Because of descriptive nonlinearity and computational inefficiency, topology optimization with fatigue life under aperiodic loads has developed slowly. A fatigue constraint topology optimization method based on bidirectional evolutionary structural optimization (BESO) under an aperiodic load is proposed in this paper. In view of the severe nonlinearity of fatigue damage with respect to design variables, effective stress cycles are extracted through transient dynamic analysis. Based on the Miner cumulative damage theory and life requirements, a fatigue constraint is first quantified and then transformed into a stress problem. Then, a normalized termination criterion is proposed by approximate maximum stress measured by global stress using a P-norm aggregation function. Finally, optimization examples show that the proposed algorithm can not only meet the requirements of fatigue life but also obtain a reasonable configuration.

For air vehicles, components or supports are exposed to aperiodic loads, and there are strict requirements for minimum service life. The weight of aircraft parts directly affects carrying travel, and lightweight research can greatly improve the power-to-weight capacity. Therefore, this paper studies the topology optimization problem with fatigue life as the constraint and the minimum volume/weight as the objective under aperiodic loading.

The field of topology optimization mainly focuses on minimizing structural compliance [

However, the aperiodicity and randomness of the aperiodic load determine that its fatigue cannot be described by stress or strain directly and can only be counted by the rain-flow method and expressed by fatigue damage. Due to the nonlinearity of fatigue damage with life history, there are few studies concerning topology optimization under aperiodic load fatigue constraints. This research aims to address the high cycle fatigue problem under aperiodic loading. If the stress value can be connected with fatigue damage [

Stress constraint topology optimization has conundrums of locality. The stress of each element must be constrained to search for the maximum stress. The sensitivity of each constraint needs to be calculated, so it leads to excessive computation whether using a direct method or adjoint method. To solve this problem, aggregation functions, such as the P-norm method [

In this paper, aiming at the severe nonlinearity of cumulative damage under aperiodic loading, a transformational relation of fatigue damage with stress is established. First, fatigue damage of the substep is calculated by the stress time history, and the problem is transformed into a stress topology optimization problem based on BESO. In the classic BESO method, volume is generally the constraint, so volume evolution is adopted to realize element addition and deletion, and the objective change value is taken as the convergence criterion. However, volume is the objective here, and if volume is taken to measure evolution and convergence simultaneously, it will be difficult to converge due to missing stress. Therefore, a normalized parameter, which comprehensively considers volume and stress, is proposed and taken as the convergence parameter. In this way, the topology lightening problem with a high cycle fatigue life as a constraint is solved.

To interpret the relationship between fatigue life, damage and stress, this section briefly introduces a fatigue analysis program for aperiodic loads. The general process is as follows: first, fatigue life is calculated by alternating stress and average stress extracted from the transient stress time history. To explain the multiaxial stress of each node, the signed von Mises stress is calculated. The reversal of symbolic von Mises stress and average stress are determined by rain flow counting. The Goodman correction method is used to calculate the amplitude of the effective stress. Then, the Basquin equation is applied to calculate the number of failure cycles for each effective stress amplitude. Finally, all the damage caused by stress reversal in the loading history is accumulated linearly by the Palmgren–Miner law [

In detail, the stress history is solved by a given dynamic system as

Equivalent parameters similar to von Mises stress are usually positive [

To consider the influence of the average stress, the Goodman correction equation is used to calculate the amplitude of the effective stress

The number of failure cycles (_{i}_{i}

Cumulative fatigue damage can be calculated by the Palmgren–Miner law as

Therefore, fatigue damage is expressed by the stress amplitude as

From

In this section, the process course of converting the fatigue lifetime requirement to a stress constraint is presented. First, the position of the danger point is obtained by transient dynamic analysis loading aperiodic cyclic force. In addition, the stress time history is obtained. Then, the rain flow counting method is used to extract the effective stress cycle. Fatigue damage is calculated with the Palmgren–Miner criterion. Finally, according to the lifetime requirements, the critical stress level is inversely calculated and taken as the constraint condition for topology optimization. The specific implementation steps are as follows:

With the requirements of fatigue design life, stress amplitude

Assuming that

If

Through the conversion above, fatigue-constrained topology optimization can be transformed into a stress-constrained problem. To reduce the quantity of stress constraints at every cycle, the P-norm aggregation function is used as
_{i}_{i}

Classic BESO uses the change of objective function as the convergence criterion. The objective parameter of optimization model in this paper is volume. If the volume change is set as the convergence criterion, the volume increment between adjacent iterations will alter drastically, and the convergence value is hard to realize. To take into account both optimization objective and constraint parameters, this paper proposes a normalized performance index PI, which can comprehensively consider stress change and volume change as

In general,

Except for sensitivity analysis, convergence criterion and fatigue constraint, the rest of the flow is based on BESO theory [

To show the effectiveness and correctness of the proposed method, we design two numerical examples: double-bar plates and L-shaped beams. Therefore, a double-bar plate is designed with no obvious stress concentration, and an L-shaped beam is designed with stress concentration. The fatigue life is designed to be 140 h to confine to elastic deformation. The stress aggregation norm in

The initial design area and boundary conditions of the double-bar plate are shown in

There is no stress concentration caused by the structure shape in the double-bar model. The iterative changes in the volume fraction and PI are shown in

The initial design area and boundary conditions of the L-shaped beam are shown in

The variation curve of the volume fraction and PI is obtained, as shown in

Curves of the maximum stress and P-norm aggregation stress are shown in

Fatigue life, strength and stiffness are three important factors that designers cannot ignore in engineering. Aiming at high cycle fatigue caused by aperiodic load, this paper realizes topology optimization method of aperiodic load based on BESO.

In order to solve the severe nonlinearity of fatigue damage to design variables, this paper uses fatigue analysis to calculate cumulative damage and then equivalent fatigue damage to critical stress by stress damage curve to transform fatigue constraint problem to stress constraint problem.

Then, P-norm is used for stress aggregation, and the dimensionless index PI is proposed to improve convergence speed. The effectiveness of proposed method in obtaining the lightest configuration which strictly satisfies life constraint is proved by two examples with and without stress concentration.