Solution Procedure Research Articles

Improving the Solution Procedure of Incremental Harmonic Balance Method for Multi-Degree-of-Freedom Self-Excited Vibration Systems

When applying the Incremental Harmonic Balance (IHB) method to solve the limit cycle in a self-excited system, the practice of fixing one of the harmonic coefficients effectively solves the problem that there are more unknowns than algebraic equations due to the presence of unknown angular frequencies in the periodic response. However, an inappropriate value for this fixed harmonic coefficient can significantly affect the convergence of subsequent Newton–Poisson iterations. To address this issue, we propose a technique that dynamically adjusts the value of the fixed harmonic coefficient, effectively improving the convergence and stability of the traditional IHB method and providing an efficient solution for the parametric study of self-excited systems. The efficiency and performance of our method in accurately predicting periodic solutions are demonstrated by its application to Van der Pol oscillators and predicting vortex-induced vibrations in cable-stayed structures. Numerical simulations show that the results obtained by the proposed method are very close to those obtained by the Runge–Kutta method. This study opens up a robust and efficient approach for analyzing and studying the dynamic behavior of complex self-excited systems.

Variable‐Coefficient Evolution Problems via the Fokas Method Part I: Dissipative Case

ABSTRACTWe derive explicit solution representations for linear, dissipative, second‐order initial‐boundary value problems (IBVPs) with coefficients that are spatially varying, with linear, constant‐coefficient, two‐point boundary conditions. We accomplish this by considering the variable‐coefficient problem as the limit of a constant‐coefficient interface problem, previously solved using the unified transform method of Fokas. Our method produces an explicit representation of the solution, allowing us to determine properties of the solution directly. As explicit examples, we demonstrate the solution procedure for different IBVPs of variations of the heat equation, and the linearized complex Ginzburg‐Landau (CGL) equation (periodic boundary conditions). We can use this to find the eigenvalues of dissipative second‐order linear operators (including non–self‐adjoint ones) as roots of a transcendental function, and we can write their eigenfunctions explicitly in terms of the eigenvalues.

Solving the combined flexible job shop scheduling and vehicle routing problem with stochastic features

ABSTRACT Today’s competitive market conditions forces companies to implement strategies for the integration of production and transportation activities, but this is studied little in the literature. This paper considers the combined production and transportation problem in which the production system is defined as a flexible jobshop (FJS) and the transportation stage is modelled as a vehicle routing problem (VRP). Stochastic production processing times and vehicle travel times are considered. A simheuristic solution procedure is proposed, as a class of simulation-optimisation approach, based on Ant Colony Optimisation (production stage) and an iterative local search (transportation stage). Randomised data sets are used to evaluate the performance of the proposed solution procedure, with metrics such as total production time and delivery time demonstrating its effectiveness. Experimental outputs show the impact of considering stochasticity of these two parameters for better decision-making. Compared to existing methods, our approach offers significant improvements in efficiency and reliability.

Solving a Fully Intuitionistic Fuzzy Transportation Problem Using a Hybrid Multi-Objective Optimization Approach

In this study, a typical transportation problem involving intuitionistic fuzzy-type variables and parameters is focused on. The approaches proposed in the literature for such transportation problems have many shortcomings, such as the use of ranking functions and obtaining an infeasible solution with negative values for variables and objective functions in the presence of non-negative unit transportation charges. To overcome such weaknesses, a new approach without a ranking function is introduced in this paper. The proposed approach first constructs an equivalent crisp multi-objective form of the intuitionistic fuzzy transportation problem and then proposes a new hybrid multi-objective solution procedure to tackle the obtained crisp multi-objective problem. The conducted computer experiments with benchmark problems from the existing studies of the literature reflect the effectiveness of the proposed solution approach of this study in terms of the quality of the results when compared to the available approaches of the literature.

A New Differential Gene Expression Based Simulated Annealing for Solving Gene Selection Problem: A Case Study on Eosinophilic Esophagitis and Few Other Gastro-intestinal Diseases.

