Volume and Mass Conservation in Lagrangian Meshfree Methods

TL;DR

The paper tackles volume and mass conservation in Lagrangian meshfree methods for free-surface flows, where traditional definitions of volume are absent and discrete mass does not guarantee volume conservation. It introduces representative masses and densities for meshfree collocation points to enable meaningful post-processing and to drive a local mass redistribution mechanism. A volume correction algorithm based on an artificial velocity divergence, derived from the difference between representative and physical densities, is proposed and validated across multiple test cases. Results show substantially improved volume conservation in both academic and industrially relevant flows without adversely affecting the physical solution, demonstrating practical impact for complex free-surface applications.

Abstract

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: unreliable volume and mass definitions. We introduce the concept of representative masses and densities, which are essential for accurate post-processing especially in meshfree collocation methods. Using these, we introduce an artificial compression or expansion in the fluid to rectify errors in volume conservation. Numerical experiments show that the introduced frameworks significantly improve volume conservation behaviour, even for complex industrial test cases such as automotive water crossing.

Figures (13)

• Figure 1: Lack of volume conservation: Sloshing of an incompressible fluid with a particle method. The gravity vector is rotated to simulate sloshing, after which the fluid is brought back to rest. Counter-clockwise from top left. The top row shows the initial state (left) and the final state (right) of the fluid, while the bottom row shows intermediate states. The red dashed line indicates the level of the fluid at the initial state. The figures shows that the volume of the final state is lower than that present initially. Since the number of particles are constant, the total numerical mass is conserved. However, the fluid volume is not conserved.
• Figure 2: Different volume definition of point cloud based meshfree methods.
• Figure 3: Adaptive refinement: Gradually refining the point cloud. The computational domain is marked in black. The stationary fluid only fills a part of the domain. The colour indicates the numerical volume of each point. The evolution of the numerical volume and the representative mass are shown in Figure \ref{['Fig:AdaptiveRefinement_MassVolume']}.
• Figure 4: Adaptive refinement test case: Evolution of the total mass ($\sum_{i=1}^N \hat{m}_i$) and volume ($\sum_{i=1}^N V_i$) in the computational domain as the point cloud is refined (left), and the evolution of the number of the points ($N$) in the domain (right).
• Figure 5: Dam breaking test case: Evolution of numerical volume for the collocation simulations, with and without the volume conservation algorithm.

Theorems & Definitions (3)

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