Data-driven methods are becoming an essential part of computational mechanics due to their unique advantages over traditional material modeling. Deep neural networks are able to learn complex material response without the constraints of closed-form approximations. However, imposing the physics-based mathematical requirements that any material model must comply with is not straightforward for data-driven approaches. In this study, we use a novel class of neural networks, known as neural ordinary differential equations (N-ODEs), to develop data-driven material models that automatically satisfy polyconvexity of the strain energy function with respect to the deformation gradient, a condition needed for the existence of minimizers for boundary value problems in elasticity. We take advantage of the properties of ordinary differential equations to create monotonic functions that approximate the derivatives of the strain energy function with respect to the invariants of the right Cauchy-Green deformation tensor. The monotonicity of the derivatives guarantees the convexity of the energy. The N-ODE material model is able to capture synthetic data generated from closed-form material models, and it outperforms conventional models when tested against experimental data on skin, a highly nonlinear and anisotropic material. We also showcase the use of the N-ODE material model in finite element simulations. The framework is general and can be used to model a large class of materials. Here we focus on hyperelasticity, but polyconvex strain energies are a core building block for other problems in elasticity such as viscous and plastic deformations. We therefore expect our methodology to further enable data-driven methods in computational mechanics
翻译:深神经网络能够学习复杂的物质反应,而不受封闭式组合近似的限制。然而,任何材料模型必须遵守的基于物理的数学要求对于数据驱动的方法来说并不是直截了当的。在本研究中,我们使用新型的神经网络类,称为神经普通差异方程式(N-ODEs),来开发数据驱动的材料模型,自动满足与传统材料模型相比的压力能量功能的多变异性。深神经网络能够学习复杂的物质反应,而不受封闭式梯度的极限值问题所需的条件。我们利用普通差异方程式的特性来创造单调功能,以近似任何材料模型的变异性。衍生物的单调能保证能源的凝固性。因此,N-ODE材料模型能够捕捉到从封闭式材料模型中产生的合成性数据,但在测试实验性软性软性软性软性结构时,在模型中可以使用高水平的硬质能量结构模型,而在模型中可以使用高水平的硬性材料框架。