Abstract
In structural optimization, the level set method is known as a well-established approach for shape and topology optimization. However, special care must be taken, if the design domains are sparsely-filled and slender. Using steepest descent-type level set methods, slender structure topology optimizations tend to instabilities and loss of structural cohesion. A sole step size control or a selection of more complex initial designs only help occasionally to overcome these issues and do not describe a universal solution. In this paper, instead of updating the level set function by solving a Hamilton–Jacobi partial differential equation, an adapted algorithm for the update of the level set function is utilized, which allows an efficient and stable topology optimization of slender structures. Including different adaptations, this algorithm replaces unacceptable designs by modifying both the pseudo-time step size and the Lagrange multiplier. Besides, adjustments are incorporated in the normal velocity formulation to avoid instabilities and achieve a smoother optimization convergence. Furthermore, adding filtering-like adaptation terms to the update scheme, even in case of very slender structures, the algorithm is able to perform topology optimization with an appropriate convergence speed. This procedure is applied for compliance minimization problems of slender structures. The stability of the optimization process is shown by 2D numerical examples. The solid isotropic material with penalization (SIMP) method is used as an alternative approach to validate the result quality of the presented method. Finally, the simple extension to 3D optimization problems is addressed, and a 3D optimization example is briefly discussed.
Highlights
The topology optimization of solid structures is one of the most challenging areas in structural optimization
The optimization algorithms presented in this work are all implemented using Matlab, and the corresponding calculations are performed on a personal computer equipped with a 3.80 GHz
It is known that the greyness in a level set-based topology optimization is limited to elements intersected by the zero-level set, whereas the solid isotropic material with penalization (SIMP) method normally leads to a higher number of grey intermediate elements
Summary
The topology optimization of solid structures is one of the most challenging areas in structural optimization. To circumvent the mentioned drawbacks of the conventional level set methods, alternative approaches have been developed, see for instance the well-established concepts proposed in Luo et al (2008), Wang et al (2007), Wang and Wang (2006a, b), Wei et al (2018) and Yamada et al (2010) In these variants, other evolution techniques rather than the explicit update schemes are used, which help to dispose of the stability limit imposed by the CFL condition. A hole creation is not hindered, and more complex topologies can be created during the optimization process, for instance, based on a natural velocity extension, see Allaire et al (2004) and van Dijk et al (2013) Following this path of alternative approaches, here an approximate level set method is utilized, which basically updates the nodal level set function values by a forward Euler scheme, and includes an approximate reinitialization.
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