An Optimization-Based Approach to Decoupling Fluid-Structure Interaction

Transcription

1 Clemson University TigerPrints All Dissertations Dissertations An Optimization-Based Approach to Decoupling Fluid-Structure Interaction Paul Allen Kuberry Clemson University Follow this and additional works at: Recommended Citation Kuberry, Paul Allen, "An Optimization-Based Approach to Decoupling Fluid-Structure Interaction" (015). All Dissertations This Dissertation is brought to you or ree and open access by the Dissertations at TigerPrints. It has been accepted or inclusion in All Dissertations by an authorized administrator o TigerPrints. For more inormation, please contact kokeee@clemson.edu.

2 An Optimization-Based Approach to Decoupling Fluid-Structure Interaction A Dissertation Presented to the Graduate School o Clemson University In Partial Fulillment o the Requirements or the Degree Doctor o Philosophy Mathematical Sciences by Paul Allen Kuberry May 015 Accepted by: Dr. Hyesuk Lee, Committee Chair Dr. Leo Rebholz Dr. Eleanor Jenkins Dr. Timo Heister

3 Abstract Fluid-structure interaction (FSI) is ubiquitous in both manuacturing and nature. At the same time, models describing this phenomenon are highly sensitive and nonlinear. Providing an analytical solution to these models or a realistic set o initial and boundary conditions has proven to be intractable. Within the domain o computational simulation, signiicant eort has been invested in developing numerical methods to approximate solutions using these models. Current eorts to date have included monolithic and partitioned schemes. In this dissertation, a novel partitioned approach is detailed in the context o Galerkin inite elements using body itted meshes. It capitalizes upon the continuity o traction orce on the interace shared between luid and structure subdomains. Introducing a eumann control in lieu o this shared traction orce, the problem is decoupled and permits solving luid and structure subproblems both independently and simultaneously. Analytical results are provided which uniormly bound and demonstrate the existence o an optimal virtual control or the decoupled weak orm o the luid-structure interaction model. The existence o Lagrange multipliers is proven using the derivation o the Fréchet derivative o the operator occurring in the state equations. Discrete solutions to the resulting optimality system are proven to converge over a single time step using the theory rom Brezzi, Rappaz, and Raviart. A steepest descent method is applied to the discrete system o equations and proven to converge under certain conditions. This work provides a theoretical oundation or the application o optimization-based decoupling to luid-structure interaction. For problems with moderate domain deormation, this method has been demonstrated to converge at optimal rates and to be competitive against state o the art methods or numerically solving luid-structure interaction. ii

4 Table o Contents Title Page i Abstract ii List o Tables v List o Figures vi 1 ITRODUCTIO I avier Stokes / Linear Elasticity 10 MODEL EQUATIOS, OTATIO, AD FRAMEWORK Model Equations otation Arbitrary Lagrangian Eulerian Framework Semidiscrete Weak Formulations OPTIMIZATIO-BASED DECOUPLIG Introduction Substitution o Traction Terms with a Control First Order Time Discretization o the Structure Subsystem Second Order Time Discretization o the Structure Subsystem Gauss ewton Algorithm umerical Results Conclusion EXISTECE PROOFS Introduction Weak Formulation o the Constraints Description o the Optimization Problem A Priori Estimates The Existence o an Optimal Solution Convergence o Vanishing Penalty Parameter The Existence o Lagrange Multipliers Lagrange Multiplier Rule iii

5 4.9 Steepest Descent Approach umerical Results Conclusion THEORETICAL COVERGECE RATES Introduction Description o the Optimization Problem Lagrange Multiplier Rule Finite Element Approximations Convergence o Steepest Descent umerical Results Conclusion II avier-stokes / onlinear St. Venant-Kirchho Elasticity APPLICATIO TO OLIEAR ELASTICITY Introduction onlinear Elasticity Description o the Optimization Problem Linearization o the First Piola Kirchho Stress Tensor umerical Results Conclusion Conclusions Bibliography iv

6 List o Tables 3.1 Error in the continuity o velocity between subsystems or each Gauss ewton iteration at three representative time steps Fluid velocity and pressure convergence results or a single time step using the steepest descent algorithm at t = 0.5 s Structure displacement and velocity convergence results or a single time step using the steepest descent algorithm at t = 0.5 s Fluid velocity and pressure convergence results or a single time step using the steepest descent algorithm at t = 0.8 s Structure displacement and velocity convergence results or a single time step using the steepest descent algorithm at t = 0.8 s Fluid velocity and pressure convergence results using the conjugate gradient algorithm rom t = 0.5 to t = 1.0 s Structure displacement and velocity convergence results using the conjugate gradient algorithm rom t = 0.5 to t = 1.0 s Iteration counts or Gauss ewton, GMRES, and luid state solves v

7 List o Figures 1.1 Dirichlet eumann coupling Optimization-based approach Fluid-structure interaction domain ALE coordinate transormation Action o the V( ) operator Domain and boundary conditions or numerical experiment Vertical displacement at three points on the interace using the irst order structure ormulation with: (1) h x = 0.1 cm, h y = 0.1 cm, () h x = 0.1 cm, h y = 0.05 cm, and (3) h x = 0.1 cm, h y = 1 30 cm Vertical displacement at three points on the interace using (1) Algorithm 3.3 with the irst order ormulation or the structure beside the vertical displacement rom () Murea and Sy Fluid pressure proiles [dyne/cm at three time steps Vertical displacement at three points on the interace using (1) irst and () second order ormulations with the optimal control algorithm beside vertical displacement using (3) Aitken s relaxation Computational domain or a manuactured solution Convergence results or the analytic problem Domain and boundary conditions or numerical experiment Vertical displacement at three points on the interace using (1) irst and () second order ormulations with the optimal control algorithm beside vertical displacement using (3) Aitken s relaxation Vertical displacement at three points on the interace using (1) δ = 10 5, () δ = 10 6, (3) δ = 5e-7, and (4) δ = Computational domain or a manuactured solution Domain and boundary conditions or numerical experiment Vertical displacement at three points on the interace using (1) optimization and () Aitken s relaxation with the St. Venant Kirchho constitutive equation and (3) optimization and (4) Aitken s relaxation with the linear elastic constitutive equation Computational domain or 3D pulsatile low through a cylinder vi

8 6.4 Snapshots o luid pressure and scaled solid deormation (by a actor o 10) using (Q 1, Q DC 0 ) elements or the luid pressure and velocity, Q 1 elements or structure and mesh updates vii

9 Chapter 1 ITRODUCTIO Modeling luid-structure interaction (FSI) problems is o great practical importance or many applications in manuacturing, energy, aeroelasticity, deense, and biology [10, 16, 0,, 40,43,48,51, 57,61,64. A ew concrete examples in which luid-structure interactions play an important role are piezoelastic print heads used in some models o inkjet printers, the design o blades or wind turbines, wing design or airplanes, combustion chambers in engines, oshore oil rigs, and blood low through vessels or arteries. Fluid-structure interaction problems are multiphysics problems with governing equations that are coupled through the condition o continuity o traction orce and velocity on the interace. For all but the strongest assumptions, namely negligible displacement, and most basic initial and boundary conditions, the FSI operators are too complex to admit tractable analytical solutions. However, in recent decades there has been success in simulating FSI numerically [11. umerical simulation o this phenomenon is not without diiculty. There is a tight coupling between the luid and structure subsystems, which makes numerical simulations diicult to perorm. As the luid exerts orce onto the structure, the traction orces must balance on the shared interace, which generally results in some movement o the structure. Again, this exerts orce on the luid. The result is that the computational domains are deorming in time and in a way that is implicitly determined by the luid and structure states. 1

10 The limitations o computational simulation are most evident in the simulating o the low o blood through an artery. In order to correctly capture the low, it is necessary to use a patient s own geometry, which requires three dimensional modeling. In order to reduce the computational workload needed, it is o particular importance that methods or solving FSI problems signiicantly reduce the number o iterations as well as the complexity o solves at each time step. There are a wide variety o methods or simulating the coupled FSI system, but each is limited by actors including computational complexity and stability. One class o approaches are monolithic ormulations enorcing the interace conditions while simultaneously solving both subproblems in a single matrix system [6, 4, 45, 67. The other class o approaches are partitioned methods, solving each subproblem separately while inding and transerring boundary conditions. Fluid-structure interaction is multidisciplinary in the sense that the governing equations in the luid subdomain involve the domain knowledge o experts in computational luid dynamics, while the governing equations in the structure subdomain involve the domain knowledge o experts in continuum mechanics. Partitioned approaches permit the use o specialized luid and structure solvers utilizing domain speciic knowledge, which is attractive because they are generally easier to implement and allow the use o legacy codes [1, 5. Solving an FSI problem using a monolithic ormulation o the problem [67 is computationally complex due to requiring many large matrix solves to converge on a solution to the nonlinear system. Additional diiculties with this method include the development o eicient and appropriate preconditioners or the matrix resulting rom the discretized system, although this is currently an active area o research [9, 33, 68. The most common approaches decouple the luid and structure subsystems, which allows or operations on a smaller matrix or each subsystem solve. In order to use existing luid or structure solvers as-is or slightly modiied to solve each subproblem, an eective approach must cleverly enorce the interace conditions so as to quickly converge upon an accurate solution. How and which interace conditions are enorced will have wide-

11 ranging eects on stability, speed, and accuracy o the algorithm. For a partitioned method, there are many options or how to transmit boundary condition inormation back and orth between the two subsystems. A strategy or the transmission o boundary conditions, such as Dirichlet eumann [6 (see Figure 1.1), can be either explicit or implicit, depending on the condition or proceeding to the next time step. I only a ew iterations between subsystems are allowed, then the method is explicit. I the approach iterates between subsystems until the interace conditions are satisied to within some tolerance, then the method is said to be implicit. The Dirichlet eumann iterations [49 will be used or comparison in the numerical results sections o this thesis. Explicit decoupling approaches have the potential to be computationally eicient, particularly in cases where the densities between the luid and structure dier greatly. For areas where there are large dierences in densities o the subsystems, such as aerospace engineering, there has been much success applying explicit methods to numerical simulation o FSI. Two possible trade-os are accuracy and stability. It has been shown that when densities o the subsystems are close, explicit sequentially staggered approaches ail outright and even implicit staggered methods may become unstable [54 due to the added mass eect [1,30,34. One stabilized explicit method uses a ormulation based on itsche s method and penalizes spurious pressure oscillations through the time penalty term on the luid pressure luctuations [19. Counterintuitively, this added mass eect is only exacerbated by decreasing the time step size. For blood low modeling, oten the density o the vessel and luid are nearly identical and the added mass eect is signiicant. Implicit decoupling approaches vary widely in how they attempt to enorce interace conditions. The most common approaches are sequentially staggered in that they pass boundary conditions back and orth between subproblems. Generally, many nonlinear subsystem solves are required and even then stability can not be guaranteed [54. By relaxing the update to the structure solve in each implicit iteration [1, 49, stability can be achieved. There are methods or dynamically changing the relaxation parameter in order to speed up convergence [34. The largest problem with using relaxation schemes is that 3

12 the closer the two densities are in magnitude, the greater the increase in computational complexity because o the additional nonlinear subsystem solves needed. However, this is still an improvement over the divergence o the algorithm. As a strong competitor among implicit methods, modiied Robin-type boundary conditions have recently been used to increase stability and decrease iterations between subsystems [5. eumann b.c. Fluid Mesh Update Structure Dirichlet b.c. Figure 1.1: Dirichlet eumann coupling All methods presented thus ar were designed with inite elements in mind, although there may be extensions to inite volume or inite dierence methods possible. However, the Immersed Boundary Method (IBM) [60, the Immersed Interace Method (IIM) [53, and variants o these make use o inite dierences or solving FSI problems on a ixed uniorm cartesian grid with an immersed interace that uses Dirac delta unctions to integrate the orce exerted by the structure on the luid. As this amily o methods are based on inite dierences, their order o convergence is low. The IBM is irst order accurate and the IIM is second order accurate. Recently, an adaptation o the IBM has been made so that inite elements may be used instead o inite dierences [14. These methods are not as general, since both the IBM or IIM treat the immersed structure as a iber, rather than a multidimensional object. Advantages exist or these methods in applications where the orcing terms are derived rom particle methods, since the Cartesian luid domain is ully Eulerian and does not move, making the mapping o particles to cells particularly straight orward. We are additionally motivated to develop algorithms or inite elements since en- 4

13 gineering disciplines are amiliar with using inite element methods or solving structural mechanics problems rom a Lagrangian ramework. This is largely due to its ability to handle complex geometry with high order accuracy. Also, the inite element method lends itsel well to analyzing stability and convergence. A partitioned approach based on optimization is introduced in this thesis that provides a stable, accurate, and eicient method or decoupling luid-structure interaction problems in the inite element setting. It is not speciic to any particular luid and structure combination, but will allow or solving the luid and structure subproblems in parallel. The algorithm will be presented in the context o solving a ewtonian incompressible luid coupled with a linear and nonlinear elastic solid. The method works by treating the FSI system as a constrained optimal control problem in which the objective is to minimize the dierence between the luid velocity and the structure velocity on the interace. Minimizing the objective is equivalent to enorcing continuity between these velocities. Two dierent deinitions o this control will be used. The deinition used in Chapters 4 5 will nearly enorce continuity o stress along the interace and will enorce it or an optimal solution but also will allow or additional analytic properties needed to show the existence o an optimal solution. The one ound in Chapter 3 and 6 will enorce continuity o stress or any choice o the control. Although this method uses partitioned solves, it is implicit and stable. It avoids the large number o iterations generally required in other partitioned methods, since a single control is used or both subsystems simultaneously (see Figure 1.), rather than iterating back and orth between subsystems. Our approach is inspired by domain decomposition methods that have been explored by Gunzburger and Lee in [39 or solving the avier Stokes equations. In their approach, the computational domain is split into two subdomains using an artiicial interace and a subproblem o the same governing equations is solved on each subdomain. The stress between the two subsystems is prescribed and updated through Gauss ewton iterations so that it minimizes the discontinuity o the luid velocity on the artiicial interace. Similarly, 5

14 Fluid g Optimizer Structure Fluid Adjoint Structure Adjoint Figure 1.: Optimization-based approach in [8, this idea was used or the Stokes Darcy equations. Because the stress is prescribed or the luid and structure subsystem as a eumann boundary condition, both subsystem solves may be made in parallel. In our adaptation o the algorithm, the interace will no longer be artiicial, but rather the natural interace between the two subdomains, each having their own respective governing equations. The use o constrained optimization or FSI has been implemented by Murea and Sy [54 or both a linear and nonlinear elastic ormulation or the structure. However, in their approach or the linear ormulation, they expand a unction along the interace by its eigenunctions and solve or coeicients to the inner product by use o optimization. This allows them to optimize a smaller number o unknowns. They use the stress on the interace as a eumann control or solving the structure subsystem. Then, enorcing continuity o velocity through a Dirichlet boundary condition or the luid subsystem, they update the control and repeat the process until the stress discontinuity on the interace is suiciently small. This process requires that the subsystem solves must be made in serial and is still sequentially staggered. In their implementation, the Broyden Fletcher Goldarb Shanno (BFGS) algorithm is used to numerically optimize over the control space. Our method diers rom that o Murea and Sy in several important ways. First and oremost, our method is not sequentially staggered and both o our subsystems may be solved or simultaneously in the state, linearized, and adjoint equations. Second, we use the analytically linearized orm o the state subsystem operators that would appear in solving the nonlinear state equations using ewton s method, rather than building a Hessian 6

15 numerically. This allows us to update our control by repeatedly solving linear problems. Lastly, by using Gauss ewton iterations, ew nonlinear state solves are necessary at each time step. There is an impetus to provide a mathematical ramework or non-ewtonian luid interaction with an elastic medium, both having domains o the same dimension. The progress made in this thesis towards providing this ramework or ewtonian luids can be used as a template or non-ewtonian luids. We intend to use the approach by constrained optimization to provide a robust oundation or numerical approximation o these problems. Improved numerical algorithms with a irm mathematical basis will beneit biomedical and polymer industries and help to improve health care outcomes. This thesis is organized as ollows. Part 1 deals with a luid modeled by avier Stokes in contact with a linear elastic structure. In Chapter, the FSI model equations or a ewtonian luid with linear elasticity are introduced in their strong orm. Ater deining the notation that will be used throughout the rest o this thesis as well as some important properties o operators that are later reerenced, the Arbitrary Lagrangian Eulerian ramework is introduced. The ully continuous variational ormulation o the FSI problem is recast in the Arbitrary Lagrangian Eulerian ramework, and then discretized in time. This chapter will provide a starting point or the application o an optimization-based decoupling algorithm. Chapter 3 begins by introducing a control into the semi-discretized weak orm o the FSI problem, and then reormulating the monolithic FSI as a constrained optimization problem. The jump in velocities o the two substructures is minimized by a eumann control enorcing the continuity o stress on the interace. A decoupling optimization algorithm is discussed, which requires ew nonlinear solves at each time step. umerical results are presented or a haemodynamic problem with parameters congruous with blood low in a human artery. Additionally, results are given or a manuactured solution on a ixed domain with nonnegligible velocities on the interace. In Chapter 4, a control is again introduced to the semi-discretized weak orm o the 7

