Topology optimization using a continuous-time high-cycle fatigue model
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Abstract
We propose a topology optimization method that includes high-cycle fatigue as a constraint. The fatigue model is based on a continuous-time approach where the evolution of damage in each point of the design domain is governed by a system of ordinary differential equations, which employs the concept of a moving endurance surface being a function of the stress and back stress. Development of fatigue damage only occurs when the stress state lies outside the endurance surface. The fatigue damage is integrated for a general loading history that may include non-proportional loading. Thus, the model avoids the use of a cycle-counting algorithm. For the global high-cycle fatigue constraint, an aggregation function is implemented, which approximates the maximum damage. We employ gradient-based optimization, and the fatigue sensitivities are determined using adjoint sensitivity analysis. With the continuous-time fatigue model, the damage is load history dependent and thus the adjoint variables are obtained by solving a terminal value problem. The capabilities of the presented approach are tested on several numerical examples with both proportional and non-proportional loads. The optimization problems are to minimize mass subject to a high-cycle fatigue constraint and to maximize the structural stiffness subject to a high-cycle fatigue constraint and a limited mass.
Keywords
Continuous-time approach Endurance surface High-cycle fatigue Topology optimization Adjoint sensitivity analysis Aggregation function1 Introduction
In mechanical components, fluctuating loads and stress concentrations lead to fatigue and possible failure. Thus, fatigue phenomena often govern the overall life of the component, quantified in terms of load cycles to failure. Concerning topology optimization (TO), fatigue constraints are either implemented as stress constraints (Holmberg et al. 2014) or with cycle-counting techniques like in Jeong et al. (2018), but these models are restricted to proportional loading histories. To incorporate a fatigue model in a TO problem that can handle a general loading history, we propose to use an evolution-based fatigue model.
Examples of other structural optimization formulations with fatigue constraints are found in Collet et al. (2017) and Oest et al. (2017b), where the fatigue prediction is done for a simplified damage model assuming a periodic load. Other examples can be found in Gerzen et al. (2017), where sizing optimization of shell structures is done with fatigue constraints at the welded joints using the commercial software SIMULIA Fe-safe, and Oest and Lund (2017a), where the TO is formulated with finite-life fatigue constraints. However, the fatigue is calculated using a cycle-counting algorithm, specifically rainflow counting (Amzallag et al. 1994), and this restricts the applicability to scalar stress measures, like signed von Mises. However, in many industrial applications, fatigue due to both proportional and non-proportional loads threaten the life of components. A recent contribution (Zhang et al. 2019) uses classical techniques, including rainflow counting, mean stress correction, and Palmgren-Miner’s rule, initially developed for unidirectionally loaded structures, to handle non-proportional loading. This is done by using a signed von Mises stress at each point of the structure as the single stress measure affecting fatigue. However, certain modes of stress reversal, including rotary stress states with constant principal stresses, give a constant signed von Mises stress, so that zero fatigue damage is predicted. This limitation of the validity of the model is a concern in TO, since the optimizer could exploit such weaknesses of the model. This motivated us to extend our previous contribution (Suresh et al. 2019), where we use a fatigue model that captures the fatigue damage from non-proportional loads as a constraint in TO problems.
The fatigue model follows a high-cycle (HC) evolution-based model developed in Ottosen et al. (2008). This model uses a continuous-time approach in the form of differential equations governing the time evolution of fatigue damage at each point in the design domain. Such evolution occurs when the stress state lies outside a so-called endurance surface, which moves in stress space depending on the current stress and a back stress tensor. The model states that the development of damage only occurs if the stress state lies outside the endurance surface while this surface evolves. The advantage of using such a model is that non-proportional loading histories can be considered in the prediction of fatigue damage. Further developments of the model can be found in (Brighenti et al. 2012), where the fatigue is assessed for complex multiaxial load histories, in Holopainen et al. (2016), where the model is extended to transversely isotropic materials, and in Ottosen et al. (2018), where the multiaxial fatigue criterion considers the stress gradient effects in critical regions like holes and notches.
Fatigue is here included in TO as a constraint on the maximum damage found in the design domain at the end of a given load history. The maximum damage is approximated by means of an aggregation function, namely the p-norm (Kennedy and Hicken 2015). The assumption of HC fatigue, in which the geometry and the material properties remain constant until failure, implies that the finite element (FE) stiffness matrix is constant throughout the load history for a given design. Therefore, the stress history required to evolve the damage can be computed at a reasonable cost. Once the stress history is obtained, we solve the spatially uncoupled ordinary differential equations for damage and back stress to get the total accumulated damage in selected points in the design domain (here the centroid of each FE) and compute the approximate maximum value.
The TO problem is solved using a gradient-based method with sensitivities of the approximated maximum fatigue determined by adjoint analysis (Haftka and Adelman 1989; Tortorelli and Michaleris 1994; van Keulen et al. 2005). Since the damage is governed by an evolution problem, computing these sensitivities requires the solution of a terminal value problem for the adjoint variables. The overall solution process is somewhat similar to that of e.g., Panagiotis et al. (1994) and Wang et al. (2017) for TO with plasticity.
