Sensitivity Analysis of Key Formulations of Topology Optimization on an Example of Cantilever Bending Beam

Abstract

Topology optimization is a modern method for optimizing the material distribution in a given space, automatically searching for the ideal design of the product. The method aims to maximize the design performance of the system regarding given conditions. In engineering practice, a given space is first described using the finite element method and, subsequently, density-based method with solid isotropic material with penalty. Then, the final shape is found using a gradient-based method, such as the optimality criteria algorithm. However, obtaining the ideal shape is highly dependent on the correct setting of numerical parameters. This paper focuses on the sensitivity analysis of key formulations of topology optimization using the implementation of mathematical programming techniques in MATLAB software. For the purposes of the study, sensitivity analysis of a simple spatial task—cantilever bending—is performed. This paper aims to present the formulations of the optimization problem—in this case, minimization of compliance. It should be noted that this paper does not present any new mathematical formulas but rather provides an introduction into the mathematical theory (including filtering methods and calculating large-size problems using the symmetry of matrices) as well as a step-by step guideline for the minimization of compliance within the density-based topology optimization and search for an optimal shape. The results can be used for complex commercial applications produced by traditional manufacturing processes or by additive manufacturing methods.

Finite Element Analysis of Upper Limb Splint Designs and Materials for 3D Printing

“…When it comes to engineering design, the topology optimization process, which relies on mathematical algorithms, greatly simplifies the task of a designer by recommending alternative design options. Considering a design space and a set of constraints (such as loads and BCs), topology optimization is an example of the computational approach that makes it possible to determine the best material distribution within that space [ 51 , 52 ]. The overall weight of the intended product can be significantly reduced while still retaining the appropriate strength and rigidity with the application of topology optimization in the design process [ 53 ].…”

Influence of topology optimization parameters on the mechanical response of an additively manufactured test structure

“…When predicting worst-case optimized designs, the elemental size dependence as well as the filtering process should be considered. Sotola et al found that solutions of a topology optimized cantilever beam differed when using different elements and filter parameters (Sotola et al, 2021). They also found that penalty factor was a key parameter in their model.…”

Reliability-Based Topology Optimization: An Extension of the SESO and SERA Methods for Three-Dimensional Structures

“…The presented article by [41] studies the effect of key formulations of the TO problem on the design performance, focusing on sensitivity analysis using the implementation of mathematical programming techniques in MATLAB software. Basically, the mesh independence filter works to avoid numerical instabilities, such as checkerboard, mesh dependence and local minima [42].…”