General Closed-form Expressions Research Articles

A closed-form expression derived from virtual testing for structural optimization of L-flange bolted connections

L-flange bolted joints are extensively used in different industrial sectors, such as wind turbine towers, rocket stages, or pipelines. Typically, the design of these connections is based on analytical expressions derived from simplified beam models. However, the influence of specific factors such as curvature, adjacent bolts, or joint separation are not addressed within conventional approaches. Finite element models offer the advantage of comprehensively accounting for these effects and the interaction between different failure mechanisms. Nevertheless, their industrial implementation requires significant computational resources, in contrast to the use of analytical formulae. In this context, this paper proposes a new framework for designing optimized L-flange bolted connections, based on general closed-form expressions that rely on validated numerical results. Three expressions, each one related to a particular failure mechanism, are derived from parametric simulations varying the most influencing parameters in the structural behavior. The generated equations can be easily implemented to guide the design process towards an optimal configuration, taking advantage of beneficial effects not addressed within classical approaches. A closer prediction of the actual load capacity of the joint can be obtained, providing a tool that allows engineers to make more reliable and economical designs.

General Solutions for MHD Motions of Ordinary and Fractional Maxwell Fluids through Porous Medium When Differential Expressions of Shear Stress Are Prescribed on Boundary

Some MHD unidirectional motions of the electrically conducting incompressible Maxwell fluids between infinite horizontal parallel plates incorporated in a porous medium are analytically and graphically investigated when differential expressions of the non-trivial shear stress are prescribed on the boundary. Such boundary conditions are usually necessary in order to formulate well-posed boundary value problems for motions of rate-type fluids. General closed-form expressions are established for the dimensionless fluid velocity, the corresponding shear stress, and Darcy’s resistance. For completion, as well as for comparison, all results are extended to a fractional model of Maxwell fluids in which the time fractional Caputo derivative is used. It is proven for the first time that a large class of unsteady motions of the fractional incompressible Maxwell fluids becomes steady in time. For illustration, three particular motions are considered, and the correctness of the results is graphically proven. They correspond to constant or oscillatory values of the differential expression of shear stress on the boundary. In the first case, the required time to reach the steady state is graphically determined. This time declines for increasing values of the fractional parameter. Consequently, the steady state is reached earlier for motions of the ordinary fluids in comparison with the fractional ones. Finally, the fluid velocity, shear stress, and Darcy’s resistance are graphically represented and discussed for the fractional model.

Bethe ansatz solutions and hidden [formula omitted] algebraic structure for a class of quasi-exactly solvable systems

The construction of analytic solutions for quasi-exactly solvable systems is an interesting problem. We revisit a class of models for which the odd solutions were largely missed previously in the literature: the anharmonic oscillator, the singular anharmonic oscillator, the generalized quantum isotonic oscillator, non-polynomially deformed oscillator, and the Schrödinger system from the kink stability analysis of ϕ6-type field theory. We present a systematic and unified treatment for the odd and even sectors of these models. We find generic closed-form expressions for constraints to the allowed model parameters for quasi-exact solvability, the corresponding energies and wavefunctions. We also make progress in the analysis of solutions to the Bethe ansatz equations in the spaces of model parameters and provide insight into the curves/surfaces of the allowed parameters in the parameter spaces. Most previous analyses in this aspect were on a case-by-case basis and restricted to the first excited states. We present analysis of the solutions (i.e. roots) of the Bethe ansatz equations for higher excited states (up to levels n=30 or 50). The shapes of the root distributions change drastically across different regions of model parameters, illustrating phenomena analogous to phase transition in context of integrable models. Furthermore, we also obtain the sl(2) algebraization for the class of models in their respective even and odd sectors in a unified way.

General Closed-Form Transfer Function Expressions for Fast Filter Bank

The existing literature focuses on the applications of fast filter bank due to its excellent frequency responses with low complexity. However, the topic is not addressed related to the general transfer function expressions of the corresponding subfilters for a specific channel. To do this, in this paper, general closed-form transfer function expressions for fast filter bank are derived. Firstly, the cascaded structure of fast filter bank is modelled by a binary tree, with which the index of the subfilter at each stage within the channel can be determined. Then the transfer functions for the two outputs of a subfilter are expressed in a unified form. Finally, the general closed-form transfer functions for the channel and its corresponding subfilters are obtained by variables replacement if the prototype lowpass filters for the stages are given. Analytical results and simulations verify the general expressions. With such closed-form expressions lend themselves easily to analysis and direct computation of the transfer functions and the frequency responses without the structure graph.

