Operator representation of an networks of PDEs modeling a coronary stent OTIND 2016, Vienna, December 20, 2016

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1 Prirodoslovno-matematički fakultet Matematički odsjek Sveučilište u Zagrebu Operator representation of an networks of PDEs modeling a coronary stent OTIND 26, Vienna, December 2, 26 Luka Grubišić, Josip Iveković and Josip Tambača Department of Mathematics, University of Zagreb luka.grubisic@math.hr Luka Grubišić / 44

2 Outline About stents Description of a Stent 2 Mixed variational formulations 3 Stent modeling D model of curved rods D stent model 4 Eigenvalue problem for D stent model Evolution problem Mixed formulation Finite element spaces on Graphs Examples 5 Some further developments Luka Grubišić 2 / 44

3 About stents carotid stenosis About stents Luka Grubišić 3 / 44

4 About stents carotid stenosis stent: solution of the problem About stents Luka Grubišić 3 / 44

5 About stents Luka Grubišić 4 / 44

6 About stents Luka Grubišić 4 / 44

7 About stents Luka Grubišić 5 / 44

8 Stent properties A network of cylindrical tubes (made by laser cuts) Typically: 36L stainless steel, but also cobalt, chrome and nickel. expanded on the place of stenosis (balloon expandable is dominant (99%) over self-expanding) properties depend on complex geometry of a stent, mechanical properties of material. metal theory of elasticity start from a conservation law constrained evolution problems About stents Luka Grubišić 6 / 44

9 Plan Mixed variational formulations are good for analyzing systems with constraints. Modal analysis gives first information on time evolution. Mixed eigenvalue problems Let us review two canonical approaches to mixed eigenvalue problems About stents Luka Grubišić 7 / 44

10 Type I Dirichlet eigenvalue problem u = λu u H = H No finite accumulation points eigenfunctions make an orthonormal system for H. A mixed formulation reads H = [ I Div Grad ], M = [ I ]. Modeli Luka Grubišić 8 / 44

11 Type II Stokes eigenvalue problem Seek u H ([, π]2 ) \ {} and λ R such that u + p = λu u =. no finite accumulation points eigenfunctions span a subspace of H = L 2 (Ω) 2. A mixed formulation reads [ ] [ Grad I H =, M = Div ]. Modeli Luka Grubišić 9 / 44

12 Some pictures Eigen Vector valeur = IsoValue e Vec Value For Type I take P P For Typer II take P2 P Modeli Luka Grubišić / 44

13 Quasidefinite block operator matrices form aproach Split the sesquilinear form h(u, v) = k(u, v ) + b(u, v 2 ) + b(v, u 2 ), as ( ) K B h = k + b B. representation theorems self-adjoint operators Ker(h) = (Ker(K) Ker(B)) Ker(B ) higher order Sobolev spaces and interpolation results? indefinite Cholesky on the level of block operator matrices e.g. Grubišić, Kostrykin, Makarov, Veselić 23 This is indeed a special structure, recall the talk of C. Tretter or A. Motovilov. With this we have access to spectral calculus to study constrained problems! Modeli Luka Grubišić / 44

14 Representation theorems for quasidefinite matrices For h we obtain a representation theorem h(u, v) = (Hu, v) H where [ ] K B H = B denotes the operator defined by h with Dom(H) = (Dom(K) Dom(B)) Dom(B ) If K is invertible then H = [ ] [ I K BK I ] [ I K B ] BK B I. Has a flavor of a second representation theorem if the domain stability condition holds. Modeli Luka Grubišić 2 / 44

15 Similarity transformations Since H = [ ] [ I K BK I ] [ I K B ] BK B I. and [ ] [ ] I I BK = I BK I if follows and so [ K BK B ] [ũ ] is an equivalent eigenvalue problem. ṽ [ = λ BK B ṽ = λmṽ M ] [ũ ] ṽ. Modeli Luka Grubišić 3 / 44

16 Similarity transformations For Type II there are various equivalent reformulations (should K be singular) like [ K + B B B B ] [ ] u p [ M = λ ] [ ] u. p since [ ] [ ] [ I B K B K + B = B B ] I B B and so M stays unchanged. Recall [ H K = K B (BK B ) BK K B (BK B ) ] (BK B ) BK (BK B ). Modeli Luka Grubišić 4 / 44

