Fatigue failure is a common failure mode under the action of cyclic loads in engineering applications, which often occurs with no obvious signal. The maximum structural stress is far below the allowable stress when the structures are damaged. Aiming at the lightweight structure, fatigue topology optimization design is investigated to avoid the occurrence of fatigue failure in the structural conceptual design beforehand. Firstly, the fatigue life is expressed by topology variables and the fatigue life filter function. The continuum fatigue optimization model is established with the independent continuous mapping (ICM) method. Secondly, fatigue life constraints are transformed to distortion energy constraints explicitly by taking advantage of the distortion energy theory. Thirdly, the optimization formulation is solved by the dual sequence quadratic programming (DSQP). And the design scheme of lightweight structure considering the fatigue characteristics is obtained. Finally, numerical examples illustrate the practicality and effectiveness of the fatigue optimization method. This method further expands the theoretical application of the ICM method and provides a novel approach for the fatigue optimization problem.
Continuum structural topology optimization is the most difficult optimization problem in structural topology optimization. Topology optimization has four important significances including searching for the best transmission path of forces, forming the optimal distribution of structural material, optimizing performance of the structure, and achieving structural lightweight. Many researchers devote themselves to topology optimization because of novel design [
Structural fatigue may occur when engineering structures are subjected to periodic or random loads. In this case, the structural stress is less than the material strength limit. Due to the absence of obvious signal and the uncertainty of the occurrence, the fatigue failure is a great safety hazard. To improve the anti-fatigue performance of the structure, it is important to design the engineering components with fatigue topology optimization. There have been some researches on fatigue topology optimization but in a few quantity. In the early researches, the crack initiation was used to indicate that the structural fatigue failure had occurred, and the fatigue life constraints were normally transformed to the stress constraints since it was convenient to calculate. For high cycle fatigue optimization, Mrzyglod et al. [
With the progress of the manufacturing industry, the service life of industrial products is increasing. More topology optimization problems are considered with high cycle fatigue life constraints [
The fatigue failure is a local phenomenon, which means the fatigue life of each unit should satisfy the fatigue life constraints. Some efficient methods were introduced to reduce computation, like the P-norm function [
The ICM method is effective to solve the structural topology optimization problems, especially in static, frequency and buckling problems [
This paper consists of seven sections. In Section 2, three features in ICM method are demonstrated. In Section 3, the fatigue life filter function is introduced to establish fatigue topology optimization model, and the fatigue life constraints are explicitly transformed to distortion energy constraints. In Section 4, the process of solving strategies with the topology optimization model is represented. In Section 5, the program flow of the optimization algorithm is presented. In Section 6, three numerical examples are presented to demonstrate the validity of the fatigue optimization method. Finally, the conclusions are obtained.
ICM method, which is a continuous topology optimization design method proposed by Professor Sui in 1996 [
Lightweight design is the consistent optimization goal of ICM method because of the high economic value in the aerospace industry, automobile industry, etc. This is of great significance for saving production cost, reducing use cost, improving product mechanical performance, etc.
The high computational efficiency is realized by two transformations of optimization model. First, the discrete optimization model is transformed into a continuous optimization model by introducing the independent continuous variable. In ICM method, the discrete variables that are 0/1 are converted to continuous variables that belong to [0,1] and inverse them back to discrete variables after optimization. Second continuous optimization model is transformed into the quadratic programming optimization model by introducing the DSQP method. The DSQP method is the combination of duality theory and the SQP algorithm. This method could transform the constraints into the objective and form a dual optimal model. Therefore, the amount of constraints is reduced, which leads to a reduction in computation.
The filter functions establish the relationship between topology variables and physic properties or geometric dimensions. The expressions of filter function will determine the establishment and solution of topology optimization. Further, it will have an impact on the performance of the optimized structure. It is the key point to establish the relationship between discrete variables and continuous variables. In mathematics concept, the filter function is the result of the continuous infinite approximation of
The differences among the filter functions directly affect the computational efficiency and the optimization results. We usually use power functions,
Mass filter function
Fatigue failure exists widely in actual projects. When fatigue failure occurs, the structural component usually fails before alternating stress reaches the allowable value of structural stress. There is no obvious warning when an accident occurs. And it is difficult to prevent in advance, resulting in great losses. To improve the structural fatigue performance. The fatigue failure is considered in structural concept design.
The lightweight fatigue topology optimization model is presented.
