This study aims at quantifying the level and changes over time of inequality in age-specific mortality and life expectancy between the 19 Norwegian counties from 1980 to 2014.Data on population and mortality by county was obtained from Statistics Norway for 1980–2014. Life expectancy and age-specific mortality rates  were estimated by year and county. Geographic inequality was described by the absolute Gini index annually.Life expectancy in Norway has increased from 75.6 to 82.0 years, and the risk of death before the age of 70 has decreased from 26 to 14% from 1980 to 2014. The absolute Gini index decreased over the period 1980 to 2014 from 0.43 to 0.32 for life expectancy, from 0.012 to 0.0057 for the age group 50–69 years, from 0.0038 to 0.0022 for the age group 5–49 years, and from 0.0009 to 0.0006 for the age group 0–4 years. It will take between 2 and 32 years  until the counties catch up with the life expectancy in the best performing county if their annual rates of increase remain unchanged.Using the absolute Gini index as a metric for monitoring changes in geographic inequality over time may be a valuable tool for informing public health policies. The absolute inequality in mortality and life expectancy between Norwegian counties has decreased from 1980 to 2014.The online version of this article (10.1186/s12939-018-0771-7) contains supplementary material, which is available to authorized users. Geographical differences in life expectancy and mortality has been shown to be present within several countries –4. Even The causes of geographic inequality are complex and only partly known, and may be amenable to interventions both within and outside the health sector. Constant or increasing geographic inequality in life expectancy have been documented in New Zealand and the European Union , 6. RegiOver the past 50 years, Norway has experienced economic growth and substantial improvements in welfare and population health. The total population of Norway is around 5.2 million and the country is administratively divided into 19 counties, with populations ranging from 75,000 in the county Finnmark located in the north to 648,000 in the capital city and county of Oslo as per January 1st 2015 . The NorDespite political interest in health inequalities, there is no systematic monitoring of geographic inequalities in health. Systematic monitoring of these inequalities will provide relevant information for policymakers concerned with reducing the inequalities. This study aims to provide an empirical illustration of how geographic inequalities can be monitored using the Gini index by applying this metric to Norwegian mortality data from the past 35 years. Additionally, this study explores a didactic way of presenting the magnitude of inequality.x to age x + 1 years, with exception of the age group 0–1 years where we estimated the distribution of the deaths according to the methodology described in the Human Mortality Database’s Methods Protocol [Data on deaths and population as of January 1st for males and females was purchased from Statistics Norway for the period 1980–2014 . Data onEstimation of life tables and statistical analysis were done using STATA IC version 12.0 . ArcMap Geographic inequality in risks of death and life expectancy at birth between the counties was measured for each year by the absolute Gini index. This is a comprehensive measure of inequality encompassing the distribution of inequality in each of the counties, and is interpreted as half the average difference between any two counties . For exaThe Gini index is closely related to the Lorenz curve, which plots the cumulative proportion of the outcome variable against the cumulative proportion of people ranked by the outcome , 24. Thet the value of the variable in the tth county, ft the county’s population share, and Rt the relative rank of the tth county. For graphical presentation of the trends over time, fitted curves using third degree polynomial regression were constructed according to the following formula:where G is the absolute Gini index, μ is the mean value of the variable, T the number of counties, μWhere χ is the year, f(χ) the estimated value of the Gini index in year χ, and a, b, c and d the constants determined by the regression model.The rates of change have been estimated to provide a didactic illustration of the magnitude of the inequalities. Even if the Gini index contains many appealing theoretical equity concerns and is a precise distributive metric, such an index may be difficult to interpret for many. As an addition, building on a previously developed model , we therLife expectancy at birth in Norway increased in all counties from 1980 to 2014. The increase over this period was from 72.3 to 79.9 years for males, from 79.0 to 84.0 years for females. The absolute difference in life expectancy at birth between the county with the highest and lowest life expectancy has decreased from 5.2 to 4.6 years for males, and increased from 3.4 to 3.9 years for females from 1980 to 2014. Figure Figure The geographic inequality, as measured by the absolute Gini index, is graphically shown for life expectancy and age-specific risks of death in Fig. The average annual rates of change over the past ten years are shown in Table Life expectancy varied substantially between counties, with a gap of 4.6 years for men and 3.9 years for women in 2014. Life expectancy and risks of death improved in all Norwegian counties from 1980 to 2014. The geographic inequality decreased; the absolute Gini index decreased from 0.43 in 1980 to 0.32 in 2014 for life expectancy and from 0.012 in 1980 to 0.0057 in 2014 for age specific risks of death in the age group 50–69 years.The magnitude of inequality and the change in inequality over time differ according to the outcome measure used. The value of the absolute Gini index relates directly to the indicator that is measured, it is therefore not meaningful to compare the magnitude of an absolute Gini index for life expectancy with the absolute Gini index for a mortality rate, but the trends can be assessed over time for each of the indicators separately. An absolute Gini index of 0.32 for life expectancy in 2014 means that the average difference between the life expectancy in two randomly chosen counties is twice the value of this estimate, namely 0.64 years. The absolute inequality in mortality is decreasing or remaining unchanged for all age groups. For life expectancy, the trend shows an increase in inequality from 1980, peaking in the early 1990s and then decreasing again to below 1980 levels. The shape of the trend line should be interpreted with caution, as the shape is influenced by the method for smoothing the graph, and the lack of data before 1980.Assessing whether the situation is getting more or less equal rely on several normative and technical choices underlying any measurement of inequality, and it is therefore important to understand the assumptions underlying the metric used . The GinIn this study an absolute measure of inequality is used, but it is not an obvious principled difference between a relative and absolute Gini measure of inequality . NeverthThe 19 Norwegian counties were chosen as geographical units for the analyses. Even though there are differences also within the counties, the counties differ from each other by population size and density, degree of urbanization, infrastructure, physical environment, main industries, and socio-economic characteristics of the population. Counties are especially relevant when it comes to the health sector as the responsibility for hospitals lied with the counties until 2002. Even with changing organizational structures, hospital services still follow county boundaries to a large degree. However, other geographical units could have been used for the study. The five health regions created in 2002 to manage secondary and tertiary levels of health care (reduced to four regions in 2007) would be a relevant geographical unit for linking the health inequalities to current and future policies governing specialized health care, but at the same time, important information would be lost if analyzing only 4–5 regions. The approximately 420 Norwegian municipalities could also have been used as units for the analysis, but estimates of life expectancy have been shown to be very uncertain if estimated for smaller administrative units than counties due to small populations . AnotherThis study adds empirical evidence at a macro level to ethical and political discussions. The inequalities quantified in this study encompass both current disparities in education, income, smoking etc., and measures aiming to compensate the effect of inequality in these underlying factors. Forces driving inequality and forces reducing inequality exist and interact in complex patterns, and it is useful for policy makers to monitor the overall effect of these factors on inequality. Information on the absolute level of inequality between counties regardless of the individuals’ socioeconomic position is therefore a useful supplement to information on specific causal pathways or subgroups of the population.There are two principal explanations of the observed associations between place of residence and mortality. First, the individual characteristics of the people living in the area, such as the individuals’ education and income, may lead to differences in mortality . Second, characteristics of the area itself, such as infrastructure and availability of health services, may influence the individuals’ health and health-related behaviour . Regional inequalities in mortality are found to persist in Norway after adjustments for factors such as income, education and marital status in other studies , 36. EffMeasuring the number of years between the points in time when the counties will have the same level of life expectancy or age-specific mortality is a way of illustrating what the inequality means , 39. EveLife expectancy in Norway has increased in all counties and risk of death before the age of 70 years has been almost halved from 1980 to 2014. Although there is a political interest in geographic health inequality, there is no systematic monitoring in place in Norway. This study shows how an absolute geographic Gini index can be used for this purpose, and explores the changes in inequality between Norwegian counties over the past 35 years. Absolute inequality has decreased both for age-specific mortality and life expectancy at birth. Even with relatively low levels of geographic inequality, it may take many years for some counties to catch up with the best performing counties, if the yearly rates of improvement remain unchanged.Additional file 1:Trendlines for risks of death in each county for selected age ranges, 1980–2014. (PDF 538 kb)Additional file 2:Absolute Gini indices for life expectancy at birth and risks of death at ages 0–4, 5–49 and 50–69, 1980–2014. (PDF 330 kb)


