守恒线性与角动量的参数化弹性问题神经网络求解器
Neural network solvers for parametrized elasticity problems that conserve linear and angular momentum
AI总结：
针对具有鞍点结构的参数化弹性问题混合公式求解成本高的问题，提出基于应力分解的策略，结合神经网络修正且不破坏动量守恒，通过含非线性模型的算例验证了潜力。
摘要：
我们研究了以应力、位移和旋转表示的参数化弹性问题的混合公式。后两个变量作为拉格朗日乘子，用于强制满足线性动量和角动量守恒。由于该系统具有鞍点结构，直接求解的计算量很大，因此我们提出了一种基于应力变量分解的高效求解策略。首先，求解一个三角方程组，得到能平衡体力与边界力的应力场。其次，利用训练好的神经网络提供修正项，且不会破坏守恒方程。如有需要，可通过后处理得到位移和旋转。我们通过三个数值算例（包括一个非线性模型）展示了该方法的潜力。
英文摘要：
We consider a mixed formulation of parametrized elasticity problems in terms of stress, displacement, and rotation. The latter two variables act as Lagrange multipliers to enforce conservation of linear and angular momentum. Due to the saddle-point structure, the resulting system is computationally demanding to solve directly, and we therefore propose an efficient solution strategy based on a decomposition of the stress variable. First, a triangular system is solved to obtain a stress field that balances the body and boundary forces. Second, a trained neural network is employed to provide a correction without affecting the conservation equations. The displacement and rotation can be obtained by post-processing, if necessary. The potential of the approach is highlighted by three numerical test cases, including a non-linear model.
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