Method and software for finite element solution of two-dimensional creep problems at large deformations

The paper presents the formulation of a two-dimensional problem of the creep theory for the case of finite strains. A description of the foundations of the calculation method presents. The method is based on the use of the generalized Lagrange-Euler (ALE) approach, in which the boundary value problem in the current solid configuration is solved by using FEM. A triangular element is involved in the numerical modeling. At each stage of creep calculations in the current configuration, the initial problem is solved numerically using the finite difference method. The preprocessing data preparation is carried out in the homemade RD program, in which two-dimensional model is surrounded by a mesh of special elements. This feature implements the ALE algorithm for the motion of material elements along the model. The examples of preprocessing as well as of the mesh rebuilding in the case of finite elements transition are given. Creep calculations are performed in the developed program, which is based on the use of the FEM Creep software package in the case of finite strains. The regular mesh is used for calculations, which allow us to use the efficient algorithm for transition between current configurations. The numerical results of the creep of specimens made from aluminum alloys are compared with the experimental and calculated ones obtained by integrating the constitutive equations. It was concluded that for material with ductile type of fracture the presented method and software allow to obtain results very close to experimental only by use of creep rate equation. Creep simulations of material with mixed brittle-ductile fracture type demand use the additional equation for damage variable.

UNIVERSAL METHOD OF CALCULATION OF MULTISPAN BEAMS

The supporting structures of machines and mechanisms of metallurgical equipment, which are characterized by large dimensions, in the calculation scheme can be reduced to multispan beams, which are statically indeterminate. Traditionally, the disclosure of their static uncertainty is based on the graphoanalytical method, which is called "Equation of 3 moments". According to this method, the equations of three points are solved by constructing intermediate epure of bending moments, finding their areas and position of the centers of gravity. These procedures lead to an unreasonable increase in time to solve the problem. It is advisable to have an analytical method for detecting static uncertainty of multispan beams, which would be free from graphical constructions and would significantly reduce the time to solve the problem. At the present stage, the calculation of complex spatial structures is carried out using the calculation and software systems "LIRA", "SCAD" and others. However, this in no way limits the need for the application in engineering practice of simple analytical methods, which include a universal method for detecting static indeterminateness of multispan beams. Additional, to the static equations, the equations of displacement are compiled using the universal equation of the bent axis of the beam, and the number of beams can be any large. Different variants of boundary conditions are considered. The results of the considered examples prove the independence of the values ​of "extra" unknown from the stiffness of the beam. The universal method is quite optimal, as it uses a very simple universal equation of the bent axis of the beam and, with each subsequent additional equation of displacement differs from the previous one only by adding another item with a new load. This approach provides the optimal solution to almost any problem, by revealing the static indeterminateness of multispan beams.

Reversible Regular Languages: Logical and Algebraic Characterisations

We present first-order (FO) and monadic second-order (MSO) logics with predicates ‘between’ and ‘neighbour’ that characterise the class of regular languages that are closed under the reverse operation and its subclasses. The ternary between predicate bet(x, y, z) is true if the position y is strictly between the positions x and z. The binary neighbour predicate N(x, y) is true when the the positions x and y are adjacent. It is shown that the class of reversible regular languages is precisely the class definable in the logics MSO(bet) and MSO(N). Moreover the class is definable by their existential fragments EMSO(bet) and EMSO(N), yielding a normal form for MSO formulas. In the first-order case, the logic FO(bet) corresponds precisely to the class of reversible languages definable in FO(<). Every formula in FO(bet) is equivalent to one that uses at most 3 variables. However the logic FO(N) defines only a strict subset of reversible languages definable in FO(+1). A language-theoretic characterisation of the class of languages definable in FO(N), called locally-reversible threshold-testable (LRTT), is given. In the second part of the paper we show that the standard connections that exist between MSO and FO logics with order and successor predicates and varieties of finite semigroups extend to the new setting with the semigroups extended with an involution operation on its elements. The case is different for FO(N) where we show that one needs an additional equation that uses the involution operator to characterise the class. While the general problem of characterising FO(N) is open, an equational characterisation is shown for the case of neutral letter languages.