Macromolecular crowding ought to stabilize folded forms of proteins, through an excluded volume effect. This explanation has been questioned and observed effects attributed to weak interactions with other cell components. Here we show conclusively that protein stability is affected by volume exclusion and that the effect is more pronounced when the crowder's size is closer to that of the protein under study. Accurate evaluation of the volume exclusion effect is made possible by the choice of yeast frataxin, a protein that undergoes cold denaturation above zero degrees, because the unfolded form at low temperature is more expanded than the corresponding one at high temperature. To achieve optimum sensitivity to changes in stability we introduce an empirical parameter derived from the stability curve. The large effect of PEG 20 on cold denaturation can be explained by a change in water activity, according to Privalov's interpretation of cold denaturation. Crowding effects—important when considering cellular environments—greatly influence protein stability. Here the authors study the impact of macromolecular crowders on high and low temperature protein unfolding, and show that volume exclusion effects are larger when the protein and crowder volumes are similar. Escherichia coli contains several macromolecules with a global concentration from 0.20 to 0.40 g ml−1 corresponding to up to 40% of a cell's volumeThe stability of proteins is generally assessed in dilute solutions that, in addition to the molecule under study, contain only few other molecular species, such as those of a buffer, whose interaction with the protein is diffusion limited. However, cellular fluids contain high concentrations of macromolecules. For instance, it has been claimed that the cytoplasm of in vivo257810119144The influence of crowding on protein stability has been the subject of heated controversies during the past few years, also in connection with its possible relevance for the function of proteins et al.The relevance of the relative dimensions of crowder and protein was only preliminarily analysed by MintonTm). This parameter is not very sensitive and, even worse, it can be completely misleading (vide infra). Here, we addressed these compelling questions by undertaking the first detailed study of the influence of crowding on the stability of yeast frataxin (Yfh1), a natural protein that undergoes cold denaturation above water freezinget al.Cp) and hence of the whole stability curve of the proteinTc) is well below water freezing, Yfh1 is an ideal system for the characterization of the cold denatured species and of the factors influencing protein thermal stability, offering the chance of using it as a tool to study protein stability in a variety of environments2122Another crucial point that makes so difficult to assess the influence of crowders on protein stability is the choice of the parameter to measure stability. It is common practice to use only the midpoint of the thermal unfolding curve . The influence of each of these crowders was studied over a range of concentrations consistent with the composition of cellular fluids. We developed an Overall, our results indicate that the most effective crowders are those whose molecular weight is closest to that of the protein under study. Careful spectroscopic investigations show that all of the synthetic crowders employed have negligible interactions with Yfh1, excluding a sizeable contribution of weak nonspecific interactions. The asymmetry of the influence of crowders on high and low temperature unfolding is consistent with a more expanded nature of the low temperature unfolded state22et al.The crowders were chosen, among other reasons, to represent a range of different sizes, starting from one comparable to that of the protein up to much larger ones. The radius of gyration of the folded state of Yfh1 has been previously estimated at 2.1 nm, typical for a globular protein of 13.8 kDa molecular weightet al.It is important to bear in mind that relative dimensions of the protein under study and crowders are not related to their molecular weights in a simple, linear way. It has been shown that synthetic crowders may have distinctly different volumes at different concentrations. In the case of several PEG polymers, Kozer The influence of PEG 20, dextran 40, Ficoll 70 and Ficoll 400 at concentrations ranging from 0 up to 20% (w/v) was investigated in the temperature range 2–70 °C.Tm) and a more pronounced decrease of the cold denaturation temperatures (Tc). Altogether, these data suggest a substantial increase in the thermal stability of Yfh1 induced by increasing concentrations of the crowders, with significant differences, particularly as far as the concentration dependence is concerned.To assess the secondary structure content of Yfh1 in the presence of different crowders at varying concentrations, and thus measure the relative populations of folded and unfolded species, we recorded thermograms, that is, thermal denaturation curves obtained from solutions of 10 μM Yfh1, by following the circular dichroism (CD) intensity at 220 nm as a function of temperature in the temperature range 2 to 70 °C and in the presence of varying concentrations of crowders , is a function of ΔG0(T), the Gibbs free energy for unfolding, that is shown as modified Gibbs–Helmholtz in With the assumption of a two-state equilibrium between folded and unfolded forms, the population of the folded species at any temperature, Tm is similar for all crowders, 12 K for PEG 20, 13 K for dextran 40 and 10 K for both Ficolls, whereas the corresponding decrease of the values of Tc ranges from ca. 44 K for PEG 20, 18 K for dextran 40 to 17 K for both Ficolls. Thus, if we assume that changes in Tc and Tm may be taken as bona fide assessments of thermal stability, for the same crowder, we must conclude that we observe a marked increase in stability at low temperature but a smaller increase in stability at high temperature.Fitting of the CD thermal unfolding curves yields stability curves . The mosA possible explanation for this asymmetric effect of crowders is the presence of weak interactions of synthetic crowders with Yfh1. To minimize this risk, we had deliberately excluded proteins as crowders since they seem to be particularly prone to interactions1430313215N heteronuclear single-quantum correlated (HSQC)spectra of Yfh1 in 10 mM HEPES and in 5% w/v solutions of representative crowders since NMR is more sensitive than CD to even slight changes of the chemical environment and can thus also reveal local weak interactions . Far-ultraviolet CD spectra of Yfh1 were recorded at 25 °C in the absence and presence of 5% w/v of each crowder, that is, a concentration that should have minimal volume exclusion effect. The spectra are very similar, although with some sign of increased secondary structure content for the solutions containing either PEG 20 or dextran 40 . The incractions . As a ruvide infra). The choice of showing spectra at 5% crowder concentration is based on the fact that, in doing so, we can disentangle effects arising from soft interactions and viscosity from those arising from volume exclusion effects, because it can be assumed that a 5% concentration is not sufficient to cause volume exclusion effects, whereas higher concentrations of crowders can strongly decrease the quality of NMR spectra owing to the increase of viscosity. Nonetheless, after observing larger effects for PEG 20, we extended the NMR study in the range 5 to 20% PEG 20. The spectra up to 10% concentration are of good quality, although with incipient broadening, due to the considerable increase in viscosity.Only few peaks show minor (<0.03 p.p.m. in the proton dimension) CSP, therefore ruling out the possibility that the crowders used here significantly alter the architecture of Yfh1. The only apparent exception is that it is possible to observe, in the centre of the spectra, a decrease in the intensity of some (sharp) peaks, typical of the unfolded species of Yfh1. Their intensity is so low that the corresponding peaks of Yfh1 in the buffer solution become dominant, bursting out from the background. This may reflect a small shift of the equilibrium in favour of the folded species that can already be observed at the concentration of 5% w/v, presumably as the precocious manifestation of a volume exclusion effect . It is common place to assume that the variation of melting temperature (ΔTm) can be used as a faithful measure of stability, implying proportionality with the change in thermodynamic stability (ΔΔG). Becktel and SchellmanTS), it is possible to relate the change in Tm to the value of the perturbation free energy at the melting temperature. The relationship between changes in melting temperature and changes in free energy is linear only if we can make the assumption that the portions of the two stability curves chosen to measure the temperature increment (ΔT) behave as parallel straight lines (T). In actual cases this is often not true: a common cause of failure stems from the simultaneous change of the enthalpy of folding and of heat content (ΔCp). Any change in curvature (−ΔCp/T) will make the two portions not parallel. For instance, as shown by McCrary et al.et al.Tm with a concomitant lowering of ΔGS, the free energy difference at the temperature of maximum stability (TS). It is also possible to have an increase in ΔGS, with little increase or even a decrease in Tm. The comparison of the stability curve of lysozyme in buffer alone and in buffer with the 5% v/v addition of propanol, calculated from the thermodynamic parameters of Velicelebi and SturtevantTm upon addition of propanol is not paralleled by a decrease in ΔGS (thermodynamic stability). On the contrary, there is a large increase of ΔGS. These contradictory results prompted us to look for a single parameter to express stability.Protein stability is difficult to define using a single parameter, to the point that, in extreme cases, it may appear as an elusive concept. As pointed out by Pucci ht lines  crossingTc and Tm.We propose the use of a single parameter to gauge thermal stability: the area of the stability curve between the two unfolding temperatures. To compare areas we can take the integral of the equation of the stability curve (1) and calculate its value between The integral of It is easy to grasp that this parameter is an assessment of the global thermodynamic stability within the entire temperature range of protein stability, rather than at a single temperature.Hm is paralleled by an increase of both Tm (of 11 K) and of the integral of the stability curve (I), as shown in In the case of the comparison between a protein from a mesophilic organism (Metmyoglobin) with a thermophilic one (Sac7d)Tm and I, point to greater thermal stability. In the case of the addition of propanol to the lysozyme solution the small decrease of Tm is accompanied by an increase in ΔGS of 30% but also by a much larger increase in the integral. It is fair to say that the large increase of I hints at a great thermal stabilization in spite of the small decrease of Tm. I) between Tc and Tm for the examples of Thus, we have no paradox: two parameters, et al.Tm and ΔGu values for some hyperthermostable proteins. The correlation is approximately linear, but only if values for mesophilic counterparts are included in the same graph. If one plots only values for hyperthermostable proteins the graph looks like that of I/Io) correlate fairly well with Tm and all values fall to an exponential curve.An interesting application of the new parameter is shown in We have also used the new parameter to analyse our data on the stabilizing influence of alcoholsTm) and low transition points (Tc) but also gives us new insights into the influence of different crowders. The comparison of the changes of Tm  to detect changes in stability. Depending on the combination of thermodynamic parameters, it may happen that the value of Tm increases very little or even decreases, whereas thermodynamic stability increases2124Several authors have remarked that experimental findings on the stabilization induced by crowders appear far smaller than those predicted by theoretical calculations2A very important result emerging from our study is that the influence of crowders on high and low temperature unfolding is dramatically different. The possibility of observing cold denaturation reveals that low temperature unfolding is much more sensitive to the presence of macromolecules that affect volume availability. The most likely interpretation of this marked asymmetry is that the volume of the low temperature unfolded species is larger than that at high temperature. This interpretation is consistent with our previous observations on the difference between the two unfolded states22The influence of PEG 20 on cold denaturation of Yfh1 is so larger than those of the other crowders that it cannot be attributed solely to the larger volume of the low temperature unfolded species. Having ruled out a direct interaction of PEG 20 with our protein, the most likely cause of the marked asymmetry observed between low and high temperature behaviour of Yfh1 in the presence of PEG must be sought in the indirect effect on water activity. The ability of PEG polymers to lower water activity is routinely exploited in protein crystallization39Cp accurately from calorimetric measurements. Thus, we can take advantage of the full stability curve and of the related integral parameter. It remains to be seen whether the low temperature unfolded species is more expanded than the high temperature one in all proteins. It is quite possible that the volume change upon unfolding, both at high and low temperature, can vary widely among different proteinsIt is fair to ask whether the behaviour of Yfh1 is