We present a new hybrid semi-implicit finite volume / finite element numerical scheme for the solution of incompressible and weakly compressible media. From the continuum mechanics model proposed by Godunov, Peshkov and Romenski (GPR), we derive the incompressible GPR formulation as well as a weakly compressible GPR system. As for the original GPR model, the new formulations are able to describe different media, from elastoplastic solids to viscous fluids, depending on the values set for the model's relaxation parameters. Then, we propose a new numerical method for the solution of both models based on the splitting of the original systems into three subsystems: one containing the convective part and non-conservative products, a second subsystem for the source terms of the distortion tensor and heat flux equations and, finally, a pressure subsystem. In the first stage of the algorithm, the transport subsystem is solved by employing an explicit finite volume method, while the source terms are solved implicitly. Next, the pressure subsystem is implicitly discretised using finite elements. Within this methodology, unstructured grids are employed, with the pressure defined in the primal grid and the rest of the variables computed in the dual grid. To evaluate the performance of the proposed scheme, a numerical convergence analysis is carried out, which confirms the second order of accuracy in space. A wide range of benchmarks is reproduced for the incompressible and weakly compressible cases, considering both solid and fluid media. These results demonstrate the good behaviour and robustness of the proposed scheme in a variety of scenarios and conditions.

翻译：本文提出了一种用于求解不可压缩与弱可压缩介质的新型半隐式混合有限体积/有限元数值格式。基于Godunov、Peshkov和Romenski（GPR）提出的连续介质力学模型，我们推导了不可压缩GPR控制方程以及弱可压缩GPR方程组。与原始GPR模型类似，新构建的方程能够通过调整松弛参数值来描述从弹塑性固体到黏性流体的多种介质。随后，我们提出了一种适用于两种模型求解的新数值方法，该方法将原始系统分裂为三个子系统：包含对流项与非守恒乘积的子系统、处理畸变张量与热流方程源项的子系统，以及压力子系统。在算法的第一阶段，采用显式有限体积法求解输运子系统，同时隐式求解源项子系统。接着，利用有限元法对压力子系统进行隐式离散。该方法采用非结构化网格，其中压力定义于主网格，其余变量计算于对偶网格。为评估所提格式的性能，我们进行了数值收敛性分析，证实了该格式具有空间二阶精度。针对不可压缩与弱可压缩情形，在固体与流体介质中复现了多种基准算例。这些结果证明了所提格式在不同场景与条件下的良好性能与鲁棒性。