FINITE ELEMENT SUPG PARAMETERS COMPUTED FROM LOCAL DOF-MATRICES FOR COMPRESSIBLE FLOWS

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1 FINITE ELEMENT SUPG PARAMETERS COMPUTED FROM LOCAL DOF-MATRICES FOR COMPRESSIBLE FLOWS Lucia Catabriga Department of Computer Science, Federal University of Espírito Santo (UFES) Av. Fernando Ferrari, s/n, Goiabeiras, , Vitória, ES, Brazil Alvaro L. G. A. Coutinho Department of Civil Engineering - COPPE, Federal University of Rio de Janeiro (UFRJ) Caixa Postal 68506, Rio de Janeiro, RJ, Brazil alvaro@nacad.ufrj.br Tayfun E. Tezduyar Team for Advanced Flow Simulation and Modeling (T*AFSM) Mechanical Engineering and Materials Science, Rice University MS Main Street, Houston TX 77005, USA tezduyar@rice.edu Abstract. For the SUPG formulation of inviscid compressible flows, we describe stabilization parameters defined based on the degree-of-freedom submatrices of the element-level matrices. In performance tests we compare these stabilization parameters with the ones defined based on the full element-level matrices. In both cases the formulation includes a shock-capturing parameter. We investigate the difference between updating the stabilization and shock-capturing parameters at the end of every time step and at the end of every nonlinear iteration within a time step. The formulation also involves activating an algorithmic feature that is based on freezing the shock-capturing parameter at its current value when a convergence stagnation is detected. Keywords: Euler equations, compressible flow, finite elements, element-by-element data structures, stabilization parameter

2 . INTRODUCTION Stabilized formulations such as the streamline-upwind/petrov-galerkin (SUPG) (Hughes and Brooks 979, Tezduyar and Hughes 982, Tezduyar and Hughes 983) and pressurestabilizing/petrov-galerkin (PSPG) (Tezduyar 99) formulations are widely used in finite element flow computations. The SUPG and PSPG formulations prevent numerical oscillations and other instabilities in solving problems with high Reynolds and/or Mach numbers and shocks and strong boundary layers, as well as when using equal-order interpolation functions for velocity and pressure and other unknowns. It was pointed out in (Tezduyar et al. 993) that these stabilized formulations also substantially improve the convergence rate in iterative solution of the large matrix systems that need to be solved at every Newton-Raphson step. The SUPG formulation for incompressible flows was first introduced in (Hughes and Brooks 979). The SUPG formulation for compressible flows was first introduced, in the context of conservation variables, in (Tezduyar and Hughes 982, Tezduyar and Hughes 983). After that, several SUPG-like methods for compressible flows were developed. Taylor Galerkin method (Donea 984), for example, is very similar, and under certain conditions is identical, to one of the SUPG methods introduced in (Tezduyar and Hughes 982, Tezduyar and Hughes 983). Another example of the subsequent SUPG-like methods for compressible flows in conservation variables is the streamline-diffusion method described in (Johnson et al. 984). Later, following (Tezduyar and Hughes 982, Tezduyar and Hughes 983), the SUPG formulation for compressible flows was recast in entropy variables and supplemented with a shockcapturing term (Hughes et al. 987). It was shown in (Le Beau and Tezduyar 99) that the SUPG formulation introduced in (Tezduyar and Hughes 982, Tezduyar and Hughes 983), when supplemented with a similar shock-capturing term, is very comparable in accuracy to the one that was recast in entropy variables. The PSPG formulation for the Navier Stokes equations of incompressible flows, introduced in (Tezduyar 99), assures numerical stability while allowing us to use equal-order interpolation functions for velocity and pressure. An earlier version of this stabilized formulation for Stokes flow was introduced in (Hughes et al. 986). A stabilization parameter that is mostly known as τ is embedded in the SUPG and PSPG formulations. This parameter involves a measure of the local length scale (also known as element length ) and other parameters such as the local Reynolds and Courant numbers. Various element lengths and τs were proposed starting with those in (Hughes and Brooks 979) and (Tezduyar and Hughes 982, Tezduyar and Hughes 983), followed by the one introduced in (Tezduyar and Park 986), and those proposed in the subsequently reported SUPG and PSPG methods. Here we will call the SUPG formulation introduced in (Tezduyar and Hughes 982, Tezduyar and Hughes 983) for compressible flows (SUP G) 82, and the set of τs introduced in conjunction with that formulation τ 82. The stabilized formulation introduced in (Tezduyar and Park 986) for advection diffusion reaction equations included a shock-capturing term and a τ definition that takes into account the interaction between the shock-capturing and SUPG terms. That τ definition precludes compounding (i.e. augmentation of the SUPG effect by the shock-capturing effect when the advection and shock directions coincide). The τ used in (Le Beau and Tezduyar 99) with (SUP G) 82 is a slightly modified version of τ 82. A shock-capturing parameter, which we will call here δ 9, was embedded in the shock-capturing term used in (Le Beau and Tezduyar 99). Subsequent minor modifications of τ 82 took into account the interaction between the shock-capturing and the (SUP G) 82 terms in a fashion similar to how it was done in (Tezduyar and Park 986) for advection diffusion reaction equations. All these slightly modified versions of τ 82 have always been used with the same δ 9, and we will categorize them here all under the label. Recently, new ways of computing the τs based on the element-level matrices and vec-

