Volume and Mass Conservation in Lagrangian Meshfree Methods

Meshfree Lagrangian frameworks for free surface flow simulations do not conserve fluid volume. Meshfree particle methods like SPH are not mimetic, in the sense that discrete mass conservation does not imply discrete volume conservation. On the other hand, meshfree collocation methods typically do not use any notion of mass. As a result, they are neither mass conservative nor volume conservative at the discrete level. In this paper, we give an overview of various sources of conservation errors across different meshfree methods. The present work focuses on one specific issue: inconsistent volume and mass definitions. We introduce the concept of representative masses and densities, which are essential for accurate post-processing in meshfree collocation methods. Using these, we introduce an artificial compressibility in the fluid to reduce errors in volume conservation. Numerical experiments show that the introduced frameworks significantly improve volume conservation behaviour in meshfree collocation methods, even for complex industrial test cases such as automotive water crossing.