Closed-form Function Research Articles

An analytical approach to predict post-cracking behavior of steel fiber reinforced concrete considering the fibers orientation

The presence of steel fiber in plain concrete improves significantly the stiffness as well as the ductility of the material. However, themechanical behavior in the post-cracking stage of steel fiber-reinforced concrete depends on many factors. Among them, the orientation of the fibers is an important factor that is difficult to quantify. This paper presents a theoretical study that describes the fiber bridging mechanism under tensile stress using a closed-form analytical function. This proposed analytical model allows estimating the stress at a crack openingwhere the fiber orientation is represented by a Gaussian-like periodic distribution function. The relevance of the proposed model is validated through several numerical analyses with both pre-defined values of different material properties and geometrical parameters of the fibers and the experimental parameters of the two directtensile tests reported in the literature. The model shows an overall reasonable estimation and presents a lot ofpotential for further development.

Performance enhancing ZNN models for time-variant equality-constraint convex optimization solving: A transition-state based attracting system approach

This paper presents designs of fixed-time zeroing neural networks (FxZNNs) for solving time-variant convex optimization problems, especially arising from repetitive motion planning of redundant manipulators. Through introducing the transition state, the bound estimation for settling time of the FxZNN models with double power-rate activation functions is carried out, and it is shown that the computing accuracy improvement can be made dramatically, in comparison to the models with the transition state being one. Novel two-phase FxZNN models with the pre-specified transition state are designed and analyzed in detail. The closed-form settling time functions are presented for each given initial condition, and the upper bounds on settling time are established in the presence of arbitrary initial conditions. The semi-global FxZNNs are designed to take account of initial conditions within the region with a specified radius. In addition, the modified FxZNN models assure predefined-time convergence also in the semi-global sense. The theoretical results demonstrate the flexibility and efficiency of the transition state introduced in the activation functions. The proposed FxZNNs are applied and compared with the existing models for solving a time-variant optimization problem and the repetitive motion planning of a redundant manipulator with initial errors, and numerical results are presented to verify effectiveness of the proposed FxZNN models.

An extended separation-of-variable method for the eigenbuckling of orthotropic open thin circular cylindrical shells

There have been numerous researches on buckling analysis of circular cylindrical shells, but still few researches on closed-form solutions for eigenbuckling of open circular cylindrical shells with non-Levy boundary conditions . This work develops an extended separation-of-variable (eSOV) method to obtain closed-form eigenbuckling solutions for orthotropic open circular cylindrical Donnell–Mushtari thin shells with arbitrary homogeneous boundary conditions. In this eSOV method, buckling mode functions are assumed to be the products of eigenfunctions of tangential and axial coordinate directions, and critical buckling loads corresponding to the two-direction eigenfunctions are assumed to be independent of each other. By employing Rayleigh’s principle, two-direction eighth-order characteristic differential equations are derived, and the two-direction eigenfunctions are expressed in terms of the eigenvalues of the characteristic differential equations, then the closed-form mode functions and explicit equations for critical buckling loads are achieved for arbitary homogeneous boundary conditions. Besides, the solutions presented in this work for open circular cylindrical shells can be simplified to those of closed circular cylindrical shells. The results of the proposed method agree well with those of FEM and other analytical methods in literatures, validating the present solutions.

Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial

This paper studies and analyzes the approximation of one-dimensional smooth closed-form functions with compact support using a mixed Fourier series (i.e., a combination of partial Fourier series and other forms of partial series). To explore the potential of this approach, we discuss and revise its application in signal processing, especially because it allows us to control the decreasing rate of Fourier coefficients and avoids the Gibbs phenomenon. Therefore, this method improves the signal processing performance in a wide range of scenarios, such as function approximation, interpolation, increased convergence with quasi-spectral accuracy using the time domain or the frequency domain, numerical integration, and solutions of inverse problems such as ordinary differential equations. Moreover, the paper provides comprehensive examples of one-dimensional problems to showcase the advantages of this approach.

