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An Inverse Geometrical Problem in Fluid Mechanics

An Inverse Geometrical Problem in Fluid Mechanics. Carlos Conca Depto. Ingenier ía Matemática Centro de Modelamiento Matem ático UMR 2071 CNRS – U. de Chile. ( with C. Alvarez, L. Friz, O. Kavian and J. H. Ortega). This lecture is organized as follows:. Setting of the Problem

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An Inverse Geometrical Problem in Fluid Mechanics

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  1. An Inverse Geometrical Problem in Fluid Mechanics Carlos Conca Depto. Ingeniería Matemática Centro de Modelamiento Matemático UMR 2071 CNRS – U. de Chile (with C. Alvarez, L. Friz, O. Kavian and J. H. Ortega) WONAPDE 2004

  2. This lecture is organized as follows: • Setting of the Problem • Introduction to Inverse Problems • Identifiability and Stability Results • Algorithm and Numerical Results WONAPDE 2004

  3. Setting of the Problem • Recover geometrical information (position and shape) about an a-priori unknown body D immersed in an incompressible viscous liquid. • To this end, we perform measurements (on velocity and stress forces) along the boundary of the cavity Ωfulfilled by the liquid. • Non-steady incompressible Stokes equations for the liquid. • Non-homogeneous Dirichlet boundary condition on Ω. • Non-slip condition on D. D Ω WONAPDE 2004

  4. What are Inverse Problems? • In an inverse boundary problem one seeks to determine the internal properties of a medium by performing measurements along the boundary of the medium. • The appropriate mathematical model of the physical situation is usually given by a PDE inside the medium. • The boundary measurements are encoded in a certain boundary map. The inverse problem is to determine the coefficients of the PDE from knowledge of this boundary map. WONAPDE 2004

  5. A Propotypical Example: Calderon´s Problem Electrical Impedance Tomography (I) • Let  be a bounded smooth domain in Rn. Let (x) be the unknown electrical conductivity of the medium fulfilling Ω. • (x) is assume to be in L1 () and strictly positive. The potential u in Ω with voltage f on Ω satisfies • div (  (x) r u) = 0 in  • u = f on  • In this case the boundary map is the so-called voltage to current map • f   (f ) =  u on  • n • Calderon´s inverse problem is to recover  from the boundary map  :    WONAPDE 2004

  6. A Prototypical Example: Calderon´s ProblemElectrical Impedance Tomography (II) • More precisely, we can divide Calderon’s Inverse problem in several parts • Injectivity of the mapping  (Identifiability) • Continuity of  and its inverse -1 if it exists (Stability) • What's the range of ? (Characterization) • Formula to recover  form  (Reconstruction) • Give a numerical algorithm to find and approximation of  (Numerical Reconstruction) WONAPDE 2004

  7. Mathematical Framework for our Inverse Problem in Fluid Mechanic (I) • Let be a smooth bounded set in Rnand let D  be an unknown rigid body immersed in the liquid. • Let 2 H1Á2( ) be a non homogeneous Dirichlet boundary data and let (v,p) be the solution of Stokes equations in ¤ : =n D • -   v +  p = 0 in ¤ • div v =0 in ¤ ….(P) • v =  on  • v = 0 on D • Let  (v, p) be the linear stress tensor, defined by •  (v,p) = -pI + 2  e(v) • and • e(v) = (r v + (r v)T) • 2 WONAPDE 2004

  8. Mathematical Framework for our Inverse Problem in Fluid Mechanic (II) • The set of admissible bodies D is • Uad = f D  : D is a smooth, open and simply connected region in g • Let  be the following boundary map, velocity to stress tensor • : D !D • defined as follows • D ( ) =  (v,p)n on   • (v,p) being the solution of the Stokes system (P). • Our Inverse Problem is to recover D from the above boundary map velocity to stress tensor  : D !D WONAPDE 2004

  9. Pioneering works Inverse Conductivity Problems • A. Calderón (1980) • R. Kohn, M. Vogelius (1984) • J. Sylvester, G. Ulhmann (1986) Inverse Geometrical Problems • S. Andrieux, A. Ben Abda y M. Jaona (1993) • E. Beretta y S. Vesella (1999) • G. Alessandrini, A. Morassi y E. Rosset (2002)

  10. Main Issues • Identifiability result, that is, the injectivity of the velocity to stress tensor map : • D1 D2) (v1, p1)n  (v2, p2)n on , 8 • Stability result, that is, the continuity of the inverse of the velocity to stress tensor map  (if two measures are close each other, then the rigid bodies are also close). • Algorithm and numerical results allow us to recover the volume and position of the unknown rigid body. WONAPDE 2004

  11. Identifiability Result Theorem (C. C., L. Friz, O. Kavian, J. Ortega) Let  be a smooth open bounded domain in ℝnand D0 , D 1 Uad . Let  be an open non empty subset of  and 2 H1Á2( ). Let (v0, p0) and (v1, p1) be solutions of -  v0 + p0 = 0 in  \ D0 div v0 =0 in \D0 v0 =  on  v0 = 0 on D0 and -  v1 + p1 = 0 in  \ D1 div v1 =0 in \D1 v1 =  on  v1 = 0 on D1 such that (v1,p1)n =(v0,p0)n on , then D 1=D0 . WONAPDE 2004

