Topology optimization with Node Density Adaptation and Geometric Multigrid solving

DL-Scale: Deep Learning for model upgrading in topology optimization

To minimize the effects of artificially chosen variables on optimization results,simultaneous topology and shape optimization( STSO) method was proposed for structural optimization of aero-engine disk at the conceptual design phase. This method was based on solid isotropic material with penalization( SIMP)method,by analyzing the sensitivities of the optimized object and constraints,sequential quadratic programming( SQP) optimized method was used to solve the problem. Then,taking shell structure for instant,the results of STSO method and multi-step optimization method were comparatively analyzed,which confirmed the advantage of STSO method. At last,the STSO method was utilized to design a conceptual disk structure.Results of disk structural optimization of STSO method were compared to that of the sole topology optimization method. Influence of frequency constraints of different vibration modes on optimization results was discussed.The results show that the structural forms of optimization results of frequency constraints of different vibration modes vary a lot. Convergence rate of STSO method is faster than that of solo topology optimization,and the consequences of STSO method are more accurate. Meanwhile,the results of STSO method may be invalid for excessive mesh distortion caused by the wrong choice of the value range of shape optimization variables.

A Physics-Informed Neural Network-based Topology Optimization (PINNTO) framework for structural optimization

Topology optimization of functionally-graded lattice structures with buckling constraints

Under the context of the dual-carbon policy, reducing energy consumption and emissions in automobiles has garnered significant attention, with automotive lightweighting being particularly important. This paper focuses on the lightweight design of automotive subframes, aiming to minimize weight while meeting performance requirements. Research has revealed that the original subframe allows further room for lightweighting and performance optimization. A topology optimization model was established using the Solid Isotropic Material with Penalization (SIMP) method and solved using the Method of Moving Asymptotes (MMA) algorithm. Integration of the SIMP method was achieved on the Abaqus-Matlab (2022x) platform via Python (3.11.0) and Matlab (R2022a) coding, forming an effective optimization framework. The optimization results provided clear load transfer paths, offering a theoretical basis for geometric model conversion. The subframe model was subsequently reconstructed in CATIA. Material redundancy was identified in the reconstructed subframe model, prompting secondary optimization. Multi-objective size optimization was conducted in OptiStruct, reducing the subframe’s mass from 33.73 kg to 17.84 kg, achieving a 47.1% weight reduction. Static stiffness and modal analyses performed in HyperMesh confirmed that results met all relevant standards. Modal testing revealed a minimal deviation of only −2.7% from the simulation results, validating the feasibility and reliability of the optimized design. This research demonstrates that combining topology optimization with size optimization can significantly reduce weight and enhance subframe performance, providing valuable support for future automotive component design.

An improved proportional topology optimization (IPTO) method is proposed in this work. The main improvement of this method is that the conventional solid isotropic material with penalization (SIMP)-based material interpolation scheme is replaced by a polarized material interpolation scheme, and the Heaviside threshold function is adopted based on the original proportional topology optimization (PTO) method. By using this approach, the minimum compliance problem can be solved without requiring the numerical derivation of the sensitivity function. To verify the feasibility and effectiveness of the proposed method, two-dimensional (2D) and three-dimensional (3D) cantilevers and L-bracket beams are used as examples. The 2D results obtained by the IPTO method are compared with those obtained by the PTO and SIMP methods. Numerical examples demonstrate that IPTO can acquire better objective function values and more ideal topology structures compared to PTO and SIMP. Furthermore, IPTO offers significant advantages over PTO and SIMP in terms of convergence speed and the ability to suppress intermediate density elements. Additionally, this method enables topology optimization design under multiple working conditions. Therefore, it provides an effective approach for structural topology optimization in research and engineering applications. With appropriate adjustment, this method can also be applied to composite material design and heat conduction design.

