Closed-form Equations Research Articles

Definition and validation of a valley amplification factor for seismic linear response of 2D homogeneous alluvial basins

The paper presents findings from a parametric study analysing geometric (e.g., shape ratio, edge inclination) and stratigraphic factors (e.g. impedance ratio) influencing ground motion in trapezoidal valleys. The study involved 2160 visco-elastic analyses, considering 180 2D models with diverse shapes and soil properties, undergoing 12 synthetic input motions. Analyses results showed that the motion at the valley centre increases with both shape and impedance ratios, while it is independent of the edge slope; on the other hand, the maximum amplification at the edges depends on their inclination and on the impedance ratio, while it is independent of the valley shape. The position and size of the zone of maximum amplification at the edges depend on all the previous parameters. A valley amplification factor (VAF) is introduced to quantify spectral acceleration increase due to 2D effects. Closed-form equations are proposed to evaluate VAF based on valley properties. The proposed VAF is then applied to predict seismic amplification in two central Italian valleys, providing results well-comparable to those obtained from 2D numerical analyses. The described approach can be easily implemented into codes of practice as a conservative design tool to estimate 2D amplification along the surface of ‘shallow valleys’ subjected to moderate seismic actions.

Tracking control of non-minimum phase systems: a kernel-based approach

Feedforward control with model inversion is a widely-used solution for high-precision output tracking. However, because inverting a non-minimum phase model generates unbounded control input, model-inversion only applies to limited types of systems. This paper presents a new non-parametric pseudo-inversion approach to design bounded optimal control input with desirable properties for arbitrary types of systems. Closed-form equations are presented for the batch (full preview) and recursive (limited preview) implementations of this approach, and its performance is compared against existing pseudo-inversion methods in benchmark numerical examples. Furthermore, the practical implementation of the proposed method is demonstrated by designing a feedforward controller for a commercial 3-Dimensional (3D) printer. The results show that the proposed approach effectively compensates for the structural vibrations of the printer, preventing layer-shifting errors that usually happen during high-speed printing.

A new modified direct method for determining the mode I delamination traction-separation law

A new equation was developed for the crack-tip separation in the double cantilever beam (DCB) specimen, which has been standardised for mode I delamination of composites. The very simple closed-form equation is based on a pseudo-elastic beam model and on the effective crack concept. Moreover, it only requires the common load–displacement data up to crack initiation and easy to obtain elastic moduli. Strain-energy release rate versus crack-tip separation curves can then be differentiated for obtaining the tractions, as in the classical direct method. Numerical results for piecewise linear traction-separation laws (TSL) showed good accuracy of strain-energy release rate versus crack-tip separation curves predicted. Application to two carbon/epoxy materials gave very promising TSL results, including cohesive strengths similar to transverse tensile strengths and post-peak steep traction decreases.

Characterizing the dispersion behavior of poly-atomic magnetic metamaterials

The propagation of magnetoinductive (MI) waves across magnetic metamaterials known as magnetoinductive waveguides (MIWs) has been an area of interest for many applications due to the flexible design and low-loss performance in challenging radio-frequency (RF) environments. Thus far, the dispersion behavior of MIWs has been limited to mono- and diatomic geometries. In this work, we present a recursive method to generate the dispersion equation for a general poly-atomic MIW. This recursive method greatly simplifies the ability to create closed-form dispersion equations for unique poly-atomic MIW geometries versus the previous method. To demonstrate, four MIW geometries that have been selected for their unique symmetries are analyzed using the recursive method. Using applicable simplifications brought on by the geometric symmetries, a closed-form dispersion equation is reported for each case. The equations are then validated numerically and show excellent agreement in all four cases. This work simultaneously aids in the further development of MIW theory and offers new avenues for MIW design in the presented dispersion equations.

Analytical realization of complex thermal meta-devices

Fourier’s law dictates that heat flows from warm to cold. Nevertheless, devices can be tailored to cloak obstacles or even reverse the heat flow. Mathematical transformation yields closed-form equations for graded, highly anisotropic thermal metamaterial distributions needed for obtaining such functionalities. For simple geometries, devices can be realized by regular conductor distributions; however, for complex geometries, physical realizations have so far been challenging, and sub-optimal solutions have been obtained by expensive numerical approaches. Here we suggest a straightforward and highly efficient analytical de-homogenization approach that uses optimal multi-rank laminates to provide closed-form solutions for any imaginable thermal manipulation device. We create thermal cloaks, rotators, and concentrators in complex domains with close-to-optimal performance and esthetic elegance. The devices are fabricated using metal 3D printing, and their omnidirectional thermal functionalities are investigated numerically and validated experimentally. The analytical approach enables next-generation free-form thermal meta-devices with efficient synthesis, near-optimal performance, and concise patterns.

