Closed-form Solution Research Articles

A computationally efficient thermo-mechanical model with temporal acceleration for prediction of residual stresses and deformations in WAAM

ABSTRACT The present paper introduces a novel temporal acceleration strategy for computationally efficient prediction of residual stresses and deformations in wire arc additive manufacturing (WAAM) components. It employs a semi-analytical approach, dividing temperature into the analytical and the complementary fields. The analytical field is obtained by a closed-form solution, and the complementary field is employed to solve the boundary conditions. A temporal acceleration factor for the heating period is applied. Meanwhile, the diffusion time in the analytical field is manipulated to guarantee accurate temperature prediction. Validation via WAAM experiments, including both a thin-wall structure and a practical engineering component with characteristic dimensions on the order of metres, indicates that the predicted stresses is 50 MPa lower than the experimental values. Moreover, the discrepancy of between the predicted deformations and the experimental measurements is less than 10%, demonstrating reasonable accuracy can be achieved with attractive computational efficiency.

Based on Symmetric Jump Risk Market: Study on the Ruin Problem of a Risk Model with Liquid Reserves and Proportional Investment

In order to deal with complex risk scenarios involving claims, uncertainty, and investments, we consider the ruin problems in a compound Poisson risk model with liquid reserves and proportional investments and study the expected discounted penalty function under threshold dividend strategies. Firstly, the integral differential equation of the expected discounted penalty function is derived. Secondly, since the closed-form solution of the equation cannot be obtained, a sinc method is used to obtain the numerical approximation solution of the equation. Finally, the feasibility and superiority of the sinc method are illustrated by error analysis. In addition, based on a symmetric jump risk market, we discuss the influence of some parameters on the ruin probability with some examples. This study can help actuaries develop more robust risk management strategies and ensure the long-term stability and profitability of insurance companies. It provides a theoretical basis for actuaries to carry out risk management.

Theoretical and Experimental Investigations of Identifying Bridge Damage Using Instantaneous Amplitude Squared Extracted from Vibration Responses of a Two-Axle Passing Vehicle

Identifying bridge damage using a movable test vehicle is highly regarded for its mobility, cost-effectiveness, and broad monitoring coverage. Previous studies have shown that the residual contact-point (CP) response between connected vehicles is free of the impact of vehicle self-vibrations and road roughness, making it particularly suitable for the indirect extraction of bridge modal properties. However, most experimental campaigns regarding contact-point (CP) responses focus on a single-axle testing vehicle within a non-moving state. This study aims to theoretically and experimentally identify bridge damage using the instantaneous amplitude squared (IAS) extracted from the residual CP response of a two-axle passing vehicle. First, the closed-form solution of the residual CP acceleration was derived for a two-axle vehicle interacting with a simply supported beam. The IAS index was constructed from the driving frequency of the residual CP acceleration. Then, numerical investigations using finite element simulation were conducted to validate using the IAS index for indirect bridge damage identification. The application scope of the approach under various vehicle speeds and road roughness grades was examined. Finally, a laboratory vehicle–bridge interaction system was tested to validate the approach. Numerical studies demonstrated that bridge damage could be directly determined by observing the IAS abnormalities, which were baseline-free. The IAS from the residual CP response outperformed the IAS from CP responses in identifying bridge damage. However, it was better to use the IAS when the vehicle speed was no greater than 2 m/s and the grade of the road surface roughness was not high. Laboratory tests showed that it was possible to identify bridge damage using the IAS extracted from the residual CP acceleration under perfect road surfaces. However, it fell short under rough road surfaces. Hence, further experiments are required to fully examine the capacity of the IAS for bridge damage identification in practical applications.