Modeling unsteady and steady 1D hydrodynamics under different hydraulic conceptualizations: Model/Software development and case studies

The stage-discharge relationship is affected by hysteresis, especially in 1D river cross-sections with very mild slopes, and/or where backwater effects occur. To model these cases, hydraulic property (HP) estimations (e.g., hydraulic radius and conveyance) are usually required as closed-form functions of stage. However, these are typically unavailable for complex cross-sections with overbanks. For no sediment-laden flows, we created the HydroHP - 1D tool to solve the 1D Saint Venant Equations aided by the cross-section HP stored in tables. It also simulates irregular and composite hydraulic cross-sections, allows hydrodynamic simulation using the single-channel method (SCM), and proposes the divided channel method (DCM) for steady and unsteady flow analysis. A wide range of applications in rivers such as the Little Washita River and the Upper Negro River, and comparisons with NOAA normal depth solver, HEC-RAS, and with an analytical solution shows a promising scenario of applying HydroHP - 1D to estimate 1D steady and unsteady hydrodynamics.

Instantaneous Frequency and Chirp Rate Estimation for Noisy Quadratic FM Signals by CNN

Deep learning and machine learning are widely employed in various domains. In this paper, Artificial Neural Network (ANN) and Convolution Neural Network (CNN) are used to estimate the Instantons Frequency (IF), Linear Chirp Rate (LCR), and Quadratic Chirp Rate (QCR) for Quadratic Frequency Modulated (QFM) signals under Additive White Gaussian (AWG) noise and Additive Symmetric alpha Stable (ASαS) noise. SαS distributions are impulsive noise disturbances except for a few circumstances, lack a closed-form Probability Density Function (PDF), and an infinite second-order statistic. Geometric SNR (GSNR) is used to determine the impulsiveness of mixture noise for Gaussian and SαS noise. ANN is a machine learning classifier with few layers that reduce FE, LCRE, and QCRE complexity and achieve high accuracy. CNN is a deep learning classifier that is built with multiple layers of FE, LCRE, and QCRE. CNN is more accurate than ANN when dealing with large amounts of data and determining optimal features. The results reveal that SαS noise is substantially more damaging to FE, LCRE, and QCRE than Gaussian noise, even when the magnitude is modest, and it is less damaging when alpha is greater than one. After training DCNN for FE, LCRE, and QCRE estimation of QFM signals. The 2D-CNN model accuracy achieved 98.7603 and 1D-CNN is 75.8678 for ten epochs. ANN model accuracy achieved 37.5 for 1000 epochs. The accuracy of TFD (spectrogram & pspectrum) for frequency estimation of QFM signals was 38.4254 by spectrogram and 38.6746 by pspectrum.

Amortized Bayesian Model Comparison With Evidential Deep Learning.

Comparing competing mathematical models of complex processes is a shared goal among many branches of science. The Bayesian probabilistic framework offers a principled way to perform model comparison and extract useful metrics for guiding decisions. However, many interesting models are intractable with standard Bayesian methods, as they lack a closed-form likelihood function or the likelihood is computationally too expensive to evaluate. In this work, we propose a novel method for performing Bayesian model comparison using specialized deep learning architectures. Our method is purely simulation-based and circumvents the step of explicitly fitting all alternative models under consideration to each observed dataset. Moreover, it requires no hand-crafted summary statistics of the data and is designed to amortize the cost of simulation over multiple models, datasets, and dataset sizes. This makes the method especially effective in scenarios where model fit needs to be assessed for a large number of datasets, so that case-based inference is practically infeasible. Finally, we propose a novel way to measure epistemic uncertainty in model comparison problems. We demonstrate the utility of our method on toy examples and simulated data from nontrivial models from cognitive science and single-cell neuroscience. We show that our method achieves excellent results in terms of accuracy, calibration, and efficiency across the examples considered in this work. We argue that our framework can enhance and enrich model-based analysis and inference in many fields dealing with computational models of natural processes. We further argue that the proposed measure of epistemic uncertainty provides a unique proxy to quantify absolute evidence even in a framework which assumes that the true data-generating model is within a finite set of candidate models.

