Challenges and recent progress in developing numerical methods for multi-material ALE Hydrocodes

Transcription

1 Challenges and recent progress in developing numerical methods for multi-material ALE Hydrocodes ICFD 25 year Anniversary Conference th September 2008 Oxford University Andrew Barlow AWE

2 Introduction This talk will attempt to describe the state of the art in numerical methods for hydrocodes. The key future challenges over the next decade will also be identified.

3 ALE and Eulerian methods ALE The two main types of hydrocode in use today are Arbitrary Lagrangian Eulerian (ALE) and Eulerian codes. This talk will concentrate on ALE methods since ALE codes offer significant advantages for multimaterial problems. However it is recognised that Eulerian codes are often used for robustness and ease of use. Eulerian AMR

4 Arbitrary Lagrangian Eulerian methods Most multi-material ALE hydrocodes employ staggered grid hydrodynamics schemes, where each timestep is divided into Lagrangian, rezone and remap steps. This talk will focus on the following key parts of the ALE algorithm that are considered most in need of further development for hydrocodes: Lagrangian methods Lagrangian treatment of material interfaces Mesh movement algorithms Interface reconstruction methods ALE/AMR

5 Compatible Hydrodynamics 1 State of the art ALE codes usually employ Compatible or Mimetic finite volume or finite element discretization methods for the Lagrangian step. Essentially this implies the internal energy is updated compatibly: where f are the corner forces calculated during the momentum step. Corner masses are also treated as Lagrangian objects like the zone masses. E. J. Caramana, D. E. Burton, M. J. Shashkov and P. P. Whalen, The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy., J. Comput. Phys. 146 (1):227, A. J. Barlow, A compatible finite element multi-material ALE hydrodynamics algorithm., Int. J. Numer. Meth. Fluids 2008; 56:

6 Compatible Hydrodynamics 2 Compatible hydro offers: Total energy conservation to round off for the Lagrangian step. Increased flexibilty in types of forces that can be allowed within a zone (e.g. sub-zonal pressures and edge artificial viscosities). This flexibility can be used to improve accuracy and robustness.

7 Sub-zonal pressures can be introduced by subdividing each computational zone into sub-elements. The initial mass for which are stored at the start of the calculation and used each time step to define sub-zonal densities. Sub-Zonal pressures Subzonal pressure perturbations can then estimated from the sound speed. 2 Δp p, z = cs, zδρ p, z E. J. Caramana and M. J. Shashkov, Elimination of Artificial Grid Distortion and Hourglass-Type Motions by Means of Lagrangian Subzonal Masses and Pressures., J. Comput. Phys. 142 (2):521, 1998.

8 Monotonic edge artificial viscosity The force exerted by the artificial viscosity between nodes 1 and 2 is: f visc 21 = ( 1 ψ 21 ) ρ ( c c, c v 21 )[ v 21. S ˆ z l s z + q Δ Δ 1 ] Δ v 21 where ρ z and c s,z are the density and sound speed; Δv 12 = v 2 -v 1 is the difference in velocity along edge 21; S i are median mesh vectors and [.] is a compression switch. E. J. Caramana, M. J. Shashkov and P. P. Whalen, Formulations of Artificial Viscosity for Multi-dimensional Shock Wave Computations., J. Comput. Phys. 144 (1):70, 1998.

9 Test Problem Results Saltzman s Piston Noh

10 Projectile impact problem 0.2 cm/μs Aluminium Steel projectile

11 Projectile impact problem

12 Challenges for Lagrangian methods 2D and 3D Symmetry preservation 2D cylindrical symmetry with non-uniform angular zoning or reduced connectivity Spherical symmetry in 3D Mesh dependence of artificial viscosity Edge artificial viscosity is less mesh dependent than a element centred scalar monotonic artificial viscosity, but still shows mesh dependence Porous and spalling material shows greatest sensitivity Sound speed dependence of sub-zonal pressure scheme A well defined sound speed is required for sub-zonal pressure schemes to be effective Treatment of disjoint node or dendritic mesh boundaries The well known artificial vorticity problem is observed when a shock crosses a mesh refinement boundary

13 Potential improvements to Lagrangian methods Tensor edge viscosity could further reduce mesh dependence and may improve symmetry A cell centred Lagrangian schemes should remove need for artificial viscosity and subzonal pressure schemes The challenge is how to move the vertices in a consistent manner. High order mixed finite element methods may provide a means of improving symmetry in 2D RZ and 3D These methods often result in node centred pressures which may provide robustness in a similar manner to the subzonal pressures schemes currently in use. J. C. Campbell and M. J. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm., J. Comput. Phys. 172 (2):739:765, P-H. Maire and J. Breil, A second-order cell-centered Lagrangian scheme for twodimensional compressible flow problems., Int. J. Numer. Meth. Fluids (in press). Published online in Wiley InterScience (

14 Lagrangian Treatment of multi-material cells State of the art multi-material ALE codes can model material interfaces with slide and contact or volume of fluid (VOF) algorithms. A VOF treatment is used for robustness to allow mesh relaxation across material interfaces when severe material deformation occurs at a material interface. This introduces multi-material cells, where the amount of each material present is given by a volume fraction variable.