16 FSI problem, but two additional terms are absorbed by the control. Through the introduction o the augmented control, stability analysis or the luid and structure subproblems becomes possible. Proos or the stability and existence o optimal solutions or the previously presented unctional minimizing the jump in velocities on the interace are given. A proo is given showing that as the Tikhonov regularization parameter goes to zero, the sequence o optimal controls converge on a solution to the ully coupled problem with interace conditions satisied within a tolerance dependent on time step. Lagrange multipliers are shown to exist and an optimality system is derived. Using the necessary condition rom the optimality system to update the control, the steepest descent approach is used to provide numerical results using the augmented control. The Brezzi Rappaz Raviart theorem is applied in Chapter 5 to prove the convergence o solutions o the discrete optimality system to a solution o the continuous optimality system over a single time step. The approximation error due to spatial discretization is rigorously proven. Attention is again ocused on the steepest descent method, and a proo is given outlining certain assumptions that must be satisied to ensure convergence o the algorithm. The manuactured solution rom Chapter 3 is again revisited with a view to demonstrating the theoretical convergence rate over a single time step. ext, the Gauss ewton with conjugate gradient algorithm described in Chapter 3 is applied to the same problem over many time steps. With a view to applying the optimization-based decoupling algorithm to more challenging and realistic applications, Part, Chapter 6 introduces the nonlinear elastic St. Venant Kirchho model or the structure in contact with a ewtonian luid again modeled by the avier Stokes equations. The strong orm o the FSI equations are presented along with a derivation o the nonlinear elastic operator on a ixed domain using the Piola transorm. The luid is recast in the Arbitrary Lagrangian Eulerian ramework and discretized in time. The semi-discretized weak orm o the FSI problem is then linearized. This linearization o the state operators are then applied to Gauss ewton outer optimization loops with BiCGstab and GMRES perorming the inner optimization loops. umerical solutions are 8

17 presented to a haemodynamic problem having the same parameters as in Chapter 3, but with more realistic physics modeling the structure. A conclusion is given summarizing the results o this thesis and detailing areas that still need attention. 9

18 Part I avier Stokes / Linear Elasticity 10

19 Chapter MODEL EQUATIOS, OTATIO, AD FRAMEWORK.1 Model Equations.1.1 avier Stokes / Linear Elastic Model The luid-structure interaction that we will consider in the irst part o this work is an incompressible ewtonian luid and an isotropic linear elastic structure. Γ s D Γ s Ä Ò Ö Ð Ø ËØÖÙØÙÖ s Γ s D Γ D Γ It0 ÁÒÓÑÔÖ Ð ÐÙ t Γ It Γ Γ D Figure.1: Fluid-structure interaction domain Let t be a bounded moving luid domain at time t in RI with the boundary Γ t such that Γ t = Γ Γ D Γ I t, where Γ It is a moving boundary. Also let s be a ixed 11

20 structure domain with the boundary Γ s such that Γ s = Γ s Γs D Γ I t0, where Γ It0 is the movable luid boundary at time t 0. Consider the system o luid and structure equations [ u ρ t + u u ν D(u) + p = in t, (.1) u = 0 in t, (.) ρ s η t ν s D(η) λ ( η) = s in s, (.3) where u denotes the velocity vector o luid, p the pressure o luid, ρ the density o the luid, ν the luid viscosity, η the displacement o structure, and ρ s the structure density. In (.1) and (.3), D( ) is the rate o the strain tensor, i.e., D(v) := ( v + v T )/. The Lamé parameters are denoted by ν s and λ, and the body orces are denoted by and s. Initial and boundary conditions or u and η are given as ollows: ν D(u)n pn = u on Γ, (.4) u = u D on Γ D, (.5) ν s D(η)n s + λ( η)n s = η on Γ s, (.6) η = 0 on Γ s D, (.7) u(x, t 0 ) = u 0 in t 0, (.8) η(x, t 0 ) = η 0 in s, (.9) η t (x, t 0 ) = η 0 in s, (.10) where η 0 = u 0 on Γ It0. For brevity, we use u = 0 on Γ D, but all o our results hold or the case where u = u D 0 on Γ D with a simple modiication. The moving boundary Γ I t is determined by the displacement η at time t (Fig..1). The interace conditions between the luid and the structure are obtained by enorcing continuity o the velocity and the 1

21 stress orce: η t = u on Γ It, (.11) [ν D(u) p I n = [ν s D(η) + λ( η) n s on Γ It. (.1). otation We use the Sobolev spaces W m,p (D) with norms m,p,d i p <, m,,d i p =. Denote the Sobolev space W m, by H m with the norm m,d. The corresponding space o vector-valued or tensor-valued unctions is denoted by H m. For the variational ormulation o the low equations (.6) (.7) in the ALE ramework, described in section.3, we deine the unction space or the reerence domain: H 1 D ( t 0 ) := {v H 1 ( t 0 ) : v = 0 on Γ D }. The unction spaces or t are then deined as H 1 D ( t ) := {v : t [t 0, T RI, v = v Ψ 1 t or v H 1 D ( t 0 )}, L ( t ) := {q : t [t 0, T RI, q = q Ψ 1 t or p L ( t 0 )}. where Ψ 1 t is the inverse ALE mapping described in section.3. For the structure displacement η, deine the unction space H 1 D (s ) := {ξ H 1 ( s ) : ξ = 0 on Γ s D}. We use (, ), (, ) ΓIt, (, ) s, and (, ) ΓIt0 to denote the L inner product over t t, Γ I t, s, and Γ It0, respectively. 13

22 In the moving luid domain, we deine the bilinear and trilinear orms a(u, v) t b(v, q) t := 1 4 := t t ( u + ( u) T ) : ( v + ( v) T ) d t u, v H 1 D ( t ), q( v) d t v H 1 D ( t ), q L ( t ), and c(u, v, w) t := 1 t u v w u w v d t. For the stationary structure domain, we deine the bilinear orms and d(η, γ) s := 1 4 It is noteworthy that s ( η + ( η) T ) : ( γ + ( γ) T ) d s η, γ H 1 D (s ), e(η, γ) s := ( η)( γ) d s s η, γ H 1 D (s ). c(u, v, v) t = 0 u, v H 1 D ( t ). (.13) Throughout this thesis, C represents a positive constant independent o time. As is well known, a(, ), b(, ), c(,, ), and d(, ) are continuous and there exist constants 14

23 C 1, C, C 3, C 4 and C 5 such that a(u, v) C 1 u t 1, v t 1, t b(v, q) C v t 1, q t 0, t u, v H 1 D ( t ), (.14) v H 1 D ( t ), q L ( t n ), (.15) c(u, v, w) C 3 u t 1, v t 1, w t 1, t u, v, w H 1 D ( t ), (.16) and d(η, γ) s C 4 η 1, s γ 1, s η, γ H 1 D (s ), (.17) e(η, γ) s C 5 η 1, s γ 1, s η, γ H 1 D (s ). (.18) There exist positive coercivity constants C 6 and C 7 such that a(u, u) t C 6 u 1, t u H 1 D ( t ), (.19) d(η, η) s C 7 η 1, s η H1 D (s ), (.0) and b(, ) has the in-sup condition sup 0 v H 1 D ( t ) b(v, q) t v 1, t C 8 q 0, t q L ( t ) (.1) where C 8 is a positive constant..3 Arbitrary Lagrangian Eulerian Framework The Arbitrary Lagrangian Eulerian (ALE) [6, 46 ramework is the most widely used description o the luid subproblem in inite element FSI simulation. In the ALE ormulation, a one-to-one coordinate transormation is introduced or the luid domain, and the luid equations can be rewritten with respect to a moving domain which is Lagrangian on its boundaries and in between Lagrangian and Eulerian on its interior. Speciically, we deine the time-dependent bijective mapping Ψ t which maps the reerence domain t 0 to 15

24 the physical domain t : Ψ t : t 0 t, Ψ t(y) = x(y, t), (.) where y and x are the spatial coordinates in t 0 and t, respectively. Ψ t -1 Ψ t t Figure.: ALE coordinate transormation The coordinate y is oten called the ALE coordinate. Using Ψ t, the weak ormulation o the low equations in t can be recast into a weak ormulation deined in the reerence domain t 0. Thus, the model equations in the reerence domain can be considered or numerical simulation and the transormation unction Ψ t needs to be determined at each time step as a part o computation. For a unction φ : t [t 0, T RI, its corresponding unction φ = φ Ψ t in the ALE setting is deined as φ : t 0 RI, φ(y, t) = φ(ψ t (y), t). (.3) The time derivative in the ALE rame is also given as φ t y: t [t 0, T RI, φ t y (x, t) = φ (y, t). (.4) t 16

25 Using the chain rule, we have φ t y= φ t x +z x φ, (.5) where z := x t y is the domain velocity. In (.5) φ t y is the so-called ALE derivative o φ. The low equations (.1) (.) can then be written in ALE ormulation as [ u ρ t y +(u z) x u ν x D x (u) + x p = in t, (.6) x u = 0 in t, (.7) where D x (u) = ( x u+ x u T )/. ote that all spatial derivatives involved in (.6) (.7), including the divergence operator, are with respect to x. Throughout this thesis we will use D x ( ) and x only when they need to be clearly speciied. Otherwise, D( ), will be used as D x ( ), x, respectively. The variational ormulation or (u, p, η) in ALE ramework is given by: or almost every t [t 0, T, ind t u(t) H 1 D ( t ), t p(t) L ( t ), and t η(t) H1 D (s ) such that ρ ( u t y ) + (u z) u, v t + ν a(u, v) t + b(v, p) t b(u, q) t (ν D(u) n pn, v) ΓIt = (, v) t + (u, v) Γ v H 1 D ( t ), (.8) = 0 q L ( t ). (.9) and ( ) η ρ s t, ξ + ν s d(η, ξ) s + λe(η, ξ) s s (ν s D(η) n s + λ( η)n s, ξ) ΓIt0 = ( s, ξ) s + (η, ξ) Γ s ξ H 1 D (s ), (.30) 17

26 where the interace conditions (.11) (.1) are imposed. In order to deine the ALE mapping Ψ t, we consider the boundary position unction h : Γ It0 [t 0, T Γ It. The ALE mapping can then determined by solving the Laplace equation y x(y) = 0 in t 0, x(y) = h(y, t) on Γ It0, x(y) = 0 on Γ t 0 /Γ It0. (.31) This method is called the harmonic extension technique, where the boundary position unction h is extended onto the whole domain [56. For a comparison o the harmonic extension with other extensions o h onto t 0, see [66. The Reynolds Transport ormula is given by d φ φ(x, t) dv = dt V (t) V (t) t y +φ x z dv (.3) or a unction φ : V (t) R, where V (t) t such that V (t) = Ψ t(v 0 ) with V 0 t 0 [58. Applying the Reynold s Transport ormula with φ = v, noting that v is a unction rom t to R, and v = v Ψ 1 t or v : t 0 R, we have that v t y = 0 and thereore [ d φ φv d = dt t t y +φ x z v d. (.33) t Using (.33), (.8) (.9) become ρ [ d dt (u, v) t + ((u z) u, v) t b(u, q) t + b(v, p) t = (, v) t (( z)u, v) t (ν D(u) n pn, v) ΓIt + (u, v) Γ + ν a(u, v) t v H 1 D ( t ), (.34) = 0 q L ( t ). (.35) 18

27 .4 Semidiscrete Weak Formulations We will implicitly deine V( ) in the ollowing way: (, V(v)) t j = (, v Ψ ti Ψ 1 t j ), (, V(v)) ΓItj t j = (, v Ψ ti Ψ 1 t j ) ΓItj, where is a unction deined on the domain t j (or Γ Itj ) and v is a unction deined on the domain t i (or Γ Iti ). ν() Ψ t i Ψ -1 tj Figure.3: Action o the V( ) operator In order to semidiscretize the continuous weak orm o the FSI problem, let us irst deine Ψ t (y) = t t n 1 Ψ (y) + t n t Ψ 1 (y) t [t n 1, t n. (.36) as in [15, where Ψ 1 and Ψ are the harmonic extensions onto t 0 o η n 1 ΓIt0 and η n ΓIt0, respectively. Here, η n 1 and η n are the time discretized displacement solutions to (.38) (.41). Also, note that as a consequence o (.36), z := Ψ t t Ψ 1 t = [ Ψ Ψ 1 Ψ 1 t t t n 1, t n. (.37) In other words, z n = z(t n ) is the mesh velocity at time step t n and or all times t t n 1, t n. Let us denote J t (y) := det[ y Ψ t(y). In Chapter 4, we will make assumptions (4.14) and (4.15), as in [15, that gives J t bounded below by a positive constant κ min and above by a constant κ max, or all t [t 0, T. For urther inormation on the mapping regularity 19

28 condition, see [55, pp The mesh velocity z n, the two domain displacements Ψ and Ψ 1, and the deormation gradient o the moving luid domain are all at least implicitly deined by η n and η n 1. While η n is an unknown we are solving or at time t n, we will treat all o its descendants as ixed and known in order to make analysis o the problem tractable. Temporal discretization o the luid subsystem (.34) (.35) by implicit Euler and o the structure subsystem (.30) by a second order midpoint scheme yields: ind (u n, p n, η n, η n ) H 1 D ( t n ) L ( t n ) H 1 D (s ) L ( s ) such that ρ [(u n, v) (u n 1, V(v)) t n 1 [ + ρ ((u n z n ) u n, v) (( z n )u n, v) [ + ν a(u n, v) + b(v, p n ) (ν D(u n ) n p n n, v) ΓIt [ = ( n, v) + (u n, v) Γ v H 1 D ( t n ), (.38) b(u n, q) = 0 q L ( t n ), (.39) and ρ s ( η n η n 1, ξ ) s + [ν s d ( η n + η n 1, ξ ) + λ e ( η n + η n 1, ξ ) ( s s ( ν s D(η n + η n 1 ) λ ( ( ) n s + η n + η n 1)) ) n s, ξ Γ It0 = [ ( n s + s n 1, ξ ) + ( η n s + η n 1, ξ) ξ H 1 Γ s D (s ), (.40) ( η n + η n 1, γ ) = ( η n η n 1, γ ) γ L ( s ). (.41) s s This ormulation will be considered in Chapters 3, 4, and 5. 0

29 Also, a irst order time discretization o the structure problem is given by ρ s (η n η n 1 + η n, ξ ) s + [ν s d (η n, ξ) s + λe (η n, ξ) s (ν s D(η n ) n s + λ( η n )n s, ξ) ΓI0 = [ ( n s, ξ) s + (η n, ξ) Γ s ξ H 1 D (s ) (.4) and will be considered in Chapter 3. The overall order o the time discretization is only irst order accurate because o using implicit Euler or the luid discretization. However, the analytical results can be easily extended to use the Crank icolson time discretization scheme or the luid instead o implicit Euler, but this was not done in order to keep the luid equations simpler. Analysis or the structure subsystem will be developed using the second order time discretization due to a necessity or higher order accuracy, which will be needed later and is explained in Remark

30 Chapter 3 OPTIMIZATIO-BASED DECOUPLIG 3.1 Introduction In this chapter, we introduce a control into the semi-discretized weak orm o the FSI equations which decouples the luid and structure subproblems. The control that we use enorces the continuity o traction orce on the interaces or every choice o control. Our goal in optimization will be to ind an optimal control which minimizes violations o the continuity o velocity on the interace. A nonlinear unction is deined whose norm is equivalent with the penalized unctional which minimizes velocity mismatches between luid and structure on the interace. Using Gauss ewton iterations based on a Taylor series truncation, updates to the control can be perormed by solving a linear least squares problem. The Fréchet derivative o the nonlinear unction is presented or both irst and second order time discretizations o the structure that were given in Chapter. A proo is urnished to prove that the adjoint o the linearized operators are correct as given. A ewton Krylov method is introduced to minimize the norm o the nonlinear unction describing the velocity jump on the interace. There are many solver choices

31 available or minimizing the linear least squares problem occurring within each Gauss ewton iteration. We choose to present the conjugate gradient or least squares algorithm, which does not require explicitly orming the normal equations in order to minimize the linear least squares problem. A numerical study is included or a haemodynamic problem having clamped artery ends which was previously simulated by Murea and Sy [54. Because the densities o the luid and structure are similar, this problem suers rom the added mass eect and is too sensitive to be computed by iterative implicit partitioned schemes unless they contain suicient relaxation. First, progressively reined meshes are used to demonstrate that the solution at the mesh resolution given in [54 is not mesh independent. ext, we compare the solution using our optimization-based approach against the result in [54. Ater this, another series o tests were perormed to compare the solutions gathered rom an implicit relaxed partitioned scheme against the solution provided by our method. 3. Substitution o Traction Terms with a Control It is desirable that an algorithm or solving FSI problems be able to stably decouple the subsystems and solve each subsystem in parallel. Stably decoupling the subsystems is o particular importance in the case when the added mass eect is present. With this in mind, we seek to exploit the shared traction (.1) orce between the coupled subsystems. We set g n = ν D(u n ) n pn as our control or the stress on the interace or the luid subproblem. Thereore ν s D(η n ) n s + λ( η n )n s I0 can be replaced by (g n Ψ 1 n )J in the structure subproblem because o interace condition (.1), ensuring continuity o stress along the interace between the two subsystems or any choice o g n. 3

