We present the augmented matrix for principal submatrix update (AMPS) algorithm, a finite element solution method that combines principal submatrix updates and Schur complement techniques, well‐suited for interactive simulations of deformation and cutting of finite element meshes. Our approach features real‐time solutions to the updated stiffness matrix systems to account for interactive changes in mesh connectivity and boundary conditions. Updates are accomplished by an augmented matrix formulation of the stiffness equations to maintain its consistency with changes to the underlying model without refactorization at each timestep. As changes accumulate over multiple simulation timesteps, the augmented solution algorithm enables tens or hundreds of updates per second. Acceleration schemes that exploit sparsity, memoization and parallelization lead to the updates being computed in real time. The complexity analysis and experimental results for this method demonstrate that it scales linearly with the number of nonzeros of the factors of the stiffness matrix. Results for cutting and deformation of three‐dimensional (3D) elastic models are reported for meshes with up to 50 000 nodes, and involve models of surgery for astigmatism and the brain.

中文翻译：

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AMPS：具有增强矩阵的实时网格切割，可用于外科手术模拟

我们提出了用于主子矩阵更新（AMPS）算法的增强矩阵，这是一种将主子矩阵更新和Schur补码技术相结合的有限元求解方法，非常适合于有限元网格的变形和切削的交互式仿真。我们的方法针对更新的刚度矩阵系统提供实时解决方案，以解决网格连接和边界条件的交互变化。通过增加刚度方程的矩阵公式来完成更新，以保持其与基础模型变化的一致性，而无需在每个时间步进行重构。随着变化在多个模拟时间步长上累积，增强型解决方案算法每秒可以进行数十或数百个更新。利用稀疏性的加速方案，备忘录和并行化导致实时计算更新。该方法的复杂度分析和实验结果表明，该方法与刚度矩阵因子的非零数目成线性比例关系。据报道，具有多达5万个结点的网格的三维（3D）弹性模型的切割和变形结果涉及散光和大脑手术模型。