Space-time Discontinuous Galerkin Methods for Compressible Flows - PowerPoint PPT Presentation

Space-time Discontinuous Galerkin Methods for Compressible Flows Jaap van der Vegt Numerical Analysis and Computational Mechanics Group Department of Applied Mathematics University of Twente Joint Work with Christiaan Klaij (UT), Daniel K¨ oster (UT), Harmen van der Ven (NLR) and Marc van Raalte (CWI) Workshop M´ ethodes Num´ eriques pour les Fluides Paris, December 18, 2007

University of Twente - Chair Numerical Analysis and Computational Mechanics 1 Space-Time Discontinuous Galerkin Finite Element Methods Motivation of research: • In many applications one encounters moving and deforming flow domains: ◮ Aerodynamics: helicopters, manoeuvering aircraft, wing control surfaces ◮ Fluid structure interaction ◮ Two-phase and chemically reacting flows with free surfaces ◮ Water waves, including wetting and drying of beaches and sand banks • A key requirement for these applications is to obtain an accurate and conservative discretization on moving and deforming meshes

University of Twente - Chair Numerical Analysis and Computational Mechanics 2 Motivation of Research Other requirements • Improved capturing of vortical structures and flow discontinuities, such as shocks and interfaces, using hp -adaptation. • Capability to deal with complex geometries. • Excellent computational efficiency for unsteady flow simulations. These requirements have been the main motivation to develop a space-time discontinuous Galerkin method.

University of Twente - Chair Numerical Analysis and Computational Mechanics 3 Overview of Lecture • Space-time discontinuous Galerkin finite element discretization for the compressible Navier-Stokes equations ◮ main aspects of discretization ◮ efficient solution techniques • Applications in aerodynamics • Concluding remarks

University of Twente - Chair Numerical Analysis and Computational Mechanics 4 Space-Time Approach Key feature of a space-time discretization • A time-dependent problem is considered directly in four dimensional space, with time as the fourth dimension

University of Twente - Chair Numerical Analysis and Computational Mechanics I n Sketch of a space-time mesh in a space-time domain. Q n t n+1 t n t 0 t � ✁ ✁ � � ✁ ✁ � ✁ � ✁ � Space-Time Domain � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � ✁ � ✁ � ✁ ✁ � ✁ � ✁ � � ✁ � ✁ � ✁ � ✁ � ✁ ✁ � ✁ � � ✁ � ✁ � ✁ ✁ � ✁ � � ✁ � ✁ � ✁ � ✁ � ✁ x Ω � ✁ ✁ � ✁ � ✁ � ✁ � � ✁ � ✁ � ✁ � ✁ � ✁ ✁ � ✁ � n+1 � ✁ ✁ � ✁ � � ✁ ✁ � ✁ � � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ � ✁ ✁ � ✁ � ✁ � ✁ � ✁ � � ✁ � ✁ ✁ � � ✁ � ✁ � ✁ ✁ � ✁ � � ✁ ✁ � ✁ � ✁ � � ✁ ✁ � ✁ � � ✁ � ✁ � ✁ Ω n y K 5

University of Twente - Chair Numerical Analysis and Computational Mechanics 6 Benefits of Space-Time Approach A space-time discretization of time-dependent problems has as main benefits • The problem is transformed into a steady state problem in space-time which makes it easier to deal with time dependent boundaries. No extrapolation or interpolation of (boundary) data • A conservative numerical discretization is obtained on deforming and locally refined meshes

University of Twente - Chair Numerical Analysis and Computational Mechanics 7 Compressible Navier-Stokes Equations • Compressible Navier-Stokes equations in space-time domain E ⊂ R 4 : + ∂F e − ∂F v ∂U i k ( U ) k ( U, ∇ U ) = 0 ∂x 0 ∂x k ∂x k • Conservative variables U ∈ R 5 and inviscid fluxes F e ∈ R 5 × 3     ρ ρu k F e  ,    U = ρu j k = ρu j u k + pδ jk ρE ρhu k

University of Twente - Chair Numerical Analysis and Computational Mechanics 8 Compressible Navier-Stokes Equations • Viscous flux F v ∈ R 5 × 3   0 F v   k = τ jk τ kj u j − q k with the total stress tensor τ ∈ R 3 × 3 defined as: τ jk = λ∂u i δ jk + µ ( ∂u j + ∂u k ) ∂x i ∂x k ∂x j and the heat flux vector q ∈ R 3 given by: q k = − κ ∂T ∂x k

University of Twente - Chair Numerical Analysis and Computational Mechanics 9 Compressible Navier-Stokes Equations • The viscous flux F v is homogeneous with respect to the gradient of the conservative variables ∇ U : ik ( U, ∇ U ) = A ikrs ( U ) ∂U r F v ∂x s with the homogeneity tensor A ∈ R 5 × 3 × 5 × 3 defined as: A ikrs ( U ) := ∂F v ik ( U, ∇ U ) ∂ ( ∇ U ) • The system is closed using the equations of state for an ideal gas.

