Prirodoslovno-matematiˇ cki fakultet Matematiˇ cki odsjek Sveuˇ ciliˇ ste u Zagrebu Operator representation of an networks of PDEs modeling a coronary stent OTIND 2016, Vienna, December 20, 2016 Luka Grubiˇ si´ c, Josip Ivekovi´ c and Josip Tambaˇ ca Department of Mathematics, University of Zagreb luka.grubisic@math.hr Luka Grubiˇ si´ c 1 / 44
Outline 1 About stents Description of a Stent 2 Mixed variational formulations 3 Stent modeling 1D model of curved rods 1D stent model 4 Eigenvalue problem for 1D stent model Evolution problem Mixed formulation Finite element spaces on Graphs Examples 5 Some further developments Luka Grubiˇ si´ c 2 / 44
About stents • carotid stenosis About stents Luka Grubiˇ si´ c 3 / 44
About stents • carotid stenosis • stent: ” solution”of the problem About stents Luka Grubiˇ si´ c 3 / 44
About stents Luka Grubiˇ si´ c 4 / 44
About stents Luka Grubiˇ si´ c 4 / 44
About stents Luka Grubiˇ si´ c 5 / 44
Stent properties • A network of cylindrical tubes (made by laser cuts) • Typically: 316L 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 Grubiˇ si´ c 6 / 44
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 Grubiˇ si´ c 7 / 44
Type I Dirichlet eigenvalue problem −△ u = λ u u ∈ H = H 1 0 • No finite accumulation points • eigenfunctions make an orthonormal system for H . A mixed formulation reads � I � � 0 � − Div H = , M = . Grad 0 − I Modeli Luka Grubiˇ si´ c 8 / 44
Type II Stokes eigenvalue problem Seek u ∈ H 1 0 ([0 , π ] 2 ) \ { 0 } and λ ∈ R such that −△ u + ∇ p = λ u ∇ · u = 0 . • no finite accumulation points • eigenfunctions span a subspace of H = L 2 (Ω) 2 . A mixed formulation reads � −△ � � I � Grad H = , M = . − Div 0 0 Modeli Luka Grubiˇ si´ c 9 / 44
Some pictures Eigen Vector 0 valeur =52.3471 IsoValue -35.5992 -30.5136 -27.1232 -23.7328 -20.3424 -16.952 -13.5616 -10.1712 -6.7808 -3.3904 -1.68677e-008 3.3904 6.7808 10.1712 13.5616 16.952 20.3424 23.7328 27.1232 35.5992 Vec Value 0 0.0900351 0.18007 0.270105 0.360141 0.450176 0.540211 0.630246 0.720281 0.810316 0.900351 0.990386 1.08042 1.17046 1.26049 1.35053 1.44056 1.5306 1.62063 1.71067 • For Type I take P 1– P 0 • For Typer II take P 2– P 1 Modeli Luka Grubiˇ si´ c 10 / 44
Quasidefinite block operator matrices – form aproach • Split the sesquilinear form h ( u , v ) = k ( u 1 , v 1 ) + b ( u 1 , v 2 ) + b ( v 1 , u 2 ) , as B ∗ � K � h = k + b ≃ . 0 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. Grubiˇ si´ c, Kostrykin, Makarov, Veseli´ c 2013 • 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 Grubiˇ si´ c 11 / 44
Representation theorems for quasidefinite matrices For h we obtain a representation theorem h ( u , v ) = ( H u , v ) H where B ∗ � K � H = 0 B denotes the operator defined by h with Dom( H ) = (Dom( K ) ∩ Dom( B )) ⊕ Dom( B ∗ ) If K is invertible then K − 1 B ∗ � � � K � � I � I 0 H = . BK − 1 − BK − 1 B ∗ 0 I I Has a flavor of a second representation theorem if the domain stability condition holds. Modeli Luka Grubiˇ si´ c 12 / 44
Similarity transformations Since K − 1 B ∗ � � � K � � I � I 0 H = . BK − 1 − BK − 1 B ∗ 0 I I and � − 1 � � � I 0 I 0 = BK − 1 − BK − 1 I I if follows � K � � ˜ � � 0 � � ˜ � u u = λ . − BK − 1 B ∗ − M ˜ v ˜ v and so BK − 1 B ∗ ˜ v = λ M ˜ v is an equivalent eigenvalue problem. Modeli Luka Grubiˇ si´ c 13 / 44
