A Novel Discrete Adjoint-based Level Set Topology Optimization Method in B-spline Space

This paper presents a novel computational scheme for sensitivity analysis of the velocity field in the level set method using the discrete adjoint method. The velocity field is represented in B-spline space, and the adjoint equations are constructed based on the discretized governing equations. The key contribution of this work is the demonstration that the velocity field in the level set method can be entirely obtained from the discrete adjoint method. This eliminates the need for shape sensitivity analysis, which is commonly used in standard level set methods. The results demonstrate the effectiveness of the approach in producing optimized results for stress and linearized buckling problems. Overall, the proposed method has the potential to simplify the way in which topology optimization problems using level set methods are solved, and has significant implications for the design of a broad range of engineering applications

Static and dynamic topology optimization: an innovative unifying approach

This paper presents a topology optimization approach that is innovative with respect to two distinct matters. First of all the proposed formulation is capable to handle static and dynamic topology optimization with virtually no modifications. Secondly, the approach is inherently a multi-input multi-output one, i.e., multiple objectives can be pursued in the presence of multiple loads. The input-to-output transfer matrix, say G, is the key ingredient that governs the algebraic mapping between applied loads and structural response. In statics G depends on the design variables only, whereas it depends on the frequency variable as well in the dynamic case. The Singular Value Decomposition (SVD) of G represents then the core of the proposed approach. Singular values are shown to be the gains of the input/output mapping and are used to compute proper norms of G that represent the goal functions to be minimized. Singular vectors provide at no extra cost the plant directions, i.e., the load combination factors that stress the structure the most. Numerical examples are discussed in much detail and open issues object of ongoing investigations are highlighted. A full Matlab code handling the static topology optimization problem is provided as an online Appendix to the manuscript. Its extension to the dynamic case may be gathered following the formulation proposed in Sect. 5

Towards solving large-scale topology optimization problems with buckling constraints at the cost of linear analyses

This work presents a multilevel approach to large--scale topology optimization accounting for linearized buckling criteria. The method relies on the use of preconditioned iterative solvers for all the systems involved in the linear buckling and sensitivity analyses and on the approximation of buckling modes from a coarse discretization. The strategy shows three main benefits: first, the computational cost for the eigenvalue analyses is drastically cut. Second, artifacts due to local stress concentrations are alleviated when computing modes on the coarse scale. Third, the ability to select a reduced set of important global modes and filter out less important local ones. As a result, designs with improved buckling resistance can be generated with a computational cost little more than that of a corresponding compliance minimization problem solved for multiple loading cases. Examples of 2D and 3D structures discretized by up to some millions of degrees of freedom are solved in Matlab to show the effectiveness of the proposed method. Finally, a post--processing procedure is suggested in order to reinforce the optimized design against local buckling.Comment: 25 pages, 15 figure

Finite Strain Topology Optimization with Nonlinear Stability Constraints

This paper proposes a computational framework for the design optimization of stable structures under large deformations by incorporating nonlinear buckling constraints. A novel strategy for suppressing spurious buckling modes related to low-density elements is proposed. The strategy depends on constructing a pseudo-mass matrix that assigns small pseudo masses for DOFs surrounded by only low-density elements and degenerates to an identity matrix for the solid region. A novel optimization procedure is developed that can handle both simple and multiple eigenvalues wherein consistent sensitivities of simple eigenvalues and directional derivatives of multiple eigenvalues are derived and utilized in a gradient-based optimization algorithm - the method of moving asymptotes. An adaptive linear energy interpolation method is also incorporated in nonlinear analyses to handle the low-density elements distortion under large deformations. The numerical results demonstrate that, for systems with either low or high symmetries, the nonlinear stability constraints can ensure structural stability at the target load under large deformations. Post-analysis on the B-spline fitted designs shows that the safety margin, i.e., the gap between the target load and the 1st critical load, of the optimized structures can be well controlled by selecting different stability constraint values. Interesting structural behaviors such as mode switching and multiple bifurcations are also demonstrated.Comment: 77 pages, 44 Figure

Topology Optimization via Machine Learning and Deep Learning: A Review

Topology optimization (TO) is a method of deriving an optimal design that satisfies a given load and boundary conditions within a design domain. This method enables effective design without initial design, but has been limited in use due to high computational costs. At the same time, machine learning (ML) methodology including deep learning has made great progress in the 21st century, and accordingly, many studies have been conducted to enable effective and rapid optimization by applying ML to TO. Therefore, this study reviews and analyzes previous research on ML-based TO (MLTO). Two different perspectives of MLTO are used to review studies: (1) TO and (2) ML perspectives. The TO perspective addresses "why" to use ML for TO, while the ML perspective addresses "how" to apply ML to TO. In addition, the limitations of current MLTO research and future research directions are examined