[PPT] - Constrained evolution problems on a metric graph c 1 Luka Grubi si PowerPoint Presentation

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Constrained evolution problems on a metric graph

Luka Grubiˇ si´ c1 Department of Mathematics, University of Zagreb luka.grubisic@math.hr Differential Operators on Graphs and Waveguides, TU Graz, 2019.

1joint work with M. Ljulj, V. Mehrmann and J. Tambaˇ

ca.

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This is the story of two discretizations

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This is the story of two discretizations

0.005 0.01 0.015 0.02 −2 −1 1 2 x 10

−3

−1.5 −1 −0.5 0.5 1 1.5 x 10

−3

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This is the story of two discretizations

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This is the story of two discretizations

Which would you rather have?

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A more comprehensive list of collaborators

Josip Tambaˇ ca, Department of Mathematics, University of Zagreb (thanks for the slides). Sunˇ cica ˇ Cani´ c, UC Berkeley David Paniagua, Baylor College of Medicine, Huston Bojan ˇ Zugec, Faculty of Organization and Informatics, University of Zagreb Mate Kosor, Maritime Department, University of Zadar Matko Ljulj, University of Zagreb Josip Ivekovi´ c, University of Zagreb Matea Galovi´ c, University of Zagreb Marko Hajba, University of Zagreb

K. Schmidt, TU Darmstadt

Volker Mehrmann, TU Berlin

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Outline

1

About stents Usage of stents Stent properties

2

FEM on the metric graph 1D curved rod model 1D stent model Weak formulation Mixed formulation

3

Time dependent problems Mixed formulation Comparison of the 1D and 3D model Examples

4

Some further developments Optimal design

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Usage of stents

carotid stenosis

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Usage of stents

carotid stenosis stent: ”solution” of the problem

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Stent properties

made of cylindrical tubes by laser cuts mostly made of metals: 316L stainless steel, lately from cobalt, chrome and nickel. expanded on the place of stenosis (balloon expandable is dominant (99%) over self-expanding) properties depend on

◮ complex geometry of stent, ◮ mechanical properties of material.

metal = ⇒ theory of elasticity

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Stent properties

made of cylindrical tubes by laser cuts mostly made of metals: 316L stainless steel, lately from cobalt, chrome and nickel. expanded on the place of stenosis (balloon expandable is dominant (99%) over self-expanding) properties depend on

◮ complex geometry of stent, ◮ mechanical properties of material.

metal = ⇒ theory of elasticity small deformation = ⇒ use linearized elasticity We are looking for stents such that response of the stented artery is closest to the response of the healthy artery.

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stent is a 3D elastic body struts thin = ⇒ very fine mesh needed

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stent is a 3D elastic body struts thin = ⇒ very fine mesh needed system very complex and computationally expensive

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stent is a 3D elastic body struts thin = ⇒ very fine mesh needed system very complex and computationally expensive stent struts are thin

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stent is a 3D elastic body struts thin = ⇒ very fine mesh needed system very complex and computationally expensive stent struts are thin use simpler model: 1D curved rod model

(rigorous justification Jurak, Tambaˇ ca (1999), (2001))

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stent is a 3D elastic body struts thin = ⇒ very fine mesh needed system very complex and computationally expensive stent struts are thin use simpler model: 1D curved rod model

(rigorous justification Jurak, Tambaˇ ca (1999), (2001))

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1D curved rod model

˜ ♣′ + ˜ ❢ = 0, ˜ q′ + t × ˜ ♣ = 0, ˜ ω′ + QH−1QT ˜ q = 0, ˜ ω(0) = ˜ ω(ℓ) = 0, ˜ ✉′ + t × ˜ ω = 0, ˜ ✉(0) = ˜ ✉(ℓ) = 0, t ♥ ❜ t ♥ ❜ ♣ q ✉

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1D curved rod model

˜ ♣′ + ˜ ❢ = 0, ˜ q′ + t × ˜ ♣ = 0, ˜ ω′ + QH−1QT ˜ q = 0, ˜ ω(0) = ˜ ω(ℓ) = 0, ˜ ✉′ + t × ˜ ω = 0, ˜ ✉(0) = ˜ ✉(ℓ) = 0, system od 12 ODE t ♥ ❜ t ♥ ❜ ♣ q ✉

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1D curved rod model

˜ ♣′ + ˜ ❢ = 0, ˜ q′ + t × ˜ ♣ = 0, ˜ ω′ + QH−1QT ˜ q = 0, ˜ ω(0) = ˜ ω(ℓ) = 0, ˜ ✉′ + t × ˜ ω = 0, ˜ ✉(0) = ˜ ✉(ℓ) = 0, system od 12 ODE Q = (t, ♥, ❜) – rotation (Frenet basis for middle curve) t – tangent, ♥ – normal, ❜ – binormal for middle curve H – properties of the cross-section geometry, material properties interpretation of unknowns ˜ ♣, ˜ q, ˜ ω, ˜ ✉ : [0, ℓ] → R3

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1D curved rod model

˜ ♣′ + ˜ ❢ = 0, ˜ q′ + t × ˜ ♣ = 0, ˜ ω′ + QH−1QT ˜ q = 0, ˜ ω(0) = ˜ ω(ℓ) = 0, ˜ ✉′ + t × ˜ ω = 0, ˜ ✉(0) = ˜ ✉(ℓ) = 0, system od 12 ODE Q = (t, ♥, ❜) – rotation (Frenet basis for middle curve) t – tangent, ♥ – normal, ❜ – binormal for middle curve H – properties of the cross-section geometry, material properties interpretation of unknowns ˜ ♣, ˜ q, ˜ ω, ˜ ✉ : [0, ℓ] → R3 1D model much easier and quicker to solve than 3D

◮ numerical approximation in 3D: minutes ◮ numerical approximation in 1D: seconds

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1D stent model

At each edge: the system of 12 ODE ˜ ♣e′ + ˜ ❢

e = 0,

˜ qe′ + te × ˜ ♣e = 0, ˜ ωe′ + Qe(He)−1(Qe)T ˜ qe = 0, ˜ ✉e′ + te × ˜ ωe = 0.

