PSI - Issue 34

Riccardo Caivano et al. / Procedia Structural Integrity 34 (2021) 221–228 Riccardo Caivano et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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the defect population paraments can be accessed only after the component production using non-destructive inspection techniques. Therefore, the fatigue limit would be evaluable only after the component realization when no further modifications are practicable. To get access to this information during the design phase, a statical model can be employed to predict the defect population in the final part volume. Murakami employed in (Yukitaka Murakami 2002) the Largest Extreme Value Distribution (LEVD) to assess the critical defect size. In particular, the probability √ to find a defect with size √ is given by: √ (√ ) = − −( √ − √ √ ) (2) where √ and √ are the location and the scale parameters of the defect size distribution, which can be estimated experimentally for a reference volume 0 . With easy passages and by rewriting Eq. 2 as a function of the component volume , times larger than 0 ( = ∙ 0 ), the fatigue limit in presence of defects for the component volume is given by Eq.(3). ̄ = 1 ∙ ( + 120) ( √ + √ ∙ (− (− ( )) + ( 0 ))) 1 6 ∙ ( 1 − 2 ) 0.226+ ∙10 −4 (3) Since HyperWorks provides the possibility to limit only the maximum first principal stress ̄ , and not the alternate one ̄ as presented in TopFat (Gao et al. 2021), Eq.(4) is used in place of Eq.(3) (S. Suresh 1998). ̄ , = 2 ̄ 1 − (4) 3. Results In the following Section, all the test cases reported and analysed with the TopFat original algorithm in (Gao et al. 2021) are reproduced with HyperWorks TO tool for benchmarking validation. The first benchmark is the L-shape geometry domain shown in Fig. 1a. The upper edge is locked whereas a downward force F of 800 N is applied to the five closest nodes to the right upper corner to avoid artificial stress concentration. The second benchmark consists of the classical Cantilever geometry domain as reported in Fig. 1b. The left edge is fully locked, and a downward force F of 1000 N is applied to the thirteen closest nodes to the middle of the right edge. Lastly, Fig. 1c shows the third benchmark geometry, i.e. the Corbel geometry domain. In this case, both the upper and lower edges are locked while a downward force F of 3500 N is applied to the eleven closest nodes to the quoted in the Figure. The mesh is obtained for all the cases with square second-order elements with a side length of 1 mm, i.e. 6400 elements for the L-shape domain, 20000 elements for the Cantilever domain and 17700 elements for the Corbel domain. In the original TopFat algorithm, a hybrid stress element model (Pian and Sumihara 1984) is used to improve the accuracy of the stress measure evaluation without adding extra nodes. However, this element model is not implemented in the HyperWorks environment, so second-order elements are used to have accurate stress measures. While the accuracy of both element models is almost identical, the use of second-order elements rises the computational cost as a side effect. The filtering radius is set equal to 1.2 mm as imposed in the TopFat original algorithm. However, in the TopFat original algorithm both a continuation method and a non-linear projection are employed to avoid local minima and obtain cleaner final topologies (Gao et al. 2020; O. Sigmund and Petersson 1998). These routines are not included in the HyperWorks environment and surely this discrepancy affects the results as it will be shown thereafter. To overcome these differences, all the optimisations in HyperWorks are carried out with a convergence parameter set equal to 10 -5 , while 10 -2 in TopFat original algorithm, and without a limit on the maximum number of iterations. Indeed, for the analysed benchmarks in HyperWorks, if the convergence parameter was set equal to 10 -2 , it would not be enough to reach a clear and definite final topology. Another difference consists of the type of programming solver: in TopFat original