Fig. 5: Computational results for 2D shell structure.

a Diagram of boundaries. ω and θ represent the displacement and rotation angle imposed on the boundaries, respectively. Note that the direction of θ is indicated as a blue arrow according to the right-handed screw rule, and the direction of ω is in the z-direction. b Distribution of collocation points. The random data points include points in the interior zone Ω and points on the boundaries S₁ ~ S₄. c Verification for case 1. Only the opposite pair of boundary displacements are constrained by ω | _{S∈S3, S4} = 0. Note that the three contour plots at the top row from left to right represent the proposed multi-level physics-informed neural network predictions (named PINN in the figure), Finite element method predictions (named FEM in the figure), and the relative error, respectively. The two plots at the bottom present the extracted results of deformation in x-directions and y-directions, respectively (as illustrated by the dotted line in the contour of FEM). d Verification for case 2. All surrounding boundary displacements are defined as ω | _{S∈S1, S2, S3, S4} = 0. e Verification for case 3. The displacement ω | _{S∈S1, S2, S3, S4} = 0 and section corner θ | _{S∈S1, S2, S3, S4} = 0 are imposed on all surrounding boundaries. f Verification for case 4. The displacement ω | _{S∈S1, S2, S3, S4} = 0 and section corner θ | _{S∈S1, S2, S3, S4} = 1 are defined, and the uniform load is given by q = −1.