Natural Frequency Constraints Research Articles

Optimal design of truss domes with frequency constraints using seven metaheuristic algorithms incorporating a comprehensive statistical assessment

In this article, seven recently developed metaheuristic algorithms are utilized for optimal design of dome-shaped trusses with natural frequency constraints. These algorithms are social network search, jellyfish search optimizer, equilibrium optimizer, teaching-learning-based optimization, grey wolf optimizer, colliding bodies optimization, and the improved grey wolf optimizer. This study focuses on the assessment and tuning of these algorithms to enhance both computational efficiency and solution quality. In particular, we introduce a hyperparameter tuning procedure that significantly improves the performance of the algorithms to make them more suitable for the solution of structural optimization problems. The performance of the selected algorithms is studied through five benchmark examples for design optimization of dome structures, from small-scale to large-scale cases, with frequency constraints. Many combination scenarios are studied for hyperparameter tuning of the algorithms in each design example in order to provide comprehensive statistical results determining the sufficient number of structural analyses as well as the number of particles to find an efficient condition of each algorithm. Indeed, this study also presents an insight into the efficient setting of hyperparameters for each algorithm, leading to a better solution with lower computational cost. Furthermore, the optimization results provide a comparison of the solutions obtained in the current and previous studies in order to demonstrate the suitability of the selected algorithms.

Topology optimization of stationary fluid–structure interaction problems considering a natural frequency constraint for vortex-induced vibrations attenuation

Topology optimization applied to fluid–structure interaction problems is challenging because the physical phenomenon in real engineering applications is usually transient and strongly coupled. This leads to costly solutions for the forward and adjoint problems, the computational bottleneck of the topology optimization method. Thus, this paper proposes a topology optimization problem formulated in the steady state with post-processing and verification in the transient state. The objective is to design a stiff structure with lower effects of vibrations induced by the transient fluid vortices. For that, the compliance minimization problem is solved subject to a natural frequency constraint (without any volume constraint). The TOBS-GT (Topology Optimization of Binary Structures with geometry trimming) method is used to solve the problem. To observe the vortex-shedding around the structure, a transient simulation is performed considering an incompressible fluid flow under a laminar regime and the structure subject to large displacements. For topology optimization, the fluid flow is at a steady state and the structure is modeled considering small displacements, i.e., a one-way coupled analysis. The finite element method is used to solve the governing equations and obtain the direct/adjoint sensitivities for the compliance and natural frequency functions. In this approach, the natural frequency of the structure is shifted away from the fluid flow vortex-shedding frequency, avoiding resonance. Numerical examples show that the proposed method can be effectively applied to design 2D structures in FSI problems with lower effects of Flow-Induced Vibration, attenuating the levels of displacement at the analyzed points of the structure.

Design of the multiphase material structures with mass, stiffness, stress, and dynamic criteria via a modified ordered SIMP topology optimization

This paper, a multi-material topology optimization method (MMTO) is presented for two-dimensional structures using modified ordered solid isotropic material with penalization (SIMP) under multiple constraints such as mass, stiffness, frequency, and stress. The element density is obtained by a density filtering and the Heaviside approximation, which are combined to avoid gray elements and numerical issues of stress. Effective improvement of multiphase material topology optimization was performed including the criteria for stiffness, stress, and free vibration. P-norm stress aggregation is employed to calculate the maximum von Mises stress and sensitivity of von Mises stress is formulated by the adjoint method. The natural frequency and stress constraints are solved simultaneously to enhance the stability and reduce the stress concentration of the optimized structures, which are not much covered in the literature. The multi-mode phenomenon in the eigenfrequency problem is treated by the simply proposed technique. Results indicate that the multi-material designs with multi-constraints can be illustrated and optimized effectively. The optimized topologies, for stress minimization, indicate that the maximum stress can be significantly reduced compared with compliance minimization. The maximum von Mises stress of the topological results considering the maximum stiffness with natural frequency and mass constraints is higher than that maximum stiffness with only mass constraints, which indicates that the optimized structure created considering the eigenvalues is safer. The proposed approach can obtain a reasonable result that effectively controls the stress level and reduces the stress concentration at the high stress regions of multi-phase material structures with multi-constraints. Benchmark numerical examples are presented to confirm the effectiveness of the presented method.

