We present a novel Finite Volume (FV) scheme on unstructured polygonal meshes that is provably compliant with the Second Law of Thermodynamics and the Geometric Conservation Law (GCL) at the same time. The governing equations are provided by a subset of the class of symmetric and hyperbolic thermodynamically compatible (SHTC) models. Our numerical method discretizes the equations for the conservation of momentum, total energy, distortion tensor and thermal impulse vector, hence accounting in one single unified mathematical formalism for a wide range of physical phenomena in continuum mechanics. By means of two conservative corrections directly embedded in the definition of the numerical fluxes, the new schemes are proven to satisfy two extra conservation laws, namely an entropy balance law and a geometric equation that links the distortion tensor to the density evolution. As such, the classical mass conservation equation can be discarded. Firstly, the GCL is derived at the continuous level, and subsequently it is satisfied by introducing the new concepts of general potential and generalized Gibbs relation. Once compatibility of the GCL is ensured, thermodynamic compatibility is tackled in the same manner, thus achieving the satisfaction of a local cell entropy inequality. The two corrections are orthogonal, meaning that they can coexist simultaneously without interfering with each other. The compatibility of the new FV schemes holds true at the semi-discrete level, and time integration of the governing PDE is carried out relying on Runge-Kutta schemes. A large suite of test cases demonstrates the structure preserving properties of the schemes at the discrete level as well.

翻译：我们提出了一种新型的非结构化多边形网格有限体积格式，该格式能够同时满足热力学第二定律和几何守恒律。控制方程由对称双曲热力学相容（SHTC）模型类中的子集提供。我们的数值方法离散了动量、总能量、畸变张量和热冲量向量的守恒方程，从而通过统一的数学形式体系涵盖了连续介质力学中的多种物理现象。通过在数值通量定义中直接嵌入两个守恒修正项，新格式被证明满足两个额外的守恒律：熵平衡定律和连接畸变张量与密度演化的几何方程。因此，经典的质量守恒方程可以被舍弃。首先，在连续层面推导了几何守恒律，随后通过引入广义势和广义吉布斯关系的新概念使其得到满足。在确保几何守恒律相容性后，以相同方式处理热力学相容性，从而实现了局部单元熵不等式的满足。这两个修正是正交的，意味着它们可以同时共存而互不干扰。新的有限体积格式在半离散层面保持相容性，控制偏微分方程的时间积分则依赖龙格-库塔格式进行。大量算例验证了该格式在离散层面上的保结构特性。