Closed-form Solution Research Articles

BDFLUX: A new bounded domain model for quantifying vertical fluxes in streambeds using heat as a tracer

The use of heat as a tracer has become a widely used approach to estimate vertical streambed fluxes (VSFs). We present BDFLUX, a new bounded domain model that can be used as a rapid-screening tool to quantify VSFs and thermal properties of streambed sediments by analyzing on-site temperature–time series. BDFLUX uses a closed-form analytical solution that takes thermal advection and thermal conduction into account. BDFLUX, unlike other one-dimensional closed-form analytical solutions, does not assume a semi-infinite half-space. The initial and boundary conditions are described by arbitrary functions that can be directly determined by fitting the measured temperature–time series in the streambed at various depths. The particle swarm optimization (PSO) method and Markov Chain Monte Carlo (MCMC) method are used to estimate VSFs using temperature–time series observed at three depths. BDFLUX is validated using synthetic temperature–time series, demonstrating that it can perform well when the top and bottom boundary conditions are determined directly by the measured data. The capabilities of BDFLUX are further evaluated using laboratory and field data. The simulation results show that BDFLUX performed well in terms of interpreting laboratory experiments and field data, with a root mean square error as low as 0.059. The simulation of laboratory experiments shows that the MCMC method has a lower computational efficiency than PSO method. The magnitude of nonvertical flow components at a field site were quantified by analyzing the results of estimated VSFs in different vertical subdomains. Overall, BDFLUX improves one-dimensional heat analytical models in terms of assessing the magnitude of the nonvertical flow component in heterogeneous streambed sediments.

Benchmark-driven investment for DC pension plans

Abstract We investigate whether a benchmark and non-constant risk aversion affect the probability density distribution of optimal wealth at retirement. We maximize the expected utility of the ratio of pension wealth at retirement to an inflation-indexed benchmark. Together with a threshold and a lower bound, we are able to generate closed-form solutions. We find that this non-constant risk aversion type of utility could shift the probability density distribution of optimal wealth more towards the benchmark, and that the probability of achieving a certain percentage of the desired benchmark could be increased. The probability density distribution generated under constant relative risk aversion (CRRA) risk preference is more widely spread along the benchmark.

A warm-start FE-dABCD algorithm for elliptic optimal control problems with constraints on the control and the gradient of the state

In this paper, elliptic control problems with the integral constraint on the gradient of the state and the box constraint on the control are considered. The optimality conditions for the problem are proved. To numerically solve the problem, a finite element duality-based inexact majorized accelerated block coordinate descent (FE-dABCD) algorithm is proposed. Specifically, both the state and the control are discretized by piecewise linear functions. An inexact majorized ABCD algorithm is employed to solve the discretized problem via its dual, which is a multi-block unconstrained convex optimization problem, but the primal variables are also generated in each iteration. Thanks to the inexactness of the FE-dABCD algorithm, the subproblems at each iteration are allowed to be solved inexactly. For the smooth subproblem, we use the preconditioned generalized minimal residual (GMRES) method to solve it. For the two nonsmooth subproblems, one of them has a closed form solution through introducing an appropriate proximal term, and another one is solved by the line search Newton's method. Based on these efficient strategies, we prove that our proposed FE-dABCD algorithm enjoys O(1k2) iteration complexity. Moreover, to make the algorithm more efficient and further reduce its computation cost, based on the mesh-independence of ABCD method, we propose an FE-dABCD algorithm with a warm-start strategy (wFE-dABCD). Some numerical experiments are done and the numerical results show the efficiency of the FE-dABCD algorithm and wFE-dABCD algorithm.

New closed-form exact solution of action induced magnetic field and fluid wall interaction on radiative mixed convective heat transfer MHD fluid flow inside a vertical walls with suction/injection and pressure gradient

The novelty of this article is not only the adaptation of a new closed-form exact method originally used to solve initial value problems but also to solve boundary value problems whose boundary conditions contain variable values. The boundary conditions of governing equations of velocity, induced magnetic, and temperature for the current study contain different parameters such as (Wall ambient temperature, Knudsen number, and Fluid wall interaction parameter) and, contain gradient of velocity, induced magnetic, and temperature. Using the Laplace transform method, the exact solution of the governing equations was obtained, and the effects of different parameters on its behavior in three different cases of wall-ambient temperature difference ratio was studied. I would also like to clarify that the reason for paying attention to such studies is that they contribute effectively and usefully in many fields and applications, such as molten metal MHD pumps, the design of small pumps, molten salt, and heat exchangers.