✉ ♣ q

♣ q

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1D stent model

At each edge: the system of 12 ODE ˜ ♣e′ + ˜ ❢

e = 0,

˜ qe′ + te × ˜ ♣e = 0, ˜ ωe′ + Qe(He)−1(Qe)T ˜ qe = 0, ˜ ✉e′ + te × ˜ ωe = 0. Junction conditions:

✉ ♣ q

♣ q

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1D stent model

At each edge: the system of 12 ODE ˜ ♣e′ + ˜ ❢

e = 0,

˜ qe′ + te × ˜ ♣e = 0, ˜ ωe′ + Qe(He)−1(Qe)T ˜ qe = 0, ˜ ✉e′ + te × ˜ ωe = 0. Junction conditions:

◮ for kinematical quantities: ˜

ω, ˜ ✉ continuity ♣ q

♣ q

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1D stent model

At each edge: the system of 12 ODE ˜ ♣e′ + ˜ ❢

e = 0,

˜ qe′ + te × ˜ ♣e = 0, ˜ ωe′ + Qe(He)−1(Qe)T ˜ qe = 0, ˜ ✉e′ + te × ˜ ωe = 0. Junction conditions:

◮ for kinematical quantities: ˜

ω, ˜ ✉ continuity

◮ for dynamical quantities: ˜

♣, ˜ q

⋆ sum of contact forces (˜

♣) at each vertex =0

⋆ sum of contact couples (˜

q) at each vertex =0

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1D stent model

At each edge: the system of 12 ODE ˜ ♣e′ + ˜ ❢

e = 0,

˜ qe′ + te × ˜ ♣e = 0, ˜ ωe′ + Qe(He)−1(Qe)T ˜ qe = 0, ˜ ✉e′ + te × ˜ ωe = 0. Junction conditions:

◮ for kinematical quantities: ˜

ω, ˜ ✉ continuity

◮ for dynamical quantities: ˜

♣, ˜ q

⋆ sum of contact forces (˜

♣) at each vertex =0

⋆ sum of contact couples (˜

q) at each vertex =0

rigorous mathematical justification by Γ-convergence in nonlinear elasticity (Tambaˇ

ca, Velˇ ci´ c (2010), Griso (2010))

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1D stent model

At each edge: the system of 12 ODE ˜ ♣e′ + ˜ ❢

e = 0,

˜ qe′ + te × ˜ ♣e = 0, ˜ ωe′ + Qe(He)−1(Qe)T ˜ qe = 0, ˜ ✉e′ + te × ˜ ωe = 0. Junction conditions:

◮ for kinematical quantities: ˜

ω, ˜ ✉ continuity

◮ for dynamical quantities: ˜

♣, ˜ q

⋆ sum of contact forces (˜

♣) at each vertex =0

⋆ sum of contact couples (˜

q) at each vertex =0

rigorous mathematical justification by Γ-convergence in nonlinear elasticity (Tambaˇ

ca, Velˇ ci´ c (2010), Griso (2010)) (Tambaˇ ca, Kosor, ˇ Cani´ c, Paniagua, SIAM J. Appl. Math., 2010) (Zunino, Tambaˇ ca, Cutri, ˇ Cani´ c, Formaggia, Migliavacca, Annals of Biomedical Engineering, 2015)

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Outline in two pictures

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1D stent model

For the stent model we need: V – vertices (nV number of vertices) t

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1D stent model

For the stent model we need: V – vertices (nV number of vertices) E – edges (nE number of edges) t

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1D stent model

For the stent model we need: V – vertices (nV number of vertices) E – edges (nE number of edges) N = (V, E) – graph t

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1D stent model

For the stent model we need: V – vertices (nV number of vertices) E – edges (nE number of edges) N = (V, E) – graph parametrization of edges Φe : [0, ℓe] → R3 (for te and Qe)

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1D stent model

For the stent model we need: V – vertices (nV number of vertices) E – edges (nE number of edges) N = (V, E) – graph parametrization of edges Φe : [0, ℓe] → R3 (for te and Qe) material and cross–section properties (for He)

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Two options

1 Place the contact conditions in the Sobolev space on a graph 2 Consider a larger product Sobolev space and leave contact conditions

in vertices as constraints

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1D stent model – in H1(N; R6)

Unknown: ❯ = (❯1, . . . , ❯nE) = ((˜ ✉1, ˜ ω1), . . . , (˜ ✉nE, ˜ ωnE)) Test function: ❱ = (❱ 1, . . . , ❱ nE) = ((˜ ✈ 1, ˜ ✇ 1), . . . , (˜ ✈ nE, ˜ ✇ nE))

❱ ❱ ❱ ❱ ✈ t ✇ ❱

❯ ✇ ❢ ✈ ♣ ✈ ♣ ✈ q ✇ q ✇ ❱

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1D stent model – in H1(N; R6)

Unknown: ❯ = (❯1, . . . , ❯nE) = ((˜ ✉1, ˜ ω1), . . . , (˜ ✉nE, ˜ ωnE)) Test function: ❱ = (❱ 1, . . . , ❱ nE) = ((˜ ✈ 1, ˜ ✇ 1), . . . , (˜ ✈ nE, ˜ ✇ nE)) Function spaces on the graph N:

H1(N; R6) =

❱ ∈

nE

i=1

H1((0, ℓei ); R6) : ❱ i((Φi)−1(v)) = ❱ j((Φj)−1(v)), v ∈ V, v ∈ ei ∩ ej , Vstent ={❱ ∈ H1(N; R6) : ˜ ✈ i ′ + ti × ˜ ✇ i = 0, i = 1, . . . , nE,

N

❱ = 0}.

