Solution Of Problem Research Articles

Towards efficient solutions for vehicle routing problems for oxygen supply chains

This work investigates an integrated production–inventory-routing problem (PIRP) for liquid oxygen supply chains consisting of multiple plants, customers, and heterogeneous vehicles. Solving such a problem is challenging, especially when dealing with large-scale industrial problems. A two-level procedure is adopted that determines decisions regarding production, inventory, and product allocation by simplifying the routing component in the first level. Then, the routing decision is considered in the lower level for which 3-index mathematical programming formulations are presented. To address the combinatorial complexity of the lower-level decision, we propose alternative multi-trip heterogeneous vehicle routing problem (MTHVRP) formulations together with a column generation-based solution strategy. A set of test instances and a real-world case study demonstrate the applicability and performance of the formulations and solution methods.

Nonlinear thermo-mechanical static stability analysis of FG-TPMS shallow spherical shells

An analytical solution for the nonlinear static stability problem of functionally graded triply periodic minimal surface (FG-TPMS) shallow spherical shells is studied in the current research for the first time. Three common TPMS structures including Primitive (P), Gyroid (G), and I-graph and Wrapped Package-graph (IWP) with three models of functionally graded porosity distribution along the thickness are considered. The shallow spherical shells (shallow SSs) are subjected to combined thermo-mechanical loadings and rested on a nonlinear elastic foundation. The fundamental formulas are expressed based on the higher-order shear deformation theory (HSDT) and von Kármán's geometrical nonlinearities. Employing the Ritz energy minimization method, the explicit relationship between load and deflection is derived. Subsequently, the static stability behavior of FG-TPMS shallow SSs is investigated. Numerical illustrations are investigated to show the superior thermo-mechanical load-carrying performance of the FG-TPMS SSs compared to corresponding isotropic structures of the same weight. The significant effects of geometric parameters, nonlinear elastic foundation parameters, and the type of FG-TPMS structures on the nonlinear static stability behavior of shallow SSs are further considered.

A polynomial collocation method for a class of singular fractional differential equations

In this work we consider a class of singular fractional differential equations (SFDEs). Using a suitable variable transformation we rewrite the SFDE as a cordial Volterra integral equation and propose a polynomial collocation method to find an approximate solution of the original problem. We provide the error analysis of the numerical method and we illustrate its feasibility and accuracy through some numerical examples.

Maintenance optimization for multi-component systems with a single sensor

We consider a multi-component system in which a single sensor monitors a condition parameter. Monitoring gives the decision maker partial information about the system state, but it does not reveal the exact state of the components. Each component follows a discrete degradation process, possibly correlated with the degradation of other components. The decision maker infers a belief about each component’s exact state from the current condition signal and the past data, and uses that to decide when to intervene for maintenance. A maintenance intervention consists of a complete and perfect inspection, and may be followed by component replacements. We model this problem as a partially observable Markov decision process. For a suitable stochastic order, we show that the optimal policy partitions in at most three regions on stochastically ordered line segments. Furthermore, we show that in some instances, the optimal policy can be partitioned into two regions on line segments. In two examples, we visualize the optimal policy. To solve the examples, we modify the incremental pruning algorithm, an exact solution algorithm for partially observable Markov decision processes. Our modification has the potential to also speed up the solution of other problems formulated as partially observable Markov decision processes.

On diagnostics of the properties of functional-gradient composites considering rheology

Abstract The solution of inverse problems on determining the variable rheological properties of functional-gradient beams using additional information about the angle of rotation of the end section is presented within the framework of the most used models: Euler–Bernoulli and Timoshenko. Within the framework of the concept of complex modules, nonlinear operator equations are obtained that connect the given and sought functions (instantaneous and long-term module, relaxation time). The solution is based on the linearization method and the formulation of an iterative process, at each step of which it is necessary to solve complex-valued boundary value problems and invert completely continuous operators with complex-valued kernels. Functions reflecting the laws of change in long-term and instantaneous modules have been restored. The results of computational experiments in some special cases are presented.

Practical Integer-Constrained Cone Construction for Conformal Parameterizations.