2 Continuous-time approach
In general, damage can affect both the material (viaE) and the geometry (viaB). However, we are only concerned with high-cycle fatigue, where these properties are assumed to be unaffected by damage until failure. This means that we will first solve (1) for all t, and then determine the fatigue-development based on the computed stresses. Henceforth, we therefore consider the stress σ = σ(t) as given and then predict fatigue damage by the continuous-time approach following Ottosen et al. (2008, 2018).
3 Numerical implementation of the fatigue model
The fatigue problem is solved numerically by dividing the time domain [0,T] into a finite number of time steps of equal length Δt, i.e., t_{i} = iΔt with i = 0,1,2,...,N. The ordinary differential equations (ODEs) in (7) and (11) are approximated using a forward Euler scheme. The stresses at the time steps are σ_{i} = σ(iΔt), \(\boldsymbol {s}\left |{~}_i\right . = \boldsymbol {s}(\boldsymbol {\sigma }_i)\) and α_{i} = α(iΔt), where \(\left |{~}_i\right .\) represents function evaluation at time step i.
Note that in this time discretization, we have replaced the time derivatives of the stress tensor by an approximate finite difference expression. However, in many applications, even treated in this paper, such time derivatives will be explicitly known. The discretization could then be simplified, but to keep the method and presentation general, we have not explicitly treated this as a special case.
4 Optimization problem formulations
Remark 1
The problem related to (\(\mathbb {P}_1\)) of minimizing mass subject to a fatigue (or stress) constraint, without a constraint on the compliance, has been previously considered in the literature (Holmberg et al. 2013, 2014; Jeong et al. 2018; Oest and Lund 2017a; Zhang et al. 2019). As shown in Appendix A, also for our fatigue model, gradient-based solution methods frequently show convergence to realistic structures without inclusion of a compliance constraint. However, note that the globally optimal solution to \((\mathbb {P}_1)\) without compliance constraint is actually x_{e} = 𝜖 for all e. This is seen by noting that x_{e} = 𝜖 for all e gives zero fatigue, because of the scaling in (19), while obviously minimizing the mass. Therefore, while the solution shown in Appendix looks plausible from an engineering point of view, it actually represents (at best) a local optimum. It makes intuitive sense that the optimal solution is a domain-spanning void; if one’s only desire is to minimize weight and avoid fatigue, then the best solution is to not create any structure in the first place. By including the compliance constraint in \((\mathbb {P}_1)\), the solution x_{e} = 𝜖 for all e is no longer feasible (provided \(\overline {C}\) is small enough). By choosing \(\bar {C}\) such that the compliance constraint is not active at the solution, we can solve the problem of minimizing mass subject to a fatigue constraint without running the risk of obtaining a solution which is degenerate from an engineering perspective.
5 Sensitivity analysis
Sensitivity analyses deal with finding derivatives of the objective function and constraints with respect to the design variables x. This forms the core of any gradient-based optimization method. We use adjoint sensitivity analysis for computational efficiency. Similar to elasto-plastic models, the predicted damage has history dependence, which is reflected in the sensitivity analysis (Panagiotis et al. 1994). The expression for the sensitivity of the compliance in both the TO problems is well known (Christensen and Klarbring 2009), while the sensitivity of the fatigue constraint is elaborated upon in the following.
5.1 Aggregation function sensitivity
5.2 Adjoint sensitivity analysis
5.3 Fatigue sensitivity
6 Numerical examples
The proposed method is implemented in the in-house finite element program TRINITAS (Torstenfelt 2012). We consider an isotropic material, namely AISI-SAE 4340 alloy steel with Young’s modulus 210 GPa and Poisson’s ratio 0.33. As mentioned in Section 2, the damage D increases from D = 0 to D = D_{crit} = F(1) and using these conditions, the fatigue model parameters are calibrated against the Wöhler curves for different mean stresses following the fitting procedure shown in Ottosen et al. (2008). The fitted material properties for AISI-SAE 4340 alloy steel are as follows: S_{0} = 490 MPa, A = 0.225, C = 1.25, K = 2.65 ⋅ 10^{− 5} and L = 14.5.
6.1 L-shaped beam with periodic load history
Since the applied load history has a constant amplitude with a zero mean stress, and also the estimated damage is a function of stresses, we expect a similar topology when solving the optimization problem with fatigue constraints as when solving with stress constraints. To demonstrate this, we compare the solution of the mass minimization TO problem with fatigue constraint \((\mathbb {P}_{1})\) to the solution of the stress-constrained TO problem, as treated in Holmberg et al. (2013).
For the problem \((\mathbb {P}_{1})\), we set the maximum fatigue damage \(\bar {{D}} = 0.4\) along with compliance bound \(\bar {C} = 0.4\text {E}-2\) Nmm. The maximum stress obtained after solving this problem is used as the stress limit in the stress-constrained problem.