3 tors were introduced in (Tezduyar and Osawa 2000) in the context of the advection diffusion equation and the Navier Stokes equations of incompressible flows. These new definitions are expressed in terms of the ratios of the norms of the relevant matrices or vectors. They automatically take into account the local length scales, advection field and the element-level Reynolds number. Based on these definitions, a τ can be calculated for each element, or even for each element node or degree of freedom or element equation. It was pointed out in (Tezduyar and Osawa 2000, Tezduyar 200) that the τs to be used in advancing the solution from time level n to n + (including the τ embedded in the LSIC stabilization term, which resembles a discontinuity-capturing term) should be evaluated at time level n (i.e. based on the flow field already computed for time level n), so that we are spared from another level of nonlinearity. In (Catabriga et al. 2002), the τ definitions based on the element-level matrices were applied to the (SUP G) 82 formulation for inviscid compressible flows supplemented with the shock-capturing term involving δ 9. In this paper, we apply to the same formulation the τ definitions based on the degree-of-freedom submatrices of the element-level matrices. We investigate the performance differences between these τ definitions, the τ definitions based on the full element-level matrices, and. We also investigate the performance differences between updating the stabilization and shock-capturing parameters at the end of every time step and at the end of every nonlinear iteration within a time step. The formulation includes activating an algorithmic feature, which was introduced earlier and is based on freezing the shock-capturing parameter at its current value when a convergence stagnation is detected. 2. EULER EQUATIONS The system of conservation laws governing inviscid, compressible fluid flow are the Euler equations. These equations, restricted to two spatial dimensions, may be written in terms of conservation variables U = (ρ, ρu, ρv, ρe), as U,t + F x,x + F y,y = 0 on Ω [0, T ] () where F x and F y are the Euler fluxes given elsewhere (Hirsh 992), Ω is a domain in IR 2 and T is a positive real number. We denote the spatial and temporal coordinates respectively by x = (x, y) Ω and t [0, T ], where the superimposed bar indicates set closure, and Γ is the boundary of domain Ω. Here ρ is the fluid density; u = (u x, u y ) T is the velocity vector; e is the total energy per unit mass. We add to equation () the ideal gas assumption, relating pressure with the total energy per unit mass and kinetic energy. Alternatively, equation () may be written as, U,t + A x U,x + A y U,y = 0 on Ω [0, T ] (2) where A i = F i. Associated to equation (2) we have proper boundary and initial conditions. U 3. STABILIZED FORMULATION AND STABILAZATION PARAMETERS Considering a standard discretization of Ω into finite elements, the (SUP G) 82 formulation for the Euler equations in conservation variables introduced by (Tezduyar and Hughes 982) and (Tezduyar and Hughes 983) is written as, n el e= ( W (A h k) T h τ Ω x e k ) [ U h. + A h i t Ω ] dω + x i ( U W h h. t n el e= + A h i Ωe δ 9 W h x i ) dω + x i. Uh x i dω = 0(3)