The mechanics and optimal design of micro-architectured stepped hexagonal lattices

This paper conceptualises and analyses a new class of stepped hexagonal lattice achieved through modifying the cross-section of the constituent beams in a controlled manner. The main idea lies in the redistribution of the mass of the constituent beam to obtain a range of equivalent elastic properties, which is not possible within the scope of regular hexagonal lattices. The mass of the stepped lattices is kept the same as the regular hexagonal lattices with equivalent geometry. The in-plane mechanics of such mass-conserved hexagonal lattices are investigated, considering stepped beams as constituent members. The mechanics of a unit cell is exploited to derive the closed-form analytical expressions of equivalent elastic properties of the lattice. The derivation utilises the stiffness elements of the constituent stepped beam members. The stiffness matrices are obtained using two different approaches. They include a semi-analytical condensed stiffness matrix approach based on static substructuring and an analytical method based Castigliano’s energy formulation. Both of these approaches are general and can handle arbitrary geometry for the constituent beams. New closed-form analytical expressions of equivalent elastic properties of the lattice are derived. The optimum geometric parameters are obtained by formulating an analytical optimisation problem. It is shown that a unique solution is possible by solving two simultaneous nonlinear equations. The general closed-form expressions of the equivalent elastic properties can be considered benchmark solutions. In the particular case of the regular lattice, they reduce to the well-known classical expressions. Numerical results show that the values of equivalent elastic properties of the lattice can change significantly by redistributing the mass of the constituent beams. It is demonstrated that up to a 37% increase in the equivalent elastic modulus can be achieved compared to the regular lattice by optimally choosing a stepped profile of the constituent elements.

Tailoring load balancing of cellular automata parallel execution to the case of a two-dimensional partitioned domain

In this paper, techniques for dynamic load balancing of the cellular automata parallel execution are presented for the case of domain space partitioned along two dimensions. Starting from general closed-form expressions that allow to compute the optimal workload assignment in a dynamic fashion when partitioning takes place along only one dimension, we tailor the procedure to allow partitioning and balancing along both dimensions. Both qualitative and quantitative experiments are carried out that assess performance improvement in applying load balancing for the case of two-dimensional partitioned domain, especially when the load balancing takes place along both dimensions.

New Statistical Results of Partial Sums of Ordered Gamma-Distributed RVs for Performance Evaluation of Wireless Communication Systems

With the increase in the complexity of communication systems, order statistics have become increasingly significant in performance evaluation, especially for advanced diversity techniques over fading channels. However, existing analytical approaches are not useful owing to their high complexity. In this paper, we present novel and insightful statistical results of the partial sums of ordered Gamma-distributed random variables. By adopting a unified analytical framework to obtain the desired joint probability density function from the joint moment-generating function, we redefine and derive the common key functions specialized for the Gamma distribution. The derived formula is presented in the form of either general closed-form expressions or at least a single integral form instead of the originally complicated multiple-integral form. In terms of a feasible application of our results, we present a complete and general closed-form analysis of the statistics of the combined received signal-to-noise ratio of the distributed cyclic delay diversity with the cyclic prefix single-carrier scheme. We also show that our analytical results can provide potential mathematical solutions for other wireless communication systems. Selected numerical examples are presented and discussed to illustrate the effectiveness of the applied approach and related results. All the derived analytical results were compared and verified by using Monte Carlo simulations to verify the accuracy of our analysis.

Average irradiance with boresight pointing errors for flat-topped beam under atmospheric turbulence

Flat-topped beams offer a promising solution to mitigate turbulence-induced spreading in free space optical communication (FSOC). Although average irradiance is fundamental to the performance of flat-topped beam-based FSOC, the impacts of FSOC boresight pointing errors on average irradiance have rarely been discussed in prior works. Herein, we establish a general closed-form expression of the average irradiance with boresight pointing errors for flat-topped beam-based FSOC. This theoretical expression has been verified through thin-phase screen. The results show that unequal standard variance and nonzero mean boresight pointing errors change the shape and central position of the average irradiance. Furthermore, the general closed-form expression of the average receiving power with boresight pointing errors is further deduced and verified. The boresight pointing errors induce an additional loss of the average received power. This work provides general closed-form expressions to analyze and design the link of flat-topped beam-based mobile FSOC in atmospheric turbulence.

