Research On ALE-DG Method Of Multi-material Fluid Dynamics

In this paper,a high-order cell-centered ALE-DG(arbitrary Lagrangian-Eulerian discontinuous Galerkin)method is developed for two-dimensional com-pressible multi-material fluid dynamics.The multi-material ALE method consists of the following phases:a Lagrangian phase;a rezoning phase,interface recon-struction in the multi-material cell and a remapping phase.The three parts of a Lagrangian phase,interface reconstruction in the multi-material cell and a remapping phase are improved,and a high-order multi-material ALE method is obtained.Numerical examples show that this method has good robustness and at least second-order accuracy,which can be used to solve complex multi-material large deformation fluid dynamics.A high-order cell-centered discontinuous Galerkin Lagrangian method is de-veloped for the Lagrangian phase.Starting from the compressible Euler equation in the Euler framework,the integral weak form in the Lagrangian framework is derived,and the spatial discretization is performed using the discontinuous fi-nite element method.Choosing a suitable basis function to make the material derivative zero,thus simplifying the discrete form of the equations and great-ly reducing the calculation cost.The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently by Maire's node solver.The time marching is implemented by a class of TVD Runge-Kutta type methods.The HWENO reconstruction algorithm is used as a limiter to eliminate spurious oscillations near discontinuities.A robust MOF(Moment of Fluid)method is developed for interface recon-struction of the multi-material cell.The essence of MOF method is to minimize an objective function.The first derivative of the objective function is continuous,so the minimum value points of the objective function must be the zero point of the first derivative.Instead of finding the zero points of the first derivative directly,we turn to calculating the minimum value points(also zero points)of the square of the first derivative,which is a convex function on a neighborhood of each zero point.Applying the properties of convex function,the neighbor of each extreme minimum point of it can be obtained efficiently.Then each zero point of the square of the first derivative can be obtained using the iterative formula in its neighbor.Finally,by comparing the values of the objective function at these zero points of the first derivative,the global minimum value point of the objective function can be found and is the desired solution.Through the above steps,the traditional MOF method is improved.By using this efficient algorithm for solving multiple roots of the nonlinear equation in large scope,a new algorith-m is obtained to enhance robustness of the MOF method.Compared with the traditional MOF method,this algorithm improves the accuracy and robustness,especially for polygon meshes with severe deformation.A high-order conservative remapping method of the cell-intersection-based type is proposed for the remapping phase.It can be divided into four stages:polynomial reconstruction,polygon intersection,integration and detection of problematic cells and limiting.Polygon intersection is based on the "clipping and projecting" algorithm to calculate the intersection of the new and old ele-ments.Detection of problematic cells depends on a troubled cell marker,and a posteriori MOOD(multi-dimensional optimal order detection)limiting strategy is used for limiting.Some minor modifications have been made to make it suit-able for multi-material remapping.The new remapping algorithm is conservation and at least second-order accuracy.