An accurate and fast regularization approach to thermodynamic topology optimization

Abstract

In a series of previous works, we established a novel approach to topology optimization for compliance minimization based on thermodynamic principles known from the field of material modeling. Hamilton's principle for dissipative processes directly yields a partial differential equation (referred to as the evolution equation) as an update scheme for the spatial distribution of density mass describing the topology. Consequently, no additional mathematical minimization algorithms are needed. In this article, we introduce a regularization scheme by penalization of the gradient of the spatial distribution of mass density. The parabolic evolution equation (owing to a similar structure to the transient heat-conduction equation) is solved most efficiently by an explicit time discretization. The Laplace operator is discretized via a Taylor series expansion yielding an operator matrix that is constant for the entire optimization process. This method shares some similarities to meshless methods and allows for an accurate application also on unstructured finite element meshes. The minimal size of the structure member can directly be controlled, a priori, by a numerical parameter introduced along with the regularization, similar to classical filter radii. KEYWORDSmeshless Laplacian, parabolic PDE, regularization, thermodynamic topology optimization, unstructured meshes, variational material modeling INTRODUCTIONOver recent decades, topology optimization has become a major field of interest in mechanical engineering. Approaches have been developed for different optimization objectives, constraints relating to physically feasible or manufacturable designs, or models for multiphysics problems. The most native approach is considered to be topology optimization with compliance minimization under a structure volume constraint for linear elastic materials, see, eg, the work of Bendsøe and Sigmund, 1 in which the elastic strain energy is subject to minimization and is referred to as (structural) compliance. Numerous solution approaches were developed of which the most popular are overviewed and compared in the work of Sigmund and Maute. 2 Usually, the topology has to be found within a given design space under mechanical boundary conditions, for which the mechanical problem is solved by a finite element approach. The design space contains material and void phases to identify the topology.This can be achieved either by tracing the phase borders or by introducing a density field that describes the material configuration in each point of the design space. The phase borders are subject to optimization in level-set approaches, 3,4 which Int J Numer Methods Eng. 2019;117:991-1017.wileyonlinelibrary.com/journal/nme

Adaptive thermodynamic topology optimization

“…For all previous results, the regularization paramater has been chosen as β = 2h 2 for the uniform meshing and a local value of β e = 2h 2 e on every mesh element e for the adaptive case. For the uniform meshing case, it has been reported in Jantos et al (2018) that smaller values lead to checkerboarding solutions. Here, we investigate the choice of the regularization parameter for the adaptive case.…”

“…The fundamentals of thermodynamic topology optimization are recalled here for convenience. For more details, we refer to Junker and Hackl (2015) for the general idea of thermodynamic topology optimization, to Jantos et al (2018) for an advanced numerical treatment, and to Jantos et al (2019) for a comparison of the method with SIMP.…”

“…Sigmund 2001), as indicated in Algorithm 1 to find an appropriate offset for the driving force. More details can be found in Jantos et al (2018).…”

“…As suggested in Jantos et al (2018), we solve a number of inner timesteps for stability reasons and also use the problem-independent control parameters η * and β * . In most cases, we employ the suggested value of β * = 2h 2 min (Jantos et al 2018) which corresponds to an internal length scale (Peerlings et al 1996;De Borst and Mühlhaus 1992) and thereby prescribes the minimal size of representable structures. This choice and the typically chosen value of η * = 15 then imply a single inner timestep.…”

“…It is obvious that accounting for the correct (non-linear) material behavior allows for using the potentials of design space and material at its best. In this regard, the so-called thermodynamic topology optimization has been introduced (Junker and Hackl 2015;Jantos et al 2018). It makes use of mathematical schemes known from the field of material modeling to derive a suitable equation for the evolution of the material density.…”