The beam finite element and molecular dynamics models are two popular methods to represent the reaction of carbon-carbon bonds in graphene. However, the wrinkles and ripples in geometrical characteristics are difficult take into consideration. The out-planar mechanical properties are neglected in classical models of graphene. This paper proposes a self-adaptive umbrella model for vibration analysis of graphene. The parameters in the umbrella model are flexible to adapting the geometrical and material characteristics of graphene. The umbrella model consists of shell and beam elements. The honeycomb beam and planar shell model of graphene are included in the self-adaptive umbrella model as particular cases. The sensitivity analysis and results confirmed the rationality and feasibility of the self-adaptive umbrella model. Due to rmations . However3 bonding using MD simulation. Javvaji [Based on diverse effective thickness and potential functions of graphene, numerous algorithms and theories have been proposed and developed, such as Quantum Mechanic (QM) calculations, molecular dynamic (MD) simulations, and continuum models (CM). Yanovsky  used the Javvaji  extensivBesides QM and MD approaches, CM methods have attracted tremendous attention. For a description of the mechanical properties of graphene sheets, Reddy  used BreIn terms of the configuration of CM, F. Scarpa proposed truss-type analytical models based on cellular material mechanics theory to describe the in-plane linear elastic properties of the single layer graphene sheets . Meo andThe analytical results in the literature vary significantly  with devIn this paper, the formation of the self-adaptive umbrella model is introduced. The model configurations are described by hybrid finite elements with corresponding parameters. The self-adaptive umbrella model extended the classical honeycomb beam finite element model to a three-dimensional out-planar model. The classical honeycomb beam finite element and planar shell element model are included in the self-adaptive umbrella model as special cases. Based on a Monte Carlo simulation, the uncertainties in parameters related with geometrical and material properties are taken into consideration. The sensitivity analysis not only selected the essential input variables in the self-adaptive umbrella model, but also proved the efficacy of the proposed model. The vibration modes of three particular cases in the self-adaptive umbrella model are compared and discussed.In the general honeycomb lattice structure of graphene, the carbon-carbon (C-C) bonds are presented by beam elements, as shown in 1R and 2R are the radii of hexagons in the bottom and top, respectively. According to the geometrical characteristics of hexagons, the radii of hexagons are equal to that of the corresponding length of each side, as demonstrated in As presented in The self-adaptive capacity in the proposed umbrella model for graphene not only contains the classic honeycomb beam finite element models and other in-plane models as special cases, but also has flexibility, in terms of material properties, to explore the relationships between parameters and different elements in the model. The three dimensional geometrical configurations  in the umbrella model are described with the introduced parameters, and the relationships between parameters corresponding to geometrical and material properties are discussed in the sensitivity analysis. The self-adaptive umbrella model is a flexible and feasible method to find appropriate descriptions for the exact microstructure of graphene. In order to precisely describe the self-adaptive umbrella model, parameters are listed in 1R, is settled as 0.27 nm, to be in agreement with the relevant literature [RR is introduced to represent the ratio between the radius of top and bottom hexagons. In addition, the sectional radius of Beam 1 and Beam 2 is set at half of the thickness of the Shell to satisfy the geometric compatibility. The interval range of the thickness of Shell is 0.001~0.04 nm with the assumption of a thin shell. However, the vertical height of the model fluctuates in a large interval range. In the three-dimensional umbrella model, more emphasis is put on the vertical height of the model. The intervals of material parameters are set according to the experimental data [The above parameters are uncertain input variables with certain interval ranges. The umbrella hybrid finite element model is meshed by Beam 1, Beam 2, and shell finite elements. By the Block Lanczos method , the resterature ,35,36. RThe Monte Carlo simulation is rooted from the integrals computation by random number generation.In the interval range A large number of samples for the variables from the probability density The integral The estimated Monte Carlo standard error can be expressed asIf the number of sample ∫0LEIFor