This paper presents a novel semi-implicit scheme for elastodynamics and wave propagation problems in nearly and truly incompressible material models. The proposed methodology is based on the efficient computation of the Schur complement for the mixed displacement-pressure formulation using a lumped mass matrix for the displacement field. By treating the deviatoric stress explicitly and the pressure field implicitly, the critical time step is made to be limited by shear wave speed rather than the bulk wave speed. The convergence of the proposed scheme is demonstrated by computing error norms for the recently proposed LBB-stable BT2/BT1 element. Using the numerical examples modelled with nearly and truly incompressible Neo-Hookean and Ogden material models, it is demonstrated that the proposed semi-implicit scheme yields significant computational benefits over the fully explicit and the fully implicit schemes for finite strain elastodynamics simulations involving incompressible materials. Finally, the applicability of the proposed scheme for wave propagation problems in nearly and truly incompressible material models is illustrated.

本文提出了一种新颖的半隐式方案，用于解决几乎真实的不可压缩材料模型中的弹性动力学和波传播问题。所提出的方法是基于对混合位移-压力公式的Schur补语的有效计算，该公式使用集总质量矩阵作为位移场。通过显式地处理偏应力和隐式地处理压力场，可以使临界时间步长受剪切波速而不是体波速的限制。通过为最近提出的LBB稳定的BT2 / BT1元素计算误差范数，证明了提出的方案的收敛性。使用以几乎完全不可压缩的Neo-Hookean和Ogden材料模型建模的数值示例，结果表明，与涉及不可压缩材料的有限应变弹性动力学模拟的完全显式和完全隐式方案相比，所提出的半隐式方案具有显着的计算优势。最后，说明了所提出的方案在几乎完全不可压缩的材料模型中对波传播问题的适用性。