Use of patellofemoral digital twins for patellar tracking and treatment prediction: comparison of 3D models and contact detection algorithms.

Introduction: Poor patellar tracking can result in painful contact pressures, patella subluxation, or dislocation. The use of musculoskeletal models and simulations in orthopedic surgeries allows for objective predictions of post-treatment function, empowering clinicians to explore diverse treatment options for patients. Although a promising approach for managing knee surgeries, the high computational cost of the Finite Element Method hampers its clinical usability. In anticipation of minimal elastic deformations in the involved bodies, the exploration of the Multibody Dynamics approach emerged as a viable solution, providing a computationally efficient methodology to address clinical concerns related to the knee joint. Methods: This work, with a focus on high-performance computing, achieved the simulation of the patellofemoral joint through rigid-body multibody dynamics formulations. A comparison was made between two collision detection algorithms employed in the simulation of contact between the patellar and femoral implants: a generic mesh-to-mesh collision detection algorithm, which identifies potential collisions between bodies by checking for proximity or overlap between their discretized mesh surface elements, and an analytical contact algorithm, which uses a mathematical model to provide closed-form solutions for specific contact problems, but cannot handle arbitrary geometries. In addition, different digital twins (3D model geometries) of the femoral implant were compared. Results: Computational efficiency was considered, and histories of position, orientation, and contact force of the patella during the motion were compared with experimental measurements obtained from a sensorized 3D-printed test bench under pathological and treatment scenarios. The best results were achieved through a purely analytical contact detection algorithm, allowing for clinical usability and optimization of clinical outcomes.

A closed-form solution for evaluating the Findley critical plane factor

The fatigue assessment of structural components is a significant topic investigated both in the academia and industry. Despite the significant progress in comprehension over the past few decades, fatigue damage remains a significant challenge, often leading to unexpected component failures. One commonly used approach for fatigue assessment is the critical plane analysis, which aids in identifying the critical location and early crack propagation direction in a component. However, the conventional method for calculating critical plane factors is computationally demanding and is typically utilized only when the critical regions of the component are already known. In situations where the critical areas are difficult to be identified due to complex geometry, loads, or constraints, a more efficient method is required for evaluating critical plane factors. This research paper introduces an analytical algorithm to efficiently evaluates the widely used Findley critical plane factor. The algorithm operates within the framework of linear-elastic material behavior and proportional loading conditions, relying on tensor invariants and coordinate transformation laws. The algorithm has been tested on different component geometries, including a box-welded joint and a tubular specimen, subjected to proportional loading conditions such as tension, torsion, and a combination of them. The analytical method allowed a significant reduction in computation time while providing the exact solution of critical plane factor and critical plane orientations.

Optimal Low-Thrust Earth Orbit Transfers with Eclipses Using Indirect Heuristic Approaches

Low-thrust propulsion allows substantial propellant saving if compared to high-thrust systems. However, multirevolution orbit transfers are affected by the unavailability of electrical power along eclipse arcs. This research addresses the identification of minimum-time low-thrust Earth orbit transfers with spacecraft eclipsing. Two different indirect heuristic approaches, based on the use of the analytical conditions for optimality, in conjunction with stochastic fractal search, are applied: 1) regularization through the hyperbolic tangent function and 2) multi-arc formulation of the problem. Both of them require no averaging, and the respective advantages and shortcomings are identified. Approach 2 leads to an extended set of multipoint conditions for optimality, which are shown to be solvable sequentially in the numerical solution process. Implicit costate transformation is proven to be a key ingredient for the purpose of obtaining closed-form solutions of the jump relations that hold for the adjoint variables at light/eclipse transitions. Some illustrative examples prove effectiveness of the two indirect heuristic approaches 1 and 2 in finding minimum-time low-thrust orbit transfers between either two coplanar or two noncoplanar Earth orbits. Approach 2 proves to be superior in terms of both computational efficiency and accuracy of the numerical results.

Cilia flow of magnetized Eyring-Powell nanofluid in a vertical thermal channel with viscous dissipation: An application of Adomian decomposition method

In this study, authors explore how magnetic fields and the buoyancy force resulting from temperature differences work together. The study is focusing at how an Eyring-Powell nanofluid behaves in a ciliated channel when both natural convection and magnetic effects are at play. A technique called the symplectic metachoronal wave has been used to analyze this analysis. A physical assumption of small Reynolds number is considered to simplify the equations describing the flow rate of the Eyring-Powell nanofluid, making it easier to understand. To get a closed form solution, we have incorporated Adomian decomposition method (ADM). It has been observed that the flow is sped up by 10% with increase in the Eyring-Powell fluid parameter M. The quantity of heat transfer increases with increasing values of the Hartmann, Eyring-Powell fluid parameters, and Brinkmann numbers. This study has implications for bio-inspired engineering, medical applications, and the design of microfluidic devices.

