Numerical Formula Research Articles

Asymptotically exact formulas for channel flows over liquid-infused surfaces

Abstract Analytical formulas are derived describing channel flows over liquid-infused surfaces. The formulas are explicit, asymptotically exact and readily evaluated; no numerical scheme beyond simple quadrature is needed to calculate the flows. The formulas are obtained using a three-stage asymptotic analysis under the assumptions that an array of aligned finite-length grooves on the lower wall of a channel are (i) slender, (ii) well separated from the upper channel wall and (iii) well separated from each other. Despite these apparently limiting assumptions it is found, by comparison with full numerical simulations, that the formulas give excellent approximations to the flow across a much broader range of operating conditions. Useful formulas for the hydrodynamic slip lengths of the liquid-infused surfaces are reported and tested against both numerical simulations and other approximate formulas appearing in the literature. The formulas are also expected to be useful in assessing the possibility of so-called shear-induced failure of liquid-infused surfaces.

Celebrating 25 Years of Optical Biometry: A Milestone in Ophthalmology

Optical biometry has fundamentally transformed cataract surgery, with 2024 marking 25 years since the introduction of the first optical biometer. In the early 1980s, Fercher and colleagues pioneered optical non-contact eye length measurement, leading to the first interferometric A-scan of the eye. This innovation, patented and later developed by Zeiss, culminated in the release of the IOLMaster in 1999, enabling more accurate and reproducible eye diagnostics. Over the years, optical biometry has evolved into advanced Swept Source OCT devices, accompanied by numerous formulas for calculating IOL power. Today, this technology is crucial not only for cataract surgeries, especially in eyes previously treated with refractive surgery, but also in advancing our understanding of diseases across fields like cardiology and oncology.

On Newton–Cotes Formula-Type Inequalities for Multiplicative Generalized Convex Functions via Riemann–Liouville Fractional Integrals with Applications to Quadrature Formulas and Computational Analysis

In this article, we develop multiplicative fractional versions of Simpson’s and Newton’s formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using generalized convex functions lies in their ability to extend results beyond traditional convex functions, encompassing a broader class of functions, and providing optimal approximations for both lower and upper bounds. These inequalities are very useful in finding the error bounds for the numerical integration formulas in multiplicative calculus. Applying these results to the Quadrature formulas demonstrates their practical utility in numerical integration. Furthermore, numerical analysis provides empirical evidence of the effectiveness of the derived findings. It is also demonstrated that the newly proven inequalities extend certain existing results in the literature.

Deformation and $K$-theoretic index formulae on boundary groupoids

Motivated by investigating K -theoretic index formulae for boundary groupoids of the form {\mathcal{H}}= M_{0} \times M_{0} \sqcup G \times M_{1} \times M_{1} \rightrightarrows M = M_{0} \sqcup M_{1}, where G is an exponential Lie group, we introduce the notion of a deformation from the pair groupoid , which makes sense for general Lie groupoids. Once there exists a deformation from the pair groupoid for a (general) Lie groupoid {\mathcal{G}}\rightrightarrows M , we are able to construct explicitly a deformation index map relating the analytic index on {\mathcal{G}} and the index on the pair groupoid M \times M , which in turn enables us to establish index formulae for (fully) elliptic (pseudo)differential operators on {\mathcal{H}} by applying the numerical index formula of M. J. Pflaum, H. Posthuma, and X. Tang. In particular, we find that the index is given by the Atiyah–Singer integral but does not involve any \eta -term in the higher codimensional cases. These results recover and generalize our previous results for renormalizable boundary groupoids via renormalized traces.

A novel finite-difference converged ENO scheme for steady-state simulations of Euler equations

The high-order shock-capturing scheme is critical for compressible fluid simulations, in particular for cases where strong shockwaves are present. However, for the steady-state solution of Euler equations, the high-order shock-capturing schemes usually suffer from the difficulty that the numerical residual hangs at a relatively high level instead of converging to machine zero. In this paper, a new paradigm of finite-difference converged ENO (CENO) scheme for steady-state simulations of Euler equations is proposed. Different from the existing methods dedicating to eliminating the slight post-shock oscillation, the unsteady effect of shock-induced numerical instability on the nonlinear weighting process as well as the resultant variation of the numerical formula for the spatially fixed cell interface flux at different time instants are demonstrated to be the core contributors to the non-convergence of high-order shock-capturing schemes. Unlike classical ENO/WENO/TENO schemes, which generate the final reconstruction based only on spatial information, evolutionary information is also exploited by a novel converged stencil selection strategy in the present scheme. With this strategy, the unsteady effect of shock-induced numerical instability on the nonlinear weighting process is incorporated and eliminated by a tailored adaptive scale-separation algorithm, while shock-capturing capabilities are not sacrificed. A set of well-established and newly designed challenging benchmark simulations involving strong shockwaves, contact discontinuities and rarefaction waves demonstrates that the new fifth-order CENO5 scheme features superior convergence properties and achieves the essentially non-oscillation property better when compared to the robust TENO5 scheme and the state-of-the-art WENO-type schemes designed for steady-state problems.