Identifying the set of genes collectively responsible for causing a disease from differential gene expression data is called gene selection problem. Though many complex methodologies have been applied to solve gene selection, formulated as an optimization problem, this study introduces a new simple, efficient, and biologically plausible solution procedure where the collective power of the targeted gene set to discriminate between diseased and normal gene expression profiles was focused. It uses Simulated Annealing to solve the underlying optimization problem and termed here as Differential Gene Expression Based Simulated Annealing (DGESA). The Ranked Variance (RV) method has been applied to prioritize genes to form reference set to compare with the outcome of DGESA. In a case study on Eosinophilic Esophagitis (EoE) and other gastrointestinal diseases, RV identified the top 40 high-variance genes, overlapping with disease-causing genes from DGESA. DGESA identified 40 gene pathways each for EoE, Crohn's Disease (CD), and Ulcerative Colitis (UC), with 10 genes for EoE, 8 for CD, and 7 for UC confirmed in literature. For EoE, confirmed genes include KRT79, CRISP2, IL36G, SPRR2B, SPRR2D, and SPRR2E. For CD, validated genes are NPDC1, SLC2A4RG, LGALS8, CDKN1A, XAF1, and CYBA. For UC, confirmed genes include TRAF3, BAG6, CCDC80, CDC42SE2, and HSPA9. RV and DGESA effectively elucidate molecular signatures in gastrointestinal diseases. Validating genes like SPRR2B, SPRR2D, SPRR2E, and STAT6 for EoE demonstrates DGESA's efficacy, highlighting potential targets for future research.

Phase transition in porous materials: effects of material parameters and deformation regime on mass conservativity

AbstractPhase transition in porous materials is relevant within different engineering applications, such as freezing in saturated soil or pancake sea ice. Mathematical descriptions of such processes can be derived based on Biot’s consolidation theory or the Theory of Porous Media. Depending on parameters such as density ratio, permeability or compressibility of the solid matrix, either small or finite deformations occur. Numerical solution procedures for the general, finite deformation case, suffers from instabilities and high computational costs. Simplifications, assuming small deformations, increases stability and computational efficiency. Within this work shortcomings of simplified theories based on Biot and linearisations of the Theory of Porous Media (TPM) are systematically studied. In order to determine the interaction of the different model parameters a non-dimensional model for poro-elasticity is presented. Based on a characteristic test-case including phase-transition and consolidation, the simplified models are compared to the fully non-linear TPM, focusing on mass errors as well as the time behaviour of the solution. Taking further into account the efficiency of discretisation based on different primal variables and finite-element-spaces, a guideline for selecting an appropriate combination of model, kinematic assumption and discretisation scheme is presented.

In-plane vibration analysis of elastically restrained FGM skew plates using variational differential quadrature method

This work presents an accurate in-plane vibration analysis of functionally graded material (FGM) skew plates with elastically restrained boundaries using the variational differential quadrature method (VDQM). The weak form of the governing equations is derived by integrating two-dimensional elasticity theory with Hamilton's principle. The differential and integral operators are directly converted into matrix forms, thereby removing the necessity for higher-order derivative approximations in the displacement field. Transformation matrices are then developed for both interior and boundary nodes to link the governing equations with the boundary conditions, leading to the formulation of the vibration eigenvalue equations for FGM skew plates. Various factors, including aspect ratios, skew angles, boundary restraints, and gradient indices, are considered to investigate the in-plane vibration mode characteristics of FGM skew plates. Detailed solution procedures are provided, and numerical examples using the proposed solutions indicate that the VDQM exhibits superior numerical convergence and stability compared to other existing methods. The study also investigates the influence of highly skewed plates (75°), where stress singularities arise at the corners. This aspect is crucial for in-plane vibration analysis and has garnered limited attention in the existing literature. Furthermore, the dynamic analysis of FGM skew plates reveals a coupling between normal and tangential vibration modes.

Buckling behavior of orthotropic thin plates using analytical and machine learning methods

Designing plate structures poses challenges due to potential instabilities and the complexities involved in obtaining accurate analytical solutions, especially for non-Lévy-type boundary conditions. This study employs the finite integral transform method to analyze the buckling behavior of orthotropic thin plates with complex boundary conditions. In solution procedure, the governing high-order partial differential equations are transformed into a system of linear algebraic equations, yielding exact and rapidly converging analytical solution. The method is simple and general and does not need to predetermine the deflection function. The correctness of the method is validated by comparison with numerical simulations performed using the ABAQUS software. Furthermore, Gene Expression Programming (GEP) is utilized to develop empirical models that predict buckling coefficients of isotropic and orthotropic plates under classical and non-classical boundary conditions. Material properties, aspect ratio, rotating fixed coefficient, and boundary conditions are used as input parameters, with simplified mathematical formulations to predict the buckling coefficient. Model performance is assessed using parametric analysis and statistical tests to ensure accuracy and generalization. Further investigation shows aspect ratio and rotating fixed coefficient are significant variables influencing buckling coefficients under classical and non-classical boundary conditions, respectively, followed by boundary conditions and material property. The performance of the GEP model is further assessed by comparing with linear and non-linear regression models. The results show that GEP outperforms the regression models, showing its higher prediction accuracy. This study not only addresses the challenges of designing thin plates with complex boundary conditions, but it also presents an effective machine-learning method for predicting buckling behavior. The analytical solution can be used as a benchmark for validating new analytical and numerical methods. The equations developed using GEP to predict the buckling coefficient of plates provide a useful tool for extrapolating results beyond the scope of this study.