32 Making this substitution, the luid subproblem (.38) (.39) becomes ρ [(u n, v) b(u n, q) (u n 1, v) t n 1 [ + = ν a(u n, v) [ + ρ ((u n z n ) u n, v) (u n ( z n ), v) + b(v, p n ) [ ( n, v) + (u n, v) Γ + (g n, v) ΓI v H 1 D ( t n ), (3.1) = 0 q L ( t n ), (3.) the irst order structure subproblem (.4) becomes ρ s (η n η n 1 + η n, ξ ) s + [ν s d (η n, ξ) s + λe (η n, ξ) s = [ ( n s, ξ) s + (η n, ξ) Γ s (g n, V(ξ)) ΓI ξ H 1 D (s ), (3.3) and the second order structure subproblem (.40) (.41) becomes ρ s ( η n η n 1, ξ ) [ s ( η n + η n 1 + ν s d ( = n s + n 1 ), ξ s ), ξ s + ( η n + η n 1 + λe ( s ) η n + ηn 1, ξ Γ s ), ξ s + (g n, V(ξ)) ΓI ξ H 1 D (s ), (3.4) (η n η n 1, γ ) ( η n + η n 1 ), γ = 0 γ H 1 s D (s ). (3.5) s 3..1 The Formal Optimization Problem ow that we see a control can be introduced which will decouple the luid and structure subsystems, we shall proceed to deine the ormal optimization problem. We seek a g n in (3.1) and (3.3) or (3.4) to be chosen as a control in each time step to enorce the 4

33 continuity o velocity (.11), i.e., we wish to minimize the penalized unctional J n (u n, η n, g n ) = 1 Γ I u n V( η n ) dγ + ɛ Γ I g n dγ, (3.6) subject to (3.1) (3.) and (3.3) or (3.1) (3.) and (3.4) (3.5), depending on the structure ormulation used. I using the irst order discretization o the structure subsystem (3.3), η n in (3.6) can be approximated by η n ηn η n 1. In (3.6), ɛ is the penalty parameter which gives relative weight to the latter term and Γ I denotes the interace at time step n, to be determined by the solution to (3.3) or (3.1) (3.) using Gauss ewton iterations described later in Algorithm 3.3. Our approach is an implicit algorithm or the luid-structure interaction problem. We use nonlinear least squares to develop a computational algorithm or the constrained optimal control problem. Deine the nonlinear operator n : L (Γ I ) L (Γ I ) L (Γ I ) by n (g n ) = (u n V( η n )) ΓI ɛ g n, where u n, η n are the luid and structure velocities satisying (3.1) (3.5) when g n is the stress unction on the interace. Then, (3.6) can be written as J n (g n ) = 1 n(g n ) L (Γ I ) L (Γ I ) (3.7) and the nonlinear least squares problem we consider is to seek g n L (Γ I ) such that (3.7) is minimized. (3.8) 5

34 We can linearize n (g n ) using the Fréchet derivative o n ( ) at g n, n(g n ), by n (g) = n (g n ) + (g n )(g n g n ) + O( g n g n L (Γ I ) L (Γ I ) ) so that solutions o the nonlinear least squares problem can be obtained by repeatedly solving the linear least squares problem 1 min h n L (Γ I ) (gn ) + n(g n )h n L (Γ I ) L (Γ I ), (3.9) where h n = g n g n. Hence, starting with arbitrary g(0) n, we can ind a sequence {gn (k) } obtained by g(k) n = gn (k 1) + hn (k), where hn (k) is a solution o the linear least squares problem (3.9). Following is the deinition o the linearized and linear adjoint problems that are to be solved in the use o the conjugate gradient algorithm and Algorithm 3.3. The linearized and linear adjoint problems are presented along with a proo o the deinition o the adjoint or the irst and second order structure ormulations in sections 3.3 and 3.4, respectively. 3.3 First Order Time Discretization o the Structure Subsystem For g n L (Γ I ), the Fréchet derivative (g n )( ) : L (Γ I ) L (Γ I ) L (Γ I ) is deined by n(g n )(h n ) = ( w n V(φn ) ɛ h n ) ΓI, 6

35 where w n, φ n are the solutions o ρ (w n, v) + ρ [(w n u n, v) (w n ( z n ), v) + ((u n z n ) w n, v) + [ ν a(w n, v) + b(v, ψ n ) = (h n, v) ΓI v H 1 D ( t n ), (3.10) b(w n, q) = 0 q L ( t n ), (3.11) and ρ s (φ n, ξ) s + [ ν s d (φ n, ξ) s + λe (φ n, ξ) s = (h n, V(ξ)) ΓI ξ H 1 D (s ), (3.1) where u n is the solution o (3.1) (3.) with g n replaced by g n. It is necessary to deine the adjoint operator o n(g n ) in order to solve the linear least squares problem (3.9). Theorem 3.1. The adjoint o ( n(g n ))( ) is ( n(g n )) ( ) : L (Γ I ) L (Γ I ) L (Γ I ), given by ( n(g n )) rn s n = ( ) β n V(ϕn ) + ɛ s n, L (Γ I ) where β n, ϕ n are the solutions o ρ (β n, v) + ρ [(v u n, β n ) (β n ( z n ), v) + ((u n z n ) v, β n ) + [ ν a(β n, v) + b(v, α n ) = (r n, v) ΓI v H 1 D ( t n ), (3.13) b(β n, q) = 0 q L ( t n ), (3.14) 7

36 and ρ s (ϕ n, ξ) s + [ ν s d (ϕ n, ξ) s + λe (ϕ n, ξ) s = (r n, V(ξ)) ΓI ξ H 1 D (s ). (3.15) ote that the linearized structure subsystem is sel-adjoint. Again, u n in (3.13) is the solution o (3.1) (3.) with the replacement o g n by g n. Proo. Let (v, q) = (β n, α n ) and ξ = ϕ n in (3.10) (3.1). Also, let (v, q) = (w n, ψ n ) and ξ = φ n in (3.13) (3.15). From this, we obtain that (h n, V(ϕ n )) ΓI = (r n, V(φ n )) ΓI and (h n, β n ) ΓI = (r n, w n ) ΓI. Thereore, n(g n )h n, rn s n = (w n V(φn ), r n ) ΓI + ɛ(h n, s n ) ΓI = (h n, β n V(ϕn ) ) ΓI + ɛ(h n, s n ) ΓI = h n, n(g n ) rn s n. 3.4 Second Order Time Discretization o the Structure Subsystem For g n L (Γ I ), the Fréchet derivative (g n )( ) : L (Γ I ) L (Γ I ) L (Γ I ) is deined by n(g n )(h n ) = (w n V( φ n )) ΓI ɛ h n, 8

37 where w n is the solution o (3.10) (3.11), and φ n is the solution o ρ s ( φn, ξ ) s + (φ n, γ) s [ ν s d (φ n, ξ) s + λ e (φn, ξ) s = (h n, V(ξ)) ΓI ξ H 1 D (s ), (3.16) ( φn, γ ) s = 0 γ H1 D (s ). (3.17) We now deine the adjoint operator o n(g n ) needed in order to solve the linear least squares problem (3.9). Theorem 3.. The adjoint o ( n(g n ))( ) is ( n(g n )) ( ) : L (Γ I ) L (Γ I ) L (Γ I ), given by ( n(g n )) rn s n = (β n V(ϕ n )) L (Γ I ) + ɛ s n, where β n is the solution o (3.13) (3.14) and ϕ n is the solution o ( ϕ n, ξ) s + [ ν s d (ϕ n, ξ) s + λ e (ϕn, ξ) s (3.18) = 0 ξ H 1 D (s ), (3.19) ρ s (ϕ n, γ) s ( ϕn, γ) s = (r n, V(γ)) ΓI γ H 1 D (s ). (3.0) Proo. Let (v, q) = (β n, α n ) and (ξ, γ) = (ϕ n, ϕ n ) in (3.10) (3.11) and (3.16) (3.17). Also, let (v, q) = (w n, ψ n ) and (ξ, γ) = (φ n, φ n ) in (3.13) (3.14) and (3.19) (3.0). From this, we obtain that (h n, V(ϕ n )) ΓI = (r n, V( φ n )) ΓI and (h n, β n ) ΓI = (r n, w n ) ΓI. 9

38 Thereore, n(g n )h n, rn s n = (w n V( φ n ), r n ) ΓI + ɛ(h n, s n ) ΓI = (h n, β n V(ϕ n )) ΓI + ɛ(h n, s n ) ΓI = h n, n(g n ) rn s n. 3.5 Gauss ewton Algorithm For problem (3.9), we adopt the conjugate gradient method or least squares (CGLS) [59. This method is mathematically equivalent to solving the normal equations, but does not require explicitly orming them. This is a variant o the conjugate gradient method which can be ound in many reerences [ For the algorithm, we use the notation A = n(g n ), A = ( n(g n )), b = n (g n ), and x = h n. The nonlinear least squares problem (3.8) can be solved using the ollowing Gauss ewton algorithm. Algorithm Choose g n (0).. For k = 1,, 3,..., a. computable in parallel: i. ind u n (k) and pn (k) on t n,(k 1) using zn (k 1) and gn (k 1), ii. ind η n (k) and ηn (k) using gn (k 1), b. update Γ (k) I, z n n (k), Ψ(k) n, and t using n,(k) ηn (k), c. i 1 Γ I (k 1) u n (k) V( ηn (k) ) dγ < ɛ tol, break, 30

39 d. compute h n (k) using CGLS to solve the least squares problem, minimizing 1 Ax b, e. set g n (k) = gn (k 1) + hn (k). Remark 3.4. Our choice o g n (0) in step 1. o Algorithm 3.3 is the inal value o gn 1 determined in the previous time step. Remark 3.5. In step.d. o Algorithm 3.3, determining h n (k) by means o the conjugate gradient algorithm is accomplished on the moving luid domain determined by the structure problem using the control g(k 1) n. Thereore, the moving luid domain must only be updated or each Gauss ewton iteration o Algorithm umerical Results Haemodynamic Experiment The irst o two numerical tests is an FSI problem using the ALE ormulation or the moving luid domain, reported in [54, using parameters that are consistent with blood low in a human body. The problem is heavily aected by the added mass eect, since the densities o the luid and structure are very close, and is thereore an excellent test candidate. This eect causes explicit decoupling without relaxation to ail, as was observed by experimentation and also reported in [54. η D = 0 η = 0 s = [0, 6 [1, 1.1 η D = 0 u = b(t) 0 = [0, 6 [0, 1 Γ I0 u = 0 u D = 0 Figure 3.1: Domain and boundary conditions or numerical experiment 31

40 A orce b(t) is applied to the let luid boundary (Fig. 3.1) at t s where ( 10 3 (1 cos πt.05 b(t) = ), 0) dyne/cm, t 0.05 (0, 0), 0.05 < t < T. The unction b(t) deines the stress on the inlet denoted by u in (.4). The volume orce or the luid and structure are (t) = (0, 0) dyne/cm. The other boundary conditions on the domain coniguration are homogeneous Dirichlet or eumann (Fig. 3.1), and the simulation begins at rest. The reerence domain or the luid subsystem has height 1 cm and length 6 cm. The density o the luid, ρ, is 1 g/cm 3 and the viscosity o the luid, ν, is g/cm s. The structure domain has height 0.1 cm and length 6 cm. The density o the structure, ρ s, is 1.1 g/cm 3. The Young s Modulus o the structure, E, is dyne/cm and its Poisson ratio, ν, is 0.3. The Lamé parameters λ and ν s are deined as ollows: λ = νe (1 ν)(1 + ν) dyne/cm, ν s = E (1 + ν) dyne/cm. The luid and structure reerence domains were spatially discretized using a uniorm mesh. Let h x and h y represent the spatial discretization in the x and y direction, respectively. We used the triangular (P 1 + bubble, P 1 ) pair or the inite element solution to (3.1) (3.) on the luid domain and P 1 inite elements or the solution to (3.3) or (3.4) (3.5) on the structure domain or all computations that will be presented. Additionally, all computations perormed used a time step o = 10 4 s and were rom T = 0 s to T = 0.1 s. All computations were perormed using FreeFEM++ [41. The irst sequence o simulations (Fig. 3.) demonstrates the strong dependence o the solution on the spatial discretization. The plots o the vertical displacement on the interace are computed using Algorithm 3.3. It is worth noting that using a spatial discretization with h x = 0.1 cm and allowing h y to range rom 0.1 cm to 1 30 cm or both computational domains gives signiicantly dierent results. There is much more agreement 3

41 using the two iner spatial discretizations which indicates that the solution is sensitive to having degrees o reedom on the interior o the structure FEM space. Displacement (cm) (1.5,1.0) (1) () (3) (3.0,1.0) (1) () (3) (1) () (3) (4.5,1.0) Time (seconds) Figure 3.: Vertical displacement at three points on the interace using the irst order structure ormulation with: (1) h x = 0.1 cm, h y = 0.1 cm, () h x = 0.1 cm, h y = 0.05 cm, and (3) h x = 0.1 cm, h y = 1 30 cm The solution to the FSI problem computed by Murea and Sy [54 is on a mesh with h x 0. cm and h y 0.1 cm or the structure domain, h x 0.1 cm and h y 0.1 cm or the luid domain. We have seen that the solution depends heavily on the spatial discretization with a mesh as coarse as is used or this comparison, so it is not reasonable or us to expect an exact match with their results. Particularly because, in Murea and Sy s work, a truncated eigenunction basis or the solution on the structure domain was used. Additionally, the uniqueness o the optimal solution is not guaranteed theoretically or Algorithm 3.3 in its continuous orm, so the numerical solution may be determined by the initial choice o the control. Regardless, we have compared our solution with Murea and Sy s (Fig. 3.3) and note that, while they dier in amplitude, they both have similar wavelike eatures. Our computation was made using h x = 0. cm and h y = 0.1 cm or the structure domain, h x = 0.1 cm and h y = 0.1 cm or the luid domain, ɛ = 0, and ɛ tol = 10 6 or Algorithm 3.3. Table 3.1 contains the norm o the jump in velocities on the interace at three time steps or the computation made using h x = 0.05 cm and h y = 0.05 cm as the spatial 33

42 Displacement (cm) (1.5,1.0) 0.05 (1) () (3.0,1.0) (1) () (1) () (4.5,1.0) Time (seconds) Figure 3.3: Vertical displacement at three points on the interace using (1) Algorithm 3.3 with the irst order ormulation or the structure beside the vertical displacement rom () Murea and Sy discretization, ɛ = 0, and ɛ tol = Observe the ast convergence o the Gauss ewton iterations. Figure 3.4 contains pressure proiles o the same solution at the same three time steps. Interace Velocity Error J n ( ) Time (s) Iter. 1 Iter. Iter e e e e e e e e e-09 Table 3.1: Error in the continuity o velocity between subsystems or each Gauss ewton iteration at three representative time steps We now veriy that the solutions ound by Algorithm 3.3 or the irst and second order ormulations o the linear elastic structure closely match the solution ound with Aitken s relaxation [1, using the same inite elements and with the same spatial discretization. Aitken s relaxation is an implicit scheme that is applied to the structure update. It works by relaxing the update to η n at each iteration o an implicit scheme. For instance, suppose η n (k) is the solution to the structure subproblem or implicit iteration k. Then, ηn (k) is updated as η n (k) = ω ηn (k) + (1 ω) ηn (k 1), ω 0, 1. (3.1) 34

43 Time s Height (cm) Length (cm) 100 Time 0.05 s Height (cm) Length (cm) Time s 0 Height (cm) Length (cm) Figure 3.4: Fluid pressure proiles [dyne/cm at three time steps See [1 or more details on Aitken s relaxation. While easy to implement, this is an incredibly expensive method to use because ω must be very small in order the system to remain stable and converge. The smaller the value o ω, the more iterations are needed at each time step, requiring many nonlinear solves on the luid domain. Using ω = 0.05, the result is reliable and useul as a reerence solution with which to compare (Fig. 3.5). Spatial discretization was made with h x = 0. cm and h y = 0.1 cm or both luid and structure domains. The stopping criteria used or ( ) 1 Aitken s relaxation was (η n ΓI0 (k) ηn (k 1) ) dγ < 10 7, while ɛ = 0 and ɛ tol = 10 6 or Algorithm 3.3. It was observed that the irst and second order ormulation or the structure made no signiicant dierence on the solution ound. Both solutions matched well the reerence result using Aitken s relaxation, and as expected, Algorithm 3.3 signiicantly reduced com- 35

44 Displacement (cm) (1.5,1.0) 0.05 (1) () (3) (3.0,1.0) (1) () (3) Time (seconds) (1) () (3) (4.5,1.0) Figure 3.5: Vertical displacement at three points on the interace using (1) irst and () second order ormulations with the optimal control algorithm beside vertical displacement using (3) Aitken s relaxation putation times. The irst order ormulation using Algorithm 3.3 ran or 135 minutes beore completion, while the Aitken s relaxation ran or 1,64 minutes with a tolerance o 10 6 on each time step and,701 minutes using a tolerance o 10 7 on each time step. We do not expect a 90% reduction in run time compared with other state-o-the-art implicit methods, but Aitken s relaxation is a benchmark against which many implicit algorithms can be compared Comparison with an Analytical Solution In order to observe the convergence and accuracy o our method, we have compared it with the analytical solution or the FSI problem presented by Astorino and Grandmont [3. The luid governing equations are or Stokes low on a stationary luid domain, but still have the same challenge o solving an FSI problem with similar densities between the luid and structure [, 5, 19. This problem also provides support that our optimal control approach is applicable or solving a broad range o FSI problems. Solving the Stokes low on a stationary domain means that when we solve (3.1) (3.), the equations will not have the nonlinear term (u n u n, v) and we will drop all terms including z, since z = 0 and z = 0 in the Eulerian ramework. The corresponding terms are removed rom the linearized and adjoint 36