General closed-form expressions for the three-dimensional vibrations of elastic bodies using the Ritz method

A novel Ritz-β approach is proposed for accurately determining the vibrational frequencies of a body based on its three-dimensional elasticity. In the conventional Ritz method, when admissible 3D displacement fields are expressed as either simple algebraic, Chebyshev, or Legendre polynomials, the volumetric integrals are transformed into beta integrals through a change of variables and then furthered simplified to beta functions and gamma functions; the resulting stiffness and mass matrices are expressed in a closed-form and the computational time is significantly decreased. The proposed approach does not involve the intensive computational procedures employed in the classical finite element, finite difference, and spectral approximation theories. Furthermore, it does not involve the kinematic constraints employed in classical beam, plate, and shell theories. The correctness and accuracy of the proposed approach were confirmed through comprehensive convergence studies and comparisons with existing results.

Energy- and Spectral- Efficiency Tradeoff in Nonlinear OFDM System of Visible Light Communications

light-emitting diode (LED) based visible light communication (VLC) provides a promising solution to enhance the speed and data rate for 5G and beyond, however, the nonlinear characteristic of LED that VLC greatly suffers from can lower the output power, hence causing degradation in the transmission rate. In this paper, the direct current (DC) biased optical orthogonal frequency division multiplexing (DCO-OFDM) is employed to investigate the nonlinear effect of LED on the tradeoff between energy efficiency (EE) and spectral efficiency (SE). The Rapps model is adopted to describe the nonlinear characteristic of commercial LED (OSRAM, LD-W5AM) and predistortion compensation algorithms are introduced to evaluate bit error rate (BER) affected by parameters of the LED. Based on which, the general closed-form expressions of EE-SE (ESE) tradeoff related to different system parameters under indoor VLC geometric scenario are derived. Next, effects of parameters in VLC system on the ESE tradeoff, such as the half power angle (HPA) of the LED and the effective area (EA) of the photodetector (PD), are discussed. Simulation results of the ESE tradeoff trend with different system parameters are provided, which are in well agreement with the theoretical derivation. This provides a good guideline on determining ESE-efficient parameters of the nonlinear DCO-OFDM VLC system.

Transformation acoustics with bulk media composed of polarized sources

Active acoustic metamaterials consisting of paired sensor-driver unit cells offer a promising path towards the practical realization of exciting transformation acoustics devices. The design of these cells is founded in a microscopic acoustic model that describes materials as collections of subwavelength polarized sources which respond to the local conditions of pressure and particle velocity. The current ability to express the polarizabilities that characterize these sources in terms of the effective macroscopic acoustic properties is limited to only a few simple cases and is not applicable to inhomogeneous bulk media of arbitrary geometries. Here, we address this challenge and derive general closed-form expressions relating the bulk modulus to the monopole polarizability and the mass density tensor to the dipole polarizability. Furthermore, we use these expressions to adapt transformation acoustics to the microscopic model. We demonstrate the accuracy of our approach by comparing the fields scattered by several devices, including cylindrical cloaks with steep property gradients and anisotropy, with the fields scattered by the devices' realizations with polarized sources.

A general analytical framework for the mechanics of heterogeneous hexagonal lattices

