SIMPLE EULER / NAVIER – STOKES SOLVERS 163 This is the same as central differencing! A stability analysis of the RHS, ri = − (ui+1 − ui−1 ), 2h (8.15) indicates that only every second node is coupled, allowing zero-energy or chequerboard modes in the solution Therefore, stabilizing terms have to be added to re-couple neighbouring nodes The two most common stabilizing terms added are as follows Terms of second order: h2 ∂ /∂x , which result in an additional RHS term of ui+1 − 2ui + ui−1 , (8.16) which equals for the (−1, 1, −1) chequerboard pattern on a uniform mesh shown in Figure 8.1 Most TVD schemes use this type of stabilization term u i-1 i i+1 i+2 x -1 Figure 8.1 Chequerboard mode on a uniform 1-D grid Terms of fourth order: h4 ∂ /∂x , which result in an additional RHS term of ui+2 − 4ui+1 + 6ui − 4ui−1 + ui−2 , (8.17) which equals 16 for the (−1, 1, −1) chequerboard pattern on a uniform mesh shown in Figure 8.1 Observe that this type of stabilization term has a much more pronounced effect than second-order terms Therefore, one may use much smaller constants when adding them The fourth-order operator can be obtained in several ways One obvious choice is to perform two ∇ -passes over the mesh (Jameson et al (1986), Mavriplis and Jameson (1990)) Another option is to first obtain the gradients of u at points, and then to approximate the third derivatives by taking a difference between first derivatives obtained from the gradients and the first derivatives obtained directly from the unknowns (Peraire et al (1992a)) The implications of choosing either of these approaches will be discussed in more depth in Chapter 10 164 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES 8.1.1 EQUIVALENCY WITH FVM Before going on, the GFEM using linear elements will be shown to be equivalent to the FVM Integration by parts in (8.8) yields ri = − ˆ N i ∇ · N j F(uj ) d ˆ (∇N i )N j · F(uj ) d = + (8.18) As seen from Section 4.2.4, for linear triangles, this integral reduces to ri = A (∇N i ) · ˆ F(uj ) = jel 1 · (si ni ) · ˆ F(uj ), (8.19) jel and in particular, for node a in Figure 8.2, = Fb + Fc Fa + 2 sa na · (8.20) On the other hand, sa na = 0, (8.21) surrel which implies that when taking the sum over all surrounding elements, the first term on the RHS of (8.20) vanishes The remaining two terms are the same as those obtained when performing a FVM line-integral approximation around node a Fb S n a a b F c c F a a Figure 8.2 Equivalency of linear GFEM and FVM The same equivalency between the GFEM with linear elements and FVMs can also be shown for 3-D tetrahedra 8.2 Lax–Wendroff (Taylor–Galerkin) All Lax–Wendroff type schemes are derived by performing a Taylor-series expansion in time (Lax and Wendroff (1960), Donea (1984), Löhner et al (1984)) u= tu,t + t2 u,tt n+ (8.22) SIMPLE EULER / NAVIER – STOKES SOLVERS 165 Then the original equation, u,t = −∇ · F, (8.23) is used repeatedly in conjunction with the Jacobians of the fluxes A, F=A· u (8.24) This results in n+ t2 ∇ ·A·∇ ·F Linearization of the second term on the RHS via u = − t∇ · F + F|n+ = Fn + A· u (8.25) (8.26) yields the final scheme: 1− t2 ∇ · A ⊗ A∇· u = − t∇ · F + t2 ∇ · A · ∇ · F (8.27) This timestepping scheme is then discretized in space using the GFEM Note that the last term in (8.27) provides a matrix dissipation (or streamline upwinding for the scalar advection equation) This is very similar to the streamline upwind Petrov–Galerkin (SUPG) techniques (Tezduyar and Hughes (1983), Hughes and Tezduyar (1984), Le Beau and Tezduyar (1991)) for linear/bilinear elements, except that in the latter case the multiplicative factor is not the timestep but a mesh and flow variables dependent variable 8.2.1 EXPEDITING THE RHS EVALUATION The appearance of the Jacobians A makes the evaluation of the RHS tedious Substantial savings may be realized by developing two-step schemes that approximate the RHS without recourse to A (Burstein (1967), Lapidus (1967), MacCormack (1969)) The simplest two-step scheme that accomplishes this goal is given by: Half-step: predict un+0.5 un+0.5 = u − t ∇ · F (8.28a) Full-step: use un+0.5 u = − t∇ · F(un+0.5 ) = − t∇ · F u − t ∇ ·F t2 ∇ · A · ∇ · F(u) (8.28b) Several possibilities exist for the spatial discretization If one chooses linear shape functions for both half-steps (N i , N i ), a five-point stencil is obtained in one dimension The scheme obtained is identical to a two-step Runge–Kutta GFEM Thus, for the steady state no damping is present Another possibility is to choose constant shape functions for the half-step solution (P e , N i ) This choice recovers the original three-point stencil in one dimension, and for the linear advection equation yields the same scheme as that given by (8.27) Observe that this scheme has a second-order damping operator for the steady state Thus, this second scheme requires no additional stabilization operator for ‘mild’ problems (e.g subsonic flows without shocks) The difference between these schemes is shown conceptually in Figure 8.3 = − t∇ · F(u) + 166 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES time t n+1 t n+0.5 tn space time t n+1 t n+0.5 tn space Figure 8.3 Two-step schemes 8.2.2 LINEAR ELEMENTS (TRIANGLES, TETRAHEDRA) Because of the widespread use of the latter scheme, a more detailed description of it will be given The spatial discretization is given by - at t n+ = t n + t: u, F piecewise constant; - at t n , t n+1 : u, F piecewise linear (a) First step When written out, the first step results in the following GATHER, ADD operations: Nn n t Nn i k n+ · · ui − N F , (8.29a) uel = Nn i=1 i=1 ,k i where Nn denotes the number of nodes of the element (b) Second step The second step involves SCATTER-ADD operations: i j N N d · t · uj = Nn i k VOLel · N,k Fel n+ , (8.29b) el and results in the matrix system Mc · un = rn The flow of information for these two steps is shown in Figure 8.4 (8.30) SIMPLE EULER / NAVIER – STOKES SOLVERS (a) 167 ( b) Figure 8.4 Two-step Taylor–Galerkin scheme 8.3 Solving for the consistent mass matrix Due to its very favourable conditioning, (8.30) is best solved iteratively as Ml · ( un − i+1 un ) = rn − Mc · i un , i = 0, , niter, un = 0 (8.31) In practice, no more than three iterations are employed (niter ≤ 3) 8.4 Artificial viscosities An observation made time and time again is that for shocks or other discontinuities the damping provided by the stabilized GFEM or the Lax–Wendroff schemes is not sufficient The simulated flowfields exhibit unphysical results, with overshoots or undershoots in pressure, density, etc In most cases, the numerical results will eventually diverge or, to use the more common term, ‘blow up’ Therefore, some additional artificial dissipation or stabilization (a more elegant term for the same thing) has to be added In a real flow, the vicinity of a shock is controlled by the viscous effects that become large over the small width of the shock The shock width is typically orders of magnitude smaller than the smallest mesh size used in current simulations Therefore, this viscosity effect has to be scaled to the current mesh size At the same time, the effects of these artificial viscosity terms should only be confined to shock or discontinuity regions Thus, a sensing function must be provided to locate these regions in space These considerations lead to artificial viscosity operators of the form t d= ∇(h2 f (u))∇u (8.32) tl Here f (u) denotes the sensing function and h the element size The ratio of timestep taken ( t) to allowable timestep ( t l ) is necessary to avoid a change in solution when shifting from local to global timesteps Some popular artificial viscosities are listed below (a) Lapidus Defined by (see Figure 8.5) d= where l= ∇|v| , |∇|v|| t· ∂ ∂u |k ll | , ∂l ∂l k ll = c1 · h2 · ∂(v · l) ∂l The salient features of this type of artificial viscosity are as follows: (8.33) (8.34a, b) 168 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES Shock Shear Layer v v l v v l Figure 8.5 Lapidus artificial viscosity - it is invariant under coordinate rotation; - it is essentially 1-D and thus fast; - it produces the desired effects at shocks; - k ll vanishes at shear and boundary layers; - the identification of shocks (k ll < 0) or expansions (k ll > 0) is simple The Lapidus artificial viscosity (Lapidus (1967), Löhner et al (1985a)) is most often employed for