Topology optimization for isotropic elastic materials using honeycomb tessell

Abstract

Topology optimization is gaining popularity as a primary tool for engineers in the initial stages of design. Essentially, the design domain is broken down into individual pixels, with the material density of each element or mesh point serving as a design variable. The optimization problem is then tackled through mathematical programming and optimization methods that rely on analytical gradient calculation. In this study, topology optimization using honeycomb tessellation elements is explored. Hexagonal elements have the ability to flexibly connect two adjacent elements. The use of the hexagonal element limits the occurrence of the checkerboard pattern to the finite elements of the quadrilateral standard Lagrangian type. A mathematical model is developed with the objective function being the minimum compliance value of the design domain. The element stiffness matrix is constructed using the strain-displacement matrix and the constitutive matrix, assuming a unit Young's modulus. Additionally, optimal conditions are established using Lagrangian multipliers. Two sensitivity and density filtering filters are employed to increase optimization efficiency, prevent the algorithm from reaching a local optimal state, and speed up convergence. If the suggested filter is employed, the objective function achieves a value of c=173,0293 and convergence is attained after 200 iterations. In contrast, without using the filter, the objective function has a larger value (c=186,7922) and convergence occurs at the 27th iteration. The results are significant for optimizing topology to meet specific boundary condition requirements. This paper proposes a novel approach using a combination of filters to advance topology optimization using hexagonal elements in future applications.