The in-plane mechanics of two-dimensional heterogeneous hexagonal lattices are investigated. The heterogeneity originates from two physically realistic considerations: different constituent materials and different wall thicknesses. Through the combination of multi-material and multi-thickness elements, the most general form of 2D heterogeneous hexagonal lattices is proposed in this paper. By exploiting the mechanics of a unit cell with multi-material and multi-thickness characteristics, exact closed-form analytical expressions of equivalent elastic properties of the general heterogeneous lattice have been derived. The equivalent elastic properties of the 2D heterogeneous lattice are Young’s modulli and Poisson’s ratios in both directions and the shear modulus. Two distinct cases, namely lattices with thin and thick constituent members, are considered. Euler–Bernoulli beam theory is employed for the thin-wall case, and Timoshenko beam theory is employed for the thick-wall case. The closed-form expressions are validated by independent finite element simulation results. The generalized expressions can be considered as benchmark solutions for validating future numerical and experimental investigations. The conventional single-material and single-thickness homogeneous lattice appears as a special case of the heterogeneous considered here. By introducing the Material Disparity Ratio (MDR) and Geometric Disparity Ratio (GDR), variability in the equivalent elastic properties has been graphically demonstrated. As opposed to classical homogeneous lattices, heterogeneous lattices significantly expand the design space for 2D lattices. Orders-of-magnitude of variability in the equivalent elastic properties is possible by suitably selecting material and geometric disparities within the lattices. The general closed-form expressions proposed in this paper open up the opportunity to design next-generation heterogeneous lattices with highly tailored effective elastic properties.

A New Compact Delay, Doppler Stretch and Phase Estimation CRB with a Band-Limited Signal for Generic Remote Sensing Applications

Since time-delay, Doppler effect and phase estimation are fundamental tasks in a plethora of engineering fields, tractable lower performance bounds for this problem are key tools of broad interest for a large variety of remote sensing applications. In the large sample regime and/or the high signal-to-noise ratio regime of the Gaussian conditional signal model, the Cramér–Rao bound (CRB) provides an accurate lower bound in the mean square error sense. In this contribution, we introduce firstly a new compact CRB expression for the joint time-delay and Doppler stretch estimation, considering a generic delayed and dilated band-limited signal. This generalizes known results for both wideband signals and the standard narrowband signal model where the Doppler effect on the band-limited baseband signal is not considered and amounts to a frequency shift. General compact closed-form CRB expressions for the amplitude and phase are also provided. These compact CRBs are expressed in terms of the baseband signal samples, making them especially easy to use whatever the baseband signal considered, therefore being valid for a variety of remote sensors. The new CRB expressions are validated in a positioning case study, both using synthetic and real data. These results show that the maximum likelihood estimator converges to the CRB at high signal-to-noise ratios, which confirms the exactness of the CRB. The CRB is further validated by comparing the ambiguity function and its 2nd order Taylor expansion where the perfect match also proves its exactness.

Information Transmission Bounds Between Moving Terminals

In networks of mobile autonomous agents, e.g. for data acquisition, we may wish to maximize data transfer or to reliably transfer a minimum amount of data, subject to quality of service or energy constraints. These requirements can be guaranteed through both offline node design/specifications and online trajectory/communications design. Regardless of the distance between them, for a stationary point-to-point transmitter-receiver pair communicating across a single link under average power constraints, the total data transfer is unbounded as time tends to infinity. In contrast, we show that if the transmitter/receiver is moving at any constant speed away from each other, then the maximum transmittable data is bounded. Although general closed-form expressions as a function of communication and mobility profile parameters do not yet exist, we provide closed-form expressions for particular cases, such as ideal free space path loss. Under more general scenarios we instead give lower bounds on the total transmittable information across a single link between mobile nodes.

Estimation of Lightning Incidence to Telecommunication Towers

General closed-form expressions for estimating lightning incidence to telecommunication towers are introduced on the basis of several lightning attachment models. The expressions consider the effects of tower height and lightning peak current distribution, as well as orographic and shielding effects. The estimated annual number of lightning strikes to towers, varying considerably among lightning attachment models, increases with tower height according to a power law and it is higher for lightning peak current distributions with high median value and standard deviation. A comparison is made between estimates and field data on lightning incidence; IEC geometric method, which is traditionally used in industry, by exaggerating the effect of tower height, overestimates lightning incidence. Literature lightning attachment models suitable for estimating lightning incidence to telecommunication towers are suggested based also on the observed failure rate of a transformer feeding a GSM base station.

Persymmetric generalised adaptive matched filter for range‐spread targets in homogeneous environment

This study deals with the problem of adaptively detecting range-spread targets in Gaussian environment, by assuming a persymmetric structure of covariance matrix. A persymmetric detector based on the two-step generalised likelihood ratio test design scheme is devised to reduce secondary data requirement, which ensures the constant false alarm rate property. Moreover, the general closed-form expressions of the probabilities of the false alarm and detection for the proposed detector are derived, in both cases of deterministic and fluctuating scatter models. Furthermore, the compact versions of forenamed expressions are presented in a special case when the joint space-time dimension of the received signal is odd. Finally, the numerical results of these derived statistical expressions are confirmed with those of Monte Carlo trials.

