Fig. 3: Multi-level physics-informed neural network (ml-PINN) for 2D shell.
a Diagrammatic sketch of shell structures. b Physical relationship between the relevant variables that transform the uniform load q into the vertical displacement ω by utilizing the three typical equations. Mx and My represent the bending moment of shell structures in the x-direction and y-direction, respectively. Mxy denotes the torque in the z-direction. The vertical displacement is in the z-direction. For the direction, curvature (κx, κy, κxy) is similar to the bending moment (Mx, My, Mxy). The intermediate variables including shear force FQ, curvature κ and displacement ω are outputted from each neural network. c Schematics of ml-PINN framework for shell structures. Lb() and Lp() represent the loss functions from boundary constraints and loss functions from PDE, respectively. Lp(ω”, κ), Lp(κ, M) and Lp(M”, q) are the loss functions corresponding to PDEs. Lb(ω), Lb(ω’), and Lb(M) are the loss functions of boundary constraints from the vertical displacement, section turning angle, and bending moment, respectively.