Computer-aided dynamic structural optimization using an advanced swarm algorithm

Natural frequencies, which are relatively easy parameters to obtain, provide efficient information about the dynamic behavior of structures. Controlling these parameters can help to minimize the destructive effect of dynamic loads on structures. However, the optimal design of the size and shape of truss structures with controlling dynamic constraints is an expensive problem. While frequency constraints are nonlinear and non-convex, handling natural frequencies as constraints is a challenging task, which prevents the phenomenon of resonance, large deformations, and destruction of a structure. Also, reducing the vibration amplitude of a structure would reduce the stress and deflections. In this paper, the Improved Marine Predators Algorithm (IMPA) is proposed and implemented for size and shape optimization of truss structures subject to natural frequency constraints. In this method, a Novel CF (NCF) factor is introduced to control the predator's step size when looking for prey. To evaluate the performance of IMPA, the optimum weight of structures under dynamic constraints was computationally investigated. To demonstrate the efficiency and robustness of IMPA, 37-bar truss bridge, 52-bar dome, 72-bar space truss, 120-bar dome, 200-bar planar truss, and 600-bar dome truss were optimized. Compared to other state-of-the-art algorithms, the results indicate that IMPA performs better in solving these nonlinear structural optimization problems.

Optimizing Truss Structures Using Composite Materials under Natural Frequency Constraints with a New Hybrid Algorithm Based on Cuckoo Search and Stochastic Paint Optimizer (CSSPO)

This article highlights the absence of published paradigms hybridized by The Cuckoo Search (CS) and Stochastic Paint Optimizer (SPO) for optimizing truss structures using composite materials under natural frequency constraints. The article proposes a novel optimization algorithm called CSSPO for optimizing truss structures made of composite materials, known as fiber-reinforced polymer (FRP) composites, to address this gap. Optimization problems of truss structures under frequency constraints are recognized as challenging due to their non-linear and non-convex search spaces that contain numerous local optima. The proposed methodology produces high-quality optimal solutions with less computational effort than the original methods. The aim of this work is to compare the performance of carbon FRP (CFRP), glass FRP (GFRP), and steel using a novel hybrid algorithm to provide valuable insights and inform decision-making processes in material selection and design. Four benchmark structure trusses with natural frequency constraints were utilized to demonstrate the efficiency and robustness of the CSSPO. The numerical analysis findings indicate that the CSSPO outperforms the classical SPO and exhibits comparable or superior performance when compared to the SPO. The study highlights that implementing CFRP and GFRP composites in truss construction leads to a notable reduction in weight compared to using steel.

Free Vibration Characteristics of Bidirectional Graded Porous Plates with Elastic Foundations Using 2D-DQM

This manuscript develops for the first time a mathematical formulation of the dynamical behavior of bi-directional functionally graded porous plates (BDFGPP) resting on a Winkler–Pasternak foundation using unified higher-order plate theories (UHOPT). The kinematic displacement fields are exploited to fulfill the null shear strain/stress at the bottom and top surfaces of the plate without needing a shear factor correction. The bi-directional gradation of materials is proposed in the axial (x-axis) and transverse (z-axis) directions according to the power-law distribution function. The cosine function is employed to define the distribution of porosity through the transverse z-direction. Equations of motion in terms of displacements and associated boundary conditions are derived in detail using Hamilton’s principle. The two-dimensional differential integral quadrature method (2D-DIQM) is employed to transform partial differential equations of motion into a system of algebraic equations. Parametric analysis is performed to illustrate the effect of kinematic shear relations, gradation indices, porosity type, elastic foundations, geometrical dimensions, and boundary conditions (BCs) on natural frequencies and mode shapes of BDFGPP. The effect of the porosity coefficient on the natural frequency is dependent on the porosity type. The natural frequency is dependent on the coupling of gradation indices, boundary conditions, and shear distribution functions. The proposed model can be used in designing BDFGPP used in nuclear, marine, aerospace, and civil structures based on their topology and natural frequency constraints.

