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# Dynamic Absorbing Boundary Conditions

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Conference given at CIMNE, Barcelona, 2010-02-19

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### Dynamic Absorbing Boundary Conditions

1. 1. Dynamic absorbing boundary conditions by M.Storti et.al. Dynamic Absorbing Boundary Conditions for Advective-Difusive Systems with Unknown Riemann Invariants by Mario Storti, Rodrigo Paz, Luciano Garelli, Lisandro Dalc´n ı ´ Centro Internacional de Metodos Computacionales en Ingenier´a - CIMEC ı INTEC, (CONICET-UNL), Santa Fe, Argentina <mario.storti at gmail.com> http://www.cimec.org.ar/mstorti ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 1 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
2. 2. Dynamic absorbing boundary conditions by M.Storti et.al. Motivation for Absorbing Boundary Conditions In wave-like propagation problems, not including ABC may lead to non-convergent solutions . utt = c2 ∆u, in Ω u = u(x, t), at Γb , u = 0, at Γ∞ v ¯ u doesn’t converge to the correct solution irradiating energy from the source, even if Γ∞ → ∞. A standing wave is always found. u is unbounded if u emits in an ¯ eigenfrequency which is a resonance mode of the closed cavity. Absorbing boundary conditions must be added to the outer boundary in order to let energy be extracted from the domain. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 2 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
3. 3. Dynamic absorbing boundary conditions by M.Storti et.al. Motivation for Absorbing Boundary Conditions (cont.) If a dissipative region is large enough so that many wavelengths are included in the region may act as an absorbing layer, i.e. as an absorbing boundary condition. dissipative However, it is a common misconception to assume that a coarse mesh adds dissipation and fine mesh coarse mesh consequently may improve absorption. For instance for the dissipative?? wave equation a coarse mesh may lead to evanescent solutions and then to act as a fully reﬂecting boundary. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 3 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
4. 4. Dynamic absorbing boundary conditions by M.Storti et.al. Boundary conditions for advective diffusive systems Well known theory and practice for advective systems say that at a boundary the number of Dirichlet conditions should be equal to the number of incoming characteristics . ∂U ∂Fc,j (U) + =0 ∂t ∂xj ∂Fc,j (U) Ac,j = , advective Jacobian ∂U Nbr. of incoming characteristics = sum(eig(A · n) < 0) ˆ ˆ n is the exterior normal. Adding extra Dirichlet conditions leads to spurious shocks, and lack of enough Dirichlet conditions leads to instability. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 4 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
5. 5. Dynamic absorbing boundary conditions by M.Storti et.al. Boundary conditions for advective diffusive systems (cont.) For simple scalar advection problems the Jacobian is the transport velocity. The rule is then to check the projection of velocity onto the exterior normal. For more complex ﬂows (i.e. with non diagonalizable Jacobians , as gas dynamics or shallow water eqs.) the number of incoming characteristics may be approx. predicted from the ﬂow conditions. subsonic flow (Minf<1) supersonic flow (Minf>1) supersonic outgoing subsonic incoming (no fields imposed) rho,u,v rho,u,v,p ck sho M>1 b ow M<1 1111111 0000000 1111111 0000000 111111 000000 111111 000000 1111111 0000000 111111 000000 1111111 0000000 111111 000000 p p subsonic outgoing subsonic outgoing ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 5 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
6. 6. Dynamic absorbing boundary conditions by M.Storti et.al. Absorbing boundary conditions However, this kind of conditions are, generally, reﬂective . Consider a pure advective system of equations in 1D, i.e., Fd,j ≡ 0 ∂H(U) ∂Fc,x (U) + = 0, in [0, L]. (1) ∂t ∂x If the system is “linear”, i.e., Fc,x (U) = AU, H(U) = CU (A and C do not depend on U), a ﬁrst order linear system is obtained ∂U ∂U C +A = 0. (2) ∂t ∂x The system is “hyperbolic” if C is invertible, C−1 A is diagonalizable with real eigenvalues. If this is the case, it is possible to make the following eigenvalue decomposition for C−1 A C−1 A = SΛS−1 , (3) where S is real and invertible and Λ is real and diagonal. If new variables are ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 6 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