PurposeStructural topology optimization is computationally expensive due to the involvement of high-resolution mesh and repetitive use of finite element analysis (FEA) for computing the structural response. Since FEA consumes most of the computational time in each optimization iteration, a novel GPU-based parallel strategy for FEA is presented and applied to the large-scale structural topology optimization of 3D continuum structures.Design/methodology/approachA matrix-free solver based on preconditioned conjugate gradient (PCG) method is proposed to minimize the computational time associated with solution of linear system of equations in FEA. The proposed solver uses an innovative strategy to utilize only symmetric half of elemental stiffness matrices for implementation of the element-by-element matrix-free solver on GPU.FindingsUsing solid isotropic material with penalization (SIMP) method, the proposed matrix-free solver is tested over three 3D structural optimization problems that are discretized using all hexahedral structured and unstructured meshes. Results show that the proposed strategy demonstrates 3.1× –3.3× speedup for the FEA solver stage and overall speedup of 2.9× –3.3× over the standard element-by-element strategy on the GPU. Moreover, the proposed strategy requires almost 1.8× less GPU memory than the standard element-by-element strategy.Originality/valueThe proposed GPU-based matrix-free element-by-element solver takes a more general approach to the symmetry concept than previous works. It stores only symmetric half of the elemental matrices in memory and performs matrix-free sparse matrix-vector multiplication (SpMV) without any inter-thread communication. A customized data storage format is also proposed to store and access only symmetric half of elemental stiffness matrices for coalesced read and write operations on GPU over the unstructured mesh.

This article presents a topology optimization method for structures consisting of multiple lattice components under a certain size, which can be manufactured with an additive manufacturing machine with a size limit and assembled via conventional joining processes, such as welding, gluing, riveting, and bolting. The proposed method can simultaneously optimize overall structural topology, partitioning to multiple components and functionally graded lattices within each component. The functionally graded lattice infill with guaranteed connectivity is realized by applying the Helmholtz PDE filter with a variable radius on the density field in the solid isotropic material with penalization (SIMP) method. The partitioning of an overall structure into multiple components is realized by applying the discrete material optimization (DMO) method, in which each material is interpreted as each component, and the size limit for each component imposed by a chosen additive manufacturing machine. A gradient‐free coating filter realizes bulk solid boundaries for each component, which provide continuous mating surfaces between adjacent components to enable the subsequent joining. The structural interfaces between the bulk solid boundaries are extracted and assigned a distinct material property, which model the joints between the adjacent components. Several numeral examples are solved for demonstration.

This article provides a comprehensive review of structural optimization employing topology methods for structures under vibration problems. Topology optimization allows creative and radical design modifications, compared to shape and size optimization techniques. Various works of structural topology optimization, which are subjected to vibration as the response function of the optimization process, are reviewed. Different types of calculus and numerical methods commonly used for solving structural topological optimization problems are briefly discussed. Moreover, different aspects of topology optimization related to vibration problems are explained. The articles reviewed are largely confined to linear systems that concern small vibration amplitudes. Accordingly, the works related to vibration topological optimization are classified according to the method employed (homogenization, evolutionary structural optimization, solid isotropic material with penalization, or level set). The reviewed works are tabulated according to their methodology, year, and the objective functions and applications of each work. Although the homogenization and evolutionary methods were common in the past, the solid isotropic material with penalization (SIMP) method is the most popular method applied in recent years. The advantages of the level set method show promise for future applications.

In this study, a novel approach is proposed to enhance the performance of the parts optimized by Solid Isotropic Material with Penalization (SIMP) method. SIMP is a topology optimization method that aims the optimum distribution of material in a design domain subjected to predefined loads, constraints and boundary conditions. The method forces every finite element composing the geometry to have a density of either 1 or 0. The main reason behind penalizing is that regions with intermediate densities are difficult to fabricate. However, including these regions in the optimization output may provide better performance results. Based on this idea, a method is proposed to utilize intermediate densities in a manufacturable form and is applied to 3D geometries. Besides, the remodeled topology is checked against any unconnected cells. In contrast to many methods, which delete the unconnected elements, the proposed method provides connectivity by adding cells. The outputs of the proposed method are fabricated by using Electron Beam Melting (EBM) and Stereolithography (SLA) technologies. EBM uses material powder and a heat source to melt and fuse the powders while SLA uses photosensitive resin and an ultraviolet light to cure the resin. A common limitation of both technologies is that powder/resin may remain inside the internal features which do not have access to outer surface of the part through the channels. The proposed method ensures the easy removal of excess powder/resin after fabrication. Performance of the method is compared with the SIMP method through test and analysis.