free vibration,The normal solution can be written askth mode. Solving the above equation is an eigenvalue problem:Then,The discretization of the governing equation can be accomplished by finite elements. For the beam finite element, there are six degrees of freedom for each node. The approximated solution for transversal displacement and rotation can be represented asConsider the natural coordinate It can be expressed in the matrix form:By substituting to the weak form of equation, the stiffness and mass matrix in element domain are written asBesides the beam finite element (Beam 1 and Beam 2), there are shell elements  in the umbrella model. The strain and displacement relation can be obtained fromWithBased on the finite elements, the resonant frequencies of graphene are computed in the umbrella model. The parameters in the self-adaptive umbrella model are supposed to be independent. Because the number of introduced parameters in the umbrella model is larger than the number of provided equations, the analytical solutions for each parameter are difficult to obtain. Therefore, specific interval ranges for parameters are used to detect the sensitivity and importance of each parameter to the resonant frequencies in the vibration of graphene. The number of the input variables in the self-adaptive umbrella model for graphene is 12. The sensitivity levels of each input variable to the resonant frequencies of graphene are different; therefore, it is necessary to distinguish the importance of the input variables. Based on the database of the Monte Carlo simulation of the self-adaptive umbrella model, the first order regression fitting coefficients of independent input variables are obtained from Equation (23), and are listed in In 1E, 3E), Poisson ratio of Beam 1, Beam 2 and Shell , vertical height of model (H), and sectional radius of Beam 1 (1T), the first to the fourth resonant frequencies are all amplified. Among them,1 E, 2V, H and 1T are the more important parameters with larger fitting coefficients in the self-adaptive umbrella model for graphene. With the enlargement of the Young’s modulus of Beam 2 (2E), physical density of Beam 1, Beam 2 and Shell , and the ratio between the radius of top and bottom hexagons (RR) cause the reduction of the first to the fourth resonant frequencies in graphene. Furthermore, 1P and RR have more significant effects on the resonant frequencies than 2E, 2P, and 3P.With the increase of the Young’s modulus of Beam 1 and Shell . H), Young’s modulus of Beam 1 (1E), section radius of Beam 1 (1T), and Poisson ratio of Beam 2 (2V) play positive roles in resonant frequencies, while in the second group, the physical density of Beam 1 (1P), the ratio between the radius of top and bottom hexagons (RR), and Poisson ratio of Shell (3V) have negative effects on the resonant frequencies of graphene. Moreover, the remaining parameters, namely Poisson ratio of Beam 1 (1V), Young’s modulus of Beam 2 and Shell  and physical density of Beam 2 and Shell , may be neglected.In order to be clearer, parameters are divided into two groups according to the regression coefficients in 1E, 2E), Poisson ratio of Beam 1 (1V), section radius of Beam 1 (1T), and physical density of Beam 1, Beam 2, and Shell  are close in independent parameter fitting and global regression. However, discrepancies appear in the other parameters, especially in the vertical height of model (H), and the Poisson ratio of Beam 2 and Shell . However, there is no doubt that the Young’s modulus of Beam 1 (1E), vertical height of model (H), Poisson ratio of Beam 2 (2V), section radius of Beam 1 (1T), the ratio between the radius of top and bottom hexagons (RR), and the physical density of Beam 1 (1P) have larger absolute values in fitting coefficients than other parameters, and play more important roles in the resonant frequencies of graphene. The results of independent parameter fitting are compared with those of global regression. There is a good agreement among some parameters in the first to fourth resonant of graphene. The coefficients of Young’s modulus of Beam 1 and Beam 2 (1E), section radius of Beam 1 (1T), the physical density of Beam 1 (1P), and the vertical height of model (H) are more sensitive to impacts on the resonant frequencies of graphene, which are consistent with the above results. The variances of resonant frequencies in subsets of the Monte Carlo-based finite element model are illustrated in 1E, 1T, 1P) except Poisson ratio are critical to the self-adaptive umbrella model for graphene. On the one hand, the self-adaptive umbrella model for graphene is compatible with the classical honeycomb beam finite element model. The honeycomb beam finite element model is an effective model to represent the important characteristics of mechanical properties in graphene. As an advanced method, the self-adaptive umbrella