Large Scale Topology Optimization Using Preconditioned Krylov Subspace Recycling and Continuous Approximation of Material Distribution

Abstract

Abstract. Large-scale topology optimization problems demand the solution of a large number of linear systems arising in the finite element analysis. These systems can be solved efficiently by special iterative solvers. Because the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems, and by using preconditioning and scaling techniques. We also provide a new implementation of the 8-node brick (B8) element for the continuous approximation of material distribution (CAMD) approach to improve designs of functionally graded materials. Specifically, we develop a B8/B8 implementation in which the element shape functions are used for the approximation of both displacements and material density at nodal locations. Finally, we evaluate the effectiveness of several solver and preconditioning strategies, and we investigate large-scale examples, including functionally graded materials, which are solved with a special version of the SIMP (solid isotropic material with penalization) model. The effectiveness of the solver is demonstrated by means of a topology optimization problem in a functionally graded material with 1.6 million unknowns on a fast PC.

A survey of subspace recycling iterative methods

“…Recycling solvers and closely related approaches have been used in lattice quantum chromodynamics [9,10,24,36,148], in particular, [9] mentions that recycling is important for handling physical regimes with very small eigenvalues. Many problems in design involve sequences of slowly changing linear systems, for which recycling is highly efficient, for example, in topology optimization and other structural optimization problems [28,34,42,153,157], and aerodynamic shape optimization [33,81]. Recycling has also been used to compute reduced order models (for a range of applications) [5,6,57,58,111].…”

Free vibration characteristics of functionally graded structures by an isogeometrical analysis approach

“…Considering the presented data, we can observe that, despite the same increase in the wall mass for the opposite combinations of n x and n y , the increases in fundamental frequency of the wall are more when n y ¼ 0. This can be easily interpreted by regarding relation (25) and the material properties of the constituent phases. Equation (25) shows that increasing each of the values n x or n y results in the same overall increment of ceramic volume fraction.…”

“…This can be easily interpreted by regarding relation (25) and the material properties of the constituent phases. Equation (25) shows that increasing each of the values n x or n y results in the same overall increment of ceramic volume fraction. However, increasing n x brings about more increase of the ceramic phase nearby the clamped edge where the role of higher stiffness seems dominant, while increasing the value of n y results in more increase of the ceramic phase in the vicinity of the lower edge of the wall where it does not play such significant role.…”

“…1 By now, this method is widely employed by many researchers for the analysis and optimization of FG structures. [19][20][21][22][23][24][25][26][27] On the other hand, so far, some researches for the study of structural vibrations in the framework of isogeometric analysis have been undertaken. In 2006, Cottrel et al 28 investigated the free vibrations of different types of structures such as bars, beams, plates and shells and compared the accuracy of the obtained results with the results of FEM.…”

Preconditioner updates for solving sequences of linear systems in matrix-free environment

“…An important class of algebraically-motivated strategies to accelerate the convergence of preconditioned iterative methods is based on constructing or improving the preconditioner by adaptive spectral information obtained directly from the Krylov subspace methods, see, e.g., [19,20,21,22,23,24,25,26]. All of these techniques have also a significant potential to be applied in the form of preconditioner updates for solving sequences of systems [27], [28], [29]. These strategies are often problem specific, but they are in general compatible with matrix-free implementations.…”