15 Closure models for multi-material cells Given the lack of information about the velocity field within multi-material cells a closure model is required to define how the volume fractions evolve during the Lagrangian step. Closure model for multi-material cells are typically based either on pressure relaxation schemes or a sub-cell dynamics model. A. J. Barlow, A new Lagrangian scheme for multi-material cells, Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering. ECCOMAS Computational Fluid Dynamics Conference 2001, Swansea, Wales, UK. M. Shashkov, Closure models for multimaterial cells in arbitrary Lagrangian-Eulerian hydrocodes., Int. J. Numer. Meth. Fluids (in press). Published online in Wiley InterScience (

16 Challenges for multi-material cell closure models To develop a closure model that fully emulates all the interface physics that can be represented by slide and contact algorithms Void closure Void opening Compatibility with material strength Slide (separate velocities required per material component) Friction

17 Potential improvements to multi-material cell closure models Recent work at AWE has produced a physics based sub-cell dynamics method which allows expansion and compression to occur simultaneously. The new method starts from the assumption of equal volumetric strain, but then uses an acoustic Riemann solver to define volume exchanges between adjacent materials. The volume exchange is limited to maintain an average cell pressure which is bounded by the maximum and minimum pressures of the multimaterial cells components

18 Modified Sod Problem L R γ U 0 0 ρ ε p Ideal gas EOS p = ( γ 1)ρε

19 Modified Sod Problem

20 Water/air problem L R γ U 0 0 ρ 1.0e+3 5.0e+2 P 1.0e+9 1.0e+6 P 6.0e+8 0 Stiffened gas EOS p = ( γ 1)ρε γp

21 Water/air problem

22 Mesh Movement 1 In most cases Lagrangian mesh motion is what is desired. Mesh movement are used as the mesh becomes distorted to improve mesh quality and to achieve robustness. Typically equipotential mesh movement algorithms like Winslow s method are used with additional constraints to limit relaxation. A. M. Winslow, Equipotential zoning of two-dimensional meshes. Technical report UCRL-7312, Lawrence Livermore National Laboratory, 1963.

23 Mesh Movement 2 Separate mesh movement algorithms are normally applied to reposition nodes along slide lines before the internal mesh movement is performed. The constraints may be physics based e.g. nodes can t relax until zone has fully detonated. Alternatively geometric criteria may be used to detect zone distortion or cell collapse e.g. measurement of corner angle or collapse of corner area.

24 Challenges for mesh movement algorithms The challenge for mesh movement algorithms is to develop algorithms that will move the mesh in such a way as maintain robustness while staying as close as possible to the Lagrangian mesh motion. Robustness must be possible without user intervention. But user needs to retain sufficient control if required to be able to focus resolution in areas of interest or follow physics of interest. P. M. Knupp, L.G. Margolin and M. J. Shashkov, Reference Jacobian optimization-based rezone strategies for Arbitrary Lagrangian-Eulerian methods., J. Comput. Phys. 176 (2002)

25 NIF Can Mesh at t=0.0 ns

26 NIF Can Mesh at t=300.0 ps

27 NIF Can Mesh at t=500.0 ps

28 NIF Can Mesh at t=700.0 ps

29 NIF Can Mesh at t=900.0 ps

30 NIF Can Mesh at t= ps

31 Interface reconstruction 1 Youngs interface reconstruction method is used most state of the art hydrocodes. Youngs method represents the material interface as an inclined straight line.

32 Interface reconstruction 2 The slope of the line is obtained from the neighbouring cell volume fractions. The line is then moved along its normal until the volume fractions in the cell are matched.

33 Interface reconstruction 3 The reconstructed interface is used to partition the donor fluxing volume for the cell. This defines the volume fraction fluxes being donated and enables first order donor cell advection of the state variables for the material components.

34 Challenges for interface reconstruction methods Youngs interface reconstruction is first order, produces discontinuous interfaces and for more than two materials interfaces can cross. A second order method is required which preserves continuous interface where possible. It must also avoid the cross over problem and handle T-junctions.

35 HELMIT Basic idea - extend interface from a single line to two connected or hinged lines: Hinged Line Method of Interface Tracking

36 Final result looks like this: Youngs interfaces HELMIT interfaces

37 Moment of fluid method Moment of fluid method looks most promising new interface reconstruction method. Makes use of centroid data in addition to local volume fractions. Tracks centroids through Lagrangian step, then adjusts interface normal until gets closest agreement between centroid for reconstructed material interface and tracked centroid position. The tracked centroid position may however need to be modified if a closure model changes volume fractions! V. Dyadechko and M. Shashkov, Moment-of-fluid interface reconstruction. Technical report LA-UR , Los Alamos National Laboratory, 2005.

38 ALE mesh adaption is efficient but mesh resolution is limited by its penalty nature. A hybrid ALE/AMR capability should maintain the benefits of the ALE, but allow extra resolution to be added locally when physics requires it e.g. reactive flow. ALE/AMR 1

39 ALE/AMR 2 A hybrid ALE/AMR method is currently under development at AWE. Cell by cell refinement strategy. Efficient data structures based on existing connectivity arrays. Disjoint nodes used at refinement boundaries for Lagrangian step and in applying mesh relaxation. Only solve on finest level. J. M. Morrell, P. K. Sweby and A. J. Barlow, A cell by cell anisotropic adaptive mesh ALE scheme for numerical solution of the Euler equations., J. Comput. Phys :1180, 2007.

40 ALE/AMR 3 Programmed burn 4 Levels of refinement Factor of 10 speedup achieved verses uniform fine

41 ALE/AMR 4

42 Challenges for ALE/AMR Parallel scalability could be degraded by AMR! Refinement and derefinement is complicated by non-orthogonal ALE mesh Error estimators need to be developed that work well for all physics of interest Extension to 3D Can AMR techniques be applied to other problems such as explicit fracture modelling!

43 Explicit Fracture modelling Ideas from 2D ALE/AMR are now being used to develop an explicit fracture modelling capability in 3D at AWE

44 Conclusions This talk has described the state of the art in numerical methods for hydrocodes used to simulate the dynamic compression of condensed matter. A view has also been given of the key challenges for the next decade in this area and some suggestions made of areas where progress is expected.