ISSN:

1556-1801

eISSN:

1556-181X

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## Networks & Heterogeneous Media

September 2014 , Volume 9 , Issue 3

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2014, 9(3): 383-416
doi: 10.3934/nhm.2014.9.383

*+*[Abstract](1936)*+*[PDF](4070.2KB)**Abstract:**

The concept of metastability has caused a lot of interest in recent years. The spectral decomposition of the generator matrix of a stochastic network exposes all of the transition processes in the system. The assumption of the existence of a low lying group of eigenvalues separated by a spectral gap has become a popular theme. We consider stochastic networks representing potential energy landscapes whose states and edges correspond to local minima and transition states respectively, and the pairwise transition rates are given by the Arrhenuis formula. Using the minimal spanning tree, we construct the asymptotics for eigenvalues and eigenvectors of the generator matrix starting from the low lying group. This construction gives rise to an efficient algorithm suitable for large and complex networks. We apply it to Wales's Lennard-Jones-38 network with 71887 states and 119853 edges where the underlying energy landscape has a double-funnel structure. Our results demonstrate that the concept of metastability should be applied with care to this system. For the full network, there is no significant spectral gap separating the eigenvalue corresponding to the exit from the wider and shallower icosahedral funnel at any reasonable temperature range. However, if the observation time is limited, the expected spectral gap appears.

2014, 9(3): 417-432
doi: 10.3934/nhm.2014.9.417

*+*[Abstract](1539)*+*[PDF](604.2KB)**Abstract:**

We present the exact finite reduction of a class of nonlinearly perturbed wave equations --typically, a non-linear elastic string-- based on the Amann--Conley--Zehnder paradigm. By solving an inverse eigenvalue problem, we establish an equivalence between the spectral finite description derived from A--C--Z and a discrete mechanical model, a well definite finite spring--mass system. By doing so, we decrypt the abstract information encoded in the finite reduction and obtain a physically sound proxy for the continuous problem.

2014, 9(3): 433-451
doi: 10.3934/nhm.2014.9.433

*+*[Abstract](1631)*+*[PDF](822.4KB)**Abstract:**

This paper describes a numerical method to optimize elastic bodies featuring a locally periodic microscopic pattern. A new idea, of optimizing the periodicity cell itself, is considered. In previously published works, the authors have found that optimizing the shape and topology of the model hole gives a limited flexibility to the microstructure for adapting to the macroscopic loads. In the present study the periodicity cell varies during the optimization process, thus allowing the microstructure to adapt freely to the given loads. Our approach makes the link between the microscopic level and the macroscopic one. Two-dimensional linearly elastic bodies are considered, however the same techniques can be applied to three-dimensional bodies. Homogenization theory is used to describe the macroscopic (effective) elastic properties of the body. Numerical examples are presented, in which a cantilever is optimized for different load cases, one of them being multi-load. The problem is numerically heavy, since the optimization of the macroscopic problem is performed by optimizing in simultaneous hundreds or even thousands of periodic structures, each one using its own finite element mesh on the periodicity cell. Parallel computation is used in order to alleviate the computational burden.

2014, 9(3): 453-476
doi: 10.3934/nhm.2014.9.453

*+*[Abstract](1729)*+*[PDF](516.8KB)**Abstract:**

We investigate connections between the continuum and atomistic descriptions of deformable crystals, using certain interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with general binary interactions in two dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. The main contribution is an explicit formula for the surface energy density as a function of the deformation gradient and boundary normal. The result is valid for a large class of domains, including faceted (polygonal) shapes and regions with piecewise smooth boundaries.

2014, 9(3): 477-499
doi: 10.3934/nhm.2014.9.477

*+*[Abstract](1420)*+*[PDF](428.4KB)**Abstract:**

Reaction-diffusion equations are treated on infinite networks using semigroup methods. To blend high fidelity local analysis with coarse remote modeling, initial data and solutions come from a uniformly closed algebra generated by functions which are flat at infinity. The algebra is associated with a compactification of the network which facilitates the description of spatial asymptotics. Diffusive effects disappear at infinity, greatly simplifying the remote dynamics. Accelerated diffusion models with conventional eigenfunction expansions are constructed to provide opportunities for finite dimensional approximation.

2014, 9(3): 501-518
doi: 10.3934/nhm.2014.9.501

*+*[Abstract](1652)*+*[PDF](441.0KB)**Abstract:**

We discuss the optimal control problem (OCP) stated as the minimization of the queues and the difference between the effective outflow and a desired one for the continuous model of supply chains, consisting of a PDE for the density of processed parts and an ODE for the queue buffer occupancy. The main goal is to consider this problem with pointwise control and state constraints. Using the so-called Henig delation, we propose the relaxation approach to characterize the solvability and regularity of the original problem by analyzing the corresponding relaxed OCP.

2014, 9(3): 519-552
doi: 10.3934/nhm.2014.9.519

*+*[Abstract](1659)*+*[PDF](942.2KB)**Abstract:**

In this paper we study a model for traffic flow on networks based on a hyperbolic system of conservation laws with discontinuous flux. Each equation describes the density evolution of vehicles having a common path along the network. In this formulation the junctions disappear since each path is considered as a single uninterrupted road.

We consider a Godunov-based approximation scheme for the system which is very easy to implement. Besides basic properties like the conservation of cars and positive bounded solutions, the scheme exhibits other nice properties, being able to select automatically a solution at network's nodes without requiring external procedures (e.g., maximization of the flux via a linear programming method). Moreover, the scheme can be interpreted as a discretization of the traffic models with buffer, although no buffer is introduced here.

Finally, we show how the scheme can be recast in the framework of the classical theory of traffic flow on networks, where a conservation law has to be solved on each arc of the network. This is achieved by solving the Riemann problem for a modified equation, and showing that its solution corresponds to the one computed by the numerical scheme.

2014, 9(3): 553-573
doi: 10.3934/nhm.2014.9.553

*+*[Abstract](2020)*+*[PDF](636.2KB)**Abstract:**

This paper addresses group consensus problems in generic linear multi-agent systems with directed information flow over (i) fixed topology and (ii) randomly switching topology governed by a continuous-time homogeneous Markov process. We propose two types of pinning control protocols to ensure group consensus regardless of the magnitude of the coupling strengths among the agents. In the case of randomly switching topology, we show that the group consensus behavior is unrelated to the magnitude of the couplings among agents if the union of the topologies corresponding to the positive recurrent states of the Markov process possesses an acyclic partition. Sufficient conditions for achieving group consensus are presented in terms of simple graphic conditions, which are easy to be checked compared to conventional algebraic criteria. Simulation examples are also presented to validate the effectiveness of the theoretical results.

2018 Impact Factor: 0.871

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