Evaluation Method of the Impact of Twin Shield Tunneling Construction on Elevated Bridges: Case Study

In urban metro construction, shield tunneling often needs to pass through building and bridge pile foundations, potentially affecting the stability of existing structures. Therefore, accurately assessing the impact of shield tunneling on bridges and buildings is crucial. This study presents a comprehensive prediction method combining numerical simulation and empirical formulas, taking the underpass project of the Shijiazhuang–Wuhan High-Speed Railway Bridge by Zhengzhou Metro Line 5 as a case study. Three-dimensional numerical model calculations were performed using finite element software to analyze the displacement and stress changes of buildings and tunnel structures at different construction stages, revealing the deformation patterns of buildings adjacent to the tunnel during shield tunneling. In particular, the ground settlement caused by twin-tunnel excavation was compared with Peck’s empirical formula to verify the reliability of the numerical simulation. The results show that twin-tunnel excavation exacerbates the horizontal displacement, uplift, and settlement of the ground, with maximum deformation rates increasing by 7.10%, 20%, and 11.4%, respectively. Comparing the ground deformation results of Peck’s empirical formulas with numerical calculations revealed similar trends in the settlement curves, with a maximum deviation of 6.67%. It can be concluded that using Peck’s empirical formula to calculate ground deformation characteristics complements the limitations of numerical simulations, making the assessment results more reliable. The findings of this study demonstrate that integrating numerical simulation with empirical formulas significantly enhances the reliability of deformation predictions in complex tunneling scenarios. This research not only offers a comprehensive safety assessment method for shield tunneling construction but also provides valuable guidance for the design and construction of similar projects, serving as a theoretical reference for future engineering endeavors.

Axial Impact Performances of Composite Aluminum Foam Tubes

An improved melt foaming method is employed to fabricate composite aluminum foam tubes (CAFTs) with varying diameter ratios, achieving metallurgical bonding between the aluminum foam and aluminum tubes. Based on the structural characteristics of aluminum foam‐filled tubes and traditional numerical formula for calculating the energy absorption properties of metal foams, a new calculation formula is proposed to accurately evaluate energy absorption performance of CAFTs under axial load, enabling more precise numerical calculation of total energy absorption. Furthermore, drop weight impact tests and finite element simulations are conducted to investigate the axial impact performance and deformation mechanism of CAFTs. The influence of diameter ratio on crashworthiness and energy absorption performance is also analyzed. As the diameter ratios decrease, both the impact resistance and total energy absorption of CAFTs increase, while the degree of tube damage gradually decreases. Moreover, finite element simulation results demonstrate that metallurgical combination between aluminum foam and aluminum tubes effectively transfers stress to thin‐walled tubes with better load‐bearing performance, preventing concentrated collapse of aluminum foam core. Additionally, the presence of aluminum foam provides robust restraint against deformation or rupture in thin‐walled tubes.

Research on Monitoring Technology for Frame Piers of Continuous Box-Girder Bridges Constructed by the Cantilever Method