idiosyncratic. How general are our findings? Is the dependence of stability on crowder's relative size a unique property of this protein or a general property of many if not all proteins? Can we extrapolate it to other proteins? The answer is yes, with the proviso that it is generally difficult to observe the influence on low temperature unfolded species directly, simply because cold denaturation transition occurs, for most proteins, at too low a temperature to allow direct observation. However, although it is not generally possible to observe the low temperature transition, it is often possible to measure ΔIt is important to emphasize that the analysis of the influence of crowders on Yfh1 was greatly facilitated by gauging stability with the aid of the integral of the stability curve. We can thus conclude that our study sets a clear answer on the question of the influence of volume exclusion, also in relation with the optimal size of a crowder and demonstrates that the use of integrals of the stability curve is a very useful addition to the parameters commonly employed to describe the thermal stability of proteins.E. coli and purified as described by He et al.Yfh1 was expressed in All crowding agents, PEG 20, dextran 40, Ficoll 70 and Ficoll 400 were purchased from Sigma Aldrich. Concentrated solutions to be used as mother solutions to prepare samples at different crowder concentrations were examined for their potential influence on the final ionic strength and pH.E. coli cells and selected for transformation in ampicillin plates, according to previously published protocolsg for 15 min at room temperature. The pellet was then resuspended in 100 ml of 15N enriched minimal medium, incubated in a rotary shaker at 37 °C for 10 min and induced with isopropylthiogalactoside (0.5 mM).Yfh1, cloned in pET21a vectors, was transformed in BL21(DE3) 2O, before transferring it into standard 5 mm NMR tubes using a Pasteur pipette and used immediately for NMR experiments.The final pellet was resuspended in 500 ml of M9 medium mixed with 50 ml of Det al.NMR spectra were recorded on a Bruker AVANCE operating at 600 MHz 1H frequency. Measurements were carried out using a 15 N uniformly labelled protein at a concentration of 70 μM in a 20 mM HEPES buffer at pH 7.0 and in the same buffer supplemented with 5% of PEG 20, dextran 40, Ficoll 70 or Ficoll 400. Water suppression was achieved by the WATERGATE pulse-sequence−1 in the temperature range 2–70 °C.Samples were prepared using a Yfh1 concentration of 10 μM in a 10 mM HEPES buffer at pH 7.0 and with varying concentrations of PEG 20, dextran 40, Ficoll 70 or Ficoll 400. Baseline correction was obtained by subtraction of the appropriate buffer spectrum. Thermal unfolding curves were obtained by monitoring the ellipticity at 220 nm using a Jasco J-815 CD spectropolarimeter equipped with a Jasco CDF-4265/15 Peltier unit. The 1 mm path length cells (Hellma) were used and a heating rate of 2 °C minG, the difference of free energy between the folded and unfolded species, as calculated from a modified Gibbs–Helmholtz Tm), as a function of temperature. For most proteins, fitting thermograms based on spectroscopic measurements, for example, CD and NMR spectroscopies, yields accurate vales for Tm and ΔHm. However, these fits are totally insensitive to the value of ΔCp, unless it is possible to observe two unfolding transitions, that is, cold denaturation and heat denaturation. The occurrence of cold denaturation is obvious from the stability curve of any proteinG, as the temperature varies from room temperature in either direction, that is, to higher or lower temperatures. All thermodynamic parameters can then be extracted from a plot of ΔG against temperature, by means of an original fitting programCp and other parameters are modest because the temperature dependence of ΔG is quite sensitive to the value of ΔCp .To have an accurate insight into protein stability, it is desirable to be able to measure its whole stability curve. This curve is the plot of ΔThe data that support the findings of this study are available from the corresponding author on reasonable request.How to cite this article: Alfano, C. et al. An optimized strategy to measure protein stability highlights differences between cold and hot unfolded states. Nat. Commun.8, 15428 doi: 10.1038/ncomms15428 (2017).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Supplementary figures and supplementary references.


A numerical model based on the smoothed particle hydrodynamics method is developed to simulate depth‐limited turbulent open channel flows over hydraulically rough beds. The 2D Lagrangian form of the Navier–Stokes equations is solved, in which a drag‐based formulation is used based on an effective roughness zone near the bed to account for the roughness effect of bed spheres and an improved sub‐particle‐scale model is applied to account for the effect of turbulence. The sub‐particle‐scale model is constructed based on the mixing‐length assumption rather than the standard Smagorinsky approach to compute the eddy‐viscosity. A robust in/out‐flow boundary technique is also proposed to achieve stable uniform flow conditions at the inlet and outlet boundaries where the flow characteristics are unknown. The model is applied to simulate uniform open channel flows over a rough bed composed of regular spheres and validated by experimental velocity data. To investigate the influence of the bed roughness on different flow conditions, data from 12 experimental tests with different bed slopes and uniform water depths are simulated, and a good agreement has been observed between the model and experimental results of the streamwise velocity and turbulent shear stress. This shows that both the roughness effect and flow turbulence should be addressed in order to simulate the correct mechanisms of turbulent flow over a rough bed boundary and that the presented smoothed particle hydrodynamics model accomplishes this successfully. © 2016 The Authors International Journal for Numerical Methods in Fluids Published by John Wiley & Sons Ltd A simple, economical and practical approach to evaluate tν is using a mixing‐length model that is known as the zero‐equation model. In this approach, the eddy‐viscosity is related to the mean strain rate from Prandtl's theory by using a turbulence characteristic length ml as follows:U is the mean streamwise velocity and ml is the mixing length. Although the mixing‐length model is easy to use, it lacks the universality and is not applicable to complicated flows (e.g. 3D non‐uniform flows with disturbed free surface) where the distribution of turbulence length scale ml is not known. A well‐known turbulence model that is commonly used for such complicated flows is the two‐equation k–ε model where a wall function technique is usually used to estimate the flow in the shear boundary layer. Although this model has the advantage of including the effect of flow history and transport on the turbulence, it meets difficulties in treating rough wall boundaries because the near‐bed logarithmic law does not hold anymore when large roughness elements exist. This has also been investigated by Nikora et al.Because all natural river flows around the world are turbulent and the channel beds are often composed of large‐scale, potentially mobile, rough elements such as sand and gravel particles, the study of turbulent open channel flows over rough beds is of significant engineering interest. This interest has motivated researchers to carry out various studies to explore the flow behaviour near the solid–fluid interface in laboratory experiments or to simulate the effect of bed roughness on the flow by numerical methods. The solution of fundamental hydrodynamic equations has become a popular numerical technique in modelling turbulent flows because it can provide time‐dependent details of the flow characteristics such as velocities, pressures and transport properties. In turbulence modelling of open channel flows, the Reynolds‐averaged Navier–Stokes (N‐S) equations or space‐filtered large eddy simulation (LES) equations have been widely used, where the large eddies are resolved and the small ones are modelled by an appropriate model, usually the eddy‐viscosity model. The eddy‐viscosity model relates the turbulent shear stress to the local strain rate through an eddy‐viscosity follows:νt=lm2dUdet al.et al.et al.Different approaches have been adopted to account for the roughness effect in numerical modelling of turbulent flow over rough walls. Some have been developed based on modifying the turbulence model near the rough boundary; while in some others, separate models have been used, for example, the roughness effect being formulated on the basis of a drag force equation. Van Driest et al.et al.et al.et al.et al.et al.Recently mesh‐free particle methods, for example, smoothed particle hydrodynamics (SPH), have been used in fluid flows because of their advantages in dealing with the large deformation of free surfaces and solid–fluid interfaces. SPH can model flows by tracking each individual particle without numerical diffusion and has been used in various applications such as wave breaking, fluid impact and flow‐structure interactions. As examples of hydraulic engineering‐related SPH studies, Gotoh et al.et al.However, very few cases have involved open channel flows, although some pioneering works in this field have been reported such as by Federico et al.et al.sC = 0.15 in their MPS simulations of open channel flows. Sahebari et al.sC = 0.12 and simulated hydraulic jumps over corrugated beds by using SPH. They treated the wall boundaries of smooth, triangular, trapezoidal and sinusoidal shapes by using lines of the particles and applied a repulsive force similar to that of Lopez et al.et al.et al.As for the shear boundary treatment, Violeau and Issa et al.et al.et al.Re number (τRe) of 1000 using SPH with the unified semi‐analytical wall boundary condition and an eddy‐viscosity model with the Smagorinsky constant sC = 0.065 for the unresolved part of the turbulence. In contrast to the DNS, the result of the LES was very poor. In order to investigate the insufficiency of their LES, they considered a Taylor–Green vortex case and stated that the failure was traced back to the SPH collocated discretization effect on the pressure–velocity interactions. Finally, they concluded the LES of a channel flow is still not possible with the present SPH formulation because of the problems inherent in the standard SPH discretizations.In two recent studies, Mayrhofer et al.et al.In grid‐based LES, a variable resolution is usually adopted so as to use a much finer mesh near the wall boundary in order to resolve the near‐wall flow scales, while in SPH, a non‐variable homogenous discretization has to be used. Hence, a wall function is usually applied, such as in the studies of Violeau and Issa et al.et al.et al.In a most recent study in this area, Kazemi 22.1t is the time, ρ is the fluid density, u is the velocity, P is the pressure, g is the gravitational acceleration, ν0 is the kinematic viscosity coefficient, τt is the turbulence stress tensor and τd is the form drag‐induced shear stress from the rough bed.The governing equations are the two‐dimensional continuity and momentum equations in the Lagrangian framework. An additional term to represent the form drag of the bed particles is included. This term as well as the turbulent shear term is not needed in a DNS. The final equation readsi and j denote the 2D coordinate components, ijτ is the component of shear stress tensor τt, ijS is the component of strain tensor S calculated by Eq. tν is the turbulence eddy‐viscosity, k is the turbulence kinetic energy calculated by Eq. ijδ is the Kronecker delta function.x and u are the position and velocity components, respectively. In SPH, the turbulence eddy‐viscosity tν is usually estimated by the Smagorinsky model et al.sC is the Smagorinsky constant, usually taken to be between 0.1 and 0.15, Δ is the characteristic length scale of eddies (filter width), which is taken as the particle spacing, and S=S:ST is the local strain rate. It should be noted that the turbulence has a three‐dimensional nature, and in particular for the spatially averaged LES‐based modelling and consideration of SPS turbulence closure, the three‐dimensional characteristics of turbulence