Three‐dimensional manufacturing tolerant topology optimization with hundreds of millions of local stress constraints

Abstract

In topology optimization, the treatment of stress constraints for very large scale problems (more than 100 million elements and more than 600 million stress constraints) has so far not been tractable due to the failure of robust agglomeration methods, that is, their inability to accurately handle the locality of the stress constraints. This article presents a three-dimensional design methodology that alleviates this shortcoming using both deterministic and robust problem formulations. The robust formulation, based on the three-field density projection approach, is extended and proved necessary to handle manufacturing uncertainty in three-dimensional stress-constrained problems. Several numerical examples are solved and further postprocessed with body-fitted meshes using commercial software. The numerical investigations demonstrate that: (1) the employed solution approach based on the augmented Lagrangian method is able to handle very large problems, with hundreds of millions of stress constraints; (2) three-dimensional stress-based results are extremely sensitive to slight manufacturing variations; (3) if appropriate interpolation parameters are adopted, voxel-based (fixed grid) models can be used to compute von Mises stresses with excellent accuracy; and (4) in order to ensure manufacturing tolerance in three-dimensional stress-constrained topology optimization, a combination of double filtering and more than three density field realizations may be required.

Local versus global stress constraint strategies in topology optimization: A comparative study

“…The maximum penalization parameter is normalized by the number of stress constraints to alleviate mesh dependence effects. 14,24 Parameters 𝛾 r and 𝛾 𝛽 are automatically chosen to achieve r =…”

“…Also, the AL method was recently employed to solve truly large-scale topology optimization problems with hundreds of millions of local stress constraints providing excellent results in reasonable computational time, proving AL to be a very promising method, and motivating its application at industrial level. 24 In this article, a numerical comparison between global solution strategies based on the P-mean function and local solution strategies based on the AL method is presented. The P-mean function is chosen instead of the P-norm as it leads to a smooth maximum approximation whose error of estimation is less dependent on the number of input arguments.…”

“…for the PDE-based filter. 24 This value of 𝛽, however, is not applied directly to the problem, but gradually approached through a continuation approach instead, as presented in Section 3.…”

An Aggregation-Free Local Volume Fraction Formulation for Topological Design of Porous Structure

“…Furthermore, the AL method introduces considerably more constraint functions than the LSM and BESO methods. However, the AL method becomes appealing when a large scale of local stress constraints can be addressed successfully, according to a recent report [ 46 ]. In this work, the Equation (6) can be solved as the solution of the unconstrained program, by establishing a sequence of AL functions in each k -th step [ 47 ]: where N Φ and N Ψ are the constants related to number of the local and global volume constraints.…”

“…In addition, the appropriate value for the aggregation parameter is also a challenge. More recently, the augmented Lagrangian (AL) method was successfully applied to handle millions of stress constraints [ 45 , 46 , 47 , 48 ]. Inspired by these facts, we reformulated the LVC constraint by virtue of the AL approach.…”