This work proposes a model-reduction approach for the material point method on nonlinear manifolds. Our technique approximates the $\textit{kinematics}$ by approximating the deformation map using an implicit neural representation that restricts deformation trajectories to reside on a low-dimensional manifold. By explicitly approximating the deformation map, its spatiotemporal gradients -- in particular the deformation gradient and the velocity -- can be computed via analytical differentiation. In contrast to typical model-reduction techniques that construct a linear or nonlinear manifold to approximate the (finite number of) degrees of freedom characterizing a given spatial discretization, the use of an implicit neural representation enables the proposed method to approximate the $\textit{continuous}$ deformation map. This allows the kinematic approximation to remain agnostic to the discretization. Consequently, the technique supports dynamic discretizations -- including resolution changes -- during the course of the online reduced-order-model simulation. To generate $\textit{dynamics}$ for the generalized coordinates, we propose a family of projection techniques. At each time step, these techniques: (1) Calculate full-space kinematics at quadrature points, (2) Calculate the full-space dynamics for a subset of `sample' material points, and (3) Calculate the reduced-space dynamics by projecting the updated full-space position and velocity onto the low-dimensional manifold and tangent space, respectively. We achieve significant computational speedup via hyper-reduction that ensures all three steps execute on only a small subset of the problem's spatial domain. Large-scale numerical examples with millions of material points illustrate the method's ability to gain an order of magnitude computational-cost saving -- indeed $\textit{real-time simulations}$ -- with negligible errors.
翻译:这项工作为非线性元体的物质点法提出了一种模型性递减方法。与典型的模型性递减技术相比,我们的技术通过使用隐含的神经代表,将变形轨迹限制在低维元数上,以近似于变形图的变形轨迹。通过明确接近变形图,可以计算出其波形梯度梯度梯度,特别是变形梯度和速度。与典型的模型性递减技术形成一个线性或非线性元数,以近于(最小数量的)自由度,以显示给定的空间离散,使用隐含的神经代表,使拟议方法将变形轨迹轨迹限制在低维维维度的图上。因此,该技术支持动态离异化,包括解析变化,在在线降序模型模拟过程中, 生成 美元或液态{动力值} 以(最小的) 自由度度度度度度度, 使一个空间变异度变异度变速率级数(最小数), 我们提议用一个全空间变速法阶的系统,每个直径直径直地计算。