[Paper Review] Robust Topology Optimization of Truss with regard to Volume

Daniel Mohr, Ina Stein+2 more|arXiv (Cornell University)|Sep 17, 2011

Topology Optimization in Engineering6 references3 citations

TL;DR

This paper proposes a robust topology optimization method for trusses that eliminates the need for finite approximations of uncertain parameters by directly solving a semi-infinite optimization problem. Using the ground structure method and matrix force formulation, it ensures structural performance under load uncertainties without relying on statistical assumptions or worst-case approximations, yielding designs with slightly increased volume but guaranteed robustness across perturbed loads.

ABSTRACT

A common problem in the optimization of structures is the handling of uncertainties in the parameters. If the parameters appear in the constraints, the uncertainties can lead to an infinite number of constraints. Usually the constraints have to be approximated by finite expressions to generate a computable problem. Here, using the example of the topology optimization of a truss, a method is proposed to deal with such uncertainties by using robust optimization techniques, leading to an approach without the necessity of any approximation. With adequately chosen load cases, the final expression is equivalent to the multiple load case. Simple numerical examples of typical problems illustrate the application of the method.

Motivation & Objective

• Address the challenge of handling parameter uncertainties in truss topology optimization without relying on finite approximations of infinite constraints.
• Develop a computationally tractable method that avoids the need for statistical assumptions or worst-case scenario approximations.
• Ensure structural robustness under load variations by directly incorporating uncertainty bounds into the optimization framework.
• Provide a black-box compatible, straightforward procedure for robust design that maintains computational efficiency and solution reliability.
• Demonstrate the method’s effectiveness through numerical examples with quantified volume increases under uncertainty.

Proposed method

• Formulates the topology optimization problem using the ground structure method, where all potential truss members are predefined in a design space.
• Applies the matrix force method to compute structural responses, enabling accurate stress and displacement analysis under various load cases.
• Models uncertainties in loading parameters as bounded intervals, avoiding reliance on probabilistic distributions.
• Transforms the robust optimization problem into a semi-infinite program by considering all possible load variations within the given bounds.
• Solves the resulting semi-infinite problem directly using numerical solvers that handle infinite constraints without approximation.
• Uses both simplex and interior-point algorithms to compute solutions, ensuring robustness and convergence across different optimization paths.

Experimental results

Research questions

• RQ1 How can truss topology optimization be made robust to load parameter uncertainties without relying on finite approximations of infinite constraints?
• RQ2 Can a direct solution to the semi-infinite robust optimization problem be achieved using standard numerical solvers without heuristic or statistical assumptions?
• RQ3 What is the trade-off in structural volume when enforcing robustness across perturbed load cases compared to nominal design?
• RQ4 How does the proposed method compare to traditional multiple load case or reliability-based approaches in terms of computational structure and solution quality?
• RQ5 To what extent can the method be implemented as a black-box procedure compatible with commercial optimization software?

Key findings

• The proposed method successfully formulates and solves a semi-infinite robust optimization problem without requiring finite approximations of the infinite constraint set.
• For Example 3, the robust solution achieved a volume of 0.0026 m³, compared to 0.0026 m³ in the nominal case, showing no increase under symmetric uncertainty.
• In Example 4 (mast structure), the robust design increased volume from 0.000514 m³ to 0.001568 m³ under 50% load perturbation, ensuring stability across all scenarios.
• In Example 5 (transmission tower with two load cases), the robust solution required 0.002562 m³, up from 0.000850 m³ in the nominal case, demonstrating consistent performance under combined and perturbed loads.
• The method produces equivalent results to multiple load case optimization but avoids the need for explicit enumeration of all load combinations.
• Numerical results confirm that the method is robust, computationally feasible, and suitable for implementation in standard optimization software without approximation errors.