Accurate Closed-Form Solutions for the Free Vibration and Supersonic Flutter of Laminated Circular Cylindrical Shells

According to the Donnell–Mushtari shell theory, this work presents a closed-form solution procedure for free vibration of open laminated circular cylindrical shells with arbitrary homogeneous boundary conditions (BCs). The governing differential equations of free vibration are derived from the Rayleigh quotient and solved by the iterative separation-of-variable (iSOV) method. In addition, considering axial aerodynamic pressure, simulated by the linear piston theory, the exact eigensolutions for the flutter of open laminated cylindrical shells with simply supported circumferential edges and closed laminated cylindrical shells are also achieved. The governing differential equations of cylindrical shell flutter are derived from the Hamilton variational principle and solved by the separation-of-variable (SOV) method. The influence of circumferential dimension on flutter speed is investigated for open cylindrical shells, which reveals that the number of circumferential waves in critical flutter mode increases with circumferential length, and there exists an infimum for flutter speed that is an invariant independent of circumferential length. The present results agree well with those obtained by the Galerkin method, the finite element method, and other analytical methods.

Damped Multistep Formulas for Solving Stiff Dynamic Problems

A novel series of damped multistep formulas of matrix coefficients are developed to find the solutions of dynamic problems with negligible stiff components or eigenmodes. They are derived from the concept of eigenmode and the development details are presented. A total of six multistep formulas is developed and they have an order of accuracy from 1 to 6 corresponding to the 1-step to 6-step multistep formula. Each formula has damping to suppress or remove the stiff components or eigenmodes that are of no importance or of no interest while the slow eigenmode solutions can be still accurately calculated. The 1-step and 2-step formulas of the series are L-stable. Besides, each formula can have a non-iterative solution procedure and hence it is of high computational efficiency in contrast to the corresponding conventional backward differentiation method. This series can give a better quality of numerical solutions for dynamic problems with negligible stiff components or eigenmodes when compared to the methods with no damping. Numerical tests reveal that this series performs well like the backward differentiation method and it can save many efforts in computing if a non-iterative solution procedure is used when compared to the backward differentiation method.

Semi-supervised multi-view feature selection with adaptive similarity fusion and learning

Existing multi-view semi-supervised feature selection methods typically need to calculate the inversion of high-order dense matrices, rendering them impractical for large-scale applications. Meanwhile, traditional works construct similarity graphs on different views and directly fuse these graphs from the view level, ignoring the differences among samples in various views and the interplay between graph learning and feature selection. Consequently, both the reliability of graphs and the discrimination of selected features are compromised. To address these issues, we propose a novel multi-view semi-supervised feature selection with Adaptive Similarity Fusion and Learning (ASFL) for large-scale tasks. Specifically, ASFL constructs bipartite graphs for each view and then leverages the relationships between samples and anchors to align anchors and graphs across different views, preserving the complementarity and consistency among views. Moreover, an effective view-to-sample fusion manner is designed to coalesce the aligned graphs while simultaneously exploiting the neighbor structures in projection subspaces to construct the joint graph compatible across views, reducing the adverse effects of noisy features and outliers. By incorporating bipartite graph fusion and learning, label propagation, and l2,0-norm multi-view feature selection into a unified framework, ASFL not only avoids the expensive computation in the solution procedures but also enhances the quality of selected features. An effective optimization strategy with fast convergence is developed to solve the objective function, and experimental results validate its efficiency and effectiveness over state-of-the-art methods.

A general solution procedure for nonlinear single freedom systems including fractional derivatives

A general solution procedure for nonlinear single freedom systems including fractional derivatives

Nonlinear phase velocities in tri-directional functionally graded nanoplates coupled with NEMS patch using multi-physics simulation