model for graphene includes the honeycomb beam finite element model as a special case. On the other hand, the vertical height of model (H) in the self-adaptive umbrella model also plays an essential role in resonant frequencies of graphene. The vertical height of model (H) describes the geometrical characteristics of ripples and wrinkles in graphene. Therefore, the self-adaptive umbrella model for graphene is a smart method, with consideration of ripples or wrinkles as well, as the function of honeycomb beam finite element model.In a word, the parameters corresponding with Beam 1  equals to 0.2 nm and the ratio between the bottom and top hexagons (RR) is set as 0.3. Case 1 is a normal self-adaptive umbrella model. In Case 2, the vertical height of the model (H) stays as 0.2 nm, but the ratio between the bottom and top hexagons (RR) is supposed to be 0.8. Case 2 is closer to the classical honeycomb beam finite element model. In Case 3, the vertical height of the model (H) changes to 0.001 nm. The ratio between the bottom and top hexagons (RR) is set as 0.02. Case 3 is an approximated plane model. The remaining parameters corresponding to the material and geometrical properties are equal in the three cases. These three cases are typical situations in the self-adaptive umbrella model.In order to observe the vibration modes of the self-adaptive umbrella model for graphene, three special cases are computed by performing finite element simulation. In Case 1, the vertical height of the model (The displacement vector sums and the rotation vector sums for the first four vibration modes are presented in Furthermore, by comparing the resonant frequencies of the umbrella model for graphene in the three special cases with those in the literatures, the results of Case 2 are closer to the results of density functional theory, molecular dynamics, and molecular mechanics, especially the results provided by Reddy , CadelanIn this paper, a self-adaptive umbrella model is proposed for vibration analysis of graphene. The classical honeycomb beam finite element model is extended into a three-dimensional out-planar model with functions to describe the wrinkles or ripples in graphene. The hybrid finite elements with corresponding parameters are applied to construct more appropriate models. The parameters related to the geometrical and material properties are discussed in the sensitivity analysis. The self-adaptive umbrella model is confirmed to have advantages in three aspects. First, the proposed model has the ability to describe the out-planar geometrical characteristics in graphene. Second, satisfactory compatibility to the classical honeycomb beam finite element and shell planar model is attested to. Third, the results of resonant frequencies in graphene are located in a reasonable interval range even, though the input variables are distributed over a wide range. Therefore, the self-adaptive umbrella model is a feasible and appropriate model for mechanical analysis of graphene.


This study investigated the consequences of a low supply of dietary Ca with or without a low dietary cation-anion difference (DCAD) during early lactation on bone mobilization and reconstitution during lactation and on the dynamics of milk Ca content. Fifteen multiparous Holstein cows were distributed among 3 treatments 5 weeks before their expected calving date. These treatments differed based on the provision of diets through the first 10 weeks of lactation. During this period, the control treatment (NCa) consisted of a diet providing 100% of the Ca requirement, with a DCAD of 200 mEq/kg dry matter (DM). The LCa (low Ca) and LCaLD  treatments consisted of diets providing 70% of the Ca requirement, with a DCAD of 200 and 0 mEq/kg DM, respectively. After 10 weeks, all cows received the same total mixed ration, which was formulated to meet 100% of the Ca requirement. LCa and LCaLD tended to decrease the body retention of Ca at 3 weeks of lactation compared with NCa but affected neither the dynamics of the blood biomarkers of bone formation and resorption during lactation nor the body retention of Ca at 17 weeks of lactation. Cows almost entirely compensated for the decrease in Ca supply caused by LCa and LCaLD by increasing their apparent digestive absorption of Ca at 3 weeks of lactation, whereas their apparent digestive absorption was unaffected by the treatments at 17 weeks of lactation. Milk production tended to be lower throughout lactation with LCa and LCaLD compared with NCa, with a mean difference of 2 kg/d. The results of this study also indicated that measuring the dynamics of milk Ca content during lactation cannot be considered effective for indirectly estimating the dynamics of bone mobilization in cows. The results also suggested that limited Ca intake at the beginning of lactation may have deleterious effects on milk