of coupled nonlinear elliptic equations. Using the method of upper and lower solutions, monotone sequences are constructed for difference schemes which approximate coupled systems of nonlinear elliptic equations. This monotone convergence leads to existence‐uniqueness theorems for solutions to problems with reaction functions of quasi‐monotone nondecreasing, quasi‐monotone nonincreasing and mixed quasi‐monotone types. A monotone domain decomposition algorithm which combines the monotone approach and an iterative domain decomposition method based on the Schwarz alternating, is proposed. An application to a reaction‐diffusion model in chemical engineering is given. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 621–640, 2012     

On quasi‐varieties of multiple valued logic models

We extend the concept of quasi‐variety of first‐order models from classical logic to multiple valued logic (MVL) and study the relationship between quasi‐varieties and existence of initial models in MVL. We define a concept of ‘Horn sentence’ in MVL and based upon our study of quasi‐varieties of MVL models we derive the existence of initial models for MVL ‘Horn theories’. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the stability of elastic-plastic systems with hardening

This paper discusses the stability of quasi-static paths for a continuous elastic-plastic system with hardening in a one-dimensional (bar) domain. Mathematical formulations, as well as existence and uniqueness results for dynamic and quasi-static problems involving elastic-plastic systems with linear kinematic hardening are recalled in the paper. The concept of stability of quasi-static paths used here is essentially a continuity property of the system dynamic solutions relatively to the quasi-static ones, when (as in Lyapunov stability) the size of initial perturbations is decreased and the rate of application of the forces (which plays the role of the small parameter in singular perturbation problems) is also decreased to 0. The stability of the quasi-static paths of these elastic-plastic systems is the main result proved in the paper.     

Convergence of coercive approximations for a model of gradient type in poroplasticity

We study the existence theory to the quasi‐static initial‐boundary‐value problem of poroplasticity. In this article the classical quasi‐static Biot model is considered for soil consolidation coupled with a nonlinear system of differential equations. This work, for the poroplasticity model of monotone‐gradient type, presents a convergence result of the coercive approximation to the solution of the original noncoercive problem. Copyright © 2008 John Wiley & Sons, Ltd.     

A modified quasi‐Newton diagonal update algorithm for total variation denoising problems and nonlinear monotone equations with applications in compressive sensing

In this paper, we present a new algorithm to accelerate the Chambolle gradient projection method for total variation image restoration. The new proposed method considers an approximation of the Hessian based on the secant equation. Combined with the quasi‐Cauchy equations and diagonal updating, we can obtain a positive definite diagonal matrix. In the proposed minimization method model, we use the positive definite diagonal matrix instead of the constant time stepsize in Chambolle's method. The global convergence of the proposed scheme is proved. Some numerical results illustrate the efficiency of this method. Moreover, we also extend the quasi‐Newton diagonal updating method to solve nonlinear systems of monotone equations. Performance comparisons show that the proposed method is efficient. A practical application of the monotone equations is shown and tested on sparse signal reconstruction in compressed sensing. Copyright © 2015 John Wiley & Sons, Ltd.     

FE‐formulation for plate and beam elements to simulate onset and propagation of discrete cracks in solid structures

The aim of this paper is to present some numerical aspects related to the modeling both the formation and the propagation of discrete cracks in solid structures. The presented formulation corresponds to the concept of embedded discontinuities [1], and will be applied to a plate and to a beam element. The failure of solid structures is often triggered by a highly localized pattern of inelastic deformation in the form of narrow bands. Characteristic examples are shear bands in metals and soils, or localized bands of cracking in brittle materials, like concrete or rocks. A well known difficulty associated with classical (local, rate‐independent) continuum theories with strain softening attributes is that numerical solutions are found to lack invariance with respect to the choice of spatial discretization. For quasi‐static boundary problems, this mathematical inconsistency causes the loss of ellipticity for the governing equations (material instability). To regularize this inconsistency, several strategies have been applied. In the presented formulation, additional degrees of freedom are considered. Within the concept of embedded discontinuities, the regular displacements are enriched by discontinuities. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Action functional and quasi‐potential for the burgers equation in a bounded interval

Consider the viscous Burgers equation u_t + f(u)_x = εu_xx on the interval [0,1] with the inhomogeneous Dirichlet boundary conditions u(t,0) = ρ₀, u(t,1) = ρ₁. The flux f is the function f(u) = u(1 − u), ε > 0 is the viscosity, and the boundary data satisfy 0 < ρ₀ < ρ₁ < 1. We examine the quasi‐potential corresponding to an action functional arising from nonequilibrium statistical mechanical models associated with the above equation. We provide a static variational formula for the quasi‐potential and characterize the optimal paths for the dynamical problem. In contrast with previous cases, for small enough viscosity, the variational problem defining the quasi‐potential admits more than one minimizer. This phenomenon is interpreted as a nonequilibrium phase transition and corresponds to points where the superdifferential of the quasi‐potential is not a singleton. © 2011 Wiley Periodicals, Inc.     