Spatial Channel Degrees of Freedom for Optimum Antenna Arrays

One of the ultimate goals of future wireless networks is to maximize data rates to accommodate bandwidth-hungry services and applications. Thus, extracting the maximum amount of information bits for given spatial constraints when designing wireless systems will be of great importance. In this paper, we present antenna array topologies that maximize the communication channel capacity for given number of array elements while occupying minimum space. Capacity is maximized via the development of an advanced particle swarm optimization (PSO) algorithm devising optimum standardized and arbitrarily-shaped antenna array topologies. Number of array elements and occupied space are informed by novel heuristic spatial degrees of freedom (SDoF) formulations which rigorously generalize existing SDoF formulas. Our generalized SDoF formulations rely on the differential entropy of three-dimensional (3D) angle of arrival (AOA) distributions and can associate the number of array elements and occupied space for any AOA distribution. The proposed analysis departs from novel closed-form spatial correlation functions (SCFs) of arbitrarily-positioned array elements for all classes of 3D multipath propagation channels, namely, isotropic, omnidirectional, and directional. Extensive simulation runs and comparisons with existing trivial solutions verify correctness of our SDoF formulations resulting in optimum antenna array topologies with maximum capacity performance and minimum space occupancy.

Solving recurrences for Legendre–Bernstein basis transformations

The change of basis matrix M from shifted Legendre to Bernstein polynomials and M−1 have applications in computer graphics. Algorithms use their properties to find the matrix elements efficiently. We give new functions for the elements of M as a summation, and a complete hypergeometric function. We find that Gosper’s algorithm does not produce closed-form expressions for the elements of either M or M−1. Zeilberger’s algorithm produces four second-order recurrences for the elements of the matrices that enable them to be computed in O(n) time, and for the derivation of closed-form functions by row and column for the elements. Two row recurrences are special cases of those found by Woźny (2013) who used a different method. We show that the recurrences for rows of M and columns of M−1 are equivalent. The recurrences for columns of M and rows of M−1 generate functions that are the Lagrange interpolation polynomials of their elements. These polynomials are equal to hypergeometric functions, which are solutions of the recurrences.

Lunar descent and landing via two-phase explicit guidance and pulse-modulated reduced-attitude control

This work considers the three-dimensional descent path of a space vehicle, from periselenium of its operational orbit to the lunar surface. The trajectory is split in two arcs: (1) descent path, up to an altitude of 50 m, and (2) terminal approach and soft touchdown. For phase 1, a new, three-dimensional locally-flat near-optimal guidance is introduced that is based on the local projection of the position and velocity variables. A minimum-time problem is defined using the locally flat coordinates of position and velocity. This leads to identifying closed-form functions of time for the two thrust angles, which identify the commanded thrust direction. During terminal approach (phase 2), correct vertical alignment, modest velocity, and negligible angular rate at touchdown are pursued. With this intent, a predictive bang-off-bang guidance algorithm is proposed that is capable of guaranteeing the desired final conditions. In both phases, the attitude control system has the objective of aligning the actual thrust direction with the commanded one, provided by the guidance algorithm. The resulting reduced-attitude control problem is addressed through the use of a new quaternion-based nonlinear control algorithm, which is proven to enjoy asymptotic stability properties. The attitude actuation system is composed of 12 monopropellant thrusters, ignited using pulse width modulation. Monte Carlo simulations are run, assuming significant displacements from the nominal initial conditions and including several harmonics of the selenopotential. The numerical results unequivocally prove that the guidance and control architecture proposed in this study is effective to achieve lunar descent and safe touchdown in nonnominal flight conditions.

A quantitative analysis procedure for solving safety factor of tunnel preliminary support considering the equivalence between Hoek–Brown and Mohr–Coulomb criteria

There is a development trend from qualitative towards quantitative analysis for tunnel stability. The widely accepted convergence-confinement method (CCM) provides a quantitative framework for tunnel stability analysis, while the conventional CCM based on closed-form functions is essentially an ideal model without considering the tunnel section shape and rock weight in plastic zone. Herein, based on the CCM framework, a quantitative analysis procedure formulated through numerical simulations for more practical situations was systematically proposed to calculate the safety factor of spray-anchor support system in non-circular tunnel. Besides, similarities and differences between the most widely used Hoek-Brown (HB) and equivalent Mohr-Coulomb (EMC) criteria were considered for practical reference. The proposed method was utilized to conduct quantitative analysis of LDPs, mechanical responses of surrounding rock and safety factors for four typical tunnels, and the consistency between tunnels based on HB criterion (TBHBC) and EMC criterion (TBEMCC) was analyzed. The results indicated that the maximum radial convergence of LDP, plastic zone area, and safety factor of TBEMCC were 34.54%, 37.29%, and 47.71% lower than those of TBHBC respectively. The reasons for differences between results were revealed. The spatial effect induced by the face excavation of TBEMCC was weakened as compared to TBHBC, leading to a drop of LDP. Consequently, the safety factor was underestimated. The method systematically provided an effective tool for quantitative stability analysis and design optimization of support system for an underground excavation.

