Preventing mass loss in the standard level set method: New insights from variational analyses
arxiv(2024)
摘要
For decades, the computational multiphase flow community has grappled with
mass loss in the level set method. Numerous solutions have been proposed, from
fixing the reinitialization step to combining the level set method with other
conservative schemes. However, our work reveals a more fundamental culprit: the
smooth Heaviside and delta functions inherent to the standard formulation. Even
if reinitialization is done exactly, i.e., the zero contour interface remains
stationary, the use of smooth functions lead to violation of mass conservation.
We propose a novel approach using variational analysis to incorporate a mass
conservation constraint. This introduces a Lagrange multiplier that enforces
overall mass balance. Notably, as the delta function sharpens, i.e., approaches
the Dirac delta limit, the Lagrange multiplier approaches zero. However, the
exact Lagrange multiplier method disrupts the signed distance property of the
level set function. This motivates us to develop an approximate version of the
Lagrange multiplier that preserves both overall mass and signed distance
property of the level set function. Our framework even recovers existing
mass-conserving level set methods, revealing some inconsistencies in prior
analyses. We extend this approach to three-phase flows for fluid-structure
interaction (FSI) simulations. We present variational equations in both
immersed and non-immersed forms, demonstrating the convergence of the former
formulation to the latter when the body delta function sharpens. Rigorous test
problems confirm that the FSI dynamics produced by our simple,
easy-to-implement immersed formulation with the approximate Lagrange multiplier
method are accurate and match state-of-the-art solvers.
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