LA-UR 14-22765 Extension of the MOOD Method to the Saurel-Petitpas-Berry Model for Multi-Material Compressible Flows Fourth-order 2D results: The good & the not-yet-good-enough a Steven Diot , a Marianne François, b Edward Dendy. a Fluid Dynamics and Solid Mechanics (T-3) b Computational Physics and Methods (CCS-2) Los Alamos National Laboratory, Los Alamos, NM, USA. SHARK-FV 2014
The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion Outline Ideas behind the MOOD method Extension to a multi-material compressible flow model Fourth-order 2D results: better accuracy and improved efficiency Fourth-order 2D results: investigating the not-so-good Conclusion and future work SHARK-FV — diot@lanl.gov — 1/23
The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion Multi-dimensional Optimal Order Detection Origins • Developed during my Ph.D. in Toulouse (FR) under S. Clain & R. Loubère • Very-high-order Finite Volume method for single-material Euler equations • Alternative to WENO limiting on multidimensional unstructured meshes • Sucessfully tested up to 6 th -order of accuracy on 3D polyhedral meshes • Papers: JCP 2011, CAF 2012, IJNMF 2013 = ⇒ public.lanl.gov/diot Main ideas • Use only one unlimited polynomial reconstruction per cell & per degree • Check after the time update (a posteriori) if the solution is acceptable • If not, locally recompute the solution with a lower-order scheme • In the worst case, use the first-order scheme as parachute SHARK-FV — diot@lanl.gov — 2/23
The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion Multi-dimensional Optimal Order Detection Origins • Developed during my Ph.D. in Toulouse (FR) under S. Clain & R. Loubère • Very-high-order Finite Volume method for single-material Euler equations • Alternative to WENO limiting on multidimensional unstructured meshes • Sucessfully tested up to 6 th -order of accuracy on 3D polyhedral meshes • Papers: JCP 2011, CAF 2012, IJNMF 2013 = ⇒ public.lanl.gov/diot Main ideas • Use only one unlimited polynomial reconstruction per cell & per degree • Check after the time update (a posteriori) if the solution is acceptable • If not, locally recompute the solution with a lower-order scheme • In the worst case, use the first-order scheme as parachute SHARK-FV — diot@lanl.gov — 2/23
The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion Multi-dimensional Optimal Order Detection Origins • Developed during my Ph.D. in Toulouse (FR) under S. Clain & R. Loubère • Very-high-order Finite Volume method for single-material Euler equations • Alternative to WENO limiting on multidimensional unstructured meshes • Sucessfully tested up to 6 th -order of accuracy on 3D polyhedral meshes • Papers: JCP 2011, CAF 2012, IJNMF 2013 = ⇒ public.lanl.gov/diot Main ideas • Use only one unlimited polynomial reconstruction per cell & per degree • Check after the time update (a posteriori) if the solution is acceptable • If not, locally recompute the solution with a lower-order scheme • In the worst case, use the first-order scheme as parachute SHARK-FV — diot@lanl.gov — 2/23
The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion Multi-dimensional Optimal Order Detection Tools • The framework to control the decrementing process developed in JCP 2011 • There is a need for Detection Criteria to define an acceptable solution Detection Criteria • PAD - Physical Admissiblity Detection ⋆ All criteria required to obtain a physical solution ⋆ Typically for Euler with perfect gas EOS — ρ > 0 and p > 0 • DMP+u2 - Discrete Maximum Principle + Relaxation ⋆ DMP is used as a relevant detector of numerical oscillations ⋆ u2 developped to overcome the 2 nd -order lock at smooth extrema ⋆ Typically for Euler with perfect gas EOS: → DMP on ρ — min ( u n i , u n i ≤ max ( u n i , u n j ) ≤ u ⋆ j ) ( j looping over neighbors) → u2 if DMP violated — comparisons of local curvatures approximations SHARK-FV — diot@lanl.gov — 3/23
