Mikhail Yu. Shashkov

ReALE: A reconnection-based arbitrary-Lagrangian-Eulerian method

Abstract We present a new reconnection-based arbitrary-Lagrangian–Eulerian (ALE) method. The main elements in a standard ALE simulation are an explicit Lagrangian phase in which the solution and grid are updated, a rezoning phase in which a new grid is defined, and a remapping phase in which the Lagrangian solution is transferred (conservatively interpolated) onto the new grid. In standard ALE methods the new mesh from the rezone phase is obtained by moving grid nodes without changing connectivity of the mesh. Such rezone strategy has its limitation due to the fixed topology of the mesh. In our new method we allow connectivity of the mesh to change in rezone phase, which leads to general polygonal mesh and allows to follow Lagrangian features of the mesh much better than for standard ALE methods. Rezone strategy with reconnection is based on using Voronoi tessellation. We demonstrate performance of our new method on series of numerical examples and show it superiority in comparison with standard ALE methods without reconnection.

Optimization-based synchronized flux-corrected conservative interpolation (remapping) of mass and momentum for arbitrary Lagrangian-Eulerian methods

A new optimization-based synchronized flux-corrected conservative interpolation (remapping) of mass and momentum for arbitrary Lagrangian-Eulerian hydro methods is described. Fluxes of conserved variables - mass and momentum - are limited in a synchronous way to preserve local bounds of primitive variables - density and velocity.

Failsafe flux limiting and constrained data projections for equations of gas dynamics

A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization/L^2 projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields are constrained using node-by-node transformations from the conservative to the primitive variables. An additional correction step is included to ensure that all the quantities of interest (density, velocity, pressure) are bounded by the physically admissible low-order values. The result is a conservative and bounded scheme with low numerical diffusion. The new failsafe FCT limiter is integrated into a high-resolution finite element scheme for the Euler equations of gas dynamics. Also, bounded L^2 projection operators for conservative interpolation/initialization are designed. The performance of the proposed limiting strategy and the need for a posteriori control of flux-corrected solutions are illustrated by numerical examples.

Symmetry- and essentially-bound-preserving flux-corrected remapping of momentum in staggered ALE hydrodynamics

We present a new flux-corrected approach for remapping of velocity in the framework of staggered arbitrary Lagrangian-Eulerian methods. The main focus of the paper is the definition and preservation of coordinate invariant local bounds for velocity vector and development of momentum remapping method such that the radial symmetry of the radially symmetric flows is preserved when remapping from one equiangular polar mesh to another. The properties of this new method are demonstrated on a set of selected numerical cyclic remapping tests and a full hydrodynamic example.

Short Note: Volume consistency in a staggered grid Lagrangian hydrodynamics scheme

Staggered grid Lagrangian schemes for compressible hydrodynamics involve a choice of how internal energy is advanced in time. The options depend on two ways of defining cell volumes: an indirect one, that guarantees total energy conservation, and a direct one that computes the volume from its definition as a function of the cell vertices. It is shown that the motion of the vertices can be defined so that the two volume definitions are identical. A so modified total energy conserving staggered scheme is applied to the Coggeshall adiabatic compression problem, and now also entropy is basically exactly conserved for each Lagrangian cell, and there is increased accuracy for internal energy. The overall improvement as the grid is refined is less than what might be expected.

Symmetry-preserving momentum remap for ALE hydrodynamics

In this paper, a symmetry-preserving remapping algorithm for vectors respecting their local bounds by components and in magnitude is presented. First, description of the Vector Image Polygon (VIP) limiter for a piece-wise linear velocity reconstruction is presented. Numerical fluxes obtained from this reconstruction lead to symmetry- and bounds- preserving remap of momentum for staggered Arbitrary Lagrangian-Eulerian (ALE) hydrodynamical methods. A novel bounds definition for vectors and corresponding modification of the VIP limiter is introduced, which fixes undershoots in a radial velocity component for polar meshes. Comparison with standard scalar-based limiters is given. Cyclic remapping is performed to numerically verify properties of the methods.

2nd workshop on new trends in numerical methods for multi-material compressible fluid flows

Abstract This workshop follows on from the successful first one held in San Diego during the SIAM Annual Meeting 2008. It provides a forum for exploring the new trends in numerical methods devoted to the simulation of multi-material compressible fluid flows.