should play an important role. However, in the present simulations of open channel uniform flow, the flow is dominated by the streamwise shear stress and vertical 2D momentum exchange, while the lateral influence is quite small so as to be reasonably neglected in this study.To model the turbulence stress, an SPS model based on the eddy‐viscosity assumption sC = 0.15 was not able to reproduce the correct shear mechanism in a uniform open channel flow over a rough wall. Also, in the study of Mayrhofer et al.sC = 0.065 in the SPH‐LES showed very poor results with an overestimation in the streamwise velocity. They pointed out that the failure was related to the pressure–velocity interactions of vortices and concluded that this problem is inherent in the standard SPH discretization.Equation sC = 0.15 to investigate this issue. The results are presented in Section sC being around 0.15. Accordingly, the eddy‐viscosity is formulated as follows:ml is calculated by the Nezu and Rodi H is the water depth, k is the von‐Karman constant and ξ = z/H is defined in which z is the vertical coordinate, and Π is the Coles parameter. Π has been introduced to describe the deviation from the log law in the outer region. This parameter comes from an empirical wake function added to the velocity log law by Coles k and/or the integration constant , can be estimated by the following integral interpolant equation as:r is the particle position, r′ denotes the particle coordinate, h is the smoothing length and W is the weighting or kernel function. The aforementioned equation can be expressed in the following discretized form to calculate A(r) at the position of particle a:a and b are the reference particle and its neighbour and bm and bρ are the mass and density of neighbouring particle b, respectively. The derivative of A(r) in the jx direction can be approximated by the following:The numerical scheme based on the weakly compressible SPH (WCSPH) method is used to discretize the governing equations. SPH is a Lagrangian particle method that was developed by Gingold and Monaghan uab = ua − ub and rab = ra − rb are defined, ∇aWab is the gradient of the kernel function between particles a and b with respect to the position of particle a and η is a small number used to prevent singularity. In the present WCSPH model, the following Eq. ρ0 is the reference density and c0 is the speed of sound. In a WCSPH numerical scheme, it is assumed that the flow is slightly compressible so the speed of sound should be chosen to be around 10 times of the bulk flow velocity to ensure the fluid compressibility being less than 1%. Finally, ρ0 and c0 are respectively taken as 1000 kg/m3 (water density) and 16 m/s as a common practice in the computations. Since the WCSPH is known to result in considerable numerical noise in the pressure field, a special treatment  could be undertaken to improve the performance. Therefore, the present WCSPH simulations have been performed using a Shepard density filter to minimize the pressure noise at every 30 computational time steps. The solution method using a predictor–corrector scheme By using the aforementioned SPH formulations, the governing equations Eqs  are disc2.3The computational domain boundaries including the free surface, rough boundary and inflow/outflow boundaries are shown in Figure 2.3.1et al.et al.et al.x should be 0 at the boundary line, represented by the following:Recently, some pioneering works have been carried out on the treatment of inflow/outflow boundary conditions in SPH, for example, Federico pd/2 from the boundary line as shown in Figure pd is the SPH particle size. Then the velocity and density of the inner‐fluid particles are averaged over a kernel area onto these points and set as the velocity and density of the corresponding inflow particles  are set equal to those of the inner‐fluid particles near the inflow boundary line. To do so, an averaging point is first defined for each inflow particle at the same elevation but inside the inner‐fluid region, with a distance of et al.At the outflow boundary, the uniform flow condition should also be satisfied to keep the uniformity of the flow through the simulation domain. The same technique used at the inflow boundary can be used for the outflow one. However, a slightly different treatment is adopted at the outlet to reduce the computational time. When an inner‐fluid particle goes across the outflow boundary line, it becomes an outflow particle, and the governing equations are not solved on the particle anymore, but its properties are kept unchanged when it moves through the outflow region. This treatment is similar to that used by Federico To check whether the inflow/outflow boundary condition satisfies the volume conservation or not, we simply calculated the volume flows inside the computational domain at the inlet as well as that flows out of the domain at the outlet boundary at every second of the simulation for several test cases, and we found out the maximum difference between the inlet and outlet volumes is less than 0.5%. This shows the validity of volume conservation on the inflow/outflow boundary condition in the present simulations. However, for a detailed modelling of inflow/outflow boundary conditions, we need to refer to Hosseini and Feng 2.3.2τd/ρ is calculated only for the fluid particles that are located between the numerical bed level and the crest of roughness zone  and is assumed to be variable for different flow conditions as according to experimental observations, the effect of bed roughness on the flow differs for different flow conditions. The numerical bed elevation that defines the base of the roughness zone can be considered as the zero‐velocity plane on which the spatial and temporally averaged flow velocity drops to 0. For this bed boundary, several layers of dummy particles  produces a drag‐induced shear on this particle. This actually exerts a force on the fluid fragment of width sd and height pd  by the following equation:A schematic view of the bed drag force model including the roughness spheres is shown in Figure This function accounts for the shape of the roughness elements that are defined as spheres in the present study to match the roughness elements used in the laboratory study.dC. According to the original work on particle modelling of porous flows using the MPS method by Gotoh and Sakai dC usually lies between 1.0 and 1.5, and thus, a value of 1.0 is simply adopted in the present study. Different values of dC have also been described in the literature for spherical particles. In the experiments of Schmeeckle et al.dWdC acts as the total drag coefficient. By assuming half of the bed grain to be the effective roughness height and dC = 1.0, the average value of dWdC for the particles inside the roughness zone would be equal to 0.785, which is close to the value found by Schmeeckle et al.Another parameter of Eq. 33.1sd is 24 mm, and the channel slope S0 ranges from 0.002 to 0.004. For this application, a rectangular computational domain is chosen with a length of L = 4H, where H is the water depth. Three layers of fixed dummy particles are used for the bottom wall, and three layers of moving particles are used for the inflow as well as outflow regions to satisfy the complete kernel area of the inner‐fluid particles near the boundary lines . The smoothing length is taken to be 1.2pd in the present study. This value has been recommended as the most appropriate SPH smoothing length in many studies as common practice. Because the interfacial boundary layer in the physical model between the bed roughness and the free flow is expected to be quite thin, a kernel function with a narrower influence domain but steeper slope near the central point is expected to be more adequate. Therefore, the cubic spline function of Monaghan and Lattanzio In this section, an SPH model is developed for uniform turbulent open channel flows over a sloping rough bed and validated by the depth and velocity data obtained from Particle Image Velocimetry (PIV) measurement in a laboratory channel with uniform‐sized spheres packed in a hexagonal pattern on the bed s Figure . BecausedR) is assumed to vary depending on the flow conditions. Therefore, six of the test cases  are used to calibrate the model in terms of dR by numerical trials when the computed mean velocity profiles achieved the best fit with the experimental data and then a semi‐empirical fitting function is obtained to establish the relationship between the flow depth and relative roughness height dR/H. Based on this, the additional test cases  are used to validate the model. The calibration tests are selected to cover most of the depth range from 40 to 100 mm and at least two cases of each bed slope. Calibration and validation tests are indicated by letters C and V, respectively, in Table As illustrated in Table dR values, and the mean absolute error (MAE) between the numerical and experimental velocity profiles is calculated for each one; then, the dR value corresponding to the minimum MAE is selected as the thickness of the roughness zone for that test case. After running the model for calibrating tests and finding the best dR with the smallest MAE, the relative roughness height dR/H is plotted against depth H , and Figure t = 70 to 90 s. It shows that the uniform flow condition has been successfully imposed by the proposed inflow/outflow boundary technique. This is also shown in Figure x = 0.25 L, 0.50 L and 0.75 L) are plotted. It is found that the depth‐averaged velocity at these three sections has a maximum difference of 0.5%. Figure t = 35, 50 and 65 s). The maximum difference of the depth‐averaged velocity between these times is 1.96%. This small change in the velocity profile over time also shows the steadiness of the flow. In the present computations, the time to reach the steady state is not exactly the same for all test cases. However, to determine a threshold, it is confirmed that it takes around 20–30 s to achieve the steady flow condition for all 12 cases. The criterion used to define if the flow reaches the steady state is that if the differences of the depth‐averaged value of the space‐averaged (but not instantaneous) velocities at the mid‐section of the channel at different times become less than 2.0%, then the flow is regarded as being steady. The bed drag‐induced shear term removes a part of the flow momentum, and this effect is transferred to the upper layers of the flow by the turbulence model. As a result, the unbalanced momentum transfer occurs during the first 20–30 s, and then the flow gradually reaches the steady state and all time‐averaged flow parameters, for example, velocity and shear stress remain unchanged. In the inflow and outflow boundaries, the flow characteristics are assumed to be unknown rather than being given prescribed values of the pressure and velocity. Therefore, the proposed SPH inflow boundary model is more general in that it does not need experimentally measured or analytically prescribed flow data at the inflow boundary and can thus be applied to more complex flow situations. In Federico et al.Figure dR are presented in Figure z is vertical coordinate, sk is the Nikuradse roughness size and rB is the logarithmic integration constant that is equal to 8.5 for rough bed uniform flow. We know that as the depth is very shallow and the bed is fully rough, the log law may not be valid. Here, the analytical profiles are used to compare with the numerical results and investigate if the model is able to predict the logarithmic velocity distribution above the roughness zone. The values of dR as well as MAE of velocity profiles of all test cases are presented in Table The numerical results of time‐averaged streamwise velocity profiles obtained by using the best‐fit values of u/∂z) is also of interest, its distribution is presented in Figure u/∂z are presented in Table To determine the error distribution over depth, the MAE is calculated separately in three parts of the depth for each test case, that is, lower 20%, middle 60% and upper 20% of the depth. The purpose of this is to investigate the hypothesis According to Figure 3.3dR values corresponding to the minimum errors, if divided by the water depth, showed a relationship with the depth based on the power function as presented in Figure dR/H) decreases. It is notable that the bed roughness