Probabilistic inference of material quantities and embodied carbon in building structures

In an effort to minimise the carbon footprint of building structures, a range of prediction tools and methods have been recently proposed, so to enable design practitioners evaluating how their design choices ultimately affect the carbon embodied in their designs. Such tools are most often targeted for use at the early stage of the design process, that is when exploration of alternative design options is usually undertaken, hence room for potential carbon reductions is greatest and at no extra cost of redesign. The overarching methodology behind existing tools predominantly relies on idealised models to characterise the structural system, usually employing closed-form design equations and/or numerical Finite Element to generate an inventory of material quantity data (that is ultimately required for embodied carbon estimates). Despite the very high level of complexity achieved by some models, the absence of any empirical reference with ‘as-built’ inventory data of material quantities leaves room for doubt on how accurate such models really are in capturing the complexities and inherent variability of the population of real building structures such models aim to represent. To bypass this limitation, a data-driven probabilistic graphical model is proposed here as alternative to existing approaches. A Bayesian Network was developed and tested as a proof of concept, trained on a dataset of 133 data-points of real building structures, leveraging on six design variables (at most) to fully characterise the entire design space of early design options. Despite the very small set of ‘explanatory’ design variables, the model exhibited a 73% accuracy (mean average absolute prediction error of 27%) when predicting the embodied carbon on a test sample of unseen real building structures. The study ultimately demonstrates the viability of adopting a probabilistic (data-driven) approach for such an inference task as an inherently robust alternative to data-blind models currently proposed in literature.

Numerical Investigation of Funicular Liquid Bridges between Three Spherical Grains in a Bidisperse Particulate System.

Appropriate capillary effects are beneficial for controlling the wet powder performance and agglomerate formation. As water content rises, the funicular regime supplants the pendular regime as the predominant state in wet granular media. The displacement of grains leading to the stretching of funicular liquid bridges until rupture is an interesting and common phenomenon. Utilizing Surface Evolver software (an energy minimization approach), this work develops an efficient and accurate numerical model to describe liquid interactions among three spherical grains. The effects of liquid volume, contact angle, grain size ratio, grain-pair gap, and separation distance on the capillary forces and rupture distances are investigated. Notably, we present a modified closed-form equation for predicting the rupture distance of funicular bridges between three grains, which reflects the coupled effects of the contact angle, grain size, and liquid volume on rupture distance. This present study provides insights for incorporating capillary effects into mechanical models relying on microassembly composed of several grains in bidisperse particulate systems. Additionally, the numerical findings confirm some findings regarding the splitting of funicular bridges.

A Closed-Form Equation for Effective Stress of Unsaturated Saline Clay Considering Capillary and Osmosis Effects

A Closed-Form Equation for Effective Stress of Unsaturated Saline Clay Considering Capillary and Osmosis Effects

Damage detection and model updating of offshore jacket platforms using a direct sensitivity relation based on structural response

One of the challenges of the finite element model updating using time domain data is the lack of a direct relationship between changes in structural parameters and structural responses. Computational techniques, such as the Finite Difference Method (FDM) and Direct Differentiation Method (DDM), are used to calculate the sensitivity of the structural response to variations of structural parameters in the time domain. The use of these methods is time-consuming for large and complex structure. In the present study, an exact sensitivity equation is presented that establishes a direct and linear relationship between changes of structural parameters and structural response variations. The proposed closed form equations, have high computational efficiency in calculating the sensitivity of time domain responses. The measured responses directly contribute to the calculation of the sensitivity matrix, so the finite element model updating procedure is improved in terms of accuracy and reliability. The proposed method updates the finite element model even using a single excitation frequency that distinguishes the presented sensitivity relationship from frequency response-based methods which use wide ranges of excitation frequencies for model updating. To evaluate the effectiveness of the proposed method, different damage scenarios with different intensities have been simulated on two offshore jacket platform models. The performance of the proposed method in identifying the location and severity of induced damages in the presence of measurement and modeling errors confirms the effectiveness of the proposed method.