The nonlinear phase velocities analysis of tri-directional functionally graded (FG) nanoplates is crucial for optimizing their performance as nano-electro-mechanical systems (NEMS). By understanding the nonlinear phase velocities of various wave modes within the nanoplate, engineers can accurately predict and enhance the efficiency of NEMS. This analysis allows for the precise tuning of the piezoelectric patches to capture vibrational energy more effectively, ensuring maximum energy conversion efficiency. Moreover, it helps in identifying the optimal placement and orientation of the piezoelectric patches, minimizing energy loss and enhancing the reliability and durability of the NEMS. Ultimately, this leads to more efficient utilization of ambient vibrations in airplanes, providing a sustainable power source for various onboard sensors and monitoring systems, contributing to reduced reliance on external power sources, and improved overall energy management. For this issue, for the first time, nonlinear phase velocity in the tri-directional functionally graded nanoplate coupled with a piezoelectric patch via COMSOL multi-physics simulation, physics-informed deep neural networks (PIDNNs), and mathematics simulation are presented. In the mathematics simulation domain, nonlocal strain gradient theory for modeling both the hardening and softening behavior of the current nanoplate is presented. The electromechanical coupling effect and the abrupt change in material properties at the interfaces will have a major influence on the mechanical performance of tri-directional functionally graded (TD-FG) nanoplate coupled with a piezoelectric patch if transverse shear deformations cannot be well modeled. Thereby, in the current simulation, a quasi-3D refined theory with 10 variables is presented. Also, for coupling the composite structure with the piezoelectric patch, compatibility conditions are considered. An analytical solution procedure is presented for solving the nonlinear partial differential equations of the current electrical system.

Flow of ferrofluid within a channel driven by a rotating magnetic field: Implications of surface magnetic stresses and magnetic body couples

This research presents a discussion of the flow of a ferrofluid in a square cross-sectional area channel driven by a rotating magnetic field. The objective was to evaluate the coupled effect of surface magnetic stresses and diffusion of the internal angular momentum as potential mechanisms for generating flow. In many analyses of flow induced by time-dependent magnetic fields, the effect of the diffusion of the internal angular momentum has been dismissed, based on the negligible value of the spin viscosity coefficient estimated from order-of-magnitude analyses. In light of these considerations, we conducted a review of the existing analyses and put forward a new approach of the order-of-magnitude analysis of the spin viscosity coefficient. The paper presents a detailed description of the solution procedure for the flow field, with and without the inclusion of the diffusion of the internal angular momentum. To justify the assumptions involved in the analytical solution of the flow, we expanded a previous scale analysis to define Reynolds and Strouhal numbers in the linear and internal angular momentum balance equations. It was proven that the maximum surface velocity predicted at the ferrofluid–air boundary closely aligns with the reported experimental data for moderate amplitudes of the magnetic field.

Enhanced Adomian Decomposition Method for Accurate Numerical Solutions of PDE Systems

This research addresses critical challenges in numerical solutions, which are vital for various engineering and physical fields. The Modified Adomian Decomposition Method (MADM) is proposed as a novel approach for solving linear and nonlinear partial differential equations (PDEs). MADM builds upon the Adomian Decomposition Method (ADM) by incorporating a new integral operator that significantly improves convergence rates and accuracy. Numerical examples demonstrate the effectiveness of MADM in handling complex nonlinear PDEs. Compared to traditional ADM, MADM consistently achieves more accurate and rapidly converging solutions. This enhancement is attributed to the novel integral operator, which addresses the limitations of ADM for intricate nonlinear problems. The paper outlines the application of MADM, its solution procedure, and its effectiveness through numerical examples. Comparisons with standard ADM solutions and exact solutions validate MADM's accuracy and superiority. The results suggest that MADM is a promising tool for expanding the applicability of Adomian methods in the field of solving PDEs.

A Finite Integral Transform-Based Generalized Eigenvalue Formulation for Buckling of Orthotropic Rectangular Plates with Rotational Restraints

Although the finite integral transform (FIT) method has been developed for buckling analysis of rectangular thin plates, the existing formulation involves solving complex nonlinear determinantal equations, leading to potentially questionable numerical results. To address this challenge, a new FIT-based formulation is established that transforms the solution of a complex nonlinear equation into that of a straightforward generalized eigenvalue problem. New analytical solutions for buckling of orthotropic rectangular thin plates with rotational restraints are obtained. The solution procedure is implemented via the following four steps: imposing the two-dimensional FIT on the governing equation; inputting deflection conditions of four edges, by which the relations between the transformed deflections and specific unknowns are provided; imposing the one-dimensional FIT on the elastic conditions to replace the unknowns with the transformed deflections, a generalized eigenvalue problem is constructed; determining the final analytical solutions by solving the generalized eigenvalue problem. Comprehensive highly accurate buckling load/mode solutions for typical rotationally restrained orthotropic plates are presented as benchmarks. The effects of boundary conditions, rotational fixity factors, and loading ratios on the buckling behaviors of orthotropic plates are expediently investigated adopting the present solutions.

