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Compatible Finite Element multi-material ALE Hydrodynamics Numerical methods for multi-material fluid flows 10-14th September 2007 Czech Technical University in Prague Andrew Barlow AWE. Introduction.
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Compatible Finite Element multi-material ALE HydrodynamicsNumerical methods for multi-material fluid flows10-14th September 2007 Czech Technical University in PragueAndrew BarlowAWE
Introduction • At the previous meeting in this series in Oxford in 2005 I showed how the ideas of compatible Lagrangian hydrodynamics developed for finite volume methods by Caramana, Shashkov, Burton et al can also be applied to a finite element Lagrangian scheme. • In this talk I will describe the extension of that compatible finite element Lagrangian scheme into a multi-material Arbitrary Lagrangian Eulerian (ALE) scheme.
Summary of Lagrangian Scheme 1 • Staggered grid, predictor corrector time discretization, the momentum equation is solved using bi-linear isoparametric finite elements with Petrov Galerkin (or area weighting) for cylindrical geometry. • The internal energy is solved compatibly: where f are the forces calculated during the momentum step. • This leads to total energy conservation to round off for the Lagrangian step and allows sub-zonal pressure schemes and edge based artificial viscosities to be used to improve accuracy and robustness.
Summary of Lagrangian Scheme 2 • The corner masses are also treated as Lagrangian objects like the zone masses. • The finite element scheme discussed here requires two different types of corner mass to be defined. • The first is used in solving the momentum equation and in order for the internal energy equation discussed above to be valid in cylindrical geometry must satisfy: • The second type of corner mass is required to use a sub-zonal pressure scheme and are defined as shown.
Extending Compatible hydro to ALE • The key issues for applying the Compatible Finite Element Lagrangian Hydro scheme as the Lagrangian step of a multi-material ALE code are how to do: • Advection of corner masses • Momentum advection • Internal energy update for multi-material cells • How best to move the mesh to exploit the benefits of the Compatible Lagrangian hydro scheme.
Advection of corner masses • Different properties are required from the two different types of corner mass, so different methods can be used for advecting them. • In advecting the corner masses associated with the sub-zonal pressures it is important to avoid the nonphysical growth of corner densities due to advection fluxes entering an acceptor cell. Some memory of the corner densities that have developed to resist hourglass modes must however be retained in donor cells. • In contrast the corner masses used in solving the momentum equationare never used to evaluate pressures, so the growth of maxima in an acceptor zone is not an issue. But it is important to make the update of these corner masses as consistent as possible with the momentum advection scheme used.
Advection of sub P corner masses 1 • The mass of an element post advection can be expressed as: where is the element mass at the end of the Lagrangian step, the mass flux leaving the zone and the mass flux entering the zone
Advection of sub P corner masses 2 • The corner masses that are used to calculate the subzonal pressures can be updated as follows: where is the corner mass at the end of the Lagrangian step, is the new corner volume after remap and is the cell volume after the remap. • This ensures that any mass remaining in a cell after advection is distributed to the corners in the same proportions as the original cell mass was prior to advection (which satisfies the limiting case of Lagrangian mesh motion). • Any new mass entering the cell during the advection step is distributed in proportion to the corner volumes (i.e. is assumed to be at constant density throughout the cell).
Momentum advection • The issue of how best to perform the momentum advection scheme is intimately connected with the issues of how to advect the corner masses used in the Lagrangian step to solve the momentum equation. • It is natural to use these corner masses for the momentum advection since by doing so momentum conservation will be guaranteed for both the Lagrangian and advection steps. • So the nodal mass at the end of the Lagrangian step used for the momentum advection is taken as the sum of the corner masses used for solving the momentum equation during the Lagrangian step of the adjacent elements of the node.
Advection of mom corner masses • The corner masses used for solving the momentum equation in the Lagrangian step are advected by defining the donor cell fluxes leaving corner masses as follows: for an element z, corner k and face j. • These fluxes are accepted by the logically equivalent corner of the element which is accepting the element mass flux . • The new corner masses are then given by:
Mass fluxes for momentum advection • The nodal mass fluxes used for the momentum advection can now be defined in a totally consistent manner as the sum of the fluxes for the corner nodes masses associated the nodes for which the mass flux is required. • The post advection nodal mass can then be defined either as the sum of the new corner masses of by applying the nodal mass fluxes to the pre-advection nodal mass. Both will give the same result. • The corner masses have also been used to define the mass coordinate system. • The rest of the momentum advection can be performed unchanged.
Internal energy update for multi-material cell components • Separate corner force components fk,p are stored for each material component in multi-material cells make use of component state variables but integrated as though they are single material values. • These are then used to update the internal energy of the multi-material cell components as follows: where Fk,z is the volume fraction for the material component k. • Note: This can be made fully consistent with pressure relaxation schemes that allow the volume fractions to vary during the Lagrangian step by replacing the volume fraction in the above expression with a relative compressibility factor.
