On network form and function

Moving from the observation that drainage network configurations minimizing total energy dissipation are stationary solutions of the general equation describing landscape evolution, we review theoretical and observational evidence on river patterns and their scale-invariant structure. Exact results complemented by numerical annealing of the basic equation in the presence of additive noise suggest that configurations at (or very close to) the global minimum of energy dissipation differ from dynamically accessible states, which have rather different scaling properties and conform much better to natural forms. Thus we argue that, at least in the fluvial landscape, Nature works through imperfect searches for dynamically accessible optimal configurations. We also show that optimal networks are spanning loopless configurations only under precise physical requirements. This is stated in a form applicable to generic networks, suggesting that other branching structures occurring in Nature (e.g. scale-free and looping) may possibly arise through optimality to selective pressures. Indeed, we show that this is the case.     

Structurally constrained protein evolution: results from a lattice simulation

We simulate the evolution of a protein-like sequence subject to point mutations, imposing conservation of the ground state, thermodynamic stability and fast folding. Our model is aimed at describing neutral evolution of natural proteins. We use a cubic lattice model of the protein structure and test the neutrality conditions by extensive Monte Carlo simulations. We observe that sequence space is traversed by neutral networks, i.e. sets of sequences with the same fold connected by point mutations. Typical pairs of sequences on a neutral network are nearly as different as randomly chosen sequences. The fraction of neutral neighbors has strong sequence to sequence variations, which influence the rate of neutral evolution. In this paper we study the thermodynamic stability of different protein sequences. We relate the high variability of the fraction of neutral mutations to the complex energy landscape within a neutral network, arguing that valleys in this landscape are associated to high values of the neutral mutation rate. We find that when a point mutation produces a sequence with a new ground state, this is likely to have a low stability. Thus we tentatively conjecture that neutral networks of different structures are typically well separated in sequence space. This result indicates that changing significantly a protein structure through a biologically acceptable chain of point mutations is a rare, although possible, event. Received 8 July 1999     

Nonlinear vibration control and energy harvesting of a beam using a nonlinear energy sink and a piezoelectric device

This paper presents an optimal design for a system comprising a nonlinear energy sink (NES) and a piezoelectric-based vibration energy harvester attached to a free–free beam under shock excitation. The energy harvester is used for scavenging vibration energy dissipated by the NES. Grounded and ungrounded configurations are examined and the systems parameters are optimized globally to both maximize the dissipated energy by the NES and increase the harvested energy by piezoelectric element. A satisfactory amount of energy has been harvested as electric power in both configurations. The realization of nonlinear vibration control through one-way irreversible nonlinear energy pumping and optimizing the system parameters result in acquiring up to 78 percent dissipation of the grounded system energy.     

Adatom mobility for the solid-on-solid model

The relaxation of a crystal surface through surface diffusion is studied within the solid-on-solid model. Two types of (conserved) dynamics are considered. ForArrhenius dynamics we show that the relevant transport coefficient, the adatom mobility, has a simple analytic form: It is independent of orientation, and depends exponentially on the inverse temperature, for any surface dimensionalityd. Together with the expression for the orientation-dependent stiffness this completely determines the macroscopic evolution equation for the surface. The predictions of the macroscopic theory are checked against simulations of profile evolution and roughening ind=1. For one-dimensionalMetropolis dynamics we provide an upper bound on the adatom mobility and obtain numerical estimates of its actual value, which indicate a nontrivial orientation dependence in this case. An alternative derivation of the macroscopic dynamics directly from the master equation is presented and discussed in relation to previous approximate work.Dedicated to Professor H. Wagner on the occasion of his 60th birthday     