Group properties and similarity transformations for generalized peakon equation with cubic nonlinearities

We perform a detailed analysis of the point transformations which leave invariant the Geng-Xue equation. We find that the Geng-Xue equation admits six Lie point symmetries which possess two three-dimensional subalgebras, the A 2,1 ⊗s A 1 and A 3,8 Lie algebras. For the Lie point symmetries we derive the one-dimensional optimal system and we perform a classification of the corresponding invariant transformations. We demonstrate the application of the Lie symmetries by deriving similarity solutions expressed by closed-form functions.

A Hierarchical Expected Improvement Method for Bayesian Optimization

The Expected Improvement (EI) method, proposed by Jones, Schonlau, andWelch, is a widely used Bayesian optimization method, which makes use of a fitted Gaussian process model for efficient black-box optimization. However, one key drawback of EI is that it is overly greedy in exploiting the fitted Gaussian process model for optimization, which results in suboptimal solutions even with large sample sizes. To address this, we propose a new hierarchical EI (HEI) framework, which makes use of a hierarchical Gaussian process model. HEI preserves a closed-form acquisition function, and corrects the over-greediness of EI by encouraging exploration of the optimization space. We then introduce hyperparameter estimation methods which allow HEI to mimic a fully Bayesian optimization procedure, while avoiding expensive Markov-chain Monte Carlo sampling steps. We prove the global convergence of HEI over a broad function space, and establish near-minimax convergence rates under certain prior specifications. Numerical experiments show the improvement of HEI over existing Bayesian optimization methods, for synthetic functions and a semiconductor manufacturing optimization problem. Supplementary materials for this article are available online.

Constitutive model for steel fiber reinforced concrete under shear and tension accounting for fiber orientation effect

This paper presents a constitutive model for the behavior of steel fiber reinforced concrete (SFRC) under mixed mode cracking taking into account the fiber orientation effect by using a micromechanical approach. Firstly, the resistant mechanisms of an inclined fiber are modeled by a closed-form analytical function. The bridging force is calculated by integrating on the cracked surface, the product of the fiber extraction force by its orientation probability. A modified model of the aggregate interlock mechanism is combined with the fiber bridging stresses to obtain the cohesive stresses. This mesoscale model is then extended using an elastic damage model and a fixed smeared cracking concept to assess the structural behavior. The model is finally implemented in a Finite Element Analysis (FEA) software, its performance is tested with some experimental results collected in the literature. The model successfully integrates the fiber orientation effect and shows a good prediction of stress transmission.

Optimize to generalize in Gaussian processes: An alternative objective based on the Rényi divergence

We introduce an alternative closed-form objective function α-ELBO for improved parameter estimation in the Gaussian process ( G P ) based on the Rényi α-divergence. We use a decreasing temperature parameter α to iteratively deform the objective function during optimization. Ultimately, our objective function converges to the exact log-marginal likelihood function of G P . At early optimization stages, α-ELBO can be viewed as a regularizer that smoothes some unwanted critical points. At late stages, α-ELBO recovers the exact log-marginal likelihood function that guides the optimizer to solutions that best explain the observed data. Theoretically, we derive an upper bound of the Rényi divergence under the proposed objective and derive convergence rates for a class of smooth and non-smooth kernels. Case studies on a wide range of real-life engineering applications demonstrate that our proposed objective is a practical alternative that offers improved prediction performance over several state-of-the-art inference techniques.

An efficient derivation of spatial Green's function of open-radiating rectangular cavity using CGF-CI technique

ABSTRACT A new technique for efficient calculation of spatial Green's function of open-radiating rectangular cavity is presented. The method is based on separability assumption of the original structure and characteristic Green's function method. Using complex image (CI) technique in spectral domain and well-known Weyl's identity, a closed-form relation for spatial Green's function can be derived. To obtain more accurate closed-form Green's function, before applying CI, discrete spectrum contribution part of the derived relation is modified by replacing more accurate reflection coefficients of surface wave modes from walls of the open-radiating cavity. Verification of the derivations of the proposed method is shown for dielectric loaded metallic box which is a separable structure. Method of moment + multilevel fast multipole method solution and also exact numerical integration approach for open-radiating rectangular cavity are considered to search the accuracy as well as speed of the proposed method. It is shown that the main advantage of the proposed method lies in its speed as well as its simple implementation while the accuracy is preserved.