4 where W h and U h, respectively the discrete weighting and test functions, are defined on standard finite element spaces. In (3) the first integral corresponds to the Galerkin formulation, the first series of element-level integrals are the SUPG stabilization terms, and the second series of element-level integrals are the shock-capturing terms added to the variational formulation to prevent spurious oscillations around shocks. The shock-capturing parameter, δ 9, is evaluated here using the approach proposed by (Le Beau and Tezduyar 99). We define the following element-level matrices: m : c : c : k : Ωe W h Uh t dω Ω e Ω e ( W h t + Wh y.ah y x.ah x ( W h.a h x x + Wh.A h y ( W h Ω x.ah xa h x e W h y.ah ya h x y x + Wh x + Wh y.ah ya h y (4) ) dω (5) t ) dω (6) x.ah xa h y y y Considering the standard finite element approximation we have: + ) dω (7) U h = Nv, W h = Nc, U h,t = Na (8) where v is the vector of nodal values of U (depending on time only), c is a vector of arbitrary constants, a is the time derivate of the v (a = dv ) and N is a matrix containing the space dt dependent shape functions. Restricting ourselves to linear triangles, the element shape functions may be represented in matrix form as, N e = [ N I N 2 I N 3 I ] (9) where N, N 2 e N 3 are the element shape functions and I is the identity matrix of order 4. It is useful to define the discrete local (element) gradient operator as, B = [ Bx B y ] = [ N x N y ] = [ y23 I y 3 I y 2 I 2A e x 32 I x 3 I x 2 I where A e is the area of the element, x ij = x i x j, and y ij = y i y j for i, j =, 2, 3. Substituting the above relations into the weak formulation and proceeding in the standard manner we may arrive to the element matrices: m = Ωe N T NdΩ e = Ae 2 2I I I I 2I I I I 2I ] (0) () c = Ωe (B T x A h xn + B T y A h yn)dω e = m m m m 2 m 2 m 2 (2) 6 m 3 m 3 m 3

5 where m, m 2 and m 3 are given by, m = y 23 A x + x 32 A y m 2 = y 3 A x + x 3 A y (3) m 3 = y 2 A x + x 2 A y c = Ωe(N T A x B x + N T A y B y )dω e = c c 2 c 3 c c 2 c 3 (4) 6 c c 2 c 3 and the submatrices c, c 2 and c 3 are, c = y 23 A x + x 32 A y c 2 = y 3 A x + x 3 A y (5) c 3 = y 2 A x + x 2 A y k = Ω e B T [ Ax A x A x A y A y A x The submatrices in Eq.(6) are: A y A y ] BdΩ e = 4A e B 2 = y 23 (y 3 A xx + x 3 A xy ) + x 32 (y 3 A yx + x 3 A yy ) B 3 = y 23 (y 2 A xx + x 2 A xy ) + x 32 (y 2 A yx + x 2 A yy ) B 2 = y 3 (y 23 A xx + x 32 A xy ) + x 3 (y 23 A yx + x 32 A yy ) B 23 = y 3 (y 2 A xx + x 2 A xy ) + x 3 (y 2 A yx + x 2 A yy ) B 3 = y 2 (y 23 A xx + x 32 A xy ) + x 2 (y 23 A yx + x 32 A yy ) B 32 = y 2 (y 3 A xx + x 3 A xy ) + x 2 (y 3 A yx + x 3 A yy ) B = (B 2 + B 3 ) B 22 = (B 2 + B 23 ) B B 2 B 3 B 2 B 22 B 23 B 3 B 32 B 33 (6) B 33 = (B 3 + B 32 ) (7) where A ij = A i A j for i, j = x, y. 3. Computing τ by element matrix norms We define the SUPG parameters from the element matrices as given in (Tezduyar and Osawa 2000), τ S = c k τ S2 = t c 2 c where t is the time step and b = max j nee { b j + b 2j b nee,j }, n ee is the number of element equations, that is, the number of element nodes times the number of degrees of freedom per node. We define the resulting SUPG parameter, as given in (Tezduyar and Osawa 2000), ( τ SUP G = + ) /r (9) τs r τs2 r where r is an integer parameter. (8)