17 Finally... [ ] [ u (K = λ K B (BK B ) BK ] )Mu p (BK B ) BK Mu. and so [ ] [ ] u T Mu = λ p SMu. which yields a reduced problem (in Ker(B)) T u = λmu. All finite eigenvalues of the reduced and the full problem coincide. Infinity is an eigenvalue of the saddle-point problem and it has associated vectors (cf. Spence et.al.). Modeli Luka Grubišić 5 / 44

18 Standard variational approach Define solution operator T : H Dom(k) and S : H Dom(B), k(tf, ũ S ) + b(sf, ũ S ) = (f, ũ S ), b(ñ S, Tf ) =, ñ S Dom(B ). ũ S Dom(k), (2.) Require only ellipticity of k S in Ker(B), Recall Ker(H) = (Ker(K) Ker(B)) Ker(B ), eg. Benzi-Liesen-Golub, for matrices Tretter, Veselić and Veselić for operators. When K invertible T = (K K B (BK B ) BK ) further information cf. Boffi Brezzi Marini Modeli Luka Grubišić 6 / 44

19 Constraint satisfaction and computability Direct computation yields BT = = B(K K B (BK B ) BK ) = BK BK B (BK B ) BK = (I (BK B )(BK B ) )BK = The action of the solution operator T is computable by solving the saddle point system. Modeli Luka Grubišić 7 / 44

20 Outline About stents Description of a Stent 2 Mixed variational formulations 3 Stent modeling D model of curved rods D stent model 4 Eigenvalue problem for D stent model Evolution problem Mixed formulation Finite element spaces on Graphs Examples 5 Some further developments Modeli Luka Grubišić 8 / 44

21 Stent modeling Consider the stent as a 3D elastic body geometry (net structure) elastic material: returns to the initial position after deformation λ, µ, E Stent modeling Luka Grubišić 9 / 44

22 Stent modeling Consider the stent as a 3D elastic body geometry (net structure) elastic material: returns to the initial position after deformation λ, µ, E for small deformation = use linearized elasticity 3D stent simulation Stent modeling Luka Grubišić 9 / 44

23 Stent modeling Consider the stent as a 3D elastic body geometry (net structure) elastic material: returns to the initial position after deformation λ, µ, E for small deformation = use linearized elasticity 3D stent simulation (very expensive!) Stent modeling Luka Grubišić 9 / 44

24 a very complex 3D structure and computationally expensive Stent modeling Luka Grubišić 2 / 44

25 a very complex 3D structure and computationally expensive stent struts are thin Stent modeling Luka Grubišić 2 / 44

26 a very complex 3D structure and computationally expensive stent struts are thin model struts with D model of curved rods (rigorous justification of this step Jurak, Tambača (999), (2)) model the whole stent as a metric graph of D rods! (mathematical justification in nonlinear elasticity, Tambača, Velčić (2), Griso (2)) Stent modeling Luka Grubišić 2 / 44

27 D model of curved rods p + f =, q + t p =, ω + QHQ T q =, ω() = ω(l) =, ũ + t ω =, ũ() = ũ(l) =, Stent modeling Luka Grubišić 2 / 44

28 D model of curved rods p + f =, q + t p =, ω + QHQ T q =, ω() = ω(l) =, ũ + t ω =, ũ() = ũ(l) =, system od 2 ODE Stent modeling Luka Grubišić 2 / 44

29 D model of curved rods p + f =, q + t p =, ω + QHQ T q =, ω() = ω(l) =, ũ + t ω =, ũ() = ũ(l) =, system od 2 ODE Q = (t, n, b) rotation (Frenet basis for middle curve) t tangent, n normal, b binormal for middle curve Stent modeling Luka Grubišić 2 / 44

30 D model of curved rods p + f =, q + t p =, ω + QHQ T q =, ω() = ω(l) =, ũ + t ω =, ũ() = ũ(l) =, system od 2 ODE Q = (t, n, b) rotation (Frenet basis for middle curve) t tangent, n normal, b binormal for middle curve H matrix, properties of the cross-section of the rod, material properties Stent modeling Luka Grubišić 2 / 44

31 D model of curved rods p + f =, q + t p =, ω + QHQ T q =, ω() = ω(l) =, ũ + t ω =, ũ() = ũ(l) =, system od 2 ODE Q = (t, n, b) rotation (Frenet basis for middle curve) t tangent, n normal, b binormal for middle curve H matrix, properties of the cross-section of the rod, material properties the equations are balance of contact forces, balance of contact couples, constitutive relationship for the material, in-extensibility and unsherability condition. Stent modeling Luka Grubišić 2 / 44

