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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (31 August 2023) | Viewed by 18521

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Special Issue Editors

Dr. Kuo Tian

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Guest Editor

Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China
Interests: composite structures; buckling analysis; digital twin; data driven method; optimization design
Special Issues, Collections and Topics in MDPI journals

Dr. Weizhu Yang

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Department of Engineering Mechanics, Northwestern Polytechnical University, Xi'an 710072, China
Interests: structural strength of aeroengines; mechanical behavior and optimization design of structures

Dr. Shiyao Lin

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Guest Editor

Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109, USA
Interests: composite materials; computational fracture mechanics; impact damage

Special Issue Information

Dear Colleagues,

The scope of this Special Issue ranges from mathematical foundations of the numerical optimization to algorithm and software development, and from benchmark examples to case studies of practical applications in a wide range of industries including aerospace, civil, automotive, architecture, mechanical, and electrical engineering. Articles that focus on developing advanced algorithms for topology optimization, shape optimization, surrogate-based optimization, data-driven optimization, reliability optimization, and multidisciplinary optimization (solid, fluid, thermal, electric and electronics, electromagnetics) are encouraged. Particularly, deep research on applications of numerical simulation and optimization in composite structure design, nanotechnology, additive manufacturing, digital twins, AI, cloud computing, electric and electronics systems, biomechanics, and hydrogen storage technology is also welcomed.

Dr. Kuo Tian
Dr. Weizhu Yang
Dr. Shiyao Lin
Guest Editors

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Keywords

Topology optimization
Shape optimization
Surrogate-based optimization
Data-driven optimization
Reliability optimization
Multidisciplinary optimization
Applications in composite structure design
Applications in additive manufacturing
Applications in aerospace engineering
Applications in civil engineering

Published Papers (13 papers)

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Research

9 pages, 551 KiB

Open AccessArticle

Filamentation of a Hollow Gaussian Beam in a Nonlinear Optical Medium

by Mir Asma, A. K. Shafeeque Ali, Abdullah Khamis Alzahrani, Malik Zaka Ullah and Stanford Shateyi

Mathematics 2023, 11(19), 4130; https://0-doi-org.brum.beds.ac.uk/10.3390/math11194130 - 29 Sep 2023

Viewed by 610

Abstract

This paper reports the filamentation of hollow Gaussian beams of the first, second, third, and fourth orders during propagation in a cubic and quintic nonlinear medium. Due to spatial modulation instability, the hollow Gaussian beams split to form either co-centric circular filaments or ultrashort pulses. It is found that the properties of the nonlinear medium used for propagation have a strong influence on certain characteristics of the formed filaments, such as peak intensity and pulse width. This correlation between the system parameters of the medium and filament characteristics represents a method for calculating the system parameters of the medium. This investigation can be helpful in the development of a hollow Gaussian beam-based artificial intelligence system that can be used to measure the system parameters of the studied nonlinear medium. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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20 pages, 11241 KiB

Open AccessArticle

A Joint Optimization Algorithm Based on the Optimal Shape Parameter–Gaussian Radial Basis Function Surrogate Model and Its Application

by Jian Sun, Ling Wang and Dianxuan Gong

Mathematics 2023, 11(14), 3169; https://0-doi-org.brum.beds.ac.uk/10.3390/math11143169 - 19 Jul 2023

Cited by 1 | Viewed by 853

Abstract

We propose a joint optimization algorithm that combines the optimal shape parameter–Gaussian radial basis function (G-RBF) surrogate model with global and local optimization techniques to improve accuracy and reduce costs. We analyze factors that affect the accuracy of the G-RBF surrogate model and use the particle swarm optimization (PSO) algorithm to determine the optimal shape parameter and control the number and spacing of the sampling points for a high-precision surrogate model. Global optimization refines the surrogate model, serving as the initial value for local optimization to further refine the problem. Our experiments show that this method significantly reduces computation costs. We optimize the section size of cantilever beams for different materials, obtaining the optimal section size and mass for each. We find that hard aluminum alloy is the optimal choice, meeting yield strength and deflection requirements through finite element analysis verification. Our work highlights the effectiveness of the joint optimization algorithm based on the surrogate model, providing valuable tools and insights into optimizing various structures. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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23 pages, 19166 KiB