Aging and energy landscapes: application to liquids and glasses

The equation of state for a liquid in equilibrium, written in the potential energy landscape formalism, is generalized to describe out-of-equilibrium conditions. The hypothesis that during aging the system explores basins associated to equilibrium configurations is the key ingredient in the derivation. Theoretical predictions are successfully compared with data from molecular dynamics simulations of different aging processes, such as temperature and pressure jumps. Received 7 August 2002 / Received in final form 8 October 2002 Published online 19 December 2002 RID="a" ID="a"Present address: Laboratoire de Physique Théorique des Liquides, Université Pierre et Marie Curie, 4 place Jussieu, Paris 75005, France e-mail: mossa@lptl.jussieu.fr     

Curvature driven flow of bi-layer interfaces

We introduce the Functionalized Cahn-Hilliard (FCH) energy, a negative multiple of the Cahn-Hilliard energy balanced against the square of its own variational derivative, as a finite width regularization of the sharp-interface Canham-Helfrich energy. Mass-preserving gradient flows associated to the FCH energy are higher-order phase field models which develop not only single-layer, or front-type interfaces, but also bi-layer, or homoclinic interfaces with associated endcap and multi-junction structures. The single-layer interfaces manifest a fingering instability which grows into endcapped bi-layers. The meandering growth of the bi-layer interfaces and the subsequent merging lead to a multi-junction dominated network that bears a striking similarity to the phase separated domains of both perfluorosulfonic membranes and amphiphilic di-block co-polymer solutions. The bi-layers generated by the gradient flows of the FCH energy have an interfacial width which scales with ε?1, however for fixed ε, there is a class of bi-layers parameterized by width and background state. Our primary result is the asymptotic derivation of the normal velocity of a closed bi-layer hypersurface in R^d (d≥2) coupled to the evolution for the surface width, curvature, and background state. We also show the convergence of the FCH energy to a scaled Canham-Helfrich type energy for both single and bi-layer interfaces, with the surface area coefficient of the limiting Canham-Helfrich energy coupling to the bi-layer width. Thus the bi-layer networks grow to maximize surface area while minimizing the square of curvature, up to the point that the increase in surface area stretches the bi-layers too thin.     

Configurational statistics of densely and fully packed loops in the negative-weight percolation model

By means of numerical simulations we investigate the configurational properties of densely and fully packed configurations of loops in the negative-weight percolation (NWP) model. In the presented study we consider 2d square, 2d honeycomb, 3d simple cubic and 4d hypercubic lattice graphs, where edge weights are drawn from a Gaussian distribution. For a given realization of the disorder we then compute a configuration of loops, such that the configurational energy, given by the sum of all individual loop weights, is minimized. For this purpose, we employ a mapping of the NWP model to the “minimum-weight perfect matching problem” that can be solved exactly by using sophisticated polynomial-time matching algorithms. We characterize the loops via observables similar to those used in percolation studies and perform finite-size scaling analyses, up to side length L = 256 in 2d, L = 48 in 3d and L = 20 in 4d (for which we study only some observables), in order to estimate geometric exponents that characterize the configurations of densely and fully packed loops. One major result is that the loops behave like uncorrelated random walks from dimension d = 3 on, in contrast to the previously studied behavior at the percolation threshold, where random-walk behavior is obtained for d ≥ 6.     

Optimal symmetric networks in terms of minimizing average shortest path length and their sub-optimal growth model

Homogeneous entangled networks characterized by small world, large girths, and no community structure have attracted much attention due to some of their favorable performances. However, the optimization algorithm proposed by Donetti et al. is very time-consuming and will lose its efficiency when the size of the target network becomes large. In this paper, an alternative optimization algorithm is provided to get optimal symmetric networks by minimizing the average shortest path length. It is shown that the synchronizability of a symmetric network is enhanced when the average shortest path length of the network is shortened as the optimization proceeds, which suggests that the optimal symmetric networks in terms of minimizing average shortest path length will be very close to those entangled networks. In order to overcome the time-consuming obstacle of the optimization algorithms proposed by us and Donetti et al., a growth model is proposed to get large scale sub-optimal symmetric networks. Numerical simulations show that the symmetric networks derived by our growth model will have small-world property, and besides, these networks will have many other similar favorable performances as entangled networks, e.g., robustness against errors and attacks, very good load balancing ability, and strong synchronizability.     