In this paper, an explicit and implicit hybrid topology optimization method is proposed for the design of multiple materials structure. The explicit topology optimization employs the moving morphable component (MMC) method to determine where the solid material is present within the design domain. The implicit topology optimization employs the solid isotropic material with penalization (SIMP) method to identify material type within the solid material region. The explicit and implicit topology optimization methods are combined through a surrogate material model, resulting in a new hybrid topology optimization framework known as the MMC–SIMP hybrid topology optimization method. The proposed method retains the advantages of both individual optimization methods, allowing for explicit boundary representation and high design freedom in material selection. The element density function and sensitivity analysis are conducted based on two-phase materials topology optimization. Finally, some numerical examples demonstrate the effectiveness of the proposed method.

This paper presents a comparison between the Solid Isotropic Material with Penalization (SIMP) approach and the Particle Swarm Optimization (PSO) method in continuous structural topology optimization. SIMP is a mature gradient-based algorithm which has stable performance and fast convergence in various topology optimization applications. PSO is a relatively new evolutionary algorithm, inspired by the nature of bird flocking. Its applications to topology optimization are only reported in recent years. In this paper, the mechanisms of these two algorithms are introduced first. Their performances in continuous topology optimization are compared through an example. Through these comparisons, improvement directions of PSO in topology optimization are outlined.

By choosing the density of particle as the design variables, a new implementation method of topology optimization is presented based on the Element-free Galerkin (EFG) method in this paper, in which the optimal objective is to minimize structural compliance. The advantage of using nodal density is that the displacement and density in the influence domain have the same approximation scheme, and the smoothness of the density field can be improved. Nodal density method presented can prevent the checkerboard from the mathematical model proposed. The topology optimal model based on EFG method under multiple loading cases and stress constraints is proposed, and the sensitivity analysis of optimal design is derived in detail. By using solid isotropic material with penalization (SIMP) method and optimality criteria (OC) method, an algorithm of topology optimization based on the EFG method is presented. The shortcoming of using nodal density can be overcome by introducing the penalty function method. Three topology optimal examples are solved successfully and test the model and algorithm proposed. The results obtained show that the checkerboard phenomenon arisen in topology optimization is not found, and the method proposed is not only effective in suppressing checkerboards but also has better convergence.

A novel topology optimization approach is proposed in this paper for the design of three rotational degree-of-freedom (DOF) spatially compliant mechanisms, combining the Jacobian isomorphic mapping matrix with the solid isotropic material with penalization (SIMP) topological method. In this approach, the isomorphic Jacobian matrix of a 3-UPC (U: universal joint, P: prismatic joint, C: cylindrical joint) type parallel prototype manipulator is formulated. Subsequently, the orthogonal triangular decomposition and differential kinematic method is applied to uncouple the Jacobian matrix to construct a constraint for topology optimization. Firstly, with respect to the 3-UPC type parallel prototype manipulator, the Jacobian matrix is derived to map the inputs and outputs to be used for initializing the topology optimization process. Secondly, the orthogonal triangular decomposition with the differential kinematic method is used to reconstruct the uncoupled mapping matrix to derive the 3-UPC type parallel prototype manipulator. Finally, a combination of the solid isotropic material with penalization (SIMP) method and the isomorphic mapping matrix is applied to construct the topological model. A typical three rotational DOF spatially compliant mechanism is reported as a numerical example to demonstrate the effectiveness of the proposed method.

In this study, we performed a three-dimensional (3D) topology optimization of a fixed offshore structure to enhance its structural stiffness. The proposed topology optimization is based on the solid isotropic material with penalization (SIMP) method, where a volume constraint is applied to utilize an equivalent amount of material as that used for the rule-based scantling design. To investigate the effects of the main legs of the fixed offshore structure on its structural stiffness, the leg region is selectively considered in the design domain of the topology optimization problem. The obtained optimal designs and the rule-based scantling design of the structure are manufactured by 3D metal printing technology to experimentally validate the topology optimization. The behaviors under compressive loading of the obtained optimal designs are compared with those of the rule-based scantling design using a universal testing machine (UTM). Based on the structural experiments, we concluded that by employing the topology optimization method, the structural stiffness of the structure was enhanced compared to that of the rule-based scantling design for an equal amount of the fabrication material. Furthermore, by effectively combining the topology optimization and rule-based scantling methods, we succeeded in enhancing the structural stiffness and improving the breaking load of the fixed offshore structure.