Numerical_Methods for Nonlinear Variable Problems Episode 7 pptx

4 Transonic Flow Calculations by Least-Squares and Finite Element Methods 227 Figure 4.8 therein) in order to eliminate nonphysical shocks; these upwinding techniques have been very effective, coupled with alternating-direction methods (implicit or semi-implicit; see Hoist [1] and Deconinck and Hirsch [1]) if the com- putational mesh is regular (finite differences or regular finite element grids). In particular, their application, combined with finite element techniques, have been limited (see Eberle [1], [2] and Deconinck and Hirsh [1]) to quadrilateral elements on quasiregular quadrangulations (fairly close to finite difference methods, in our opinion). In Sec. 4.6.3.2 we would like to discuss a method (due to M. O. Bristeau) which also makes use of an upwinding of the density; this method can be used with simplicial 10 finite elements (triangles in two dimension, tetrahedra in three dimensions) and has been very effective for computing flows at high Mach numbers and around complicated two- and three-dimensional geo- metries. 4.6.3.2. A modified discrete continuity equation by upwinding of the density in the flow direction. Following Jameson [l]-[4] and Bristeau [2], [3], we may write the continuity equation (4.14) in a system of local coordinates {s, n}, where (see Fig. 4.8) for a two-dimensional flow, s is the unit vector of the stream direction (i.e., s = u/1 u | if u # 0) and n is the corresponding normal unit vector (conventionally oriented). Using {s, n} and setting 11 we obtain (from (4.14)) 10 We use here the terminology of Ciarlet [1], [2]. 1 ' U is the Mach number M*. 228 VII Least-Squares Solution of Nonlinear Problems the elliptic-hyperbolic aspect of the problem is clear from (4.61). Actually (4.61) can also be written 0 + V * V O (462) [we have 1/2/ca = (y + l)/2]. EXERCISE 4.3. Prove (4.61), (4.62). We use (4.62) to modify the discrete continuity equation (4.27) as follows: Find 4> h e V h such that Vv h dx + ^- £ (^ h s {U 2 h - l) + \<P h • \p k ^v h dx = \g h v h dT, Vv h eV h . (4.63) The approximate problem (4.63) has been introduced by M. O. Bristeau and is a finite element variant of a finite difference scheme due to Hoist [1]. In (4.63): (i) h s is a measure of the local size of the finite element mesh in the flow direction, (ii) (d/ds) h is an approximation of (V^/| V(/> A |) • V, (iii) U h , p h are upwinded approximations of U and p, respectively. More precisely, we write the second integral in the left-hand side of (4.63) as follows. = (-1) Z ^Z^ E m(T)h s (T)(U 2 h - 1) + V^-V^-^, (4.64) where (a) l, h = {P^o is the set of the vertices of ST h , with P o = T.E.; (b) W; is the basis function of HI (cf. (4.24)) associated with P t by w, e Hi Vi = 0, ,N h , Wi (Pd = 1, WiiPj) = 0, \/jjt i; (4.65) (c) ^ is the subset of 2T h consisting of those triangles having P i as a common vertex; (d) m(T) = meas(T); 4 Transonic Flow Calculations by Least-Squares and Finite Element Methods Figure 4.9 229 (e) {iiji}j= i is an approximation of \(/)J| \<f> h | at vertex P ; obtained, for j = 1, 2, by the following averaging formulas: I '50» (others averaging methods are possible); (f) h s (T) is the length of T in the s direction, i.e., K(T) = =i v kT ds (4.66) (4.67) where w fcr , k = 1, 2, 3 are the basis functions associated with the vertices P kT of T; actually (4.67) can also be written (4.68) (g) We define C/j as follows: With each vertex P t o{ST h we associate an inflow triangle T { which is the triangle of ?7~ h having P t as vertex and which is crossed by the vector u ; — i u ij}j=I pointing to P t (as shown on Fig. 4.9); we then define Uf as Uf = cl 230 VII Least-Squares Solution of Nonlinear Problems and for each triangle T of 3~ h , k=l 3 I k=l dw kT ds dw kT ds T T (4.69) (h) We finally obtain p h e Hi as follows: As for Uf, we define p t by Pi = P(<t>h)\ Ti , and then Ph = (4.70) 4.6.3.3. Some brief comments on the least-squares sohrtion of (4.63). For solving the discrete upwinded continuity equation (4.63), we can use the following least- squares formulation: (4.71) where with y h (i; h ) (=y h ), the solution of the following linear variational equation: Find y h e V h such that f f f Vj> h • \v h dx = p((j) h )S/4> h • Vv h dx - \ g h v h dT Jn Jn Jr + 2iL I (Is W * " 1} + V</> "' Vp "), t; ' 1 JX ' V "* £ Fft ' (472) Since J h is a nondifferentiable functional of £,,, to solve (4.71) we have used (instead of algorithm (4.33)-(4.40)) a generalization of the conjugate gradient method due to Lemarechal [1], [2] which also applies to the minimization of nondifferentiable functional (actually good results are also obtained if one uses algorithm (4.33)-(4.40) with J h ( •) denned as in (4.71); however, more iterations are needed). 