❯ ✇ ❢ ✈ ♣ ✈ ♣ ✈ q ✇ q ✇ ❱

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1D stent model – in H1(N; R6)

Unknown: ❯ = (❯1, . . . , ❯nE) = ((˜ ✉1, ˜ ω1), . . . , (˜ ✉nE, ˜ ωnE)) Test function: ❱ = (❱ 1, . . . , ❱ nE) = ((˜ ✈ 1, ˜ ✇ 1), . . . , (˜ ✈ nE, ˜ ✇ nE)) Function spaces on the graph N:

H1(N; R6) =

❱ ∈

nE

i=1

H1((0, ℓei ); R6) : ❱ i((Φi)−1(v)) = ❱ j((Φj)−1(v)), v ∈ V, v ∈ ei ∩ ej , Vstent ={❱ ∈ H1(N; R6) : ˜ ✈ i ′ + ti × ˜ ✇ i = 0, i = 1, . . . , nE,

N

❱ = 0}.

Add weak formulations for rods: ❯ ✇ ❢ ✈ ♣ ✈ ♣ ✈ q ✇ q ✇ ❱

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1D stent model – in H1(N; R6)

Unknown: ❯ = (❯1, . . . , ❯nE) = ((˜ ✉1, ˜ ω1), . . . , (˜ ✉nE, ˜ ωnE)) Test function: ❱ = (❱ 1, . . . , ❱ nE) = ((˜ ✈ 1, ˜ ✇ 1), . . . , (˜ ✈ nE, ˜ ✇ nE)) Function spaces on the graph N:

H1(N; R6) =

❱ ∈

nE

i=1

H1((0, ℓei ); R6) : ❱ i((Φi)−1(v)) = ❱ j((Φj)−1(v)), v ∈ V, v ∈ ei ∩ ej , Vstent ={❱ ∈ H1(N; R6) : ˜ ✈ i ′ + ti × ˜ ✇ i = 0, i = 1, . . . , nE,

N

❱ = 0}.

Add weak formulations for rods: find ❯ ∈ Vstent such that

nE

i=1

ℓi QiHi(Qi)T(˜ ωi)′ · ˜ ✇ ′dx1 =

nE

i=1

ℓi ˜ ❢

i · ˜

✈dx1 +

nE

i=1

˜ ♣i(ℓi)˜ ✈(ℓi) − ˜ ♣i(0)˜ ✈(0) + ˜ qi(ℓi) ˜ ✇(ℓi) − ˜ qi(0) ˜ ✇(0), ❱ ∈ Vstent

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1D stent model – in H1(N; R6)

Unknown: ❯ = (❯1, . . . , ❯nE) = ((˜ ✉1, ˜ ω1), . . . , (˜ ✉nE, ˜ ωnE)) Test function: ❱ = (❱ 1, . . . , ❱ nE) = ((˜ ✈ 1, ˜ ✇ 1), . . . , (˜ ✈ nE, ˜ ✇ nE)) Function spaces on the graph N:

H1(N; R6) =

❱ ∈

nE

i=1

H1((0, ℓei ); R6) : ❱ i((Φi)−1(v)) = ❱ j((Φj)−1(v)), v ∈ V, v ∈ ei ∩ ej , Vstent ={❱ ∈ H1(N; R6) : ˜ ✈ i ′ + ti × ˜ ✇ i = 0, i = 1, . . . , nE,

N

❱ = 0}.

Add weak formulations for rods: find ❯ ∈ Vstent such that

nE

i=1

ℓi QiHi(Qi)T(˜ ωi)′ · ˜ ✇ ′dx1 =

nE

i=1

ℓi ˜ ❢

i · ˜

✈dx1, ❱ ∈ Vstent

(ˇ Cani´ c & Tambaˇ ca, IMA Journal of Applied Mathematics, 2012)

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1D stent model: properties for the forms

a(❯, ❱ ) =

nE

i=1

ℓi QiHi(Qi)T(˜ ωi)′ · ˜ ✇ ′dx1, f (❱ ) =

nE

i=1

ℓi ˜ ❢

i · ˜

✈ idx1,

Problem (W)

Find ❯ ∈ Vstent such that a(❯, ❱ ) = f (❱ ), ❱ ∈ Vstent.

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1D stent model: properties for the forms

a(❯, ❱ ) =

nE

i=1

ℓi QiHi(Qi)T(˜ ωi)′ · ˜ ✇ ′dx1, f (❱ ) =

nE

i=1

ℓi ˜ ❢

i · ˜

✈ idx1,

Problem (W)

Find ❯ ∈ Vstent such that a(❯, ❱ ) = f (❱ ), ❱ ∈ Vstent. Vstent is a Hilbert space (b is continuous) a is Vstent-elliptic f is continuous on Vstent

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1D stent model: properties for the forms

a(❯, ❱ ) =

nE

i=1

ℓi QiHi(Qi)T(˜ ωi)′ · ˜ ✇ ′dx1, f (❱ ) =

nE

i=1

ℓi ˜ ❢

i · ˜

✈ idx1,

Problem (W)

Find ❯ ∈ Vstent such that a(❯, ❱ ) = f (❱ ), ❱ ∈ Vstent. Vstent is a Hilbert space (b is continuous) a is Vstent-elliptic f is continuous on Vstent Lax-Milgram implies

Theorem (form a is Vstent-elliptic)

There exits a unique solution of Problem (W).

