Closed-form Solution Research Articles

Anisotropic Hyperelastic Strain Energy Function for Carbon Fiber Woven Fabrics

The present paper introduces an innovative strain energy function (SEF) for incompressible anisotropic fiber-reinforced materials. This SEF is specifically designed to understand the mechanical behavior of carbon fiber-woven fabric. The considered model combines polyconvex invariants forming an integrity basisin polynomial form, which is inspired by the application of Noether’s theorem. A single solution can be obtained during the identification because of the relationship between the SEF we have constructed and the material parameters, which are linearly dependent. The six material parameters were precisely determined through a comparison between the closed-form solutions from our model and the corresponding tensile experimental data with different stretching ratios, with determination coefficients consistently reaching a remarkable value of 0.99. When considering only uniaxial tensile tests, our model can be simplified from a quadratic polynomial to a linear polynomial, thereby reducing the number of material parameters required from six to four, while the fidelity of the model’s predictive accuracy remains unaltered. The comparison between the results of numerical calculations and experiments proves the efficiency and accuracy of the method.

Large displacement model for the nonlinear behaviour of the face plate component

This paper deals with the face plate component in steel joints and addresses the full characterization of the nonlinear behaviour of the face plate component. An equivalent beam strip model is proposed that tackles the connection of a beam to the column web of open I-sections in a minor axis joint or the face of tubular columns, covering endplate or fin plate joint typologies. Closed-form analytical solutions are obtained for the elastic large displacement and the elastic-plastic large displacement behaviour of the equivalent beam strip. Criteria for the establishment of the design resistance using the continuous strength method are also proposed. It was concluded that the model is easy to apply, was validated against a large parametric study using finite element beam models, leads to accurate solutions and demonstrates the need to consider membrane effects in design.

Closed-form kinematics solutions for redundant medical robots with joint limits and singularity avoidance

Closed-form kinematics solutions for redundant medical robots with joint limits and singularity avoidance

A new method for determining crack propagation

This paper presents a new method for obtaining approximate closed-form solutions for crack propagation problems. In general, it is not possible to obtain explicit solutions of the parameters and elementary functions of these problems, due to their terms complexity, present in their definitions. In this way, we used numerical methods to obtain approximate numerical solutions. The method presented in this article provides a proposal, which, when defined by differential equations, is applicable and extensible to other problems. The approximate solutions are obtained as an exact solution of a modified initial value problem. The performance is evaluated in relation to problems with known exact solution.

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.

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 .

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.

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

AbstractThis 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.

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.

Effect of crack surface friction on stress intensity factor for double-edge cracked Brazilian disk specimens under parabolic distribution load

The weight function method was employed to derive the closed-form solutions of stress intensity factors (SIFs) for double-edge cracked Brazilian disk (DECBD) specimens subjected to parabolic distribution loads, taking into account the influence of crack surface friction. Additionally, the accuracy and validity of these solutions were discussed in comparison with previous numerical results. Furthermore, the impact of friction coefficient and load distribution angle on SIFs was further investigated. The results indicate that the effect degree of load distribution angle on SIFs is closely correlated with both the loading angle θ and crack length α. Both the mode I and mode II SIFs increase with increasing load distribution angle γ for θ = 15° and α < 0.4. However, when α ≥0.60, the impact of load distribution angle γ on SIFs becomes negligible. Furthermore, the friction between crack surfaces of DECBD specimens significantly affects the dimensionless mode II SIF YII, leading to a decrease in dimensionless mode II SIF YII as the friction coefficient μ increases. Moreover, it further demonstrates that both the relative crack length α and friction coefficient μ play important roles in determining the critical loading angle θIIc for crack sticking. The critical loading angle θIIc for crack sticking increases with an increasing relative crack length; however, an increase in the friction coefficient leads to a reduction of this critical loading angle θIIc.

Joint Phase Shift Design and Resource Management for a Non-Orthogonal Multiple Access-Enhanced Internet of Vehicle Assisted by an Intelligent Reflecting Surface-Equipped Unmanned Aerial Vehicle

This paper integrates intelligent reflecting surfaces (IRS) with unmanned aerial vehicles (UAV) to enhance the transmission performance of the Internet of Vehicles (IoV) through non-orthogonal multiple access (NOMA). It focuses on strengthening the signals from cell edge vehicles (CEVs) to the base station by optimizing the wireless propagation environment via an IRS-equipped UAV. The primary goal is to maximize the sum data rate of CEVs while satisfying the constraint of the successive interference cancellation (SIC) decoding threshold. The challenge lies in the non-convex nature of jointly considering the power control, subcarrier allocation, and phase shift design, making the problem difficult to optimally solve. To address this, the problem is decomposed into two independent subproblems, which are then solved iteratively. Specifically, the optimal phase shift design is achieved using the deep deterministic policy gradient (DDPG) algorithm. Furthermore, the graph theory is applied to determine the subcarrier allocation policy and derive a closed-form solution for optimal power control. Finally, the simulation results show that the proposed joint phase shift and resource management scheme significantly enhances the sum data rate compared to the state-of-the-art schemes, thereby demonstrating the benefits of integrating the IRS-equipped UAV into NOMA-enhanced IoV.