The role of d-orbital polarization on rhodium cluster collisions

The effects due to d-orbital polarization in collision processes between a single rhodium atom and a 12 atom rhodium cluster are investigated by means of ab initio molecular dynamics. For the initial configuration of the 12 atom rhodium targets we adopt two different low energy structures. The kinetic energy and impact parameter of the projectile are chosen in such a way that fusion, scattering and fragmentation of the cluster do occur. The collision is treated by means of density functional theory molecular dynamics (DFT-MD). Both spin unpolarized and polarized treatments are implemented in order to clearly distinguish the effects that are due to d-orbital polarization. We find a novel block dynamics, of parts of the cluster, which is due to the directional nature of d-bonds. In addition, the treatment of the collisions by means of high temperature DFT dynamics yields promising minimal energy configurations, which are target dependent but are difficult to obtain otherwise.     

Social power and opinion formation in complex networks

In this paper we investigate the effects of social power on the evolution of opinions in model networks as well as in a number of real social networks. A continuous opinion formation model is considered and the analysis is performed through numerical simulation. Social power is given to a proportion of agents selected either randomly or based on their degrees. As artificial network structures, we consider scale-free networks constructed through preferential attachment and Watts–Strogatz networks. Numerical simulations show that scale-free networks with degree-based social power on the hub nodes have an optimal case where the largest number of the nodes reaches a consensus. However, given power to a random selection of nodes could not improve consensus properties. Introducing social power in Watts–Strogatz networks could not significantly change the consensus profile.     

Three-dimensional phase field microelasticity theory of a multivoid multicrack system in an elastically anisotropic body: Model and computer simulations

The phase field microelasticity theory of a three-dimensional, elastically anisotropic system of voids and cracks is proposed. The theory is based on the equation for the strain energy of the continuous elastically homogeneous body presented as a functional of the phase field, which is the effective stress-free strain. It is proved that the stress-free strain minimizing the strain energy of this homogeneous modulus body fully determines the elastic strain and displacement of the body with voids and/or cracks. The proposed phase field integral equation describing the elasticity of an arbitrary system of voids and cracks is exact. The geometry and evolution of multiple voids and/or cracks are described by the phase field, which is the solution of the time-dependent Ginzburg-Landau equation. Other defects, such as dislocations and precipitates, are trivially integrated into this theory. The proposed model does not impose a priori constraints on possible void and crack configurations or their evolution paths. Examples of computations of elastic equilibrium of systems with voids and/or cracks and the evolution of cracks under applied stress are considered.     

Damage and fracture evolution in brittle materials by shape optimization methods

This paper is devoted to a numerical implementation of the Francfort–Marigo model of damage evolution in brittle materials. This quasi-static model is based, at each time step, on the minimization of a total energy which is the sum of an elastic energy and a Griffith-type dissipated energy. Such a minimization is carried over all geometric mixtures of the two, healthy and damaged, elastic phases, respecting an irreversibility constraint. Numerically, we consider a situation where two well-separated phases coexist, and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy. In the context of interface variations (Hadamard method) and using a steepest descent algorithm, we compute local minimizers of this quasi-static damage model. Initially, the damaged zone is nucleated by using the so-called topological derivative. We show that, when the damaged phase is very weak, our numerical method is able to predict crack propagation, including kinking and branching. Several numerical examples in 2d and 3d are discussed.     

Least Action Principle for an Integrable Shallow Water Equation

Abstract

For an integrable shallow water equation we describe a geometrical approach showing that any two nearby fluid configurations are successive states of a unique flow minimizing the kinetic energy.     