Theoretical modeling of a bottom-raised oscillating surge wave energy converter structural loadings and power performances

This study presents theoretical formulations to evaluate the fundamental parameters and performance characteristics of a bottom-raised oscillating surge wave energy converter (OSWEC) device. Employing a flat plate assumption and potential flow formulation in elliptical coordinates, closed-form equations for the added mass, radiation damping, and excitation forces/torques in the relevant pitch-pitch and surge-pitch directions of motion are developed and used to calculate the system's response amplitude operator and the forces and moments acting on the foundation. The model is benchmarked against numerical simulations using WAMIT and WEC-Sim, showcasing excellent agreement. The sensitivity of plate thickness on the analytical hydrodynamic solutions is investigated over several thickness-to-width ratios ranging from 1:80 to 1:10. The results show that as the thickness of the benchmark OSWEC increases, the deviation of the analytical hydrodynamic coefficients from the numerical solutions grows from 3 % to 25 %. Differences in the excitation forces and torques, however, are contained within 12 %. While the flat plate assumption is a limitation of the proposed analytical model, the error is within a reasonable margin for use in the design space exploration phase before a higher-fidelity (and thus more computationally expensive) model is employed. A parametric study demonstrates the ability of the analytical model to quickly sweep over a domain of OSWEC dimensions, illustrating the analytical model's utility in the early phases of design.

Bond behavior of deformed rebar in concrete containing recycled brick aggregate and waste rubber

The reuse of aggregates and crumb rubber (CR) in concrete is gaining interest globally for reducing the adverse effects on the environment and maintaining construction sustainability. The bond strength of reinforcing rebars in such concrete is one of the key issues to be addressed before their practical application, particularly in structural elements. Pull-out tests were performed on 80 specimens to assess the bond performance of deformed rebars in rubberized recycled brick aggregate concrete (RRBAC). After investigating the strength properties of RRBAC, pull-out tests were performed on two sizes of cylindrical specimens. RBA%, CR%, rebar diameter (d), embedment length (le), and concrete cover-to-rebar diameter ratio (c/d) were among the test variables in this bond behavior investigation. Bond stress-slip relationships were obtained, examined, and average bond strengths were compared with the available codes. Concrete split failure was observed in 79% of the tested specimens, whereas rebar fracture was not seen in any of the tests. The bond strength was found to be decreased with the inclusion of RBAs and CRs though relative improvement was observed for higher percentage RBAs. Additionally, regression analysis was performed on the experimental data to produce a closed-form equation for forecasting the bond strength for RRBAC. With a strong coefficient of determination (R2 = 0.97) and low values of error terms, the predicted bond strengths were seen to be a good agreement with the test results.

Unified AI-Based Predictive Models for the Ultimate Capacity of Multi-Planar Gapped KK Steel Pipe Joints

The multi-planar steel pipe joints are widely used in communication towers, industrial structures, and offshore platforms. The current design formulas consider this joint as a uniplanar joint and account for the multi-planar effect using empirical correction factors. Recent studies deal with this multi-planar joint as a 3D joint but considering certain loading conditions. Hence, the aim of this research is to develop more general AI-based predictive models for the ultimate capacity of multi-planar gapped KK steel pipe joints, considering both symmetric and asymmetric loading conditions. Three AI techniques were applied to a database of previously published works. These techniques are “Genetic Programming” (GP), “Artificial Neural Network” (ANN), and “Evolutionary Polynomial Regression” (EPR). The prediction accuracies of the developed AI models were compared against two previously published formulas. The results indicated that the developed AI models are much more accurate than the previously published formulas. Also, the results showed that both the ANN and EPR models have almost the same level of accuracy (about 92%), but the EPR model has the advantage of presenting a closed-form equation that could be implemented either manually or using software. Doi: 10.28991/CEJ-SP2024-010-07 Full Text: PDF

Closed-form equations for statically analyzing shaft-gear systems

Closed-form equations for statically analyzing shaft-gear systems

Outdoor-to-Indoor mmWave Relaying with Massive MIMO: Impact of Imperfect Channel Estimation