  12. Sketch of the proof (I) • Let us define D=D0[ D1 . On n D we set • v = v1 – v0 and p = p1 – p0 • The couple(v,p) solves • -  v + p = 0 in  \ D • div v = 0 in  \ D • v = 0 on  • and it satisfies •  (v,p) n = 0 on . • From the unique continuation property (C. Fabre & G. Lebeau, 1990), we conclude that • v = 0 and p = constant in n D. • Then v0 =v1 in n D. D0 D1 WONAPDE 2004

  13. Sketch of the proof (II) D1 D0 • If A0 = D1n D0 is non empty, we can write • -  v0 + p0 = 0 in A0 • div v0 = 0 in A0 • v0 = 0 on A0 = ( D1Å (D0)c) [ ( D0Å D1) • Multiplication by v0 yields • sA0 |r v0 |2 = 0 ) v0 =0 in A0 • Applying the unique continuation property we conclude • v0 =0 in nD0 • This is a contradiction because • v0 =  0 2 • Thus A0 = ;. Analogously, one can prove D0n D1 = ; , andhence • D0=D1. WONAPDE 2004

  14. Stability Question • We want to prove that if the external boundary measurements are close, then the rigid bodies are close. • The main tool we use to tackle stability is the so-called shape differentiation which involves small perturbations of the domain. • We consider a reference domain D and a deformed domain D+u. • The deformation u is assumed to be smooth and such that u = 0 in a neighborhood of . D D +u  WONAPDE 2004

  15. Smoothness of the velocity and pressure with respect to deformations of the rigid body • For each given smooth deformation u and for any Dirichlet boundary condition 2 H1Á2()n, such that s¢ n =0, we consider the problem: •  vu +  pu = 0 in ¤+u = n (D-u) • div vu = 0 in ¤+u ……(Pu) • vu=  on , • vu= 0 on  (D – u) • There exists a unique solution (vu , pu) 2 H1() n£ L2(), of the above problem (Pu). WONAPDE 2004

  16. Regularity of the solutions • Lemma (C. C., L. Friz, O. Kavian, J. Ortega) • There exists a neighborhood W around u=0 such that the mapping u ! ( vu , pu ) ± ( I + u ), which is defined in W and takes values in H1( )n £ L2( ), is analytic in W. Here, (vu , pu) is the unique solution of the problem (Pu). • The proof is based on the implicit function theorem. WONAPDE 2004

  17. Stability Issue Theorem (C.C., L. Friz, O. Kavian, J. Ortega) Let u02 W3,1(; Rn) be given. Assume that u=tu0. Then there exists a strictly positive constant C=C(uo,¤,D,) and an integer m = m(uo,¤,D,) 2 N, such that || D ( ) - D-tuo () ||H-1Á2()¸ C | t |m where D () = (vo , po)n on  and D-tuo () = (vtuo, ptuo)n on . WONAPDE 2004

  18. Sketch of the proof (I) • Assume that the deformed domain has the form ¤+u = nD-u. • Let 2 H1Á2() be given and let (,q) be solution of • - +  q = 0 in  • div  = 0 in  • =  on , • Multiplying (P0) and (Ptuo) by  and integrating by parts we have WONAPDE 2004

  19. Sketch of the proof (II) • Subtracting both identities before we obtain • Therefore WONAPDE 2004

  20. Sketch of the proof (III) Recall that u = t u0.. Thanks to regularity results, one can expand the right hand side of the above inequality as follows (Murat-Simon´s approach to shape differentiation (1974)) WONAPDE 2004

  21. Sketch of the proof (IV) • The previous identity yields the existence of a non negative constant C=C(¤,D, u0,  ) such that for  =  we can write • If C>0, this completes the stability result with m=1. • If C=0, we need to go further and come back to the previous asymptotic expansion analyzing the second order term. A similar calculation to the done above yields the existence of a nonnegative constant C=C(¤,D,u0,) such that … and so on …. WONAPDE 2004

  22. Algorithm and Numerical Experiments Test Problem vu=(0,0) 0 2 (vu,pu)n =0 vu=(0,0) 1 -  vu + pu = 0 in ¤ + u div vu = 0 in ¤ + u vu =  on  = 0[1[2  (vu , pu) n =0 on n  vu = 0 on D-u. WONAPDE 2004

  23. Algorithm and Numerical Experiments A Suitable Objective Function • Let D be the stress linear tensor on the external boundary Ω measured for the unknown body D. • For numerical purposes, a suitable formulation of our inverse problem is to write it down as the following minimization problem which has a unique global minimum u=0, due to the identifiability result. WONAPDE 2004

  24. Explicit Formula for the Gradient • A classical formula of shape differentiation shows that the gradient of our objective function is (,q) being the unique solution of the adjoint problem -   + q = 0 in ¤ + u div  = 0 in ¤ + u  = 2[σ(vu, pu) – σD]n on   ( , q) n =0 on n  = 0 on D-u. WONAPDE 2004

  25. Numerical Method Main Features • Non convex and nonlinear objective function. • Steepest descent algorithm (SD) and non linear conjugate gradient (NLCG) method (an explicit formula for the gradient is available). • Existence of multiple local minimum (Heuristic initialization by means of a Simulated Annealing type algorithm). WONAPDE 2004

  26. Velocity Fields WONAPDE 2004

  27. Vorticity WONAPDE 2004

  28. Difference of Vorticities WONAPDE 2004

  29. Iterations WONAPDE 2004

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