Closed-form Formulation Research Articles

Effects of multiple reflections on nonlinear absorption measurements.

Accurate measurement of nonlinear absorption coefficients by transmission-based techniques, such as Z-scan and its modifications, remains challenging due to high measurement errors. In this Letter, we claim that one of the reasons for that could lie in ignoring multiple reflections of light in plane-parallel samples in the presence of multi-photon excitation. Closed-form formulations for transmittance and reflectance coefficients, considering both linear and multiphoton absorption, are obtained in the geometrical optics approximation. The contributions of multiple reflections to the total transmittance are calculated for several II-VI materials reported in the literature. The results show a nonlinear dependence on incident intensity, ranging from 15.7% down to 7.9% for CdS, from 10.3% down to 5.8% for CdSe, and from 18.1% down to 9.1% for ZnSe. The dependence of the contribution of multiple reflections on the sample thickness is shown to exhibit a near exponential decay.

Spectrum Sharing Environment Using Deep Learning Techniques

Recently, the satellite communication has main demand but still facing main problems which are the spectrum limited, both cognitive-radio (CR) and non-orthogonal-multiple-access (NOMA) methods are identified as a one of the potential solutions for these issues. Using the deep learning techniques with the CR and NOMA systems will be improvement in the spectrum effectiveness. One of the performance enhancements of the NOMA, the integration with the satellite-terrestrial-relay-network (ISTRN) in a spectrum-sharing context of several main-users (PUs) is examined. We specifically presented and given the interference limitation imposed by several nearby pus and its effect on each other’s. The closed-form formulations are investigated the ergodic-capacity (EC) and outage-probability (OP). The asymptotic- analysis for OP at high SNRs is achieved to get additional insights. To support our research and reveal the effects of the significant characteristics on the quality of the system; the numerical data and analysis are provided with deep discussions.

Research on Vehicle Vibration Fatigue Damage Potential under Non-Gaussian Road Profile Excitation

The amplitude modulation method was used to generate a non-Gaussian road profile with prescribed power spectral density (PSD) and kurtosis. The vehicle vibration fatigue damage potential has been proven to be closely related to the amplitude modulation signal (AMS) and kurtosis of vehicle response. In this paper, the iterative method of AMS modelling based on absolute standard Gaussian distribution is first reviewed. To address the long iteration time problem, a closed-form formulation is presented to construct the AMS directly. Furthermore, by proving that the vehicle response under a slowly varying non-Gaussian road profile excitation can be regarded as the product of the same AMS and vehicle response under a Gaussian road profile excitation with the same PSD, the theoretical relationship between fatigue damage spectrum (FDS) of vehicle response under non-Gaussian and corresponding Gaussian road profiles is formulated based on the AMS. A case study is used to verify the proposed approach. The results show that a wide range of specified kurtosis of road profile can be achieved and the kurtosis of vehicle response is the same as for the road profile. Given kurtosis and fatigue exponent, the extra fatigue damage caused by non-Gaussian road profile can be derived easily from the corresponding Gaussian road profile without calculating the vehicle response, which lays the foundation for a significantly simplified and more accurate fatigue test of vehicle vibration under non-Gaussian road profile.

Investigation of local soil resistance on suction caissons at capacity in undrained clay under combined loading

Winkler modelling offers a flexible and computationally efficient framework for estimating suction caisson capacity. However, there is a limited understanding of the local soil resistance acting on caissons at capacity under combined six degrees-of-freedom (6DoF) loading, which is essential for accurately estimating caisson failure envelopes. Furthermore, existing simplified design models for caissons cannot assess capacity under non-planar lateral and moment loading, which is common in offshore wind applications. To address these limitations, this paper presents a comprehensive three-dimensional (3D) finite element analysis (FEA) study, which investigates the local soil resistance acting on the caisson at capacity in undrained clay under combined 6DoF loading. The paper introduces the concept of ‘soil reaction failure envelopes’ to characterise the interactions between soil reactions at capacity. Closed-form formulations are derived to approximate these soil reaction failure envelopes. An elastoplastic Winkler model is then developed, incorporating linear elastic perfectly plastic soil reactions based on these formulations. The results demonstrate that the Winkler model can provide efficient and reasonably accurate estimations of caisson capacity under combined 6DoF loading, even for irregular soil profiles that pose much uncertainty and challenges to existing macro-element models.