7. 7. Dynamic absorbing boundary conditions by M.Storti et.al. deﬁned V = S−1 U, then equation (2) becomes ∂V ∂V +Λ = 0. (4) ∂t ∂x Now, each equation is a linear scalar advection equation ∂vk ∂vk + λk = 0, (no summation over k ). (5) ∂t ∂x vk are the “characteristic components” and λk are the “characteristic velocities” of propagation. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 7 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
8. 8. Dynamic absorbing boundary conditions by M.Storti et.al. Linear 1D absorbing boundary conditions Assuming λk = 0, the absorbing boundary conditions are, depending on the sign of λk , if λk > 0: vk (0) = vk0 ; ¯ no boundary condition at x =L (6) if λk < 0: vk (L) = vkL ; no boundary condition at x = 0 ¯ This can be put in compact form as ¯ Π+ (V − V0 ) = 0; at x =0 V (7) Π− (V V ¯ − VL ) = 0; at x =L ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 8 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
9. 9. Dynamic absorbing boundary conditions by M.Storti et.al. Linear 1D absorbing boundary conditions (cont.) Π± are the projection matrices onto the right/left-going characteristic modes V in the V basis,  1; if j = k and λ > 0 k Π+ = V,jk 0; otherwise, (8) Π+ + Π− = I. It can be easily shown that they are effectively projection matrices, i.e., Π± Π± = Π± and Π+ Π− = 0. Coming back to the boundary condition at x = L in the U basis, it can be written Π− S−1 (U − UL ) = 0 V ¯ (9) or, multiplying by S at the left Π± (U − U0,L ) = 0, at x = 0, L, U ¯ (10) where Π± = S Π± S−1 , U V (11) ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 9 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
10. 10. Dynamic absorbing boundary conditions by M.Storti et.al. Linear 1D absorbing boundary conditions (cont.) Π± (U − U0,L ) = 0, at x = 0, L, U ¯ (12) Π± U = S Π± V −1 S , These conditions are completely absorbing for 1D linear advection system of equations (2). + The rank of Π is equal to the number n+ of positive eigenvalues, i.e., the number of right-going waves. Recall that the right-going waves are incoming at the x = 0 boundary and outgoing at the x = L boundary. Conversely, the − rank of Π is equal to the number n− of negative eigenvalues, i.e., the number of left-going waves (incoming at x = L and outgoing at the x = 0 boundary). ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 10 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
11. 11. Dynamic absorbing boundary conditions by M.Storti et.al. ABC for nonlinear problems First order absorbing boundary conditions may be constructed by imposing exactly the components along the incoming characteristics. Π− (Uref ) (U − Uref ) = 0. Π− is the projection operator onto incoming characteristics. It can be obtained straightforwardly from the projected Jacobian. This assumes linearization of the equations around a state Uref . For linear problems Ac,j do not depend on U, and then neither the projection operator, so that absorbing boundary conditions coefﬁcients are constant. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 11 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
12. 12. Dynamic absorbing boundary conditions by M.Storti et.al. ABC for nonlinear problems (cont.) For non-linear problems the Jacobian and projection operator may vary and then the above mentioned b.c.’s are not fully absorbing. In some cases the concept of characteristic component may be extended to the non-linear case: the “Riemann invariants” . Fully absorbing boundary conditions could be written in terms of the invariants: wj = wref,j , if wj is an incoming R.I. R.I. are computed analytically. There are no automatic (numerical) techniques to compute them. (They amount to compute an integral in phase space along a speciﬁc path ). R.I. are known for shallow water, channel ﬂow (for rectangular √ √ wj = u · n ± 2 gh, and triangular channel shape wj = u · n ± 4 gh). ˆ ˆ For gas dynamics the well known R.I. in fact are invariants only under isentropic conditions (i.e. not truly invariant). ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 12 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