Assuming incomplete knowledge of the channel state information (CSI), we investigate two scenarios involving millimeter wave (mmWave) relaying to support outdoor-to-indoor communications. We proceed to derive the average signal-to-noise ratio (SNR) expressions for two relaying scenarios and quantify the asymptotic SNR. The performance of the two relaying scenarios is evaluated using the outage probability—for which we have derived closed-form equations—the end-to-end channel capacity, and the energy efficiency. The obtained results are compared with those derived assuming complete knowledge of the CSI. The effect of the imperfect CSI is therefore assessed in relation to the reference of perfect CSI. In these scenarios, an outside base station (BS) in an urban cellular network serves several indoor users. In the context of a two-hop full-duplex (FD) relaying scheme, we initially suggest a method in which the base station (BS) utilizes zero-forcing (ZF) precoding, and we take into account the overall channel response. Furthermore, we make the assumption that the base station (BS) engages in precoding only depending on the response of the channel in the first hop; in this second design, the relay precodes (using the response of the second-hop channel), amplifies, and sends the signals. Both techniques utilize massive multiple-input–multiple-output (mMIMO) arrays to permit transmission. We also present Monte Carlo simulation results to assess the accuracy of our analytical results. Finally, the two systems are compared in terms of channel estimation and precoding complexity, the number of antennas, as well as the number of users. Practical deployment recommendations are formulated at the end of this work.

Microstrip Quasi-Elliptic Absorptive Bandpass Filter with Ultra-Wide Reflectionless Range and Compact Size

Absorptive bandpass filters (ABPFs) are highly attractive in modern microwave communication systems due to their ability to internally absorb the harmful stopband RF-power reflections. This paper reports an approach to designing quasi-elliptic ABPFs with ultra-wide reflectionless range, enhanced selectivity, and compact size. The method is realized based on a fourth-order quasi-elliptic absorptive lowpass filter (ALPF) prototype with a simplified structure. This ALPF prototype exhibits both good impedance-matching over the whole normalized frequency domain and an adjustable transmission zero close to the passband. By applying an equivalent impedance transformer model, a coupled-line-based ABPF scheme is devised from the ALPF prototype, which eliminates conventional dispersive transmission line inverters, resulting in an ultra-wide reflectionless range and a compact size. Closed-form equations are derived to support the filter synthesis. A 2.45 GHz microstrip ABPF with 30% fractional bandwidth is designed for verification. The measured minimum in-band insertion loss is 0.83 dB and the reflectionless range of return loss better than 10 dB is from DC to 12.88 GHz. Both the upper and lower stopband suppression exceed 20 dB, with the upper stopband extending up to 6.80 GHz. The upper and lower out-of-band roll-off rates are 93.9 and 121.4 dB/GHz, respectively. The overall circuit size is 0.12 λg2.

Anomalous diffusion of synaptic vesicles and its influences on spontaneous and evoked neurotransmission.

Neurons in the central nervous system communicate with each other by activating billions of tiny synaptic boutons distributed along their fine axons. These presynaptic varicosities are very crowded environments, comprising hundreds of synaptic vesicles. Only a fraction of these vesicles can be recruited in a single release episode, either spontaneous or evoked by action potentials. Since the seminal work by Fatt and Katz, spontaneous release has been modelled as a memoryless process. Nevertheless, at central synapses, experimental evidence indicates more complex features, including non-exponential distributions of release intervals and power-law behaviour in their rate. To describe these features, we developed a probabilistic model of spontaneous release based on Brownian motion of synaptic vesicles in the presynaptic environment. To account for different diffusion regimes, we based our simulations on fractional Brownian motion. We show that this model can predict both deviation from the Poisson hypothesis and power-law features in experimental quantal release series, thus suggesting that the vesicular motion by diffusion could per se explain the emergence of these properties. We demonstrate the efficacy of our modelling approach using electrophysiological recordings at single synaptic boutons and ultrastructural data. When this approach was used to simulate evoked responses, we found that the replenishment of the readily releasable pool driven by Brownian motion of vesicles can reproduce the characteristic binomial release distributions seen experimentally. We believe that our modelling approach supports the idea that vesicle diffusion and readily releasable pool dynamics are crucial factors for the physiological functioning of neuronal communication. KEY POINTS: We developed a new probabilistic model of spontaneous and evoked vesicle fusion based on simple biophysical assumptions, including the motion of vesicles before they dock to the release site. We provide closed-form equations for the interval distribution of spontaneous releases in the special case of Brownian diffusion of vesicles, showing that a power-law heavy tail is generated. Fractional Brownian motion (fBm) was exploited to simulate anomalous vesicle diffusion, including directed and non-directed motion, by varying the Hurst exponent. We show that our model predicts non-linear features observed in experimental spontaneous quantal release series as well as ultrastructural data of synaptic vesicles spatial distribution. Evoked exocytosis based on a diffusion-replenished readily releasable pool might explain the emergence of power-law behaviour in neuronal activity.