45 problems deined in sections 3.3 and 3.4. Γ s D Γ s D s Γ s D Γ I0 Γ Γ Γ D Figure 3.6: Computational domain or a manuactured solution Parameters or the problem are: ρ = 1.0 g/cm 3, ν = g/cm s, ρ s = 1.9 g/cm 3, ν s = 3 dyne/cm, and λ = 4.5 dyne/cm. Initial conditions, body orces, and boundary conditions are determined by the analytical solution according to the method o manuactured solutions: On t = = [0, 1 [0, 1 and s = [0, 1 [1, 1.5 (Fig. 3.6), u 1 = cos(x + t) sin(y + t) + sin(x + t) cos(y + t), u = sin(x + t) cos(y + t) cos(x + t) sin(y + t), p = ν (sin(x + t) sin(y + t) cos(x + t) cos(y + t)) + ν s cos(x + t) sin(y + t), η 1 = sin(x + t) sin(y + t), η = cos(x + t) cos(y + t). (3.) As in [3, we used a uniorm mesh with a spatial discretization o h = 0.05 cm. The Taylor Hood inite element pair, (P, P 1 ), were used or solutions on the luid domain, while P inite elements were used to approximate the structure displacement. The FSI 37

46 problem was repeatedly solved by Algorithm 3.3 using decreasing time steps ( = , , , s) and compared with the exact solution (3.). For Algorithm 3.3, ɛ = 0 and ɛ tol = 10 6 were used. orms used to compute the error between the solution ound by means o our optimal control algorithm and the exact solution are as ollows: or u and p, L (0.5, 1; L ( )); or η, L (0.5, 1; H 1 ( s )). Results are plotted in logarithmic scale ormat as a unction o in Figure 3.7 and are approximately linear. The results indicate that our algorithm or solving the FSI problem converges upon the exact solution. Providing error estimation or the optimal control algorithm will be the ocus o Chapter Conclusion The monolithic luid-structure interaction problem was reormulated as an optimal control problem where violation o continuity o velocity on the interace o two subsystems is minimized using a control on the interace representing the shared traction orce. A nonlinear unction was deined whose norm is the nonnegative Tikhonov regularized unctional. The Gauss ewton outer optimization loop was detailed using the nonlinear unction as well as its Fréchet derivative. Linearized versions or the luid and both time discretized versions o the structure along with their adjoints were derived in sections 3.3 and 3.4 and proos were urnished showing that the adjoint was as deined. Steps or the Gauss ewton algorithm were outlined along with the related roles o the Fréchet derivative and adjoint o the nonlinear unction. The problems described in and were implemented in FreeFEM++ using the optimal control algorithm described in section 3.5 and computations were perormed on the Palmetto cluster. The presented algorithm successully decoupled the FSI problem into two subproblems. Very ew nonlinear solves were needed or each time step because o the ast convergence o the Gauss ewton algorithm in determining the optimal virtual control. It was 38

47 not necessary to introduce relaxation parameters or updating either the structure or luid subsystems. Additionally, all problems including the linearized and adjoint problems or the luid and structure subsystem can be solved in parallel, which is increasingly important as more eort is being expended on building large computational clusters or distributed computing. Although Figure 3. indicates that the solution to the irst FSI problem tested was very sensitive to spatial discretization and inite element choice, our algorithm was able to ind solutions that enorced continuity o stress always and continuity o velocity within a speciied tolerance. Further, Figure 3.5 indicates that the solution o the optimization algorithm matched the results o using a much more computationally expensive implicit method and also shows very strong agreement between both irst and second order ormulations or the structure subproblem. Computation time was reduced by over 90% when compared with a naive but easy to implement Aitken s relaxation method. Using a Stokes-linear elastic structure FSI problem on a ixed domain with an analytical solution, we were able to show nearly linear convergence in the very strict L norm with respect to time. This gives us conidence that the algorithm by optimization is converging upon the true solution. 39

48 1 The dependence o error on max.5 t 1 u u ex 0 max.5 t 1 p p ex 0 max.5 t 1 η η ex Error (seconds) Figure 3.7: Convergence results or the analytic problem 40

49 Chapter 4 EXISTECE PROOFS 4.1 Introduction A new and alternative implicit decoupling approach to recast the FSI problem as an optimal control problem is presented in this chapter. It is similar in orm to the approach in Chapter 3, using each subsystem as a constraint. The control introduced in this chapter diers in that it has additional terms included and approximately and progressively enorces the continuity o stress on the interace while minimizing any violation o the continuity o velocity on the interace. This is opposed to constantly enorcing continuity o traction orce or any choice o control, as was the case with the control rom Chapter 3. The two subsystems are again decoupled and this allows or them to be solved simultaneously and independently, but also allows or additional analysis to be perormed on the optimization problem. Using a penalized unctional with Tikhonov regularization and some a priori stability estimates proved in Section 4.4, the existence o an optimal solution is shown to exist. A proo is given that with some strong assumptions on the regularity o the strong orm o the FSI problem, the limit o the sequence o optimal solutions ormed by taking the limit as the Tikhonov parameter approaches zero exists and satisies the continuity o velocity condition to within a speciied tolerance. The existence o Lagrange multipliers, and the 41

50 derivation o an optimality system are presented next. A steepest descent algorithm is detailed in Section 4.9 ollowed by a numerical experiment in Section The algorithm is applied to a haemodynamic numerical experiment, which demonstrates that the approach by optimization is aster than a Dirichlet eumann implicit decoupled approach augmented by Aitken s relaxation. A parameter study is also included to show the eects o the Tikhonov regularization parameter on the optimal solution ound. 4. Weak Formulation o the Constraints Assuming n, n 1 s, n s L ( t n ), g n L (Γ I ), u n L (Γ ), ηn L (Γ s ), u n 1 H 1 D ( t n 1 ), and η n 1 H 1 D (s ), (.38) (.39) can be rewritten as ρ [(u n, v) (u n 1, V(v)) + ρ [c(u n, u n, v) t + 1 n 1 ((un n )u n, v) ΓI + 1 ((un n )u n, v) Γ 1 ((zn n )u n, v) ΓI 1 (( zn )u n, v) c(z n, u n, v) + ν a(u n, v) + b(v, p n ) = ( n, v) + (u n, v) Γ (4.1) b(u n, q) + (ν D(u n ) n pn, v) ΓI v H 1 D ( t n ), = 0 q L ( t n ), (4.) using (u n u n, v) (z n u n, v) = (u n v, u n ) (( u n )v, u n ) + ((u n n )v, u n ) ΓI Γ, = (z n v, u n ) (( z n )v, u n ) + ((z n n )v, u n ) ΓI Γ, by Green s Theorem and ((z n n )v, u n ) Γ = 0 since z n Γ = 0 because Γ is ixed. 4

51 On the interace, we replace (ν D(u n ) n pn 1 ((un z n ) n )u n, v) ΓI (4.3) with (g n, v) ΓI where g n ΓI will become our control. We similarly replace (ν s D(η n ) n s + λ( η n )n s, ξ) with (V(g n )J, ξ) ΓIt0. The terms replaced by the control dier rom the terms selected in Chapter 3. The purpose in adding terms to the control is to make analysis o the problem possible. A detailed explanation o the justiication or this substitution is given in Section 4.3. Making these substitutions, (4.1) and (4.) become ρ [(u n, v) (u n 1, V(v)) + ρ [c(u n, u n, v) t + 1 n 1 ((un n )u n, v) Γ b(u n, q) 1 (( zn )u n, v) c(z n, u n, v) + ν a(u n, v) + b(v, p n ) = ( n, v) + (u n, v) Γ + (g n, v) ΓI v H 1 D ( t n ), (4.4) = 0 q L ( t n ). (4.5) Also, (.40) and (.41) can be rewritten as ρ s [( η n, ξ) s ( η n 1, ξ) s + ν s d(η n + η n 1, ξ) s + λe(ηn + η n 1, ξ) s = ( s n + s n 1, ξ) s + (ηn + η n 1, ξ) Γ s (V(gn )J + V(g n 1 )J 1, ξ) ΓIt0 ξ H 1 D (s ), (4.6) [( ηn, γ) s + ( η n 1, γ) s = (η n, γ) s (η n 1, γ) s γ L ( s ). (4.7) 4.3 Description o the Optimization Problem For any choice o g n, (4.4) (4.5) and (4.6) (4.7) can be solved independently. Thereore, using g n as a control at each time step permits the luid and structure subsystems to 43

52 be solved simultaneously using partitioned solvers. For any g n chosen arbitrarily, we should not expect that the solution will satisy the interace conditions (.11) (.1), which are necessary to satisy the system (.38) (.39) and (.40) (.41). In order to attempt to enorce the continuity o velocity (.11) and continuity o stress (.1) along the interace, we wish to minimize the penalized unctional Jn δ (u n, p n, η n, η n, g n ) = 1 u n V( η n ) dγ I + δ g n dγ I, (4.8) Γ I Γ I subject to (4.4) (4.5) and (4.6) (4.7), where Γ I denotes the interace in the n-th time step and δ is a positive constant penalty parameter that is chosen to dictate the importance o the last term in (4.8). This optimal control problem is to be solved at each time step. We anticipate that it will not be possible to get a stability estimate or η n in H 1 D (s ) and the existence o an optimal ˆ η n can be shown only in L ( s ). In this case, the unctional (4.8) is not well-deined since the trace o an optimal η n on Γ I is not well-deined. In order to avoid this diiculty, we will replace η n with its irst order approximation, ηn η n 1. As will be seen in Remark 4.6, using a irst order approximation in the unctional will cause no greater loss in accuracy than using a higher order approximation. With u n 1, η n 1, and η n 1 as known data obtained at the previous time step, we wish to minimize J δ n (u n, p n, η n, η n, g n ) = 1 Γ I un V(ηn ) V(η n 1 ) dγ I + δ g n dγ I, Γ I (4.9) subject to (4.4) (4.5) and (4.6) (4.7), which will enorce continuity o velocity and stress along the interace (.11) (.1). The optimization problem to be solved is ind u n, p n, η n, η n, and g n such that (4.9) is minimized subject to (4.4) (4.5) and (4.6) (4.7) or a given u n 1, η n 1, and η n 1, (4.10) 44

53 and our admissibility set is S := {(u n, p n,η n, η n, g n ) H 1 D ( t n ) L ( t n ) H 1 D (s ) L ( s ) L (Γ I ) : J δ n (u n, p n, η n, η n, g n ) < and (4.4) (4.5) and (4.6) (4.7) are satisied.} (4.11) that Then (û n, ˆp n, ˆη n, ˆ η n, ĝ n ) S is called an optimal solution i there exists ɛ > 0 such J δ n (û n, ˆp n, ˆη n, ˆ η n, ĝ n ) J δ n (u n, p n, η n, η n, g n ) (u n, p n, η n, η n, g n ) S satisying g n ĝ n 0,ΓI ɛ. (4.1) ow that we have an objective to minimize, we can explain the rationale or our substitution o the control into the semidiscrete weak ormulation o the FSI problem. When replacing terms with the control in the luid equations, 1 ((un z n ) n )u n ΓI was added to (4.3) in order to make a stability result or the resulting weak orm o the partial dierential equation possible. I we choose a g n which results in a solution that satisies u n ΓI ( ηn η n 1 ) Ψ 1 t n ΓI, then u n ΓI z n ΓI by the deinition o z n in (.37). This means that 1 ((un z n ) n )u n ΓI will be almost zero and thereore the control g n will represent ν D(u n ) n pn ΓI in the luid equations and (g n Ψ n )J will represent ν s D(η n ) n s + λ( η n )n s ΓIt0 in the structure equations. Since g n then holds the place o the interacial stress common to both subsystems, the continuity o stress on the interace (.1) will be satisied. While many solutions may exist such that 1 ((un z n ) n )u n ΓI will be approximately zero, this will certainly be the case at an optimal solution. 45

54 4.4 A Priori Estimates We make the ollowing assumptions to obtain a priori bounds or solutions to the weak ormulations (4.4) (4.7) and or analysis throughout the rest o this chapter: the eumann boundary Γ is an outlow boundary, (4.13) t = Ψ t( t 0 ) is a Lipschitz domain, (4.14) Ψ t W 1, ( t 0 ) and Ψ 1 t W 1, ( t ) t [t 0, T, (4.15) z, z t W1, ( t ) t [t 0, T. (4.16) Assumption (4.13) is necessary or stability o any avier Stokes low with nonhomogeneous eumann boundary conditions. Assumptions (4.14) (4.16) are reasonable or the movement and shape o the moving domain [3, 56. As a result o (4.14) and (4.15), proposition.1 o [56 urther gives κ min, κ max R + such that 0 < κ min J t κ max < t [t 0, T. (4.17) The ollowing estimates will be used or analysis o the optimal control problem. Theorem 4.1. Stability o u n I C 9 z i L, t i < 1 or i = 1,..., n where C 9 = y V(z i ) L, t 0 y Ψ t L, t 0, then ρ u n 0, + [ n ν C 6 u i 1, t i i=1 C n [ i 0, + u i t i i=0 0,Γ + g i 0,Γ Iti + ρ u 0 0, t 0. (4.18) 46

55 Proo. Let (v, q) = (u n, p n ) in (4.4) (4.5). From this, ρ [ u n 0, (u n 1, V(u n )) t n 1 + ρ [c(u n, u n, u n ) c(z n, u n, u n ) + 1 ((un n ), u n ) Γ 1 (( zn ), u n ) + ν a(u n, u n ) + b(u n, p n ) = ( n, un ) + (u n, u n ) Γ + (g n, u n ) ΓI, (4.19) b(u n, p n ) = 0. (4.0) Using c(u n, u n, u n ) = 0, c(z n, u n, u n ) assumption u n n > 0 (4.13), and also using b(u n, p n ) = 0, dropping 1 ((un n ), u n ) Γ = 0 in (4.19) gives by the ρ u n 0, ρ (( zn ), u n ) + ν a(u n, u n ) ( n, un ) + (u n, u n ) Γ + (g n, u n ) ΓI + ρ (u n 1, V(u n )). t n 1 (4.1) Combining terms and using (.19) and Cauchy-Schwarz inequality in (4.1) yields ρ u n 0, ρ (( zn ), u n ) + ν C 6 u n 1, n 1, u n 1, + u n 0,Γ u n 0,Γ + g n 0,ΓI u n 0,ΓI + ρ u n 1 0, t n 1 V(u n ) 0, t n 1. (4.) The trace theorem ollowed by Young s inequality with an appropriate choice o parameters 47

56 gives ρ u n 0, ρ (( zn ), u n ) + ν C 6 u n 1, [ C n 1, + u n 0,Γ + g n 0,Γ I + ν C 6 u n 1, [ 1 + ρ u n 1 0, + 1 t n 1 V(un ) 0,. (4.3) t n 1 We rewrite and combine like terms in (4.3) to get ρ [ u n 0, u n 1 0, t n 1 + ρ [ u n 0, V(u n ) 0, t n 1 ρ (( zn ), u n ) + ν C 6 u n 1, [ C n + u n 0,Γ + g n 0,Γ I. (4.4) 0, Since u n is not time dependent, we can integrate both sides o (.3) rom t n 1 to t n where φ(x, t) = (u n (x)) as in [15, using t n 1 d u n d dt t dt = t t n 1 t V((u n ) ) y + ( x z(x, t)) V(u n ) d t t dt with z(x, t) = z n (x) t [t n 1, t n by (.37) and V((un ) ) t y = 0, to get u n 0, V(u n ) 0, t n 1 (( x z n ), u n ) 0, = t n 1 (( x z n ), V(u n ) ) 0, t dt (( x z n ), u n ) 0,. Because u n and V(u n ) have the same values or corresponding points on the moving domain, and J t J C 9 where C 9 = C y V(z n ) L, t 0 y Ψ t L, t 0 and C is a positive 48

57 constant that does not depend on or the ALE mapping (see [15), u n 0, V(u n ) 0, t n 1 (( x z n ), u n ) 0, t n 1 (( x z n ), V(u n ) (J t J )) t 0 C 9 ( z n, u n ) 0, dt C 9 z n L, u n 0, (4.5) Substituting (4.5) into (4.4), ρ [ u n 0, [ C n u n 1 0, t n 1 + ν C 6 u n 1, 0, + u n 0,Γ + g n 0,Γ I + ρ C 9 z n L, u n 0,. Observe that with z n L ( t n ), an implication o (4.16), we can derive two stability results. The irst is or boundedness at a single time step: ρ [ 1 C 9 z n L, u n 0, + ν C 6 u n 1, [ C n + u n 0,Γ + g n 0,Γ I + u n 1 0,. t n 1 0, The second is or boundedness over all time steps using a discrete Gronwall lemma ater 49

58 irst multiplying by and summing over time steps. ρ u n 0, + C + This yields n ν C 6 u i 1, t i i=1 n [ i 0, + u i t i i=0 0,Γ + g i 0,Γ Iti n C 9 z i L, ρ u i t 0, + ρ u 0 i t 0,. i t 0 i=0 ρ u n 0, + [ n ν C 6 u i 1, t i i=1 C n [ i 0, + u i t i i=0 0,Γ + g i 0,Γ Iti + ρ u 0 0, t 0 when C 9 z i L, t i < 1 or i = 1,..., n, according to the discrete Gronwall lemma [44. Thereore, must be chosen suiciently small. Theorem 4.. Stability o p n I C 9 z i L, t i < 1 or i = 1,..., n, then p n 0, C P ( 0 0,, g 0 t 0,ΓIt0, u 0 0 0,Γ,..., n 0,, g n 0,ΓI, u n 0,Γ ), (4.6) where P (,..., ) is a quadratic polynomial. Proo. Rearranging (4.4), we get b(v, p n ) [ = ρ (u n, v) (u n 1, V(v)) t n (( zn )u n, v) 1 ((un n )u n, v) Γ + ρ [ c(u n, u n, v) + c(z n, u n, v) ν a(u n, v) + ( n, v) + (u n, v) Γ + (g n, v) ΓI v H 1 D ( t n ). (4.7) 50