sphere size is fixed in the present study (sd = 24 mm). Therefore, dR/H decreases with a decrease in the ratio of roughness size to water depth (sd/H) and vice versa. In this work, the bed roughness configuration is kept constant to study its effect under different flow conditions. As the depth is not the only parameter affecting the flow condition and the bed slope is also involved, we also explored the relationship between the relative roughness height (dR/H) and the shear velocity u*. The result is shown in Figure During the calibration/validation process Section , the Rd dR/S0H of all cases are plotted against u* in Figure u*), smaller relative roughness height is required to simulate the experimental condition. In other words, as the ratio of bed roughness to water depth (sd/H) becomes smaller, that is, when the flow depth becomes deeper, a weaker bed effect is generated by the proposed drag force model. However, the magnitude of the form‐drag shear term could be larger for the cases with higher u* because the near‐bed flow velocity is faster.It can also be seen that an increase in the shear velocity causes the numerical relative roughness height to become milder for all bed slopes. To provide a single relationship between the relative roughness height and the flow condition, 3.4dτ/ρ) in the effective roughness zone is presented in Figure dτ/ρ is larger for cases with higher u* or Re number. In other words, where the flow depth is deeper and/or the bed slope is steeper, the form‐drag shear term is larger because of the higher velocity. In most tests, the streamwise dτ/ρ increases with depth to some distance above the wall (zero‐velocity plane) and then decreases to the crest of the roughness zone although the velocity increases in this zone. This decrease can be the result of the shape function in Eq. The distribution of the streamwise form‐drag shear term . Here, the eddy‐viscosity is higher when the thickness of the roughness zone (dR) is larger; thus, the shear stress calculated by Eq. dR could affect both the drag force and the turbulence models, and thus, the simulated flows will change. It is also notable that a small change in the mixing length, on the crest of the roughness zone, could have a considerable effect on the eddy‐viscosity (Eq. S|) is at a maximum on this interface.Using a variable sity Eq.  because τ0 is the shear stress at the bed that is estimated by ρgHS0.For six of the 12 test cases shown in Table τ0, and the vertical one is normalized by the flow depth H. As can be seen, the numerical computations underestimate the experimental shear stresses, although they are in a fairly good agreement with the analytical solution. It is notable that the experimental data are taken from a 3D flow over a rough bed surface, which could lead to that they exceed the analytical shear stresses by about 20–30%. Besides, the underestimation of the experimental shear stress by the numerical model is also related to the dimensional differences, as in the present 2D model, the shear stress in the lateral direction is neglected. The width‐wise shear stress is the result of steady streaming in the form of flow circulations in the lateral direction. In spite of this, the 2D SPS model is still able to give satisfactory results in the uniform flow because the effect of the lateral shear stress is very much smaller compared with the streamwise one. Moreover, the close collapses of six SPH data along almost a single line indicate the consistency and convergence of the numerical simulations.To better illustrate the data, the horizontal axis is normalized by sC = 0.15. The result is presented in Figures As mentioned before, the eddy‐viscosity model with a Smagorinsky constant in the range of 0.1–0.15 is not able to estimate the correct amount of turbulent shear stress in a uniform open channel flow over a rough bed boundary. To investigate this issue, here, we repeat the simulations of three test cases S004H50, S003H70 and S002H60 by using the Smagorinsky model with et al.Re of 1000, where an eddy‐viscosity model was used with a Smagorinsky constant sC = 0.065. They pointed out that the correct representation of energy redistribution between Reynolds stress components in an SPH‐LES framework would require 16 times finer resolution than needed in a classic Eulerian LES one. They stated that the most obvious solution is an increase in the resolution but it also highly increases the computational cost. Finally, they concluded that the underperformance of their LES was due to the problems inherent in the standard SPH discretizations related to the pressure–velocity interactions.In contrast to their DNS results with good agreement with the reference data, Mayrhofer Re is even higher than that in the study of Mayrhofer et al.sC.In the present study, the friction Nonetheless, one shortcoming of the proposed turbulence model is that the eddy‐viscosity coefficient is physically defined so it is not dependent on the computational resolution. In other words, if one uses a smaller particle size (higher resolution), the resolved part of the turbulence stress would become higher, but the mixing‐length product that is the representative of the unresolved part would not decrease with the discretization and/or filtering size. Thus, the total turbulent stresses could be overestimated in the cases with higher resolution. Accordingly, the flow velocities would be expected to be underestimated in such a situation. It is promising to note that the present mixing‐length approach works quite effectively with the SPH when relatively coarse particle resolution is used, which is the case in most practical engineering applications.4sC = 0.15 was unable to reproduce the correct shear mechanisms in uniform open channel flows. Therefore, a mixing‐length model has been applied to calculate the turbulent eddy‐viscosity. A drag force model has been developed to account for the bed roughness effect, in which a shape function is introduced to consider the geometry of the bed surface roughness elements. Meanwhile, an efficient inflow/outflow boundary treatment has been proposed and demonstrated to generate a stable flow simulation without the need to use prescribed velocities at the flow inlet.In this study, an SPH model has been developed to simulate the turbulent open channel flows over rough bed boundaries based on the solution of 2D N‐S equations including two additional stress terms to account for the flow turbulence and bed roughness effect. As shown, the standard Smagorinsky‐based SPS model with a fixed dR) is assumed to be flow dependent, such as being related to the flow depth H and the shear velocity on bed u*. The model results showed good agreement with the experimental data as well as the analytical solutions in view of the velocity and shear stress profiles. This confirms that the bed roughness effect has been successfully addressed by the drag force model and the transport of this effect to the upper layers of the flow has been correctly reproduced by the proposed turbulent mixing‐length approach. Because the governing equations, as well as the computational domain, are discretized at a macroscopic scale in the roughness zone, the physical dispersion in the vertical direction is disregarded. Thus, the flow shear is dominantly driven in the streamwise direction but transported vertically by the turbulence closure. The computed streamwise velocity and shear stress profiles suggested that this assumption has not caused substantial errors for the 12 flow test cases and the macro flow behaviours have been well reproduced. This is due to the turbulence model correctly modelling the shear transfer from the roughness layer to the free flow.Twelve test cases of different bed slopes and water depths have been simulated to investigate the effect of bed roughness under various flow conditions. A roughness zone is defined near the rough bed boundary where a form‐induced drag shear term is applied on the SPH particles. The thickness of this zone (Whether the inaccuracy of the SPH‐LES approach in wall‐bounded channel flows is related to the pressure–velocity interactions (as addressed by


A fracture strength model applied at room temperature for embedded elliptical crack in brittle solid was obtained. With further research on the effects of various physical mechanisms on material strength, a thermo-damage strength model for ultra-high-temperature ceramics was applied to each temperature phase. Fracture strength of TiC and the changing trends with elliptical crack shape variations under different temperatures were studied. The study showed that under low temperature, the strength is sensitive to the crack shape variation; as the temperature increases, the sensitivities become smaller. The size of ellipse’s minor axes has great effect on the material strength when the ratio of ellipse’s minor and major axes is lower than 0.5, even under relatively high temperatures. The effect of the minor axes of added particle on material properties thus should be considered under this condition. As the crack area is set, the fracture strength decreases firstly and then increases with the increase of ratio of ellipse’s minor and major axes, and the turning point is 0.5. It suggests that for the added particles the ratio of ellipse’s minor and major axes should not be 0.5. All conclusions significantly coincided with the results obtained by using the finite element software ABAQUS. Ultra-high temperature ceramics (UHTCs) such as the transition metal borides and carbides have melting points higher than 3000 °C, which can be used in the high temperature and oxidizing environments and have good chemical and physical stabilities. These materials have been developed for the leading edge and nose cap materials in hypersonic vehicles. The addition of SiC particles or sheets has been proved to be the most promising way for the improvement of the oxidation resistance and mechanical performances of UHTCs ,2,3,4. Tet al. [et al. [The study on embedded crack, particularly on embedded elliptical crack, has always been important in linear elastic fracture mechanics. In recent decades, mechanical workers have attempted to develop applicable theories for embedded elliptical crack in brittle solid, initially establishing the widely accepted theoretical frameworks under normal or slightly higher temperatures. Green and Sneddon  analyzedet al.  evaluate [et al.  proposedHowever, the above methods do not consider the effect of temperature. Mechanical performances and fracture mechanisms under high temperatures differ significantly from those under low temperatures. Thus, many fracture theories are not applicable at high temperatures. Although several such fracture theories for high temperatures have achieved some progress ,11,12, fIn this work, a fracture strength theoretical model was applied to an embedded elliptical crack in brittle solid at room temperature. The model was based on the stress intensity factor model proposed by Irwin and the fracture criterion proposed by Griffith. Through further research on effects of various physical mechanisms on material strength, a thermo-damage strength theoretical model for UHTCs was established and applied to each temperature phase. Fracture strength of TiC and the changing trends with the elliptical crack shape variations under different temperatures were studied and then compared with the results obtained by using the finite element software ABAQUS .KI is the stress intensity factor of an embedded elliptical crack; E(k) is an elliptical integral of the second kind; σ is the uniform tensile stress; a and b are one-half of the ellipse’s major and minor axes, respectively; φ is the polar angle from the ellipse center; α is the integral angle; k is a function of a and b. Floor plans of the embedded elliptical crack are as shown in The following is the stress intensity factor model for an embedded elliptical crack in the elastic matrix subjected to uniform tensile stress as proposed by Irwin : (1)KI=σKmax is removed. Then, letting KIC is the fracture toughness of materials; E0 is the Young’s modulus; ν is the Poisson’s ratio; γ is the fracture surface energy; φm is the polar angle of the point where the stress intensity factor is largest from the ellipse center.The Griffith fracture criterion states that the fracture of brittle materials initiates