Selection of beam, plate, and shell theories using an axiomatic/asymptotic method and neural networks

Using 1D and 2D theories for structural problems remains desirable to reduce computational costs. Beam (1D), plate, and shell (2D) elements require fewer degrees of freedom than 3D ones as their aspect ratio constraints are less demanding. The accuracy of 1D and 2D theories depends on the assumptions retained to build the primary unknown fields, e.g., the classical models for composites are the Classical Lamination Theory (CLT) and the First-Order Shear Deformation Theory (FSDT) do not consider transverse stretching. Over the last decades, many refined 1D and 2D models have been developed to overcome the limitations of classical theories and reduce the accuracy gap with 3D elements. The present work presents the latest developments concerning the synergistic use of the Carrera Unified Formulation (CUF), the Axiomatic/Asymptotic Method (AAM), and Neural Networks (NN) to develop 1D and 2D models. CUF generates any-order theory's governing equations and finite element (FE). AAM evaluates the influence of each generalized unknown variable on the solution to retain only those variables influencing the solution, reducing the number of degrees of freedom. NNs are used to explore the multitude of combinations of generalized variables and problem parameters, e.g., geometry, boundary conditions, material systems, and obtain the best theories for given input parameters. For the first time, this paper extends the approach to beam models for dynamic and static analyses. Various cross-section geometries are considered, and natural frequencies, modal shapes, and stress fields are evaluated. A displacement-based formulation is adopted in which the three displacement components are expanded over the cross-section using polynomials and CUF. AAM is then used to generate reduced models as accurate as 3D elements, and such models are then used to train NN. Trained NN accepts as input parameters, e.g., cross-section geometry or elastic properties, and provides the structural theory to use to fulfill the 3D accuracy requirement.