59 In (4.7), (u n 1, V(v)) = u n 1 V(v) d t n 1 t n 1 t n 1 = J 1 V(u n 1 ) V(v) d t 0 t 0 J 1 V(u n 1 ) V(v) J t n d t κ 0 t min 0 1 κ min ( J 1 L J 1 L 1, t0, t0 ) 1 ( J 1 (V(u n 1 )) d t 0 J V(v) d t t 0 0 t 0 1 κ min J 1 L J 1 L 1 u n 1 0, v, t0, t 0, n 1 t0 κ max κ min u n 1 0, t n 1 v 1, C u n 1 0, v t 1,. (4.8) n 1 ) 1 Applying Hölder s inequality, the Sobolev imbedding o W 1, (Γ I ) W 0,4 (Γ I ) [1, p. 85, and then the trace theorem, ((u n n )u n, v) Γ C u n L 4,Γ u n 0,Γ v L 4,Γ C u n 1,Γ u n 0,Γ v 1,Γ C u n 3 u n 1 1, v 0, 1, (4.9) b(v, p n ) v 1, Applying Cauchy Schwarz, using (.14) (.16) and (4.8) (4.9) in (4.7), C [ u n 1, + u n 1 [ 0, + ρ u n t 1, C 3 u n n 1 1, + C 3 z n 1, + 1 zn L, + C un 1 u n 1 0, 1, + ν C 1 u n 1, + n 0, + u n 0,Γ + g n 0,ΓI v H 1 D ( t n ). (4.30) 51

60 Thereore, using (4.18) with suiciently small, the in-sup condition (.1), and z n H 1 D ( t n ) by (4.16), p n 0, sup 0 v b(v, p n ) v 1, C P ( 0 0,, g 0 t 0,ΓIt0, u 0 0,Γ,..., n 0,, g n 0,ΓI, u n 0 0,Γ ), where P (,..., ) is a quadratic polynomial. Theorem 4.3. Stability o η n and η n η n + η n 1 + η n + η n 1 0, s 1, [ s ( C s n 1, s + s n 1 ) ( + η n 1, s 0,Γ s + η n 1 ) 0,Γ s + ( g n 0,ΓI + g n 1 ) 0,ΓI 1 + η 0 0,s + η 0 1,s. (4.31) Proo. Letting ξ = ηn η n 1 in (4.6), γ = η n η n 1 in (4.7), and substituting, ρ s ( η n η n 1, ηn + η n 1 ) s + ν s d (η n + η n 1, ηn η n 1 ) + λ (η e n + η n 1, ηn η n 1 ) s s = 1 ( s n + s n 1, ηn η n 1 ) + 1 ( η n + η n 1, ηn η n 1 ) s Γ s 1 (V(g n )J + V(g n 1 )J 1, ηn η n 1 ). (4.3) Γ It0 Adding ρ s ( η n 1, η ) n + η n 1, ν s d(η n + η n 1, ηn 1 ) s, and λ e(ηn + η n 1, ηn 1 ) s to s 5

61 both sides o the equation, ρ s ( η n + η n 1, ηn + η n 1 ) s + ν s d (η n + η n 1, ηn + η n 1 ) + λ (η e n + η n 1, ηn + η n 1 ) s = 1 ( s n + s n 1, ηn η n 1 ) + 1 ( η n + η n 1, ηn η n 1 ) (V(g s n )J + V(g n 1 )J 1, ηn η n 1 ) Γ s s 1 ( η + ρ s n 1, ηn + η n 1 ) + λ e ( η n + η n 1, ηn 1 Γ It0 ) + ν s d (η n + η n 1, ηn 1 ) s s. (4.33) Using (.0), dropping a positive term, multiplying by, and simpliying, (4.33) becomes ρ s η n + η n 1 + 4ν s C 7 η n + η n 1 0, s 1, s = 1 [ (s n + s n 1, η n η n 1 ) s + (η n + η n 1, ηn η n 1 ) Γ s (V(g n )J + V(g n 1 )J 1, η n η n 1 ) ΓIt0 s + ρ s ( η n 1, η n + η n 1) s + ν s d(η n + η n 1, η n 1 ) s + λe(η n + η n 1, η n 1 ) s. Applying Cauchy Schwarz inequality, the trace theorem, 0 < J t < κ max <, and then 53

62 Young s inequality, ρ s η n + η n 1 + 4ν s C 7 η n + η n 1 0, [ s ( C s n 1, s + s n 1 ) + 1, s 1, s ( η n 1,Γs + ( g n 1 + g n 1 ) + ν sc 7,ΓI 1,ΓI ρ s η n 1 0, s 0, s η n + η n 1 + 3ν s C 7 + ρ s η n + η n 1 1, s + [ νs C 4 C 7 + λ C 5 ν s C 7 + η n 1 1,Γs η n η n 1 ) 1, s η n 1 1, s. (4.34) Adding and subtracting terms with the triangle inequality or norms, Cauchy Schwarz inequality, and combining like terms, ρ s η n + η n 1 + ν sc 7 η n + η n 1 0, s 1, [ s ( C s n 1, s + s n 1 ) ( + η n 1, s 1 + η n 1 ),Γs 1,Γs + ( g n 1 + g n 1 ) n 1 + 4nρ s η j + η j 1,ΓI 1,ΓI 1 [ νs C 7 + 4n + ν sc4 + λ C n 1 5 η j + η j 1 C 7 ν s C 7 j=1 1, s [ + C η 0 + 0, η 0. (4.35) s 1, s j=1 0, s Let α = min {ρ s, νsc 7 } and β = max {ρs, νsc 7 + νsc 4 C 7 + λ C 5 ν sc 7 }. With the discrete Gronwall lemma [44 and no restriction on, (4.35) is bounded by η n + η n 1 + η n + η n 1 0, s 1, s C ( 4n ) [ α exp β ( s n 1, α s + s n 1 ) ( + η n 1, s 1 + η n 1,Γs + ( g n 1 + g n 1 ) + η 0 0,s + η 0 1,s.,ΓI 1,ΓI 1 1,Γs ) 54

63 Multiplying by and summing (4.35) over time steps, [ n η i + η i 1 n + η i + η i 1 i=1 0, s i=1 1, s n [ C s i + η i 1, s + g i 1,Γ + η 0 + η 0. I ti 0, s 1, s i=0 1,Γs 4.5 The Existence o an Optimal Solution Theorem 4.4. There exists a (û n, ˆp n, ˆη n, ˆ η n, ĝ n ) S such that (4.10) is minimized. Proo. Clearly, the set S is nonempty. Let {(u n (k), pn (k), ηn (k), ηn (k), gn (k))} be a sequence in S such that lim J n δ (u n (k), k pn (k), ηn (k), ηn (k), gn (k) ) = in J (u n,p n,η n, η n,g n n δ (u n, p n, η n, η n, g n ). ) S By the deinition o (4.11), g n (k) is uniormly bounded in L (Γ I ). ρ u n (k) From (4.18), (4.6), and (4.31), we have, 0, p n 0, + ν C 6 [ u n (k) C [ [ g n (k) 1, n 1 + u i 1, i=1 n 1 + g i + 0,Γ 0,Γ Iti I +ρ u 0 0, t 0 C P ( 0 0,, i=0, g 0 t 0,ΓIt0, u 0 0 0,Γ n [ i i=0,..., n 0, 0, + u i 0,Γ i, g n (k), u n 0,ΓI 0,Γ ), 55

64 and η n (k) η + n (k) 0, s 1, [ s ( C s n 1, s + s n 1 ) ( + η n 1, s 0,Γ s + η n 1 ) 0,Γ s ( g n + (k) + g n 1 ) + η 0 + η 0 0,Γ 0,Γ I 1 0, s 1, s I + η n 1 + η n 1. 0, s 1, s Thereore, (u n (k), pn (k), ηn (k), ηn (k), gn (k) ) H1 D ( t n ) L ( t n ) H 1 D (s ) L ( s ) L (Γ I ) is uniormly bounded. Then, there must exist subsequences such that u n (k j ) û n in H 1 ( t n ), η n (k j ) ˆη n in H 1 ( s ), p n (k j ) ˆp n in L ( t n ), η n (k j ) ˆ η n in L ( s ), u n (k j ) û n in L ( t n ), g(k n j ) ĝ n in L (Γ I ), and u n (k j ) Γ Γ û n I in L (Γ Γ I ), or some (û n, ˆp n, ˆη n, ˆ η n, ĝ n ) S. The last two statements with strong convergence are a result o the compact imbeddings H 1 ( t n ) L ( t n ) and H 1 (Γ Γ I ) L (Γ Γ I ). ow we wish to show that by passing to the limit, (û n, ˆp n, ˆη n, ˆ η n, ĝ n ) satisies (4.4) (4.7). Equations (4.4) (4.7) are linear with the exception o c(u n (k), un (k), v) and 1 ((un (k) n )u n (k), v) Γ in (4.4). lim k c(un (k), un (k), v) + 1 ((un (k) n )u n (k), v) Γ = lim k (un (k) v, un (k) ) + 1 ((un (k) n )u n (k), v) Γ Γ I + 1 ((un (k) n )u n (k), v) Γ = (û n v, û n ) + 1 ((ûn n )û n, v) Γ Γ + 1 I ((ûn n )û n, v) Γ = c(û n, û n, v) + 1 ((ûn n )û n, v) Γ v C ( t n ) (4.36) since u n (k j ) û in L ( t n ) and u n (k j ) û Γ I in L (Γ I ). Then, because C ( t n ) is dense 56

65 in H 1 D ( t n ), lim k c(un (k), un (k), v) + 1 ((un (k) n )u n (k), v) Γ = c(û n, û n, v) + 1 ((ûn n )û n, v) Γ v H 1 D ( t n ). Additionally, since J δ n is lower semicontinuous, Jn δ ( uˆ n, pˆ n, ηˆ n, ˆ η n, ĝ n ) = in J (u n,p n,η n, η n,g n n δ (u n, p n, η n, η n, g n ), ) S so there exists a solution to the optimal control problem, although we can not show in general that (û n, ˆp n, ˆη n, ˆ η n, ĝ n ) is unique. 4.6 Convergence o Vanishing Penalty Parameter In this section, we show that as δ 0, the optimal solution to (4.10) converges to a solution that satisies (.38) (.39) and (.40) (.41) and satisies the interace conditions (.11) (.1) within a tolerance on the order o 3. In addition to (4.13) (4.16), we make the ollowing assumptions on regularity o the strong solution satisying (.1) (.1) that are needed or the theorem that ollows: u, du dt y L (t 0, t n ; H ( t )), p, dp dt y L (t 0, t n ; H 1 ( t )), η L (t 0, t n ; H ( s )), η t L (t 0, t n ; H 1 ( s )), η tt L (t 0, t n ; H 1 ( s )), the body orces and boundary conditions are suiciently smooth, and is suiciently small. We will assume no time discretization error o g n in the L sense. Future work will provide a detailed proo using the entire derived optimality system. Theorem 4.5. Let (u n δ, pn δ, ηn δ, ηn δ, gn δ ) denote an optimal solution satisying (4.4) (4.5) and (4.6) (4.7) or δ > 0, where u n 1 = u(t n 1 ) H 1 D ( t n 1 ), η n 1 = η(t n 1 ) H 1 D (s ), and u(t n 1 ) and η(t n 1 ) are solutions to (.1) (.1) at t = t n 1. Then, the solution (ũ n, p n, η n, η n ) exists as a solution o (4.4) (4.7) at t = t n such that u n δ ũn 1, + p n δ pn 0, + η n δ ηn 1, s + η n δ η n 0, s 0 as δ 0. Also, 57

66 in the limit, J n (ũ n, p n, η n, η n, g n ) C 3. Proo. Consider the solution to the semidiscrete weak ormulations (.38) (.39) and (.40) (.41) satisying the boundary conditions (.11) (.1), and denote this (u n sd, pn sd, ηn sd, ηn sd ). Letting g n sd = [ν D(u n sd ) n p n sd n 1 ((un sd zn sd ) n )u n sd Γ I, we see that (u n sd, pn sd, ηn sd, η n sd, gn sd ) is also a solution to (4.4) (4.7). Also, un sd, ηn sd, and ηn 1 are bounded in H 1 D ( t n ), H 1 D (s ), and H 1 D (s ), respectively, so Jn δ (u n sd, pn sd, ηn sd, ηn sd, gn ) = 1 un sd V(ηn sd ) V(ηn 1 ) Γ I dγ I + δ gsd n dγ I Γ I [ C u n sd 1, + η n sd 1, s + η n 1 + δ g n 1, s sd 1, <. Thereore, we can conclude that (u n sd, pn sd, ηn sd, ηn sd, gn sd ) S. S, By the deinition o an optimal solution in (4.1) and since (u n sd, pn sd, ηn sd, ηn sd, gn sd ) J δ n (u n δ, pn δ, ηn δ, ηn δ, gn δ ) = in J (u n,p n,η n, η n,g n n δ (u n, p n, η n, η n, g n ) ) S Jn δ (u n sd, pn sd, ηn sd, ηn sd, gn sd ). (4.37) In (4.37), Jn δ (u n sd, pn sd, ηn sd, ηn sd, gn sd ) = 1 [ Γ I un sd V(ηn sd ) V(ηn 1 ) dγ I + δ gsd n dγ I Γ I C u n sd u(t n) 0,Γ I + u(t n ) V(η t (t n )) 0,Γ I + η t(t n ) ηn sd ηn 1 + δ gn sd 0,Γ I (4.38) 0,Γ It0 where u(t n ) and η t (t n ) are solutions to (.1) (.1) at time t = t n. Since u(t n ) and η t (t n ) satisy (.11), u(t n ) V(η t (t n )) 0,Γ I = 0. 58

67 By adding and subtracting η() to η t (t n ) ηn sd ηn 1 that η n 1 = η(t n 1 ), and using the triangle inequality or norms, 0,Γ It0 in (4.38), recognizing J δ n (u n sd, pn sd,ηn sd, ηn sd, gn sd ) δ gn sd 0,Γ I + C [ u n sd u(t n) 0,Γ I + η t(t n ) η(t n) η(t n 1 ) 0,Γ It0 + η(t n ) η n sd 0,Γ It0. (4.39) Since t is Lipschitz continuous by (4.14), we use the trace theorem to get u n sd u(t n) 0,Γ I C u n sd u(t n) 0, u n sd u(t n) 1,. (4.40) We use a result similar to the error estimate derived in [5, except in our case it is continuous in space and has a constant viscosity. Along with this, we include the assumptions on u, made in the statement o the theorem, to get u n sd u(t n) 0,Γ I C 3. (4.41) The second order time discretization [55, pp or the linear elasticity equations along with the assumptions on η and η t gives η(t n ) η n sd 0,Γ It0 C η(t n ) η n sd 0, s η(t n ) η n sd 1, s C 7. (4.4) Using (4.4), η(t n ) η n sd C 3. (4.43) 0,Γ It0 59

68 Substituting (4.41) and (4.43) into (4.39) gives Jn δ (u n sd, pn sd, ηn sd, ηn sd, gn sd ) δ gn sd 0,Γ I + C 3 + η t (t n ) η(t n) η(t n 1 ) 0,Γ It0. (4.44) Using Taylor series expansion along with η tt L (t n 1,t n;h 1 ( s )) < gives η t(t n ) η(t n) η(t n 1 ) C η tt L (t n 1,t n;h 1 ( s )) C. (4.45) 0,Γ It0 We now substitute (4.45) into (4.44) and (4.44) into (4.37) to get J δ n (u n δ, pn δ, ηn δ, ηn δ, gn δ ) J δ n (u n sd, pn sd, ηn sd, ηn sd, gn sd ) C 3 + δ gn sd 0,Γ I. (4.46) From (4.46), we can see that g n δ is uniormly bounded. Combining this with (4.18), (4.6), and (4.31), we know that u n δ, pn δ, ηn δ, and ηn δ are uniormly bounded as well. Then, there must exist subsequences such that u n δ ũ n in H 1 ( t n ), η n δ η n in H 1 ( s ), p n δ p n in L ( t n ), η n δ η n in L ( s ), u n δ ũ n in L ( t n ), gδ n g n in L (Γ I ), and u n δ Γ Γ ũ n I in L (Γ Γ I ), as δ 0 or some (ũ n, p n, η n, η n, g n ) S, such that J n (ũ n, p n, η n, η n, g n ) = V( ηn ) V(η(t n 1 )) C 3. 0,Γ I (4.47) Remark 4.6. For any choice o inite dierence ormula in the unctional, note rom (4.43) that we will lose two powers o rom (4.4). This is the reason that, rom an analytical 60

69 perspective, we should choose to use a second order time discretization or the structure despite only using a irst order time-ormula in the unctional. Additionally, with respect to (4.43), we see that by using a second order time discretization or the structure, there would be no improvement by increasing the order o the inite dierence ormula used in the unctional. 4.7 The Existence o Lagrange Multipliers We desire to change the optimization problem rom constrained to unconstrained. This can be done by means o the Lagrange multiplier rule ater proving the existence o Lagrange multipliers. We use the same approach to show the existence o Lagrange multipliers as was used in [7, 31. Lemma 4.7. Let X and Y be two real Banach spaces, J a unctional on X, and M a mapping rom X to Y. Assume u is a solution o the ollowing constrained minimization problem: ind u X such that J (u) = in{j (v) v X, M(v) = y 0 }, where y 0 is some ixed element o Y. Additionally, assume the ollowing three conditions are satisied: 1. M is Fréchet-dierentiable in an open neighborhood o u and its Fréchet derivative M is continuous at u.. J : eighborhood(u) X R is Fréchet-dierentiable at u with Fréchet derivative J. 3. M : X Y is onto. Then there exists a µ Y satisying J (u) w + µ, M (u) w = 0 w X. 61