Open AccessArticle

A Hybrid Full-Discretization Method of Multiple Interpolation Polynomials and Precise Integration for Milling Stability Prediction

by Xuefeng Yang, Wenan Yang and Youpeng You

Mathematics 2023, 11(12), 2629; https://0-doi-org.brum.beds.ac.uk/10.3390/math11122629 - 08 Jun 2023

Cited by 1 | Viewed by 694

Abstract

As milling chatter can lead to poor machining quality and limit the efficiency of productivity, it is of great significance to learn about milling chatter stability and research the effective and fast prediction of milling stability. In this study, a hybrid full-discretization method of multiple interpolation polynomials and precise integration (HFDM) is proposed for milling stability prediction. Firstly, the third-order Newton interpolation polynomial, third-order Hermite interpolation polynomial and linear interpolation are applied to approximate the state term, delay term and periodic coefficient matrix, respectively. Meanwhile, the matrix exponentials can be calculated based on the precise integration algorithm, which can improve computational accuracy and efficiency. The numerical simulation results indicate that the proposed method can not only effectively generate a stability lobe diagram (SLD) but also obtain better prediction accuracy and computation efficiency. A milling experiment is offered to demonstrate the feasibility of the method. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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35 pages, 686 KiB

Open AccessArticle

Adaptive Hybrid Mixed Two-Point Step Size Gradient Algorithm for Solving Non-Linear Systems

by Eltiyeb Ali and Salem Mahdi

Mathematics 2023, 11(9), 2102; https://0-doi-org.brum.beds.ac.uk/10.3390/math11092102 - 28 Apr 2023

Cited by 1 | Viewed by 1024

Abstract

In this paper, a two-point step-size gradient technique is proposed by which the approximate solutions of a non-linear system are found. The two-point step-size includes two types of parameters deterministic and random. A new adaptive backtracking line search is presented and combined with the two-point step-size gradient to make it globally convergent. The idea of the suggested method depends on imitating the forward difference method by using one point to estimate the values of the gradient vector per iteration where the number of the function evaluation is at most one for each iteration. The global convergence analysis of the proposed method is established under actual and limited conditions. The performance of the proposed method is examined by solving a set of non-linear systems containing high dimensions. The results of the proposed method is compared to the results of a derivative-free three-term conjugate gradient CG method that solves the same test problems. Fair, popular, and sensible evaluation criteria are used for comparisons. The numerical results show that the proposed method has merit and is competitive in all cases and superior in terms of efficiency, reliability, and effectiveness in finding the approximate solution of the non-linear systems. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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14 pages, 2554 KiB

Open AccessFeature PaperArticle

Novel Approach of Airfoil Shape Representation Using Modified Finite Element Method for Morphing Trailing Edge

by Martynas Lendraitis and Vaidas Lukoševičius

Mathematics 2023, 11(9), 1986; https://0-doi-org.brum.beds.ac.uk/10.3390/math11091986 - 23 Apr 2023

Viewed by 1672

Abstract

This study presents a novel approach to parameterize the geometry of a morphing trailing-edge flap that allows its aerodynamics to be optimized while capturing the expected structural behavior of the flap. This approach is based on the finite frame element method, whereby the initial flap surface is defined as a structure with constraints that are similar to those of a morphing flap with passive skin. The initial shape is modified by placing a series of distributed loads on the surface. The finite frame element method is modified with rigid rotation corrections to maintain the initial element length without requiring nonlinear calculations and to achieve accurate surface-length results by only solving the linear FEM equations twice. The proposed method enables the shape of the morphing flaps to be rapidly formulated while maintaining the initial upper surface-length and trailing-edge angle. The constraints are inherently integrated into the algorithm, eliminating the need for unnecessary feasibility checks during the aerodynamic optimization. By using the proposed airfoil parameterization method, a case study was conducted by using a genetic algorithm to optimize the lift-to-drag ratio of the NACA 23012 airfoil flap starting at 0.7c with 10 degrees of deflection. The optimizer resulted in a structurally feasible morphing flap that achieved a 10% increase in the lift-to-drag ratio in the optimized angle of attack range. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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20 pages, 515 KiB