production. Dairy cows excrete a large amount of Ca during lactation due to the high Ca content in milk , and thiQuestions remain about the variations in the amplitude and completeness of these cycles of bone mobilization and reconstitution, their causes, and their consequences for the health and productivity of dairy cows. Incomplete bone reconstitution at the end of lactation can result in a higher susceptibility of cows to the restricted supply of P during the subsequent lactation, resulting in suboptimal production performance, as highlighted for sucker cows by Dixon et al. , or possThe effects of dietary Ca and P contents and Ca and P supplementation strategies on the amplitude of the bone formation-resorption cycle have been quantified in several experiments in lactating ruminants –13,18,19This lack of studies on the effects of dietary Ca content on bone mobilization and reconstitution throughout lactation in dairy cows could be related to a lack of fast and cost-efficient methods for evaluating the amplitude and eventually the completeness of the cycles of bone mobilization and reconstitution during lactation and gestation in a significant number of cows. Measurements of retained Ca and P at the animal scale  and boneThe objective of this experiment was to quantify the effect of a limited dietary supply of Ca (70% of the French recommendation ) in addiThe 3 compared treatments consisted of 3 mineral supplements fed between 5 days after calving and 10 weeks after the beginning of lactation. The Ca content of the supplement was calculated either to fully meet the Ca requirements of the cows according to the INRA feeding system , with a st, 2016, to June 30th, 2017. During the experiment, the cows were housed in a free-stall barn cubicle covered with rubber carpet, except during 3 periods of 3 weeks for measurements of Ca retention, during which they were transferred to individual tie stalls (1.4 × 2.0 m). The individual stalls were also covered with a rubber mat and supplied with individual troughs and individual water bowls. During lactation, a total mixed ration (TMR) was distributed twice a day in 2 equal-sized meals at 0830 h and 1630 h. The TMR was distributed by an automatic dispenser into the individual troughs specific to each animal in the free-stall barn and directly by animal technicians in the individual tie stalls. Cows always had free access to the trough and to water during the day. Lactating cows were fed ad libitum and offered quantities calculated to allow 10% orts. Orts were weighed daily before the morning feeding. Cows were milked twice a day, at 0630 h and 1630 h. Milk production and DM intake were recorded daily for each individual. Milk composition (fat and protein contents) and somatic cell count were measured twice a week at the evening and morning milkings. Before calving, each cow was fed a fixed amount of diet, which was distributed once per day. Procedures related to the care and use of animals for the experiment were approved by the animal care committee of the French Ministry of Agriculture in accordance with French regulations (project number 7096–20 16082515505689v2).The experiment was conducted at the INRA experimental farm of Méjusseaume  from September 1+ and K+ contents minus the sum of the dietary Cl- and S2- contents, which are expressed in mEq/kg of DM. Five days after calving and until the end of the 10th week of lactation, cows were assigned to one of 3 TMRs corresponding to the compared treatments. The LCa and LCaLD TMRs were formulated to meet the requirements of cows for NEL, PDI, and all macrominerals and trace minerals, except for Ca, according to the INRA recommendations [During the experiment, cows were fed 4 or 5 successive diets according to their physiological stage and the treatment to which they were assigned . During ndations . The trex g for 12 min within 30 min of sampling and stored at -80°C for OC analysis and at -20°C for the other analyses. Milk samples were collected during the morning milking from total milk before blood sampling. The samples were stored at 4°C for analyses of fat and protein contents and for separation of the N, crude protein, soluble and nonsoluble Ca and P fractions , noncasein nitrogen (NCN), urea, soluble Ca and P) and frozen at −20°C for analysis of the total Ca and P contents. Additional samples of milk were also collected at the morning and evening milkings and twice per week for determination of milk fat and protein contents , and at weeks 1, 3, 6, 8, 10, 12, 14, 17, 19, 22, 24, 27, 29 and 31 of lactation, samples were collected for analyses of the milk total Ca and P contents .For each group, blood was sampled 3 weeks before the average expected date of calving in the group and at 1, 3, 8, 12, 17, 22, 27 and 31 weeks of lactation (average stage of lactation of the group). After being milked, before being