70 Proo. See [69. In the ollowing two theorems, we will veriy the assumptions o Lemma 4.7. We begin by veriying the irst part o Assumption 1 o Lemma 4.7 in Theorem 4.8, namely that M is Fréchet-dierentiable in an open neighborhood o u. Let X = H 1 D ( t n ) L ( t n ) H 1 D (s ) L ( s ) L (Γ I ), Y = H 1 D ( t n ) L ( t n ) H 1 D (s ) L ( s ), J = J δ n deined in (4.9), and M : X Y be the nonlinear mapping representing the generalized constraint equations: M(u n, p n, η n, η n, g n ) = ( n, φ 1, n s, φ ) or (u n, p n, η n, η n, g n ) X and ( n, φ 1, n s, φ ) Y, i and only i, ρ (u n, v) + ρ [c(u n, u n, v) + 1 ((un n )u n, v) Γ c(z n, u n, v) + ν a(u n, v) + b(v, p n ) (g n, v) ΓI b(u n, q) = ( n, v) = (φ 1, q) v H 1 D ( t n ), (4.48) q L ( t n ), (4.49) ρ s ( η n, ξ) s + ν s d(η n + η n 1, ξ) s + λe(ηn + η n 1, ξ) s + (V(gn )J, ξ) ΓIt0 = ( n s, ξ) s ξ H 1 D (s ), (4.50) ( ηn, γ) s (η n, γ) s = (φ, γ) s γ L ( s ). (4.51) Theorem 4.8. Let (u n, p n, η n, η n, g n ) be an optimal solution to Problem (4.10) and M be deined as in (4.48) (4.51). The Fréchet derivative o M, denoted M, exists in an open neighborhood o (u n, p n, η n, η n, g n ). M is deined by M (u n, p n, η n, η n, g n ) (w n, r n, θ n, ϕ n, h n ) = ( n, φ 1, n s, φ ) or (w n, r n, θ n, ϕ n, h n ) X and ( n, φ n 1, n s, φ n ) Y, i and only i 6

71 ρ (w n, v) + ρ [c(u n, w n, v) + c(w n, u n, v) + 1 ((un n )w n, v) Γ + 1 ((wn n )u n, v) Γ 1 (( zn )w n, v) c(z n, w n, v) + ν a(w n, v) + b(v, r n ) (h n, v) ΓI = ( n, v) v H 1 D ( t n ), (4.5) b(w n, q) = ( φ 1, q) q L ( t n ), (4.53) ρ s (ϕ n, ξ) s + ν s d(θ n, ξ) s + λe(θn, ξ) s + (V(hn )J, ξ) ΓIt0 = ( n s, ξ) s ξ H 1 D (s ), (4.54) (ϕn, γ) s (θ n, γ) s = ( φ, γ) s γ L ( s ). (4.55) Proo. We begin by showing that or ɛ > 0, δ > 0 such that i (u n 1 un, pn 1 pn, ηn 1 ηn, ηn 1 η n, g1 n gn ) X < δ, then M(u n 1, pn 1, ηn 1, ηn 1, g1 n) M(un, pn, ηn, ηn, g n) M (u n, p n, η n, η n, g n ) (u n 1 un, pn 1 pn, ηn 1 ηn, ηn 1 η n, g1 n gn ) Y ɛ. 63

72 Let ɛ > 0. For any (v, q, ξ, γ) Y, M(u n 1, p n 1, η n 1, η n 1, g n 1 ) M(u n, p n, η n, η n, g n ) M (u n, p n, η n, η n, g n ) (u n 1 u n, p n 1 p n, η n 1 η n, η n 1 η n, g n 1 g n ), (v, q, ξ, γ) = c(u n 1, u n 1, v) + 1 ((un 1 n )u n 1, v) Γ c(u n, u n, v) 1 ((un n )u n, v) Γ c(u n, u n 1 u n, v) c(u n 1 u n, u n, v) 1 ((un n )(u n 1 u n ), v) Γ 1 (((un 1 u n ) n )u n, v) Γ = c(u n 1 u n, u n 1, v) + 1 (((un 1 u n ) n )u n 1, v) Γ + c(u n, u n 1 u n, v) + 1 ((un n )(u n 1 u n ), v) Γ c(u n, u n 1 u n, v) c(u n 1 u n, u n, v) 1 ((un n )(u n 1 u n ), v) Γ 1 (((un 1 u n ) n )u n, v) Γ C u n 1 u n 1, v 1,. Thereore, M(u n 1, pn 1, ηn 1, ηn 1, g n 1 ) M(un, pn, ηn, ηn, g n ) M (u n, p n, η n, η n, g n ) (u n 1 un, pn 1 pn, ηn 1 ηn, ηn 1 η n, g n 1 gn ) Y C u n 1 un 1, ɛ when (u n 1 un, pn 1 pn, ηn 1 ηn, ηn 1 η n, g n 1 gn ) X is suiciently small. It is straight orward to veriy Assumption o Lemma 4.7, showing that J δ n is Fréchet dierentiable at (u n, p n, η n, η n, g n ) with Fréchet derivative (J δ n ). ext, we will prove Assumption 3 and the second part o Assumption 1 o Lemma 4.7. Theorem 4.9. The operator M : X Y is onto Y or a suiciently small and u 0 0,, and M is continuous at (u n, p n, η n, η n, g n ). Proo. In Theorem 4.8, it was shown that M exists and maps X Y. With h n = 0 and v = w n, (4.5) (4.53) is a time-discretized linearized avier Stokes operator with our additional terms: c(w n, u n, w n ), (( z n )w n, w n ), ((w n n )u n, w n ) Γ, and ((u n n )w n, w n ) Γ. Coercivity o the linearized avier Stokes operater can be proven without a time step restriction [65, p. 334, i.e, we are let with proving that or any 64

73 w n H 1 D ( t n ), ρ wn 0, + C 9 ν w n 1, + ρ c(w n, u n, w n ) ρ (( zn )w n, w n ) + ρ ((un n )w n, w n ) Γ + ρ ((wn n )u n, w n ) Γ C 10 w n 1,, where C 9 and C 10 are some positive constants. Here, we show that ρ c(w n, u n, w n ) ρ c(w n, u n, w n ) = ρ Cρ C 11 ρ 4 u n C 3 9 ν3 4 1, [ (w n u n, w n ) (w n w n, u n ) [ w n L 4, u n 0, + w n L 4, w n L 4, w n 0, u n L 4, by applying Hölder s inequality with p=4, q=, r=4, since t n R [50, p. 11, Cρ w n 1 w n 3 0, u n 1, 1, C ρ 4 wn 0, u n 4 1, applying Young s inequality with p=4 and q= 4 3. Thereore, w n 0, + C 9ν (3ρ ) 3 (C 9 ν ) 3 + 3C 9ν 3 4 wn 1, 4 w n 1,. ρ c(w n, u n, w n ) C 11 ρ 4 u n 4 1, C 3 9 ν3 w n 0, + C 9ν 4 w n 1,. The term u n is uniormly bounded by (4.18) independent o, i the additional assumption is made that u 0 0, < 4. This is a reasonable assumption to make i the simulation begins at rest. Because z n W 1, ( t n ) by assumption (4.16), we can bound ρ (( z n )w n, w n ) C 1 ρ z n L, w n 0, using Hölder s inequality with p=, q=, and r=. Applying the Sobolev imbedding o W 1, (Γ I ) W 0,4 (Γ I ) [1, p. 85 ollowed by Hölder s inequality and the trace theorem along with the boundedness o u n rom (4.18), 65

74 we get ρ ((w n n )u n, w n ) Γ Cρ n L,Γ I w n L 4,Γ I u n L 4,Γ I w n 0,ΓI Cρ w n 1 W u n, 1,Γ I W w n 1 w n 1,,Γ I 0, 1, Cρ w n 1 w n 3 0, u n 1, 1, C 13 ρ w n 0, u n 4 1, C 13ρ 4 C 3 9 ν3 u n 4 1, w n 0, ρ 3 C 3 9 ν3 + C 9ν + C 9ν w n 1, w n 1,. We apply the inequalities and imbeddings in the same way or ρ ((u n n )w n, w n ) Γ. Again, C 11, C 1, and C 13 are positive constants. ow we have that w n 0, 1 [C 11 + C 13 ρ 4 u n 4 1, C 3 9 ν3 ρ zn L, + C 9ν 4 w n 1, C 10 w n 1,, where C 10 is a positive constant. This result yields the coercivity o (4.5) (4.53) when is suiciently small. The operator or (4.5) (4.53) is a time-discretized linearized avier Stokes operator with ive additional terms: c(w n, u n, v), (( z n )w n, v), ((w n n )u n, v) Γ, ((u n n )w n, v) Γ, and (h n, v) ΓI. We begin by noting that the linearized avier Stokes operator is continuous [65, p. 334 and now turn our attention to these ive additional terms. For the irst term, apply Hölder s inequality and the regularity o u n. For the second term, apply Hölder s inequality and use the regularity o z n (see (4.16)). Then, apply the Sobolev imbedding o W 1, (Γ I ) W 0,4 (Γ I ) and Hölder s inequality ollowed by the trace theorem along with the boundedness o u n to the third and ourth terms. For the ith term, apply Cauchy Schwarz inequality. In this way, it can be shown that (4.5) (4.53) is continuous at (u n, p n, η n, η n, g n ). Since (4.54) (4.55) is linear, it retains the same continuity 66

75 as the system (4.6) (4.7). Using the Lax Milgram theorem and the continuity, coercivity, and in-sup condition satisied by the operator (4.5) (4.53), or each element in Y, there is a unique solution in X. Similarly, with h n = 0, (4.54) (4.55) have the same properties o well-posedness as (4.50) (4.51). Since a unique solution to M (u n, p n, η n, η n, g n ) (w n, r n, θ n, ϕ n, h n ) = ( n, φ 1, n s, φ ) exists ( n, φ n 1, n s, φ n ) Y, M (u n, p n, η n, η n, g n ) is onto Y. Theorem Let (u n, p n, η n, η n, g n ) X denote an optimal solution to Problem (4.10). Then there exists a nonzero Lagrange multiplier (ū n, p n, η n, η n ) Y such that (J δ n ) (u n, p n, η n, η n, g n ) (w n, r n, θ n, ϕ n, h n ) + M (u n, p n, η n, η n, g n ) (w n, r n, θ n, ϕ n, h n ), (ū n, p n, η n, η n ) = 0 (w n, r n, θ n, ϕ n, h n ) X, (4.56) where Y = H 1 D ( t n ) L ( t n ) H 1 D (s ) L ( s ). Proo. We have shown that M is onto Y, that M exists and is continuous in a neighborhood about (u n, p n, η n, η n, g n ) X, and we also know that Jn δ is Fréchet dierentiable. We now apply Lemma 4.7 to see that there exists a solution (ū n, p n, η n, η n ) Y satisying (4.56). 4.8 Lagrange Multiplier Rule The optimality system will now be derived using the Lagrange multiplier rule. Again, we treat the deormation and velocity o t as known. For an example o perorming such a derivation with the deormation and velocity o t treated as unknowns, see [63, where optimization was applied to a luid-structure interaction problem or the purpose o parameter estimation. 67

76 Let us begin by deining the Lagrangian L(u n, p n, η n, η n, g n, ū n, p n, η n, η n ) = Jn δ (u n, p n, η n, η n, g n ) + ρ [(u n, ū n ) (u n 1, V(ū n )) t n 1 + ρ [c(u n, u n, ū n ) + 1 ((un n )u n, ū n ) Γ 1 (( zn )u n, ū n ) c(z n, u n, ū n ) + ν a(u n, ū n ) + b(ū n, p n ) ( n, ūn ) (u n, ū n ) Γ (g n, ū n ) ΓI + b(u n, p n ) + ρ s [( η n, η n ) s ( η n 1, η n ) s + ν s d(η n + η n 1, η n ) s + λe(ηn + η n 1, η n ) s ( s n + s n 1, η n ) s (ηn + η n 1, ηn ) Γ s + (V(gn )J + V(g n 1 )J 1, η n ) ΓIt0 + [( ηn, η n ) s + ( η n 1, η n ) s [(η n, η n ) s (η n 1, η n ) s or any (u n, p n, η n, η n, g n, ū n, p n, η n, η n ) X Y. We will now seek to ind stationary points o L over the product space X Y. Variations in the Lagrange multipliers ū n, p n, η n, and η n yield the state equations (4.4) (4.7). Variations in the state variables u n, p n, η n, and η n yield the adjoint equations ρ (ū n, v) + ρ [c(u n, v, ū n ) + c(v, u n, ū n ) + 1 ((un n )v, ū n ) Γ b(ū n, q) + 1 ((v n )u n, ū n ) Γ 1 (( zn )v, ū n ) c(z n, v, ū n ) + ν a(ū n, v) + b(v, p n ) ( = u n V(ηn ) V(η n 1 ) ), v v H 1 D ( t n ), (4.57) Γ I = 0 q L ( t n ), (4.58) 68

77 ( η n, ξ) s + ν s d( η n, ξ) s + λe( ηn, ξ) s = 1 (J [V(u n ) ηn η n 1 ), ξ ξ H 1 D (s ), (4.59) Γ It0 ( η n, γ) s + ρ s ( η n, γ) s = 0 γ L ( s ). (4.60) Also, variations in the control g n yield the necessary condition δ(g n, c) ΓI = (c, ū n ) ΓI + (V(c)J t n, η n ) ΓIt0 c L (Γ I ). (4.61) The adjoint problem (4.57) (4.60) is well-posed, similar to the linearized problem (4.5) (4.55). Also, (4.56) can be rewritten as the adjoint equations (4.57) (4.60) and the necessary condition (4.61). It is obvious that an optimal solution to (4.8) will satisy the state equations (4.48) (4.51), and thereore we see that an optimal solution satisies the optimality system (4.4) (4.7), (4.57) (4.60), and (4.61). 4.9 Steepest Descent Approach The optimality system is oten large in practice, and so it is common practice to decouple the state equations, adjoint equations, and the necessary condition. This is reerred to as a reduced space method [1. As in [31, we will use a gradient method or minimizing the penalized unction (4.9). Accordingly, g(k) n = gn (k 1) ω k dj n δ dg(k) n, where ω k is a step-size appropriately chosen. The necessary condition (4.61) allows us to solve or dj n δ dg(k) n, yielding [39 dj δ n dg n (k) = δg n + ū n ΓI ηn (Ψ n (k) ) 1 ΓI. Let ω k have the orm α k δ, and now the algorithm has the orm g n (k+1) = (1 α k)g n (k) α k δ (ūn (k) Γ I ηn (k) (Ψn (k) ) 1 ΓI ). (4.6) Algorithm Steepest Descent Algorithm 69

78 1. Choose an initial control g n (0). For k = 0, 1,... (a) Solve (4.4) (4.7) or (u n (k), pn (k), ηn (k), ηn (k) ) (b) I u n Γ I (k) V(ηn ) V(η n 1 ) dγ < ɛ tol, then break (c) Solve (4.57) (4.60) or (ū n (k), pn (k), ηn (k), η n (k) ) (d) Update the control using (4.6) (e) Solve (.31), and then update z n (k) and the mesh deormation gradient 4.10 umerical Results An FSI problem using the ALE ormulation or the moving luid domain, reported in [54 and subsequently reproduced in Chapter 3, uses parameters that are consistent with blood low in a human body. η D = 0 η = 0 s = [0, 6 [1, 1.1 η D = 0 u = b(t) 0 = [0, 6 [0, 1 Γ I0 u = 0 u D = 0 Figure 4.1: Domain and boundary conditions or numerical experiment A orce b(t) is applied to the let luid boundary (Fig. 4.1) at t s where ( 10 3 (1 cos πt.05 b(t) = ), 0) dyne/cm, t 0.05 (0, 0), 0.05 < t < T. The unction b(t) deines the stress on the inlet denoted by u in (.4). For numerical tests, we impose the eumann condition on both the inlow and outlow boundaries in order to use the same conditions and parameters as in the literature. The volume orce or the luid and structure are (t) = (0, 0) dyne/cm. The other boundary conditions on the domain 70

79 coniguration are homogeneous Dirichlet or eumann (Fig. 4.1), and the simulation begins at rest. The reerence domain or the luid subsystem has height 1 cm and length 6 cm. The density o the luid, ρ, is 1 g/cm 3 and the viscosity o the luid, ν, is g/cm s. The structure domain has height 0.1 cm and length 6 cm. The density o the structure, ρ s, is 1.1 g/cm 3. The Young s Modulus o the structure, E, is dyne/cm and its Poisson ratio, ν, is 0.3. The Lamé parameters λ and ν s are deined as ollows: λ = νe (1 ν)(1 + ν) dyne/cm, ν s = E (1 + ν) dyne/cm. In Figure 4., we compare the vertical displacement over time o three points along the interace. Comparison is made between the solution ound using Aitken s relaxation [1 and the solution ound using the optimization Algorithm See Section or [1 or more details on Aitken s relaxation. The result using Aitken s relaxation is reliable and useul as a reerence solution with which to compare. Spatial discretization was made in the x direction with h x = 0. cm and in the y direction with h y = 0.1 cm or both luid and structure domains on a uniorm mesh. The simulation was perormed with = 10 4 s rom T = 0 s to T = 0.1 s. Computations were perormed in FreeFEM++ [41 using the triangular (P 1 + bubble, P 1 ) inite element pair or the luid and triangular P 1 elements or the structure. The stopping ( ) 1 criteria used or Aitken s relaxation was (η n ΓIt0 (k) ηn (k 1) ) dγ < 10 7, while δ = and ɛ tol = 10 4 or Algorithm There is strong agreement between the solution computed by Aitken s relaxation and the two solutions computed using Algorithm The optimization algorithm has been implemented or both the irst and second order ormulations o the structure subsystem to demonstrate the similarity in result and that requiring a second order structure subsystem ormulation is needed or analysis, but not necessarily or computation. A study was also made o varying the Tikhonov regularization parameter δ in order 71