Cooling load prediction of a double-story terrace house using ensemble learning techniques and genetic programming with SHAP approach

Since the cooling systems used in buildings in hot climates account for a significant portion of the energy consumption, it is very important for both economy and environment to accurately predict the cooling load and consider it in building designs. This study aimed to maximize energy efficiency by appropriately selecting the features of a building that affect its cooling load. To this end, data-driven, accurate, and accessible tools were developed that enable the prediction of the cooling load of a building by practitioners. The study involves simulating the energy consumption of a mid-rise, double-story terrace house in Malaysia using building information modeling (BIM) and estimating the cooling load using ensemble machine learning models and genetic programming. Categorical Boosting (CatBoost), eXtreme Gradient Boosting (XGBoost), Light Gradient Boosting Machine (LightGBM), and Random Forest (RF) models have been developed and made available as an online interactive graphical user interface on the Streamlit platform. Furthermore, the symbolic regression technique has been utilized to obtain a closed-form equation that predicts the cooling load. The dataset used for training the predictive models comprised 94,310 data points with 10 input variables and the cooling load as the output variable. Performance metrics such as the coefficient of determination (R2), root mean squared error (RMSE), and mean absolute error (MAE) were used to measure the predictive model performances. The results of the machine learning models indicated successful prediction, with the CatBoost model achieving the highest score (R2 = 0.9990) among the four ensemble models and the predictive equation. The SHAP analysis determined the aspect ratio of the building as the most impactful feature of the building.

Developing Closed-Form Equations of Maximum Drag and Moment on Rigid Vegetation Stems in Fully Nonlinear Waves

Developing Closed-Form Equations of Maximum Drag and Moment on Rigid Vegetation Stems in Fully Nonlinear Waves

On the Link between Plastic Wake Induced Crack Closure and the Fatigue Threshold

This purpose of this paper is to examine the relationship between crack growth equations based on Elber’s original plastic wake induced crack closure concept and the fatigue threshold as defined by the American Society for Testing and Materials (ASTM) fatigue test standard ASTM E647-15el. It is shown that, for a number of conventionally manufactured metals, the function U(R), where R is the ratio of the minimum to maximum applied remote stress, that is used to relate the stress intensity factor ΔK to the effective stress intensity factor ΔKeff is inversely proportional to the fatigue threshold ΔKth(R). This finding also results in a simple closed form equation that relates the crack opening stress intensity factor Ko(R) to ΔK, Kmax, and the fatigue threshold terms ΔKth(R) and ΔKeff,th. It is also shown that plotting da/dN as function of ΔK/ΔKth(R) would appear to have the potential to help to identify the key fracture mechanics parameters that characterise the effect of test temperature on crack growth. As such, for conventionally manufactured metals, plotting da/dN as function of ΔK/ΔKth(R) would appear to be a useful addition to the tools available to assess the fracture mechanics parameters affecting crack growth.

Stochastic model for enhanced PET image reconstruction

In this paper, we present a detailed stochastic model of crystal-to-crystal response for PET scanners, which is crucial for addressing their physical constraints. We present both a closed-form solution and several accurate approximations to manage the complexity of the exact mathematical expressions. Through experimental and analytical validation, we demonstrate that the disparities between the best approximations and actual crystal-to-crystal response are negligible. These responses are utilized to develop a compensation model termed a white image. This model is expressed in a single closed-form equation, simplifying its integration into established reconstruction methods. We modify the maximum likelihood expectation maximization (MLEM) algorithm and integrate our white image model into it. Unlike the conventional approach that relies on the system matrix, the modified MLEM algorithm operates on ray-driven projections and back-projections. The system matrix, being large and requiring complex methods and simulations for evaluation, contrasts with the white image, which is smaller and enables efficient computation. Our compensation model provides comprehensive information about the PET system. We validate our approach with both synthetic and real data. For real-world data acquisition, we utilize the Raytest ClearPET camera for small animals and the NEMA NU 4-2008 phantom. The proposed modification outperforms competing uncompensated reconstruction methods, delivering superior Peak Signal-to-Noise Ratio (PSNR) and reduced execution time. This research deepens our understanding of real PET measurement systems, providing improved input for subsequent reconstruction algorithms and ultimately producing high-quality images.