13. 13. Dynamic absorbing boundary conditions by M.Storti et.al. ABC for nonlinear problems (cont.) Search for an absorbing boundary condition that should be fully absorbent in non-linear conditions, and can be computed numerically (no need of analytic expressions like R.I.) Solution: Use last state as reference state, ULSAR . Uref = Un , n = time step number. Π− (Un ) (Un+1 − Un ) = 0. As Un+1 − Un is usually small, linearization is valid. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 13 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
14. 14. Dynamic absorbing boundary conditions by M.Storti et.al. ABC for nonlinear problems (cont.) Disadvantage: Flow conditions are only determined from the initial state. No external information comes from the outside. Solution: use a combination of linear/R.I. b.c.’s on incoming boundaries and use fully non-linear a.b.c’s with previous state as reference state at the outlet. subsonic/supersonic flow n Uref=Uinf Uref=U 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 14 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
15. 15. Dynamic absorbing boundary conditions by M.Storti et.al. Dynamic boundary conditions As the ﬂow is computed it may happen that the number of characteristics changes in time. Two examples follow. Think at transport of a scalar on top of a velocity ﬁeld obtained by an incompressible Navier-Stokes solver (not truly that in the example since both systems are coupled). If the exterior ﬂow is not modeled then ﬂow at the top opening may be reverted. ventilacion perfil de viento inyeccion de carbon ventilacion ventilacion ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 15 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
16. 16. Dynamic absorbing boundary conditions by M.Storti et.al. Dynamic boundary conditions (cont.) If the interior is modeled only, then it’s natural to leave concentration free at the top opening. However the ﬂow can revert at some portions of the opening producing a large incoming of undetermined values (in practice large negative concentrations are observed). Imposing a value at the opening is stable but would lead to a spurious discontinuity at the outlet. The ideal would be to switch dynamically from one condition to the other during the computation. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 16 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
17. 17. Dynamic absorbing boundary conditions by M.Storti et.al. Nozzle chamber ﬁll The case is the ignition of a rocket launcher nozzle in a low pressure ◦ atmosphere. The ﬂuid is initially at rest (143 Pa, 262 K). At time t = 0 a membrane at the throat is broken. Behind the membrane there is a reservoir at ◦ 5 6×10 Pa, 4170 K. A strong shock (intensity p1 /p2 >1000) propagates from the throat to the outlet. The gas is assumed as ideal (γ = 1.17). In the steady state a supersonic ﬂow with a max. Mach of 4 at the outlet is found. The objective of the simulation is to determine the time needed to ﬁll the chamber (< 1msec) and the ﬁnal steady ﬂow. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 17 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
18. 18. Dynamic absorbing boundary conditions by M.Storti et.al. Nozzle chamber ﬁll (cont.) We impose density, pressure and tangential velocity at inlet (assuming subsonic inlet), slip condition at the nozzle wall. The problem is with the outlet boundary. Initially the ﬂow is subsonic (ﬂuid at rest) there, and switches to supersonic. The rule dictaminates to impose 1 condition, as a subsonic outlet (may be pressure, which is known) and no conditions after (supersonic outlet). If pressure is imposed during the wall computation, then a spurious shock is formed at the outlet. This test case has been contrasted with experimental data obtained at ESTEC/ESA (European Space Research and Technology Centre-European Space Agency, Noordwijk, Holanda). The predicted mean velocity was 2621 m/s to be compared with the experimental value of 2650 ± 50 m/sec. Again, the ideal would be to switch dynamically from one condition to the other during the computation . ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 18 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