H∞ closed-form solution of tuned mass damper enhanced with negative stiffness element (TMD-NS) for structural vibration control

Negative stiffness element has been applied to improve the control performance of tuned mass dampers (TMDs) recently, and two tuned mass dampers enhanced with negative stiffness (TMD-NS) element, namely KDamper (Without loss of generality, it is referred to as TMD-NS I in the present study) and Extended KDamper (EKD, TMD-NS II), have been developed. Previous studies have demonstrated the control effectiveness of TMD-NS I and II. However, there still exist some issues to be addressed: (1) previous studies normally optimize TMD-NS via the intricate and time-consuming numerical methods, the analytical solutions for the optimal design parameters of TMD-NS II remain unknown; (2) a comprehensive and exhaustive evaluation that compares the control effectiveness of TMD-NS I and II is absent from existing literature. To fill these research gaps, this study derives closed-form optimum solutions for TMD-NS II using the H∞ approach. The control effectiveness of TMD-NS I and II in suppressing the seismic responses of structures is investigated systematically. Specifically, the analytical model of an undamped SDOF system equipped with TMD-NS I or II is first developed within a unified framework, and corresponding dynamic equations of motion are formulated. Subsequently, the optimal parameters of TMD-NS I and II are derived based on the classical “fixed-point” theory. Based on the derived optimal parameters, the control effectiveness of TMD-NS I and II are examined by using a damped SDOF system subjected to harmonic excitations and real earthquake ground motions. Finally, a 5-storey isolated benchmark building model is adopted to further investigate the effectiveness of TMD-NS in the seismic protection of engineering structures. The results reveal that the derived closed-form solutions are accurate in capturing the optimal parameters of TMD-NS. In addition, both the optimized TMD-NS I and II outperform the conventional TMD in reducing the seismic responses of structures. Furthermore, TMD-NS I proves more effective in reducing the absolute acceleration of the isolated building, whereas TMD-NS II demonstrates better performance in mitigating the isolating deformation. In a nutshell, both the TMD-NS I and II are highly effective alternatives to conventional TMDs, showcasing superior performance in vibration reduction and robustness.

Finite-element assembly approach of optical quantum walk networks

We present a finite-element approach for computing the aggregate scattering matrix of a network of linear coherent scatterers. These might be optical scatterers or more general scattering coins studied in quantum walk theory. While techniques exist for two-dimensional lattices of feed-forward scatterers, the present approach is applicable to any network configuration of any collection of scatterers. Unlike traditional finite-element methods in optics, this method does not directly solve Maxwell’s equations; instead it is used to assemble and solve a linear, coupled scattering problem that emerges after Maxwell’s equations are abstracted within the scattering matrix method. With this approach, a global unitary is assembled corresponding to one time step of the quantum walk on the network. After applying the relevant boundary conditions to this global matrix, the problem becomes non-unitary and possesses a steady-state solution that is the output scattering state. We provide an algorithm to obtain this steady-state solution exactly using a matrix inversion, yielding the scattering state without requiring a direct calculation of the eigenspectrum. The approach is then numerically validated on a coupled-cavity interferometer example that possesses a known, closed-form solution. Finally, the method is shown to be a generalization of the Redheffer star product, which describes scatterers on one-dimensional lattices (2-regular graphs) and is often applied to the design of thin-film optics, making the current approach an invaluable tool for the design and validation of high-dimensional phase-reprogrammable optical devices and study of quantum walks on arbitrary graphs.

Efficient subspace clustering and feature extraction via [formula omitted]-norm and [formula omitted]-norm minimization

The nuclear norm-based Latent Low-Rank Representation (LatLRR) has gained much attention due to its success in subspace clustering and feature extraction. However, it suffers from high computational costs due to the calculation of singular value decomposition for large matrices. To this end, we develop an efficient subspace clustering and feature extraction method (ESCFE) which substitutes the nuclear norm with the ℓ2,1-norm and ℓ1,2-norm respectively. Theoretically proof shows both the ℓ2,1-norm and ℓ1,2-norm can serve as the convex surrogates of the nuclear norm while can derive closed-form solutions. Furthermore, the ℓ2,1-norm (or ℓ1,2-norm) regularization promotes column (or row) structure sparsity due to the discriminative nature inherited from the ℓ1-norm. Thus the proposed ESCFE is robust to outliers in data and can extract features with joint sparsity. Extensive experiments on multiple benchmark datasets demonstrate the superiority of our method in both efficiency and effectiveness.

