We continue our investigation of finite deformation linear viscoelastodynamics by focusing on constructing accurate and reliable numerical schemes. The concrete thermomechanical foundation developed in the previous study paves the way for pursuing discrete formulations with critical physical and mathematical structures preserved. Energy stability, momentum conservation, and temporal accuracy constitute the primary factors in our algorithm design. For inelastic materials, the directionality condition, a property for the stress to be energy consistent, is extended with the dissipation effect taken into account. Moreover, the integration of the constitutive relations calls for an algorithm design of the internal state variables and their conjugate variables. A directionality condition for the conjugate variables is introduced as an indispensable ingredient for ensuring physically correct numerical dissipation. By leveraging the particular structure of the configurational free energy, a set of update formulas for the internal state variables is obtained. Detailed analysis reveals that the overall discrete schemes are energy-momentum consistent and achieve first- and second-order accuracy in time, respectively. Numerical examples are provided to justify the appealing features of the proposed methodology.

翻译：本文继续研究有限变形线性粘弹性动力学，重点构建精确可靠的数值格式。前序研究建立的具体热力学基础为保持关键物理和数学结构的离散格式奠定了基础。能量稳定性、动量守恒和时间精度构成算法设计的主要因素。对于非弹性材料，考虑耗散效应后扩展了应力满足能量一致性的方向性条件。此外，本构关系的积分要求对内部状态变量及其共轭变量进行算法设计。引入共轭变量的方向性条件作为确保物理正确数值耗散不可或缺的要素。利用构型自由能的特殊结构，获得了内部状态变量的更新公式组。详细分析表明，整体离散格式满足能量-动量一致性，并分别实现时间一阶和二阶精度。数值算例验证了所提方法的优良特性。