80 Displacement (cm) (1.5,1.0) 0.05 (1) () (3) (3.0,1.0) (1) () (3) Time (seconds) (1) () (3) (4.5,1.0) Figure 4.: Vertical displacement at three points on the interace using (1) irst and () second order ormulations with the optimal control algorithm beside vertical displacement using (3) Aitken s relaxation Displacement (cm) (1.5,1.0) (1) () (3) (4) (3.0,1.0) (1) () (3) (4) Time (seconds) (1) () (3) (4) (4.5,1.0) Figure 4.3: Vertical displacement at three points on the interace using (1) δ = 10 5, () δ = 10 6, (3) δ = 5e-7, and (4) δ = to see its eect on the solution. In Figure 4.3, the vertical displacement at three points along the interace are shown or δ = {10 5, 10 6, 5e-7, }. It can be observed that with a larger penalty parameter, the optimization ocuses more on minimizing the stress on the interace rather than minimizing the discontinuity o the velocities along the interace. For δ < 10 10, the results were indistinguishable. The computation times were similar or simulations made with δ < For choices o δ larger than 10 6, the step size or the steepest descent algorithm was required to be decreased in order or the algorithm not to diverge. This results in a less aggressive optimization, requiring more time to reach an optimal solution. When determining stopping criteria or each time step, it is obvious that 7

81 one should stop i the objective value is smaller than ɛ tol. At an optimal solution or a larger δ value, the objective may still be larger than ɛ tol. This requires an additional stopping criteria. An example o such a criteria could be to stop when more than a predetermined number o optimization iterations are executed without relative improvement o more than 1% in the objective Conclusion We have introduced a control that has allowed the luid-structure interaction problem to be decoupled into subsystems that may be solved in parallel. This control allowed us to demonstrate the uniorm boundedness o each optimization variable in order to show the existence o an optimal solution or a given δ. Then, it was shown that as δ 0, the optimal solutions or each given δ converge to an optimal solution with no penalty parameter, and that this optimal solution satisies the constraint equations and minimizes the unctional to within a C 3 target tolerance. The existence o Lagrange multipliers was proved, allowing us to derive the optimality system and to show that an optimal solution satisies the optimality system. The steepest descent algorithm was then introduced or updating the control in order to decouple the optimality system. The numerical results conirm that Algorithm 4.11 accurately simulates the luidstructure interaction or a blood low problem o great computational diiculty, due to the added mass eect. This computation was made over 1000 time steps while maintaining the correct solution proile, despite the analysis supporting only a single time step. In uture work, we hope to provide an analytical ramework or the optimal control problem over all time steps and then to extend this work to other luid-structure conigurations. 73

82 Chapter 5 THEORETICAL COVERGECE RATES 5.1 Introduction Building o o previous analytical results showing the existence o an optimal solution and Lagrange multipliers in Chapter 4, an a priori error estimate is given or the optimality system by means o Brezzi Rappaz Raviart (BRR) [18 theory. The convergence o the steepest descent method is proven in a discrete setting or a suiciently small time step and mesh size. A numerical study is included supporting the theoretical rate o convergence over a single time step. Additional numerical results demonstrate optimal convergence in space and time over several time steps. 5. Description o the Optimization Problem By applying Green s theorem and replacing the term (ν D(u n ) n pn 1 ((un z n ) n )u n, v) ΓI with (g n, v) ΓI and (ν s D(η n ) n s + λ( η n )n s, ξ) with (V(g n )J, ξ) ΓIt0 on the interace, we are let with seeking a g n that will minimize the velocity jump on the interace between the luid and structure subsystems. As explained in Chapter 3, the lack 74

83 o a H 1 bound or the structure velocity requires us to minimize an objective unction which uses some inite dierence approximation o the structure velocity on the interace, e.g., we seek to minimize J δ n (u n, p n, η n, η n, g n ) = 1 Γ I un V(ηn ) V(η n 1 ) dγ I + δ g n dγ I, Γ I (5.1) rather than Jn δ (u n, p n, η n, η n, g n ) = 1 u n V( η n ) dγ I + δ g n dγ I. (5.) Γ I Γ I Thereore, we seek to ind u n, p n, η n, η n, and g n such that (5.1) is minimized or a given u n 1, η n 1, and η n 1, subject to (5.3) ρ [(u n, v) (u n 1, V(v)) + ρ [c(u n, u n, v) t + 1 n 1 ((un n )u n, v) Γ b(u n, q) 1 (( zn )u n, v) c(z n, u n, v) + ν a(u n, v) + b(v, p n ) = ( n, v) + (u n, v) Γ + (g n, v) ΓI v H 1 D ( t n ), (5.4) = 0 q L ( t n ), (5.5) 75

84 and ρ s [( η n, ξ) s ( η n 1, ξ) s + ν s d(η n + η n 1, ξ) s + λe(ηn + η n 1, ξ) s = ( s n + s n 1, ξ) s + (ηn + η n 1, ξ) Γ s (V(gn )J + V(g n 1 )J 1, ξ) ΓIt0 ξ H 1 D (s ), (5.6) [( ηn, γ) s + ( η n 1, γ) s = (η n, γ) s (η n 1, γ) s γ L ( s ). (5.7) For any choice o g n, (5.4) (5.5) and (5.6) (5.7) can be solved independently. Thereore, using g n as a control at each time step permits the luid and structure subsystems to be solved simultaneously using partitioned solvers. 5.3 Lagrange Multiplier Rule With the existence o an optimal solution and Lagrange multipliers proven in Chapter 4, we may proceed in deining the Lagrangian and transorming our constrained optimization problem into an unconstrained one. The optimality system below is derived using the Lagrange multiplier rule, treating the deormation and velocity o t as known. Again, it should be noted that when the Fréchet derivative was taken to orm the linearized operator in Theorem 4.8, the determinant o the deormation gradient and the deormation gradient are all implicitly deined by η n and η n 1, but are treated as known. 76

85 We now deine the Lagrangian as L(u n, p n, η n, η n, g n, ū n, p n, η n, η n ) = Jn δ (u n, p n, η n, η n, g n ) + ρ [(u n, ū n ) (u n 1, V(ū n )) t n 1 + ρ [c(u n, u n, ū n ) + 1 ((un n )u n, ū n ) Γ 1 (( zn )u n, ū n ) c(z n, u n, ū n ) + ν a(u n, ū n ) + b(ū n, p n ) ( n, ūn ) (u n, ū n ) Γ (g n, ū n ) ΓI + b(u n, p n ) + ρ s [( η n, η n ) s ( η n 1, η n ) s + ν s d(η n + η n 1, η n ) s + λe(ηn + η n 1, η n ) s ( s n + s n 1, η n ) s (ηn + η n 1, ηn ) Γ s + (V(gn )J + V(g n 1 )J 1, η n ) ΓIt0 + [( ηn, η n ) s + ( η n 1, η n ) s [(η n, η n ) s (η n 1, η n ) s or any (u n, p n, η n, η n, g n, ū n, p n, η n, η n ) H 1 D ( t n ) L ( t n ) H 1 D (s ) L ( s ) L (Γ I ) H 1 D ( t n ) L ( t n ) H 1 D (s ) L ( s ). We will now seek to ind stationary points o L over this product space. Variations in the control g n yield the necessary condition, δ(g n, c) ΓI = (c, ū n ) ΓI + (V(c)J t n, η n ) ΓIt0 c L (Γ I ). (5.8) Since H 1 ( t n ) is compactly embedded in L (Γ I ) and H 1 ( s ) is compactly embedded in L (Γ I ), we can substitute the necessary condition (5.8) into the state equation (5.4) and (5.6) by considering c = v ΓI and c = ξ ΓI which can be substituted or any v H 1 ( t n ) and ξ H 1 ( s ), respectively. Variations in the state variables, Lagrange multipliers, and the control g n along 77

86 with the substitution just described yield the optimality system: ρ [(u n, v) (u n 1, V(v)) + ρ [c(u n, u n, v) t + 1 n 1 ((un n )u n, v) Γ b(u n, q) 1 (( zn )u n, v) c(z n, u n, v) + ν a(u n, v) + b(v, p n ) = ( n, v) + (u n, v) Γ δ (ū n V( ηn ), v) ΓI v H 1 D ( t n ), (5.9) = 0 q L ( t n ), (5.10) ρ s [( η n, ξ) s ( η n 1, ξ) s + ν s d(η n + η n 1, ξ) s + λe(ηn + η n 1, ξ) s = ( s n + s n 1, ξ) s + (ηn + η n 1, ξ) Γ s + δ (V(ūn )J 1 ηn J, ξ) ΓIt0 (V(gn 1 )J 1, ξ) ΓIt0 ξ H 1 D (s ), (5.11) [( ηn, γ) s + ( η n 1, γ) s = (η n, γ) s (η n 1, γ) s γ L ( s ), (5.1) ρ (ū n, w) + ρ [c(u n, w, ū n ) + c(w, u n, ū n ) + 1 ((un n )w, ū n ) Γ b(ū n, r) + 1 ((w n )u n, ū n ) Γ 1 (( zn )w, ū n ) c(z n, w, ū n ) + ν a(ū n, w) + b(w, p n ) = (u n V(ηn ) V(η n 1 ), w) ΓI w H 1 D ( t n ), (5.13) = 0 r L ( t n ), (5.14) ( η n, φ) s + ν s d( η n, φ) s + λe( ηn, φ) s = 1 [ (J t n V(u n ) ηn η n 1, φ) ΓIt0 φ H 1 D (s ), (5.15) ( η n, θ) s + ρ s ( η n, θ) s = 0 θ L ( s ), (5.16) 78

87 5.4 Finite Element Approximations We select inite element spaces W h, Q h, D h, and V h such that W h H 1 D ( t n ), Q h L ( t n ), D h H 1 D (s ), and V h H 1 ( s ). We assume that the inite element spaces satisy the standard approximation properties, i.e., there exists an integer k and a constant C such that in v h W h v vh 1, Ch m v m+1, v H 1 D ( t n ), 1 m k, (5.17) in q h Q h q qh 0, Ch m q m, q L ( t n ), 1 m k, (5.18) in η h D h η η h 1, t 0 Ch m η m+1, t 0 η H 1 D ( t 0 ), 1 m k, (5.19) and in γ h V h γ γ h 0, t 0 Additionally, we assume that the in-sup condition holds, i.e., Ch m γ m, t 0 γ H 1 D ( t 0 ), 1 m k. (5.0) in sup 0 q h Q h 0 v h W h (v h, q h ) v h 1, q h 0, where C is a positive constant independent o h; see [4, 17, 36, 65. C, (5.1) The inite element approximations o solutions o the optimality system can now be deined as: ind (u n,h, p n,h, η n,h, η n,h, ū n,h, p n,h, η n,h, η n,h ) W h Q h D h V h W h Q h D h V H ) such that 79

88 ρ [(u n,h, v h ) (u n 1,h, V(v h )) t n 1 b(u n,h, q h ) + ρ [c(u n,h, u n,h, v h ) + 1 ((un,h n )u n,h, v h ) Γ 1 (( zn,h )u n,h, v h ) c(z n,h, u n,h, v h ) + ν a(u n,h, v h ) + b(v h, p n,h ) = ( n, vh ) + (u n, v h ) Γ δ (ū n,h V( ηn,h ), v h ) ΓI v h W h, (5.) = 0 q h Q h, (5.3) ρ s [( η n,h, ξ h ) s ( η n 1,h, ξ h ) s + ν s d(η n,h + η n 1,h, ξ h ) s + λe(ηn,h + η n 1,h, ξ h ) s = ( s n + s n 1, ξ h ) s + (ηn + η n 1, ξh ) Γ s + δ (V(ūn,h )J 1 ηn,h J, ξ h ) ΓIt0 (V(gn 1,h )J 1, ξ h ) ΓIt0 ξ h D h, (5.4) [( ηn,h, γ h ) s + ( η n 1,h, γ h ) s = (η n,h, γ h ) s (η n 1,h, γ h ) s γ h V h, (5.5) ρ (ū n,h, w h ) + ρ [c(u n,h, w h, ū n,h ) + c(w h, u n,h, ū n,h ) b(ū n,h, r h ) + 1 ((un,h n )w h, ū n,h ) Γ + 1 ((wh n )u n,h, ū n,h ) Γ 1 (( zn,h )w h, ū n,h ) c(z n,h, w h, ū n,h ) + ν a(ū n,h, w h ) + b(w h, p n,h ) = (u n,h V(ηn,h ) V(η n 1,h ), w h ) ΓI w h W h, (5.6) = 0 r h Q h, (5.7) ( η n,h, φ h ) s + ν s d( η n,h, φ h ) s + λe( ηn,h, φ h ) s = 1 [ (J t n V(u n,h ) ηn,h η n 1,h, φ h ) ΓIt0 φ h D h, (5.8) ( η n,h, θ h ) s + ρ s ( η n,h, θ h ) s = 0 θ h V h. (5.9) 80

89 We will now use BRR Theory to obtain error estimates or inite element approximation o the nonlinear optimality system [18. Let X and Y denote Banach spaces and Λ a positive compact interval o R containing λ, λ > 0. We consider a nonlinear problem o the type F (λ, ψ) = ψ + T G(λ, ψ) = 0, (5.30) where G is a C mapping rom Λ X to Y and T L(Y, X). I F (λ, ψ(λ)) = 0 or λ Λ and the mapping λ ψ(λ) is a continuous unction rom Λ to X, then {(λ, ψ(λ)) : λ Λ} is called a branch o solutions o (5.30). Additionally, i the Frèchet derivative D ψ F (λ, ψ(λ)) is an isomorphism rom X into X or all λ Λ, then the branch o solutions {(λ, ψ(λ)) : λ Λ} are called a regular branch. Let X h X be a inite element space and T h L(Y, X h ) be a discretization o the operator T. The approximation problem or (5.30) seeks ψ h X h such that F h (λ, ψ h ) = ψ h + T h G(λ, ψ h ) = 0. (5.31) Assume that there exists a Banach space Z continuously embedded in Y such that D ψ G(λ, ψ(λ)) L(X, Z) λ Λ, ψ X. (5.3) We also assume the ollowing approximation properties or the operator T h : lim (T h T )r = 0 r Y (5.33) X h 0 and lim T h T = 0. (5.34) L(Z,X) h 0 Under these assumptions, we now quote the ollowing theorem o [18: Theorem 5.1. Let X and Y be Banach spaces and Λ a compact interval o the real line R. 81

90 Assume that G is a C mapping rom Λ X into Y and that D G is bounded on all bounded sets o Λ X. Assume that (5.3) (5.34) hold and that {(λ, ψ(λ)) : λ Λ} is a branch o regular solutions o (5.30). Then, there exists a neighborhood O o the origin in X and, or h h 0 small enough, a unique C unction λ ψ(λ) X h such that {(λ, ψ h (λ)) : λ Λ} is a branch o regular solutions o (5.31) and ψ h (λ) ψ(λ) O or all λ Λ. Furthermore, there exists a positive constant C, independent o h and λ, such that ψ(λ) h ψ(λ) C (T h T )G(λ, ψ(λ)) λ Λ. (5.35) X X We will use Theorem 5.1 to in order to obtain error estimates between the optimality system (5.9) (5.16) and the discretized optimality system (5.) (5.9). Let X = (H 1 D ( t n ) L ( t n ) H 1 D (s ) L ( s )), Y = H 1 D ( t n ) H 1/ (Γ ) H 1/ (Γ I ) H 1 D (s ) H 1/ (Γ s ) H 1/ (Γ It0 ) H 1 D (s ) H 1 D ( t n ) H 1/ (Γ I ) H 1 D (s ) H 1/ (Γ It0 ) H 1 D (s ) Z = L 3/ ( t n ) L (Γ ) L (Γ I ) L 3/ ( s ) L (Γ s ) L (Γ It0 ) L 3/ ( s ) L 3/ ( t n ) L (Γ ) L (Γ I ) L 3/ ( s ) L (Γ It0 ) L 3/ ( s ) X h = (W h Q h D h V h ). Deine the operator T L(Y, X) by T (σ 1, σ, σ 3, Φ 1, Φ, Φ 3, Π, ϱ 1, ϱ, ϱ 3, Ξ 1, Ξ, Υ) = (u n, p n, η n, η n, ū n, p n, η n, η n ) i and only i, 8