We propose a practical method to construct sparse integer-constrained cone singularities with low distortion constraints for conformal parameterizations. Our solution for this combinatorial problem is a two-stage procedure that first enhances sparsity for generating an initialization and then optimizes to reduce the number of cones and the parameterization distortion. Central to the first stage is a progressive process to determine the combinatorial variables, i.e., numbers, locations, and angles of cones. The second stage iteratively conducts adaptive cone relocations and merges close cones for optimization. We extensively test our method on a data set containing 3885 models, demonstrating practical robustness and performance. Our method achieves fewer cone singularities and lower parameterization distortion than state-of-the-art methods.

Solution of the plane problem of thermoelasticity with variable properties with a new variant of the derivation of equations in stresses

The interaction of thermal and mechanical fields in modern technologies for the synthesis and processing of materials attracts more and more attention. This is due to the need to predict the behavior and properties of new materials and products depending on the conditions of their creation. In the present work, attention is focused on the comparison of the solutions to the thermoelasticity plane problems with and without accounting for the temperature dependence of the material properties. It is assumed that an external heat source traveling over the surface with a specific velocity and following a given trajectory is responsible for a significant temperature increase. It may correspond to a laser source, for example. The new way of problem formulation in stresses is used. The temperature dependence of the characteristics leads to nonlinear thermoelasticity equations when the equilibrium equations, the requirement of strain compatibility, and the Duhamel-Neuman relation for the plane stress state case are combined. To determine the temperature field, a nonlinear thermal conductivity equation is used. To solve the problem numerically, the use of an implicit finite-difference scheme and coordinate splitting for the thermal conductivity equation and the finite-difference method of successive over-relaxation with Chebyshev acceleration for the equations for stresses was made. As a result, the difference in the solution of linear and nonlinear problems is demonstrated for the considered cases. Using the example of a mixture of titanium and aluminum powders, numerical modeling demonstrated that some stress components can reach values up to 3.5 times higher than those obtained in the case of temperature-independent material properties when there is a 500-degree local temperature increase.

Preliminary Numerical Modelling of a Dynamic Spring-Mounted Wing System to Reduce the Drag of Vehicles at Higher Speeds

The dynamic behaviour of a spring-mounted symmetrical NACA0012 wing in a freestream flow of air is studied in the pre-stall region, over 0° to 12° angles of incidence. The primary aim of this work is for use within the automotive sector to reduce drag and fuel emissions. However, this work will also be of interest in the motorsport sector to improve performance, and also have some applications within the aerospace and renewable energy sectors. The general operation of the concept has previously been verified at these low angles in the pre-stall region with that of a theoretical estimation using finite and infinite wings. This paper provides a numerical solution of the same problem and is compared with the previous experimentation. At these low angles, the computations yield a dynamic response settling into a static equilibrium. The stable solutions match the start of a steady regime well, when compared with the experiment. The trends are also comparable with the experiment, but the velocities at which they occur are underestimated in the computation. The computations demonstrate a drag reduction of 59% when compared to a fixed wing, whereas the lift remains stable at a near constant value with increasing wind speed. Thence, downforce is maintained whilst drag is reduced, which will facilitate higher speeds on the straight whilst maintaining vehicle direction stability. Limitations to this proof-of-concept work are highlighted and future development work is suggested to achieve even further increases in performance.

Solution of the modified Helmholtz equation using mixed boundary conditions in an equilateral triangle

The modified Helmholtz equation qxx+qyy−4β2q=0, is one of the basic equations of classical mathematical physics. In this paper we have obtained the solution of the boundary-value problems for the modified Helmholtz equation in an equilateral triangle. An additional mixed boundary condition related to the symmetry of the solution is taken into consideration. We have analysed the Global relation and only used the algebraic techniques to obtain the explicit solution of modified Helmholtz equation bypassing the Riemann Hilbert approach. This solution is applied to the problem of diffusion-limited coalescence, A+A⇌A.