transient simulations in conjunction with more sophisticated FCT (Löhner et al (1987)) or TVD schemes (Woodward and Colella (1984)) in order to add some background damping (b) Pressure-based Given by d= t ∇(h2 f (p))∇u tl (8.35) The salient features of this type of artificial viscosity are as follows: - it is invariant under coordinate rotation; - ∇ may be approximated by (Ml − Mc ), yielding a fast scheme; - because the pressure p is near constant in shear and boundary layers, f (p) should vanish there; - usually, the enthalpy is taken for the energy equation in u Many variations have been proposed for the sensing function f (p) The three more popular ones are as follows: (b1) Jameson–Schmidt–Turkel: f (p) = c1 |∇h2 ∇p| , p ¯ (8.36) which is very good for transonic flows (Jameson et al (1981), Löhner et al (1985b)); SIMPLE EULER / NAVIER – STOKES SOLVERS 169 (b2) Hassan–Morgan–Peraire: f (p) = c2 |∇h2 ∇p| , |h∇p| (8.37) which is more suitable for hypersonic flows (Hassan et al (1990)); and (b3) Swanson–Turkel: f (p) = c3 |∇h2 ∇p| , α|h∇p| + (1 − α)p ¯ (8.38) which is a blend of the first two (Swanson and Turkel (1992)) Typically, α = 0.5 8.5 Boundary conditions No numerical scheme is complete without proper ways of imposing boundary conditions The Euler equations represent a hyperbolic system of PDEs Linearization around any arbitrary state yields a system of advection equations for so-called characteristic variables (Usab and Murman (1983), Thomas and Salas (1985)) The proper handling of these boundary conditions will be discussed below The following notation will be employed: ρ: : vt : p: c: the density; the normal velocity component at the boundary (pointing inwards); the tangential velocity component at the boundary; the pressure; and the velocity of sound Furthermore, use will be made of the Bernoulli equation, given by the total pressure relation: v2 γ p v2 γ p0 + = const., + = γ − ρ0 γ −1ρ and the isentropic relation p p0 = γ ργ ρ0 (8.40) (8.41) Given a velocity, and assuming constant total pressure along the streamlines, the pressure and density may be computed as p0 γ −1 p = [v0 − v ], + ρ ρ0 2γ (8.42) and from the isentropic relation (8.41), γ ρ= ρ0 p p0 ρ 1/(γ −1) (8.43) With ρ, p is obtained again from (8.41): p= p0 γ γρ ρ0 (8.44) 170 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES Suppose that an explicit evaluation of the fluxes (i.e the RHS) has been performed and the unknowns have been updated These predicted unknowns, that have to be corrected at boundaries, will be denoted by ‘∗’, e.g the predicted density as ρ∗ A linearized characteristics analysis can be used to correct the predicted (∗) values; performing a 1-D analysis at the boundary, in the direction normal to it, we have, with the inward normal n (see Figure 8.6), the following set of eigenvalues and eigenvectors: v λ= n , vn + c − c ρ − p/c2 vt W= [vn + p/(ρc)] , [−vn + p/(ρc)] (8.45) where u denotes the average between the values at the previous timestep and the predicted ¯ state for unknown u n Subsonic Inflow Wing Wall Subsonic Outflow n Ma > P − , otherwise (9.9) For the elements, the final value taken is the most conservative: Cel = min(element nodes) R+ R− if AEC > 0, if AEC < (9.10) max The allowed value uimin is taken between each point and its nearest-neighbours For elementbased schemes, it may be obtained in three steps as follows: (a) maximum (minimum) nodal unknowns of un and ul : u∗ = i max (ul , un ); i i (9.11) (b) maximum (minimum) nodal value of element: u∗ = el max (u∗ , u∗ , , u∗ ); A B C (9.12) 178 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES (c) maximum (minimum) unknowns of all elements surrounding node i: max uimin = max (u∗ , u∗ , , u∗ ) m (9.13) max A number of variations are possible for uimin For example, the so-called ‘clipping limiter’ is obtained by setting u∗ = max ul in (a), i.e by not looking back to the solution at the previous i i timestep or iteration, but simply comparing nearest-neighbour values at the new timestep or iteration for the low-order scheme As remarked before, the limiting is based solely on the unknowns u, not on a ratio of differences as in most TVD schemes 9.2 Steepening The