The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion Multi-dimensional Optimal Order Detection Tools • The framework to control the decrementing process developed in JCP 2011 • There is a need for Detection Criteria to define an acceptable solution Detection Criteria • PAD - Physical Admissiblity Detection ⋆ All criteria required to obtain a physical solution ⋆ Typically for Euler with perfect gas EOS — ρ > 0 and p > 0 • DMP+u2 - Discrete Maximum Principle + Relaxation ⋆ DMP is used as a relevant detector of numerical oscillations ⋆ u2 developped to overcome the 2 nd -order lock at smooth extrema ⋆ Typically for Euler with perfect gas EOS: → DMP on ρ — min ( u n i , u n i ≤ max ( u n i , u n j ) ≤ u ⋆ j ) ( j looping over neighbors) → u2 if DMP violated — comparisons of local curvatures approximations SHARK-FV — diot@lanl.gov — 3/23
The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion Multi-dimensional Optimal Order Detection Tools • The framework to control the decrementing process developed in JCP 2011 • There is a need for Detection Criteria to define an acceptable solution Detection Criteria • PAD - Physical Admissiblity Detection ⋆ All criteria required to obtain a physical solution ⋆ Typically for Euler with perfect gas EOS — ρ > 0 and p > 0 • DMP+u2 - Discrete Maximum Principle + Relaxation ⋆ DMP is used as a relevant detector of numerical oscillations ⋆ u2 developped to overcome the 2 nd -order lock at smooth extrema ⋆ Typically for Euler with perfect gas EOS: → DMP on ρ — min ( u n i , u n i ≤ max ( u n i , u n j ) ≤ u ⋆ j ) ( j looping over neighbors) → u2 if DMP violated — comparisons of local curvatures approximations SHARK-FV — diot@lanl.gov — 3/23
The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion The MOOD concepts — Today’s state & future Already done - papers in press or in preparation • OpenMP parallelization of the 3D Euler code G. Moebs • Application into a VHO ADER scheme M. Dumbser, R. Loubère • Extension to convection-diffusion & shallow-water The Portuguese Team • Used to design entropy-preserving 2 nd -order schemes V. Desveaux, C. Berthon • Design of a MOOD method for compr. multi-mat. flows with diffuse interface On-Going & future • Development of a VHO remapping process for ALE R. Loubère, M. Kucharik • Design of a MOOD method for multi-material flows with sharp interface • Other systems of PDE’s: elastic-plastic, Navier-Stokes, etc. • Any original/useful idea of applications = ⇒ Supports the fact that the a posteriori approach of MOOD is relevant SHARK-FV — diot@lanl.gov — 4/23
The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion The MOOD concepts — Today’s state & future Already done - papers in press or in preparation • OpenMP parallelization of the 3D Euler code G. Moebs • Application into a VHO ADER scheme M. Dumbser, R. Loubère • Extension to convection-diffusion & shallow-water The Portuguese Team • Used to design entropy-preserving 2 nd -order schemes V. Desveaux, C. Berthon • Design of a MOOD method for compr. multi-mat. flows with diffuse interface On-Going & future • Development of a VHO remapping process for ALE R. Loubère, M. Kucharik • Design of a MOOD method for multi-material flows with sharp interface • Other systems of PDE’s: elastic-plastic, Navier-Stokes, etc. • Any original/useful idea of applications = ⇒ Supports the fact that the a posteriori approach of MOOD is relevant SHARK-FV — diot@lanl.gov — 4/23
The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion 6-eqation model of Saurel-Petitpas-Berry (SPB) Reference: J. Comput. Phys. 228 (2009) 1678-1712. The non-conservative hyperbolic system is given by ∂ t U + ∇ · F ( U ) = 0 ∂ t ( α 1 ) + V · ∇ ( α 1 ) = µ ( p 1 − p 2 ) ∂ t ( α 1 ρ 1 e 1 ) + ∇ · ( α 1 ρ 1 e 1 V ) + α 1 p 1 ∇ · V = − ¯ p I µ ( p 1 − p 2 ) ∂ t ( α 2 ρ 2 e 2 ) + ∇ · ( α 2 ρ 2 e 2 V ) + α 2 p 2 ∇ · V = ¯ p I µ ( p 1 − p 2 ) • Conservation of material masses, mixture momenta and mixture total energy α 1 ρ 1 α 1 ρ 1 V α 2 ρ 2 α 2 ρ 2 V U = F ( U ) = and ρ V ρ V ⊗ V + pI ρ E ( ρ E + p ) V Stiffened gas EOS: p k = ρ k e k ( γ k − 1 ) − Π k γ k SHARK-FV — diot@lanl.gov — 5/23
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