This paper focuses on the analysis of the stress state of a large-span frame pier-continuous box girder bridge with pier crossbeams anchored by pier crossbeams on the main pier of the Guangfo-Zhao Expressway. The bridge is constructed by the cantilever method, and a refined finite element model of the entire bridge is established using the finite element software Midas/FEA to analyze the stress state of the frame pier during the cantilever construction process. It is found that under the possible combined action of an unbalanced load during construction, the torsional resistance of the frame pier crossbeam does not meet the requirements of the design code. In order to eliminate the torsion of the frame piers, counterweights were used to monitor the frame piers during the construction of the box girders. In this paper, the theoretical calculation formula of the inclination angle of the end section of the frame pier crossbeam with the change of unbalanced bending moment, the calculation formula of the relationship between the horizontal displacement of the frame pier and the unbalanced bending moment, and the calculation formula corresponding to the relationship with the water tank counterweight are derived using the structural mechanics method. Two monitoring methods for the frame pier are proposed. In the construction monitoring of the bridge, the numerical fitting formula obtained by finite element numerical analysis calculation is compared with the calculated formula obtained by substituting the design parameters of the frame pier into the theoretical formula. The basic constants in both formulas are basically equal, verifying the correctness of the monitoring calculation formula proposed in this paper for the torsional resistance of the frame pier crossbeam. The applicability of the two monitoring methods is also compared and analyzed. This paper takes the main pier of Chaoyang overpass’s mainline bridge as the engineering background, which adopts the framework pier with a large-span prestressed concrete continuous box girder bridge. It analyzes the torsional state of the beam of the framework pier during the bridge construction process and conducts research on the construction monitoring of the framework pier crossbeam, providing valuable references for the construction monitoring of framework pier crossbeams in the construction of large-span framework pier continuous bridges in the future. The research results of this paper can provide assistance for the construction monitoring of similar projects. This paper’s innovation primarily resides in employing structural mechanics methods to compute the torsion of frame piers. On this basis, a simplified beam torsion calculation formula is proposed to strengthen its practical application in construction monitoring. The findings of this paper can help in the construction monitoring of similar projects.

Review of Particle Physics

The summarizes much of particle physics and cosmology. Using data from previous editions, plus 2,717 new measurements from 869 papers, we list, evaluate, and average measured properties of gauge bosons and the recently discovered Higgs boson, leptons, quarks, mesons, and baryons. We summarize searches for hypothetical particles such as supersymmetric particles, heavy bosons, axions, dark photons, etc. Particle properties and search limits are listed in Summary Tables. We give numerous tables, figures, formulae, and reviews of topics such as Higgs Boson Physics, Supersymmetry, Grand Unified Theories, Neutrino Mixing, Dark Energy, Dark Matter, Cosmology, Particle Detectors, Colliders, Probability and Statistics. Most of the 120 reviews are updated, including many that are heavily revised. The is divided into two volumes. Volume 1 includes the Summary Tables and 97 review articles. Volume 2 consists of the Particle Listings and contains also 23 reviews that address specific aspects of the data presented in the Listings. The complete (both volumes) is published online on the website of the Particle Data Group () and in a journal. Volume 1 is available in print as the . A with the Summary Tables and essential tables, figures, and equations from selected review articles is available in print, as a web version optimized for use on phones, and as an Android app. The 2024 edition of the Review of Particle Physics should be cited as: S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)© 20242024

A phase field method for predicting hydrogen-induced cracking on pipelines

An accurate determination of the threshold conditions to initiate cracks on aged hydrogen pipelines is paramount for ensuring energy transport safety. In this work, a finite element-based phase field method was developed to assess the crack initiation on dented pipelines while considering the hydrogen (H) impact. Theoretical and multi-physics numerical formulas were derived for prediction of the elastic-plastic fracture behavior of H-contained steel. A critical phase field parameter, ϕ=0.69, is defined for predicting crack initiation at the dent on pipelines. The presence of H within the steel decreases the threshold dent depth for initiating H-induced cracks. When the initial H concentration increases from 0 to 0.5 wppm, the maximum dent depth for crack initiation reduces from 17.5 mm to 10.7 mm. The maximum dent depth required for crack initiation reduces from 17.5 mm to 7.8 mm when an internal pressure of 8 MPa is applied on the steel pipe. The site with the maximum phase field parameter changes during indentation, implying that the location initiating cracks depends on the dent dimension. The existing criteria in ASME B31.12 standard are not applicable for predicting H-induced crack initiation on dented pipelines. This study proposes a new method to predict hydrogen-induced cracking on aged pipelines when transporting hydrogen.

Numerical Calculation and Analytic Formulas of Yarkovsky Effect Parameter of Circumsolar Asteroids

Numerical Calculation and Analytic Formulas of Yarkovsky Effect Parameter of Circumsolar Asteroids

Hydraulic Model Experiments and Performance Analysis of Existing Empirical Formulas for Overtopping Discharge on Tetrapod Armored Rubble Mound Structures with Low Relative Freeboard

In coastal structure design incorporating revetments, the assessment of wave overtopping discharge relies on hydraulic model experiments. Numerous empirical formulas have been developed to predict overtopping discharge based on quantitative data from these experiments. Typically, for revetment structures aimed at mitigating wave overtopping, crest height is determined by considering the maximum amplitude of the design wave, resulting in a relatively high freeboard compared to wave heights. However, achieving complete prevention of all wave overtopping would require the crown wall to have substantial crest heights, rendering it economically impractical. Therefore, the concept of limiting discharge has been introduced in the design of revetment structures, aiming to restrict wave overtopping discharge to an acceptable level. Consequently, many coastal structures in real-world settings feature relatively lower freeboard heights than incident wave heights. This study investigated wave overtopping discharge on rubble-mound breakwaters with relatively low freeboard heights through hydraulic model experiments. Furthermore, it conducted a comparative analysis of the predictive capabilities of existing empirical formulas for estimating overtopping discharge using experimental data.