fed, and while restrained in self-locking head gates at the feedline, the cows were sampled for blood by venipuncture of the tail vessels. The samples were collected in vacutainer tubes  coated with lithium heparin for Ca and inorganic P (Pi) analyses and in tubes coated with EDTA for osteocalcin (OC) and carboxy-terminal crosslinking telopeptide of type I collagen (CTX) analyses. OC and CTX are biomarkers of bone formation and resorption, respectively . Plasma For each group, all input (feed and water intake) and output  flows of Ca and P were measured 3 times during the experiment, i.e., 3 weeks before the average expected calving date of the group and 3 and 17 weeks after the average calving date. For each measurement, cows were moved from the free-stall barn 2 weeks before beginning the measurements and sent to individual tie stalls for habituation. The individual tie stalls were located in the same building as the free-stall barn. All cows were kept in the same room and were able to smell and hear each other. The feeding modalities remained similar to those applied in the free-stall barn. To determine the fecal excretion of Ca and P, large trays were positioned behind the cows on day 15 after the entry of the cows into the stall at 0900 h. Gross fecal output was weighed and sampled from day 15 at 0900 h to day 19 at 0900 h. Two representative samples (500 g of fresh feces each) were dried in a forced-air oven  to determine the daily amount of fecal DM excreted. These dried samples were pooled by cow and period for Ca and P determination. The daily volume of excreted urine was measured from day 15 at 0900 h to day 19 at 0900 h by equipping cows with urinary catheters connected by a Tygon tube to a 25-L container, which was closed with a rubber plug. To prevent urine deterioration, 250 mL of sulfuric acid (20% vol/vol) was added to the container. The urine was weighed and emptied daily at 0900 h. Each day and for each cow, a sample of 1% of the daily excreted volume was stored at −20°C. At the end of the experiment, these samples were pooled by animal and by period for further Ca and P content analyses.Samples of the offered diets, refusals, and feces were ground with a 3-blade knife mill through a 0.8-mm screen. Ash was determined by calcination at 550°C for 5 h in a muffle furnace. Nitrogen concentration was determined by the Dumas method according to the Association Française de Normalisation  on a LECO apparatus . The dietary, fecal, urine and milk Ca contents were measured by atomic absorption spectrophotometry  with the use of lanthanum chloride solution to dilute the sample and after calcination of the solid samples (500°C for 12 h). Phosphorus contents were determined using a KONE PRO multiparameter analyzer  by the Allen method for P . Milk faijkY was a dependent variable of cow k within treatment i at stage of lactation j. Treatment, stage of lactation and their interaction were fixed factors, and cow was a random factor. Measurements repeated over a stage of lactation were considered by using a covariance matrix. The choice of the structure of the matrix was determined according to the structure of the data and by using the Akaike information criterion for analyses of all variables. The average flows of Ca and P during the 4 days of measurements were also analyzed independently during each stage of lactation with a generalized linear model using PROC GLM in SAS and included only the fixed effect of treatment. These analyses supplemented the previous analyses performed with the complete model because it has been shown that with a smaller quantity of data, the inclusion of repeated measurements in a generalized linear mixed model can limit the detection of significant effects by increasing the type II error [Data were analyzed using PROC GLIMMIX in SAS , with a II error . Data otth week of lactation, one cow in the NCa treatment died due to bowel obstruction, and its data were removed from the data set. Despite these events, the pre-experimental characteristics of the 3 experimental lots remained similar. The average parities were 2.4 for the LCaLD and 3.0 for the NCa and LCa. Milk production during the first 32 weeks of the previous lactation was not affected by the treatments . During this period, the average daily intake of Ca was 136.0 ± 4.98 g/d and 126.5 ± 4.98 g/d for LCa and LCaLD, respectively, and 184.1 ± 5.06 g/d for NCa, leading to a Ca intake that was 31% lower for the LCa and LCaLD treatments compared with the NCa treatment, as expected. During the same period, the difference between absorbable Ca intake and Ca requirements calculated according the INRA feeding system [th day of lactation, when cows received the same diet, the Ca intake was not affected by the treatments, and the difference between absorbable Ca intake and Ca requirements always remained positive, with an