Reptile search algorithm and kriging surrogate model for structural design optimization with natural frequency constraints

Abstract This study explores the use of a recent metaheuristic algorithm called a reptile search algorithm (RSA) to handle engineering design optimization problems. It is the first application of the RSA to engineering design problems in literature. The RSA optimizer is first applied to the design of a bolted rim, which is constrained optimization. The developed algorithm is then used to solve the optimization problem of a vehicle suspension arm, which aims to solve the weight reduction under natural frequency constraints. As function evaluations are achieved by finite element analysis, the Kriging surrogate model is integrated into the RSA algorithm. It is revealed that the optimum result gives a 13% weight reduction compared to the original structure. This study shows that RSA is an efficient metaheuristic as other metaheuristics such as the mayfly optimization algorithm, battle royale optimization algorithm, multi-level cross-entropy optimizer, and red fox optimization algorithm.

Multi-material topology optimization considering natural frequency constraint

PurposeThe purpose of this paper is to propose an effective and efficient numerical method that can consider natural frequency in multi-material topology optimization (MMTO) and which is scalable for complex three-dimensional (3D) problems.Design/methodology/approachThe optimization algorithm is developed by combining custom FORTRAN code for MMTO with the open-source software Mystran, which is used as a finite element analysis (FEA) solver. The proposed algorithm allows the designer to shift the fundamental frequency of the design beyond a defined frequency spectrum from the initial designing phase. The methodology is formulated in a smooth and differentiable manner, with the sensitivity expressions, required by gradient-based optimization solvers, presented.FindingsNatural frequency constraint has been successfully implemented into MMTO. The use of open-source software Mystran as an FEA solver in the algorithm provides ability to solve complex problems. Mystran offers powerful built-in functions for eigenvalue extraction using methods like Givens, modified Givens, inverse power and the Lanczos method, which provide the ability to solve complex models. The algorithm is successfully able to solve both two- and three-material MMTO jobs for two-dimensional and 3D geometries.Originality/valueNatural frequency constraint consideration into topology optimization is very challenging due to three common issues: localized eigenmodes, mode switching and high computational cost. The proposed algorithm addresses these inherent issues, implements natural frequency constraint to MMTO and solves for complex models, which is hardly possible using conventional methods.

Frequency-Constrained Optimization of a Real-Scale Symmetric Structural Using Gold Rush Algorithm

The optimal design of real-scale structures under frequency constraints is a crucial problem for engineers. In this paper, linear analysis, as well as optimization by considering natural frequency constraints, have been used for real-scale symmetric structures. These structures require a lot of time to minimize weight and displacement. The cyclically symmetric properties have been used for decreasing time. The structure has been decomposed into smaller repeated portions termed substructures. Only the substructure elements are needed when analyzing and designing with the concept of cyclic symmetries. The frequency constrained design of real-scale structures is a complex optimization problem that has many local optimal answers. In this research, the Gold Rush Optimization (GRO) algorithm has been used to optimize weight and displacement performances due to its effectiveness and robustness against uncertainties. The efficacy of the concept of cyclic symmetry to minimize the time calculated is assessed by three examples, including Disk, Silo, and Cooling Tower. Numerical results indicate that the proposed method can effectively reduce time consumption, and that the GRO algorithm results in a 14–20% weight reduction of the problems.

Improved slime mould algorithm with elitist strategy and its application to structural optimization with natural frequency constraints

An improved slime mould algorithm (ISMA) is proposed and applied to the size optimization of truss structures with natural frequency constraints. The slime mould algorithm is a recently proposed metaheuristic inspired by the morphological changes of the acellular slime mould Physarum polycephalum while foraging. This algorithm has been successfully applied to some real-world optimization problems in science and industry. However, it is found that the classical SMA suffers from slow convergence and often converges prematurely to non-optimal solutions, especially for large-scale optimization problems. Compared to the classical SMA, two main improvements are introduced in the proposed ISMA: (1) In the replacement phase of the ISMA, an elitist strategy is adopted to replace the generational replacement strategy of the classical SMA. The elitist strategy aims to increase the convergence rate of the ISMA. (2) A slight modification is made to the exploration phase of the classical SMA to ensure a vast exploration of the search space. The suggested modification enables the ISMA to overcome the shortcoming of the premature convergence of the classical SMA. Finally, the efficiency and robustness of the ISMA are demonstrated through three large-scale benchmark dome trusses with natural frequency constraints. To our knowledge, this is the first time to apply SMA to structural optimization. It is shown that the proposed ISMA overcomes the problems of premature convergence and slow convergence rate of the classical SMA. Numerical results show that the ISMA outperforms the classical SMA and shows superior or comparable performance to other reported methods.