A linearized heat‐conducting fluid and its quasi‐static approximation

A linearized flow of a compressible inviscid heat‐conducting fluid is considered and a comparison is made with its coupled/quasi‐static approximation. Copyright © 2005 John Wiley & Sons, Ltd.     

Local stability analysis of a stochastic evolutionary financial market model with a risk-free asset

This paper introduces and analyzes an evolutionary model of a financial market with a risk-free asset. Focus is on the study of local stability of the wealth dynamics through the application of recent results on the linearization and stability of random dynamical systems (Evstigneev et al. Proc Am Math Soc 139:1061–1072, 2011). Conditions are derived for the linearization of the model at an equilibrium state which ensure local convergence of sample paths to this equilibrium. The paper also shows that the concept of local stability is closely related to the notion of evolutionary stability. A locally evolutionarily stable investment strategy in the evolutionary model with a risk-free asset is derived, extending previous research. The method illustrated here is applicable for the analysis of manifold economic and financial dynamic models involving randomness.     

A new efficient approach to the characterization of D‐stable matrices

The concept of D‐stability is relevant for stable square matrices of any order, especially when they appear in ordinary differential systems modeling physical problems. Indeed, D‐stability was treated from different points of view in the last 50 years, but the problem of characterization of a general D‐stable matrix was solved for low‐order matrices only (ie, up to order 4). Here, a new approach is proposed within the context of numerical linear algebra. Starting from a known necessary and sufficient condition, other simpler equivalent necessary and sufficient conditions for D‐stability are proved. Such conditions turn out to be computationally more appealing for symbolic software, as discussed in the reported examples. Therefore, a new symbolic method is proposed to characterize matrices of order greater than 4, and then it is used in some numerical examples, given in details.     

Numerical solution to a one‐dimensional thermoplastic problem with unilateral constraint

In this article, we present a numerical simulation of one‐dimensional problem of quasi‐static contact with an elastic obstacle. A finite difference scheme is derived by the method of reduction of order on uniform meshes. The stability and convergence are proved. The convergence order is of O(τ² + h²), where τ and h are the time step size and the space step size, respectively. Some numerical examples demonstrate the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

The scattering of plane elastic waves by a one‐dimensional periodic surface

The two‐dimensional scattering problem for time‐harmonic plane waves in an isotropic elastic medium and an effectively infinite periodic surface is considered. A radiation condition for quasi‐periodic solutions similar to the condition utilized in the scattering of acoustic waves by one‐dimensional diffraction gratings is proposed. Under this condition, uniqueness of solution to the first and third boundary‐value problems is established. We then proceed by introducing a quasi‐periodic free field matrix of fundamental solutions for the Navier equation. The solution to the first boundary‐value problem is sought as a superposition of single‐ and double‐layer potentials defined utilizing this quasi‐periodic matrix. Existence of solution is established by showing the equivalence of the problem to a uniquely solvable second kind Fredholm integral equation. Copyright © 1999 John Wiley & Sons, Ltd.     

Introductory Statement

This work continues the account given in Part I of the paper¹ by presenting a short summary of some of the mathematical techniques employed in the wave front analysis of quasi‐linear hyperbolic partial differential equations. Starting from a number of important physical examples, the classification of quasi‐linear first‐order systems is discussed and followed by a simple account of the theory of characteristics for systems involving n dependent and two independent variables. A special example is discussed showing how discontinuities arise in solutions, and the paper is concluded by an account of wave front analysis as applied to the piston problem of gas dynamics.     

Hodograph Transformations and Cauchy Problem to Systems of Nonlinear Parabolic Equations

In this paper, we provide a method to solve the Cauchy problem of systems of quasi‐linear parabolic equations, such systems can be transformed to the systems of linear parabolic equations with variable coefficients via the hodograph transformations. Our approach to solve the linear systems with variable coefficients is to use their fundamental solutions, which are constructed by using the Lie's symmetry method. In consequence, we can derive explicit solutions to the Cauchy problem of the quasi‐linear systems in terms of the solutions of the linear systems and the hodograph transformations relating to the quasi‐linear and the linear systems.