average value of 10.9 ± 1.12 g/d . The milk protein content was numerically higher for cows fed the LCa and LCaLD treatments compared with those fed the NCa treatment between 5 and 9 weeks of lactation and after 20 weeks of lactation, but the treatment effect was not significant , with no effect on the dynamics of the blood biomarkers of formation and resorption throughout lactation. Even at 3 weeks of lactation, the Ca retention remained close to 0 in the LCa and LCaLD treatments and was not clearly negative. For the LCa and LCaLD treatments, the dietary Ca supply was limited to 70% of the recommended supply needed to fulfill the cows’ requirements according to the expected milk production and intake according to the INRA feeding system . Other slization .In our experiment, as in those of Taylor et al.  and Brai2D3 synthesis at the onset of lactation [We observed a clear, strong effect of cow physiological stage on the apparent digestibility of Ca, with higher digestibility at 3 or 17 weeks of lactation compared with 3 weeks before calving. The increase in Ca absorption capacity of the digestive tract between gestation and lactation agrees with the increase in PTH release and 1,25-(OH)actation .Experiments with lactating mice have demonstrated that a decrease in Ca intake can induce an increase in bone resorption and a concomitant decrease in the milk Ca to protein ratio . These eThe cows affected by the LCa and LCaLD treatments tended to have a higher milk Ca content after 10 weeks than those in the control treatment, which did not agree with our hypothesis for 2 reasons. First, this difference appeared when the diet was similar for all treatments. A lower milk Ca content was expected for those treatments. Because genetics is known to be a major determinant of milk Ca content in cows  and becaThe difference of 2 kg/d in milk production between the NCa treatment in one direction and the LCa and LCaLD treatments in the other direction was an unexpected result that could not be attributed to the pre-experimental measurements of the cows. The discrepancy between the treatments appeared approximately 2 weeks after the differentiation of the diets according to the treatment, i.e., at 3 weeks of lactation, and lasted until the end of the experiment, i.e., largely after 10 weeks of lactation, when all cows began to receive the same diet. It is impossible to draw firm conclusions regarding the effect of the dietary Ca supply on milk production from our experiment, given the small number of cows involved. However, whether the low Ca intake impaired potential milk secretion by the mammary gland at peak lactation by altering either the proliferation of the mammary epithelial cells or their exfoliation merits further investigation. Ca has been shown to function as an important messenger in cell proliferation, notably for the breakdown of the nuclear envelope . It is pAnother unexpected observation in our experiment was that the culling rate of cows before the next calving was clearly numerically higher in the LCa and LCaLD treatments than in the NCa treatment. In the LCa treatment, 3 of the 5 cows were culled before the next lactation, 1 because of the absence of estrus detection and 2 because of claw disorders. In the LCaLD treatment, 1 of the 5 cows was culled because of artificial insemination failure. All cows in the NCa treatment were kept for subsequent lactation without health or reproductive problems before the calving. Due to the small number of cows involved in this experiment, affirmation of an effect of the dietary Ca content on cow reproduction and health from our results was not possible. Subclinical hypocalcemia during the first three days of lactation has a negative effect on the reproductive performance of cows. Our results suggest that a Ca supply lower than the required supply during the first weeks of lactation could also have a detrimental effect , but thiLowering the dietary Ca content to between 0.8 and 0.6 g/kg DM clearly increased the apparent digestive absorption of Ca of the cows at 3 weeks of lactation but marginally affected the body retention of Ca, which remained nearly zero. This result suggests that bone mobilization in cows at the beginning of lactation can be unaffected by the supply of Ca, as long as the source of Ca is available for absorption and the supply is not lower than 70% of the Ca requirement. The low supply of Ca did not clearly affect the milk Ca content. However, the possibility remains that the low Ca level did not lower the milk production level and did not affect the reproductive performances of the cows and their probability of undergoing a subsequent lactation. These results need to be confirmed using more animals and suggest that Ca supplementation must be carefully checked at the beginning of lactation.S1 Table(XLSX)Click here for additional data file.