Analytically pricing variance and volatility swaps under a Markov-modulated model with liquidity risks

This paper determines strike prices of discretely sampled variance/volatility swaps taking into account stochastic liquidity risks and the switching of economic conditions. We adopt nonlinear regime switching volatility to reflect how asset prices are affected by economic cycles, and market prices of assets are discounted according to the level of market liquidity. We then establish a risk-neutral measure under regime switching Esscher transform, so that analytical valuation of variance/volatility swaps can be completed based on the closed-form forward characteristic function. The limiting behavior of discretely sampled variance/volatility swaps is also considered through the investigation of pricing continuously sampled variance/volatility swaps. Finally, based on the results from numerical implementation, we confirm that the new model is very flexible in reflecting different influence associated with common real market observations.

Analytical Green's function for the acoustic scattering by a flat plate with a serrated edge

An analytical Green's function is developed to study the acoustic scattering by a flat plate with a serrated edge. The scattered pressure is solved using the Wiener–Hopf technique in conjunction with the adjoint technique. It is shown that the kernel decomposition proposed in recent literature appears only valid at high frequencies. We focus on this high-frequency regime and obtain the scattered pressure in the form of a contour integral. We show that such an integral, although complicated, can be evaluated exactly for any arbitrary piecewise linear serrations, for which closed-form analytical Green's functions are obtained. The derivation is validated by performing numerical integration of the contour integral showing excellent agreement. The Green's function is shown to agree well with the numerical results obtained using the finite element method at high frequencies. The noise directivity patterns are studied as a function of the frequency, serration amplitude, source position and Mach number. It is found that noise is often enhanced at low observer angles and may be slightly reduced at high observer angles, which may be understood from the perspective of an extended or removed rigid reflection surface. It is found that increasing the mean-flow Mach number leads to increasingly evident noise amplification at side angles, a seemingly strange Doppler behaviour exhibited in source-fixed coordinate frames. The analytical Green's function is applicable to both leading- and trailing-edge scatterings, and is particularly suitable for developing a three-dimensional trailing-edge noise model that is not only highly efficient but also capable of including non-frozen turbulence effects.

Optimal trauma care network redesign with government subsidy: A bilevel integer programming approach

Trauma presents a prominent health problem worldwide. However, trauma centers are often clustered in urban areas and sparsely located in rural areas. The geographic maldistribution of trauma centers leads to system-related mistriage errors. While some local governments offer subsidy to incentivize the affiliated hospital group to redesign the trauma care network, the approach is ad hoc. To address this issue, we propose a bilevel integer programming model to investigate the subsidized trauma care network redesign problem, which considers the government as the leader and the hospital group as the follower. To solve the resultant problem efficiently, we propose a branching idea to exclude additional infeasible solutions and suboptimal solutions, in turn speeding up the branch-and-bound algorithm. In a case study, we redesign a trauma care network in the midwestern area of the U.S. based on closed-form approximate functions of system-related mistriage errors. The results show that the optimal network redesign redistributes the network by slightly reducing the number of trauma centers to relieve the crowded trauma care resource, and achieves an overall improvement of about 11% over the original network.

An optimization method for frequency-invariant beamforming with arbitrary sensor arrays

This paper proposes a beampattern optimization method for frequency-invariant beamforming with arbitrary sensor arrays, which is an extension of the previously proposed method for circular sensor arrays. Based on the criterion of minimizing the mean square error between the synthesized beampatterns at each frequency and the desired beampattern, the relationship between the weighting vectors and the desired weighting vector can be accurately obtained. Then, the relevant performance parameters, such as the directivity factor, error sensitivity function and minimum mean square error, can all be expressed as the accurate closed-form functions of the desired weighting vector. With the above conclusions, a multi-constraint optimization problem is formulated to calculate the optimal desired weighting vector under different constraints. The optimal frequency-invariant beampatterns and their performance parameters can be readily achieved using the derived functions. Thus, the proposed method can not only obtain the optimal desired weighting vector, but also achieve the broadband frequency-invariant beampatterns with only one optimization calculation of the desired weighting vector. Simulations and experimental results show that the proposed method can provide a good trade-off among the directivity, robustness, and frequency invariance.