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Optimal analytical Michell frame structures have been extensively used as benchmark examples in topology optimization, including truss, frame, homogenization, density and level-set based approaches. However, as we will point out, partly the interpretation of Michell’s structural continua as discrete frame structures is not accurate and partly, it turns out that limiting structural topology to frame-like structures is a rather severe design restriction and results in structures that are quite far from being stiffness optimal. The paper discusses the interpretation of Michell’s theory in the context of numerical topology optimization and compares various topology optimization results obtained with the frame restriction to cases with no design restrictions. For all examples considered, the true stiffness optimal structures are composed of sheets (2D) or closed-walled shell structures (3D) with variable thickness. For optimization problems with one load case, numerical results in two and three dimensions indicate that stiffness can be increased by up to 80 % when dropping the frame restriction. For simple loading situations, studies based on optimal microstructures reveal theoretical gains of +200 %. It is also demonstrated how too coarse design discretizations in 3D can result in unintended restrictions on the design freedom and achievable compliance.

In this article, a density-driven unified multi-material topology optimization framework is suggested for functionally graded (FG) structures under static and dynamic responses. For this, two-dimensional solid structures and plate-like structures with/without variable thickness are investigated as design domains using multiple in-plane bi-directional FG materials (IBFGMs). In the present approach, a generally refined interpolation scheme relying upon Solid Isotropic Material with Penalization is proposed to deal with equivalent properties of IBFGMs. This methodology’s topological design variables are totally independent of all material phases. Therefore, the present method can yield separate material phases at their contiguous boundaries without intermediate density materials. The assumption of mixed interpolation of tensorial components of the 4-node shell element is employed to analyze plate elements, aiming to tackle the shear-locking phenomenon encountered as the optimal plate thickness becomes thinner. The mesh-independence filter is utilized to suppress the checkerboard formation of the material distribution. The method of Moving Asymptotes is used as an optimizer to update design variables in the optimization process. Several numerical examples are presented to evaluate the efficiency and reliability of the current approach.

This study firstly presents a multi-material topology optimization approach for thin plates with variable thickness based on Kirchhoff plate theory. For this purpose, an alternating active-phase algorithm in conjunction with the block Gauss-Seidel method is utilized to transform a multiphase topology optimization problem with multiple volume fraction constraints to many binary phase topology optimization sub-problems with only one volume fraction constraint. Accordingly, the number of design variables depends only on one active phase in each of those sub-problems no matter how many phases the original problem are. In addition, moved and regularized Heaviside function (MRHF) that plays the role of a filter is also investigated in the framework of multiple materials field. The mathematical formulations of stiffness and complaint sensitivity with respect to multi-directional variable thickness linked to thin plate potential energy are derived in terms of multiphase design variables. Numerical examples demonstrate interactions of variables thickness and multiple materials to thin mid-plates with the same amount of volume fraction and total structural volume.

To prevent excessive vibrations in the rotor system at critical rotational speeds caused by unbalanced excitation, a nonlinear vibration absorber is proposed in this paper. The absorber consists of a rigid mass ring and four circumferentially distributed variable thickness circular arches. It is mounted in the grooves of the disk and rotates with it. The variable thickness circular arches in the absorber is used to provide linear positive and nonlinear stiffness. The absorber achieves rotor vibration reductions by absorbing vibrational energy. When the attachment does not contain linear positive stiffness, the absorber becomes a nonlinear energy sink. The motion equations of the rotor system with the absorber are derived using Lagrange’s equation. The differential evolution algorithm has the advantages of being easy to implement and high optimization efficiency, and it also can avoid getting trapped in local optima during the optimization process. In this study, the DE algorithm is employed to investigate the vibration reduction capability of the designed nonlinear vibration absorber. The present results indicate that with the equivalent stiffness to a composite of linear positive and cubic nonlinear stiffness, the absorber demonstrates a pronounced suppression effect on the vibration of the rotor system. This study offers a novel approach to the vibration reduction of rotors.

The thin film hybrid nanofluid (HNF) flow over a rotating stretching disk is considered for heat transfer (HT) enhancement applications. The water-based HNF consists of Ag and TiO 2 nanoparticles (NPs). The film thickness is kept variable and the flow medium is also considered porous and variable. The variable porous space for the variable thickness of the thin film is the main focus of the present analysis. The variable thickness of the thin film improves heat transfer (HT) between the rotating disk and the surrounding fluid. Thicker film layers provide increased thermal resistance, reducing the (HT) rate, while thinner film layers enhance (HT) by minimizing the thermal resistance. Controlling the thickness of the film through the variable porous space allows for the optimization of (HT) in various engineering applications. Additionally, the presence of Ag and TiO 2 nanoparticles (NPs) in the water-based HNF further enhances the heat transfer properties, making it an attractive option for heat transfer (HT) enhancement applications. Moreover, the stability of the variable thin film is more adjustable in the variable porous space. The artificial neural network is used to solve the problem and validate the obtained results, through Training, Testing, and error estimations.