Systematic failure mechanism of an FGMs polyhedral arched liner under a fire disaster environment

This paper investigated the failure performance of an arched polyhedral liner yielding a fire disaster environment. The functionally graded materials (FGMs) are used in the liner to increase the ability of fire resistance. A concentrated load may occur at the invert of the arched liner due to the debris, which is considered in the study. The polyhedral arched liner deforms in a single-lobe shape, and this deformation is expressed by a trigonometric function. The nonlinear equilibrium equations are presented by applying the thin-walled shell theory and the principle of minimum potential energy, respectively. Based on the above investigations, two innovative contributions of the present study are addressed: (1) a novel FGMs polyhedral liner is introduced to rehabilitate the cracked non-circular tunnel in the combination of the mechanical loading and fire environment; and (2) the critical buckling load, as well as the load-displacement nonlinear equilibrium curves, is derived explicitly by solving the nonlinear equilibrium equations. The effect of the fire disaster on the buckling load is quantized to describe the destabilization behavior of the arched liner in a fire environment, respectively. Then, the present study is compared efficiently with other closed-form formulations when the polyhedral FGM arched liner simplifies to a circular homogeneous liner. It is shown that a smaller number of polyhedral edges and a higher content of ceramic are beneficial to resist external loadings and fire disasters in engineering practices. Finally, the effects of geometric parameters, volume fraction exponents, and temperature variations on the nonlinear buckling behavior of the polyhedral arched liner are evaluated.

Thermoelastic modeling of cubic lattices from granular materials to atomic crystals

When a cubic lattice is confined by a surface layer, the effective thermoelastic properties can be tailored by the prestress produced by the surface. The thermal expansion coefficient, temperature derivative of elasticity, and the equation of state (EOS) of the solid depend on the potential of each bond and the lattice structure, which can be predicted by the recently developed singum model. This paper first uses a granular lattice confined by a spherical shell to demonstrate singum modeling of the thermoelastic behavior of the cubic lattices and then extends it to atomic crystal lattices by considering the surface tension and long-range interactions. Given the elasticity and the EOS of a cubic crystal, the interatomic potential can be inversely derived. As the bond length changes with thermal expansion and pressure, the singum model predicts the temperature- and pressure-dependent elasticity. Using the orientational average, isotropic elastic constants can be obtained for polycrystals. The case study of copper (Cu) demonstrates the versatility of the model for different cubic lattices and predicts the experimental results of pressure- and temperature-dependent elasticity. The singum model is general for different lattice types and EOS forms and provides clear physical and mechanical meanings to correlate the interatomic potential, EOS, and elasticity in the closed-form formulation, which is very useful in engineering design and analysis of metal structural members in fire, geothermal, and space applications without the needs of large-scale numerical simulations.

Composite curved hourglass cellular structures: Design optimization for stiffness response and crashworthiness performance

Auxetic lattice structures are known for their superior stiffness-to-density and strength-to-density ratios. Re-entrant hourglass exhibits an acceptable equivalent elastic modulus and remarkable energy absorption capacity among various auxetic configurations due to their flexibility and hinge deflecting during crushing loads. This study introduces a curved hourglass honeycomb which can be produced using 3D printing with pure or reinforced filaments incorporating continuous fibers. New closed-form formulations are analytically derived using an energy method by considering an extracted unit cell to predict the equivalent in-plane mechanical properties for the first time. The plateau stress of the proposed honeycomb is evaluated at the densification state of a unit cell using the energy conservation theory, equating external work with plastic energy dissipation. The accuracy of these extended relations is validated against experimental outcomes, demonstrating good agreement between analytical and experimental results. Additionally, robust linear and nonlinear finite element analyses are conducted to assess the accuracy of the obtained relations, including the equivalent stiffness and plateau stress for structures produced with reinforced filaments. The nonlinear finite element code employs appropriate subroutines to model plasticity/damage events. Based on the established analytical relations for stiffness and plateau stress, multi-objective optimization using Genetic Algorithm is applied to define optimum values for the Pareto front chart's objective functions of stiffness and plateau stress. The optimization process involves exploring the design space and defining the values of geometrical parameters to achieve the desired objectives. In conclusion, this study presents a novel approach to developing curved re-entrant honeycomb structures, with analytical formulations validated against experimental and numerical results. Optimization helps to identify optimal configurations concerning stiffness and plateau stress, offering potential applications in lightweight and high-strength materials design. According to the obtained results, for selection of the best value for curved strut angle which results in the optimum value of stiffness and plateau stress, a range between 10°-34° can be considered depending on the volume fraction of fibers.

