预测脆性材料裂纹路径的物理变分DeepONet
A physics-informed variational DeepONet for predicting the crack path in brittle materials
摘要：
故障轨迹，识别可能的故障区，和损伤统计是脆性断裂应用中的一些关键的相关量。存在可靠地估计这些相关量的高保真数值求解器，但它们在计算上要求很高，需要高分辨率的裂纹。此外，即使对于域参数和/或材料性质的微小变化，也需要进行独立的密集模拟。因此，需要快速和通用的代理模型，以减轻计算负担，但断裂力学的不连续性提出了一个重大的挑战，发展这样的模型。我们提出了一个物理信息变分公式的DeepONet（V-DeepONet）的脆性断裂分析。V-DeepONet被训练以将缺陷的初始配置映射到相关的感兴趣领域（例如，损伤和位移场）。一旦网络被训练，整个全球解决方案可以迅速获得任何初始裂纹配置和加载步骤上的域。虽然原始的DeepONet完全是数据驱动的，但我们采取了不同的路径来训练V-DeepONet，通过以变分形式施加控制方程，并且我们还使用一些标记数据。我们通过脆性断裂的两个基准测试证明了V-DeepOnet的有效性，并使用高保真解算器的结果验证了其准确性。考虑到裂缝建模对波动非常敏感，对物理定律和一些数据进行编码以训练网络使得代理模型能够准确地执行内插和外推任务。V-DeepONet的混合训练优于最先进的方法，可以应用于各种具有复杂响应的动态系统。
英文摘要：
Failure trajectories, identifying the probable failure zones, and damage statistics are some of the key quantities of relevance in brittle fracture applications. High-fidelity numerical solvers that reliably estimate these relevant quantities exist but they are computationally demanding requiring a high resolution of the crack. Moreover, independent intensive simulations need to be carried out even for a small change in domain parameters and/or material properties. Therefore, fast and generalizable surrogate models are needed to alleviate the computational burden but the discontinuous nature of fracture mechanics presents a major challenge to developing such models. We propose a physics-informed variational formulation of DeepONet (V-DeepONet) for brittle fracture analysis. V-DeepONet is trained to map the initial configuration of the defect to the relevant fields of interests (e.g., damage and displacement fields). Once the network is trained, the entire global solution can be rapidly obtained for any initial crack configuration and loading steps on that domain. While the original DeepONet is solely data-driven, we take a different path to train the V-DeepONet by imposing the governing equations in variational form and we also use some labelled data. We demonstrate the effectiveness of V-DeepOnet through two benchmarks of brittle fracture, and we verify its accuracy using results from high-fidelity solvers. Encoding the physical laws and also some data to train the network renders the surrogate model capable of accurately performing both interpolation and extrapolation tasks, considering that fracture modeling is very sensitive to fluctuations. The proposed hybrid training of V-DeepONet is superior to state-of-the-art methods and can be applied to a wide array of dynamical systems with complex responses.
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