Imbalanced complemented subspace representation with adaptive weight learning

Class imbalance problems pose significant challenges in the field of data mining. The skewed distribution of classes in imbalanced datasets often leads conventional classification methods to neglect the importance of minority classes, favoring the majority ones. In this paper, we propose an imbalanced complemented subspace representation model with adaptive weight learning to address imbalanced classification challenges. This approach incorporates a novel regularization technique and an adaptive weighting mechanism. The regularization technique integrates a complemented subspace term based on space theory into the efficient collaborative representation-based classification (CRC) framework, enhancing the overall accuracy of existing CRC-based methods. Furthermore, it takes into account both intra-class and inter-class density information to assign greater weights to minority classes. This highlights the contribution of minority classes and mitigates bias towards majority classes. Additionally, it is demonstrated that the proposed model can be efficiently solved with a closed-form solution. Extensive experiments conducted on diverse imbalanced datasets showcase the superiority of our method over most state-of-the-art imbalanced classification algorithms.

On micropolar elastic foundations

The modelling of heterogeneous and architected materials poses a significant challenge, demanding advanced homogenisation techniques. However, the complexity of this task can be considerably simplified through the application of micropolar elasticity. Conversely, elastic foundation theory is widely employed in fracture mechanics and the analysis of delamination propagation in composite materials. This study aims to amalgamate these two frameworks, enhancing the elastic foundation theory to accommodate materials exhibiting micropolar behaviour. Specifically, we present a novel theory of elastic foundation for micropolar materials, employing stress potentials formulation and a unique normalisation approach. Closed-form solutions are derived for stress and couple stress reactions inherent in such materials, along with the associated restoring stiffness. The validity of the proposed theory is established through verification using the double cantilever beam configuration. Concluding our study, we elucidate the benefits and limitations of the developed theory by quantifying the derived parameters for materials known to exhibit micropolar behaviour. This integration of micropolar elasticity into the elastic foundation theory not only enhances our understanding of material responses but also provides a versatile framework for the analysis of heterogeneous materials in various engineering applications.

The Post-Buckling Behavior of a Beam Constrained by Nonlinear Springy Walls

Abstract The post-buckling behavior of a beam that is subjected to lateral constraints is of relevance to a range of medical and engineering applications, such as endoscopic examination of internal organs, the insertion of a guidewire into an artery in-stent procedures, root growth, deep-drilling, and more. In this paper, we address a disconnect between the existing literature and the reality of these systems, in which the lateral constraints are flexible and experience nonlinear deformations. As a step towards bridging this gap, we consider a beam undergoing planar deformations that is laterally constrained by a nonlinear springy wall, i.e., a wall that is laterally pushed by the beam against a nonlinear spring. Based on a simplified mathematical model, we obtain closed-form analytical solutions, which provide valuable insights and intuition. For example, we show that important features of the behavior, such as the transition from point contact to line contact and switching to the next mode, are dictated solely by a non-dimensional force, regardless of all other parameters of the system, and that the full description of the behavior is possible by means of two non-dimensional quantities that describe the relative stiffness of the nonlinear spring compared to that of the beam. The results also highlight the fundamental differences between the behavior with a stiffening spring or with a softening spring, such as the number of attainable modes and the monotonicity of the overall force–displacement relation. These results are then validated by experiments.

Singular integral based closed-form solutions for modified EMPS models in semipermeable magneto-electro-elastic materials

The singular integral equation method is applied to present analytical solutions for modified electric-magnetic-polarization saturation (EMPS) models subjected to semipermeable center cracked magneto-electro-elastic (MEE) materials. A generalized methodology is presented to explicitly solve the EMPS model under any arbitrary saturated electric and magnetic conditions using the distributed dislocation method (DDM) and finding the analytical solutions for developed singular integral equations. The explicit expressions are derived for distributed dislocation densities, crack opening displacement (COD), crack opening potential (COP), crack opening induction (COI), saturated zone lengths, and local stress intensity factor (LSIF) for particular cases of the generalized EMPS model such as constant, linear, quadratic and cubic polynomial varying saturated electric and magnetic conditions. Using an iterative scheme and the explicit solutions of COD, COP, and COI, the exact semipermeable crack-face conditions are evaluated at the center of the crack, and the same applies throughout the crack-surface for the study of semipermeable modified EMPS models. A comparative analysis of the fracture parameters has also been carried out between the results of exact semipermeable crack-face conditions and the results of semipermeable crack-face conditions of the center crack problem to analyze the impact of polynomial varying saturated conditions on the crack-face conditions. Numerical studies performed under different electromagnetic-mechanical loadings, volume fractions, and crack-face conditions demonstrate the increasing effects in the fracture parameters' values with the degree of the polynomial varying saturated conditions.

