Research paperFatigue-resistance topology optimization of continuum structure by penalizing the cumulative fatigue damage
Introduction
Topology optimization (TO) has experienced tremendous development from pure academic research to complex engineering applications, which can be traced back to the seminal work introduced by Bendsøe and Kikuchi [1]. One of the most prevalent methodologies for TO is the variable density approach, also known as the solid isotropic material with penalization (SIMP) method [2,3]. The typical SIMP interpolation scheme describes Young's modulus as the simple power-law function of the pseudo density. Nowadays, the SIMP approach and other methods have been successfully applied to a wide range of practical problems in the past few decades [4], [5], [6], [7].
Imposing stress constraint in TO formulation is certainly crucial in structural design. However, the overwhelming literature focuses on minimum compliance and other global criteria. There exist three unavoidable challenges related to stress constraint [8]. Firstly, the singularity phenomenon stems from the degeneracy or irregularity of the design space, when the elemental density approaches zero. The typical relaxation techniques for alleviating this difficulty include ε-relaxed formulation, qp approach [9,10] and introduction of the relaxed stress indicator [8,11]. Secondly, the local nature of stress constraint on each element in the design domain leads to enormous constraint functions, which inevitably yields prohibitive burdens for the optimization solver. To solve the dilemma, Yang and Chen initially utilized the p-norm measure and Kreisselmeier-Stenhauser measure as the surrogate function to approximate the maximum value of the stress in the concerned region [12]. Another choice to reduce the number of constraints is the regional or block strategies controlling the maximum stress level via covering the design region [13,14]. Le et al. proposed the adaptive calibration method to overcome the inherent weaknesses of the aggregate function, which might overestimate or underestimate the maximum stress value [8]. Another extremely striking problem is the nonlinearity associated with local stress. The optimized results are highly sensitive to the penalty factor, filtering scheme and algorithm parameters etc. The stability transformation method is proposed to calculate the adjustment coefficient. Yang et al. proposed the violated set approach to replace the maximum stress constraint. To stabilize the convergence, static compliance constraint function is appended to the optimization formulation [15]. More recently, a three-phase robust formulation with the evaluation at jagged boundaries is proposed to achieve the topological design considering manufacturing uncertainties [16]. Other than SIMP method, stress-constrain problems can be also solved in the framework of level set method [17], [18], [19], [20], bi-directional evolutionary structural optimization (BESO) approach [20], [21], [22], [23] and moving morphable void-based explicit approach [24] nowadays.
Meanwhile, TO by controlling the allowable stress has been furthered for other engineering problems. For instance, Lee et al. investigated the design-dependent loading and into the stress-constrain TO problem [25]. Luo and Kang extended the SIMP approach applicable to continuum structure possessing asymmetrical strength behaviors [26]. Jeong et al. proposed a separable stress interpolation, which restricts maximum stress in continua containing multiple phases [27]. Collet et al. introduced stress limitation in the microstructure design for porous material [28]. Lee et al. presented the TO approach for composite laminates with the help of layerwise theory [29]. Long et al. presented alternative choice of solving the stress-constrain optimization problem, by setting up a series of quadratic programming subproblems [30].
Associated with stress criteria, the TO for fracture and fatigue resistance has received ever-increasing attention in recent years. The J integral principle for evaluating the fracture behavior is initially introduced as part of the objective function by Kang et al. [31]. Xia et al. proposed the fracture-resistance method of quasi-brittle composite structures via the phase field approach [32]. More recently, Da et al. present a fracture resistance design of bio-inspired periodic composite structures [33,34], by incorporating phase field method and topology optimization. Holmberg et al. initially discussed the optimized results including fatigue and static stress [35]. A dynamic fatigue and static failure constraint TO method under proportional loads is developed and evaluated by Jeong et al. [36]. Collet et al. [37] presented the optimization tool coping with lightweight design accompanied by compliance and fatigue constraints. Through using a quasi-static analysis and rainflow-counting method, Oest and Lund [38] investigated fatigue-based TO. Jeong et al. [39] also proposed a layout optimization methodology resisting fatigue failure subjected to variable amplitude and proportional load. Meanwhile, two different TO methods to assess dynamic fatigue failure are proposed by Lee et al. [40] and Zhao et al. [41], respectively. Except for the SIMP method, the fatigue-based TO issue has also been addressed by BESO approach [42].
Despite the significance of fatigue resistance in TO community, the existing works considering fatigue failure in TO are rather limited [43,44]. To this crucial research topic, this paper aims to develop effective constraint schemes for performing the fatigue damage constraint in structural TO. To this end, the penalization scheme of the fatigue damage is suggested, also as the main novelty of the present work. Moreover, the TO problem is well-posed by density-filter SIMP method and adjoint sensitivity analysis. The method of moving asymptotes (MMA) is called as the optimizer [45]. In the end, several representative 2D and 3D examples are provided to investigate the effect of the penalized factor, mean stress, proportional and non-proportional loads on optimized results.
The rest of the paper is organized as follows: Section 2 introduces the penalized fatigue damage and TO formulation. Section 3 presents derivations of sensitivity information. In Section 4, several numerical examples are provided to validate the effectiveness of the proposed method. Finally, the conclusive remarks of this paper are given in Section 5.
Section snippets
Topology optimization problem statement
To begin with, the whole design domain is discretized as finite elements and each element is assigned the design variable ρi (0 ≤ ρi ≤ 1). For ensuring a well-posed optimization problem [46,47], the original designs variables are mapped into a set of physical densities :where Ne is the set of elements whose center-to-center distance d(e, i) to the eth element is within the filter radius rmin. The weight factor ωei is defined asIn this work, the eth
Sensitivity analysis
For the reason of the utilization of a gradient-based optimization algorithm, the derivation of the objection V and the aggregate function Dpn will be conducted in this section.
Numerical examples and discussions
In this section, several numerical examples including 2D and 3D structures are provided to validate the effectiveness of the proposed method. The design variables in Eq. (9) are updated by MMA algorithm and we refer to the default MMA parameter for all examples as those in literature [45]. A rather conservative move limit 0.02 is adopted to stabilize the optimization iteration. The design domain is discretized using four-node plane stress elements and eight-node solid elements for 2D and 3D
Conclusion
In this paper, a TO approach is proposed for the design of fatigue-resistance structure sustaining time varying fatigue loads. After quasi-static finite element analysis, structural displacement can be efficiently obtained. The penalization of elemental damage is proposed as major theory contribution of the current study. A mass of constraints imposed on elemental fatigue damage is aggregated into single index by a p-norm measurement. The derivation of sensitivity expression is detailed.
CRediT authorship contribution statement
Zhuo Chen: Conceptualization, Methodology. Kai Long: Writing - original draft, Methodology, Software, Formal analysis, Supervision, Funding acquisition. Pin Wen: Conceptualization, Funding acquisition. Saeed Nouman: Conceptualization.
Declaration of Competing Interest
The authors declare that they have no conflict of interest.
Acknowledgments
This work was financially supported by the National Natural Science Foundation of Beijing (2182067), the National Natural Science Foundation of China (11902232), the Fundamental Research Funds for the Central Universities (2018ZD09). We thank Professor Krister Svanberg for providing the Matlab code of MMA. This paper was partly completed during the epidemic of Covid-19. The authors dedicate this paper to Chinese doctors and nurses due to their great efforts and even their lives against disease.
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