Control stability analysis of dynamic model with time delay for quadrotor UAV

For the first-order unstable object with time delay, the PD control parameter tuning method based on the traditional definition lacks consideration of the lower boundary of amplitude margin, resulting in a certain deviation between the result and the reality. In this paper, through analyzing the control loop of quadrotor UAV (QUAV), the integral link is moved from the controlled object to the controller, and the QUAV becomes a first-order unstable object, realizing the conversion of PD to PI in the hovering state. The system stability margin remains unchanged, and the position and attitude control objectives can be realized synchronously. By means of parameter setting method, the upper boundary of stability margin is obtained. Pade approximation is carried out for the time delay link, and then the lower margin of the amplitude margin is deduced according to Routh stability criterion, so as to obtain a more accurate reachable stability margin region. To avoid the high gain feedback of attitude angular velocity of QUAV, a numerical analytical setting formula for PD attitude control is derived to meet the requirements of gain and phase margin. The graph shows that the reachable region obtained by the strict definition of stability margin analysis decreases significantly. The proposed method can not only give the reachable region intuitively, but also give the tuning formula of the control parameters concisely, which has important engineering application value.

Probability density of the solution to nonlinear systems driven by Gaussian and Poisson white noises

A new method is proposed to compute the probability density of the multi-dimensional nonlinear dynamical system perturbed by a combined excitation of Gaussian and Poisson white noises. We first deduce a probability-density solver from the Euler–Maruyama scheme of the stochastic system and the corresponding Chapman–Kolmogorov equation. This solver actually is an explicit numerical formula of the probability density of the solution to this stochastic system. To compute the probability density, we propose an efficient algorithm for this solver, which actually is the implementation of a numerical integration. Furthermore, we prove this solver is an approximated solution of the corresponding forward Kolmogorov equation. Numerical examples are conducted to illustrate our probability-density solver.

Some Simpson- and Ostrowski-Type Integral Inequalities for Generalized Convex Functions in Multiplicative Calculus with Their Computational Analysis

Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus.

Modeling and simulation of ceramic tiles linear shrinkage variation during the sintering process

Linear shrinkage (SL) is utilized as a criterion to modify the size of the sintered ceramic tiles, and the sintering cycle affects the shrinkage variation. The exact sintering cycle associated with the SL of ceramic tiles is difficult to verify experimentally, and it affects the quality of the final ceramic tiles. Production costs also increased due to the large amount of experimental work. This study investigated a powerful numeric model of the sintering process within ceramic tile, using computational fluid dynamics (CFD) to evaluate the variation of the SL in ceramic tiles during the sintering process. The sintering process was simultaneous and integrated with energy and mass transport phenomena. Numerical formulas were developed for the sintering operation, and the SL behavior of ceramic tiles during sintering was examined using a kinetic model. An unsteady 3D model was established, and simulation was performed using the CFD tool by varying the temperature profile. The results of CFD simulation can ascertain the temporal and spatial changes in the SL of ceramic tiles through sintering. To validate the model, the green body mixture of ceramic tile was prepared and analyzed. Ceramic tiles were shaped by a powder pressing technique on a laboratory scale. Dried tiles were sintered in a laboratory furnace by adjusting the maximum sintering temperatures. The structure of sintered ceramic tiles was analyzed. The SL variation of each sintered tile was determined and compared with the results of CFD simulations. The results of the CFD simulation were validated and concurred with experimental outcomes, and the R2 value of the results was 0.9. The developed CFD model is capable of predicting the temporal and spatial changes in SL of ceramic tiles through sintering, and it is a great help to find the exact sintering cycle associated with SL. Finally, the quality of the tile is increased, and production costs are also reduced.

Building a New Platform for Significantly Improving Performance of Hartree-Fock and CCSD(T) Correlation Energy Based on Two-Point Complete Basis Set Extrapolation Schemes.