Why Lévy α-stable distributions lack general closed-form expressions for arbitrary α.

The ubiquitous Lévy α-stable distributions lack general closed-form expressions in terms of elementary functions-Gaussian and Cauchy cases being notable exceptions. To better understand this 80-year-old conundrum, we study the complex analytic continuation p_{α}(z), z∈C, of the symmetric Lévy α-stable distribution family p_{α}(x), x∈R, parametrized by 0<α≤2. We first extend known but obscure results, and give a new proof that p_{α}(z) is holomorphic on the entire complex plane for 1<α≤2, whereas p_{α}(z) is not even meromorphic on C for 0<α<1. Next, we unveil the complete complex analytic structure of p_{α}(z) using domain coloring. Finally, motivated by these insights, we argue that there cannot be closed-form expressions in terms of elementary functions for p_{α}(x) for generalα.

Multi-Hop D2D Communications With Network Coding: From a Performance Perspective

Multi-hop device-to-device (D2D) communications play an important role in expanding D2D coverage. In this paper, we study a relay-based and network-coding-assisted (in particular, XOR coding) multi-hop D2D communication system. In the system, toward jointly considering the impact of interference and network traffic conditions on the quality of D2D communications, various channel fading models and traffic models are investigated, and the packet loss probability of D2D links is meticulously computed using these models. With the packet loss probability of D2D links, the general closed-form expressions of end-to-end packet loss probability (E2EPLP) of the system with the presence (or absence) of XOR coding are subsequently derived. Our experiments reveal that both the number of relays and the mechanism of XOR coding can affect the system performance. Specifically, the increase in the number of relays will lower the overall system performance (e.g., an increase in the E2EPLP and end-to-end completion time, and a decrease of the end-to-end rate may follow as a result). On the other hand, although the presence of XOR coding unfortunately raises the system E2EPLP, it can effectively improve the end-to-end completion time and end-to-end rate. It is our belief that the analytical approach proposed in this paper and the results found in our work can be useful to peer studies in the context of applying network coding in multi-hop D2D networks.

Non‐orthogonal multiple access in full‐duplex relaying system with Nakagami‐ m fading

This study presents a full-duplex (FD) relaying non-orthogonal multiple access (NOMA) scheme with self-interference at relay taken into account, where a sender communicates with multiple terminals simultaneously with the help of an FD amplify-and-forward relay. The relay is equipped with two antennas, one for transmission and the other for reception. Moreover, with a direct link as a useful information transmission link, selection combining is applied to merge the information from the sender and the relay. General closed-form expressions of the outage probability are deduced and the upper bound of the ergodic sum rate is derived, including half-duplex (HD) mode relay as a special case. Compared with the HD relaying scheme, the authors can observe that the FD relaying scheme can achieve higher ergodic sum rate in the moderate- and high-SNR regions and that the outage behaviour can be improved in most cases (if the self-interference is not extremely severe). More importantly, the use of NOMA in FD relaying system is shown to outperform conventional orthogonal multiple access (OMA) network in terms of terminal fairness. Extensive simulation results are presented to confirm their conclusions and provide a significant reference for the system design.

Closed Form Expressions of Linear Continuous Time System Responses

General closed form expressions of linear continuous time system responses of an arbitrary order are derived, by first relating them to basic responses, i.e., responses corresponding to unity numerator transfer functions. Those are then related to the fundamental solutions of the underlying differential equations. These expressions apply to all systems without any restrictions on the poles or the zeros, further the systems may be noncausal. We derive responses for all regular types of inputs, impulse, step, ramp, parabola, etc., in addition for all generalized derivatives of the impulse. All the presented results have a direct counterpart in results presented in Sigurðsson et al. (2017) on discrete time systems based on the fundamental solution of the associated difference equation. Efficient evaluations of the fundamental solutions along with their derivatives and integrals can thus be extended to the responses and are readily implemented, e.g., as Matlab functions. Such results may be presented symbolically as functions of time or evaluated numerically at any sequence of times without time stepping.