Deformation clustering methods for topologically optimized structures under crash load based on displacement time series

Multi-objective Topology Optimization has been receiving more and more attention in struc- tural design recently. It attempts to maximize several performance objectives by redistributing the ma- terial in a design space for a given set of boundary conditions and constraints, yielding many Pareto- optimal solutions. However, the high number of solutions makes it difficult to identify preferred designs. Therefore, an automatic way of summarizing solutions is needed for selecting interesting designs ac- cording to certain criteria, such as crashworthiness, deformation, and stress state. One approach for summarization is to cluster similar designs and obtain design representatives based on a suitable met- ric. For example, with Euclidean distance of the objective functions as the metric, design groups with similar performance can be identified and only the representative designs from different clusters may be analyzed. However, previous research has not dealt with the deformation-related time-series data of structures with different topologies. Since the non-linear dynamic behavior of designs is important in various fields such as vehicular crashworthiness, a clustering method based on time-dependent behav- ior of structures is proposed here. To compare the time-series displacement data of selected nodes in the structure and to create similarity matrices of those datasets, euclidean metrics and Dynamic Time Warping (DTW) are introduced. This is combined with clustering techniques such as k-medoids and Ordering Points To Identify the Clustering Structure (OPTICS), and we investigate the use of unsuper- vised learning methods to identify and group similar designs using the time series of nodal displacement data. In the first part, we create simple time-series datasets using a mass-spring system to validate the proposed methods. Each dataset has predefined clusters of data with distinct behavior such as different periods or modes. Then, we demonstrate that the combination of metrics for comparison of time series (Euclidean and DTW) and the clustering method (k-medoids and OPTICS) can identify the clusters of similar behavior accurately. In the second part, we apply these methods to a more realistic, engineer- ing dataset of nodal displacement time series describing the crash behavior of topologically-optimized designs. We identify similar structures and obtain representative designs from each cluster. This reveals that the suggested method is useful in analyzing dynamic crash behavior and supports the designers in selecting representative structures based on deformation data at the early stages of the design process.