We present the numerical methods and GPU-accelerated implementation underlying a Total Lagrangian finite element framework for finite-deformation flexible multibody dynamics, introduced in the companion paper [1]. The framework supports 10-node quadratic tetrahedral (T10) elements and ANCF beam and shell elements, with quadrature-based hyperelastic response (St. Venant-Kirchhoff and Mooney-Rivlin) and an optional Kelvin-Voigt viscous stress contribution. Time stepping employs a velocity-based implicit backward-Euler scheme, yielding a nonlinear residual in velocity that couples inertia, internal and external forces, and bilateral constraints. Constraints are enforced via an augmented Lagrangian method (ALM), structured as an outer loop alternating an inner velocity solve with a dual-ascent multiplier update. We introduce a two-stage GPU parallelization strategy for internal force and tangent stiffness evaluation, and provide two inner solvers: a first-order AdamW optimizer and a second-order Newton solver that assembles and factorizes a sparse global Hessian on the GPU using cuDSS. A fixed-sparsity matrix strategy eliminates repeated symbolic analysis and enables efficient numerical refactorization across Newton iterations. For collision detection, we present a GPU-native two-thread asynchronous algorithm operating on triangle soups, avoiding bounding-volume hierarchies entirely. Systematic scaling benchmarks across all three supported element types and six mesh resolutions show that the Newton solver achieves approximately one order of magnitude reduction in real-time factor relative to CPU baselines at the largest resolutions tested. The frictional contact model is validated against closed-form rigid-body predictions through quasi-static and dynamic impact unit tests.

翻译：本文介绍了伴随论文[1]中提出的用于有限变形柔性多体动力学的总拉格朗日有限元框架的数值方法及GPU加速实现。该框架支持10节点二次四面体（T10）单元以及ANCF梁和壳单元，采用基于正交的超弹性响应（圣维南-基尔霍夫和穆尼-里夫林模型），并可选配开尔文-沃伊特粘性应力贡献。时间推进采用基于速度的隐式向后欧拉格式，形成耦合惯性力、内力、外力及双边约束的速度非线性残差。通过增广拉格朗日方法（ALM）施加约束，该算法结构化为外层循环，交替执行内层速度求解和对偶上升乘子更新。我们提出了一种两阶段GPU并行化策略用于内力和切线刚度评估，并提供两种内层求解器：一阶AdamW优化器和二阶牛顿求解器，后者使用cuDSS在GPU上组装并分解稀疏全局黑塞矩阵。固定稀疏性矩阵策略避免了重复的符号分析，并在牛顿迭代中实现了高效的数值重分解。针对碰撞检测，我们提出了一种基于GPU的双线程异步算法，直接操作三角形网格，完全避免了包围体层次结构。对三种支持单元类型及六种网格分辨率的系统缩放基准测试表明，在最大测试分辨率下，牛顿求解器相比CPU基线实现了约一个数量级的实时因子缩减。通过准静态和动态冲击单元测试，基于封闭形式刚体预测验证了摩擦接触模型。