This study investigates the application of machine learning (ML) techniques for predicting vibration frequencies of thin rectangular plates with variable thickness. Traditional optimization methods, such as genetic algorithms, require repeated solutions of the plate vibration eigenproblem using finite element (FE) analysis, which is computationally expensive. To reduce this cost, a surrogate model based on artificial neural networks (ANNs) is proposed as an efficient alternative. The dataset includes variations in plate geometry, boundary conditions, and thickness distribution, encoded numerically for model training. ANN architecture and hyperparameters—such as the number of hidden layers, neurons per layer, and activation functions—were systematically tuned to achieve high prediction accuracy while avoiding overfitting. Data preprocessing steps, including standardization and scaling, were applied to improve model stability. Performance was evaluated using metrics such as RMSE and R2. The results demonstrate that ANNs can accurately predict eigenvalues with significantly reduced computational effort compared to FE analysis. This approach offers a practical solution for integrating machine learning into structural optimization workflows.

Variable thickness sheet and homogenization-based topology optimization often result in spread-out, non-well-defined solutions that are difficult to interpret or de-homogenize to sensible final designs. By extensive numerical investigations, we demonstrate that such solutions are due to non-uniqueness of solutions or at least very flat minima. Much clearer and better-defined solutions may be obtained by adding a measure of non-void space to the objective function with little if any increase in structural compliance. We discuss various alternatives for cleaning up solutions and propose two efficient approaches which both introduce an auxiliary field to control non-void space: one approach based on a cut element based auxiliary field (hybrid approach) and another approach based on an auxiliary element based field (density approach). At the end, we demonstrate significant qualitative and quantitative improvements in variable thickness sheet and de-homogenization designs resulting from the proposed cleaning schemes.

Variable-thickness rolled blank (VRB) structures can offer excellent crashworthiness and weight reduction potential with its large-scale applications with satisfying manufacturing constraints, whose crashworthiness optimization is classified into the high-dimensional expensive problem including explicit and implicit constraints. Therefore, an efficient parallel constrained Bayesian optimization (PCBO) algorithm is proposed to improve the global searching accuracy and efficiency from three aspects: (1) the bilog transformation for implicit constraints is introduced to reduce the difficulty of identifying the feasibility of \"expensive\" sample points near constraint boundaries; (2) the trust region updating strategy is introduced to balance the exploration and exploitation of the searching process by dynamically updating the searching space; (3) the parallel high-quality points addition strategy based on multiple acquisition functions (PPA-MAF) is proposed, which not only increases the diversity of the optimal solutions but also achieves the multi-task parallel computation. Seven classical cases are adopted to validate the convergence and robustness of PCBO algorithm by comparing with several popular algorithms. Finally, the crashworthiness optimization of a VRB bumper system is performed by the proposed algorithm which can get better lightweight case under satisfying the manufacturing and performance constraints.

Composite components with variable-thickness structures often suffer from insufficient forming pressure during curing due to complex pressure transfer in regions with abrupt thickness changes, which easily causes void defects and degrades component performance. In this study, a mechanical vibration-assisted double vacuum bag process is proposed. Finite element analysis of the vibration energy field in saturated porous composites is conducted, and curing experiments for variable-thickness specimens are designed. The effects of vibration, vacuum, and their synergy on void characteristics and mechanical properties are studied using microscopic characterization and mechanical tests. The results indicate that vibration can effectively facilitate gas discharge and accelerate resin flow, while the double vacuum bag process reduces gas discharge resistance in the early curing stage by delaying the vacuum negative pressure application, yet it also results in insufficient resin flow due to this delay. Through the synergistic optimization of vibration-assisted energy field parameters and the double vacuum bag process, gas-induced and flow-induced voids can be effectively suppressed while ensuring curing efficiency, reducing the macroscopic porosity of variable-thickness regions from 8.34% (single vacuum bag process) to 0.43%. This study provides a new approach for the high-quality curing and manufacturing of variable-thickness composite components.

This paper investigates the stress fields at the onset of plastic yield of variable thickness rotating orthotropic disk, which is rigidly fixed on an inclusion. In the analytical modeling of the problem, two different analytical solution methods have been displayed where small deformations have been considered with the application of plane stress conditions. Well-known power law is considered for the disk's thickness variation, and Hill’s yield criterion is applied to obtain the elastic limits. Four parameters have been utilized while analyzing the limit fields: geometric parameter to manipulate the disk thickness, orthotropy parameter from the ratio between Young's modulus in radial and tangential directions, and two parameters owing to the applied yield criteria. The effects of these parameters on the limit fields have been comprehensively examined in the numerical examples, and possible outcomes have been discussed. Additionally, using Autodesk Inventor Nastran, finite element solution of the disk is generated, analytical and numerical results have been compared, and consequently, closely matching results have been achieved.