Limit-state solutions for the active earth pressure behind walls rotating about the base

Determining earth pressure-induced moments in the active limit state is critical for the safety assessment of retaining structures. While traditional design methods assume a Coulomb's resultant force acting at one-third of the wall height, the literature suggests that the earth pressure distribution depends on the failure mode. This paper presents a rigorous kinematic solution for the ultimate moment acting on a wall undergoing rotation about its base (due to bending failure or overturning). In addition, an approximate static solution is considered. These solutions give a good approximation to the ultimate load. While the kinematic solution is closer to the exact numerical solution, the approximate static solution provides a reliable conservative approximation for common geometric and soil parameters. Its advantage is that it has a closed-form formulation. The kinematic solution is successfully validated against experimental data and is further used to investigate the peculiarities of the rotational failure mode and to evaluate traditional methods. It is found that traditional methods are close to the rigorous solutions and, therefore, reasonably safe. A useful by-product of this study: the formulation of the kinematic solution is versatile and can be applied to various geotechnical problems involving rotation.

Safe Trajectory Generation for Wheel-Leg Hybrid Mechanism Using Discrete Mechanics and Optimal Control

Abstract The wheel-legged robot inherits the merit of both the wheeled robot and the legged robot, which can not only adapt to the complex terrain but also maintain the driving efficiency on the flat road. This article presents an optimization-based approach that leverage ideas from computational geometric mechanics to generate safe and high-quality wheel-leg hybrid motions among obstacles. The formulation of the proposed motion optimization problem incorporates the Lagrange–d’Alembert principle as the robot’s dynamic constraints and an efficient closed-form formulation of collision-free constraints. By discretizing the variational mechanics principle directly, rather than its corresponding forced Euler–Lagrange equation, the continuous trajectory optimization problem is transformed into a nonlinear programming (NLP) problem. Numerical simulations and several real-world experiments are conducted on a wheel-legged robot to demonstrate the effectiveness of the proposed trajectory generation approach.

Modeling virus transmission risks in commuting with emerging mobility services: A case study of COVID-19

Commuting is an important part of daily life. With the gradual recovery from COVID-19 and more people returning to work from the office, the transmission of COVID-19 during commuting becomes a concern. Recent emerging mobility services (such as ride-hailing and bike-sharing) further deteriorate the infection risks due to shared vehicles or spaces during travel. Hence, it is important to quantify the infection risks in commuting. This paper proposes a probabilistic framework to estimate the risk of infection during an individual’s commute considering different travel modes, including public transit, ride-share, bike, and walking. The objective is to evaluate the probability of infection as well as the estimation errors (i.e., uncertainty quantification) given the origin-destination (OD), departure time, and travel mode. We first define a general trip planning function to generate trip trajectories and probabilities of choosing different paths according to the OD, departure time, and travel mode. Then, we consider two channels of infections: 1) infection by close contact and 2) infection by touching surfaces. The infection risks are calculated on a trip segment basis. Different sources of data (such as smart card data, travel surveys, and population data) are used to estimate the potential interactions between the individual and the infectious environment. A first-order approximation is used to simplify the computational complexity. We also derive the closed-form formulation for the estimation errors, enabling us to quantify the uncertainty of the estimation results. The model is implemented in the MIT community as a case study. We evaluate the commute infection risks for employees and students. Results show that most of the individuals have an infection probability close to zero. The maximum infection probability is around 1.3%, implying that the probability of getting infected during the commuting process is low. Individuals with larger travel distances, traveling in transit, and traveling during peak hours are more likely to get infected. Practical implementations of the model are also discussed.