Analytical modeling and simulations of dynamic mode-III fracture in pre-stressed dry sandy elastic continuum

A mathematical model has been developed to investigate the SIF (stress intensity factor) and COD (crack opening displacement) for the propagation of mode-III fracture influenced by propagating SH-wave in a pre-stressed dry sandy elastic continuum. The Wiener-Hopf mathematical method along with the Fourier integral transform (FIT) technique have been successively implemented to obtain the solution of the boundary value problem associated with the mathematical model. Furthermore, the closed form solutions of SIF and COD for non-static and static conditions of mode-III fracture have been obtained, and are found to be in well agreement with the existing results present in the literature. The sound effects of various affecting parameters viz. horizontal and vertical (compressive/tensile) pre-stresses, crack depth, and crack velocity on SIF and COD have been explored. In order to delineate the influences of these aforementioned parameters on SIF and COD graphically, numerical simulation has been accomplished.

Dynamic Home Care Routing and Scheduling with Uncertain Number of Visits per Referral

Despite the rapid growth of the home care industry, research on the scheduling and routing of home care visits in the presence of uncertainty is still limited. This paper investigates a dynamic version of this problem in which the number of referrals and their required number of visits are uncertain. We develop a Markov decision process (MDP) model for the single-nurse problem to minimize the expected weighted sum of the rejection, diversion, overtime, and travel time costs. Because optimally solving the MDP is intractable, we employ an approximate linear program (ALP) to obtain a feasible policy. The typical ALP approach can only solve very small-scale instances of the problem. We derive an intuitively explainable closed-form solution for the optimal ALP parameters in a special case of the problem. Inspired by this form, we provide two heuristic reduction techniques for the ALP model in the general problem to solve large-scale instances in an acceptable time. Numerical results show that the ALP policy outperforms a myopic policy that reflects current practice, and is better than a scenario-based policy in most instances considered. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2018-05225 and RGPIN-2020-210524] and by the Telfer School of Management SMRG Postdoctoral Research Fellowship Support [Grant 2020]. Supplemental Material: The electronic companion is available at https://doi.org/10.1287/trsc.2023.0120 .

Thermal buckling analysis of a new type of rectangular FG sandwich plates using a new higher-order shear deformation theory

This article presents a novel higher-order shear deformation Theory to investigate the thermal buckling behavior of a new type of functionally graded material (FGM) sandwich structure. The study focuses on a specific configuration of rectangular FGM sandwich plate, consisting of FGM face layers and FGM core layer subjected to temperature gradients varying linearly and non-linearly across the thickness. By employing the principle of virtual work, the governing equations are derived, involving only four unknown functions. The Navier procedure is then utilized to obtain closed-form solutions. In this study, the material properties and thermal expansion coefficient of the two face layers of the sandwich plate are assumed to vary in the thickness direction according to a simple power-law distribution. In contrast, the core layer is assumed to have an exponential variation along the thickness direction. The critical thermal buckling is influenced by various factors, including the thickness, and aspect ratios of the sandwich plate, the gradient index, the type of loading, and the type of sandwich plate. The effects of these variables on the critical buckling temperature are thoroughly examined and analyzed.

Multi-Channel Cooperative Spectrum Sensing and Scheduling for Cognitive IoT Networks