4.7. Numerical experiments In this section we shall present some of the numerical results obtained using the above methods. The results of Sec. 4.7.1 are related to a NACA 0012 airfoil; those of Sec. 4.7.2 (resp., 4.7.3) to a Korn airfoil (resp., a two-piece airfoil). 4 Transonic Flow Calculations by Least-Squares and Finite Element Methods 231 4.7.1. Simulation of flows around a NACA 0012 airfoil. As a first example we have considered flows around a NACA 0012 airfoil at various angles of attack and Mach numbers at infinity. The corresponding pressure distributions on the skin of the airfoil are shown on Figs. 4.10-4.17 in which the isomach lines (in the supersonic region only on Figs. 4.10-4.14) are also shown. The results shown in Figs. 4.10-4.14 have been obtained using the interior penalty method of Sec. 4.6.2; those of Fig. 4.15-4.17 have been obtained using the upwinding method of Sec. 4.6.3. We observe that the physical shocks are quite neat and also that the transition (without shock) from the subsonic to the supersonic region is smoothly restituted, implying that the entropy condition has been satisfied. The above numerical results are very close to those obtained by various authors using finite difference methods (see, particularly, Jameson [1]). 4.7.2. Flow around a Horn's airfoil In Fig. 4.18 we have represented the pressure distribution corresponding to the flow around a Korn's airfoil atM 0O = 0.75 and a. — 0.11; the computation method is the interior penalty method of Sec. 4.6.2. The agreement with a finite difference solution is good, as indicated in Fig. 4.18. 4.7.3. Flows around a two-piece airfoil The tested two-piece airfoil is shown on Figs. 4.19 and 4.20. Each piece is a NACA 0012 airfoil (the body No. 1 is the upper body). The pressure distribution and the isomach lines (computed by the interior penalty method of Sec. 4.6.2) are shown on Figs. 4.19 and 4.20. We observe that the region between the two airfoils acts as a nozzle; we also observe supersonic regions, in particular, between the two airfoils. 4.8. Transonic flow simulations on large bounded computational domains Consider the situation depicted in Fig. 4.3; if the supersonic zone extends far from the airfoil, it is necessary to use a very large computational domain. Let 0 be the origin of coordinates; it is then reasonable to take the circle {x eU 2 ,r = R x } for r^, with r = y/x{ + xf. Now suppose that the disk of center 0 and radius R o is sufficiently large to contain the airfoil B in its interior; we then introduce Qi = {x e U 2 , 0 < r < R o , x $ B}, Q 2 = {xeR 2 ,J? 0 <r<RJ. o 5 o Figure 4.10. NACA 0012 airfoil. M a = 0.6; a = 6. (Interior penalty method.) -1 3 o 9 5' §' tn tr o a Figure 4.11. NACA 0012 airfoil. M» = 0.78; a = 1. (Interior penalty method.) 234 VII Least-Squares Solution of Nonlinear Problems B o < U < z 4 Transonic Flow Calculations by Least-Squares and Finite Element Methods 235 -II •g < -1 Figure 4.14. NACA 0012 airfoil. M. = 0.85; a = 0. (Interior penalty method.) I Z o 3 5 o [...]... transformation v -+ v: H -+H1(Q): 1 v1(x1,x2) ifx = {x1; u(x) = (4 .74 ) v2\R0n+Log~\e Using the above properties, the continuity equation (4.19) is now formulated as follows: Find | e f l ' such that V P E H 1 2lc (4 .75 ) with {«loo>«2oo} = "ao 244 VII Least-Squares Solution of Nonlinear Problems and dS ^i2 + R-2 0 2 \ ~| i/ (7- D ^2 \e2(R0- S,)/Ro Kl I J ' (4 .76 ) The discretization of (4 .75 ), (4 .76 ) can... problem (3.1) On the other hand, problems (5.54) and (5. 