6 3.2 Computing τ by dof element submatrix norms We can calculate a separate τ for each element matrix degree of freedom. The resulting stabilization tensor is, τ SUP G = τ ρ τ u τ v τ e (20) where the subindexes (ρ, u, v, e) are the primitive variables. Each τ i can be calculated by τ i = ( + ) /r τ τs r i τ = c i S2 r Si i k i τ S2i = t c i 2 c i (2) where t is the time step. c i, k i and c i are the submatrices of the element matrices for each degree of freedom i = ρ, u, v, e. Theses submatrices are defined by element matrix coefficients. Let a general degree of freedom be denoted by i. Then the corresponding element submatrix b i can be written as, b p, b p,2 b p,3... b p,0 b p, b p,2 b i = b p2, b p2,2 b p,3... b p2,0 b p2, b p2,2 (22) b p3, b p3,2 b p3,3... b p3,0 b p3, b p3,2 where if i = ρ then (p, p 2, p 3 ) (, 5, 9) if i = u then (p, p 2, p 3 ) (2, 6, 0) if i = v then (p, p 2, p 3 ) (3, 7, ) if i = e then (p, p 2, p 3 ) (4, 8, 2) (23) The norms of these submatrices, used in (2) are computed by b i = max j nee { b p,j + b p2,j + b p3,j }. 4. NUMERICAL RESULTS In this section we report numerical test results obtained with various stabilization parameters. The symbol τ g represents τ calculated based on the element-level matrices, and the symbol represents τ calculated based on the degree-of-freedom submatrices of the element-level matrices. We also test different update strategies for τs and δ 9. In all test computations the number of multicorrections is 3. For the GMRES iterations, the dimension of the Krylov subspace is 5 and the tolerance is. All solutions are initialized with the free-stream values. 4. Oblique Shock The first problem consists of a two-dimensional steady problem of a inviscid, Mach 2, uniform flow, over a wedge at an angle of 0 with respect to a horizontal wall, resulting in the occurrence of an oblique shock with an angle of 29.3 emanating from the leading edge of the wedge, as shown in Fig.. The computational domain is the square 0 x and 0 y. Prescribing the following flow data at the inflow, i.e., on the left and top sides of the shock, results in the

7 y x = 0.9 M = 2.0 M = ο x Figure : Oblique shock problem description. solution with the flow data past the shock: M = 2.0 ρ =.0 Inflow u = cos0 0 Outflow u 2 = sin0 0 p = 7857 M = ρ = u = u 2 = 0.0 p = (24) where M is the Mach number, ρ is the flow density, u and u 2 are the horizontal and vertical velocities respectively, and p is the pressure. Four Dirichlet boundary conditions are imposed on the left and top boundaries; the slip condition u 2 = 0 is set at the bottom boundary; and no boundary conditions are imposed on the outflow (right) boundary. A mesh with 800 linear triangles and 44 nodes is employed. Figure 2(a) and 2(c) show, respectively, density along line x = 0.9 and the evolution of density residual for 300 steps, computed with, τ g and, with r = 2. Here we update τ g, and δ 9 at every nonlinear iteration of a time level (i.e. iteration update). We also tested what happens if we update, τ g and δ 9 only at every time step at the beginning of a time step (i.e. time-step update), the results are in Fig. 2(b) and 2(d). The evolution of density residual for time-step update is faster than iteration update. Figure 3 shows the action of a procedure (Catabriga and Coutinho 2002) detecting convergence stagnation and freezing the shock-capturing parameter, with the two update strategies for, τ g and stabilization parameters and δ 9 shock-capturing parameter. Table shows the number of GMRES iterations (N GMRES ) and number of steps (N steps ) for both update schemes. The results with need less steps than the other alternatives, but more GMRES iterations. Note too that needs more steps than all others. Results with time-step update are similar, but less steps are required. 4.2 Reflected Shock This two-dimensional steady problem consists of three regions (R, R2 and R3) separated by an oblique shock and its reflection from a wall, as shown in Fig. 4. Prescribing the following Mach 2.9 flow data at the inflow, i.e., the first region on the left (R), and requiring that the