Principle of minimal singularity for Green’s functions

Analytic continuations of integer-valued parameters can lead to profound insights, such as angular momentum in Regge theory, the number of replicas in spin glasses, the number of internal degrees of freedom, the spacetime dimension in dimensional regularization, and Wilson’s renormalization group. In this work, we consider a new kind of analytic continuation of correlation functions, inspired by two recent approaches to underdetermined Dyson-Schwinger equations in D-dimensional spacetime. If the Green’s functions Gn=⟨ϕn⟩ admit analytic continuation to complex values of n, the two different approaches are unified by a novel principle for self-consistent problems: Singularities in the complex plane should be minimal. This principle manifests as the merging of different branches of Green’s functions in the quartic theories. For D=0, we obtain the closed-form solutions of the general gϕm theories, including the cases with complex coupling constant g or noninteger power m. For D=1, we derive rapidly convergent results for the Hermitian quartic and non-Hermitian cubic theories by minimizing the complexity of the singularity at n=∞. Published by the American Physical Society 2024

Asymptotically correct isoenergetic formulation of geometrically nonlinear anisotropic plates

In the present work, a new displacement-based Equivalent Single Layer (ESL) plate theory has been developed using the Variational Asymptotic Method (VAM) and the principle of isoenergetics. In contrast to the existing assumption-based ESL plate theories, the proposed VAM-based plate theory is devoid of any presuppositions. The VAM decouples the 3D plate problem into a 1D through the thickness analysis and a 2D planar problem. Closed form solutions have been obtained for the 1D problem in terms of 2D displacement variables and their higher-order derivatives. The complexity involving higher-order derivatives of 2D displacement variables has been simplified through the isoenergetic principle ensuring a similar rate of convergence for strains as well as displacements. Solutions for various field variables have been compared against the benchmark problems available in the literature and 3D FEA. These comparisons demonstrate the accuracy, efficiency, and versatility of the present methodology. Key contributions of the present work are (a) First principles-based derivation of the reduced-order 2D plate model from the 3D model energy. (b) The plane stress condition as well as the quadratic variation of transverse shear stress and strain are natural outcomes of the present mathematical framework. (c) This leads to zero tangential traction boundary conditions on the plate surface. (d) The effectiveness of capturing higher-order behavior at lower-order results in simplified analysis, reduced computational complexity and increased efficiency.

Vibration analysis of functionally graded carbon nanotubes reinforced composite nanoplates

This work presents the analytical analysis for free linear vibration behavior of functionally graded-carbon nanotubes reinforced composite (FG-CNTRC) nanoplates in the framework of nonlocal strain gradient theory (NSGT) and the first-order shear deformation plate theory (FSDPT). The nanoplate is considered made of a mixture of an isotropic polymer matrix and reinforced carbon nanotubes (CNTs). Four different distributions of CNTs are examined including uniformly distributed and FG reinforcements (FG-O, FG-X, and FG-V). The governing equations of motion are established based on the Hamilton’s principle. The closed-form analytical solution for the natural frequency of FG-CNTRC nanoplates with simply supported all edges is carried out by using the Navier-type solution. The impact of some key parameters on the natural frequencies of FG-CNTRC nanoplates is also studied and discussed. The result shows that FG-CNTRC nanoplates reveal the softening- or hardening-stiffness effects depending on the relationship between the nonlocal parameter and the material length scale parameter. The aspect ratios of FG-CNTRC nanoplates, the volume fraction, and the distribution pattern of CNTs have also an important impact on the vibration behavior of FG-CNTRC nanoplates.

A closed-form analytical variational approach to thermally induced thin-film delamination

Thin polymeric films applied on stiff substrates are prone to delamination upon thermal loading. Different thermal expansion coefficients and different elastic properties can cause localized stress concentrations in the interface between applied layer and substrate. The present work proposes a variational approach to compute residual stresses and strains within the film. Using finite element calculations, it is shown that the presented closed-form analytical solution provides accurate interfacial stresses. Employing a critical distances failure criterion, these stresses are used to predict thermally induced crack nucleation. The proposed model is validated using experimental data of thermally loaded epoxy–glass specimens. It is in good agreement with the test data and provides predictions for thin-film delamination.

