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：我们提出了伴随论文[1]中介绍的、用于有限变形柔性多体动力学的总拉格朗日有限元框架所依托的数值方法与GPU加速实现。该框架支持10节点二次四面体（T10）单元以及ANCF梁与壳单元，采用基于数值积分的超弹性本构响应（圣维南-基尔霍夫与穆尼-里夫林模型），并可选择添加开尔文-伏伊特粘性应力贡献。时间推进采用基于速度的隐式后向欧拉格式，生成一个耦合惯性力、内力、外力及双侧约束的速度非线性残差。约束通过增广拉格朗日方法（ALM）施加，其结构为外层循环，交替进行内层速度求解与对偶上升乘子更新。我们提出了一种用于内力和切线刚度评估的两阶段GPU并行化策略，并提供两种内层求解器：一阶AdamW优化器与二阶牛顿求解器（后者使用cuDSS在GPU上组装并分解稀疏全局海森矩阵）。固定稀疏度矩阵策略避免了重复的符号分析，并可在牛顿迭代过程中实现高效的数值重新分解。针对碰撞检测，我们提出了一种基于GPU的双线程异步算法，该算法直接作用于三角面片集合，完全避免了包围体层次结构的构建。对所有三种支持单元类型及六种网格分辨率的系统规模基准测试表明，在测试的最大分辨率下，牛顿求解器相较于CPU基线实现了约一个数量级的实时因子缩减。通过准静态与动态冲击单元测试，利用闭式刚体预测对摩擦接触模型进行了验证。