at the crack edge with the largest stress intensity factor. When this largest stress intensity factor is equal to the fracture toughness of materials, the fracture of materials occurs. In this work, the stress intensity factor is predicted by using Equation (1), and the largest stress intensity factor th(T) is the temperature dependent fracture strength of materials; E(T) is the temperature dependent Young’s modulus; Cp(T) is the specific heat capacity for constant p pressure and temperature T; Tm is the melting point; ∆HM is the latent heat of melting.In our previous work , based ol corresponds to the damage term.As is well known, fracture is a very complex process that involves the combined effects of temperature and microstructures. There has been no one set theory “set in stone” to handle all of the factors in fracture, especially at high temperatures. The studies have shown that under low temperatures the fracture of brittle materials is sensitive to the single crack size, while as the temperature increases the sensitivity decreases. It can be observed that for the fracture mechanisms of UHTCs under different temperatures the temperature and damage should be first concerned problems. Based on the above temperature dependent fracture strength model, we developed a simple method to consider both the effects of temperature and damage on the material strength, which can be expressed as follows : (5)σth. The biggest stresses σ  on the crack edge under different temperatures were calculated by ABAQUS. Using the biggest stress σ , the trend of the largest stress intensity factor on the crack edge with b/a ratio changes can be determined. Then the trend of the fracture strength with changes in b/a ratio can be obtained. This is because the trend of the fracture strength with crack shape variation is just completely opposite to the trend of the biggest stress intensity factor with crack shape variation.In this work, the finite element software ABAQUS was used to verify the above trends. We consider the model to be an infinite elastic matrix with an embedded elliptical crack. According to the Saint-Venant’s principle, the ratio of the largest half of the ellipse’s major axis and the side of the matrix paralleling to the major axis was set to 1:20. The geometry sizes were set to 400 × 400 × 400 (thickness × width× length). The b increases. The size of b has great effect on the material strength when the b/a ratio is lower than 0.5, even under relatively high temperatures. While the current research usually did not consider the effect of the minor axes of added particle on material properties, which just considered the effect of major axes [b/a ratio is higher than 0.5, the change of b has little effect on the material strength, even under low temperatures. Under this condition, the effect of the minor axes of added particle can be negligible. It also can be seen form b when b/a ratio is lower than 0.5. As the temperature increases, the sensitivity of material strength to the size of b decreases. Under super high temperatures, the effects of the size of b on material strength compared to temperature are negligible. Comparing jor axes . If undea increases. The relationship between the fracture strength of materials and the size of a is nearly linear. It can be observed that whatever the value of b/a is the size of a is always the main control mechanism of material strength; even under the relatively high temperatures (about 2000 °C). It suggests that when preparing the UHTCs reducing the major axes of added elliptical particle should be firstly considered. This coincides with the experimental results reported by Watts et al. [a. While under high temperatures; the sensitivity is little; and the effects of the size of a on fracture strength of materials compared to temperature are negligible. Comparing s et al. . Moreoveb/a ratio as the crack area is set. It can be observed that fracture strength decreases as temperature increases. The strength of material initially decreases with the increase of b/a ratio. The strength of material reaches its lowest point when the b/a ratio is 0.5, coinciding with the literature stating that on the same crack edge the stress intensity factor of the crack is largest when the b/a ratio is 0.5. However, when the b/a ratio increases from 0.5, the fracture strength starts increasing. Therefore, the fracture strength appears to have two maxima, when the b/a ratios are very small and equal to 1. It suggests that for the added particles the ratio of ellipse’s minor and major axes should not equal to 0.5. b/a ratio lower than 0.5. However, when the b/a ratio is higher than 0.5, the trend of material strength with shape variation is more relaxed. The material strength is not sensitive to the b/a ratio under super high temperatures. Comparing a increases, the fracture strength of materials decreases lineally. When preparing the UHTCs reducing the major axes of added elliptical particle should be firstly considered. The size of b also has great effect on the material strength when the b/a ratio is lower than 0.5, even under relatively high temperatures. The effect of the minor axes of added particle on material properties thus should be considered under this condition. As the crack area is set, the fracture strength decreases firstly and then increases with the increase of b/a ratio, and the turning point is 0.5. It suggests that for the added particles the b/a ratio should not be equal to 0.5. This study will provide a theoretical basis and guidance on the design and preparation of the UHTCs.In this work, a fracture strength theoretical model applied to room temperature for embedded elliptical crack in brittle solid was obtained. The model is based on the stress intensity factor model for embedded elliptical crack proposed by Irwin and the fracture criterion proposed by Griffith. Further research on the effects of various physical mechanisms on material strength established a thermo-damage strength theoretical model for UHTCs applied to each temperature phase. This model was verified by comparison with results obtained by using the finite element software ABAQUS. The study showed that under low temperature, the strength is sensitive to the crack shape variation; as the temperature increases, the sensitivity decreases; under super high temperatures, the effects of crack shape on fracture strength of materials compared to temperature are negligible. As the size of


We develop a theory for transit times and mean ages for nonautonomous compartmental systems. Using the McKendrick–von Förster equation, we show that the mean ages of mass in a compartmental system satisfy a linear nonautonomous ordinary differential equation that is exponentially stable. We then define a nonautonomous version of transit time as the mean age of mass leaving the compartmental system at a particular time and show that our nonautonomous theory generalises the autonomous case. We apply these results to study a nine-dimensional nonautonomous compartmental system modeling the terrestrial carbon cycle, which is a modification of the Carnegie–Ames–Stanford approach model, and we demonstrate that the nonautonomous versions of transit time and mean age differ significantly from the autonomous quantities when calculated for that model. Compartment models play an important role in the modeling of many biological systems ranging from pharmacokinetics to ecology Anderson . Key valWe are motivated by an interest in studying the dynamics of the terrestrial carbon cycle which is typically modeled as a number of discrete pools of carbon in plant biomass, litter and soil organic matter. Many of the best studied models of the dynamics of carbon are linear, which reflects the fact that changes in carbon pools are proportional to the pool size  of carbon inputs and losses from the system change over time as a function of changes in climate, such as temperature. For example, increases in temperature are normally assumed to increase the rates of soil decomposition  model  Anderson  and withB have negative real parts . This means that  Consider the two compartmental systemsr, or by considering the fact that particles can only leave from pool 1 in  Let nonautonomous compartmental system with d pools is given by an inhomogeneous linear nonautonomous differential equationB(t) satisfy(transition operator of the corresponding homogeneous equation Let Explicitly solvable nonautonomous two-pool model) The nonautonomous compartmental system(ii)(iii)Then the linear system  Consider the linear nonautonomous differential equationI is bounded below, and consider the linear systemsAssume first that rator of , p. 540 rator of  is boundestimate . In casePötzsche , p. 540 forward attracting, and in case the interval I is unbounded below, then there also exists a unique pullback attracting solutionThe estimate  for the s system , for whis system  holds. Td Wanner . This sofined in  and \docasmussen , Rasmussasmussen  for an imean age system. We derive this result from the evolution of age distributions, given by the well-known McKendrick–von Förster equation  Consider the nonautonomous compartmental system  with a fBy using d-dimensional ordinary differential equation of skew product type, i.e. the x-equation does not depend on x. Note that (Combining the Eqs.  and 15)15) yieldWe will show now that under additional weak assumptions, the mean age equation is exponentially stable.for all Then the mean age system  Consider the nonautonomous compartmental system  with a fmentclass2pt{minime system  is exponA) has the same block decomposition as the matrix B(t), which is described in (We show now that the three conditions (i)–(iii) of Theorem i-th row of the matrix A) equals to x(t) of ) equals to i and j chosen as in (b), will not be considered in the sum in condition (iii) of Theorem Condition (i) of Theorem mentclass2pt{minimmentclass2pt{minimmentclass2pt{minimI is unbounded below. If the interval I is unbounded below, this will be the only bounded solution of the system, i.e. the norm of all other solutions converges to I is bounded below, then all solutions of the nonautonomous compartmental system  Consider the skew product system  Let  Consider the autonomous compartmental system . The trat, and this finished the proof of this proposition.Using Definition R and mean age M in Proposition R and M in the limit R and M.Note that derivation of the autonomous quantities for transit time B(t) and s(t) in (s(t) through carbon dioxide fertilization. They also directly increase mean global temperatures. This increases the carbon loss rates from some of the carbon pools, changing components of B(t), and also has an effect on s(t). Thus increased Here we illustrate predicted changes in the mean age of carbon leaving and remaining in the system for a terrestrial carbon model under a climate change scenario. We consider a modification of the CASA model as used in Buermann et al.  globallyt is years since the year 1850. This represents a plausible time course of atmospheric We simulated changes in atmospheric mentclass2pt{minims(t) to The effect of r et al. . Changesand Wang . The solmentclass2pt{minimmentclass2pt{minimmentclass2pt{minimx(0) has reached the positive equilibrium solution of the resulting system of autonomous equations. The model is then simulated forward from this initial condition using (x(t)).To define the model initial conditions we assume that on using  as the fmentclass2pt{minimM is the mean age of the equilibrium solution at Calculations of the transit time entclass1pt{minima}Mt Fig. . This inB(t) and s(t) translates into non-monotonic effects on Perhaps surprisingly, the monotonic forcing of R and M  case when parameters do not change in time. However, the formulae for the nonautonomous case also highlight the fact that even in the constant parameter case the transit time and mean age do depend on initial conditions; some of the standard formulae do not include this dependence.The difference between a transit time or mean age that is computed based on the parameters at a given instant and the better approach of taking into account the history of the system can be substantial as we illustrate using a variant of the CASA model. Thus, the approach we develop here is not just of mathematical interest but is of substantial practical importance as well.