A Novel Analytical Formulation of the Magnetic Field Generated by Halbach Permanent Magnet Arrays

This paper presents an analytical study of the air-gap magnetic field of a surface permanent magnet (SPM) linear, slot-less machine with a Halbach PM configuration, under the no-load condition. While other analytical formulations of the magnetic field generated by PMs are available, they exhibit some drawbacks, such as only providing a Fourier series, or being suitable to determine magnetic field average values, but not local magnetic field distributions. On the contrary, the proposed approach allows the determination of a unique, closed-form formulation for the slot-less machine air-gap field. This is obtained starting from the complex expression of the magnetic field of a conductor, inside the air gap, between two parallel smooth iron surfaces, obtained by means of the method of images. The magnetic field due to an infinitesimal conductor belonging to a current sheet is then integrated along a segment, providing the expression of the magnetic field due to the corresponding linear current density distribution, for current sheets perpendicular or parallel to the iron surfaces. Any Halbach PM segment disposition can, hence, be obtained via a suitable combination of field distributions generated by couples of current sheets with perpendicular and parallel orientation. Lastly, the no-load magnetic field expression with a Halbach array of PMs is retrieved. The proposed analytical model provides an accurate representation of the magnetic field distribution produced by any Halbach array, with an arbitrary number of segments and orientations. Additionally, the results obtained from the proposed analytical expressions are compared with FEM simulations realized by commercial software, and show an excellent agreement.

Rocking stiffness of electrical cabinets with tubular base in nuclear power plants

Safe operation of nuclear power plants during an earthquake requires continued operation of safety related electrical equipment as intended. Such equipment are mounted in electrical cabinets and are seismically qualified by using an in-cabinet response spectra (ICRS). Generation of ICRS requires a good understanding of the cabinet’s dynamic characteristics. Past studies have shown that the rotational stiffness associated with the cabinet base and the mounting arrangement plays a primary role in the dynamic behavior of the cabinet. This rotational stiffness is needed to evaluate and characterize the global rocking mode of cabinet vibration. Existing studies have developed formulations to calculate the rocking stiffness for a few mounting arrangements that are typically found in United States and some other western countries. However, existing formulations could not be applied to cabinets in South Korean nuclear plants due to the differences in cabinet mounting type. This study is focused on consideration of tubular mounting arrangement that is typically found in South Korean plants. First, closed-form formulations are developed using fundamental principles of mechanics for two separate cases of this mounting arrangement. Then, detailed nonlinear finite element (FE) analyses are conducted for different mounting arrangements corresponding to different properties and dimensions of the tubular base. The FE models are used to calibrate the proposed closed-form formulations for various different configurations and structural or geometrical properties of tubular members. The proposed formulations provide a simple and accurate estimation of cabinet rocking stiffness which otherwise is difficult to evaluate without a shake table test or a detailed finite element analysis.

Moment curvature response of composite UHPC filled hollow structural steel cross-sections

A mechanics based analytical method for obtaining the complete moment–curvature response of composite hollow structural steel (HSS) and ultra high performance concrete (UHPC) cross-sections is presented. Flexural response at the cross-sectional level is traced past the local buckling of the steel elements and up to the exhaustion of UHPC and steel material capacity. The proposed method is validated using experimental data from seven flexural tests on composite HSS-UHPC beams. Flexural failure mode is classified as either a UHPC compression-controlled failure, UHPC fiber tension-controlled failure, or a balanced failure. Behavioral differences between the cross-sections that fall into these three categories are discussed in terms of flexural capacity, curvature ductility, and relative contributions of HSS and UHPC to flexural capacity. Flexural response of the composite HSS-UHPC cross-section is compared to that of the bare HSS cross-section. The enhancement in flexural capacity and curvature ductility, for UHPC fiber tension-controlled sections compared to the bare HSS cross-section, was 52% and 79%, respectively. For composite cross-sections that feature a balanced failure, the increase in flexural capacity and curvature ductility, compared to the bare HSS cross-section, was 37% and 35%, respectively. Closed form formulations are proposed to identify the flexural failure mode and predict flexural capacity. The influence of various parameters, such as HSS material grade, size of cross-section, and UHPC material characteristics on the complete flexural response of the cross-section is investigated. The average ratio and corresponding coefficient of variation (COV) of tested to computed flexural strength obtained using the proposed procedure was 1.06 and 21.28%, respectively. Similarly, the average ratio and COV of tested to calculated flexural strength obtained using the proposed closed form formulations was 1.07 and 22.18%, respectively.

