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This paper presents a graph-manifold iterative algorithm to predict the configurations of geometrically exact shells subjected to external loading. The finite element solutions are first stored in a weighted graph where each graph node stores the nodal displacement and nodal director. This collection of solutions is embedded onto a low-dimensional latent space through a graph isomorphism encoder. This graph embedding step reduces the dimensionality of the nonlinear data and makes it easier for the response surface to be constructed. The decoder, in return, converts an element in the latent space back to a weighted graph that represents a finite element solution. As such, the deformed configuration of the shell can be obtained by decoding the predictions in the latent space without running extra finite element simulations. For engineering applications where the shell is often subjected to concentrated loads or a local portion of the shell structure is of particular interest, we use the solutions stored in a graph to reconstruct a smooth manifold where the balance laws are enforced to control the curvature of the shell. The resultant computer algorithm enjoys both the speed of the nonlinear dimensional reduced solver and the fidelity of the solutions at locations where it matters. 

This study evidences that the particle surface-area-to-volume ratio (A/V) and the particle volume (V) have the key information of particle geometry and the ‘signature’ is realized by a power-law relationship between A/V and V in a form of V = (A/V)^α × β. We find that the power value α is influenced by the shape-size relationship while the β* term (β evaluated with a fixed value of α = -3) informs the average particle shape of a granular material regarding the overall angularity. This study also discusses how the particle shape can be retrieved in terms of Wadell’s true sphericity using the A/V and V. This concept is linked to another shape index M that interprets the particle shape as a function of surface area A, volume V, and size L. This paper explains the analytical aspects of geometric ‘signature’ and examines the idea using the example particles to address the DEM modelling-related questions.