In fatigue topology optimization, the unit fatigue life of the structure should be greater than the unit extended allowable fatigue life:
And
Above all, the unit extends allowable fatigue life is identified by the filter function of the unit allowable fatigue life in the fatigue topology optimization. Then we get the lightweight fatigue topology optimization model as follows:
To establish the sign of fatigue failure, the structural damage accumulation theory is introduced to represent the structural damage. The fatigue damage can be linearly accumulated with the Miner rule and defined as follows:
With the Miner rule, the sign of fatigue failure is obtained. The mathematical expression of structural failure is established. But in the physical phenomenon, the fact of fatigue failure is the release of structure energy. It is important to obtain the physical expression of structural failure. The relationship between fatigue life and distortion energy is presented by the S-N curve and distortion energy theory. First, the fatigue life is transformed to the structural peak stress with S-N curve. Then the structural peak stress is transformed to distortion energy by distortion energy theory.
The fatigue life is transferred to structural peak stress with S-N curve and structural damage accumulation theory. The form of the S-N curve is formulated by the power function and defined.
The S-N curve is used to make the fatigue topology optimization constraint explicitly. And the optimized model in
The distortion energy theory is presented as follows:
In this inequation,
The unit distortion energy
In order to keep the results safe, the unit structural strain energy is the substitute for the unit distortion energy.
The cyclic load in fatigue topology optimization is dynamic. The unit structural strain energy
Based on the relation between the unit structural strain energy
Then the dynamic stiffness matrix filter function is introduced as follows:
And then, we can get:
The hypothesis of static determination is introduced. The internal forces
To obtain the explicit expression of the unit dynamic strain energy, the stress constraints in
When fatigue failure occurs, the unit structural peak stress is smaller than the unit allowable structure stress, that is:
Based on the distortion energy theory, we transform
The constraint in
The unit structure distortion energy corresponding to
Both sides of
The filter functions
For the convenience of proof, we set:
The constraints can be explicitly expressed as
where
The second-order Taylor expansion is introduced to standardize the objective. According to the topology optimization formulation, the objective is shown as
Then we set
The objective is expanded by quadratic Taylor:
The constant terms can be ignored, so the
Therefore, a standard sequential quadratic programming model can be obtained:
where
The number of design variables is larger than constraints in fatigue topology optimization. According to dual theory, the above topology optimization formulation programming is transformed into dual programming, as shown in
where
In
Then
where
So, we can get
The quadratic programming model is obtained after
After the quadratic programming has been solved, then the
Then the iteration can be terminated. The
The fatigue optimization method is applied to MSC.Patran software platform with MSC.Natran and MSC.Fatigue solver. PCL language is used to realize the continuum fatigue topology optimization. The details of the fatigue topology optimization procedure are given, and the algorithm flowchart is shown in
Step 1: Establish the continuum fatigue topology optimization model based on MSC.Patran.
Step 2: Set an optimized objective, fatigue constraints. Initialize the element topological values.
Step 3: Carry out the fatigue analysis with MSC.Patran.
Step 4: Form the topology optimization with dynamic strain energy constraints.
Step 5: Solve the topology optimization with the dual sequence quadratic programming (DSQP) method. Get the continuous topology optimization structure.
Step 6: Judge convergence of the optimized structural mass. If the results satisfy
Step 7: Obtain the discrete topological variables with the inverse threshold. Form the discrete topological structure.
Step 8: Carry out the fatigue analysis with MSC.Patran.
Step 9: Judge fatigue life of the optimized results. If it satisfies the fatigue life constraints, obtain the optimized structure, and end the calculation. Otherwise, modify the inverse threshold and go to Step 7.
Three examples are presented to test the fatigue optimization method. The form of cyclic load in numerical examples is the sine function, which is shown in
The design domain is a cantilever with the size of
The iterative history of the mass is shown in
Comparing the mass iteration history in
In
In Example 2, the basic structure is a simply supported beam with the size of
From
From
The design domain is a beam structure of
To compare the differences between fatigue topology optimization and traditional stress optimization [
From
In this paper, the fatigue topology optimization is presented based on ICM method and fatigue analysis method. The lightweight topology optimization model is established, which uses fatigue life as constraint. The fatigue life constraints are transformed into distortion energy constraints with the S-N curve and the distortion energy theory. The effectiveness and validity of fatigue optimization method are verified by the comparation between the ICM method and the SIMP method. The numerical examples demonstrate the lightweight topology optimization design with the fatigue constraint can be achieved by the presented method.
In addition, the Miner rule and the S-N curve is carried out in this paper. In the future, we can discuss effect on the structural topological configuration according to other different fatigue failure criteria.