anti-diffusive step was designed to steepen the low-order solution obtained at the new timestep or iteration In some cases, particularly for high-order schemes of order greater than two, the anti-diffusive step can flatten the profile of the solution even further, or lead to an increase of wiggles and noise This is the case even though the solution remains within its allowed limits A situation where this is the case can be seen from Figure 9.2 AEC x u un ul un+1 ( = 0) un+1 ( =-1) x Figure 9.2 Steepener for FCT This type of behaviour can be avoided if the anti-diffusive flux is either set to zero or reversed A simple way to decide when to reverse the anti-diffusive fluxes is to compute the scalar product of the low-order solution at the new timestep or iteration and the anti-diffusive element contributions: ∇ul · AEC < ⇒ AEC = −α · AEC, (9.14) with < α < Values of α greater than unity lead to steepening This can be beneficial in some cases, but is highly dangerous, as it can lead to unphysical solutions (e.g spurious contact discontinuities) in complex applications FLUX - CORRECTED TRANSPORT SCHEMES 179 9.3 FCT for Taylor–Galerkin schemes For the explicit Taylor–Galerkin schemes, FCT takes a particularly simple form We have: - high-order: consistent mass Taylor–Galerkin, Mc · uh = r ⇒ Ml · uh = r + (Ml − Mc ) · uh ; (9.15a) - low-order: lumped mass Taylor–Galerkin + diffusion, Ml · ul = r + d = r − cτ (Ml − Mc ) · u, (9.15b) where cτ denotes a diffusion coefficient Thus AEC = [(Ml − Mc ) · (cτ u + uh )]el , (9.16) implying that no physical flux terms are left in AEC This is typical of FCT schemes, and reflects the fact that monotonicity may be imposed without knowledge of the physics described by the fluxes This separation of numerics and physics has several advantages, among them speed and generality The low-order solution will be monotone for a diffusion coefficient cτ = For cases where a large variation of possible timesteps is encountered throughout the mesh, the allowable diffusion factor may be much lower The ‘optimal’ choice (in the sense of least diffusive yet monotone low-order scheme) is to take cτ = t g / t l , where t g is the global timestep chosen (i.e the minimum over all elements) and t l the allowable local timestep ( t l ≥ t g ) 9.4 Iterative limiting Given that the anti-diffusive element contributions surrounding a point can have different signs, and that a ‘most conservative’ compromise has to be reached for positive and negative contributions, the remaining (i.e not yet added) anti-diffusive element contributions may still be added to the new solution without violating monotonicity constraints This may be achieved by the following iterative limiting procedure: For iterations j = 1, k: - perform limiting procedure, un+1 = ul + Cel · AEC; el - update remaining anti-diffusive element contributions, AEC ← (1 − Cel )AEC; - re-define the low-order solution at t n+1 , ul = un+1 Experience indicates that, for explicit schemes, the improvements obtained by this iterative limiting procedure are modest However, for implicit schemes, the gains are considerable and well worth the extra computational effort (Kuzmin (2001), Kuzmin and Turek (2002), Kuzmin et al (2003, 2005)) 180 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES 9.5 Limiting for systems of equations The results available in the literature (Boris and Book (1973), Book and Boris (1975), Boris and Book (1976), Zalesak (1979), Parrott and Christie (1986), Löhner et al (1987, 1988), Zalesak and Löhner (1989)) indicate that with FCT results of excellent quality can be obtained for a single PDE, e.g the scalar advection equation However, in the attempt to extend the limiting process to systems of PDEs no immediately obvious or natural limiting procedure becomes apparent Obviously, for 1-D problems one could advect each simple wave system separately, and then assemble the solution at the new timestep However, for multi-dimensional problems such a splitting is not possible, as the acoustic waves are circular in nature FDM–FCT codes used for production runs (Fry and Book (1983), Fyfe et al (1985)) have so far limited each equation separately, invoking