The leading cause of high expense in gold standard coupled cluster theory is that calculations of electronic energies converge exceedingly slowly with an increased basis set size. Extrapolation principally allows for achieving higher-quality outcomes at reduced costs. Numerous extrapolation formulas have been developed, with attempts to predict energies up to the complete basis set limit. Unfortunately, since the intricate shape of the function hinges on the molecular properties with the highest angular momentum of the basis set, the accuracy of the extrapolated energies highly depends on the fitted empirical parameters, which rely on the quality of the data sets for fitting. In this work, to overcome the extrapolation deficiency caused by the very limited data sets and smaller basis sets in the early stages, we constructed a new benchmark platform that includes a broader data set of 183 species (containing open-shell, closed-shell, ionic, and neutral species) and a larger basis set up to aug-cc-pV6Z. The newly optimized parameters can significantly improve the energy-predictive abilities of ten published formulas. Notably, all ten formulas perform quite similarly under the new platform with the reoptimized parameters. Finally, we built an online calculator for researchers to use for these extrapolation schemes. Our work would reignite the interest and applications of the underestimated formulas.

A Study on the Predictions of Wave Breaker Index in a Gravel Beach Using Linear Machine Learning Model

To date, numerous empirical formulas have been proposed through hydraulic model experiments to predict the wave breaker index, including wave height and depth of wave breaking, due to the inherent complexity of generation mechanisms. Unfortunately, research on the characteristics of wave breaking and the prediction of the wave breaker index for gravel beaches has been limited. This study aims to forecast the wave breaker index for gravel beaches using representative linear-based machine learning techniques known for their high predictive performance in regression or classification problems across various research fields. Initially, the applicability of existing empirical formulas for wave breaker indices to gravel seabeds was assessed. Various linear-based machine learning algorithms were then employed to build prediction models, aiming to overcome the limitations of existing empirical formulas in predicting wave breaker indices for gravel seabeds. Among the developed machine learning models, a new calculation formula for easily computable wave breaker indices based on the model was proposed, demonstrating high predictive performance for wave height and depth of wave breaking on gravel beaches. The study validated the predictive capabilities of the proposed wave breaker indices through hydraulic model experiments and compared them with existing empirical formulas. Despite its simplicity as a polynomial, the newly proposed empirical formula for wave breaking indices in this study exhibited exceptional predictive performance for gravel beaches.

Приложение биоинспирированных алгоритмов глобальной оптимизации в задаче подбора коэффициентов модели усталостной деградации жесткости композиционного материала

<p>The problem of finding the fatigue characteristics of a material based on the test results of a composite material is considered. The initial data are the properties of the material, load parameters and the tabulated dependence of the elastic modulus on the number of tests performed. A differential mathematical model has been formed that describes the change in the elastic modulus with increasing number of loading cycles. Its parameters are found by approximating the rate of change of the elastic modulus using numerical differentiation formulas of various orders and solving the parametric identification problem. The model coefficients are determined by using a method that simulates the behavior of a swarm of moths, related to bio–inspired global optimization algorithms. A solution to the problem for a specific composite material is given.</p>

Mechanical properties, microstructure, and environmental assessment of recycled concrete from aggregate after fire

In terms of ever-increasing fire accidents in cities together with upgrades in the amount of construction and demolition (C&D) waste, the rapid modernization and urbanization process have caused an adverse effect on environmental protection. The lack of natural resources in concrete provides a driving force for the reuse of C&D waste caused by fire accidents. Aiming to conserve natural resources and reduce environmental pollution, the feasibility of using aggregate after fire as an alternative material to produce recycled aggregate concrete (RAC) is investigated experimentally. The materials and environmental properties of the RAC are evaluated with the change of four different replacement ratios (25%, 50%, 100%, including a control group of 0%), and five different heating temperatures (200 °C, 400 °C, 600 °C, and 800 °C, including a control group of 20 °C). The results showed that with the increase of the substitution ratio, the strength, elastic modulus and ascending-branch slope for the stress–strain curves decrease gradually, which the brittle failure phenomenon becomes more obvious, which may have a bearing on the presence of old adhered mortar (AM) on recycled coarse aggregate (RCA). Nevertheless, the utilization of waste concrete treated by heating in RAC can mitigate the adverse effect of increasing the replacement ratio, which is attributed to the decreasing content of old AM in RCA. Numerous calculation formulas related to the material properties and constitutive model are proposed by considering different factors. Based on SEM tests and X-ray diffraction techniques, it was identified that the strengthening mechanism in the mechanical properties and uniaxial compressive behavior for RAC prepared with waste concrete after heating is mainly ascribed to the decreasing possibility of penetrating cracks in concrete under compressive loading. The cost and EI assessment shows that RAC with a 25% replacement ratio produced by waste concrete after 400 °C can probably be a suitable selection in concrete materials from an economic and environmental perspective.