8 (a) Density profile at x = 0.9 iteration update (b) Density profile at x = 0.9 time-step update (c) Evolution of density residual iteration update (d) Evolution of density residual time-step update Figure 2: Oblique shock solutions and residuals with iteration update and time-step update using, τ g and. Table : Oblique shock computational costs (in number of GMRES iterations and time steps) with iteration update and time-step update (both with freezing shock-capturing parameter) using, τ g and. Update τ g N GMRES N steps N GMRES N steps N GMRES N steps Iteration 5, , , Time-step 5, , ,857 89

9 (a) Density profile at x = 0.9 iteration update e-06 e-07 e-08 e-09 e-0 e (c) Evolution of density residual iteration update (b) Density profile at x = 0.9 time-step update e-06 e-07 e-08 e-09 e-0 e (d) Evolution of density residual time-step update Figure 3: Oblique shock solutions and residuals with iteration update and time-step update (both with freezing shock-capturing parameter) using, τ g and.

10 incident shock to be at an angle of 29, leads to the solution (R2 and R3): M = 2.9 M = M = ρ =.0 ρ =.7 ρ = R u = 2.9 R2 u = R3 u = u 2 = 0.0 u 2 = u 2 = 0.0 p = p =.5289 p = (25) y M=2.378 M=2.9 y= ο 2 ο M= x Figure 4: Reflected shock problem description. We prescribe density, velocities and pressure on the left and top boundaries; the slip condition is imposed on the wall (bottom boundary); and no boundary conditions are set on the outflow (right) boundary. We consider a structured mesh with cells, where each cell was divided into two triangles (28 nodes and 2400 elements) and an unstructured mesh with,837 nodes and 3,429 elements covering the domain 0 x 4. and 0 y. Figure 5(a) and 5(b) show the density along line y = 0.25 for the structured mesh, computed by a parameter, τ g parameter and, respectively, for iteration update and timestep update, both with r = 2. The density profile with and time-step update shows some discrete oscillations. The evolution of density residual for 300 steps can be observed in Fig. 5(c) and 5(d). We observe that the density residual computed updating τ g, and δ 9 with time-step update is smaller than the one computed by iteration update. Figure 6 shows the action of the convergence stagnation detection procedure, with the two updating strategies for τ g, and δ 9. In Fig. 6(a) and 6(b) we can observe the density profiles computed with the convergence stagnation procedure switched on are very similar to the previous cases. However, here lead to a better evolution of the density residual (6(c) and 6(d)). We may see in Table 2 that here solution required less steps for both update schemes. It is also interesting to note that τ g and solutions were more sensitive to the particular update scheme, with better results for time-step update. Table 2: Reflected shock (with structured mesh) computational costs (in number of GMRES iterations and time steps) with iteration update and time-step update (both with freezing shock-capturing parameter) using, τ g and. Update τ g N GMRES N steps N GMRES N steps N GMRES N steps Iteration 3, , , Time-step 3, , , Figures 7 and 8 show the results for the unstructured mesh, computed by using the same τs and update strategies of the previous case. The action of the convergence stagnation detection

11 (a) Density profile at y = 0.25 iteration update (b) Density profile at y = 0.25 time-step update (c) Evolution of density residual iteration update (d) Evolution of density residual time-step update. Figure 5: Reflected shock (with structured mesh) solutions and residuals with iteration update and time-step update using, τ g and.