Similarity transformations For Type II there are various equivalent reformulations (should K be singular) like � K + B ∗ B B ∗ � � u � � M � � u � = λ . 0 0 B p p since B ∗ B ∗ � K + B ∗ B B ∗ � I � � K � � = 0 I B 0 B 0 and so M stays unchanged. Recall � K − 1 − K − 1 B ∗ ( BK − 1 B ∗ ) − 1 BK − 1 K − 1 B ∗ ( BK − 1 B ∗ ) − 1 � H − 1 = . ( BK − 1 B ∗ ) − 1 BK − 1 − ( BK − 1 B ∗ ) − 1 Modeli Luka Grubiˇ si´ c 14 / 44
Finally ... � ( K − 1 − K − 1 B ∗ ( BK − 1 B ∗ ) − 1 BK − 1 ) Mu � u � � = λ . ( BK − 1 B ∗ ) − 1 BK − 1 Mu p and so � u � � TMu � = λ . p SMu which yields a reduced problem (in Ker ( B )) T − 1 u = λ Mu . 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 Grubiˇ si´ c 15 / 44
Standard variational approach Define solution operator T : H → Dom( k ) and S : H → Dom( B ), k ( Tf , ˜ u S ) + b ( Sf , ˜ u S ) = ( f , ˜ u S ) , u S ∈ Dom( k ) , ˜ (2.1) n S ∈ Dom( B ∗ ) . b (˜ n S , Tf ) = 0 , ˜ 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, Veseli´ c and Veseli´ c for operators. When K invertible T = ( K − 1 − K − 1 B ∗ ( BK − 1 B ∗ ) − 1 BK − 1 ) further information cf. Boffi–Brezzi–Marini Modeli Luka Grubiˇ si´ c 16 / 44
Constraint satisfaction and computability Direct computation yields BT = = B ( K − 1 − K − 1 B ∗ ( BK − 1 B ∗ ) − 1 BK − 1 ) = BK − 1 − BK − 1 B ∗ ( BK − 1 B ∗ ) − 1 BK − 1 = ( I − ( BK − 1 B ∗ )( BK − 1 B ∗ ) − 1 ) BK − 1 = 0 The action of the solution operator T is computable by solving the saddle point system. Modeli Luka Grubiˇ si´ c 17 / 44
Outline 1 About stents Description of a Stent 2 Mixed variational formulations 3 Stent modeling 1D model of curved rods 1D stent model 4 Eigenvalue problem for 1D stent model Evolution problem Mixed formulation Finite element spaces on Graphs Examples 5 Some further developments Modeli Luka Grubiˇ si´ c 18 / 44
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 Grubiˇ si´ c 19 / 44
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 Grubiˇ si´ c 19 / 44
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 Grubiˇ si´ c 19 / 44
• a very complex 3D structure and computationally expensive Stent modeling Luka Grubiˇ si´ c 20 / 44
• a very complex 3D structure and computationally expensive • stent struts are thin Stent modeling Luka Grubiˇ si´ c 20 / 44
• a very complex 3D structure and computationally expensive • stent struts are thin • model struts with 1D model of curved rods (rigorous justification of this step Jurak, Tambaˇ ca (1999), (2001)) • model the whole stent as“a metric graph”of 1D rods! ( mathematical justification in nonlinear elasticity, Tambaˇ c (2010), Griso (2010) ) ca, Velˇ ci´ Stent modeling Luka Grubiˇ si´ c 20 / 44
1D model of curved rods p ′ + ˜ f = 0 , ˜ q ′ + t × ˜ ˜ p = 0 , ω ′ + QHQ T ˜ q = 0 , ω (0) = ˜ ω ( ℓ ) = 0 , ˜ ˜ u ′ + t × ˜ ˜ ω = 0 , ˜ u (0) = ˜ u ( ℓ ) = 0 , Stent modeling Luka Grubiˇ si´ c 21 / 44
1D model of curved rods p ′ + ˜ f = 0 , ˜ q ′ + t × ˜ ˜ p = 0 , ω ′ + QHQ T ˜ q = 0 , ω (0) = ˜ ω ( ℓ ) = 0 , ˜ ˜ u ′ + t × ˜ ˜ ω = 0 , ˜ u (0) = ˜ u ( ℓ ) = 0 , • system od 12 ODE Stent modeling Luka Grubiˇ si´ c 21 / 44
1D model of curved rods p ′ + ˜ f = 0 , ˜ q ′ + t × ˜ ˜ p = 0 , ω ′ + QHQ T ˜ q = 0 , ω (0) = ˜ ω ( ℓ ) = 0 , ˜ ˜ u ′ + t × ˜ ˜ ω = 0 , ˜ u (0) = ˜ u ( ℓ ) = 0 , • system od 12 ODE • Q = ( t , n , b ) – rotation (Frenet basis for middle curve) • t – tangent, n – normal, b – binormal for middle curve Stent modeling Luka Grubiˇ si´ c 21 / 44
Recommend
More recommend