19. 19. Dynamic absorbing boundary conditions by M.Storti et.al. Object falling at supersonic speed 11 00 11 00 11 00 11 00 11 00 11 00 Consider, for simplicity, a two 11 00 11 00 11 00 dimensional case of an homogeneous ellipse in free fall. As A the body accelerates, the pitching 11111 00000 11111 00000 11111 00000 11111 00000 moments tend to increase the angle of attack until it stalls (A), and then 111 000 111 000 111 000 the body starts to fall towards its 111 000B 111 000 111 000 other end and accelerating etc... 111 000 C 111 000 11111 00000 ( “ﬂutter” ). However, if the body 11111 00000 11111 00000 11111 00000 has a large angular moment at (B) 111 000 111 000 then it may happen that it rolls on 111 000 111 000 111 000 111 000 111 000 111 000 itself, keeping always the same 111 000 111 000 111 000 111 000 111 000 111 000 111 000 sense of rotation. This kind of falling 111 000 11111 00000 1111 0000 11111 00000 1111 0000 mechanism is called “tumbling” 11111 00000 1111 0000 11111 00000 1111 0000 and is characteristic of less slender and more massive objects. flutter tumble ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 19 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
20. 20. Dynamic absorbing boundary conditions by M.Storti et.al. Object falling at supersonic speed (cont.) Under certain conditions in size and density relation to the surrounding atmosphere it reaches supersonic speeds. In particular as form drag grows like L2 whereas weight grows like L3 , larger bodies tend to reach larger limit speeds and eventually reach supersonic regime. At supersonic speeds the principal source of drag is the shock wave, we use slip boundary condition at the body in order to simplify the problem. We also do the computation in a 11 00 11 00 non-inertial system following the body, 11 00 11 00 11 00 11 00 so that non-inertial terms (Coriolis, 11 00 11 00 centrifugal, etc...) are added. In this 11 00 frame some portions of the boundary are alternatively in all the conditions (subsonic incoming, subsonic 11111 00000 11111 00000 outgoing, supersonic incoming, 11111 00000 supersonic outgoing). 11111 00000 11111 00000 Again, the ideal would be to switch 11111 00000 11111 00000 dynamically from one condition to 11111 00000 11111 00000 the other during the computation. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 20 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
21. 21. Dynamic absorbing boundary conditions by M.Storti et.al. Object falling at supersonic speed (cont.) ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 21 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
22. 22. Dynamic absorbing boundary conditions by M.Storti et.al. Object falling at supersonic speed (cont.) Whether RI or ULSAR based boundary conditions are used, if the number of incoming/outgoing characteristics vary in time this requires to change the proﬁle of the system matrix during time evolution. In order to do this we add dynamic boundary conditions either via Lagrange multipliers or penalization. However this techniques add extra bad conditioning to the system of equations so that special iterative methods are needed. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 22 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
23. 23. Dynamic absorbing boundary conditions by M.Storti et.al. Object falling at supersonic speed (cont.) ∂U ∂U C +A = 0. ∂t ∂x Consider for simplicity a linear system of advective equations discretized with centered (i.e. no upwind) ﬁnite differences Un+1 − Un 0 0 Un+1 − Un+1 C +A 1 0 = 0; ∆t h Un+1 − Un Un+1 − Un+1 C k k + A k+1 k−1 = 0, k ≥ 1 ∆t 2h k ≥ 0 node index, n ≥ 0 time index, h = mesh size, C, A =enthalpy and advective Jacobians. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 23 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
24. 24. Dynamic absorbing boundary conditions by M.Storti et.al. Object falling at supersonic speed (cont.) Using Lagrange multipliers for imposing the boundary conditions leds to the following equations Π+ (U0 − Uref ) + Π− λ= 0, Un+1 − Un Un+1 − Un+1 C 0 0 +A 1 0 +CΠ+ λ = 0; ∆t h Un+1 − Un Un+1 − Un+1 C k k + A k+1 k−1 = 0, k ≥ 1. ∆t 2h Π± is the projection operator onto incoming/outgoing waves, λ are the Lagrange multipliers. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 24 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