32 D model of curved rods p + f =, q + t p =, ω + QHQ T q =, ω() = ω(l) =, ũ + t ω =, ũ() = ũ(l) =, system od 2 ODE Q = (t, n, b) rotation (Frenet basis for middle curve) t tangent, n normal, b binormal for middle curve H matrix, properties of the cross-section of the rod, material properties the equations are balance of contact forces, balance of contact couples, constitutive relationship for the material, in-extensibility and unsherability condition. D model much easier and quicker to solve than 3D Stent modeling Luka Grubišić 2 / 44

33 D model of curved rods p + f =, q + t p =, ω + QHQ T q =, ω() = ω(l) =, ũ + t ω =, ũ() = ũ(l) =, system od 2 ODE Q = (t, n, b) rotation (Frenet basis for middle curve) t tangent, n normal, b binormal for middle curve H matrix, properties of the cross-section of the rod, material properties the equations are balance of contact forces, balance of contact couples, constitutive relationship for the material, in-extensibility and unsherability condition. D model much easier and quicker to solve than 3D numerical approximation in 3D: minutes numerical approximation in D: seconds Stent modeling Luka Grubišić 2 / 44

34 D stent model At each edge: the system of 2 ODE p e + f e =, q e + t e p e =, ω e + Q e H e (Q e ) T q e =, ũ e + t e ω e =, Stent modeling Luka Grubišić 22 / 44

35 D stent model At each edge: the system of 2 ODE p e + f e =, q e + t e p e =, ω e + Q e H e (Q e ) T q e =, ũ e + t e ω e =, Junction conditions for kinematical quantities: ω, u continuity Stent modeling Luka Grubišić 22 / 44

36 D stent model At each edge: the system of 2 ODE p e + f e =, q e + t e p e =, ω e + Q e H e (Q e ) T q e =, ũ e + t e ω e =, Junction conditions for kinematical quantities: ω, u continuity for dynamical quantities: p, q sum of contact forces ( p) at each vertex = 2 sum of contact couples ( q) at each vertex = Stent modeling Luka Grubišić 22 / 44

37 D stent model At each edge: the system of 2 ODE p e + f e =, q e + t e p e =, ω e + Q e H e (Q e ) T q e =, ũ e + t e ω e =, Junction conditions for kinematical quantities: ω, u continuity for dynamical quantities: p, q sum of contact forces ( p) at each vertex = 2 sum of contact couples ( q) at each vertex = rigorous mathematical justification in nonlinear elasticity (Tambača, Velčić (2), Griso (2)) Stent modeling Luka Grubišić 22 / 44

38 D stent model Unknown: U = (U,..., U n E ) = ((ũ, ω ),..., (ũ n E, ω n E )) Test function: V = (V,..., V n E ) = ((ṽ, w ),..., (ṽ n E, w n E )) Stent modeling Luka Grubišić 23 / 44

39 D stent model Unknown: U = (U,..., U n E ) = ((ũ, ω ),..., (ũ n E, ω n E )) Test function: V = (V,..., V n E ) = ((ṽ, w ),..., (ṽ n E, w n E )) Function spaces: { H (N ; R 6 ) = V n E i= H ((, l e ); R 6 ) : V i ((Φ i ) (v)) = V j ((Φ j ) (v)), v V, v e i e j}, V stent ={V H (N ; R 6 ) : ṽ i + t i w i =, i =,..., n E, V = }. N Stent modeling Luka Grubišić 23 / 44

40 D stent model Unknown: U = (U,..., U n E ) = ((ũ, ω ),..., (ũ n E, ω n E )) Test function: V = (V,..., V n E ) = ((ṽ, w ),..., (ṽ n E, w n E )) Function spaces: { H (N ; R 6 ) = V n E i= H ((, l e ); R 6 ) : V i ((Φ i ) (v)) = V j ((Φ j ) (v)), v V, v e i e j}, V stent ={V H (N ; R 6 ) : ṽ i + t i w i =, i =,..., n E, V = }. Sum up weak formulations for rods: N Stent modeling Luka Grubišić 23 / 44