Matrix completion via modified schatten 2/3-norm

Low-rank matrix completion is a hot topic in the field of machine learning. It is widely used in image processing, recommendation systems and subspace clustering. However, the traditional method uses the nuclear norm to approximate the rank function, which leads to only the suboptimal solution. Inspired by the closed-form formulation of L_{2/3} regularization, we propose a new truncated schatten 2/3-norm to approximate the rank function. Our proposed regularizer takes full account of the prior rank information and achieves a more accurate approximation of the rank function. Based on this regularizer, we propose a new low-rank matrix completion model. Meanwhile, a fast and efficient algorithm are designed to solve the proposed model. In addition, a rigorous mathematical analysis of the convergence of the proposed algorithm is provided. Finally, the superiority of our proposed model and method is investigated on synthetic data and recommender system datasets. All results show that our proposed algorithm is able to achieve comparable recovery performance while being faster and more efficient than state-of-the-art methods.

Analytical model to predict axial stress-strain behavior of heat-damaged unreinforced concrete columns wrapped by FRP jacket

Although there are several confinement models to obtain analytically the axial stress-strain response (fc-εc) of concrete columns wrapped with fiber-reinforced-polymer (FRP) jacket at ambient conditions, a reliable design-oriented model to determine the fc-εc of heat-damaged concrete columns post-confined with FRP is still lacking in the literature. This study aims to address this research gap, by proposing a formulation that predicts the favourable effects of FRP confinement on concrete elements previously exposed to high temperatures. This model proposes a closed-form formulation to derive a fc-εc expression, including a set of strength and strain sub-models to calculate the stress/strain information at transition and ultimate points defining the stress-strain response. To develop the model and calibrate its key components by data analysis of statistical treatment techniques, a large test database of FRP confined unheated/heat-damaged concrete of circular/square cross-section consisting of 1914 specimens was collected. The proposed design-oriented model is able to demonstrate the influence of pre-existing thermal damage on the axial fc-εc relationship, whose reliability is revealed comprehensively through predicting data from several experimental heat-damaged concrete specimens confined with FRP systems.

Analysis of LRFD factors for a linear limit-state equation with correlated and time-independent random variables: Closed-Form formulation and numerical algorithm

The calibration of load and resistance factors (LRFs) of multiple loads can be a demanding process, where multiple loads must be considered in a potentially large set of combination scenarios to maintain consistent reliability in structural design. In this study, a set of closed-form formulas of the LRFs are developed with the aid of the coordinates of the design point, under the assumption that the limit-state equation is linear, and the load and resistance variables are mutually correlated and time-independent. These formulas can be used to efficiently calculate LRFs for a combination of multiple loads. Each load ratio is defined in a way that the derived equations can explicitly show the relationship between the LRFs and the conventional safety factor used in the allowable strength design or allowable stress design (ASD). All parameters used in the LRF formulas are dimensionless, making them more general and capable of facilitating comparative observations in a direct and quantitative manner. The closed-form formulas are kept simple by limiting the scope to linear limit-state equations with normally distributed load and resistance variables. For cases where one or more variables are nonnormal, an efficient single-loop iteration algorithm based on the developed formulas and the Rackwitz-Fiessler method is proposed. Four numerical examples are provided to demonstrate the feasibility of the presented formulas and algorithm used for LRFD calibration, followed by discussions of several interesting observations. For example, the fundamental advantage of the calibrated LRFD over the conventional, but calibrated ASD as load ratios change, and the effect of correlation between resistance and loads to the LRFs are illustrated.

The optimum enhanced viscoelastic tuned mass dampers: Exact closed-form expressions

