Multi-probability constrained reliability-based topology optimization using volume control

Abstract

To overcome the limitation of deterministic topology optimization (DTO) which ignores uncertainties, this paper proposes a multi-probability-constrained reliability-based topology optimization (MRBTO) model for structures under multiple displacement constraints. The MRBTO model treats loads and material properties as random variables and uses a series system to represent multiple failure modes. System failure probability is calculated using the first-order reliability method (FORM). An outer loop adaptively adjusts the structural volume to meet a target failure probability, while an inner loop employs a modified SIMP method to optimize the material layout. A two-stage dynamic Gaussian sensitivity filtering (DGSF) method eliminates checkerboards and gray elements. The framework is validated on a 2D MBB beam and a cantilever beam using Monte Carlo simulation. Compared with DTO, MRBTO reduces the failure probability from approximately 50% to target levels (e.g. 5% or 1%) with high precision. The volume increase is modest – 5% for the cantilever beam even at the strictest target (1%), and about 7% for the MBB beam, which is a slight increase that is acceptable given the large reliability gain. Combined with DGSF, the discreteness rate (grayness ratio) drops from over 30% to nearly 0%, producing crisp boundaries and well-controlled displacements. MRBTO successfully handles multiple independent displacement constraints and different target failure probabilities simultaneously. A novel volume-controlled MRBTO model that handles system-level multi-failure probabilities is introduced, integrating DGSF to eliminate gray elements and boundary blurring. The dual-loop solution strategy efficiently couples reliability analysis with topology optimization, offering a practical and robust design tool under uncertainties.