Keywords: topology optimization, stress limit, variable volume constraint.

This paper proposes a new topology optimization algorithm that seeks the minimum-compliance structural configuration of which the maximum stress will not exceed the imposed maximum stress limit. Traditional topology optimization methods such as homogenization methods or material distribution methods distribute a user-predetermined allowable amount of material on the design domain so as to create the minimum-compliance structural configuration. The magnitude of the maximum stress in the resultant structure cannot be predicted and controlled. The optimum topology thereby attained may become impractical or unsafe in cases where the allowable stress limit for the structure is largely different from the allowable stress limit. It is the main goal of this study to develop strategies to extend the capabilities of the traditional topology optimization methods so that the minimum-compliance structural topology thereby attained can also meet the predefined allowable stress limit. This new algorithm will iteratively adjust the allowable volume constraint according to the difference between the maximum stress limit and the actual maximum stress value in the current structural topology.

The detail procedure of the stress-limit based topology optimization method is defined as follows:

STEP 1:

Define design domain, loading conditions, boundary conditions, and a maximum stress limit. Provide an initial design, and an initial volume constraint.

STEP 2:

Conduct a traditional topology optimization process but the optimization process is temporarily stopped after two or three iterations of optimization, then the maximum stress and the compliance of the current structure are read. (During first ten iterations of the topology optimization, the maximum stress is checked every two iterations; after that the maximum stress is checked every three iterations.)

STEP 3:

Check if both stress and compliance convergence criteria are met:
Stress criterion: one of the following four conditions must be met:

(i)

The normalized difference between the maximum stress in the structure and the maximum stress limit is less than 0.008;

(ii)

The normalized difference between the maximum stress in the structure and the maximum stress limit is less than 0.040 for five consecutive checks;

(iii)

The volume constraint is adjusted to a number larger than 8020% for five consecutive checks;

(iv)

The volume constraint has been adjusted more than 30 times.

Compliance criterion: the variation percentage of the compliance values in two consecutive checkouts must be less than 0.001.

STEP 4:

If both criteria are met, the SLB topology optimization ends. Otherwise, go on to STEP 5.

STEP 5:

Adjust the volume constraint. Increase or decrease the volume constraint by 1/5 of the difference (in percentage) between the actual maximum stress in the structure and the allowable maximum stress, with a maximum cap of 5%. Go back to STEP 2.

The stress-limit based topology optimization proposed in this work provides a simple algorithm to be used with traditional topology optimization methods for seeking the minimum-compliance optimal topology of a structure of which the maximum stress can be controlled under the prescribed maximum stress limit. The fixed volume constraint for traditional topology optimization methods is replaced by the variable volume constraint re-adjustable after two or three iterations of optimization based on the difference between the maximum stress in the structure and the allowable maximum stress limit. Two illustrative examples proved that the stress-limit based topology optimization algorithm could effectively adjust the volume constraint along the course of the optimization and lead to a final minimum- compliance structure that meets the maximum stress limit. This simple SLB algorithm significantly increases the practicability of traditional topology optimization methods. The drawback of this variable volume constraint approach is the extra computational cost needed to conduct the extended topology optimization with dual convergence criteria.

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