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1D stent model: a look into constraints

Problem: to construct functions within Vstent! Vstent = {❱ ∈ H1(N; R6) : ˜ ✈ i ′ + ti × ˜ ✇ i = 0, i = 1, . . . , nE,

N

❱ = 0}, ❱ P ♣ ✈ t ✇ ✈ ✇ P ♣ ♣ ❱ ❱

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1D stent model: a look into constraints

Problem: to construct functions within Vstent! Vstent = {❱ ∈ H1(N; R6) : ˜ ✈ i ′ + ti × ˜ ✇ i = 0, i = 1, . . . , nE,

N

❱ = 0}, b(❱ , P) :=

nE

i=1

ℓi ˜ ♣i · (˜ ✈ i ′ + ti × ˜ ✇ i)dx1 + α ·

nE

i=1

ℓi ˜ ✈ idx1 + β ·

nE

i=1

ℓi ˜ ✇ idx1, P ♣ ♣ ❱ ❱

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1D stent model: a look into constraints

Problem: to construct functions within Vstent! Vstent = {❱ ∈ H1(N; R6) : ˜ ✈ i ′ + ti × ˜ ✇ i = 0, i = 1, . . . , nE,

N

❱ = 0}, b(❱ , P) :=

nE

i=1

ℓi ˜ ♣i · (˜ ✈ i ′ + ti × ˜ ✇ i)dx1 + α ·

nE

i=1

ℓi ˜ ✈ idx1 + β ·

nE

i=1

ℓi ˜ ✇ idx1, P = (˜ ♣1, . . . , ˜ ♣nE, α, β), M := L2(N; R3) × R3 × R3 =

nE

i=1

L2(0, ℓi; R3) × R3 × R3 Then Vstent = {❱ ∈ H1(N; R6) : b(❱ , Θ) = 0, ∀Θ ∈ M}.

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1D stent model: a look into constraints

Problem: to construct functions within Vstent! Vstent = {❱ ∈ H1(N; R6) : ˜ ✈ i ′ + ti × ˜ ✇ i = 0, i = 1, . . . , nE,

N

❱ = 0}, b(❱ , P) :=

nE

i=1

ℓi ˜ ♣i · (˜ ✈ i ′ + ti × ˜ ✇ i)dx1 + α ·

nE

i=1

ℓi ˜ ✈ idx1 + β ·

nE

i=1

ℓi ˜ ✇ idx1, P = (˜ ♣1, . . . , ˜ ♣nE, α, β), M := L2(N; R3) × R3 × R3 =

nE

i=1

L2(0, ℓi; R3) × R3 × R3 Then Vstent = {❱ ∈ H1(N; R6) : b(❱ , Θ) = 0, ∀Θ ∈ M}. Solution: mixed formulation

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Mixed formulation

Problem (M)

Find (❯, P) ∈ H1(N; R6) × M such that a(❯, ❱ ) + b(❱ , P) = f (❱ ), ❱ ∈ H1(N; R6), b(❯, Θ) = 0, Θ ∈ M.

❱

❱ ❱

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SLIDE 49

Mixed formulation

Problem (M)

Find (❯, P) ∈ H1(N; R6) × M such that a(❯, ❱ ) + b(❱ , P) = f (❱ ), ❱ ∈ H1(N; R6), b(❯, Θ) = 0, Θ ∈ M.

Theorem

If b satisfies the inf-sup condition: inf

Θ∈L2(N;R3)×R3×R3

sup

❱ ∈H1(N;R6)

b(❱ , Θ) ❱ H1(N;R6)ΘL2(N;R3)×R3×R3 ≥ β > 0 the Problem (M) has unique solution. Then the Problem (W) is equivalent to Problem (M)!

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SLIDE 50

Alternatively – in the direct product space

We will also consider direct product space formulation ⇛ Let linear algebra solver do the heavy lifting We build H1(N; R6) by eliminating constraints ❱ i((Φi)−1(v)) = ❱ j((Φj)−1(v)) Alternative introduce new variables and extend the restriction form. We then we get

V = L2(N; R3) × L2(N; R3) × R3nE × R3nE × R3nE × R3nE × R3 × R3, M = L2

H1(N; R3) × L2 H1(N; R3) × R3nV × R3nV.

Projection by interpolation theory will be easier, since this is just a large product space We pay by considerably increasing the dimension of the problem – is it to expensive?

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SLIDE 51

Recall

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SLIDE 52

inf–sup for stents

Lemma

All stents in class S satisfy inf–sup condition. Class S contains: stents with all curved struts stents with straight struts which are linearly independent in all vertices they meet. Proof, essentially LA (Grubiˇ si´ c, Ivekovi´ c, Tambaˇ ca, ˇ Zugec, Rad HAZU, 2017) Proof for the V space formulation (Grubiˇ si´ c, Ljulj, Mehrmann, Tambaˇ ca, 2018) Direct product space formulation adds additional constraints (continuity at vertices of displacements and couples)

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SLIDE 53

FEM for mixed formulation

Let us take finite dimensional subspaces V h ⊂ H1(N; R6), Mh ⊂ M.

Problem (Mh)

Find (❯h, Ph) ∈ V h × Mh such that a(❯h, ❱ h) + b(❱ h, Ph) = f (❱ h), ❱ h ∈ V h, b(❯h, Θh) = 0, Θh ∈ Mh. ❯ P ❯ ❯ ❱ ❱ ❯ ❱ ❱ ❱

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SLIDE 54

FEM for mixed formulation

Let us take finite dimensional subspaces V h ⊂ H1(N; R6), Mh ⊂ M.