The Pathologically Evolving Aggregation-State of Cells in Cancerous Tissues as Interpreted by Fractal and Multi-Fractal Dispersion Theory in Saturated Porous Formations

A recent author’s fractal fluid-dynamic dispersion theory in porous media has focused on the derivation of the associated nonergodic (or effective) macrodispersion coefficients by a 3-D stochastic Lagrangian approach. As shown by the present study, the Fickian (i.e., the asymptotic constant) component of a properly normalized version of these coefficients exhibits a clearly detectable minimum in correspondence with the same fractal dimension (d ≅ 1.7) that seems to characterize the diffusion-limited aggregation state of cells in advanced stages of cancerous lesion progression. That circumstance suggests that such a critical fractal dimension, which is also reminiscent of the colloidal state of solutions (and may therefore identify the microscale architecture of both living and non-living two-phase systems in state transition conditions) may actually represent a sort of universal nature imprint. Additionally, it suggests that the closed-form analytical solution that was provided for the effective macrodispersion coefficients in fractal porous media may be a reliable candidate as a physically-based descriptor of blood perfusion dynamics in healthy as well as cancerous tissues. In order to evaluate the biological meaningfulness of this specific fluid-dynamic parameter, a preliminary validation is performed by comparison with the results of imaging-based clinical surveys. Moreover, a multifractal extension of the theory is proposed and discussed in view of a perspective interpretative diagnostic utilization.

Møller-Plesset Adiabatic Connection at Large Coupling Strengths for Open-Shell Systems.

We study the adiabatic connection that has as weak-coupling expansion the Møller-Plesset perturbation series, generalizing to the open-shell case previous closed-shell results for the large-coupling limit. We first focus on the hydrogen atom with fractional spins, providing results along the adiabatic connection from small to large coupling strengths. We reveal an intriguing phase diagram and an equation for the large-coupling leading order that has closed-form solutions for specific choices of its relevant quantum numbers. We then show that the hydrogen atom results provide variational estimates for the large-coupling leading terms for the general many-electron open-shell case in terms of functionals of the Hartree-Fock α-spin and β-spin densities.

Analytical buckling scheme of a functionally graded porous liner reinforced by nanocomposites encased in an egg-shaped pipe

This study uses the FGP-GPLs (functionally graded porous metal reinforced by graphene platelets) to rehabilitate the egg-shaped liner-pipe system. The water penetrates the cracked pipe and surrounds the thin-walled liner, resulting in hydrostatic pressure that may contribute dominantly to the buckling of the thin-walled liner. The critical buckling pressure is derived analytically by combining the thin-walled shell theory, a refined displacement function, and the energy scheme. It is found the egg-shaped FGLs liner sustains higher buckling pressure than the circular FGP-GPLs one. Furthermore, a finite element model is developed to verify the theoretical solutions. The results indicate the numerical equilibrium paths agree well with the analytical ones. In addition, the present analytical predictions are compared successfully with other close-formed solutions. Finally, parametric analyses indicate that the addition of GPLs may significantly improve the stability of the egg-shaped liner, while the increase of pores decreases the stability of the liner. The thickness of the GPLs has a slight effect on the stability of the liner when the volume of GPLs is constant. Thus, thick GPLs may be recommended in engineering practices to reduce the manufacturing cost.

Efficient image restoration via non-convex total variation regularization and ADMM optimization

This article presents a novel approach to image restoration utilizing a unique non-convex l1/2-TV regularization model. This model integrates the l1/2-quasi norm as a regularization function, introducing non-convexity to promote sparsity and unevenly penalize elements, thereby enhancing restoration outcomes. To tackle this model, an efficient algorithm named the Alternating Direction Method of Multipliers, based on the Lagrangian multiplier, is introduced. This effectively prevents the penalty parameter from reaching infinity and ensures excellent convergence behavior. The proposed algorithm decomposes the optimization problem into subproblems, for which closed-form solutions are derived, particularly addressing the challenging l1/2 regularization problem. To validate its effectiveness, a comprehensive set of experiments are conducted to compare its performance with existing methods. The experimental results demonstrate that the proposed model performs well in both qualitative and quantitative evaluations. Consequently, the proposed model is not only efficient and stable but also exhibits excellent convergence behavior.