Saltzman‘s Piston Problem 1 1.0 cm/s • Piston moves with unit velocity from left to right across a grid that is skewed with respect to the vertical with a half sinwave perturbation. The right end treated as reflecting boundary. • Ideal gas (=1.66) with initial density unity and internal energy zero. • Compare mesh quality and density behind before and after reflection at t=0.8 s. • A density of 4 g/cc should be observed behind first shock and 10 g/cc behind reflected shock.
Saltzman‘s Piston Problem 2 • A good solution is obtained with the compatible finite element hydro scheme without subzonal pressures when the monotonic edge artificial viscosity is used.
Saltzman‘s Piston Problem 3 • The solution obtained with compatible finite element hydro is further improved by using the subzonal pressure scheme in addition to the edge viscosity.
Saltzman‘s Piston with ALE 1 • ALE calculation of Saltzman’s piston problem with limited 10-4 x equipotential mesh relaxation without subzonal pressures some mesh distortion is observed. Lagrangian with subzonal pressures ALE without subzonal pressures
Saltzman‘s Piston with ALE 2 • ALE calculation of Saltzman’s piston problem with limited 10-4 x equipotential mesh relaxation using subzonal pressures and corner mass advection is in very close agreement with pure Lagrangian solution. ALE with subzonal pressures Lagrangian with subzonal pressures
Saltzman‘s Piston with ALE 3 • If the calculations are performed without the modifications to the momentum advection and both types of corner mass are updated using procedure described for the corner masses associated with the subzonal pressures then significantly worse solutions are obtained. ALE with subzonal pressures ALE without subzonal pressures
ALE Projectile Impact Problem 0.2 cm/s Aluminium Steel projectile
ALE Projectile Impact Problem at 20.0 s • Compatible finite element ALE hydro is robust and total energy conservation without KE fix up is significantly better than original PdV scheme (2.3% energy loss compared to 12.4% for PdV scheme).
ALE Projectile Impact Problem at 20.0 s • If the calculation is performed without the modifications to the momentum advection and both types of corner mass are updated using procedure described for the corner masses associated with the subzonal pressures then there is less improvement to total energy conservation compared to the PdV scheme. Without modifications to momentum advection With modifications to momentum advection
Mesh Movement for compatible hydro 1 • In order to fully exploit the benefits of the compatible hydro scheme it is desirable that the mesh motion is kept as close to Lagrangian as possible, while still retaining robustness for high material deformation. • A local mesh movement strategy has been developed which has been motivated by corner volumes used in the compatible hydro scheme to achieve this. • This should be of great benefit for radiation hydrodynamics simulations where high aspect ratio zones and significant variations in resolution are often used.
Mesh Movement for compatible hydro 2 • The local ALE mesh movement scheme has the following key ingredients: • A limit on mesh velocity which controls how far a node can be moved each timestep. • A series of geometric criteria are used to determine whether a node can relax. • A more stringent criterion is applied to nodes on material interfaces.
Mesh velocity limit • The timestep is determined from: where c is the sound speed u are the nodal velocities. • The magnitude of the mesh velocity w at each node i is constrained to not exceed the distance that information is allowed to propagate by the following:
Geometric criteria 1 • An internal node not on a material interface will move with Lagrangian motion unless one of its corners satisfies one of two criteria: • Area of corner/(0.25 * zone area) < user defined parameter (~0.5) (This detects the collapse of corner volumes, working with area ensures symmetry is preserved for cylindrical geometry) • Sin (corner angle) < user defined parameter (~0.87) (Detects shear.) • Once this criteria is reached at an internal node then that node will remain relaxed from then on.
Geometric criteria 2 • Interface nodes are defined as nodes on material number boundaries in order to capture both Lagrangian and multi-material interfaces. • It is desirable to limit relaxation across material interfaces to keep resolution in materials of interest. • In order to achieve this interface nodes only relax if relaxation criterion is met that time step. • In addition the corner area criterion will only trigger relaxation if it is met for a corner in both the two materials either side of the interface. • While the angle test need only be met in one of the two materials.
Geometric criteria 3 • Initial zoning that would immediately trigger relaxation can be preserved by modifying the criteria for relaxation as follows: • Area of corner/(0.25 * zone area) < Initial area of corner/(0.25 * initial zone area) * user defined parameter (~0.5) • Sin (corner angle) < Initial Sin (corner angle) * user defined parameter (~0.87)
Local ALE Projectile Impact Problem • The local ALE method keeps the material interface Lagrangian in regions of low deformation and introduces multi-material cells in regions of high deformation.
Conclusion • A method has also been proposed for extending a compatible finite element Lagrangian scheme into a multi-material ALE hydrodynamics scheme. • A method has been proposed for how to advect the two types of corner masses required by the compatible hydro scheme. • A new momentum advection scheme has also been devised which is fully consistent with the corner mass advection and offers further improvements to total energy conservation. • A local ALE mesh motion scheme has also been presented which attempts to exploit the advantages of the compatible Lagrangian hydro scheme. • It has also been demonstrated that compatible finite element ALE hydro scheme provides a robust and hi-fidelity capability for modelling radiation hydrodynamics problems.
Acknowledgements • I would like to thank Ed Caramana (LANL) for the many useful discussions we have had on this topic. • I also like to thank Stephen Hughes (AWE) who performed the radiation hydrodynamics simulations.