operator-splitting arguments This approach does not always give very good results, as may be seen from Sod’s (1978) comparison of schemes for the Riemann problem, and has been a point of continuing criticism by those who prefer to use the more costly Riemann-solver-based, essentially 1-D TVD schemes (van Leer (1974), Roe (1981), Osher and Solomon (1982), Harten (1983), Sweby (1984), Colella (1990), Woodward and Colella (1984)) An attractive alternative is to introduce ‘system character’ for the limiter by combining the limiters for all equations of the system Many variations are possible and can be implemented, giving different performance for different problems Some of the possibilities are listed here, with comments where empirical experience is available (a) Independent treatment of each equation as in operator-split FCT This is the least diffusive method, tending to produce an excessive amount of ripples in the non-conserved quantities (and ultimately also in the conserved quantities) (b) Use of the same limiter (Cel ) for all equations This produces much better results, seemingly because the phase errors for all equations are ‘synchronized’ This was also observed by Harten and Zwaas (1972) and Zhmakin and Fursenko (1981) for a class of schemes very similar to FCT We mention the following possibilities (i) Use of a certain variable as ‘indicator variable’ (e.g density, pressure, entropy) (ii) Use of the minimum of the limiters obtained for the density and the energy (Cel = min(Cel (ρ), Cel (ρe))): this produces acceptable results, although some undershoots for very strong shocks are present This option is currently the preferred choice for strongly unsteady flows characterized by propagating and/or interacting shock waves (iii) Use of the minimum of the limiters obtained for the density and the pressure (Cel = min(Cel (ρ), Cel (p))): this again produces acceptable results, particularly for steadystate problems 9.5.1 LIMITING ANY SET OF QUANTITIES A general algorithm to limit any set of quantities may be formulated as follows: - define a new set of (non-conservative) variables u ; - transform u, ul → u , u l and see how much u can change at each point ⇒ u |max ; FLUX - CORRECTED TRANSPORT SCHEMES 181 - define a mean value of ul in the elements and evaluate u = A(ul ) · - limit the transformed increments u; u ⇒ Cel ⇒ u = lim( u ); (9.17) - transform back the variables and add u∗ = A−1 (ul ) · u = A−1 (ul ) · Cel · A(ul ) · u (9.18) Excellent results using limiting based on characteristic variables have been reported by Zalesak (2005) (see also Kuzmin et al (2003, 2005)) 9.6 Examples 9.6.1 SHOCK TUBE This is a classic example, which was used repeatedly to compare different Euler solvers (Sod (1978)) Initially, a membrane separates two fluid states given by ρ1 = 1.0, v1 = 0.0, p1 = 1.0 and ρ2 = 0.1, v2 = 0.0, p2 = 0.1 The membrane ruptures, giving rise to a shock, a contact discontinuity and a rarefaction wave The results were obtained for an FCT run with limiter synchronization on the density and energy Figure 9.3 shows the surface mesh and the surface contours of the density A line cut through the 3-D mesh is compared to the exact solution in Figure 9.4 Note that the number of points appearing here corresponds to the faces of the tetrahedra being cut, i.e., it is two to three times the number of actual points One can see that the shock and contact discontinuities are captured over two or four elements, respectively Contact Discontinuity Shock Figure 9.3 Shock tube: surface mesh and density 182 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES 1.1 1.2 Exact FEFLO Exact FEFLO 0.9 0.8 0.7 VELOCITY DENSITY 0.8 0.6 0.5 0.6 0.4 0.4 0.2 0.3 0.2 0.1 0 -0.2 10 X-COORDINATE 10 X-COORDINATE 1.1 Exact FEFLO 0.9 PRESSURE 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 X-COORDINATE Figure 9.4 Shock tube: comparison to exact values 9.6.2 SHOCK DIFFRACTION OVER A WALL The second example shown is typical of some of the large-scale blast simulations carried out with FCT schemes over the last decade (Baum and Löhner (1991), Sivier et al (1992), Baum et al (1993, 1995), Baum and Löhner (1994), Löhner et al (1999)), and is taken from