41 D stent model Unknown: U = (U,..., U n E ) = ((ũ, ω ),..., (ũ n E, ω n E )) Test function: V = (V,..., V n E ) = ((ṽ, w ),..., (ṽ n E, w n E )) Function spaces: { H (N ; R 6 ) = V n E i= H ((, l e ); R 6 ) : V i ((Φ i ) (v)) = V j ((Φ j ) (v)), v V, v e i e j}, V stent ={V H (N ; R 6 ) : ṽ i + t i w i =, i =,..., n E, V = }. Sum up weak formulations for rods: find U V stent such that N n E l i i= Q i H i (Q i ) T ( ω i ) w dx = n E l i i= f i ṽdx, V V stent (Tambača, Kosor, Čanić, Paniagua, SIAM J. Appl. Math., 2) Stent modeling Luka Grubišić 23 / 44

42 D stent model: weak formulation revisited a(u, V) = b(v, Ξ) = n E l i i= n E l i i= + α Q i H i (Q i ) T ( ω i ) w dx, ξ i (ṽ i + t i w i )dx n E l i i= Ξ = (ξ,..., ξ n E, α, β), f (V) = n E l i i= f i ṽ i dx, M = L 2 (N ; R 3 ) R 3 R 3 = ṽ i dx + β n E i= n E l i i= w i dx, L 2 (, l i ; R 3 ) R 3 R 3 K = V stent = {V H (N ; R 6 ) : b(v, Θ) =, Θ M}. Stent modeling Luka Grubišić 24 / 44

43 D stent model: weak formulation revisited a(u, V) = n E l i i= Q i H i (Q i ) T ( ω i ) w dx, b(v, Ξ) = unsherability/inextensibility +the nonslip condition. Ξ = (ξ,..., ξ n E, α, β), f (V) = n E l i i= f i ṽ i dx, M = L 2 (N ; R 3 ) R 3 R 3 = n E i= L 2 (, l i ; R 3 ) R 3 R 3 K = V stent = {V H (N ; R 6 ) : b(v, Θ) =, Θ M}. Stent modeling Luka Grubišić 25 / 44

44 Evolution problem p i + f i = ρ i A i tt ũ i, q i + t i p i =, ω i + Q i H i (Q i ) T q i =, ũ i + t i ω i = θ i, i =,..., n E + junction conditions Eigenvalue problem for D stent model Luka Grubišić 26 / 44

45 Evolution problem p i + f i = ρ i A i tt ũ i, q i + t i p i =, ω i + Q i H i (Q i ) T q i =, ũ i + t i ω i = θ i, i =,..., n E + junction conditions Let (limit of 3D linearized Antman-Cosserat model) m(u, V) = Problem (EvoP) n E i= Find U L 2 (, T ; V stent ) such that l i ρ i A i ũ i ṽ i dx, d 2 m(u, V) + a(u, V) = f (V), dt2 V V stent. Eigenvalue problem for D stent model Luka Grubišić 26 / 44

46 Convergence theory by Boffi Brezzi Marini Recall solution operator T : H Dom(k) and S : H Dom(B), k(tf, ũ S ) + b(sf, ũ S ) = (f, ũ S ), b(ñ S, Tf ) =, ñ S Dom(B ). ũ S Dom(k), (4.2) Disscretization For a discretization take X h i M h as piecewise polynomial spaces on N. Then the discrete solution operators are k(t h f h, ũ S ) + b(s h f, ũ S ) = (f, ũ S ), ũ S X h, b(ñ S, Tf ) =, ñ S M h. (4.3) The space K h K is the space ov polynomials which satisfy the constraints. Eigenvalue problem for D stent model Luka Grubišić 27 / 44

47 Convergence theory by Boffi Brezzi Gastaldi Proposition Let K h be a nonempty sequence of subspaces for which we have inperpolation estimates and let k be elliptic on K. Then there is ω(h) = o() such that Tf T h f VS ω(h) f L 2 (N ;Z 3 ). Bounded compact operators T h norm converge to T. If the resolvent is converging somewhere say at z = then it converges for every z in resolvent set ρ(t ) = C \ Spec(T ). Eigenvalue problem for D stent model Luka Grubišić 28 / 44

48 Convergence rate coments Let λ, u and λ h and u h be eigenvalues and eigenvectors from X h Then λ λ h C T T h Dom(k) = O(ω(h)) u u h Dom(k) C T T h Dom(k) = O(ω(h)). eigenfuction c.rate optimal, eigenvalue c.rate not. If S h exists, then λ λ h C T T h Dom(k) S S h. λ h is a Ritz value of h but not a Ritz value of the solution operator T. Eigenvalue problem for D stent model Luka Grubišić 29 / 44

