On the Gauss–Legendre quadrature rule of deep energy method for one-dimensional problems in solid mechanics

Deep energy method (DEM) has shown its successes to solve several problems in solid mechanics recently. It is known that determining proper integration scheme to precisely calculate total potential energy (TPE) value is crucial to achieve high-quality training performance of DEM but it has not been discovered satisfactorily in previous related works. To shed light on this matter, this study focuses on investigating the application of Gauss–Legendre (GL) quadrature rule in training DEM to solve one-dimensional (1D) solid mechanics problems. The technical idea of this work is (1) to design a theoretical polynomial regression (PR) model via Taylor series expansion that could well-approximate multi-layer perceptron (MLP) output and its derivatives for fully capturing the representation of DEM solution, and then (2) to extract the polynomial order of the TPE loss function via the devised PR to calculate the necessary number of GL points for training DEM. To do so, mathematical analyses are firstly developed to find out the representability of DEM for geometrically nonlinear beam bending problem as a case study and the convergence of the alternative PR to the MLP with tanh activation function, providing theoretical foundations for utilizing the PR to take the place of DEM network. Subsequently, minimum number of GL points are analytically extracted and a technical framework for estimating the maximin required GL points is devised to accurately compute the TPE loss function for ensuring DEM training convergence. Several 1D linear and nonlinear beam bending examples using both Euler–Bernoulli (EB) and Timoshenko theories with various types of boundary conditions (BCs) are selected to examine the proposed method in practice. The numerical results validate the preciseness of the developed theory and the empirical effectiveness of the devised framework.