We use the alternating direction method to simulate implicit dynamics. Our spatial discretization uses isogeometric analysis. Namely, we simulate a (hyperbolic) wave propagation problem in which we use tensor-product B-splines in space and an implicit time marching method to fully discretize the problem. We approximate our discrete operator as a Kronecker product of one-dimensional mass and stiffness matrices. As a result of this algebraic transformation, we can factorize the resulting system of equations in linear (i.e., $O\left(N\right)$) time at each step of the implicit method. We demonstrate the performance of our method in the model P-wave propagation problem. We then extend it to simulate the linear elasticity problem once we decouple the vector problem using alternating triangular methods. We prove theoretically and experimentally the unconditional stability of both methods.

我们使用交替方向方法来模拟隐式动力学。我们的空间离散化使用等几何分析。即，我们模拟了一个（双曲）波传播问题，其中我们在空间中使用张量积B样条和隐式时间行进方法来完全离散化该问题。我们将离散算子近似为一维质量和刚度矩阵的Kronecker乘积。代数变换的结果是，我们可以分解线性方程组（即，$Ø（ñ）$）时间在隐式方法的每个步骤中。我们证明了我们的方法在模型P波传播问题中的性能。一旦我们使用交替三角法将向量问题解耦，就可以将其扩展为模拟线性弹性问题。我们在理论上和实验上证明了这两种方法的无条件稳定性。