25. 25. Dynamic absorbing boundary conditions by M.Storti et.al. Using penalization Add a small regularization term and then eliminate the Lagrange multipliers. − λ + Π+ (U0 − Uref ) + Π− λ = 0, Un+1 − Un Un+1 − Un+1 C 0 0 +A 1 0 + CΠ+ λ = 0; ∆t h Eliminating the Lagrange multipliers λ we arrive to a boundary equation Un+1 − Un Un+1 − Un+1 C 0 0 +A 1 0 +(1/ )CΠ+ (U0 − Uref ) = 0. ∆t h ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 25 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
26. 26. Dynamic absorbing boundary conditions by M.Storti et.al. Absorbing boundary conditions and ALE When using Arbitrary Lagrangian-Eulerian formulations: ∂U ∂Fc,j (U) + −vmesh U = 0 ∂t ∂xj ∂Fc,j (U) AALE,j = −vmesh,j I, ALE advective Jacobian ∂U Nbr. of incoming characteristics = sum(eig(A · n) − vmesh · n < 0) ˆ ˆ ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 26 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
27. 27. Dynamic absorbing boundary conditions by M.Storti et.al. ALE invariance test case. Sudden stop of gas container shock wave expansion fan t t x x density container initially moving container initially at rest at constant speed u0 is suddenly put in movement is suddenly stopped with constant negative speed −u0 ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 27 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
28. 28. Dynamic absorbing boundary conditions by M.Storti et.al. ALE invariance. SUPG Stabilization term ∂Uc ∂Fc,x ∂Fd,x + = ; (gov. eqs. in cons. form) ∂t ∂x ∂x (13) ∂U ∂U ∂2U C + A = K 2 ; (gov. eqs. in quasi-linear form) ∂t ∂x ∂x ˜ Sufﬁcient conditions for ALE invariance (A = A − vmesh C) P = ˜ N · Aτ C−1 ; (SUPG pert. function) ∂U ∂U (14) τ transform as U × U, i.e. τ = τ . ∂U ∂U ˜ This last is veriﬁed if τ is f (C−1 A), for instance (inviscid case): h ˜ τ = I, λj = eig(C−1 A), (max. eigenv.) max |λj | (15) ˜ τ = h|C−1 A|−1 . (| · | in matrix sense) ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 28 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
29. 29. Dynamic absorbing boundary conditions by M.Storti et.al. ALE invariance. Sudden stop of a gas container γ = 1.4, u0 /c0 = 0.5. Results in both reference systems are equivalent to machine precision. 1.2 pressure 1.0 x shock wave 0.8 0.6 1 0.4 0 t 0 A B C 1 D E 2 F G 3 4 expansion fan ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 29 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
30. 30. Dynamic absorbing boundary conditions by M.Storti et.al. ALE invariance. Sudden stop of a gas container (cont.) 2 1.8 1.6 Eint Ek+E int 1.4 1.2 Energy Ek 1 0.8 0.6 Ek 0.4 0.2 0 0 2 4 6 8 10 12 14 AB C D E F G t 2 (Up) Energy balance in 1.5 reference system ﬁxed w.r.t container) Energy 1 Ek E W (Down) Energy balance in int Ek+E int−W reference system ﬁxed 0.5 w.r.t initial gas at rest) 0 0 2 4 6 8 10 12 14 AB C D E F G t ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 30 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
31. 31. Dynamic absorbing boundary conditions by M.Storti et.al. Open channel ﬂow. Flow in a channel can be cast in advective form as follows Up = [h, u]T , w T U = Uc = [A, Q] , H(U) = Uc ,   A h Q F = . Q2 /A + F where h and u are water depth and velocity (as in the shallow water equations). A(h) is the section of the channel occupied by water for a given water height h. It then deﬁnes the geometry of the channel. Q = Au is the water ﬂow rate. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 31 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
32. 32. Dynamic absorbing boundary conditions by M.Storti et.al. Open channel ﬂow. (cont.) For instance Rectangular channels: A(h) = wh, w =width. Triangular channels: A(h) = 2h2 tan θ/2; with θ=angle opening. Circular channel: h A(h) = 2Rh − h2 dh h =0 2 = θR − w(h)(R − h)/2 √ w(h) = 2 2Rh − h2 , θ = atan[w/(2(R − h))] is angular aperture, h F (h) = h =0 A(h ) dh Double rectangular channel. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 32 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
33. 33. Dynamic absorbing boundary conditions by M.Storti et.al. Comparison with RI for rectangular channel |u_ulsar-u_RI| (at x=0) |h_ulsar-h_RI| (at x=0) ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 33 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
