# MAIDESC Project

## Adaptive mesh refinement for unsteady interfaces.

*Meshing*

### Summary

Mesh adaptive numerical methods have been recently developed with a considerable success. They allow computations which are not possible without mesh adaptation. The teams of the proposing consortium are among those who have contributed to important recent advances in the field. They have converged towards a common framework and are in position, by combining their competence, to make breakthroughs, in both methods and applications. A fundamental factor is that this framework offers high-order convergence in singular cases where the other approximation methods do not. We address in the proposed research several well identified main obstacles in order to maintain a high-order convergence for unsteady Computational Mechanics involving moving interfaces separating and coupling continuous media. A priori and a posteriori error analysis of Partial Differential Equations on static and moving meshes will be developed from interpolation error, goal-oriented error, and norm-oriented error. From the minimization of the chosen error, an optimal unsteady metric is defined. The optimal metric is then converted into a sequence of anisotropic unstructured adapted meshes by means of mesh regeneration, deformation, high stretching, and curvature. A particular effort will be devoted to build an accurate representation of physical phenomena involving curved boundaries and interfaces. In association with curved boundaries, a part of studies will address third-order accurate mesh adaption. Mesh optimality produces a nonlinear system coupling the physical fields (velocities, etc.) and the geometrical ones (unsteady metric, including mesh motion). Parallel solution algorithms for the implicit coupling of these different fields will be developed.

### Test case: Dam breaking 2D

The dam breaking 2D test case is a well-known CFD study. A liquid body initially at rest is released instantaneously by removal of a vertical barrier. The liquid is subject to gravity (g). It leads to an unsteady flow over a horizontal bed. This test case can be represented as in the $Figure 1$.

This test case has been studied experimentally by Koshizuka et al. (1995) and Martin and Moyce (1952). Experimental measurements give the position of the bottom right point in term of dimensionless variables given by:

$$x^{*} = \frac{x}{L}$$ $$t^{*} = t \sqrt{\frac{2g}{L}}$$* where $L$ is the initial width of the liquid body as presented in $Figure 1$. The domain size is $4L \times 4L$.*

### Test case simulation

We have conducted the simulation of this test case using Aeromines. The mesh is dynamically adapted to the speed and to the filtered distance to the liquid. You can run this test case by yourself in the free simulation section of your personal dashboard (Dam breaking simulation). The domain boundaries are considered as slipping boundaries without friction. The main characteristics of the simulation are given in the following table:

Domain size | $4L \times 4L$ |

Reference length | $0.146$ $m$ |

Fluid density | $998$ $kg.m^{-3}$ |

Fluid viscosity | $10^{-8}$ $m^2.s^{-1}$ |

Gas density | $1.2$ $kg.m^{-3}$ |

Gas viscosity | $10^{-8}$ $m^2.s^{-1}$ |

Gravity | $9.81$ $m.s^{-2}$ |

$Figure 2$ demonstrates that the general displacement of the liquid body follows a curve similar to the experimental profile. The displacement is too important at the beginning of the simulation however the slope of the curve is respected then (after 1.2 in the $Figure 2$). The interesting part is that the number of nodes does not yield to a better result thanks to mesh adaptation. It means that with a number of 2 500 elements the general displacement is well captured.

### Illustrations

### Acknowledgments

Aeromines would like to thank the Association Nationale de la Recherche (ANR) for financing this MAIDESC #ANR-13-MONU-0010 project.