Revisiting non-convexity in topology optimization of compliance minimization problems

PurposeThis is an attempt to better bridge the gap between the mathematical and the engineering/physical aspects of the topic. The authors trace the different sources of non-convexification in the context of topology optimization problems starting from domain discretization, passing through penalization for discreteness and effects of filtering methods, and end with a note on continuation methods.Design/methodology/approachStarting from the global optimum of the compliance minimization problem, the authors employ analytical tools to investigate how intermediate density penalization affects the convexity of the problem, the potential penalization-like effects of various filtering techniques, how continuation methods can be used to approach the global optimum and how the initial guess has some weight in determining the final optimum.FindingsThe non-convexification effects of the penalization of intermediate density elements simply overshadows any other type of non-convexification introduced into the problem, mainly due to its severity and locality. Continuation methods are strongly recommended to overcome the problem of local minima, albeit its step and convergence criteria are left to the user depending on the type of application.Originality/valueIn this article, the authors present a comprehensive treatment of the sources of non-convexity in density-based topology optimization problems, with a focus on linear elastic compliance minimization. The authors put special emphasis on the potential penalization-like effects of various filtering techniques through a detailed mathematical treatment.

Stress minimization for lattice structures. Part I: Micro-structure design

Lattice structures are periodic porous bodies which are becoming popular since they are a good compromise between rigidity and weight and can be built by additive manufacturing techniques. Their optimization has recently attracted some attention, based on the homogenization method, mostly for compliance minimization. The goal of our two-part work is to extend lattice optimization to stress minimization problems two-dimensionally. The present first part is devoted to the choice of a parametrized periodicity cell that will be used for structural optimization in the second part of our work. In order to avoid stress concentration, we propose a square cell microstructure with a super-ellipsoidal hole instead of the standard rectangular hole often used for compliance minimization. This type of cell is parametrized two-dimensionally by one orientation angle, two semi-axis and a corner smoothing parameter. We first analyse their influence on the stress amplification factor by performing some numerical experiments. Second, we compute the optimal corner smoothing parameter for each possible microstructure and macroscopic stress. Then, we average (with specific weights) the optimal smoothing exponent with respect to the macroscopic stress. Finally, to validate the results, we compare our optimal super-ellipsoidal hole with the Vigdergauz microstructure which is known to be optimal for stress minimization in some special cases. This article is part of the theme issue ‘Topics in mathematical design of complex materials’.

A NON-GRADIENT APPROACH FOR THREE DIMENSIONAL TOPOLOGY OPTIMIZATION

The present work is devoted to the extension of the non-gradient approach, namely Proportional Topology Optimization (PTO), for compliance minimization of three-dimensional (3D) structures. Two schemes of material interpolation within the framework of the solid isotropic material with penalization (SIMP), i.e. the power function and the logistic function are analyzed. Through a comparative study, the efficiency of the logistic-type interpolation scheme is highlighted. Since no sensitivity is involved in the approach, a density filter is applied instead of sensitivity filter to avoid checkerboard issue

Multiscale structural optimization with concurrent coupling between scales

A robust three-dimensional multiscale structural optimization framework with concurrent coupling between scales is presented. Concurrent coupling ensures that only the microscale data required to evaluate the macroscale model during each iteration of optimization is collected and results in considerable computational savings. This represents the principal novelty of this framework and permits a previously intractable number of design variables to be used in the parametrization of the microscale geometry, which in turn enables accessibility to a greater range of extremal point properties during optimization. Additionally, the microscale data collected during optimization is stored in a reusable database, further reducing the computational expense of optimization. Application of this methodology enables structures with precise functionally graded mechanical properties over two scales to be derived, which satisfy one or multiple functional objectives. Two classical compliance minimization problems are solved within this paper and benchmarked against a Solid Isotropic Material with Penalization (SIMP)–based topology optimization. Only a small fraction of the microstructure database is required to derive the optimized multiscale solutions, which demonstrates a significant reduction in the computational expense of optimization in comparison to contemporary sequential frameworks. In addition, both cases demonstrate a significant reduction in the compliance functional in comparison to the equivalent SIMP-based optimizations.

Smooth Design of 3D Self-Supporting Topologies Using Additive Manufacturing Filter and SEMDOT

The smooth design of self-supporting topologies has attracted great attention in the design for additive manufacturing (DfAM) field as it cannot only enhance the manufacturability of optimized designs but can obtain light-weight designs that satisfy specific performance requirements. This paper integrates Langelaar’s AM filter into the Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT) algorithm—a new element-based topology optimization method capable of forming smooth boundaries—to obtain print-ready designs without introducing post-processing methods for smoothing boundaries before fabrication and adding extra support structures during fabrication. The effects of different build orientations and critical overhang angles on self-supporting topologies are demonstrated by solving several compliance minimization (stiffness maximization) problems. In addition, a typical compliant mechanism design problem—the force inverter design—is solved to further demonstrate the effectiveness of the combination between SEMDOT and Langelaar’s AM filter.