Coupled time discretization of dynamic damage models at small strains

The dynamic damage model in viscoelastic materials in Kelvin–Voigt rheology is discretized by a scheme that is coupled, suppresses spurious numerical attenuation during vibrations and has a variational structure with a convex potential for small time steps. In addition, this discretization is numerically stable and convergent for the time step going to zero. When combined with a finite-element spatial discretization, it leads to an implementable scheme and to that iterative solvers (e.g., the Newton–Raphson) used for the nonlinear algebraic systems at each time level have guaranteed global convergence. Models that are computationally used in some engineering simulations in a nonreliable way are thus stabilized and theoretically justified in this viscoelastic rheology. In particular, this model and algorithm can be used in a reliable way for a dynamic fracture in the usual phase-field approximation.

Analysing Mechanical Systems in Domestic and Foreign CAD Systems

The article considers applications of foreign CAD-systems in creating the challenging projects at domestic enterprises and design bureaus. As stated in the article "... presently, there is no domestic CAD-system that could completely replace such foreign products as NX, CATIA, Credo". Besides, due to international cooperation in creating the challenging projects (for example, the project to create a modern wide-body aircraft, proposed jointly with China), it makes sense to use the worldwide known and popular CAD systems (the aforementioned NX, CATIA, Credo). Therefore, in the foreseeable future, we will still have to use foreign software products. Of course, there always remains a question of the reliability of the results obtained. Actually, this question is always open regardless of what software product is used - domestic or foreign. This question has been haunting both developers and users of CAD systems for the last 30 to 40 years. But with using domestic systems, it is much easier to identify the cause of inaccurate results and correct the mathematical models used, the methods of numerical integration applied, and the solution of systems of nonlinear algebraic systems. Everything is much more complicated if we use a foreign software product. All advertising conversations that there is a tool to make the detected errors available to the developers, remain only conversations in the real world. It is easily understandable to domestic users, and, especially, to domestic developers of similar software products. The existing development rates and competition for potential buyers dictate a rigid framework of deadlines for releasing all new versions of the product and introducing the latest developments into commercial product, etc. As a result, the known errors migrate from version to version, and many users have accepted it long ago. Especially, this concerns the less popular tools rather than the most popular applications (modules) of a CAD system. For example, in CAD systems, the "Modeling" module where geometric models of designed parts and assembly units are created has been repeatedly crosschecked. But most of the errors are hidden in applications related to the design of parts from sheet material and to the pipeline design, as well as in applications related to the analysis of moving mechanisms and to the strength or gas dynamic analysis by the finite element method.The article gives a concrete example of a moving mechanism in the analysis of which an error was detected using the mathematical model of external influence (a source of speed) in the NX 10.0 system of Siemens.

ALGEBRAIC DESCRIPTION OF SHAPE INVARIANCE REVISITED

<span>We revisit the algebraic description of shape invariance method in one-dimensional quantum mechanics. In this note we focus on four particular examples: the Kepler problem in flat space, the Kepler problem in spherical space, the Kepler problem in hyperbolic space, and the Rosen-Morse potential problem. Following the prescription given by Gangopadhyaya et al., we introduce new nonlinear algebraic systems and solve the bound-state problems by means of representation theory.</span>

Existence of Positive Solutions for a Class of Nonlinear Algebraic Systems

Based on Guo-Krasnoselskii’s fixed point theorem, the existence of positive solutions for a class of nonlinear algebraic systems of the formx=GFxis studied firstly, whereGis a positiven×nsquare matrix,x=col⁡(x1,x2,…,xn), andF(x)=col⁡(f(x1),f(x2),…,f(xn)), where,F(x)is not required to be satisfied sublinear or superlinear at zero point and infinite point. In addition, a new cone is constructed inRn. Secondly, the obtained results can be extended to some more general nonlinear algebraic systems, where the coefficient matrixGand the nonlinear term are depended on the variablex. Corresponding examples are given to illustrate these results.