On the Possibility of Chaos in a Generalized Model of Three Interacting Sectors

In this study we extend a model, proposed by Dendrinos, which describes dynamics of change of influence in a social system containing a public sector and a private sector. The novelty is that we reconfigure the system and consider a system consisting of a public sector, a private sector, and a non-governmental organizations (NGO) sector. The additional sector changes the model’s system of equations with an additional equation, and additional interactions must be taken into account. We show that for selected values of the parameters of the model’s system of equations, chaos of Shilnikov kind can exist. We illustrate the arising of the corresponding chaotic attractor and discuss the obtained results from the point of view of interaction between the three sectors.

FaIRv2.0.0: a generalised impulse-response model for climate uncertainty and future scenario exploration

Here we present an update to the FaIR model for use in probabilistic future climate and scenario exploration, integrated assessment, policy analysis and education. In this update we have focussed on identifying a minimum level of structural complexity in the model. The result is a set of six equations, five of which correspond to the standard Impulse Response model used for greenhouse gas (GHG) metric calculations in the IPCC's fifth assessment report, plus one additional physically-motivated additional equation to represent state-dependent feedbacks on the response timescales of each greenhouse gas cycle. This additional equation is necessary to reproduce non-linearities in the carbon cycle apparent in both Earth System Models and observations. These six equations are transparent and sufficiently simple that the model is able to be written in standard tabular data analysis packages, such as Excel; increasing the potential user base considerably. However, we demonstrate that the equations are flexible enough to be tuned to emulate the behaviour of several key processes within more complex models from CMIP6. The model is exceptionally quick to run, making it ideal for integrating large probabilistic ensembles. We apply a constraint based on the current estimates of the global warming trend to a one million member ensemble, using the constrained ensemble to make scenario dependent projections and infer ranges for properties of the climate system. Through these analyses, we reaffirm that simple climate models (unlike more complex models) are not themselves intrinsically biased hot or cold: it is the choice of parameters and how those are selected that determines the model response, something that appears to have been misunderstood in the past. This updated FaIR model is able to reproduce the global climate system response to GHG and aerosol emissions with sufficient accuracy to be useful in a wide range of applications; and therefore could be used as a lowest common denominator model to provide consistency in different contexts. The fact that FaIR can be written down in just six equations greatly aids transparency in such contexts.

Hybrid Method Based on FV Concept for Lean Blowout Limits Predictions for Aero Engine Combustors With Low Inlet Temperature

Lean blowout (LBO) limit is one of the most important parameters for aero engines. The LBO must be avoided during the whole flight envelope and modes. Thus the LBO limit should be predicted accurately during the research and development (R&D) process of an aero engine combustor. Some prediction models had been proposed by the previous studies. Lefebvre’s LBO model is widely used among these models due to its good accuracy and generality. Based on Lefebvre’s LBO model, the flame volume (FV) model is proposed to improve the prediction accuracy as well as keep the good generality. However, an additional equation is needed when the FV model is applied to predict the LBO limit. Based on our previous studies, the volume in the combustor liner enclosed by the iso-surface whose temperature is 900K could be chosen as the flame volume. The flame volume could be served as the additional equation and further makes the FV model available for the LBO limit prediction. This method could be named as the hybrid method based on the FV concept in this paper. On the other hand, the LBO limit at high altitude is attracting more and more attentions due to the extension of the flight envelope and modes in the recent years. Low inlet temperature is one of the most typical characteristics for the operating condition at high altitude. The temperature could reach less than −40 °C at high altitude. Compared with the normal condition, the characteristics of flow, atomization and combustion are different at the low temperature condition. In this paper, the hybrid method based on the FV concept is applied to predict the LBO limits of 11 combustors with low inlet temperature. The inlet temperature is 236.2K (corresponding to the altitude of 8km). The results show that the flame volumes obtained by the numerical simulation vary near linearly with the fuel/air ratio when the fuel/air ratio is close to the LBO limits. For all these 11 combustor configurations, the LBO limits with low inlet temperature (236.2K) obtained from the hybrid method based on the FV concept are significantly larger than those with normal inlet temperature (300K) obtained from experiments. The maximum and minimum increases are 29% and 2%, respectively. The LBO limits increases are more than 20% and 10% for 6 out of the 11 combustors and 9 out of the 11 combustors, respectively. The reasons for the increase of the LBO limits at low temperature include the decreasing of chemical reaction rate and evaporation rate, the increasing of the mean drop size.