Truss optimization with natural frequency constraints using generalized normal distribution optimization

Truss optimization with natural frequency constraints using generalized normal distribution optimization

Dynamic Arithmetic Optimization Algorithm for Truss Optimization Under Natural Frequency Constraints

Metaheuristic algorithms have successfully been used to solve any type of optimization problem in the field of structural engineering. The newly proposed Arithmetic Optimization Algorithm (AOA) has recently been presented for mathematical problems. The AOA is a metaheuristic that uses the main arithmetic operators’ distribution behavior, such as multiplication, division, subtraction, and addition in mathematics. In this paper, a dynamic version of the arithmetic optimization algorithm (DAOA) is presented. During an optimization process, a new candidate solution change to regulate exploration and exploitation in a dynamic version in each iteration. The most remarkable attribute of DAOA is that it does not need to make any effort to preliminary fine-tuning parameters relative to the most present metaheuristic. Also, the new accelerator functions are added for a better search phase. To evaluate the performance of both the AOA and its dynamic version, minimizing the weight of several truss structures under frequency bound is tested. These algorithms ’ efficiency is obtained by five classical engineering problems and optimizing different truss structures under various loading conditions and limitations.

Non-probabilistic reliability-based robust design of micro-scale topology optimization (NRRD-MTO) for structural vibro-acoustic problem under harmonic excitation and natural frequency constraints

Non-probabilistic reliability-based robust design of micro-scale topology optimization (NRRD-MTO) for structural vibro-acoustic problem under harmonic excitation and natural frequency constraints

Crash analysis of sandwich composite electric microbus

Sandwich composite is ideal for lightweight monocoque structure owing to not only its superiority in strength-to-weight ratio and specific flexural properties but also its manufacturability to any desired complex shape. In this work, an optimum lightweight design of monocoque composite sandwich-structure microbus subjected to bending and torsion stiffness criteria and natural frequency constraints is further investigated under crash conditions through finite element analysis. A composite sandwich panel made of 5.4-mm woven glass fabric-epoxy face sheets and 50-mm rigid polyurethane foam core is tested under low-velocity impact with a hemisphere headed impactor following ASTM D7136 to characterize the failure characteristics. An enhanced composite damage material model so-called MAT54 in LS-DYNA with damage progression in each lamina utilizing strength-based Chang-Chang criteria is implemented with various formulations to validate and compare for their efficiency and accuracy under impact test. The reduced-integration Balytschko-Tsay shell element with 5 through-thickness integration points is applied to the monocoque composite microbus model. The structural crashworthiness under the frontal crash of the bus at 20, 40 and 50 km/h to a rigid wall barrier is examined. The front structure absorbs the most frontal impact energy up to 60% and shows severe damage when the crash velocity is 40 km/h or higher. For the bus impact velocity of 50 km/h, the maximum deformation of the bus front is 809 mm resulting in the intrusion of the structure into the specified driver survival space. Design improvement of the sandwich structure should be further explored to enhance its energy absorption and safety of the bus under crash events.

On the effect of boundary condition uncertainty on robust topology optimization of aerospace structures

Uncertainty quantification (UQ) within topology optimization (TO) is a growing trend, with designers recognizing that deterministic analysis does not reflect the natural variabilities found in real-world structural performance. Thus far the majority of works have dealt with uncertain loading and material properties; however, minimal research has considered uncertain boundary conditions (BCs), which play a vital role for static and dynamic analysis involving compliance and natural frequency objectives and/or constraints. BCs uncertainty is studied in this paper by assuming a random percentage of a cantilever can be freed to rotate and/or translate in space. This is firstly demonstrated through a case study on a flat plate wing where the frequency gap between two modes is maximized, and secondly through optimizing the wing of a sensorcraft where compliance is minimized. A robust objective is formulated via a weighted sum of the mean and standard deviation, which is approximated efficiently using a Non-Intrusive Polynomial Chaos Expansion (NIPC). Both studies demonstrate the pitfalls of a deterministic optimum in dealing with uncertain BCs, and how using Robust TO (RTO) can effectively reduce the variance and worst-case performance of the objective. Experimental validation for the flat plate wing is also undertaken to ensure practical feasibility of the RTO topologies and to provide motivation to study BCs uncertainty further.