An on-line interface to the program for quick testing purposes can be found at http://www.susy-ai.org/.A key research question at the Large Hadron Collider is the test of models of new physics. Testing if a particular parameter set of such a model is excluded by LHC data is a challenge: it requires time consuming generation of scattering events, simulation of the detector response, event reconstruction, cross section calculations and analysis code to test against several hundred signal regions defined by the ATLAS and CMS experiments. In the BSM-AI project we approach this challenge with a new idea. A machine learning tool is devised to predict within a fraction of a millisecond if a model is excluded or not directly from the model parameters. A first example is SUSY-AI, trained on the phenomenological supersymmetric standard model (pMSSM). About 300, 000 pMSSM model sets – each tested against 200 signal regions by ATLAS – have been used to train and validate SUSY-AI. The code is currently able to reproduce the ATLAS exclusion regions in 19 dimensions with an accuracy of at least  The ATLAS and CMS experiments at the Large Hadron Collider (LHC) have analyzed the full Run 1 and a small fraction of the Run 2 data set and no evidence of new physics has been found. In particular, there is no trace of supersymmetry (SUSY) in conventional searches.natural SUSY scenarios addressing the emerging little hierarchy problem [Both collaborations have explored intensively the impact of the null results in the context of simplified models –3 as wel problem , 5. Fina problem , 7.The estimation of the number of expected signal events for a fixed point in the SUSY parameter space can take from several minutes to hours in CPU time when full detector simulations with GEANT are performed. In addition, it requires one to model all LHC SUSY searches. Therefore any attempt to use LHC data on models like the pMSSM  with 19 In order to overcome this problem, several tools have been developed in the past few years to recast the LHC results: for simplified models, Fastlim  and SModMachine learning (ML) is becoming a powerful tool for the analysis of complex and large datasets, successfully assisting scientists in numerous fields of science and technology. An example of this is the use of boosted decision trees  in the aIn this work we propose the use of ML methods to explore in depth LHC constraints on the rich SUSY phenomenology. In particular, we investigate the use of classifiers to predict whether a point in the pMSSM parameter space is excluded or not in light of the results of the full set of ATLAS Run 1 data, avoiding time consuming MC simulations. We show that decision tree classifiers like the Random Forest (RF) algorithm perform very well in the pMSSM. Similar results have been obtained for other MSSM realizations such as the natural SUSY model and the cMSSM. The method discussed here allows for a quick analysis of large datasets and can be coupled with recasting tools to resolve the remaining ambiguities by generating more training data. It could also be used in projects aiming at fits of the multidimensional parameter space of the pMSSM or derived models, like e.g. , 25.The paper is structured as follows. In Sect. R-parity conservation is uniquely described by its particle spectrum and the superpotential [SU(2) tensor with m and n. The chiral superfields have the following gauge quantum numbers under the Standard Model (SM) group G:G. Since the origin of supersymmetry breaking is unknown, one approach to addressing this issue is avoiding explicit assumptions as regards a SUSY-breaking mechanism. It is then common to write down the most general supersymmetry breaking terms consistent with the gauge symmetry and the R-parity conservation [The MSSM with otential ,2.1\docuervation ,2.4\docuigned} g:,1,0,\quaThe sfermion mass sector is described by the first and second generation universal squark masses After the application of the electroweak symmetry breaking conditions, the Higgs sector can be fully described by the ratio of the Higgs vacuum expectation values, The final ingredients of our model are the three gaugino masses: the bino mass B decays [Z boson width [The ATLAS study  considerentclass1pt{minimaument}Δρ , the braB decays –57, the }Δ(g-2)μ , 59, andon width  and LEP on width . Furtheron width . Finallyon width , 64 was All models with production cross sections larger than a threshold were further processed. Matched truth level MC event samples with up to one additional parton in the matrix element were generated and efficiency factorsk-nearest neighbors [In terms of ML the problem considered in this paper is a classification problem and there are several methods for addressing it. We will focus on decision tree classifiers and in particular on the random forest classifier , which weighbors  and suppeighbors , amongstIn the following we present an introduction to decision trees and the random forest classifier, aiming at providing a basic understanding of the algorithms. For more complete texts on the subject, the reader is referred to Refs. , 70. A mC, corresponding to the class it belongs to. The procedure starts with training a classifier by presenting parameter sets and the corresponding class labels, in order to learn patterns that the input data follow. Though this basic principle is the same for all classification algorithms, the specific implementation differs depending on the particular problem.In a classification problem the goal is to classify a parameter set (attribute set), A or class B.Decision trees are often used as a method to approach a classification problem. An example of a decision tree is shown in Fig. A decision tree consists of multiple nodes. Every node specifies a test performed on the attribute set arriving at that node. The result of this test determines to which node the attribute set is sent next. In this way, the attribute set moves down the tree. This process is repeated until the final leaf node is reached, i.e. the node with no further nodes connected to it. At the final node no test is performed, but a class label is assigned to the set, specifying its class according to the classifier. The depth of the tree is the maximum number of nodes, as shown in Fig. Because the tree works on the entire parameter space, every test performed in each node can also be interpreted as a cut in this space. By creating a tree with multiple nodes, the parameter space is split into disjunct regions, each having borders defined by the cuts in the root and internal nodes, and a classification defined by a leaf node.overtraining: they have a tendency to learn every single data point as an expression of a true feature of the underlying pattern, yielding decision boundaries with more detailed features than actually present in the underlying pattern. Although this overtraining may cause a better prediction when classifying the training data, such classifiers generally perform poorly on new data sets.One of the drawbacks of decision trees is that they are prone to pruning: training the entire tree, but cutting away all nodes beyond a certain maximum depth. This effectively reduces the amount of details the tree can distinguish in the learned data pattern, since fewer cuts are made in the parameter space. With a maximum depth set to a certain value, classification will not be perfect and mistakes will be made in predictions for the training. Lone data points in sparse regions or individual data points with a classification different from a classification of the data points around it will therefore be learned less efficiently, thus reducing their influence on the trained classifier.A simple, yet crude method to fix this problem is known as C and the total number of trees is taken as a measure of probability the attribute set belongs to class C. The class with the highest score is assigned to the attribute set. This method averages out fluctuations that cause overtraining in individual trees.In the Random Forest algorithm, multiple decision trees are combined into a single classifier, creating an ensemble classifier. Classification of an attribute set is then decided by a majority vote: the ratio of trees that predict class Another method to overcome overtraining is to implement a random attribute selection, meaning that at each node the cuts are applied only to a subset of attributes. In a single decision tree this would introduce a massive error in predictions, but since the random forest is an ensemble classifier, its predictions actually improve.bagging to reduce overtraining even more. In bagging, each decision tree is trained on a random selection of n model points out of N available in the training set. The sampling is done with replacement, meaning a single model point can be selected multiple times. In bagging, n is conventionally chosen to be equal to the total amount of model points available, which means each tree is trained on approximately N [In addition to these two methods, random forests also use  large N . By usingeneralization error. A high generalization error is a possible indicator of overtraining. Normally the check on this error is performed by splitting the available data into training and testing subsets. The training set is used to train the classifier, while the testing set is used to test the predictions of the algorithm. This splitting of the dataset is, however, not the procedure we followed here. Since random forests employ bagging, a single data point is used only in training of a part of the trees. Testing of the algorithm can thus be done by letting all data points be predicted by those trees that did not use a particular data point in their training. As with a test set, one can now obtain a fraction incorrectly predicted data points, and thus estimate the generalization error. This method is called out-of-bag estimation (OOB). The obvious advantage of this procedure is that all the data can be used in training without the need to split the sample into the training and testing sets, hence improving the general prediction quality of the algorithm. It was shown in Ref. [Overtraining of a classifier is difficult to express quantitatively, but it can be tested qualitatively using an independent test set, with which one can estimate the fraction of incorrectly predicted data points for general datasets (i.e. datasets other than the training set). This fraction is called the  in Ref.  that thiThough random forest is in general not very susceptible to overtraining due to the bagging procedure, its performance depends on the number of trees it contains: the more trees, the less overtraining will occur, because predictions are averaged over many individual trees thus reducing undesired fluctuations. The number of decision trees inside the forest is a configuration parameter that has to be set before starting a training, as are the maximum depth of the decision trees and the number of features used in the random attribute selection at each node for example.In this work we used the RF implementation in the scikit-learn Python package  . If the true classification is positive and the prediction by the classifier is positive, then the attribute set is counted as a true positive (TP). If the classifier classifies the set as negative, it is counted as false negative (FN). If, on the contrary, the attribute set is truly negative