Vibration confinement and energy harvesting in flexible structures using collocated absorbers and piezoelectric devices

We propose an optimal design for supplementing flexible structures with a set of absorbers and piezoelectric devices for vibration confinement and energy harvesting. We assume that the original structure is sensitive to vibrations and that the absorbers are the elements where the vibration energy is confined and then harvested by means of piezoelectric devices. The design of the additional mechanical and electrical components is formulated as a dynamic optimization problem in which the objective function is the total energy of the uncontrolled structure. The locations, masses, stiffnesses, and damping coefficients of these absorbers and capacitances, load resistances, and electromechanical coupling coefficients are optimized to minimize the total energy of the structure. We use the Galerkin procedure to discretize the equations of motion that describe the coupled dynamics of the flexible structure and the added absorbers and harvesting devices. We develop a numerical code that determines the unknown parameters of a pre-specified set of absorbers and harvesting components. We input a set of initial values for these parameters, and the code updates them while minimizing the total energy in the uncontrolled structure. To illustrate the proposed design, we consider a simply supported beam with harmonic external excitations. Here, we consider two possible configurations for each of the additional piezoelectric devices, either embedded between the structure and the absorbers or between the ground and absorbers. We present simulations of the harvested power and associated voltage for each pair of collocated absorber and piezoelectric device. The simulated responses of the beam show that its energy is confined and harvested simultaneously.     

Statistics of two-dimensional point vortices and high-energy vortex states

Results from the theory ofU-statistics are used to characterize the microcanonical partition function of theN-vortex system in a rectangular region for largeN, under various boundary conditions, and for neutral, asymptotically neutral, and nonneutral systems. Numerical estimates show that the limiting distribution is well matched in the region of major probability forN larger than 20. Implications for the thermodynamic limit are discussed. Vortex clustering is quantitatively studied via the average interaction energy between negative and positive vortices. Vortex states for which clustering is generic (in a statistical sense) are shown to result from two modeling processes: the approximation of a continuous inviscid fluid by point vortex configurations; and the modeling of the evolution of a continuous fluid at high Reynolds number by point vortex configurations, with the viscosity represented by the annihilation of close positive-negative vortex pairs. This last process, with the vortex dynamics replaced by a random walk, reproduces quite well the late-time features seen in spectral integration of the 2d Navier-Stokes equation.     

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

We consider a modification of the three-dimensional Navier–Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as e^|k|/k_d{{{\rm e}^{|k|/k_{\rm d}}}} at high wavenumbers |k|. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than e^{-C(k/k_d) ln(|k|/k_d)}{{{\rm e}^{-C(k/k_{\rm d})\,{\rm ln}(|k|/k_{\rm d})}}} for any C < 1/(2 ln 2). The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with C = C _* = 1/ ln 2. The same behavior with a universal constant C _* is conjectured for the Navier–Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier–Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.     

Impact of drifts in the ASDEX upgrade upper open divertor using SOLPS-ITER

SOLPS-ITER L-mode-like simulations with the full set of currents and drift velocities activated, and fluid neutrals have been carried out to interpret experimental results obtained in AUG. Drifts are critical to quantitatively reproduce the experimental results; however, simulations without drifts can also reproduce some trends qualitatively. The magnitude and dependence of the peak heat flux onto both targets on the upstream collisionality are, in general, in quantitative agreement within uncertainties with infrared thermography measurements in favourable field direction. The onset of power detachment is observed. In unfavourable toroidal field direction, a more symmetrical inner/outer target solution with regards to the power distribution is predicted, in agreement with experimental observations. However, also in unfavourable toroidal field direction, insufficient power is dissipated in the simulations and therefore q_{peak, inn} is overpredicted by up to a factor of 4 and q_{peak, out} by up to a factor of 1.5. The largest contribution to the sources due to radial transport in the energy balance equation is the radial divergence of the energy flux due to V_{E × B}.     

Master equation — The information gain minimum approximation

Approximate time-dependent solutions of a master equation having unique stationary solution can be obtained by minimizing the information gain functional subject to constraints for mean values of a number of chosen observables. We study mathematical properties of such an approximation. We find the region of applicability, prove that the approximate solutions are globally asymptotically stable, and show how the approximation is related to some exact integrodifferential equation governing the time evolution of the mean values of the chosen observables.