49 Finite element spaces x x For finite element spaces take X (n) h = {u S H (N ; R 6 ) : u S e i P n (e i ; R 6 ), i =,..., n E }, M (m) h = {n S L 2 (N ; R 6 ) : n S e i P m (e i ; R 3 ), i =,..., n E }. We expect the error behaving as in D interpolation. Piecewise polynomial approximation of the curved middle line? Eigenvalue problem for D stent model Luka Grubišić 3 / 44

50 Convergence rates Sobolev spaces on graphs nice review by O. Post Good interpolation operators for lower order spaces hard because of the contact conditions in junctions! Geometry of the graph plays a role. For X (n) h, M(n ) h approximation we obtain (#DOF) (n+) decay rates for the eigenvalue errors. for second order problems see Arioli and Benzi 25. The interplay of geometry and constraints Doing interpolation on each edge and then assembling into a graph fails for first order polynomials since the constraint is to restrictive. Using higher order polynomials introduces numerical integration problems in matrix assembly. Eigenvalue problem for D stent model Luka Grubišić 3 / 44

51 Matrix formulation For stiffness we have A A 2 A m+ K = , B = A m+ A m+2 A m+m+ B B n+ C C n+.. B m+ B m+n+ C m+ C m+n+. and for mass M = [ ] M = D D 2 D m D m+ D m+2 D m+m+. translations have mass but no stifness micro-rotations have stifness but no mass graph connectivity is in B stifness positive definite on K. Eigenvalue problem for D stent model Luka Grubišić 32 / 44

52 Examples of eigenproblem for D stent model Four stents considered x 3 x x x x 3 x x x Eigenvalue problem for D stent model Luka Grubišić 33 / 44

53 Leading eigenvalues Palmaz Cypher Express Xience Eigenvalue problem for D stent model Luka Grubišić 34 / 44

54 Palmaz 3 x x 3.5 x x 3 3 x x 3 3 x x 3 3 x x 3 2 x x 3 Eigenvalue problem for D stent model Luka Grubišić 35 / 44

55 Cypher x x 3 x x x 3 3 x x 3 3 x x 3.5 x x 3 Eigenvalue problem for D stent model Luka Grubišić 36 / 44

56 Express 3 x x 3 2 x x 3 3 x x 3 3 x x 3 3 x x 3 2 x x 3 Eigenvalue problem for D stent model Luka Grubišić 37 / 44

57 Xience 3 x x 3 2 x x 3 3 x x 3 3 x x 3 3 x x 3 2 x x 3 Eigenvalue problem for D stent model Luka Grubišić 38 / 44

58 Convergence rates dominated by geometry errors Eigenvalue problem for D stent model Luka Grubišić 39 / 44

59 Exponential integrators Recall T u = ω 2 Mu. so let s use, for u Ker(B), u = cos((tm) /2 t)u + (TM) /2 sin((tm) /2 t)u for vibration analysis. Some further developments Luka Grubišić 4 / 44

60 Exponential integrators Recall so let s use, for u Ker(B), T u = ω 2 Mu. u = cos((tm) /2 t)u + (TM) /2 sin((tm) /2 t)u for vibration analysis. For the heat equation see Emmrich and Mehrmann Some further developments Luka Grubišić 4 / 44

61 Exponential integrators Recall so let s use, for u Ker(B), T u = ω 2 Mu. u = cos((tm) /2 t)u + (TM) /2 sin((tm) /2 t)u for vibration analysis. For the heat equation see Emmrich and Mehrmann Grimm, Hochbruck, Goeckler (trigonometric integrators) use resolvent Krylov subspace projection to propagate a very stiff second order system. Some further developments Luka Grubišić 4 / 44

62 Trigonometric integrators Action of T is accessible by solving the saddle-point problem. For u, so that u + Gu < we have u W n (τg /2 )u τ C n Gu where W n is the rational approximation of the sine or cosine operator from the (resolvent) Krylov space D stent simulation K n = span{tu, T 2 u,, T n u } Some further developments Luka Grubišić 4 / 44

63 Some conclusions Can detect undesirable vibration modes (meshy structures are stiffer, but can unexpectedly buckle). For a numerical integration we would like to approximate curved rod by splitting it in a chain of straight rods which approximate the middle line Questions? Discrete inf-sup Good linear for Schur complements? Some further developments Luka Grubišić 42 / 44

64 Acknowledgment Research supported by the Croatian Science Foundation grant nr. HRZZ 9345 Some further developments Luka Grubišić 43 / 44

65 Thank you for your attention! Some further developments Luka Grubišić 44 / 44