and it is classified as negative, it is counted as a true negative (TN) and if it is classified as positive, it is counted as a false positive (FP).With this, one can define the true positive rate (TPR) as the ratio of the positives correctly classified and the actual positive data points. The false positive rate (FPR) is the ratio of negatives incorrectly classified and the total truly negative data points.A receiver operating characteristic (ROC) graph is a 2-dimensional plot in which the TPR is plotted on the vertical axis and the FPR is plotted on the horizontal axis.The ROC curve is a 2-dimensional representation of a classifier performance. A common method used to compare classifiers is the area under the ROC curve (AUC) . Its valThe classifier was trained using the data points generated by ATLAS as discussed in Sect. The class labels were generated by the exclusion analysis performed in Ref. . From thIn order to find an optimal configuration of the classifier, we used a grid search as an automated investigation method, which varies the hyperparameters  to find a configuration with a maximal value for a figure of merit, which is the OOB score in our case. The result of this search yielded the following parameters: 900 decision trees, a maximal depth of 30 nodes and a maximum number of features considered at each node of 12. The training was performed including the out-of-bag estimation technique for the creation of an estimate for the generalization accuracy.Figure The desired location of the cut depends on the required properties of the classifier. For example, when one would like to avoid false positives the preferred value should be close to 1.0, while the value close to 0.0 will result in many true positive points being classified as positive for a price of many true negative points wrongly classified as positives. Typically, the neutral choice would be at a point of intersection of red and green histograms, in our example at 0.535. It is assumed that newly added points will follow the same distribution as a function of the classifier output.Another possibility, which we adopt here, is to plot the ratio of the majority class and the total number of points for each bin, as showed in Fig. Using this confidence level (CL) method, one can use the trained classifier to provide both a classification and a measure of confidence in that classification. By demanding a specific CL (and so the probability of the wrong classification) one determines the classifier output, which can be read from Fig. In order to demonstrate how SUSY-AI outperforms a simple decision tree we make a comparison of both methods in Fig. In this section we study the performance of SUSY-AI on a sample that was initially used for its training and on two specific SUSY models. The first one is a natural SUSY model that focuses on only several chosen parameters (with the rest effectively decoupled) of the 19-dimensional pMSSM parameter space, but fulfills almost all the constraints of the original sample. The second one is the constrained MSSM defined by high-scale parameters. It generally contains all particles from the pMSSM spectrum, but with the constraints from dark matter relic abundance and Higgs physics being relaxed. In the last subsection we discuss validation performance on two additional ad hoc models and provide a general discussion of SUSY-AI validation and applicability for models not specified in this paper, which users could try on their own.To validate the performance of SUSY-AI, all possible 2-dimensional projections of the 19-dimensional pMSSM parameter space have been searched for differences between the classification by SUSY-AI and ATLAS. Figures One could expect that misclassified points are primarily located at a border of allowed and excluded regions of the 19-dimensional parameter space. To test this hypothesis, we bin every possible 2-dimensional projections of parameter space and calculate the ratio of allowed points to the total number of points in each bin. We compare the true classification and out-of-bag prediction.The classification has been studied for different cases: including all points, including only points within the 95% CL limit and for points within the 99% CL limit, as shown in the different rows of Figs. incorrect exclusions is at 66%. With the confidence level cut of 95%, corresponding to 70% of the full dataset, the correct classification by the RF is increased to 99.0%. Finally the confidence level cut of 99%, corresponding to 50% of the full dataset, yields a correct classification of 99.7%.Without a confidence level cut, SUSY-AI classifies 93.2% of the data correctly at the working point with the classifier output cut of 0.535. This can be compared with the performance of the simple decision tree in Fig. Z and Higgs boson pole, have a higgsino or wino NLSP for co-annihilation or have colored scalars  or staus as the NLSP candidate. Usually, benchmark points with a wino and higgsino LSP are constrained to masses below 1.5 TeV.One might think that the volume of the 19-dimensional parameter space is so large that the data points become too sparse to make reliable classification possible. Our results show otherwise. The first reason for this is that the size of the parameter space in the pMSSM is inherently reduced by sampling restrictions. Of the Secondly, the experimental constraints on the pMSSM significantly reduce the allowed SUSY parameter space. Only less than 1 out of 100 randomly sampled SUSY parameter points were selected after the constraints applied by ATLAS. The 300, 000 training points, therefore, represent a much larger set of randomly selected parameters. The classifier remains valid nevertheless, since one only needs to sample the part of parameter space where the decision boundary shape will change as a function of a particular feature ‘X’. This happens in the low-energy range therefore justifies the upper cut of <4 TeV. Another relevant issue here is the coverage of the compressed spectrum region where one might expect poor performance. The ATLAS scan, however, covers fairly well compressed spectra and provides training data also in these regions.The final reason is that not all 19 dimensions of the pMSSM are phenomenologically relevant. For example, the production of gluinos and squarks, which is the main search channel at the LHC, depends mainly on the squark masses, the gluino mass, and the electroweakino mass parameters impurity. In our case, we implement the Gini impurity which is given byC is the total number of classes and i in this node. The smaller the Gini impurity, the purer the dataset at the given node. Minimizing this value during training guarantees that model points will be split according to their class label.This can be exposed by investigating features’ importance. Every node in a decision tree is a condition on a single feature and splits the dataset into two parts. The locally optimal condition is chosen based on a measure called j decreases the weighted impurity: the impurity change weighted with the fraction of model points it influences, summed over all nodes making a split on feature j in that tree:After training a tree, it can be computed for that tree how much each feature One can see that a subset of all features have a significantly higher contribution to the final prediction by SUSY-AI. We investigated a reduction of the number of features taken as an input by SUSY-AI and the reduction yielded classifiers with systematically lower quality.N, where N is the number of points in each dimension. The error actually scales with V is the allowed volume. Let us assume From the above discussion one can see that the effective dimensionality of the problem is significantly reduced. With this in mind let us make several remarks about the uncertainty of a decision boundary. Firstly, we note that it does not scale with 1 / In addition, the separation of the excluded and allowed regions close to the decision boundary becomes better defined when applying confidence level cuts that remove model points not classified with a high enough certainty. We note that in the future the probability for correct classification by SUSY-AI will be improved with more training data.In order to further test the performance of the trained classifier, a cross-check has been performed on two models: the 6-dimensional natural SUSY  and the b-physics experiments have been imposed. In summary, our natural SUSY sample fulfills the ATLAS pMSSM constraints, except the b-physics limits.In Ref.  limits warameter , LEP2 coarameter –80 and harameter , i.e. thThe event generation was performed with Pythia 8.185  as well S is the number of signal events, r is only computed for the best expected signal region, in order to avoid exclusions due to downward fluctuations of the experimental data, which is expected considering the large number of signal regions. CheckMATE does not statistically combine signal regions nor combine different analyses. It considers a parameter point to be excluded at 95r defined in Eq.  –94. The mentclass2pt{minimIn this plot, only the data points within the 95entclass1pt{minimaIn the previous subsections, we have shown that SUSY-AI indeed performs very well despite the fact that the training sample is relatively small. However, here we want to discuss the current limitations of SUSY-AI. Some regions of parameter space of the pMSSM-19 are poorly sampled by the ATLAS data since in these corners of the parameter space it is difficult to satisfy all phenomenological constraints. For example, there are only a few parameter points with very light stops since this would require the maximal mixing scenario with a very heavy The lack of data points, but also the improvement on the difference between the true classification and the predicted classification, can be observed in Fig. Figure Although introducing a cut on the confidence level removes data points on which a prediction can be made from a testing sample, both Figs. Another limitation of the current version is that it only uses the combined classification from all searches, without making a distinction which particular analysis excluded a given parameter point. While this may underpower some of the analyses, e.g. electroweak searches, the validation plots in Figs. R-parity violation or the gravitino LSP. Finally, as an additional functionality, SUSY-AI issues and automated warning when a tested point lies outside the limits specified in Table The pMSSM sample used for training of SUSY-AI meets the set of constraints discussed in Sect. A random forest classifier has been trained on over SUSY-AI will be continuously updated with future LHC results as a part of the BSM-AI project. When possible, the publicly available ATLAS and CMS data will be used as in the current work. Additionally, we plan to generate our own MC data samples and recast them to produce limits using CheckMATE based on the existing and future LHC analyses. Classifiers and regressors for other models of new physics are also planned so that the whole project could cover a broad range of theories.http://susyai.hepforge.org, where we also provide installation instructions, more detailed technical information, frequently asked questions, example codes and updates of SUSY-AI using results of the ongoing Run 2, currently based on Refs. [SUSY-AI can be downloaded from the web page on Refs. , 97.