PSI - Issue 28

Florian Vlădulescu et al. / Procedia Structural Integrity 28 (2020) 637–647 Vl ă dulescu and Constantinescu / Structural Integrity Procedia 00 (2019) 000–000

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same geometry. Since it is not always possible to accurately estimate the yielding of the optimized component, it may be necessary to start the optimization with increasingly tighter constraints before obtaining a desired result from the lattice optimization. The finite element model is easily obtained, by using ANSYS (2019a); the finite element type Solid 185 of the first order allows several geometric shapes and thus a hybrid finite element model of equal element size. Performing a mesh convergence study, Vlădulescu and Constantinescu (2020) increased the element size from 2 to 14 mm. For an element size of 8 mm compared with the one of 2 mm the number of nodes decreased almost 43 times (from 767630 to 17945) and the number of elements about 41 times (from 816801 to 19962); at the same time, increasing the element size, the fundamental frequency increased only with less than 2%, from 823.59 Hz to 839.06 Hz. Therefore, we decided to choose further the 8 mm equal size for the solid element. The initial meshing of the bracket is shown in Fig. 2 in a lateral view and in a cross section.

Fig. 2. Initial finite element model.

4. Modal analysis Determination of the natural frequencies and the corresponding modes shape of the structural components can be realized through a modal analysis (ANSYS 2019b). Modal analysis is a linear analysis. Any nonlinearities, such as material plasticity or nonlinear contacts, are ignored, even if defined. The established fixing conditions of the bracket are used for the modal analysis. We used a Block Lanczos algorithm for computing a few of the smallest eigenvalues and the corresponding eigenvectors of the symmetric matrix. The Block shifted Lanczos algorithm is a variation of the classical Lanczos (1950) algorithm, where the Lanczos recursions are performed using a block of vectors, as opposed to a single vector. The Block Lanczos eigenvalue extraction method can be used for large symmetric eigenvalue problems and employs an automated shift strategy to extract the number of eigenvalues requested. This method is especially powerful when searching for eigenfrequencies in a given part of the eigenvalue spectrum. The convergence rate of the eigenfrequencies will be about the same when extracting modes in the midrange or in the higher range of the spectrum, as when extracting the lowest modes. First 6 natural frequencies were extracted for this unoptimized model. As mesh size is reduced the natural frequencies decrease a little bit, with no more than 2%, but we should mention that a warning was given during solving as mesh size was too small. So, we kept the 8×8×8 mm element size for further analysis. As already mentioned, the first natural frequency has a value of 839 Hz (Fig. 3) with a maximum total deformation at the top of the bracket of 14.2 mm. 5. Lattice type optimization and homogenization A special type of the topological optimization process is the lattice type optimization, considered here as being in the first stage of analysis. In this type of optimization, the model is "filled" with a structure optimized with lattices, the solid elements being replaced by the interconnected lattices forming a common body. Lattice type optimization works only for solids. As mentioned by Jansen and Pierard (2020), incorporating lattice structures in a model of the