基于深度学习技术的薄板非线性弯曲多项式级数解法

Polynomial series solution method for nonlinear bending of thin plates based on deep learning technique

摘要

基于弹性力学理论，结合深度学习技术，发展了一种求解薄板几何非线性弯曲分析的多项式解法。为满足边界条件并实现快速求导，首先基于符号微分计算并搭建好位移试函数模型，该模型的输入为求解域内任意一点的坐标，并输出与之对应的位移，再考虑基于变分原理的几何非线性弯曲理论，以薄板的系统势能构造目标函数，最后结合自动微分计算目标函数的梯度，并结合Adam优化算法进行参数优化，直至得到最优的参数向量，进而得到薄板位移显式的多项式级数解答。两种微分技术的结合提高了程序的易用性与准确性，本研究求解了不同形状、边界下薄板的几何非线性弯曲问题，分析了不同阶次下多项式级数解答的收敛性，并将本研究解与Abaqus有限元解及文献解进行对比，证明了本方法的有效性和适用性。

Abstract

Based on the theory of elastic mechanics and combined with deep learning technique， a polynomial method for solving geometric nonlinear bending analysis of thin plates is developed. In order to satisfy the boundary conditions and realize rapid derivation， trial function model of the displacement is built based on symbolic differential calculation. The input of the model is the coordinates of any point in the solution domain， and the output is the corresponding displacement. Considering the geometric nonlinear bending theory based on variational principle， the objective function is constructed by the systemic potential energy of thin plate. Last，the gradient of the objective function is calculated by the automatic differentiation， and the parameters are optimized by the gradient descent method until the optimal parameter vector is obtained. Then the explicit polynomial series solution of the displacement is obtained. The combination of the two differential techniques improves the ease of use and accuracy of the program. In this paper， the geometric nonlinear bending problems of thin plates with different shapes and boundaries are solved， and the convergence of polynomial series solutions under different orders is analyzed. The solution is compared with the Abaqus finite element solution and the literature solution， which proves the effectiveness and applicability of the method.

关键词

几何非线性 / 符号微分 / 自动微分 / 多项式级数 / 深度学习

Key words

geometric nonlinearity / symbolic differentiation / automatic differentiation / polynomial series / deep learning

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基金资助

国家自然科学基金资助项目(12162004)

国家重点研发计划资助项目(2019YFC1511103)

广西科技重大专项资助项目(桂科AA18118029)

广西重点研发计划资助项目(桂科AB22036007)