Rice et al (2000) The outline of the domain is shown in Figure 9.5 Surface pressures for an axisymmetric and 3-D run, together with a comparison to experimental photographs are given in Figures 9.6 and 9.7 The comparison to experimental results at different stations is shown in Figure 9.8 As one can see, FCT schemes yield excellent results for this class of problem Pressure Gages Hm HoB Wg W Hg L2 L3 Figure 9.5 Shock diffraction over a wall Lg FLUX - CORRECTED TRANSPORT SCHEMES 183 Figure 9.6 Results at time t = 146 µs Figure 9.7 Results at time t = 327 µs 9.7 Summary The notion of limiting was introduced by reviewing FCT schemes As seen above, these schemes not require gradient evaluations for the limiting procedure, and in some instances 184 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES 2.5 3D Simulation 2D Simulation Experiment Pressure (bar) 1.5 0.5 –0.5 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 Time (s) 5.5 4.5 3D Simulation 2D Simulation Experiment Pressure (bar) 3.5 2.5 1.5 0.5 –0.5 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 Time (s) 5.5 4.5 3D Simulation 2D Simulation Experiment Pressure (bar) 3.5 2.5 1.5 0.5 –0.5 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 Time (s) Figure 9.8 Comparison to experimental values avoid the recalculation of fluxes while limiting This makes them extremely fast as compared to TVD schemes Due to their generality and ease of extension to other physical problems that require monotonicity preserving schemes, FCT has been used extensively in many fields, among them simulation of shock propagation (Fyfe et al (1985), Löhner et al (1987, 1988), FLUX - CORRECTED TRANSPORT SCHEMES 185 Baum and Löhner (1991), Sivier et al (1992), Baum et al (1993, 1995), Baum and Löhner (1994), Löhner et al (1999)), detonation and flame propagation (Patnaik et al (1987, 1989), magnetohydrodynamics (Scannapieco and Ossakow (1976)) and plasma physics Tests have shown that FCT schemes compare very well, or in some cases are superior, to standard TVD schemes (Zalesak and Löhner (1989), Luo et al (1993)) or even spectral schemes (deFainchtein et al (1995)) This is surprising, considering that no detailed wave analysis is invoked during the limiting process 10 EDGE-BASED COMPRESSIBLE FLOW SOLVERS Consider the typical formation of a RHS using a finite element approximation with shape functions N i The resulting integrals to be evaluated are given by ri = N i r(u) d = N i r(N j uj ) d el (10.1) el These integrals operate on two sets of data: (a) point data, for ri , ui ; and (b) element data, for volumes, shape functions, etc The flow of information is as follows: GATHER point information into the element (e.g ui ); operate on element data to evaluate the integral in (10.1); and SCATTER-ADD element RHS data to point data to obtain ri For many simple flow solvers the effort in step may be minor compared to the cost of indirect addressing operations in steps and A way to reduce the indirect addressing overhead for low-order elements is to change the element-based data structure to an edgebased data structure This eliminates certain redundancies of information in the elementbased data structure To see this more clearly, consider the formation of the RHS for the Laplacian operator on a typical triangulation Equation (10.1) may be recast as ri = K ij uj = Kel uel = el rel (10.1a) el This immediately opens three possibilities: - obtain first the global matrix K ij and store it in some optimal way (using so-called sparse storage techniques); - perform a loop over elements, obtaining rel and adding to r; and - obtain the off-diagonal coefficients K ij , i = j , perform a loop over edges i, j , obtaining rij and adding to r Applied Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods, Second Edition Rainald Lưhner © 2008 John Wiley & Sons, Ltd ISBN: 978-0-470-51907-3 188 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES The most economic choice will depend on the number of times the RHS needs to be evaluated, how often the mesh changes, available storage, etc The indirect addressing overhead incurred when performing a RHS evaluation for the Laplacian using linear elements, as well