Non-linear analysis of rectangular cross-sections with different reinforcements: dimensionless closed form solution

This paper presents an algorithm to solve a non-linear rectangular cross-section analysis by means of closed form equations. The equations are provided in a dimensionless form in order to provide generalizable results. The algorithm is based on the direct integration of the non-linear constitutive behaviour of the materials into the stress-block constitutive behaviour. The algorithm is based on masonry rectangular cross-sections characterized by an external axial load similar to Reinforced-Concrete cross-sections. The failure of the strengthening system and the masonry are analysed in the proposed algorithm, in an unconventional manner, analysing the behaviour even after the attainment of the ultimate strains, accounting for progressive damage. The algorithm allows to prevent the cross-section fiber analysis and the consequential iterative procedure to retrieve the neutral axis depth. Furthermore, the direct integration allows to evaluate the exact solution of the problem without numerical approximations. The results are discussed by means of several parametric analyses in terms of P–M domains and bending moment–curvature diagrams in dimensionless form. The proposed approach aims to simplify the cross-section analysis process in complex algorithms as in the risk analysis based on a large scale approach, where a large number of structures and structural members are analysed in the vulnerability process.

Improved-RSSI-based indoor localization by using pseudo-linear solution with machine learning algorithms

With the rapid advancement of the Internet of Things and the popularization of mobile Internet-based applications, the location-based service (LBS) has attracted much attention from commercial developers and researchers. Received signal strength indicator (RSSI)-based indoor localization technology has irreplaceable advantages for many LBS applications. However, due to multipath fading, noise, and the limited dynamic range of the RSSI measurements, precise localization based on a path-loss model and multiliterate becomes highly challenging. Therefore, this study proposes a machine learning (ML)-based improved RSSI-based indoor localization approach in which RSSI data is first augmented and then classified using ML algorithms. In addition, we implement an experimental testbed to collect the RSSI value based on Wi-Fi using various reference and target nodes. The received RSSI measurements undergo pre-processing using pseudo-linear solution techniques for closed-form solutions, approximating the original system of nonlinear RSSI measurement equations with a system of linear equations. Finally, the RSSI measurement are trained using ML models such as linear regression, polynomial regression, support vector regression, random forest regression, and decision tree regression. Consequently, the experimental results express in terms of root mean square error and coefficient of determinant compared with various machine learning models with hyper-parameter tuning.

Stagnation-Point Brinkman Flow of Nanofluid on a Stretchable Plate with Thermal Radiation

The study is an analytical exploration of hybrid nanofluid flow at a stagnation-point with Brinkman effect on a stretchable plate with thermal radiation. All of the aforementioned factors were taken into account when developing the mathematical model based on the Navier–Stokes equations for nanofluids, leading to a system of partial differential equations. Using suitable scaling, these equations are reduced to system of ordinary differential equations. The outcome of the system of ordinary differential equations are solved analytically and closed-form solutions are obtained in terms of incomplete error function. The results are analysed for the many significant flow characteristics with the profiles of velocity and temperature explored graphically. The amount of the heat transfer is increased due to the interaction between nanoparticles and the wall, and the wall surface is cooled when wall suction is present.

Exact closed-form dynamic stiffness matrix of damaged frames comprising Timoshenko–Ehrenfest beams

The present work proposes the extension of an exact distributional model for cracked beams to include the local influence on axial deformation on shear deformable beams. Axial, shear and flexural concentrated flexibilities are taken into account to model the damaged sections. Shear deformability is accounted for by adopting the Timoshenko–Ehrenfest model and exact closed-form solution of the mode shapes for multiple cracked beams are derived. On this ground, the final achievement of the work consists in the original formulation of the Dynamic Stiffness Matrix (DSM) with multiple cracked sections in exact closed form. The closed-form expression of the DSM allows the assemblage of the global DSM of a structure to conduct the free vibration analysis of damaged framed structures as well as for the study of frames with semi-rigid joints. This approach has the advantage to avoid any numerical procedure for the construction of the DSM as well as the introduction of further degrees of freedom (dofs) in correspondence of the damaged sections, keeping the size of the problem the same as in the intact configuration. Natural frequencies of the frames are determined by means of the Wittrick–Williams algorithm (W–W), then the mode shapes and modal forces are subsequently obtained in closed form. The proposed approach is validated with results available in literature for two portal frames characterized by the presence of cracks and semi-rigid joints, respectively. Finally, the potentiality of the approach, also in view of practical applications, is demonstrated analysing a multi-storey frame with semi-rigid connections.