Problem (Mh)

Find (❯h, Ph) ∈ V h × Mh such that a(❯h, ❱ h) + b(❱ h, Ph) = f (❱ h), ❱ h ∈ V h, b(❯h, Θh) = 0, Θh ∈ Mh. If (❯h, Ph) solves Problem (Mh) then ❯h solves weak forulation

Problem (Wh)

Find ❯h ∈ V h

stent = {❱ h ∈ V h : b(❱ h, Θh) = 0, Θh ∈ Mh} such that

a(❯h, ❱ h) = f (❱ h), ❱ h ∈ V h

stent.

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SLIDE 55

Geometry matters!

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SLIDE 56

Existence for ❯h

Note: V h

stent ⊂ Vstent

✉ ❯

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SLIDE 57

Existence for ❯h

Note: V h

stent ⊂ Vstent=

⇒ We can not use Vstent–ellipticity of a ✉ ❯

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SLIDE 58

Existence for ❯h

Note: V h

stent ⊂ Vstent=

⇒ We can not use Vstent–ellipticity of a Discretization: V h – piecewise polynomials of order n (denoted by Pn) Mh – piecewise polynomials of order m

Lemma

The form a is V h

stent–elliptic if and only if m ≥ n − 1.

✉ ❯

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SLIDE 59

Existence for ❯h

Note: V h

stent ⊂ Vstent=

⇒ We can not use Vstent–ellipticity of a Discretization: V h – piecewise polynomials of order n (denoted by Pn) Mh – piecewise polynomials of order m

Lemma

The form a is V h

stent–elliptic if and only if m ≥ n − 1.

Discrete inextensibility: ℓi

0 ˜

θ

i · (˜

✉i)′ = 0, for all ˜ θ

i ∈ Pm.

❯

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SLIDE 60

Existence for ❯h

Note: V h

stent ⊂ Vstent=

⇒ We can not use Vstent–ellipticity of a Discretization: V h – piecewise polynomials of order n (denoted by Pn) Mh – piecewise polynomials of order m

Lemma

The form a is V h

stent–elliptic if and only if m ≥ n − 1.

Discrete inextensibility: ℓi

0 ˜

θ

i · (˜

✉i)′ = 0, for all ˜ θ

i ∈ Pm.

Theorem

If m ≥ n − 1, ❯h, exists.

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SLIDE 61

Existence for ❯h

Note: V h

stent ⊂ Vstent=

⇒ We can not use Vstent–ellipticity of a Discretization: V h – piecewise polynomials of order n (denoted by Pn) Mh – piecewise polynomials of order m

Lemma

The form a is V h

stent–elliptic if and only if m ≥ n − 1.

Discrete inextensibility: ℓi

0 ˜

θ

i · (˜

✉i)′ = 0, for all ˜ θ

i ∈ Pm.

Theorem

If m ≥ n − 1, ❯h, exists.

Remark

We miss discrete inf-sup for the formulation in H1(N; R6). However, for the direct product space formulation inf-sup follows readily!

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SLIDE 62

Note – compare with information science PDE graph models.

Information science:

◮ Huge graph, but a scalar function ◮ “Finite difference” discretization

Structural optimization:

◮ Highly structured graph, nodes of low incidence degree ◮ Constrained vector valued functions. ◮ FEM discretization

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SLIDE 63

Convergence of FEM for stents

introduce new vertices (with the same junction conditions) ❯ P ❯ ❯ ❯ ❯ ❯ ✉ t

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SLIDE 64

Convergence of FEM for stents

introduce new vertices (with the same junction conditions) the same problem with struts of length ≤ h ❯ P ❯ ❯ ❯ ❯ ❯ ✉ t

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SLIDE 65

Convergence of FEM for stents

introduce new vertices (with the same junction conditions) the same problem with struts of length ≤ h we use P2 approximation for ❯h and P1 for Ph ❯ ❯ ❯ ❯ ❯ ✉ t

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SLIDE 66

Convergence of FEM for stents

introduce new vertices (with the same junction conditions) the same problem with struts of length ≤ h we use P2 approximation for ❯h and P1 for Ph The error estimate is based on the interpolation estimate for interpolation operators I2 and I1 ❯ ❯ ❯ ❯ ❯ ✉ t

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SLIDE 67

Convergence of FEM for stents

introduce new vertices (with the same junction conditions) the same problem with struts of length ≤ h we use P2 approximation for ❯h and P1 for Ph The error estimate is based on the interpolation estimate for interpolation operators I2 and I1 interpolation is strut by strut (this simplifies analysis) ❯ ❯ ❯ ❯ ❯ ✉ t

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SLIDE 68

Convergence of FEM for stents

introduce new vertices (with the same junction conditions) the same problem with struts of length ≤ h we use P2 approximation for ❯h and P1 for Ph The error estimate is based on the interpolation estimate for interpolation operators I2 and I1 interpolation is strut by strut (this simplifies analysis) I2 : Vstent → V h

stent is defined by

I2❯|e ∈ P2, (I2❯)|ei(0) = ❯|ei(0), (I2❯)|ei(0) = ❯|ei(ℓ), and discrete inextensibility ℓi ˜ θ

i · ((˜

✉i)′ + ti × ˜ ωi)dx1 = 0, ˜ θ

i ∈ P1.