This paper presents a novel multi-channel cooperative spectrum sensing and scheduling (MC$_2$S$_3$) framework for spectrum and energy harvesting cognitive Internet of Things (IoT) networks. We address the challenge of maximizing network throughput by formulating a combinatorial problem that jointly optimizes the sensing scheduling of primary channels (PCs), the assignment of IoT devices for sensing scheduled PCs, and the clustering and allocation of IoT nodes to efficiently use discovered idle PCs; subject to spectrum utilization and collision avoidance constraints. Recognizing the inherent complexity of the underlying NP-hard mixed-integer non-linear programming (MINLP) problem, we propose a decomposition strategy that decouples the master problem into PC exploration and exploitation sub-problems. In the exploration phase, we derive closed-form solutions for optimal sensing durations and detection thresholds that satisfies spectrum utilization and collision avoidance constraints, which are then used to develop a priority metric to rank PCs. The proposed PC ranking informs a sequential PC scheduling and IoT sensing assignment approach that exploits a linear bottleneck assignment (LBA) strategy, proceeding until further scheduling does not enhance network utility. For the exploitation phase, we leverage a non-orthogonal multiple access (NOMA) strategy to multiplex multiple IoT nodes on a single PC, employing an iterative linear sum assignment (LSA) method for efficient allocation to maximize utilization of idle PCs. Numerical results validate the efficacy of our proposed methodologies, reaching an accuracy of approximately 99% in the order of milliseconds, significantly outperforming time complexity of brute-force benchmarks.

Closed-form P-I diagram based on flextural and shear failure for ptotective wall

Protective walls are crucial structures in mitigating damage from explosions, particularly in environments such as gas refueling stations. However, evaluating their performance typically involves intricate numerical analyses to derive Pressure-Impulse (P–I) diagrams. This paper presents a novel closed-form solution for deriving P–I diagrams specifically tailored for protective walls, considering flexural and shear resistance. By assuming perfect plastic resistance, the proposed solution offers an efficient alternative to numerical analysis, particularly beneficial in preliminary design stages. The validity of the closed-form solution is demonstrated through comparisons with numerically derived P–I diagrams, showing promising results across various design parameters. This approach enhances the accessibility and ease of designing protective walls for blast resistance, contributing to improved safety measures in critical infrastructure.

Long-term solutions of calibrated and generalised Macdonald’s model for pedestrian-induced lateral forces

The inverted pendulum pedestrian model (IPM) for walking on laterally-oscillating structures, originally proposed by Macdonald, has been recently calibrated based on data from pedestrians walking on a laterally-oscillating instrumented treadmill. It was then generalised to be suitable for the application in a predictive manner, i.e. in simulations of lateral structural response for any population of pedestrians. The generalised IPM captures the fundamental pedestrian-structure interaction mechanism and pedestrian-generated lateral forces onto a structure, including the self-excited forces which are critical from the point of view of structural stability. However, since the closed-form solutions of the generalised IPM are currently unavailable, its application requires numerical integration. To address this shortcoming, closed form solutions for the long-term average lateral self-excited forces are derived in this study based on the framework introduced by McRobie. A parametric study is conducted revealing complex interdependencies between pedestrian and bridge behaviour and their influence on the self-excited forces. The presented results are related to the measurements from full-scale bridges and laboratory investigations.

Evaluation of T-stress in an infinite plate with double-crack under compression and shear

The T-stress at the crack tip under compression and shear has received more and more attention, but most of the theoretical researches have focused on single-crack, with little reported on the closed form solutions of T-stress at the double-crack or multi-crack tips. Therefore, the single-crack problem under compression and shear is first analyzed in this paper, the formulas for calculating the three components of T-stress (Tx, Ty and Txy) are obtained. According to the principle of superposition, the origins of the three components of T-stress are examined. Then the basic equations for solving the double-crack problems under compression and shear are established, leading to the formulas for calculating the three components of T-stress at the double-crack tips. Finally, the effects of the spacing between two cracks, the ratio of double-crack half-length, the dip angle of rock-bridge and the absolute angle between two cracks on the three components of T-stress are discussed. The results indicate that the Tx and Ty are unaffected by the friction on the crack surface, but the Txy is significantly correlated with it. Each component of T-stress at the inner and outer tips of double-crack is influenced by different geometrical factors and shows a different variation curve, which may be a potential factor contributing to the complexity of the double-crack or multi-crack problems under compression shear.