Multiobjective optimal control problems. Stationary Navier–Stokes equations

This paper deals with the solution of some multi-objective optimal control problems for stationary Navier–Stokes equations. More precisely, we look for Pareto and Nash equilibria associated to standard cost functionals. First, we prove the existence of equilibria and we deduce appropriate optimality systems. Then, we analyse the existence and characterization of Pareto and Nash equilibria for the Navier–Stokes equations. Here, we use the formalism of Dubovitskii and Milyoutin., see [Girsanov FV. Lectures on mathematical theory of extremum problems. Berlin: Springer-Verlag; 1972. (Notes in economics and mathematical systems; vol. 67)]. Finally, we also present a finite element approximation of the bi-objective problem, we illustrate the techniques with several numerical experiments and we compare the Pareto and Nash equilibria.

Imbalanced ensemble learning leveraging a novel data-level diversity metric

Ensemble learning is one of the best solutions for imbalanced classification problems. Diversity is a key factor that affects the performance of ensemble learning. Most existing diversity metrics such as Q-statistics measure diversity based on the outputs of the base classifiers, incurring high complexity on model training due to the need to re-train base classifiers to achieve satisfactory diversity. We propose a new diversity measure, named Instance Euclidean Distance metric (IED), to evaluate diversity directly based on the training data without training base classifiers, which can significantly cut down the time costs of diversity measuring. A new imbalanced ensemble learning algorithm named P-EUSBagging is proposed to reduce training complexity and improve learning performance by combining IED with population-based incremental learning to generate training datasets with the maximal data-level diversity. Experimental results demonstrate that the diversities measured by IED and three classifier-based diversity measures exhibit a mean absolute correlation coefficient of 0.94, and P-EUSBagging significantly reduces training time and improves learning performance on both Geometric Mean (G-Mean) and Area Under the Curve (AUC).

SLAPE solar panels: a possible solution for the future energy problems of our society

Today, the silicon photovoltaic cell (SPVC) solar panels are being deployed across the globe by all most all the governments to generate renewable carbon neutral electricity to use in place of the one generated at thermal power plants by burning fossil fuels. These SPVC solar panels generate electricity with less than 20% efficiency, and heat energy with more than 30% efficiency. This 10% excess heat energy released into the atmosphere can cause a very severe damage to the environment and ecology as it is million times higher than the one caused by the equivalent CO2 gas in the atmosphere with a concentration of about 421 ppm. To avoid the harmful and danger effects of heat energy released by these SPVC solar panels into the atmosphere, either they have to be stopped using or the heat generated by them has to be trapped to use for the beneficial purposes of the society or any other alternate safe solar panels have to be developed. In this study, for the first time, the SLAPE (Semiconductor and Liquid Assisted Photothermal Effect) solar panels are introduced to generate electricity from sunlight. In these new solar panels, a semiconducting material along with a stable organic solvent is employed to capture the sunlight reaching the earth surface and to convert it into the heat energy. This in situ generated heat energy is then converted into a rotational mechanical energy with the help of an heat engine, and then into electricity with the help of suitable electric generator. To establish the Proof of Concept (poc), in this study, SPV cells as semiconducting material and γ-butyrolactone were employed to fabricate a one square meter area SLAPE solar panel and generated about 17 V AC electricity with the help of a custom manufactured reciprocally moved steam engine (RMSE) and an electric generator. Thus obtained results are presented and discussed along this article.

Higher-order multi-scale physics-informed neural network (HOMS-PINN) method and its convergence analysis for solving elastic problems of authentic composite materials

The limitations of prohibitive computation and Frequency Principle remain difficult for deep learning methods to effectively resolve multi-scale problems. In this work, a novel higher-order multi-scale physics-informed neural network (HOMS-PINN) framework is developed to compute the elastic problems of authentic composite materials with strongly discontinuous and high-contrast material parameters, which inherits the advantages of higher-order multi-scale method and physics-informed neural network, and halves the computational expense remarkably in comparison with direct PINN simulation. In the numerical framework of HOMS-PINN method, two types of network structures including single neural networks and independent neural networks are designed to mesh-free solve lower-order and higher-order microscopic unit cell (UC) functions, and macroscopic homogenization equations of multi-scale composites, which are then assembled into higher-order multi-scale solutions for multi-scale elastic problems by using automatic differentiation technology. Moreover, transfer learning is introduced to extend HOMS-PINN framework accelerating simulation of multi-scale elastic problems for composite materials with high-contrast mechanical parameters. Furthermore, the error estimate of the proposed HOMS-PINN method is demonstrated under some assumptions. Finally, numerous numerical experiments including high-contrast, multi-inclusion and three-dimensional composites are presented to validate the accuracy and effectiveness of HOMS-PINN method, especially for accurately capturing the sharply oscillatory behaviors in micro-scale. This study offers a robust HOMS-PINN computational framework that enables the simulation and analysis of large-scale mechanical problems of authentic composite materials.