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SLIDE 69

Convergence of FEM for stents

introduce new vertices (with the same junction conditions) the same problem with struts of length ≤ h we use P2 approximation for ❯h and P1 for Ph The error estimate is based on the interpolation estimate for interpolation operators I2 and I1 interpolation is strut by strut (this simplifies analysis) I2 : Vstent → V h

stent is defined by

I2❯|e ∈ P2, (I2❯)|ei(0) = ❯|ei(0), (I2❯)|ei(0) = ❯|ei(ℓ), and discrete inextensibility ℓi ˜ θ

i · ((˜

✉i)′ + ti × ˜ ωi)dx1 = 0, ˜ θ

i ∈ P1.

Note that full inextensibility poses to big restriction for the error estimate!

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SLIDE 70

Convergence of FEM for stents

The interpolation satisfies the estimate ❯ − I2❯H1(0,ℓi;R6) ≤ Ch2❯′′′L2(0,ℓi). ❯ P ❯ P ❯ ❯ ❯ P

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SLIDE 71

Convergence of FEM for stents

The interpolation satisfies the estimate ❯ − I2❯H1(0,ℓi;R6) ≤ Ch2❯′′′L2(0,ℓi). Adapting an argument from Boffi, Brezzi and Gastaldi we obtain

Theorem

Let h > 0 denotes the size of the discretization mesh, (❯, P) is the solution of the mixed formulation and (❯h, Ph) solution of the discretized problem piecewisely (P2)6 × (P1)3 polynomials. Then ❯ − ❯hH1(N;R6) ≤ Ch2(❯′′′L2(N;R6) + P′′L2(N;R3)). no error estimate for multipliers! yields resolvent estimates for the spectral problem!

(Grubiˇ si´ c, Tambaˇ ca, in review NLAA)

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SLIDE 72

Numerical examples – convergence validate estimates

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 −2 −1.5 −1 −0.5 0.5 1 1.5 2 x 10

−3

0.01 0.02 −2 −1 1 2 x 10

−3

−2 −1.5 −1 −0.5 0.5 1 1.5 2 x 10

−3

constant radial forcing

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 −5 5 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

0.01 0.02 −3 −2 −1 1 2 3 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

x2

1 radial forcing

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SLIDE 73

Numerical rate of convergence

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 1 1.5 2 2.5 3 3.5 Rates of convergence rate of H1 error for u rate of L2 error for n rate of L2 error for u

E(h) = Chα for radial forcing depending as x2

1 on the longitudinal variable.

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SLIDE 74

Numerical rate of convergence

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 1 1.5 2 2.5 3 3.5 Rates of convergence rate of H1 error for u rate of L2 error for n rate of L2 error for u

E(h) = Chα for radial forcing depending as x2

1 on the longitudinal variable.

numerical approximations for 2, 4, 8, 16, 32, 64 division of struts are compared with solution for 128 divisions of every strut. L2 errors are one order better then H1. coincide with theoretical estimates.

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SLIDE 75

A paradox!

direct product for. in H1(N) splitting # time(s) size of matrix time(s) size of matrix 8 22 105198 2 38958 16 47 211182 42 78702 32 108 423150 152 158190 64 288 847086 629 317166 128 903 1694958 4183 635118

Tablica: Times and matrix sizes for the old and new numerical scheme

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SLIDE 76

Evolution problem

˜ ♣i ′ + ˜ ❢

i = ρiAi∂tt ˜

✉i, ˜ qi ′ + ti × ˜ ♣i = 0, ˜ ωi ′ + QiHi(Qi)T ˜ qi = 0, ˜ ✉i ′ + ti × ˜ ωi = θi, i = 1, . . . , nE + junction conditions ❯ ❱ ✉ ✈ ❯ ❯ ❱ ❯ ❱ ❱ ❱

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SLIDE 77

Evolution problem

˜ ♣i ′ + ˜ ❢

i = ρiAi∂tt ˜

✉i, ˜ qi ′ + ti × ˜ ♣i = 0, ˜ ωi ′ + QiHi(Qi)T ˜ qi = 0, ˜ ✉i ′ + ti × ˜ ωi = θi, i = 1, . . . , nE + junction conditions Let (limit of 3D linearized Antman-Cosserat model) m(❯, ❱ ) =

nE

i=1

ρiAi ℓi ˜ ✉i · ˜ ✈ idx1,

Problem (EvoP)

Find ❯ ∈ L2(0, T; Vstent) such that ∂ttm(❯, ❱ ) + a(❯, ❱ ) = f (❱ ), ❱ ∈ Vstent.

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SLIDE 78

Eigenvalue problem

Problem (EigP)

Find (λ, ❯) ∈ R × Vstent, ❯ = 0 such that a(❯, ❱ ) = λ2m(❯, ❱ ), ❱ ∈ Vstent. ❯ ❯ ❯ ❱ ❱ ❯ ❱ ❱ ❯

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SLIDE 79

Eigenvalue problem

Problem (EigP)

Find (λ, ❯) ∈ R × Vstent, ❯ = 0 such that a(❯, ❱ ) = λ2m(❯, ❱ ), ❱ ∈ Vstent.

Problem (EigQ)

Find (λ, (❯, Ξ)) ∈ R × (H1(N; R6) × L2(N; R3) × R3 × R3), (❯, Ξ) = 0 such that a(❯, ❱ ) + b(❱ , Ξ) = λ2m(❯, ❱ ), ❱ ∈ H1(N; R6), b(❯, Θ) = 0, Θ ∈ L2(N; R3) × R3 × R3. ⇛ continuous inf − sup condition guarantees that the resolvent set is nonempty

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SLIDE 80

Convergence theory by Boffi–Brezzi–Marini

Recall solution operators T : H → Dom(a) and S : H → Dom(B), a(T❢ , ❱ ) + b(S❢ , ❱ ) = (❢ , ❱ ), ❱ ∈ Dom(a), b(Θ, T❢ ) = 0, Θ ∈ Dom(B∗). (1)

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SLIDE 81

Convergence theory by Boffi–Brezzi–Marini

Recall solution operators T : H → Dom(a) and S : H → Dom(B), a(T❢ , ❱ ) + b(S❢ , ❱ ) = (❢ , ❱ ), ❱ ∈ Dom(a), b(Θ, T❢ ) = 0, Θ ∈ Dom(B∗). (1) ⇛ Bounded compact operators Th norm converge to T.