Open AccessArticle

Optimal Power Dispatch of PV Generators in AC Distribution Networks by Considering Solar, Environmental, and Power Demand Conditions from Colombia

by Luis Fernando Grisales-Noreña, Oscar Danilo Montoya, Brandon Cortés-Caicedo, Farhad Zishan and Javier Rosero-García

Mathematics 2023, 11(2), 484; https://0-doi-org.brum.beds.ac.uk/10.3390/math11020484 - 16 Jan 2023

Cited by 3 | Viewed by 1647

Abstract

This paper deals with the problem regarding the optimal operation of photovoltaic (PV) generation sources in AC distribution networks with a single-phase structure, taking into consideration different objective functions. The problem is formulated as a multi-period optimal power flow applied to AC distribution grids, which generates a nonlinear programming (NLP) model with a non-convex structure. Three different objective functions are considered in the optimization model, each optimized using a single-objective function approach. These objective functions are (i) an operating costs function composed of the energy purchasing costs at the substation bus, added with the PV maintenance costs; (ii) the costs of energy losses; and (iii) the total CO₂ emissions at the substation bus. All these functions are minimized while considering a frame of operation of 24 h, i.e., in a day-ahead operation environment. To solve the NLP model representing the studied problem, the General Algebraic Modeling System (GAMS) and its SNOPT solver are used. Two different test feeders are used for all the numerical validations, one of them adapted to the urban operation characteristics in the Metropolitan Area of Medellín, which is composed of 33 nodes, and the other one adapted to isolated rural operating conditions, which has 27 nodes and is located in the department of Chocó, Colombia (municipality of Capurganá). Numerical comparisons with multiple combinatorial optimization methods (particle swarm optimization, the continuous genetic algorithm, the Vortex Search algorithm, and the Ant Lion Optimizer) demonstrate the effectiveness of the GAMS software to reach the optimal day-ahead dispatch of all the PV sources in both distribution grids. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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13 pages, 5794 KiB

Open AccessArticle

Numerical Simulation and Structure Optimization of Multilayer Metamaterial Plus-Shaped Solar Absorber Design Based on Graphene and SiO₂ Substrate for Renewable Energy Generation

by Haitham Alsaif, Shobhit K. Patel, Naim Ben Ali, Ammar Armghan and Khaled Aliqab

Mathematics 2023, 11(2), 282; https://0-doi-org.brum.beds.ac.uk/10.3390/math11020282 - 05 Jan 2023

Cited by 9 | Viewed by 1680

Abstract

Renewable energy is the energy for future generations as it is clean and widely available. The solar absorber is a sustainable energy source that converts solar energy into heat energy. The structural optimization is analyzed to enhance the absorption of the multilayer design. The proposed efficient solar absorber is made of a multilayer plus-shaped resonator supported by a SiO₂ substrate with a graphene spacer. The multilayer approach is utilized to enhance the absorption of the overall structure. The absorption of the multilayer solar absorber design is presented with AM 1.5 response observing the amount of energy absorbed from solar radiation. The different structural parameters are optimized to obtain the efficiency plus-shaped absorber design. The results of a different angle of incidence clearly show that the absorber is giving high absorption over a wide-angle range. The design results are also being analyzed with other similar works to show the improvement. The proposed absorber with high efficiency will be a good choice for solar thermal energy conversion applications. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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28 pages, 916 KiB

Open AccessArticle

Optimal Operation of PV Sources in DC Grids for Improving Technical, Economical, and Environmental Conditions by Using Vortex Search Algorithm and a Matrix Hourly Power Flow

by Luis Fernando Grisales-Noreña, Andrés Alfonso Rosales-Muñoz, Brandon Cortés-Caicedo, Oscar Danilo Montoya and Fabio Andrade