Algebraic Conditions for Stability in Runge-Kutta Methods and Their Certification via Semidefinite Programming

In this work, we present approaches to rigorously certify A- and A(α)-stability in Runge-Kutta methods through the solution of convex feasibility problems defined by linear matrix inequalities. We adopt two approaches. The first is based on sum-of-squares programming applied to the Runge-Kutta E-polynomial and is applicable to both A- and A(α)-stability. In the second, we sharpen the algebraic conditions for A-stability of Cooper, Scherer, Türke, and Wendler to incorporate the Runge-Kutta order conditions. We demonstrate how the theoretical improvement enables the practical use of these conditions for certification of A-stability within a computational framework. We then use both approaches to obtain rigorous certificates of stability for several diagonally implicit schemes devised in the literature.

Landau levels for charged particles with anisotropic mass

The problem of the two-dimensional motion of a charged particle with constant mass in the presence of a uniform constant perpendicular magnetic field features in several undergraduate and graduate quantum physics textbooks. This problem is very important to studies of two-dimensional materials that manifest quantum Hall behavior, as evidenced by several major discoveries over the last few years. Many real experimental samples are more complicated due to the anisotropic mass of the electrons. In this work, we provide the exact solution to this problem by means of a clever scaling of coordinates. Calculations are done for a symmetric gauge of the magnetic field. This study allows a broad audience of students and teachers to understand the mathematical techniques that lead to the solution of this quantum problem.

Approximate Bayesian Computation for structural identification of ancient tie-rods using noisy modal data

Masonry arches and vaults are common historic structural elements that frequently experience asymmetric loading due to seismic action or abutment settlements. Over the past few decades, numerous studies have sought to enhance our understanding of the structural behaviour of these elements for the purpose of preventive conservation. The assessment of the structural performance of existing constructions typically relies on effective numerical models guided by a set of unknown input parameters, including geometry, mechanical characteristics, physical properties, and boundary conditions. These parameters can be estimated through deterministic optimization functions aimed at minimizing the discrepancy between the output of a numerical model and the measured dynamic and/or static structural response. However, deterministic approaches overlook uncertainties associated with both input parameters and measurements. In this context, the Bayesian approach proves valuable for estimating unknown numerical model parameters and their associated uncertainties (posterior distributions). This involves updating prior knowledge of model parameters (prior distributions) based on current measurements and explicitly considering all sources of uncertainties affecting observed quantities through likelihood functions. However, two significant challenges arise: the likelihood function may be unknown or too complex to evaluate, and the computational costs for approximating the posterior distribution can be prohibitive. This study addresses these challenges by employing Approximate Bayesian Computation (ABC) to handle the intractable likelihood function. Additionally, the computational burden is mitigated through the use of accurate surrogate models such as Polynomial Chaos Expansions (PCE) and Artificial Neural Networks (ANN). The research focuses on setting up numerical models for simple structural systems (tie-rods) and inferring unknown input parameters, such as mechanical properties and boundary conditions, through Bayesian updating based on observed structural responses (modal data, strains, displacements). The main novelties of this research regard, on the one hand, the proposal of a methodology for obtaining a reliable estimate of the axial force in ancient tie-rods accounting for different sources of uncertainty and, on the other hand, the application of ABC to obtain the structural identification inverse problem solution.

