Rigorous Upper Bound for the Discrete Bak–Sneppen Model

Fix some $$p\in [0,1]$$ p ∈ [ 0 , 1 ] and a positive integer n. The discrete Bak–Sneppen model is a Markov chain on the space of zero-one sequences of length n with periodic boundary conditions. At each moment of time a minimum element (typically, zero) is chosen with equal probability, and it is then replaced alongside both its neighbours by independent Bernoulli(p) random variables. Let $$\nu ^{(n)}(p)$$ ν ( n ) ( p ) be the probability that an element of this sequence equals one under the stationary distribution of this Markov chain. It was shown in Barbay and Kenyon (in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, DC, 2001), pp. 928–933, SIAM, Philadelphia, PA, 2001) that $$\nu ^{(n)}(p)\rightarrow 1$$ ν ( n ) ( p ) → 1 as $$n\rightarrow \infty $$ n → ∞ when $$p>0.54\dots $$ p > 0.54 ⋯ ; the proof there is, alas, not rigorous. The complimentary fact that $$\displaystyle \limsup _{n\rightarrow \infty } \nu ^{(n)}(p)< 1$$ lim sup n → ∞ ν ( n ) ( p ) < 1 for $$p\in (0,p')$$ p ∈ ( 0 , p ′ ) for some $$p'>0$$ p ′ > 0 is much harder; this was eventually shown in Meester and Znamenski (J Stat Phys 109:987–1004, 2002). The purpose of this note is to provide a rigorous proof of the result from Barbay and Kenyon (in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, DC, 2001), pp. 928–933, SIAM, Philadelphia, PA, 2001), as well as to improve it, by showing that $$\nu ^{(n)}(p)\rightarrow 1$$ ν ( n ) ( p ) → 1 when $$p>0.45$$ p > 0.45 . (Our method, in fact, shows that with some finer tuning the same is true for $$p>0.419533$$ p > 0.419533 .)

The effect of mesh parameters on computational cost and results in simulation of milling in Inconel 718

The Finite Element Method analysis of machining processes has become a ubiquitous feature to the area, however, there sometimes occur considerable deviations between experimental and simulated results due to the inherent complexity of the process. The basis for such may conceivably be related to imprecisions in the material and friction modelling, besides improper setup of mesh parameters. Elements should be small enough to allow for the proper representation of the chip formation, but taking into account that the computational time increases accordingly with mesh downsizing. Simulations of the milling process of Inconel 718 were conducted using the software Thirdwave AdvantEdge under different cutting conditions for three different meshes. Power and temperature output were compared to experimental results, most of which were measured via Hall-effect sensors and thermographic camera, respectively. The tool cutting edge radius was found to be an important factor and was estimated using Scanning Electron Microscope images. The influence of the finite element mesh size was higher for harsher cutting conditions, with effects felt on machining power only. In this case, finer mesh produced results that showed a higher agreement with experimental data, but at higher computational cost as shown by analysis of elapsed processing time. Although errors higher than 40% were observed, power and temperature trends from simulations were always in accordance with that found in experimental tests. Comparisons with experimental data from other studies showed the errors tend to grow for higher feed and cutting speed, which indicates the constitutive model of the material is more adequate for softer machining conditions. Simulation time seemed to be exponentially proportional to the inverse of minimum element size, and measured values might serve as a reference for other users.

An analytical method for solving the problem of heat network load flow

The author aimed to develop an analytical solution to the problem of the load flow of a six-, eleven- and twelve-circuit heat network, as well as to solve the problem of optimisation of a multi-circuit heat network, including the choice of the objective function and the determination of a number of variable technical parameters. For accelerating the optimisation process, the method of decomposition of the heat network graph was used. Decomposition involves is cutting the network graph at some nodes for the transition of a multi-circuit scheme to a branched scheme in the form of a tree. Optimisation of each branched circuit was carried out by the dynamic programming method, as a result of which new values of the variable parameters were obtained at the current iteration. Next, the author returned to the multi-circuit scheme to solve the load flow problem and calculate the value of the objective function. The iterative convergence of the decomposition method was not mathematically proven. The author proposed a method for splitting the graph, which eliminates the decomposition procedure when optimising a heat network. The following methods were applied: mathematical modelling of the hydraulic circuit, graph splitting method and the analytical method for solving the algebraic equation of the fourth degree. The following results were achieved: a scheme of the minimum element of a multi-circuit heat network was determined, the possibility of series and parallel circuits of minimum elements was shown, and analytical dependencies for the problem of load flow of a heat network of these schemes were obtained. The proposed analytical solution of the load flow problem for a multi-circuit heat network allows the problem of calculating a complex network to be reduced to the calculation of several minimum elements, which significantly reduces the amount of computational work when modelling a hydraulic circuit. The provided examples show that the calculation error does not exceed 3%.

