Elastic–plastic behavior of a semicircular frame being pressed against a rigid plane

As a simplified structural model, a semicircular frame is used to study the crashworthiness behavior of an aircraft fuselage. The quasi-static large elastic-plastic deformation of a semicircular frame in the process of its being pressed against a rigid ground is analyzed. First, based on the linear elastic assumption, the quasi-static large deformation contact process of the frame can be divided into three phases, i.e., point contact, line contact and post-buckling. By means of a shooting method, the relations between the displacement and contact force as well as the distribution of bending moment in the three phases are obtained. Then, by assuming an elastic, perfectly-plastic moment-curvature relationship for the semi-circular frame, the contact process is analyzed in detail to reveal the plastic collapse mechanism, the traveling of plastic hinge and the force-displacement relationship. In order to verify the analysis, a preliminary experiment was conducted, in which two types of half rings with clamped ends were pressed by a rigid plate. In addition, a numerical simulation is also conducted by employing ABAQUS to analyze both rectangular cross-sectional beam and I-beam. Finally, the theoretical predictions are compared with the experimental results and numerical solutions, showing that the elastic-plastic analysis can predict the contact process very well.     

Nonlinear waves in diatomic crystals

A possible improvement of a continuum model for diatomic crystals is examined using continuum limit of a discrete diatomic model. For this purpose, various discrete models of diatomic lattice are compared at the linearized and weakly nonlinear levels. The suitable numbering of the atoms in the lattice is found which is better adopted for continualization than the familiar pair numbering introducing two sub-lattices. The coupled governing partial nonlinear differential equations for longitudinal strain and relative distance between the atoms are obtained in the continuum limit that allows us to describe localization of the strains due to the presence of the atoms of two kinds. It is found, that the equations obtained possess two kinds of localized wave solutions, one related to the acoustical branch and the other one related to the optical branch.     

Some remarks about a simple history dependent nonlinear viscoelastic model

A simple model for history dependent nonlinear viscoelasticity is considered. The determining equation governing shear motions is derived and investigated in the quasistatic approximation and under the traveling waves ansatz. Traveling waves are possible only if an inequality involving the constitutive parameters is satisfied. This fact is in contrast to what happens in viscoelasticity of the Kelvin–Voigt type. On the other hand, in the quasi-static approximation (classical creep and recovery experiments) the behavior of the history dependent model is similar to analogous rate dependent models.     

Traveling wave solutions for the mKdV equation and the Gardner equations by new approach of the generalized (G′/G)-expansion method

In this paper, we demonstrate the effectiveness of the new generalized (G′/G)-expansion method by seeking more exact solutions via the mKdV equation and the Gardner equations. The method is direct, concise and simple to implement compared to other existing methods. The traveling wave solutions obtained by this method are expressed in terms of hyperbolic, trigonometric and rational functions. The method shows a wide application for handling nonlinear wave equations. Moreover, the method reduces the large amount of calculations.     

Exact algorithms for the traveling salesman problem with draft limits

This paper deals with the Traveling Salesman Problem (TSP) with Draft Limits (TSPDL), which is a variant of the well-known TSP in the context of maritime transportation. In this recently proposed problem, draft limits are imposed due to restrictions on the port infrastructures. Exact algorithms based on three mathematical formulations are proposed and their performance compared through extensive computational experiments. Optimal solutions are reported for open instances of benchmark problems available in the literature.     

Improved bounds for the traveling umpire problem: A stronger formulation and a relax-and-fix heuristic

Given a double round-robin tournament, the traveling umpire problem (TUP) consists of determining which games will be handled by each one of several umpire crews during the tournament. The objective is to minimize the total distance traveled by the umpires, while respecting constraints that include visiting every team at home, and not seeing a team or venue too often. We strengthen a known integer programming formulation for the TUP and use it to implement a relax-and-fix heuristic that improves the quality of 24 out of 25 best-known feasible solutions to instances in the TUP benchmark. We also improve all best-known lower bounds for those instances and, for the first time, provide lower bounds for instances with more than 16 teams.     

On the integrality ratio of the subtour LP for Euclidean TSP

A long standing conjecture says that the integrality ratio of the subtour LP for metric TSP is 4/3$4/3$. A well known family of graphic TSP instances achieves this lower bound asymptotically. For Euclidean TSP the best known lower bound on the integrality ratio was 8/7$8/7$. We improve this value by presenting a family of Euclidean TSP instances for which the integrality ratio of the subtour LP converges to 4/3.     

Physical forces exerted by microbubbles on a surface in a traveling wave field

The effect of a wave with a varying traveling component on the bubble activity as well as the physical force generated by microbubbles on a surface has been studied. The acoustic emission from a collection of bubbles is measured in a 928 kHz sound field. Particle removal tests on a surface, which actually measures the applied physical force by the bubbles on that surface, indicate a very strong dependence on the angle of incidence. In other words, when the traveling wave component is maximized, the average physical force applied by microbubbles reaches a maximum. Almost complete particle removal for 78 nm silica particles was obtained for a traveling wave, while particle removal efficiency was reduced to only a few percent when a standing wave was applied. This increase in particle removal for a traveling wave is probably caused by a decrease in bubble trapping at nodes and antinodes in a standing wave field.     

The dynamics of a forced coupled network of active elements

This paper presents the derivation and analysis of mathematical models motivated by the experimental induction of contour phosphenes in the retina. First, a spatially discrete chain of periodically forced coupled oscillators is considered via reduction to a chain of scalar phase equations. Each isolated oscillator locks in a 1:2 manner with the forcing so that there is intrinsic bistability, with activity peaking on either the odd or even cycles of the forcing. If half the chain is started on the odd cycle and half on the even cycle (“split state”), then with sufficiently strong coupling, a wave can be produced that can travel in either direction due to symmetry. Numerical and analytic methods are employed to determine the size of coupling necessary for the split state solution to destabilize such that waves appear. Taking a continuum limit, we reduce the chain to a partial differential equation. We use a Melnikov function to compute, to leading order, the speed of the traveling wave solution to the partial differential equation as a function of the form of coupling and the forcing parameters and compare our result to the numerically computed discrete and continuum wave speeds.     

Finding low cost TSP and 2-matching solutions using certain half-integer subtour vertices

Consider the traveling salesman problem (TSP) defined on the complete graph, where the edge costs satisfy the triangle inequality. Let TOUR denote the optimal solution value for the TSP. Two well-known relaxations of the TSP are the subtour elimination problem and the 2-matching problem. If we let SUBT and 2M represent the optimal solution values for these two relaxations, then it has been conjectured that TOUR/SUBT ≤4/3, and that 2M/SUBT ≤10/9.In this paper, we exploit the structure of certain 1/2-integer solutions for the subtour elimination problem in order to obtain low cost TSP and 2-matching solutions. In particular, we show that for cost functions for which the optimal subtour elimination solution found falls into our classes, the above two conjectures are true. Our proofs are constructive and could be implemented in polynomial time, and thus, for such cost functions, provide a 4/3 (or better) approximation algorithm for the TSP.