34. 34. Dynamic absorbing boundary conditions by M.Storti et.al. Comparison with RI for rectangular channel (cont.) x=L (outgoing)  5.8 w+  5.7  5.6  5.5 right-going RI  5.4  5.3  5.2  5.1  0 2  4  6  8 10  12  14  16 x=0 (ingoing) time -4.3 x=0 (outgoing) w- -4.4 -4.5 left-going RI -4.6 x=L (ingoing) -4.7 -4.8  0 2  4  6  8 10  12  14  16 time ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 34 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
35. 35. Dynamic absorbing boundary conditions by M.Storti et.al. Comparison with RI for rectangular channel (cont.) We restrict here to the case of constant channel section and depth. For rectangular channels the equations are reduced to those for 1D shallow water equations. Channel ﬂow is very interesting since it is in fact a family of different 1D hyperbolic systems depending on the area function A(h). Riemann invariants are only known for rectangular and triangular channel shapes. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 35 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
36. 36. Dynamic absorbing boundary conditions by M.Storti et.al. Stratiﬁed shallow water ﬂows Another physical model where Riemann invariants have not a mathematical closed form is the ﬂow of a multi-layer ﬂuid in channels. This kind of physical model exists for instance when ﬂow takes place on a mountainous terrain over plain areas or dense distribution of torrents combined with heavy rainfall. Each layer may have different density, velocity in a two-dimensional domain. ∂Uc ∂Fc,j (Uc ) ∂Uc + + Bj (Uc ) = G(Uc ); j = 1, 2. (16) ∂t ∂xj ∂xj Uc = [h1 , h2 , h1 u1 , h1 v1 , h2 u2 , h2 v2 ]T is the vector of conservation variables and Bj (Uc ) is the Jacobian matrix of the non-conservative products in the j direction. h1 (x, t) and h2 (x, t) are the thickness of each layer while the height of the bottom is h0 (x, t). [u, v]i is the velocity vector of the layer i. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 36 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
37. 37. Dynamic absorbing boundary conditions by M.Storti et.al. Startiﬁed shallow water eqs (two strata) ∂h1 ∂ ∂ + (h1 u1 ) + (h1 v1 ) = 0, ∂t ∂x ∂y ∂h2 ∂ ∂ + (h2 u2 ) + (h2 v2 ) = 0, ∂t ∂x ∂y ∂ ∂ 1 ∂ ∂ ρ2 (h1 u1 ) + h1 (u1 )2 + gh2 + (h1 u1 v1 ) + gh1 h0 + h2 = 0, ∂t ∂x 2 1 ∂y ∂x ρ1 ∂ ∂ ∂ 2 1 2 ∂ ρ2 (h1 v1 ) + (h1 u1 v1 ) + h1 (v1 ) + gh1 + gh1 h0 + h2 = 0, ∂t ∂x ∂y 2 ∂y ρ1 ∂ ∂ 2 1 2 ∂ ∂ (h2 u2 ) + h2 (u2 ) + gh2 + (h2 u2 v2 ) + gh2 (h0 + h1 ) = 0, ∂t ∂x 2 ∂y ∂x ∂ ∂ ∂ 1 ∂ (h2 v2 ) + (h2 u2 v2 ) + h2 (v2 )2 + gh2 + gh2 2 (h0 + h1 ) = 0, ∂t ∂x ∂y 2 ∂y ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 37 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
38. 38. Dynamic absorbing boundary conditions by M.Storti et.al. Stratiﬁed shallow water ﬂows ρ1,2 are constant layer densities. Vertical velocity is averaged. Hydrostatic pressure assumption in each layer. In the general case of n layers and 2D model, the system of equations has 3n waves that propagate inside the domain. It could happen that, depending on the densities ratio ρ2 /ρ1 , and the velocities the system becomes non-hyperbolic. In fact if the velocities of the ﬂuid are the same, the ﬂow is unstable for ρ2 > ρ1 . Riemann-Invariants for stratiﬁed ﬂow are not known. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 38 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
39. 39. Dynamic absorbing boundary conditions by M.Storti et.al. Stratiﬁed shallow water ﬂows (cont.) ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 39 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))
40. 40. Dynamic absorbing boundary conditions by M.Storti et.al. Acknowledgment This work has received ﬁnancial support from Consejo Nacional de ´ Investigaciones Cient´ﬁcas y Tecnicas (CONICET, Argentina, PIP 5271/05), ı Universidad Nacional del Litoral (UNL, Argentina, grants CAI+D 2005-10-64) ´ ´ and Agencia Nacional de Promocion Cient´ﬁca y Tecnologica (ANPCyT, ı Argentina, grants PICT PME 209/2003, PICT-1141/2007, PICT-1506/2006). We made extensive use of Free Software (http://www.gnu.org) as GNU/Linux OS, MPI, PETSc, GCC/G++ compilers, Octave, Open-DX, VTK, Python, Git, among many others. In addition, many ideas from these packages have been inspiring to us. ´ Centro Internacional de Metodos Computacionales en Ingenier´a ı 40 ((version texstuff-1.0.31-36-g29c1d2e Fri Feb 19 08:14:56 2010 +0100) (date Fri Feb 19 11:35:45 2010 +0100))