Global storm tide modeling with ADCIRC v55: unstructured mesh design and performance

This paper details and tests numerical improvements to the ADvanced CIRCulation (ADCIRC) model, a widely used finite-element method shallow-water equation solver, to more accurately and efficiently model global storm tides with seamless local mesh refinement in storm landfall locations. The sensitivity to global unstructured mesh design was investigated using automatically generated triangular meshes with a global minimum element size (MinEle) that ranged from 1.5 to 6 km. We demonstrate that refining resolution based on topographic seabed gradients and employing a MinEle less than 3 km are important for the global accuracy of the simulated astronomical tide. Our recommended global mesh design (MinEle = 1.5 km) based on these results was locally refined down to two separate MinEle values (500 and 150 m) at the coastal landfall locations of two intense storms (Hurricane Katrina and Super Typhoon Haiyan) to demonstrate the model's capability for coastal storm tide simulations and to test the sensitivity to local mesh refinement. Simulated maximum storm tide elevations closely follow the lower envelope of observed high-water marks (HWMs) measured near the coast. In general, peak storm tide elevations along the open coast are decreased, and the timing of the peak occurs later with local coastal mesh refinement. However, this mesh refinement only has a significant positive impact on HWM errors in straits and inlets narrower than the MinEle and in bays and lakes separated from the ocean by these passages. Lastly, we demonstrate that the computational performance of the new numerical treatment is 1 to 2 orders of magnitude faster than studies using previous ADCIRC versions because gravity-wave-based stability constraints are removed, allowing for larger computational time steps.

Global Storm Tide Modeling with ADCIRC v55: Unstructured Mesh Design and Performance

This paper details and tests numerical improvements to ADCIRC, a widely used finite element method shallow water equation solver, to more accurately and efficiently model global storm tides with seamless local mesh refinement in storm landfall locations. The sensitivity to global unstructured mesh design was investigated using automatically generated triangular meshes with a global minimum element size (MinEle) that ranged from 1.5 km to 6 km. We demonstrate that refining resolution based on topographic seabed gradients and employing a MinEle less than 3 km is important for the global accuracy of the simulated astronomical tide. Our recommended global mesh design (MinEle = 1.5 km) based on these results was locally refined down to two separate MinEle (500 m and 150 m) at the coastal landfall locations of two intense storms (Hurricane Katrina and Super Typhon Haiyan) to demonstrate the model's capability for coastal storm tide simulations and to test the sensitivity to local mesh refinement. Simulated maximum storm tide elevations closely follow the lower envelope of observed high water marks (HWMs) measured near the coast. In general, peak storm tide elevations along the open coast are decreased and the timing of the peak occurs later with local coastal mesh refinement. However, this mesh refinement only has a significant positive impact on HWM errors in straits and inlets narrower than the MinEle, and in bays and lakes separated from the ocean by these passages. Lastly, we demonstrate that the computational performance of the new numerical treatment is one-to-two orders of magnitude faster than studies using previous ADCIRC versions because gravity-wave based stability constraints are removed allowing for larger computational time steps.

Bidirectional Coupling Model of Electromagnetic Field and Thermal Field Applied to the Thermal Analysis of the FSPM Machine

The paper aimed to find an accurate and fast model to study the electromagnetic (EM) thermal (TH) filed coupling calculation for the TH analysis in the flux switching permanent magnet (FSPM) machine. It is extremely important to know the coupling mechanism between the EM field and TH field for the designers and users of the FSPM machines. Firstly, in order to study the EM properties of the silicon steel sheet with the temperature, the Epstein frame experiment was set up, where the effect of dc magnetic bias on the core loss is also considered. In order to save the computation time, the bidirectional coupling model based on 2D finite element (FE) EM field and 3D asymmetric minimum element TH field is established, and the steady state and transient TH fields are calculated, respectively. For the transient bidirectional coupling of EM field and TH field, a method based on the adaptive adjustment calculation step is adopted to improve the computing speed. The temperature rise experiment of the prototype was carried out to verify the accuracy of the proposed coupling model. The experimental results are in good agreement with the simulation results.

Position-only synthesis of uniformly excited elliptical antenna arrays with minimum element spacing constraint

Pattern synthesis of non-uniform elliptical antenna arrays is presented in this paper. Only the element positions of the antenna arrays are optimized by the combination of differential evolution (DE) and invasive weed optimization (IWO) to reduce the peak side lobe level (PSLL) of the radiation pattern. In order to avoid the overlap of the array elements, the minimum spacing of the adjacent elements is constrained. Also, the beam width of the radiation pattern can be constrained effectively. Three elliptical antenna arrays that have 8, 12, and 20 elements are investigated. The synthesis results show that the introduced method can present a good side lobe reduction for the radiation pattern. Compared with other optimization methods, the method proposed in this paper can obtain better performance.