Stress-based topology optimization using spatial gradient stabilized XFEM

Sharma, Ashesh; Maute, Kurt

doi:10.1007/s00158-017-1833-y

Cited by 50 publications

(24 citation statements)

References 47 publications

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“…the necessity of using high dense meshes in the design domain involving millions of elements and the application to new topological design problems (e.g. multiscale topological design [14,19,26,37], immersed boundary methods combined with XFEM techniques [41], convected methods [52], etc.). This is an issue to be explored in a subsequent research.…”

Section: Discussionmentioning

confidence: 99%

Variational approach to relaxed topological optimization: Closed form solutions for structural problems in a sequential pseudo-time framework

Oliver

Yago

Cante

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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The work explores a specific scenario for structural computational optimization based on the following elements: (a) a relaxed optimization setting considering the ersatz (bi-material) approximation, (b) a treatment based on a nonsmoothed characteristic function field as a topological design variable, (c) the consistent derivation of a relaxed topological derivative whose determination is simple, general and efficient, (d) formulation of the overall increasing cost function topological sensitivity as a suitable optimality criterion, and (e) consideration of a pseudo-time framework for the problem solution, ruled by the problem constraint evolution. In this setting, it is shown that the optimization problem can be analytically solved in a variational framework, leading to, nonlinear, closed-form algebraic solutions for the characteristic function, which are then solved, in every time-step, via fixed point methods based on a pseudo-energy cutting algorithm combined with the exact fulfillment of the constraint, at every iteration of the non-linear algorithm, via a bisection method. The issue of the ill-posedness (mesh dependency) of the topological solution, is then easily solved via a Laplacian smoothing of that pseudo-energy. In the aforementioned context, a number of (3D) topological structural optimization benchmarks are solved, and the solutions obtained with the explored closed-form solution method, are analyzed, and compared, with their solution through an alternative level set method. Although the obtained results, in terms of the cost function and topology designs, are very similar in both methods, the associated computational cost is about five times smaller in the closedform solution method this possibly being one of its advantages. Some comments, about the possible application of the method to other topological optimization problems, as well as envisaged modifications of the explored method to improve its performance close the work.

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Section: Discussionmentioning

confidence: 99%

Variational approach to relaxed topological optimization: Closed form solutions for structural problems in a sequential pseudo-time framework

Oliver

Yago

Cante

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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“…Aggregation approaches have been investigated since the early contributions of Yang and Chen 47 and Duysinx and Sigmund 48 . Block aggregation of the stresses 49,50 and promising results by Le et al 51 have created a new surge of investigations and implementations, leading to many high‐quality results over the last few years (including References 52‐61, among others). As mentioned in Section 1, the main contribution herein is the investigation of approximate solves of the state and adjoint problems, and the p ‐norm formulation is utilized just as a prototype for stress‐based topology optimization.…”

Section: Problem Statementmentioning

confidence: 99%

Efficient stress‐constrained topology optimization using inexact design sensitivities

Amir

2021

Numerical Meth Engineering

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An efficient computational approach to stress‐constrained topology optimization is presented. Using a multigrid‐preconditioned Krylov solver for the state and adjoint problems, the number of Krylov iterations is reduced significantly by enforcing early termination of the iterative solves. The criterion for early termination is based on the convergence of the design sensitivities with respect to Krylov iterations. Consequently, the progress of optimization is not affected by the inexact resolution of the state and adjoint problems. The proposed approach is demonstrated on several design problems that possess different characteristics of the maximum stress. Optimization results obtained with early termination show very good agreement with those obtained with an accurate direct solver—in terms of objective value, constrained maximum stress and overall number of design cycles. Savings of 80% in the number of Krylov iterations are achieved, compared to the number of iterations required to satisfy the force residual convergence criterion to a common tolerance. The approach is directly applicable to high‐resolution problems that are solved on parallel computational environments. A MATLAB code for reproducing the results is freely available at https://github.com/odedamir/topopt-stress-inexact-sensitivities

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“…Moreover, X/GFEM may result in ill-conditioned matrices, in which case Stable Generalized FEM (SGFEM) (Babuška and Banerjee 2012; Gupta et al 2013;Kergrene et al 2016) or advanced preconditioning schemes (Lang et al 2014) are needed. Furthermore, the approximation of stresses can be highly overestimated near material boundaries (Van Miegroet and Duysinx 2007;Noël and Duysinx 2017;Sharma and Maute 2018). Finally, as the enriched functions are associated with original mesh nodes, the accuracy of the approximation may degrade in blending elements, i.e., elements that do not have all nodes enriched (Fries 2008).…”

Section: Introductionmentioning

confidence: 99%

An interface-enriched generalized finite element method for level set-based topology optimization

Boom

Zhang

Keulen

et al. 2020

Struct Multidisc Optim

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During design optimization, a smooth description of the geometry is important, especially for problems that are sensitive to the way interfaces are resolved, e.g., wave propagation or fluid-structure interaction. A level set description of the boundary, when combined with an enriched finite element formulation, offers a smoother description of the design than traditional density-based methods. However, existing enriched methods have drawbacks, including ill-conditioning and difficulties in prescribing essential boundary conditions. In this work, we introduce a new enriched topology optimization methodology that overcomes the aforementioned drawbacks; boundaries are resolved accurately by means of the Interface-enriched Generalized Finite Element Method (IGFEM), coupled to a level set function constructed by radial basis functions. The enriched method used in this new approach to topology optimization has the same level of accuracy in the analysis as the standard finite element method with matching meshes, but without the need for remeshing. We derive the analytical sensitivities and we discuss the behavior of the optimization process in detail. We establish that IGFEM-based level set topology optimization generates correct topologies for well-known compliance minimization problems.

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Stress-based topology optimization using spatial gradient stabilized XFEM

Cited by 50 publications

References 47 publications

Variational approach to relaxed topological optimization: Closed form solutions for structural problems in a sequential pseudo-time framework

Variational approach to relaxed topological optimization: Closed form solutions for structural problems in a sequential pseudo-time framework

Efficient stress‐constrained topology optimization using inexact design sensitivities

An interface-enriched generalized finite element method for level set-based topology optimization

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