Mathematics 2023, 11(1), 93; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010093 - 26 Dec 2022

Cited by 9 | Viewed by 1318

Abstract

This document presents a master–slave methodology for solving the problem of optimal operation of photovoltaic (PV) distributed generators (DGs) in direct current (DC) networks. This problem was modeled using a nonlinear programming model (NLP) that considers the minimization of three different objective functions in a daily operation of the system. The first one corresponds to the minimization of the total operational cost of the system, including the energy purchasing cost to the conventional generators and maintenance costs of the PV sources; the second objective function corresponds to the reduction of the energy losses associated with the transport of energy in the network, and the third objective function is related to the minimization of the total emissions of CO

_{2}

by the conventional generators installed on the DC grid. The minimization of these objective functions is achieved by using a master–slave optimization approach through the application of the Vortex Search algorithm combined with a matrix hourly power flow. To evaluate the effectiveness and robustness of the proposed approach, two test scenarios were used, which correspond to a grid-connected and a standalone network located in two different regions of Colombia. The grid-connected system emulates the behavior of the solar resource and power demand of the city of Medellín-Antioquia, and the standalone network corresponds to an adaptation of the generation and demand curves for the municipality of Capurganá-Choco. A numerical comparison was performed with four optimization methodologies reported in the literature: particle swarm optimization, multiverse optimizer, crow search algorithm, and salp swarm algorithm. The results obtained demonstrate that the proposed optimization approach achieved excellent solutions in terms of response quality, repeatability, and processing times. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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17 pages, 15995 KiB

Open AccessArticle

Research on the Lightweight Design of an Aircraft Support Based on Lattice-Filled Structures

by Zhou Huang, Yong Liu, Xin Huang and Dong Mu

Mathematics 2022, 10(23), 4576; https://0-doi-org.brum.beds.ac.uk/10.3390/math10234576 - 02 Dec 2022

Cited by 1 | Viewed by 1252

Abstract

This work studied the lightweight design of an aircraft support based on lattice-filled structures. Different from the traditional design process of lattice-filled structures, this work combined several approaches, including topology optimization, homogenization analysis, and Non-Uniform Rational B-splines (NURBS) surface modeling, to reduce the structural weight more effectively. The theories and implementations involved in the design process are introduced in this work. The new lattice-filled design of the aircraft support component reduced the weight by 40% compared with the original value, and its additive manufacturability was verified. Finally, the structural responses of the lattice-filled design from both a detailed model and homogenization model were determined and compared, considering both the static responses and dynamic characteristics. The results revealed that the homogenization method efficiently and accurately obtained the structural displacements and natural frequencies of the complex lattice-filled design. This indicates that the homogenization method can effectively reduce the calculation burden of the design process of lattice-filled structures, which opens a new channel for the structural optimizations of lattice-filled structures. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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16 pages, 4354 KiB

Open AccessArticle

Relaxed Variable Metric Primal-Dual Fixed-Point Algorithm with Applications

by Wenli Huang, Yuchao Tang, Meng Wen and Haiyang Li

Mathematics 2022, 10(22), 4372; https://0-doi-org.brum.beds.ac.uk/10.3390/math10224372 - 20 Nov 2022

Viewed by 1098

Abstract

In this paper, a relaxed variable metric primal-dual fixed-point algorithm is proposed for solving the convex optimization problem involving the sum of two convex functions where one is differentiable with the Lipschitz continuous gradient while the other is composed of a linear operator. Based on the preconditioned forward–backward splitting algorithm, the convergence of the proposed algorithm is proved. At the same time, we show that some existing algorithms are special cases of the proposed algorithm. Furthermore, the ergodic convergence and linear convergence rates of the proposed algorithm are established under relaxed parameters. Numerical experiments on the image deblurring problems demonstrate that the proposed algorithm outperforms some existing algorithms in terms of the number of iterations. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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18 pages, 4221 KiB