This paper introduces the inertial amplifier viscoelastic tuned mass dampers (IAVTMD). The viscoelastic materials are implanted inside the core material of the inertial amplifier tuned mass dampers. The standard linear solid models are applied mathematically to formulate the viscoelastic material. H2 and H ∞ optimization mechanisms apply to derive the exact mathematical closed-form formulations for optimal design parameters for novel inertial amplifier viscoelastic tuned mass dampers. The optimum IAVTMD installs at the top of the structures to mitigate the dynamic responses, determining the dynamic responses analytically through transfer function formation. At first, IAVTMD’s dynamic response reduction capacity compares with the conventional tuned mass damper’s (CTMD) dynamic response reduction capacity. As a result, H2 optimized IAVTMD’s dynamic response reduction capacity is significantly 20.87% and 26.47% superior to two well-established H2 optimized conventional tuned mass damper’s dynamic response reduction capacity. In addition, H ∞ optimized IAVTMD has 15.48% more dynamic response reduction capacity than H ∞ conventional tuned mass damper. H2 and H ∞ optimized tuned mass damper inerters with optimal closed-form solutions are introduced in this paper. A higher damper mass ratio and a lower inerter mass ratio are recommended to produce H2 and H ∞ optimized tuned mass damper inerter (TMDI) with a lower frequency and damping ratio in an affordable range. Accordingly, H2 and H ∞ optimized IAVTMD’s dynamic response reduction capacities are significantly 6.94% and 23.29% superior to H2 and H ∞ optimized TMDI’s dynamic response reduction capacity. The closed-form expressions for optimal design parameters of inertial amplifier viscoelastic tuned mass dampers and tuned mass damper inerters are mathematically correct and effective for practical applications.

Vertical dynamic response of single floating piles in poroelastic soil

This paper presents a closed-form formulation for the time-harmonic response of single floating piles that are embedded in a saturated soil layer of finite thickness. The frictional force of surrounding soil to pile vertical motion is first established, considering the soil layer as homogeneous isotropic poroelastic medium that obeys Biot’s governing equations. The pile and the soil column underlying the pile tip are modelled using one-dimensional rod theory, and then closed-form solution is obtained from the boundary and continuity conditions, instead of numerically solving complicated integral equations that are used in the common boundary integral methods. Finally, we use arithmetic examples to identify the sensitivity of the dynamic pile impedance and of the soil reactions to the main geometrical and material problem parameters.

Analytical Modeling of Magnetic Field Distribution at No Load for Surface Mounted Permanent Magnet Machines

This paper presents an analytical study of the air-gap magnetic field of a Surface Permanent Magnet (SPM) linear machine under no load. By means of the method of images, the complex expression of the magnetic field of a conductor, inside the air gap between two smooth iron surfaces, is retrieved. Then, integrating the conductor expression, the formulation of the magnetic field of a current sheet and thus the one of a SPM, using two vertical current sheets, is obtained. At last, the no-load magnetic field expression, for a generic time instant, of a slotless machine is retrieved. The novelty of the proposed approach is the availability, due to a different calculation approach, of a unique closed-form formulation for the slotless machine air-gap field, a quantity that, in literature, is usually present in Fourier series formulation. Additionally, as a means to calculate integral quantities and show the goodness of the method a complex slotting function is introduced to account for the slotted geometry. Finally, starting from Lorenz’s force formulation, the expression of the Maxwell tensor in complex form is retrieved and the contribution of forces, integral of the complex stress tensor quantity, will be calculated and compared with FEM simulations, showing a good agreement also with the analytical slotted model.

On the Mathematical Modeling and Optimization for the Energy Efficiency Performance of CSMA-NOMA Random Access Networks With Channel Inversion

The paradigm of upcoming 5G and beyond is the massive machine type communications (mMTC), where a large number of devices automatically operate wireless communications. Due to their automatic and energy-intensive operations, energy efficiency (EE) becomes crucial, but there is a lack of investigation for EE of random access networks, which is the underlying platform for mMTC. In this paper, we focus on the EE of carrier sense multiple access-based non-orthogonal multiple access (NOMA) random access networks. We first construct mathematical models. Instead of investigating all combinations regarding successful decoding events as in previous works, by pivoting around the decoding process of a specific signal, a hidden pattern of NOMA decoding process is unveiled, which can largely decrease analytical complexity. Then, together with this feature, by adopting Markov chain and Q-function approximation, closed-form formulation for EE is derived. Subsequently, to efficiently solve the complicated non-convex EE maximization problem built via the constructed models, we employ an approach that unifies complementary geometric programming (CGP) and difference of convex programming (DCP) to optimize all the controllable parameters at device side, namely, transmission probability, power, and data rate, with a tightest lower bound strategy to guarantee seamless EE improvement and very fast convergence to local optimal points even in the worst case. Simulation experiments verify the accuracy of mathematical models and efficiency of optimization scheme.