Weighted superposition attraction-repulsion (WSAR) algorithm for truss optimization with multiple frequency constraints

Structural optimization of truss structures under multiple frequency constraints is a highly nonlinear and complex optimization problem with non-convex solution space. The optimization method used to solve mentioned problem is expected to provide a very good balance between solution accuracy and computational cost. In this work, the Weighted Superposition Attraction-Repulsion (WSAR) algorithm, which is a recent swarm intelligence based metaheuristic algorithm, is proposed for effective solution of truss optimization problems with multiple natural frequency constraints. The effectiveness and robustness of the WSAR algorithm is studied by solving several planar/space truss structures optimization problems. The optimization results reveal that the successfulness and effectiveness of WSAR in solving truss optimization problems under multiple frequency constraints where WSAR is able to generate the best results in terms of optimized weight and standard deviation compared to the other state-of-the-art metaheuristic algorithms.

Parameter free Jaya algorithm for truss sizing-layout optimization under natural frequency constraints

In this study, the parameter free Jaya algorithm (PFJA) is developed for sizing and layout optimization of truss structures subject to natural frequency constraints. The distinctive feature of PFJA is that it uses neither algorithm-specific parameters nor common parameters in the search process. Besides using an elitist strategy where new structural analyses are performed only if strictly necessary, PFJA adaptively changes population size in the optimization process. The validity of proposed PFJA is demonstrated by solving eight classical truss weight minimization problems including up to 59 sizing and layout design variables. The results obtained by the PFJA are compared with those of standard JA, modified Jaya algorithm (MJA) and other state-of-art metaheuristic algorithms in terms of optimized weight, convergence speed and several statistical parameters. Optimization results prove the superiority of PFJA over standard JA, MJA and other metaheuristic optimizers available in the literature.

Truss Optimization with Natural Frequency Constraints Using Modified Social Engineering Optimizer

Truss Optimization with Natural Frequency Constraints Using Modified Social Engineering Optimizer

Set theoretical variants of the teaching–learning-based optimization algorithm for optimal design of truss structures with multiple frequency constraints

In this paper, a set theoretical framework is proposed for the population-based metaheuristic algorithms. Using the proposed framework, two new set theoretical variants of the teaching–learning-based optimization (TLBO) algorithm are developed. These algorithms are named as OST-TLBO and STMP-TLBO, which are acronyms for the “ordered set theoretical teaching–learning-based optimization” and “set theoretical multi-phase teaching–learning-based optimization”, respectively. The present framework can be applied to other population-based metaheuristic algorithms. In order to verify the stability and robustness of the presented algorithms, some optimization problems are examined. These problems include four truss optimization problems with multiple natural frequency constraints. Comparing the results obtained from the proposed algorithms with those of the standard version of the TLBO algorithm shows that the proposed set theoretical framework improves the performance of the standard TLBO in terms of robustness and convergence characteristics.

Total and Partial Updating Technique: A Swift Approach for Cross-Section and Geometry Optimization of Truss Structures

In this paper, a novel two-phase method called Total and Partial Updating Technique, is employed for the cross section and geometry optimization of truss structures. The goal of optimizing such structures is to minimize their weights under natural frequency constraints. In Total and Partial Updating Technique, in order to find the global optimum point, only the best two existed solution points at each repetition move together in the search space defined in each phase. Each time, the first phase is destined to find a solution near the global optimum point although searching in the unfeasible region. In the second phase, it is forced jumping into the feasible region to reach the global optimum point. The comparison of the results with those in the literature on two numerical examples demonstrates that due to its relatively significant low computational costs and its assured non-violated constrained optimum solutions at the end of the process, the method may be considered as more reliable and efficient technique on such problems.