12 (a) Density profile at y = 0.25 iteration update e-06 e-07 e-08 e-09 e-0 e (c) Evolution of density residual iteration update (b) Density profile at y = 0.25 time-step update e-06 e-07 e-08 e-09 e-0 e (d) Evolution of density residual time-step update Figure 6: Reflected shock (with structured mesh) solutions and residuals with iteration update and time-step update (both with freezing shock-capturing parameter) using, τ g and.

13 procedure can be observed. We see that the steady-state density distributions are in good agreement. We may see im Table 3 that for the unstructured mesh results with are slightly better than all others for both update schemes. (a) Iteration update using (b) Time-step update using (c) Iteration update using τ g (d) Time-step update using τ g (e) Iteration update using (f) Time-step update using Figure 7: Reflected shock density computed with iteration update and time-step update (both with freezing shock-capturing parameter) using, τ g and. e-06 e-06 e-07 e-07 e-08 e-08 e-09 e-09 e-0 e-0 e (a) Evolution of density residual iteration update. e (b) Evolution of density residual time-step update. Figure 8: Reflected shock (with unstructured mesh) residuals with iteration update and time-step update (both with freezing shock-capturing parameter) using, τ g and. 5. CONCLUDING REMARKS We described, for the SUPG formulation of inviscid compressible flows, stabilization parameters defined based on the degree-of-freedom submatrices of the element-level matrices. These definitions are expressed in terms of the ratios of the norms of the relevant matrices, and take automatically into account the flow field, the local length scales, and the time step size. By inspecting the solution quality and convergence history, we compared these stabilization

14 Table 3: Reflected shock (with unstructured mesh) computational costs (in number of GMRES iterations and time steps) with iteration update and time-step update (both with freezing shock-capturing parameter) using, τ g and. Update τ g N GMRES N steps N GMRES N steps N GMRES N steps Iteration 9, , , Time-step 9, , , parameters with the ones defined based on the full element-level matrices. In both cases the formulation includes a shock-capturing parameter. Also by inspecting the solution quality and convergence history, we investigated the performance difference between updating the stabilization and shock-capturing parameters at the end of every time step and at the end of every nonlinear iteration within a time step. The formulation also involves activating an algorithmic feature that is based on freezing the shock-capturing parameter at its current value when a convergence stagnation is detected. We observe that in all cases the solution qualities are very comparable. In terms of computational efficiency, τ definitions based on the degree-of-freedom submatrices and full element-level matrices show comparable performances. 6. ACKNOWLEDGMENTS This work is partly supported by grant CNPq/MCT /95-8. The third author was supported by the US Army Natick Soldier Center and NASA. REFERENCES Catabriga, L., A. L. G. A. Coutinho and T.E. Tezduyar (2002). Finite element supg parameters computed from local matrices for compressible flows. In: 9th Brazilian Congress of Thermal Engineering and Sciences CD-ROM. Caxambu, Minas Gerais, Brazil. pp.. Catabriga, L. and A. L. G. A. Coutinho (2002). Improving convergence to steady-state of implicit SUPG solution of Euler equations. Communications in Numerical Methods in Engineering 8, Donea, J. (984). A Taylor-Galerkin method for convective transport problems. International Journal for Numerical Methods in Engineering 20, Hirsh, C. (992). Numerical Computation of Internal and External Flows - Computational Methods for Inviscid and Viscous Flows, Vol. 2. John Wiley and Sons Ltd. Chichester. Hughes, T. J. R. and A. N. Brooks (979). A multi-dimensional upwind scheme with no crosswind diffusion. In: Finite Element Methods for Convection Dominated Flows (T.J.R. Hughes, Ed.). Vol. 34. ASME. New York. pp Hughes, T. J. R., L. P. Franca and M. Balestra (986). A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška Brezzi condition: A stable Petrov Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Computer Methods in Applied Mechanics and Engineering 59, Hughes, T. J. R., L. P. Franca and M. Mallet (987). A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formu-

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