隐式拉格朗日流体力学与高阶有限元
Implicit Lagrangian Hydrodynamics with High-Order Finite Elements
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中文总结 AI 辅助
本文提出一种基于高阶曲线有限元拉格朗日流体力学的隐式时间积分方法，利用MFEM和自动微分实现牛顿-克雷洛夫求解，在多个基准问题上验证了高阶收敛性和强激波行为，并显著提升三重点问题的精度-时间比。
中文摘要 AI 辅助
我们提出了一种用于高阶曲线有限元拉格朗日流体力学的隐式时间积分能力。从现有的显式公式出发，隐式处理直接建立在原始离散化和算子结构之上，不改变底层空间公式或物理模型。我们使用基于MFEM有限元库构建的Laghos迷你应用来演示我们的实现。为了支持基于梯度的非线性求解方法，我们利用MFEM的$\partial$FEM接口结合基于Enzyme的自动微分自动计算雅可比作用，并在牛顿-克雷洛夫求解器中以无矩阵或完全组装的方式应用所得的雅可比矩阵。为了确保在存在激波时稳健且可微的非线性求解，我们引入了一种基于不可微逐点运算的光滑近似的平滑人工粘性处理。目前可微的人工粘性缺乏限制器以确保远离激波时的高阶缩放，但足以说明隐式拉格朗日流体力学的优势。隐式方法的性能和表现通过几个标准基准问题进行了演示。我们在无人工粘性的光滑泰勒-格林涡上验证了高阶收敛性，在Sedov爆炸问题上显示了正确的强激波行为，并在三重点问题上获得了显著的每时间到解精度的改进，其中显式稳定性约束对高阶离散化变得越来越严格。
英文摘要
We present an implicit time integration capability for high-order curvilinear finite element Lagrangian hydrodynamics. Starting from an existing explicit formulation, the implicit treatment builds directly on the original discretization and operator structure and does not alter the underlying spatial formulation or physics model. We demonstrate our implementation using the Laghos miniapp, which is built on the MFEM finite element library. To support gradient-based nonlinear solution methods, we compute Jacobian actions automatically using MFEM's $\partial$FEM interface together with Enzyme-based automatic differentiation, and apply the resulting Jacobian in a matrix-free or fully-assembled manner within a Newton-Krylov solver. To ensure robust and differentiable nonlinear solves in the presence of shocks, we introduce a smooth artificial viscosity treatment based on smooth approximations of non-differentiable pointwise operations. The differentiable artificial viscosity presently lacks a limiter to ensure high-order scaling away from shocks, but is sufficient for illustrating the benefits of implicit Lagrangian hydrodynamics. The behavior and performance of the implicit method are demonstrated on several standard benchmark problems. We verify high-order convergence on the smooth Taylor-Green vortex in the absence of artificial viscosity, show correct strong-shock behavior on the Sedov blast problem, and obtain significant improvements in accuracy-per-time-to-solution on the Triple Point problem where explicit stability constraints become increasingly severe for high-order discretizations.
发表机构
- Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
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