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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.

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.

PurposeIn the current manuscript, the authors examine the Belleville spring with the variable thickness. The thickness is assumed to be variable along the meridional and parallel coordinates of conical coordinate system. The calculation of the Belleville springs includes the cases of the free gliding edges and the edges on cylindric curbs, which constrain the radial movement. The equations developed here are based on common assumptions and are simple enough to be applied to the industrial calculations.Design/methodology/approachIn the current manuscript, the authors examine the Belleville spring with the variable thickness. The calculation of the Belleville springs investigates the free gliding edges and the edges on cylindric curbs with the constrained radial movement. The equations developed here are based on common assumptions and are simple enough to be applied to the industrial calculations.FindingsThe developed equations demonstrate that the shift of the inversion point to the inside edge does not influence the bending of the cone. On the contrary, the character of the extensional deformation (circumferential strain) of the middle surface alternates significantly. The extension of the middle surface of free gliding spring occurs outside the inversion. The middle surface of the free gliding spring squeezes inside the inversion point. Contrarily, the complete middle surface of the disk spring on the cylindric curb extends. This behavior influences considerably the function of the spring.Research limitations/implicationsA slotted disk spring consists of two segments: a disk segment and a number of lever arm segments. Currently, the calculation of slotted disk spring is based on the SAE formula (SAE, 1996). This formula is limited to a straight slotted disk spring with freely gliding inner and outer edges.Practical implicationsThe equations developed here are based on common assumptions and are simple enough to be applied to the industrial calculations. The developed method is applicable for disk springs with radially constrained edges. The vertical displacements of a disk spring result from an axial load uniformly distributed on inner and outer edges. The method could be directly applied for calculation of slotted disk springs.Originality/valueThe nonlinear governing equations for the of Belleville spring centres were derived. The equations describe the deformation and stresses of thin and moderately thick washers. The variation method is applicable for the disc springs with free gliding and rigidly constrained edges. The developed method is applicable for Belleville spring with radially constrained edges. The vertical displacements of a disc spring result from an axial load uniformly distributed on inner and outer edges.

The main aim of the present paper is to solve numerically the free vibration problem of sandwich shell structures with variable thickness and made of Functionally Graded Materials (FGMs). Several Higher-order Shear Deformation Theories (HSDTs), defined by a unified formulation, are employed in the study. The FGM structures are characterized by variable mechanical properties due to the through-the-thickness variation of the volume fraction distribution of the two constituents and the arbitrary thickness profile. A four-parameter power law expression is introduced to describe the FGMs, whereas general relations are used to define the thickness variation, which can affect both the principal coordinates of the shell reference domain. A local scheme of the Generalized Differential Quadrature (GDQ) method is employed as numerical tool. The natural frequencies are obtained varying the exponent of the volume fraction distributions using higher-order theories based on a unified formulation. The structural models considered are two-dimensional and require less degrees of freedom when compared to the corresponding three-dimensional finite element (FE) models, which require a huge number of elements to describe the same geometries accurately. A comparison of the present results with the FE solutions is carried out for the isotropic cases only, whereas the numerical results available in the literature are used to prove the validity as well as accuracy of the current approach in dealing with FGM structures characterized by a variable thickness profile.

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.