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SLIDE 82

Convergence theory by Boffi–Brezzi–Marini

Recall solution operators T : H → Dom(a) and S : H → Dom(B), a(T❢ , ❱ ) + b(S❢ , ❱ ) = (❢ , ❱ ), ❱ ∈ Dom(a), b(Θ, T❢ ) = 0, Θ ∈ Dom(B∗). (1) ⇛ Bounded compact operators Th norm converge to T. ⇛ If the resolvent is converging somewhere – say at z = 0 –then it converges for every z in resolvent set ρ(T) = C \ Spec(T).

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SLIDE 83

Convergence theory by Boffi–Brezzi–Marini

Recall solution operators T : H → Dom(a) and S : H → Dom(B), a(T❢ , ❱ ) + b(S❢ , ❱ ) = (❢ , ❱ ), ❱ ∈ Dom(a), b(Θ, T❢ ) = 0, Θ ∈ Dom(B∗). (1) ⇛ Bounded compact operators Th norm converge to T. ⇛ If the resolvent is converging somewhere – say at z = 0 –then it converges for every z in resolvent set ρ(T) = C \ Spec(T). ⇛ Analogous definition of Th.

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SLIDE 84

Convergence rate – comments

⇛ Let λ, u and λh and uh be eigenvalues and eigenvectors from V h ⇛ Then |λ − λh| ≤ CT − ThDom(a) = O(I − I2) = O(h2) ❯ − ❯hDom(a) ≤ CT − ThDom(a) = O(I − I2) = O(h2) . eigenfuction c.rate optimal, eigenvalue c.rate not. ⇛ If Sh exists, then |λ − λh| ≤ CT − ThDom(a)S − Sh . ⇛ λh is not a Ritz value of the solution operator T.

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SLIDE 85

Convergence rate – comments

⇛ Let λ, u and λh and uh be eigenvalues and eigenvectors from V h ⇛ Then |λ − λh| ≤ CT − ThDom(a) = O(I − I2) = O(h2) ❯ − ❯hDom(a) ≤ CT − ThDom(a) = O(I − I2) = O(h2) . eigenfuction c.rate optimal, eigenvalue c.rate not. ⇛ If Sh exists, then |λ − λh| ≤ CT − ThDom(a)S − Sh . ⇛ λh is not a Ritz value of the solution operator T. ⇛ Note that here we also have the “singular mass” operator, and so we are actually studying the convergence of ThMh!

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SLIDE 86

Structure recapitulation

⇛ Jordan structure as a consequence of algebraic constraints.

Lemma

Consider (EigQ) and let (E, K) be its block operator matrix representation. Then there exists a nonsingular V with the property that

ˆ E = V∗EV =      M      , V−1③ =      ˆ ③1 ˆ ③2 ˆ ③3 ˆ ③4 ˆ ③5      ˆ K = V∗KV =       ˆ BT

51

ˆ A22 ˆ BT

42

ˆ A33 ˆ B42 ˆ B51       , V∗❋ =      ˆ ❋ 4     

where ˆ A33 = ˆ AT

33, ˆ

B42, and ˆ B51 are invertible, and ˆ ❋ 4 = ❋ 3.

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SLIDE 87

Recapitulation

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 second order problems see Arioli and Benzi 2015.

The interplay of geometry and constraints

Doing interpolation on each edge and then assembling into a graph sometimes fails.

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SLIDE 88

Verification of the 1D model

Compare 1D and 3D solutions (jointly with K. Schmidt and A. Semin) geometry discretization

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SLIDE 89

Verification of the 1D model – COMSOL, CONCEPTS

Error between 1D reduced model and CONCEPTS 3D model is 2%.

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SLIDE 90

Examples of eigenproblem for 1D stent model

Four stent meshes considered (similar to Palmaz, Express; Cypher and Xience by Cordis)

0.005 0.01 0.015 0.02 −2 −1 1 2 x 10

−3

−1.5 −1 −0.5 0.5 1 1.5 x 10

−3

0.01 0.02 −2 −1 1 2 x 10

−3

−1.5 −1 −0.5 0.5 1 1.5 x 10

−3

0.005 0.01 0.015 0.02 −2 −1 1 2 x 10

−3

−1.5 −1 −0.5 0.5 1 1.5 x 10

−3

0.01 0.02 −1.5 −1 −0.5 0.5 1 1.5 x 10

−3

−1.5 −1 −0.5 0.5 1 1.5 x 10

−3

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SLIDE 91

Leading eigenvalues

Palmaz Cypher Express Xience 1. 1.033 0.8894 0.06014 0.05488 2. 1.033 0.8895 0.06014 0.05488 3. 5.265 1.3683 0.32504 0.28767 4. 7.499 3.5328 0.33972 0.32201 5. 7.499 3.5329 0.33973 0.32201 6. 11.329 3.6604 0.58740 0.58038

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SLIDE 92

Palmaz

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−1.5 −1 −0.5 0.5 1 1.5 x 10

−3

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−2 −1.5 −1 −0.5 0.5 1 1.5 2 x 10

