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

翻译：本文从数值分析角度研究用于求解线弹性方程的深度Ritz方法。为处理（本质）狄利克雷边界条件，我们针对线弹性方程引入并分析了一种采用$H^{1/2}(\Gamma_D)$范数的改进Ritz形式。研究表明，由此产生的深度Ritz方法能够在深度神经网络函数集合中，就"能量"范数而言提供最优逼近。此外，我们证明了深度Ritz模拟的总误差由网络逼近误差与数值积分误差之和界定（忽略代数误差）。为有效控制数值积分误差，我们提出了一种基于残差局部误差指标的自适应求积数值积分技术。该方法能够高效逼近改进后的能量泛函。通过与光滑问题、奇异问题及应力集中问题的数值实验，验证了所提出的自适应求积深度Ritz方法的有效性与高效性。