Odyssean forward guidance in normal times

We present a systematic treatment of dynamic effects of Odyssean forward guidance – explicit announcements about future policy rates. We focus on normal times unlike the mainstream of the literature that focuses on episodes of an effective (zero) lower bound on nominal interest rates. We present novel closed-form solutions to both deterministic and stochastic versions of a stylized new Keynesian model with fully anticipated future shocks. We establish a theorem that delineates the parameter space into real and complex-root regions characterized by different characteristic roots. We formulate a simple recipe for testing the effectiveness of forward guidance: the size of the smallest root is a single sufficient statistics that determines whether or not forward guidance generates significant contemporaneous effects.

Slender phoretic loops and knots

We present an asymptotic theory for solving the dynamics of slender autophoretic loops and knots. Our formulation is valid for nonintersecting three-dimensional center lines, with arbitrary chemical patterning and varying (circular) cross-sectional radius, allowing a broad class of slender active loops and knots to be studied. The theory is amenable to closed-form solutions in simpler cases, allowing us to analytically derive the swimming speed of chemically patterned tori, and the pumping strength (stresslet) of a uniformly active slender torus. Using simple numerical solutions of our asymptotic equations, we then elucidate the behavior of many exotic active particle geometries, such as a bumpy uniformly active torus that spins and a Janus trefoil knot, which rotates as it swims forwards. Published by the American Physical Society 2024

Gauge-Optimal Approximate Learning for Small Data Classification.

Small data learning problems are characterized by a significant discrepancy between the limited number of response variable observations and the large feature space dimension. In this setting, the common learning tools struggle to identify the features important for the classification task from those that bear no relevant information and cannot derive an appropriate learning rule that allows discriminating among different classes. As a potential solution to this problem, here we exploit the idea of reducing and rotating the feature space in a lower-dimensional gauge and propose the gauge-optimal approximate learning (GOAL) algorithm, which provides an analytically tractable joint solution to the dimension reduction, feature segmentation, and classification problems for small data learning problems. We prove that the optimal solution of the GOAL algorithm consists in piecewise-linear functions in the Euclidean space and that it can be approximated through a monotonically convergent algorithm that presents-under the assumption of a discrete segmentation of the feature space-a closed-form solution for each optimization substep and an overall linear iteration cost scaling. The GOAL algorithm has been compared to other state-of-the-art machine learning tools on both synthetic data and challenging real-world applications from climate science and bioinformatics (i.e., prediction of the El Niño Southern Oscillation and inference of epigenetically induced gene-activity networks from limited experimental data). The experimental results show that the proposed algorithm outperforms the reported best competitors for these problems in both learning performance and computational cost.

Optimal additional voluntary contribution in DC pension schemes to manage inadequacy risk

AbstractIn defined contribution pension schemes the member bears the investment risk and her main concern is to obtain an inadequate fund at retirement. To address inadequacy risk, flexibility is often given to the member to pay additional voluntary contributions (AVCs) into the fund. In many countries the AVC schemes allow members of the workplace pension plan to increase the amount of retirement benefits by paying extra contributions. In this paper, we define a target-based optimization problem where the member of an AVC scheme can choose at any time the investment strategy and the AVCs to the fund. In setting the problem, the member faces a trade-off between the importance given to the stability of payments during the accumulation phase and the achievement of the desired annuity at retirement. We derive closed-form solutions via dynamic programming and prove that (i) the optimal fund never reaches the target final fund, (ii) the optimal amount invested in the risky asset is positive, and (iii) the optimal AVC is higher than the target one. We run numerical simulations to allow for different member’s preferences, and perform sensitivity analyses to assess the controls’ robustness.

