Towards a space reduction approach for efficient structural shape optimization

A manifold learning-based reduced order model for springback shape characterization and optimization in sheet metal forming

“…2), so that the set of all admissible level set functions are connected together. This 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 fundamental hypothesis was previously made by Raghavan et al [13,14]. The desired final shape (by the manufacturer, etc.)…”

“…In the field of metal forming, [29] presented an approach for displacement field approximation using the Proper Orthogonal Decomposition (POD) combined with kriging interpolation of projection coefficients. The authors have presented shape space meta-models for a variety of industrial problems [30,31,13] ending finally in the α-manifold [14]. Around the same time, [32,33] developed similar ideas of "slow manifolds" for the reduction of the output space of a problem in elastodynamics.…”

“…In order to numerically evaluate the springback and to be able to characterize complex shapes, we need a universal and case-independent technique to find the smallest number of parameters needed to fully describe the final shape obtained regardless of complexity, and easily compare it with the desired geometry. The first effort was made by the authors in [12] using their previously introduced "shape manifold" concept [13,14] for the simple NUMISHEET 93 benchmark problem of 2-D draw bending. Here, the concept of an "admissible shape" for a forming process was introduced for the first time to distinguish beween realizable/attainable post-springback shapes and idealized shapes for a given drawing process, and the notion of interpolation between admissible shapes was introduced.…”

A volumetric geometry and topology parameterisation for fluids-based optimisation

“…The first approach, proposed by Diez et al [4,5], is to randomly sample a wide range of geometries constructed via the chosen parameterisation and use a Karhunen-Loève expansion (although other matrix approaches may also be used [6]) to find a series of eigenvectors and values ordered in terms of decreasing importance. From this, a subset of the 'early' eigenvectors may then be used as design variables, and this approach has been applied for marine hull design [4,5] using free-form deformation.…”

Identification of material parameters using indentation test —study of the intrinsic dimensionality of P-h curves and residual imprints

“…More recently, due to the significant development of manifold learning algorithms and its applications in the field of computational and applied mechanics, biomechanics, and especially in material science [14][15][16][17][18], our most recent work [19] recommended that the identification procedure be carried out within a α-space constructed by using only the residual imprint of an indentation test. The core idea is based on the fundamental hypothesis that, there exists a low-dimensional space in which the original higherdimensional data could be embedded and the identification is implemented over a manifold defined within the constructed shape space.…”