Fatigue constraint vs stress constraint: mass minimization problem with fatigue constraint \((\mathbb {P}_{1})\) and stress constraint
Topology | Stress | Damage | |
---|---|---|---|
Fatigue constraint | |||
\(\sigma _{\max \limits } = 615\) MPa | \(D_{\max \limits } = 0.31\) | ||
Stress constraint | |||
\(\sigma _{\max \limits } = 616\) MPa | \(D_{\max \limits } = 0.36\) |
Optimization results of the L-shaped beam with periodic load history
Topology | Objective function | Fatigue constraint | |
---|---|---|---|
\((\mathbb {P}_{1})\) | |||
Mass = 22.6 × 10^{− 3} kg | D^{PN} = 0.39 | ||
\((\mathbb {P}_{2})\) | |||
Compliance = 0.33E − 2 Nmm | D^{PN} = 0.5 |
6.2 L-shaped beam with non-periodic load history
For the next example, we again use the L-shaped beam shown in Fig. 2. However, for the second load case, a non-periodic load history, σ(t) = σ(Q_{1})S_{f}(t), t = 0,0.1,...,100, is used for fatigue estimation. Here S_{f}(t) is a pseudorandom function depicted in Fig. 3.
Optimization results of the L-shaped beam with non-periodic load history
Topology | Objective function | Fatigue constraint | |
---|---|---|---|
\((\mathbb {P}_{1})\) | |||
Mass = 21 × 10^{− 3} kg | D^{PN} = 0.501 | ||
\((\mathbb {P}_{2})\) | |||
Compliance = 0.42E − 2 Nmm | D^{PN} = 0.62 |
6.3 MBB beam with out-of-phase load history
Problem \((\mathbb {P}_{2})\) is solved for three different phase angles ϕ = 0^{∘},45^{∘}, and 90^{∘} and fatigue bounds \(\bar {{D}} = 0.35, 0.35\), and 0.50, respectively. The reason for setting a higher fatigue bound for ϕ = 90^{∘} is that, with out-of-phase loading history, the model predicts that the fatigue strength increases, i.e., fatigue damage decreases, with the phase difference; thereafter, a decrease of the fatigue limit is predicted, i.e., fatigue damage increases. This tendency is in accordance with experimental data (Liu and Zenner 2003).
Optimization results of MBB beam with different phase angles for the problem \((\mathbb {P}_{2})\)
ϕ = 0^{∘} | ϕ = 45^{∘} | ϕ = 90^{∘} | ||
---|---|---|---|---|
Optimized results | ||||
Objective function | ||||
Fatigue constraint | ||||
Compliance | 0.1E-1 Nmm | 0.86E-2 Nmm | 0.68E-2 Nmm |
7 Conclusion and outlook
A new gradient-based TO formulation with fatigue constraint based on a continuous-time approach for fatigue calculation was proposed. As the fatigue damage is integrated for the whole stress history, cycle-counting techniques like rainflow counting were not used. The presented model can handle a general load history, that also includes non-proportional loads.
The fatigue sensitivities were derived using an adjoint method. Since this approach has history dependence, the adjoint variables are obtained by solving a discrete terminal value problem. The validity of the fatigue sensitivities derived by the adjoint method were verified by comparing against a finite difference calculation. The proposed method was tested on several numerical examples, namely, the L-shaped beam with periodic and non-periodic loads, and the MBB beam geometry with out-of-phase loading. Although we have numerically solved only 2-D models, the theoretical presentation is not restricted to these cases. However, the high computational cost we experience makes it presently difficult to treat 3-D models in practice. An important extension of the work is, therefore, to develop acceleration techniques that can improve the overall performance and shorten the computational time. One possibility is to discretize the time domain using a higher order scheme, which could enable us to use larger time steps. However the sensitivity analysis becomes more elaborate, increasing the number of floating-point operations needed for each time step. Thus, it is difficult to make an a priori prediction of the relative efficiency of different order implementations.
8 Replication of results
To reproduce the above optimized results, a first step is to implement the fatigue model following the box in Section 3. The optimization problems \((\mathbb {P}_{1})\) and \((\mathbb {P}_{2})\) can be solved using the MMA method from Svanberg (1987). The sensitivities are derived in Section 5 and the verification study is shown in Appendix . Relevant parameters are given in Section 6.
Notes
Funding information
Open access funding provided by Linköping University. The work was performed within the AddMan project, funded by the Clean Sky 2 joint undertaking under the European Unions Horizon 2020 research and innovation programme under grant agreement No 738002 and within the Centre for Additive Manufacturing-Metal (CAM^{2}) financed by Sweden’s innovation agency under grant agreement No 2016-05175.
Compliance with ethical standards
Conflict of interests
The authors declare that there is no conflict of interest with the publication of this manuscript.
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