Temperature-dependent thermal buckling and free vibration behavior of smart sandwich nanoplates with auxetic core and magneto-electro-elastic face layers

This article addresses the thermomechanical thermal buckling and free vibration response of a novel smart sandwich nanoplate based on a sinusoidal higher-order shear deformation theory (SHSDT) with a stretching effect. In the proposed sandwich nanoplate, an auxetic core layer with a negative Poisson’s ratio made of Ti-6Al-4V is sandwiched between Ti-6Al-4V rim layers and magneto-electro-elastic (MEE) face layers. The MEE face layers are homogenous volumetric mixtures of cobalt ferrite (CoFe2O4) and barium titanate (BaTiO3). The mechanical and thermal material properties of the auxetic core and MEE face layers are temperature-dependent. Using Hamilton’s principle, governing equations are constructed. To characterize the size-dependent behavior of the nanoplate, governing equations are adapted with the nonlocal strain gradient theory (NSGT). By applying the principles of Navier’s technique, closed-form solutions are obtained. Parametric simulations are carried out to examine the effects of auxetic core parameters, temperature-dependent material properties, nonlocal parameters, electric, magnetic, and thermal loads on the free vibration and thermal buckling behavior of the nanoplate. According to the simulation results, it is determined that the auxetic core parameters, temperature-dependent material properties, and nonlocal factors significantly affect the thermomechanical behavior of the nanoplate. The outcomes of this investigation are expected to contribute to the advancement of smart nano-electromechanical systems, transducers, and nanosensors characterized by lightweight, exceptional structural integrity and temperature sensitivity. Also, the auxetic core with a negative Poisson’s ratio provides a metamaterial feature, and thanks to this feature, the proposed model has the potential to be used as an invisibility technology in sonar and radar-hiding applications.

Elastic solids under frictionless rigid contact and configurational force

A homogeneous elastic solid, bounded by a flat surface in its unstressed configuration, undergoes a finite strain when in frictionless contact against a rigid and rectilinear constraint, ending with a rounded or sharp corner, in a two-dimensional formulation. With a strong analogy to fracture mechanics, it is shown that (i.) a path-independent J–integral can be defined for frictionless contact problems, (ii.) which is equal to the energy release rate G associated with an infinitesimal growth in the size of the frictionless constraint, and thus gives the value of the configurational force component along the sliding direction. Furthermore, it is found that (iii.) such a configurational sliding force is the Newtonian force component exerted by the elastic solid on the constraint at the frictionless contact. Assuming the kinematics of an Euler–Bernoulli rod for an elastic body of rectangular shape, the results (i.)–(iii.) lead to a new interpretation from a nonlinear solid mechanics perspective of the configurational forces recently disclosed for one-dimensional structures of variable length. Finally, approximate but closed-form solutions (validated with finite element simulations) are exploited to provide further insight into the effect of configurational forces. In particular, two applications are presented which show that a transverse compression can lead to Eulerian buckling or to longitudinal dynamic motion, both realizing novel examples of soft actuation mechanisms. As an application to biology, our results may provide a mechanical explanation for the observed phenomenon of negative durotaxis, where cells migrate from stiffer to softer environments.

Joint Transmit and Receive Beamforming Design for DPC-Based MIMO DFRC Systems

This paper proposes an optimal beamforming strategy for a downlink multi-user multi-input–multi-output (MIMO) dual-function radar communication (DFRC) system with dirty paper coding (DPC) adopted at the transmitter. We aim to achieve the maximum weighted sum rate of communicating users while adhering to a predetermined transmit covariance constraint for radar performance assurance. To make the intended problem trackable, we leverage the equivalence of the weighted sum rate and the weighted minimum mean squared error (MMSE) to reframe the issue and devise a block coordinate descent (BCD) approach to iteratively calculate transmit and receive beamforming solutions. Through this methodology, we demonstrate that the optimal receive beamforming aligns with the traditional MMSE approach, whereas the optimal transmit beamforming design can be cast into a quadratic optimization problem defined on a complex Stiefel manifold. Based on the majorization–minimization (MM) method, an iterative algorithm is then developed to compute the optimal transmit beamforming design by solving a series of orthogonal Procrustes problems (OPPs) that admit closed-form optimal solutions. Numerical findings serve to validate the efficacy of our scheme. It is demonstrated that our approach can achieve at least 73% higher spectral efficiency than the existing methods in a high signal-to-noise ratio (SNR) regime.