Gradient-Based Form Finding for Thin-Walled Engineering Structures
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This work proposes and demonstrates a highly flexible shape optimization/form finding workflow and applies it to thin-walled structures in a lightweight-engineering context. While many previous works build on Finite Element Analysis and the Force Density Method [1], this work starts from an even more general approach: Adaptive gradient descent optimization is performed directly on the vertex coordinates of shell meshes. PyTorch, a mature framework for automatic differentiation and backpropagation, is used to implement the procedure. Thereby, the optimization is seamlessly compatible with current machine learning techniques. This is demonstrated by implementing design space restrictions as Signed Distance Fields (SDF) via neural networks as a particularly efficient general representation [2]. The training loop is shown in the left part of Figure 1, while the right part shows the actual shape optimization loop including inference as part of the loss calculation. Figure 1: The proposed workflow including two optimization loops; shown in blue: inputs; green: optimization results Such an integration of an SDF model with form finding has not been demonstrated before. Also, this work demonstrates the utility of current adaptive gradient descent optimizers. After investigating examples with known solutions based on catenoids, a more realistic engineering example is demonstrated: A housing assembly for a hypothetical differential gearbox is to be designed given the essential components (gears, shafts, bearings). The total mass of the assembly including the mass of the flange connecting the cover to the housing is to be minimized. Given clearances must be maintained between the relevant components during assembly and operation. The proposed workflow solves the problem without posing notable requirements on the initial mesh and computing resources. The resulting mesh can be converted to B-Spline faces and conveniently integrated into a CAD design as shown in Figure 2. Finally, the integration of added manufacturing requirements (exemplified by demolding constraints) is demonstrated by adding further constraints to the problem (not shown in picture). Figure 2: CAD design based on optimization REFERENCES [1] K.-U. Bletzinger and E. Ramm, Structural optimization and form finding of light weight structures, Computers & Structures, Volume 79, Issues 22–25, 2001, Pages 2053-2062, ISSN 0045-7949, https://doi.org/10.1016/S0045-7949(01)00052-9. [2] J.J. Park, P. Florence, J.