Parametric identification of coefficients for a model of fatigue stiffness degradation of a composite material

The problem of finding the fatigue characteristics of a composite material based on test results is considered. The results of endurance tests of unidirectional polymer composite materials with different initial stiffness, breaking stress and working cycle stress were used as the initial data. As a mathematical model of stiffness degradation, a nonlinear ordinary differential equation with five unknown parameters is used, reflecting characteristic changes in the properties of the material. It is required to find such parameter values that the solution of the differential equation should describe the available test results with sufficient accuracy. The solution procedure is reduced to the problem of optimizing the objective function, the value of which characterizes the achieved accuracy. As optimization methods, a method simulating the behavior of a flock of moths and a method of sequential reduction of the search set were used. A step-by-step algorithm for finding unknown model parameters is proposed, and numerical results of processing input data containing information on changing the elasticity modulus of the composite material in the course of applying load cycles are presented.

Flow of a biomagnetic fluid embedded by magnetic dipole over a continuously moving sheet

AbstractThis paper is concerned with the investigation of steady, two‐dimensional, and laminar boundary layer flow of a biomagnetic fluid over a continuously moving sheet in the presence of a magnetic dipole. The magnetic field resulting from the dipole is contemplated to be strong enough to saturate the biofluid. The magnetization of the fluid is regarded to be a linear function of temperature. The solution procedure involves the reduction of a nonlinear system of coupled PDEs into ODEs that comprise five parameters. The transformed ODEs along with the boundary conditions are then solved numerically by introducing an efficient numerical technique based on the finite difference algorithm. The velocity, as well as temperature profiles within the boundary layer, are illustrated at specified values of free stream velocity (), wall velocity () and ferrohydrodynamic interaction parameter (). The demonstration of the Nusselt number and friction factor are achieved for various governing parameters. Considering a specified Prandtl number () and normalized velocity difference , higher values of the friction factor are obtained for increasing and than that of . Whereas for the Nusselt number, we attain higher values for decreasing and . Moreover, an increase in the velocity ratio results in a decrease in both heat transfer rate as well as friction factor. Furthermore, as increases, the heat transfer rate decreases yet we get higher values for higher and . In the case of friction factor, it increases with increasing and gives higher values for lower and higher . We have also depicted the streamlines for the 2‐D boundary layer flow of the biomagnetic fluid for different . The graphical results manifest that the flow field is greatly impacted by the ferrohydrodynamic field, which could be of interest in medical as well as bioengineering implementations, like, magnetic drug delivery in blood cells, separating RBCs as well as controlling the flow of blood during surgical procedures.

Urban freight distribution with electric vehicles: comparing some solution procedures

The Vehicle Routing Problem (VRP) is a well-known discrete optimization problem that has an impact on theoretical and practical applications. In this paper, a freight distribution model that includes a charging system located at the depot, making it feasible for real world-implementation, is proposed. Two different solution methods are proposed and compared: a genetic algorithm (GA) and a population-based simulated annealing (PBSA) with the number of moves increasing during the iterations. Among the variety of algorithm used to solve the VRP, population-based search methods are the most useful, due to the ability to update the memory at each iteration. To demonstrate the practical aspects of the proposed solution a case study is solved using travel time on a real network to evaluate the potentiality for a real-world application.

Three-dimensional buckling analysis of stiffened plates with complex geometries using energy element method

A novel numerical method, energy element method (EEM), is proposed for the three-dimensional (3D) buckling analysis of stiffened plates with complex geometries. The problem is formulated in a cuboidal domain, and any complex geometric stiffened plate is modeled by assigning cutouts within the cuboidal domain. The stiffened plate is considered as an energy body and is discretized using Gauss points with variable stiffness properties to simulate its energy distribution. Incorporating the extended interval integration, Gauss quadrature, variable stiffness properties, and Chebyshev polynomials, the strain energy of stiffened plates with complex geometries can be numerically simulated by putting the stiffness and thickness of Gauss points in the cutouts to zero in the cuboidal domain. Using the principle of minimum potential energy and Ritz solution procedure, the deformation and buckling behaviors of stiffened plates with complex geometries can be captured. As a result of the new formulations in EEM, new standard energy functionals and solving procedures have been developed. In addition, Gauss points are generated within the energy elements accounting for the geometric boundaries of the stiffened plate, which are characterized by level set functions. EEM is employed to investigate complex-shaped stiffened plates with straight or curvilinear stiffeners, and the results are compared to those obtained using FEM or mesh-free method. The precision, generalization, and stability of EEM are demonstrated.