Open AccessArticle

Effective Optimization Based on Equilibrium Optimizer for Dynamic Cutting Force Coefficients of the End-Milling Process

by Minh-Quang Tran, Mahmoud Elsisi, Viet Q. Vu, Fahad Albalawi and Sherif S. M. Ghoneim

Mathematics 2022, 10(18), 3287; https://0-doi-org.brum.beds.ac.uk/10.3390/math10183287 - 10 Sep 2022

Cited by 5 | Viewed by 1356

Abstract

This study aims to develop an accurate dynamic cutting force model in the milling process. In the proposed model, the estimated cutting force tackles the effect of the self-excited vibration that causes machining instability during the cutting process. In particular, the square root of the residual cutting force between the prediction and the actual cutting force is considered as an objective function for optimizing the cutting force coefficients using the equilibrium optimizer (EO) approach instead of the trial-and-error approach. The results confirm that the proposed model can provide higher prediction accuracy when the EO is applied. In addition, the proposed EO has a minimum integral square error (ISE) of around 1.12, while the genetic algorithm (GA) has an ISE of around 1.14 and the trial-and-error method has an ISE of around 2.4. Moreover, the proposed method can help to investigate the cutting stability and to suspend the chatter phenomenon by selecting an optimal set of cutting parameters. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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15 pages, 5539 KiB

Open AccessArticle

A High-Precision Surrogate Modeling Method Based on Parallel Multipoint Expected Improvement Point Infill Criteria for Complex Simulation Problems

by Shande Li, Jian Wen, Jun Wang, Weiqi Liu and Shuai Yuan

Mathematics 2022, 10(17), 3088; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173088 - 27 Aug 2022

Cited by 2 | Viewed by 1426

Abstract

In order to overcome the problem of the low fitting accuracy of the expected improvement point infill criteria (EI) and the improved expected improvement point infill criteria (IEI), a high-precision surrogate modeling method based on the parallel multipoint expected improvement point infill criteria (PMEI) is presented in this paper for solving large-scale complex simulation problems. The PMEI criterion takes full advantage of the strong global search ability of the EI criterion and the local search ability of the IEI criterion to improve the overall accuracy of the fitting function. In the paper, the detailed steps of the PMEI method are introduced firstly, which can add multiple sample points in a single iteration. At the same time, in the process of constructing the surrogate model, it is effective to avoid the problem of the low fitting accuracy caused by adding only one new sample point in each iteration of the EI and IEI criteria. The numerical examples of the classical one-dimensional function and two-dimensional function clearly demonstrate the accuracy of the fitting function of the proposed method. Moreover, the accuracy of the multi-objective optimization surrogate model of a truck cab constructed by the PMEI method is tested, which proves the feasibility and effectiveness of the proposed method in solving high-dimensional modeling problems. All these results confirm that the Kriging model developed by the PMEI method has high accuracy for low-dimensional problems or high-dimensional complex problems. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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15 pages, 4965 KiB

Open AccessArticle

Numerical Simulation of Failure Analysis of Storage Tank with Partition Plate and Structure Optimization

by Wenxian Su and Xiao Feng

Mathematics 2021, 9(24), 3230; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243230 - 14 Dec 2021

Cited by 1 | Viewed by 2371

Abstract

Storage tanks with partition plates are widely used in the petrochemical industry. However, relevant standards do not propose corresponding design criteria and methods for this type of structure, and theoretical design formulas cannot be applied to ensure the reliability of its structure. Therefore, it is necessary to analyze and design the storage tank with a partition plate by using finite elements. This paper studies the problem of buckling depression and cracks in the welded parts of the S-shaped tank with a partition plate during its operation. We used the finite element software ANSYS to analyze the overall strength and stability of the structure and obtain the larger stress area. Based on this, a safe and economical optimization plan is proposed: under the condition of strictly controlling the liquid level difference on both sides of the partition, the tank structure is optimized by adding stiffeners and tie rods. The study revealed that the measure effectively improves the overall rigidity of the tank body and reduces the maximum stress of the structure and enhances the safety performance of storage tank. Additionally, it provides a reference for the structural strength design of storage tanks with partition plates. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization: Methods and Applications)

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