57) , (5.59) are closely related to the discrete Stokes problem (Ph) (cf Sec 5.3.3.2) whose numerical solution will be discussed in Sec 5 .7 Remark 5.3 With respect to storage requirements, the optimal values 17 of 9 are 9 = £ for (5.53)-(5.55) and 6 = i for (5.56)-(5.59); these choices will be justified in Sec 5 .7 and are directly related to the... least-squares formulation of 260 VII Least-Squares Solution of Nonlinear Problems (5.66) and in Sec 5.5.3 a conjugate gradient algorithm for solving the leastsquares problem The discrete variants of the methods described in Sees 5.5.2 and 5.5.3 will be described in Sec 5.6 5.5.2 A least-squares formulation of (5.66) A natural least squares formulation of (5.66) is Find u € Vg such that J(u) < J(\), (5. 67) with... and also that the formulation (5. 67) -(5.69) is a natural generalization of (3. 37) in Sec 3.3.2 The above least-squares formulation is justified by the following obvious proposition Proposition 5.1 Suppose that {a, p} e Vg x (L2(Q)/R) is a solution of (5.66); then u is also a solution of the least-squares problem (5. 67) , and we have, for the corresponding pair {!;, n}, % = 0, n = -p; (5 .70 ) we also have... equations, which are nonlinearity and incompressibility In both methods, the nonlinear problems solved at each time step (namely (5.55) and (5.58), respectively) are in fact finite-dimensional variants of the nonlinear problem (3.56) (cf Sec 3.4, Exercise 3.1), and therefore they can be solved by least-squares conjugate gradient methods very close to those methods used in Sec 3.4 for solving the model... O VII Least-Squares Solution of Nonlinear Problems 238 CP4 25 5 75 Figure 4.16 NACA 0012 airfoil Mm = 0.85;a = 0 (Computation with upwinding of the density.) 4 Transonic Flow Calculations by Least-Squares and Finite Element Methods 239 -1 25 5 75 Figure 4. 17 NACA 0012 airfoil M = 0.9; a = 0 (Computation with upwinding of the density.) Figure 4.18 Korn airfoil Ma — 0 .75 ; a = 0.11 (Interior penalty method.)... where Re is the Reynold's number), (iv) f is the density of external forces; 12 13 14 One also says stationary One also says nonstationary Q a W, N = 2, 3 in practice 246 VII Least-Squares Solution of Nonlinear Problems in (5.1), (5.3), (u • V)u is a symbolic notation for the nonlinear (vector) term: Boundary conditions have to be added; for example, in the case of the airfoil, B of Fig 4.1 of Sec 4.3.2... kinds of steady nonlinear problems (cf Chapter IV, Sec 2.6.6 and also Sec 3.2.4 of this chapter); actually these methods are also very useful for solving timedependent problems and most particularly the unsteady Navier-Stokes equations as indicated by the two methods described below (and the corresponding numerical experiments) 5.4.4.1 A Peaceman-Rachford alternating-direction method for solving the... a$n\zm\2dx + v$n\\z\dx ' ( ' and finally w m + 1 = z m + 1 + ym+1w'", (5 .78 ) m = m + 1 go to (5 .74 ) Remark 5.8 To obtain z m + 1 from u m + 1 , via (5 .76 ), we also have to solve a Stokes problem (written in a variational form) 5.5.3.2 Calculation ofJ'(am+ J ) and zm+1 A most important step in order to use algorithm (5 .71 )-(5 .78 ) is the calculation of J'(u m+ x ); due to the importance of this step,... obtain z m + 1 from (5 .76 ), (5.84) To summarize, at each iteration of the conjugate gradient algorithm (5 .71 )(5 .78 ), we have to solve several Stokes problems, namely: (i) the Stokes problem (5.69), (5.80) with v = u m + 1 to obtain £ m + 1 from um+l (ii) the Stokes problem (5 .76 ) to obtain (via (5.84)) z m + 1 from u m+ \ ^ m+ x; (iii) the solution of the one-dimensional problem (5 .74 ) requires several . (4 .71 ); however, more iterations are needed). 4 .7. Numerical experiments In this section we shall present some of the numerical results obtained using the above methods. The results of Sec. 4 .7. 1. VPEH 1 2lc (4 .75 ) with {«loo>«2oo} = "ao 244 VII Least-Squares Solution of Nonlinear Problems and ^i 2 dS 2 ~| i/ (7- D + R -2 ^2 e 2(R 0 - S,)/Ro 0 Kl I J ' (4 .76 ) The discretization. airfoil; those of Sec. 4 .7. 2 (resp., 4 .7. 3) to a Korn airfoil (resp., a two-piece airfoil). 4 Transonic Flow Calculations by Least-Squares and Finite Element Methods 231 4 .7. 1. Simulation of flows