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：本文介绍了伴侣论文[1]提出的有限变形柔性多体动力学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)单元及ANCF梁壳单元，采用基于求积的超弹性响应模型（圣维南-基尔霍夫和穆尼-瑞林模型），并包含可选的凯尔文-沃伊特粘性应力贡献项。时间步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。约束通过增广拉格朗日法(ALM)施加，采用外循环交替进行内速度求解与对偶上升乘子更新的结构。我们提出了用于内力和切线刚度评估的两阶段GPU并行化策略，并提供两种内部求解器：一阶AdamW优化器和二阶牛顿求解器，后者利用cuDSS在GPU上组装并分解稀疏全局海森矩阵。固定稀疏性矩阵策略消除了重复的符号分析，并能在牛顿迭代过程中实现高效的数值重组。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。系统性的缩放基准测试覆盖了所有三种支持单元类型和六种网格分辨率，结果表明：在最大测试分辨率下，牛顿求解器相对于CPU基线实现了约一个数量级的实时因子降低。通过准静态和动态冲击单元测试，摩擦接触模型得到了与闭式刚体预测相一致的验证。