Geometrically Nonlinear Dynamic Analysis of FG-TPMS Plates on a Pasternak Foundation Subjected to Moving Loads using the MEM Method

A Geometrically Nonlinear (GNL) dynamic model, for Functionally Graded Triply Periodic Minimal Surface (FG-TPMS) plates influenced by moving loads on a Pasternak foundation, is developed in this paper. The FG-TPMS core is described by Primitive, Gyroid, and I-graph/Wrap-Package (IWP) topologies with through-thickness relative-density gradation, enabling spatial tuning of effective elastic and shear modulus. Plate kinematics follow Mindlin theory. Within a total Lagrangian framework, Green–Lagrange strains and second Piola–Kirchhoff stresses are used to formulate the governing equations. Geometric stiffness terms are included to capture large-displacement effects. Mesh-load relocation and domain-size limitations of the Finite Element Method (FEM) are circumvented by a load-attached coordinate transformation within the Moving Element Method (MEM). Nine-node isoparametric elements are used, with the time integration and stepwise nonlinear equilibrium achieved via Wilson and Newton–Raphson schemes. The formulation accommodates foundation normal/shear stiffness, and a concentrated moving load with general velocity. The approach is designed to quantify the influence of density-gradation parameters, TPMS topology, plate thickness, and load velocity on the GNL dynamic response, and to benchmark FG-TPMS plates. The resulting framework provides an efficient and robust tool for analyzing lightweight architected plates under realistic moving-load conditions.