Tracking Control for $(x,u)$-Flat Systems by Quasi-Static Feedback of Classical States

It is well known that for flat systems the tracking control problem can be solved by utilizing a linearizing quasi-static feedback of generalized states. If measurements (or estimates) of a so-called generalized Brunovský state are available, a linear, decoupled and asymptotically stable tracking error dynamics can be achieved. However, from a practical point of view, it is often desirable to achieve the same tracking error dynamics by feedback of a classical state instead of a generalized one. This is due to the fact that the components of a classical state typically correspond to measurable physical quantities, whereas a generalized Brunovský state often contains higher order time derivatives of the (fictitious) flat output which are not directly accessible by measurements. In this paper, a systematic solution for the tracking control problem based on quasi-static feedback and measurements of classical states only is derived for the subclass of $(x,u)$-flat systems.

The eigenfunctions method in relation to diffuse reflection problems of the radiative energy transfer theory

Using the new method proposed earlier by the author, a solution of diffuse reflection problem is given in this report. A case of diffuse reflection of radiation from a semi-infinite absorbing and isotropically reflecting medium is presented, when in the elementary act of scattering there is a general law of redistribution of radiation by frequencies. In the approach presented here, conventionally called the method of ”decomposition of the resulting field” - DRF, in contrast to the previously widely used traditional method of ”decomposition of a single act” of scattering - DSA, the problem of diffuse reflection is solved without the use of any decomposition or special representation of the characteristic of the elementary act of scattering, i.e., the redistribution function of radiation by frequencies. The resulting radiation field, formed in the medium by means of multiple scatterings, is sought and constructed directly in the form of a bilinear series through a specially derived system of its own eigenfunctions. The physical basis of this approach lies in the fact that after each successive act of scattering, the field formed in the medium becomes more and more smoothed, so its direct representation in a decomposed form is more expedient than the traditionally used decomposition of a pre-known ”unsmoothed” characteristic of the elementary act of scattering.

PRECIPITATION DATA FROM GOOGLE EARTH ENGINE (GEE) IN REVETMENT DRAINAGE EVALUATION IN KALIANDA

Floods can occur frequently if there are no organized and constant changes in land use. Flood issues in Waymuli, South Lampung, has been occurring due to changes in land use in the area. Prior studies regarding drainage of coastal revetments may have been done before. However, the analysis was done using precipitation data from GEE and therefore, new findings as the solution for the flood problems in the region were necessary. The main objective of this research is to assess the extent of success of drainage system management using rainfall information obtained via GEE. The method used was by evaluating the capability of the drainage system using data from GEE, which involve identifying and measuring the drainage capacity with the HEC-RAS application. This study conclude that the use of GEE can be accurately used to evaluate revetment drainage. Rainfall in the last two years were 54.0340 mm, 81.8905 mm in the last five years, and 104.0784 mm in the last ten years. However, GEE results show that the highest rainfall recorded in September 2022 was 46.857 mm. Based on the HEC-RAS evaluation, it was also found that the drainage system does not have the capacity to drain rain within a 2-year period. It can be concluded that the function of the drainage system in this area is to support the coastal revetment system, not as the main drainage system.

Forward and Inverse Modeling of Ice Sheet Flow Using Physics‐Informed Neural Networks: Application to Helheim Glacier, Greenland

AbstractPredicting the future contribution of the ice sheets to sea level rise over the next decades presents several challenges due to a poor understanding of critical boundary conditions, such as basal sliding. Traditional numerical models often rely on data assimilation methods to infer spatially variable friction coefficients by solving an inverse problem, given an empirical friction law. However, these approaches are not versatile, as they sometimes demand extensive code development efforts when integrating new physics into the model. Furthermore, this approach makes it difficult to handle sparse data effectively. To tackle these challenges, we use the Physics‐Informed Neural Networks (PINNs) to seamlessly integrate observational data and governing equations of ice flow into a unified loss function, facilitating the solution of both forward and inverse problems within the same framework. We illustrate the versatility of this approach by applying the framework to two‐dimensional problems on the Helheim Glacier in southeast Greenland. By systematically concealing one variable (e.g., ice speed, ice thickness, etc.), we demonstrate the ability of PINNs to accurately reconstruct hidden information. Furthermore, we extend this application to address a challenging mixed inversion problem. We show how PINNs are capable of inferring the basal friction coefficient while simultaneously filling gaps in the sparsely observed ice thickness. This unified framework offers a promising avenue to enhance the predictive capabilities of ice sheet models, reducing uncertainties, and advancing our understanding of poorly constrained physical processes.