as the associated FLOPS, are given in Table 10.1 Recall that for a typical mesh of triangles nelem=2*npoin, nedge=3*npoin, npsup=6*npoin, whereas for a typical mesh of tetrahedra nelem=5.5*npoin, nedge=7*npoin, npsup=14*npoin Table 10.1 (a) Laplacian: Gather/Scatter overhead for linear elements (b) FLOPS overhead: Laplacian/linear elements Element-Based Edge-Based Sparse 3*3*nelem=18*npoin 3*4*nelem=66*npoin 3*2*nedge=18*npoin 3*2*nedge=42*npoin 6*npoin 14*npoin 3*6*nelem= 36*npoin 4*8*nelem=176*npoin 4*nedge=12*npoin 4*nedge=28*npoin 12*npoin 28*npoin (a) 2-D 3-D (b) 2-D 3-D The indirect addressing overhead incurred when performing a RHS evaluation for the Galerkin approximation to the advective fluxes and the associated FLOPS are given in Table 10.2 Table 10.2 (a) Gather/Scatter overhead for linear elements (b) FLOPS overhead: Galerkin fluxes for linear elements Element-Based Edge-Based 2*3*nelem=12*npoin 2*4*nelem=44*npoin 2*2*nedge=12*npoin 2*2*nedge=28*npoin (a) 2-D 3-D (b) 3-D 22*nelem=121*npoin 7*nedge=49*npoin As one can see, a considerable reduction of indirect addressing operations and FLOPS has been achieved by going to an edge-based data structure Given their importance in generalpurpose CFD codes, the present chapter is devoted to edge-based solvers Starting from the by now familiar element integrals obtained for Galerkin weighted residual approximations, the resulting edge-based expressions are derived Thereafter, a brief taxonomy of possible solvers is given 10.1 The Laplacian operator The geometrical expressions required when going from an element-based data structure to an edge-based data structure will now be derived For simplicity and clarity we start with the Laplacian operator The RHS in the domain is given by ri = − ∇N i · ∇N j d ∇N i · ∇N j d uj = − ˆ el uj ˆ (10.2) EDGE - BASED COMPRESSIBLE FLOW SOLVERS 189 This integral can be split into those shape functions that are the same as N i and those that are different (j = i): ∇N i · ∇N j d ri = − j =i ∇N i · ∇N i d uj − ˆ el ui ˆ (10.3) el The conservation property of the shape functions i N,k = − j j =i (10.4) N,k , allows (10.3) to be re-written as ∇N i · ∇N j d ri = − j =i ∇N i · uj + ˆ el el ∇N j d ui , ˆ (10.5) j =i or, after interchange of the double sums, r i = k ij (ui − uj ), ˆ ˆ ∇N i · ∇N j d , k ij = j = i (10.6) el One may observe that: - from a change in indices (ij versus ji) we obtain k ji = k ij ; this is expected from the symmetry of the Laplacian operator; - the edge and its sense of direction have to be defined; this is accomplished by - a connectivity array that stores the points of an edge inpoed(1:2,nedge):= ip1, ip2, and - taking ip1 < ip2 to define the orientation; - to build the stiffness coefficients k ij we require the edges of an element, i.e a list of the form inedel(nedel,nelem):= ied1,ied2, The rapid construction of both inpoed and inedel have been described in Chapter A typical edge-based evaluation of the Laplacian RHS would look like the following: rhspo(1:npoin)=0 ! Set points-RHS to zero iedge=1,nedge ! Loop over the edges ipoi1=inpoed(1,iedge) ! 1st point of edge ipoi2=inpoed(2,iedge) ! 2nd point of edge ! Evaluate the edge-RHS redge=cedge(iedge)*(unkno(ipoi2)-unkno(ipoi1) rhspo(ipoi1)=rhspo(ipoi1)-redge ! Subtract from ipoi1 rhspo(ipoi2)=rhspo(ipoi2)+redge ! Add to ipoi2 enddo The conservation law expressed by the Laplacian as ∇ · ∇u = is reflected on each edge: whatever is added to a point is subtracted from another The flow of information and the sequence of operations performed in the loop are shown in Figure 10.1 ... density 18 2 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES 1. 1 1. 2 Exact FEFLO Exact FEFLO 0.9 0 .8 0.7 VELOCITY DENSITY 0 .8 0.6 0.5 0.6 0.4 0.4 0.2 0.3 0.2 0 .1 0 -0 .2 10 X-COORDINATE 10 X-COORDINATE... et al (19 85 ), Löhner et al (19 87 , 19 88 ), FLUX - CORRECTED TRANSPORT SCHEMES 18 5 Baum and Löhner (19 91) , Sivier et al (19 92), Baum et al (19 93, 19 95), Baum and Löhner (19 94), Löhner et al (19 99)),... r Applied Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods, Second Edition Rainald Lưhner © 20 08 John Wiley & Sons, Ltd ISBN: 97 8- 0 -4 7 0-5 19 0 7-3 18 8 APPLIED