University of Twente - Chair Numerical Analysis and Computational Mechanics 10 Space-Time Discontinuous Galerkin Discretization Main features of a space-time DG approximation • Basis functions are discontinuous in space and time • Weak coupling through numerical fluxes at element faces • Discretization results in a coupled set of nonlinear equations for the DG expansion coefficients

University of Twente - Chair Numerical Analysis and Computational Mechanics 11 Space-Time Slab x 0 Ω (T) T E n+1 K t j n+1 I Q n K n n n j Q j j t n n K j x 1 Space-time slab with elements in a space-time domain.

University of Twente - Chair Numerical Analysis and Computational Mechanics 12 Benefits of Space-Time DG Discretization Main benefits of a space-time DG approximation • The space-time DG method results in a very local discretization, which is beneficial for: ◮ hp -mesh adaptation ◮ parallel computing • The space-time discretization is conservative on moving and deforming meshes and also on locally adapted meshes

University of Twente - Chair Numerical Analysis and Computational Mechanics 13 Discontinuous Finite Element Approximation Approximation spaces • The finite element space associated with the tessellation T h is given by: W h := � W ∈ ( L 2 ( E h )) 5 : W | K ◦ G K ∈ ( P k ( ˆ � K )) 5 , ∀K ∈ T h • We will also use the space: V h := � V ∈ ( L 2 ( E h )) 5 × 3 : V | K ◦ G K ∈ ( P k ( ˆ � . K )) 5 × 3 , ∀K ∈ T h • Note the fact that ∇ h W h ⊂ V h is essential for the discretization.

University of Twente - Chair Numerical Analysis and Computational Mechanics 14 First Order System • Rewrite the compressible Navier-Stokes equations as a first-order system using the auxiliary variable Θ : + ∂F e ∂U i ik ( U ) − ∂ Θ ik ( U ) = 0 , ∂x 0 ∂x k ∂x k Θ ik ( U ) − A ikrs ( U ) ∂U r = 0 . ∂x s

University of Twente - Chair Numerical Analysis and Computational Mechanics 15 Weak Formulation • Weak formulation for the compressible Navier-Stokes equations Find a U ∈ W h , Θ ∈ V h , such that for all W ∈ W h and V ∈ V h , the following holds: � � ik − Θ ik ) � d K � ∂W i U i + ∂W i ( F e − ∂x 0 ∂x k K K∈T h � � W L F e Θ ik ) n L i ( � U i + � ik − � + k d ( ∂ K ) = 0 , ∂ K K∈T h � � � � ∂U r V ik Θ ik d K = V ik A ikrs d K ∂x s K K K∈T n K∈T n h h � � V L ik A L U r − U L n L ikrs ( � + r )¯ s d Q Q K∈T n h

University of Twente - Chair Numerical Analysis and Computational Mechanics 16 Geometry of Space-Time Element 2 n+1 [−1,1] K t 2 ∆ t n G n ∆ t K K − S S 2 n= 1 ∆ ξ x 0 ξ 1 t ∆ n x K x F n K Geometry of 2D space-time element in both computational and physical space.

University of Twente - Chair Numerical Analysis and Computational Mechanics 17 Transformation to Arbitrary Lagrangian Eulerian form • The space-time normal vector on a grid moving with velocity � v is:   (1 , 0 , 0 , 0) T at K ( t − n +1 ) ,   ( − 1 , 0 , 0 , 0) T at K ( t + n = n ) ,    n ) T at Q n . ( − v k ¯ n k , ¯ • The boundary integral then transforms into: � � W L F e Θ ik ) n L i ( � U i + � ik − � k d ( ∂ K ) ∂ K K∈T h � � � � � W L W L i � i � = U i dK + U i dK K ( t − K ( t + n +1) n ) K ∈T h � � W L F e n L i ( � ik − � U i v k − � + Θ ik )¯ k d Q Q K ∈T h

University of Twente - Chair Numerical Analysis and Computational Mechanics 18 Numerical Fluxes • The numerical flux � U at K ( t − n +1 ) and K ( t + n ) is defined as an upwind flux to ensure causality in time: � U L at K ( t − n +1 ) , � U = U R at K ( t + n ) , • At the space-time faces Q we introduce the HLLC approximate Riemann solver as numerical flux: F e U i v k )( U L , U R ) = H HLLC ( U L , U R , v, ¯ n k ( � ik − � ¯ n ) i

University of Twente - Chair Numerical Analysis and Computational Mechanics 19 ALE Weak Formulation • The ALE flux formulation of the compressible Navier-Stokes equations transforms now into: Find a U ∈ W h , such that for all W ∈ W h , the following holds: � � � ∂W i � U i + ∂W i ( F e − ik − Θ ik ) d K ∂x 0 ∂x k K K∈T n h � � � � � W L i U L W L i U R + i dK − i dK K ( t − K ( t + n +1) n ) K ∈T n h � � W L i ( H HLLC ( U L , U R , v, ¯ n L n ) − � + Θ ik ¯ k ) d Q = 0 . i Q K∈T n h