Sensitivity of the lower edge of the pair-instability black hole mass gap to the treatment of time-dependent convection

ABSTRACT Gravitational-wave detections are now probing the black hole (BH) mass distribution, including the predicted pair-instability mass gap. These data require robust quantitative predictions, which are challenging to obtain. The most massive BH progenitors experience episodic mass ejections on time-scales shorter than the convective turnover time-scale. This invalidates the steady-state assumption on which the classic mixing length theory relies. We compare the final BH masses computed with two different versions of the stellar evolutionary code $\tt{MESA}$: (i) using the default implementation of Paxton et al. (2018) and (ii) solving an additional equation accounting for the time-scale for convective deceleration. In the second grid, where stronger convection develops during the pulses and carries part of the energy, we find weaker pulses. This leads to lower amounts of mass being ejected and thus higher final BH masses of up to ∼$5\, \mathrm{M}_\odot$. The differences are much smaller for the progenitors that determine the maximum mass of BHs below the gap. This prediction is robust at $M_{\rm BH, max}\simeq 48\, \mathrm{M}_\odot$, at least within the idealized context of this study. This is an encouraging indication that current models are robust enough for comparison with the present-day gravitational-wave detections. However, the large differences between individual models emphasize the importance of improving the treatment of convection in stellar models, especially in the light of the data anticipated from the third generation of gravitational-wave detectors.

Comments on “On a Continuum Model for Avalanche Flow and Its Simplified Variants” by S. S. Grigorian and A. V. Ostroumov

This note first summarizes the history of the manuscript “On a Continuum Model for Avalanche Flow and Its Simplified Variants” by Grigorian and Ostroumov—published in this Special Issue—since the early 1990s and explains the guiding principles in editing it for publication. The changes are then detailed and some explanatory notes given for the benefit of readers who are not familiar with the early Russian work on snow avalanche dynamics. Finally, the editor’s personal views as to why he still considers this paper of relevance for avalanche dynamics research today are presented in brief essays on key aspects of the paper, namely the role of simple and complex models in avalanche research and mitigation work, the status and possible applications of Grigorian’s stress-limited friction law, and non-monotonicity of the dynamics of the Grigorian–Ostroumov model in the friction coefficient. A comparison of the erosion model proposed by those authors with two other models suggests to enhance it with an additional equation for the balance of tangential momentum across the shock front. A preliminary analysis indicates that continuous scouring entrainment is possible only in a restricted parameter range and that there is a second erosion regime with delayed entrainment.

Interaction between curvature-driven width oscillations and channel curvature in evolving meander bends

We study the morphodynamics of channel width oscillations associated with the planform development of river meander bends. With this aim we develop a novel planform evolution model, based on the framework of the classical bend theory of river meanders by Ikeda et al. (J. Fluid Mech., vol. 112, 1981), that accounts for local width changes over space and time, tied to the local hydro-morphodynamics through a two-way feedback process. We focus our attention on ‘autogenic’ width variations, which are forced by flow nonlinearities driven by channel curvature dynamics. Under the assumption of regular, sinusoidal width and curvature oscillations, we obtain a set of ordinary differential equations, formally identical to those presented by Seminara et al. (J. Fluid Mech., vol. 438, 2001, pp. 213–230), with an additional equation for the longitudinal oscillation of the channel width. The proposed approach gives insight into the interaction between autogenic width variations and curvature in meander development and between forcing and damping effects in the formation of width variations. Model outcomes suggest that autogenic width oscillations mainly determine wider-at-inflection meandering river patterns, and affect their planform development particularly at super-resonant aspect ratios, where the width oscillation reaches its maximum and reduces meander sinuosity and lateral floodplain size. The coevolution of autogenic width oscillation and curvature occurs through temporal hysteresis cycles, whereby the peak in channel curvature lags behind that of width oscillation. Width oscillation amplitudes predicted by the model are consistent with those extracted from remotely sensed data.