−3

−2 −1 1 2 x 10

−3

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SLIDE 93

Cypher

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 −2 −1 1 2 x 10

−3

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−4 −3 −2 −1 1 2 3 4 x 10

−3

−1.5 −1 −0.5 0.5 1 1.5 x 10

−3

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SLIDE 94

Express

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−2 −1 1 2 x 10

−3

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−2 −1 1 2 x 10

−3

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SLIDE 95

Xience

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−2 −1 1 2 x 10

−3

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−3 −2 −1 1 2 3 x 10

−3

−5 5 10 15 20 x 10

−3

−2 −1 1 2 x 10

−3

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SLIDE 96

Convergence rates

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SLIDE 97

Evolution problems

Structure of the pencil 0 is in the resolvent set Problem is index 2 (∞ eigenvalue has Jordan chain of length 2) Imaginary eigenvalues are semisimple use deflation and exponential integrators, or backward differentiation schemes like BDF-2, or Volker’s favorite Radau2a.

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SLIDE 98

Evolution problems

Structure of the pencil 0 is in the resolvent set Problem is index 2 (∞ eigenvalue has Jordan chain of length 2) Imaginary eigenvalues are semisimple use deflation and exponential integrators, or backward differentiation schemes like BDF-2, or Volker’s favorite Radau2a. ⇛ Perhaps another time in more detail :)

Work so far Grubiˇ si´ c, Tambaˇ ca, Quasi-semidefinite eigenvalue problem and applications. Nanosystems: Physics, Chemistry, Mathematics, 2017 Christian Mehl, Volker Mehrmann, Michal Wojtylak, Linear algebra properties of dissipative Hamiltonian descriptor systems, ArXiv.org, 2018 Luka Grubiˇ si´ c, Matko Ljulj, Volker Mehrmann, Josip Tambaˇ ca, Modeling and discretization methods for the numerical simulation of elastic stents, ArXiv.org, 2018

L. Grubiˇ

si´ c Constrained evolution 25.2.2019 46 / 49

SLIDE 99

Evolution problems

Structure of the pencil 0 is in the resolvent set Problem is index 2 (∞ eigenvalue has Jordan chain of length 2) Imaginary eigenvalues are semisimple use deflation and exponential integrators, or backward differentiation schemes like BDF-2, or Volker’s favorite Radau2a. ⇛ Perhaps another time in more detail :)

Work so far Grubiˇ si´ c, Tambaˇ ca, Quasi-semidefinite eigenvalue problem and applications. Nanosystems: Physics, Chemistry, Mathematics, 2017 Christian Mehl, Volker Mehrmann, Michal Wojtylak, Linear algebra properties of dissipative Hamiltonian descriptor systems, ArXiv.org, 2018 Luka Grubiˇ si´ c, Matko Ljulj, Volker Mehrmann, Josip Tambaˇ ca, Modeling and discretization methods for the numerical simulation of elastic stents, ArXiv.org, 2018

⇛ For the “heat” equation also see Emmrich and Mehrmann

L. Grubiˇ

si´ c Constrained evolution 25.2.2019 46 / 49

SLIDE 100

Evolution problems

Structure of the pencil 0 is in the resolvent set Problem is index 2 (∞ eigenvalue has Jordan chain of length 2) Imaginary eigenvalues are semisimple use deflation and exponential integrators, or backward differentiation schemes like BDF-2, or Volker’s favorite Radau2a. ⇛ Perhaps another time in more detail :)

Work so far Grubiˇ si´ c, Tambaˇ ca, Quasi-semidefinite eigenvalue problem and applications. Nanosystems: Physics, Chemistry, Mathematics, 2017 Christian Mehl, Volker Mehrmann, Michal Wojtylak, Linear algebra properties of dissipative Hamiltonian descriptor systems, ArXiv.org, 2018 Luka Grubiˇ si´ c, Matko Ljulj, Volker Mehrmann, Josip Tambaˇ ca, Modeling and discretization methods for the numerical simulation of elastic stents, ArXiv.org, 2018

⇛ For the “heat” equation also see Emmrich and Mehrmann

L. Grubiˇ

si´ c Constrained evolution 25.2.2019 46 / 49

SLIDE 101

The time stepper actually reads

⇛ In the phase space implicit midpoint rule reads E K uk+1 − uk vk+1 − vk

=

K −K uk+1 + uk vk+1 + vk h 2 + f (·)

h

⇛ Port Hamiltionian formulation since K is invertible ⇛ Time stepper in action

L. Grubiˇ

si´ c Constrained evolution 25.2.2019 47 / 49

SLIDE 102

Optimal design of stents

Minimization of maximal radial displacement – minimize the discrepancy to the artery without stenosis.

0.5 1 1.5 2 2.5 3 x 10

−3

−1.5 −1 −0.5 0.5 1 1.5 x 10

−3

0.5 1 1.5 2 2.5 3 x 10

−3

−2 2 x 10

−3

−1.5 −1 −0.5 0.5 1 1.5 x 10

−3

Original configuration fminsearch ”Optimal” configuration

L. Grubiˇ

si´ c Constrained evolution 25.2.2019 48 / 49

SLIDE 103

Thank you for your attention!

L. Grubiˇ

si´ c Constrained evolution 25.2.2019 49 / 49

SLIDE 104

Acknowledgment

Research supported in part by the grant: “Asymptotic and algebraic analysis of nonlinear eigenvalue pro- blems in contact mechanics and elec- tro magnetism” Administered jointly by DAAD and MZO.

Acknowledgment

Research supported in part by the Croatian Science Foundation grant nr. HRZZ 9345

L. Grubiˇ

si´ c Constrained evolution 25.2.2019 49 / 49