Deep Ritz method with adaptive quadrature for linear elasticity

In this paper, we study the deep Ritz method for solving linear elasticity equation from a numerical analysis perspective. A modified formulation using $H^{1/2}(\Gamma_D)$ norm is introduced and analyzed in order to deal with (essential) Dirichlet boundary condition. We show that resulting provides best approximation among set of neural network (DNN) functions respect ``energy'' norm. Furthermore, demonstrate total error simulation bounded by sum integration error, disregarding algebraic error. To effectively control propose an adaptive quadrature-based technique residual-based local indicator. This approach enables efficient energy functional. Through experiments involving smooth singular problems, as well problems stress concentration, validate effectiveness efficiency proposed quadrature.