91 ν a(u n, v) + b(v, p n ) b(u n, q) ν a(ū n, w) + b(w, p n ) = (σ 1, v) + (σ, v) Γ + (σ 3, v) ΓI v H 1 D ( t n ), (5.36) = 0 q L ( t n ), (5.37) ν s d(η n, ξ) s = (Φ 1, ξ) s + (Φ, ξ) Γ s + (Φ 3, ξ) ΓIt0 ξ H 1 D (s ), (5.38) 1 ( ηn, γ) s = (Π, γ) s γ L ( s ), (5.39) b(ū n, r) = (ϱ 1, w) + (ϱ, w) Γ + (ϱ 3, w) ΓI w H 1 D ( t n ), (5.40) = 0 r L ( t n ), (5.41) ν s d( η n, φ) s = (Ξ 1, φ) s + (Ξ, φ) ΓIt0 φ H 1 D (s ), (5.4) 1 ( η n, θ) s = (Υ, θ) s θ L ( s ). (5.43) Likewise, the discrete operator T h L(Y, X h ) is deined by: or (σ 1, σ, σ 3, Φ 1, Φ, Φ 3, Π, ϱ 1, ϱ, ϱ 3, Ξ 1, Ξ, Υ) Y and (u n,h, p n,h, η n,h, η n,h, ū n,h, p n,h, η n,h, η n,h ) X h, T h (σ 1, σ, σ 3, Φ 1, Φ, Φ 3, Π, ϱ 1, ϱ, ϱ 3, Ξ 1, Ξ, Υ)= (u n,h, p n,h, η n,h, η n,h, ū n,h, p n,h, η n,h, η n,h ) i and only i, ν a(u n,h, v h ) + b(v h, p n,h ) b(u n,h, q h ) = (σ 1, v h ) + (σ, v h ) Γ + (σ 3, v h ) ΓI v h W h, (5.44) = 0 q h Q h, (5.45) ν s d(η n,h, ξ h ) s = (Φ 1, ξ h ) s + (Φ, ξ h ) Γ s + (Φ 3, ξ h ) ΓIt0 ξ h D h, (5.46) 1 ( ηn,h, γ h ) s = (Π, γ h ) s γ h V h, (5.47) 83

92 ν a(ū n,h, w h ) + b(w h, p n ) b(ū n,h, r h ) = (ϱ 1, w h ) + (ϱ, w h ) Γ + (ϱ 3, w h ) ΓI w h W h, (5.48) = 0 r h Q h, (5.49) ν s d( η n,h, φ h ) s = (Ξ 1, φ h ) s + (Ξ, φ h ) ΓIt0 φ h D h, (5.50) 1 ( η n,h, θ h ) s = (Υ, θ h ) s θ h V h. (5.51) Let Λ be a compact interval containing 1 δ. Let us deine the nonlinear operator G : Λ X Y by: or (σ 1, σ, σ 3, Φ 1, Φ, Φ 3, Π, ϱ 1, ϱ, ϱ 3, Ξ 1, Ξ, Υ) Y and ( 1 δ, (un, p n, η n, η n, ū n, p n, η n, η n )) Λ X, G( 1 δ, (un, p n, η n, η n, ū n, p n, η n, η n )) = (σ 1, σ, σ 3, Φ 1, Φ, Φ 3, Π, ϱ 1, ϱ, ϱ 3, Ξ 1, Ξ, Υ) i and only i, (σ 1, v) = ρ [(un, v) + (u n 1, V(v)) + ρ [c(u n, u n, v) t n (( zn )u n, v) + c(z n, u n, v) ( n, v) v H 1 D ( t n ), (σ, v) Γ (σ 3, v) ΓI = = ρ 1 ((un n )u n, v) Γ (u n, v) Γ v H 1 D ( t n ), δ (ūn V( ηn ), v) ΓI v H 1 D ( t n ), (Φ 1, ξ) s = ρs [( ηn, ξ) s + ( η n 1, ξ) s + ν s d(η n 1, ξ) s 1 λe(ηn η n 1, ξ) s 1 ( n s + n 1 s, ξ) s ξ H 1 D (s ), (Φ, ξ) Γ s = 1 (ηn + η n 1, ξ) Γ s ξ H1 D (s ), (Φ 3, ξ) ΓIt0 = δ (V(ūn )J 1 ηn J, ξ) ΓIt0 + 1 (V(gn 1 )J 1, ξ) ΓIt0 ξ H 1 D (s ), (Π, γ) s = 1 ( ηn 1, γ) s + 1 [ (η n, γ) s + (η n 1, γ) s γ L ( s ), 84

93 (ϱ 1, w) = ρ (ūn, w) + ρ [c(u n, w, ū n ) + c(w, u n, ū n ) + 1 (( zn )w, ū n ) + c(z n, w, ū n ) w H 1 D ( t n ), (ϱ, w) Γ = ρ [ 1 ((un n )w, ū n ) Γ + 1 ((w n )u n, ū n ) Γ w H 1 D ( t n ) (ϱ 3, w) ΓI = 1 ( u n V(ηn ) + V(η n 1 ) ), w w H 1 D ( t n ) Γ I (Ξ 1, φ) s = 1 ( η n, φ) s 1 λe( ηn, φ) s φ H 1 D (s ), (Ξ, φ) ΓIt0 = 1 (J [V(u n ) ηn + η n 1 ), φ φ H 1 D (s ), Γ It0 (Υ, θ) s = ρs ( ηn, θ) s θ L ( s ). ow, the optimality system (5.9) (5.16) is equivalent to (u n, p n, η n, η n, ū n, p n, η n, η n ) + T G( 1 δ, (un, p n, η n, η n, ū n, p n, η n, η n )) = 0, and the discrete optimality system (5.) (5.9) is equivalent to (u n,h, p n,h,η n,h, η n,h, ū n,h, p n,h, η n,h, η n,h ) + T h G( 1 δ, (un,h, p n,h, η n,h, η n,h, ū n,h, p n,h, η n,h, η n,h )) = 0. ow the optimality system and the discrete optimality system have the orm o (5.30) and (5.31), respectively. In the next theorem, we set λ = 1 δ to simpliy the notation. Theorem 5.. Assume that {(λ, ϕ(λ)) = (u n, p n, η n, η n, ū n, p n, η n, η n ) : λ Λ} is a branch o regular solutions o the optimality system (5.9) (5.16), where Λ is a compact interval in R. Assume that W h, Q h, D h, and V h satisy (5.17) (5.1). Then or h h 0 small enough, there exists a unique branch o solutions o the discrete optimality system (5.) (5.9), {(λ, ϕ h (λ)) = (u n,h, p n,h, η n,h, η n,h, ū n,h, p n,h, η n,h, η n,h ) : λ Λ}, such that 85

94 φ φ h O, a neighborhood about the origin in X. Additionally, as h 0 uniormly in X, ϕ(λ) ϕ h (λ) = X u n (λ) u n,h (λ) 1, + p n (λ) p n,h (λ) 0, + η n (λ) η n,h ū (λ) + n (λ) ū n,h (λ) 0, s + η n (λ) η n,h (λ) + 1, s η n (λ) η n,h (λ) 1, 0, s + η n (λ) η n,h (λ) 1, s + p n (λ) p n,h (λ) 0, 0. (5.5) With the additional assumption that (u n (λ), p n (λ), η n (λ), η n (λ), ū n (λ), p n (λ), η n (λ), η n (λ)) (H m+1 ( t n ) H m ( t n ) H m+1 ( s ) H m ( s )), there exists a constant C, independent o h, such that u n (λ) u n,h (λ) 1, + p n (λ) p n,h (λ) 0, + η n (λ) η n,h ū (λ) + n (λ) ū n,h (λ) 0, s + η n (λ) η n,h (λ) + 1, s η n (λ) η n,h (λ) + η n (λ) η n,h (λ) 1, 0, s 1, s + p n (λ) p n,h (λ) 0, Ch m [ u n (λ) m+1, + p n (λ) m, + η n (λ) m+1, s + η n (λ) m, s + ū n (λ) m+1, + p n (λ) m, + η n (λ) m+1, s + η n (λ) m, s (5.53) Proo. Λ is compact and G is a C mapping rom Λ X into Y. Thereore D G is bounded on all bounded set o Λ X by bound (.16). G ϕ, the Fréchet derivative o G, is deined by: or (ṽ, q, ξ, γ, w, r, φ, θ) X, G ϕ (λ, (u n, p n, η n, η n, ū n, p n, η n, η n )) (ṽ, q, ξ, γ, w, r, φ, θ) = ( σ 1, σ, σ 3, Φ 1, Φ, Φ 3, Π, ϱ 1, ϱ, ϱ 3, Ξ 1, Ξ, Υ) i and only i, ( σ 1, v) = ρ (ṽ, v) + ρ [c(ṽ, u n, v) + c(u n, ṽ, v) + 1 (( zn )ṽ, v) + c(z n, ṽ, v) v H 1 D ( t n ), 86

95 ( σ, v) Γ ( σ 3, v) ΓI = δ = ρ [((ṽ n )u n, v) Γ + ((u n n )ṽ, v) Γ v H 1 D ( t n ), V( φ) ( w, v) Γ I v H 1 D ( t n ), ( Φ 1, ξ) s = ρs ( γ, ξ) s 1 λe( ξ, ξ) s ξ H 1 D (s ), ( Φ, ξ) Γ s = 0 ξ H 1 D (s ), ( Φ 3, ξ) ΓIt0 = δ (V( w)j t n 1 φj, ξ) ΓIt0 ξ H 1 D (s ), ( Π, γ) s = 1 ( ξ, γ) s γ L ( s ), ( ϱ 1, w) ( ϱ, w) Γ = ρ ( w, w) + ρ [c(ṽ, w, ū n ) + c(u n, w, w) + c(w, ṽ, ū n ) + c(w, u n, w) + 1 (( zn )w, w) + c(z n, w, w) w H 1 D ( t n ), = ρ [((ṽ n )w, ū n ) Γ + ((u n n )w, w) Γ + ((w n )ṽ, ū n ) Γ + ((w n )u n, w) Γ w H 1 D ( t n ) ( ) ( ϱ 3, w) ΓI = 1 ṽ V( ξ), w w H 1 D ( t n ) w H 1 D ( t n ) Γ I ( Ξ 1, φ) s = 1 ( θ, φ) s 1 λe( φ, φ) s φ H 1 D (s ), [ ( Ξ, φ) ΓIt0 = (J 1 V(ṽ) ξ ), φ φ H 1 D (s ), Γ It0 ( Υ, θ) s = ρs ( φ, θ) s θ L ( s ). Thereore, D ϕ G(λ, (u n, p n, η n, η n, ū n, p n, η n, η n )) L(X, Y ). By the Sobolev embedding theorem [1, u n, u n, ṽ, and w L 6 ( t n ), and u n Γ, u n Γ, ṽ Γ and w Γ L 4 (Γ ). Additionally, and un, u n, ṽ, and w L ( t n ). As a result o these embeddings, ( σ 1, σ, σ 3, Φ 1, Φ, Φ 3, Π, ϱ 1, ϱ, ϱ 3, Ξ 1, Ξ, Υ) Z and thereore we can write that D ϕ G(λ, (u n, p n, η n, η n, ū n, p n, η n, η n )) L(X, Z). Also, note that Z Y is a compact embedding. Considering the operators T and T h, equations (5.36) (5.37), (5.38), (5.39), (5.40) 87

96 (5.41), (5.4), and (5.43) are all uncoupled rom one another. This is likewise the case or (5.44) (5.45), (5.46), (5.47), (5.48) (5.49), (5.50), and (5.51). Using well known results or Stokes low, Poisson s equation, and projections, the dierence in solutions can be shown to be û n (λ) û n,h (λ) 1, + ˆp n (λ) ˆp n,h (λ) 0, + ˆ η n (λ) ˆ η n,h (λ) + 0, s ˆū n (λ) ˆū n,h (λ) + ˆη n (λ) ˆη n,h (λ) 1, + ˆ η n (λ) ˆ η n,h (λ) + 1, s ˆ η n (λ) ˆ η n,h (λ) 0 0, s 1, s + ˆ p n (λ) ˆ p n,h (λ) 0, as h 0 uniormly, where (û n, ˆp n, ˆη n, ˆ η n, ˆū n, ˆ p n, ˆ η n, ˆ η n ) is a solution to (5.36) (5.43) and (û n,h, ˆp n,h, ˆη n,h, ˆ η n,h, ˆū n,h, ˆ p n,h, ˆ η n,h, ˆ η n,h ) is a solution to (5.44) (5.51); see [65. This means that (5.33) holds and we have already seen that D ϕ G(λ, (u n, p n, η n, η n, ū n, p n, η n, η n )) L(X, Z) where Z Y is compactly embedded, so (5.34) also holds. This satisies all o the assumptions o Theorem 5.1. Also, we have the approximation result or the Stokes operator and projections, namely that û n (λ) û n,h (λ) 1, + ˆp n (λ) ˆp n,h (λ) 0, + ˆ η n (λ) ˆ η n,h (λ) + 0, s ˆū n (λ) ˆū n,h (λ) + ˆ η n (λ) ˆ η n,h (λ) + 1, s ˆ η n (λ) ˆ η n,h (λ) Ch m [ û n (λ) m+1, + ˆp n (λ) m, + ˆū n (λ) m+1, + ˆ p n (λ) m, + ˆη n (λ) ˆη n,h (λ) 1, 0, s 1, s + ˆ p n (λ) ˆ p n,h (λ) 0, + ˆη n (λ) m+1, s + ˆ η n (λ) m, s + ˆ η n (λ) m+1, s + ˆ η n (λ) m, s or some constant C independent o h. I it is also true that (u n (λ), p n (λ), η n (λ), η n (λ), ū n (λ), p n (λ), η n (λ), η n (λ)) (H m+1 ( t n ) H m ( t n ) 88

97 H m+1 ( s ) H m ( s )), then (T T h )G(λ, ϕ(λ) X Ch m [ u n (λ) m+1, + p n (λ) m, + η n (λ) m+1, s + η n (λ) m, s + ū n (λ) m+1, + p n (λ) m, + η n (λ) m+1, s + η n (λ) m, s. From this, the error estimate (5.53) ollows. 5.5 Convergence o Steepest Descent Using the steepest descent method described in Section 4.9, it will be shown that a sequence o solutions generated by this algorithm converge on an optimal solution i the time step is suiciently small and the strong solution to the PDE is suiciently smooth. Theorem 5.3. Let X be a Hilbert space equipped with the inner product (, ) X and norm X. Suppose M is a unctional on X such that 1. M has a local minimum at ˆx and is twice dierentiable in an open ball B centered at ˆx;. M (u), (x, y)) m a x X y X, u B, x X, y X; 3. M (u), (x, x)) m b x X, u B, x X, where m a and m b are positive constants. Let R denote the Riesz map, i.e., x = R, x X or all x X and all X. Choose x (0) suiciently close to ˆx and choose a sequence {ω n } such that 0 < ω ω n ω < m b /m a. Then, the sequence {x (n) } deined by x (n) = x (n 1) ω n RM (x (n 1) ), or n = 1,,... (5.54) converges to ˆx. Furthermore, i B = X and ˆx is a global minimum, then the sequence generated by (5.54) converges to ˆx or any initial guess x (0). 89

98 Proo. See, e.g., [4. Theorem 5.4. Let (u n,h (m), pn,h (m), ηn,h (m), ηn,h (m), ūn,h (m), pn,h (m), ηn,h (m), η n,h (m)) be a sequence obtained by Algorithm 4.11 and (u n,h, p n,h, η n,h, η n,h, ū n,h, p n,h, η n,h, η n,h ) be a solution o the optimality system (5.) (5.9), where M is unctional (5.1). I α m 4 δ Cδh 8 1 C, h is suiciently small, and g n,h (0) gn,h < 1 where g n,h is a local minimum, then u n,h (m) un,h, p n,h (m) 0,ΓI p n,h, η n,h (m) ηn,h, η n,h (m) ηn,h, ū n,h (m) ūn,h, p n,h (m) pn,h, η n,h (m) ηn,h, η n,h (m) η n,h, as m. Proo. For a given g n L (Γ I ), the second Fréchet derivative D M(ũ n ( g n ), η n ( g n ), g n ) is deined by D M(ũ n ( g n ), η n ( g n ), g n ) (ḡ 1, ḡ ) = ( ū 1 η 1, ū η ) + (ũ Γ I + δ(ḡ 1, ḡ ) ΓI, ) η ηn 1 η, ū Γ I (5.55) where ũ is the solution o ρ [(ũ, v) (u n 1, v) + ρ [c(ũ, ũ, v) + 1 ((ũ n )ũ, v) Γ 1 (( zn )ũ, v) c(z n, ũ, v) + ν a(ũ, v) + b(v, p) = ( n, v) + (u n, v) Γ + ( g n, v) ΓI v H 1 D ( t n ), b(ũ, q) = 0 q L ( t n ), (5.56) 90

99 and ( η, η) is the solution o ρ s [( η, ξ) s ( η n 1, ξ) s + ν s d( η, ξ) s + λe( η, ξ) s = ( n s + s n 1, ξ ) + s ( η n + η n 1, ξ) Γ s (V( gn )J + V(g n 1 )J 1, ξ) ΓIt0 ξ H 1 D (s ), [( η, γ) s + ( η n 1, γ) s = ( η, γ) s (η n 1, γ) s γ H 1 D (s ). (5.57) The irst variations ū i, i = 1,, are the solutions o ρ (ū i, v) b(ū i, q) + ρ [c(ū i, ũ i, v) + c(ũ i, ū i, v) + 1 ((ū i n )ũ i, v) Γ + 1 ((ũ i n )ū i, v) Γ 1 (( zn )ū i, v) c(z n, ū, v) + ν a(ū, v) + b(v, p i ) = (ḡ i, v) ΓI v H 1 D ( t n ), = 0 q L ( t n ), (5.58) and the irst variations ( η i, η i ), i = 1,, are the solutions o ρ s ( η i, ξ) s + ν s d( η i, ξ) s + λe( η i, ξ) s = (V(ḡ i)j, ξ) ΓIt0 ξ H 1 D (s ), (5.59) ( η i, γ) s ( η i, γ) s = 0 γ H 1 D (s ). 91

100 The second variation ū is a solution o ρ (ū, v) + ρ [c(ū, ũ, v) + c(ũ, ū, v) + 1 ((ū n )ũ, v) Γ + 1 ((ũ n )ū, v) Γ 1 (( zn )ū, v) c(z n, ū, v) + ν a(ū, v) + b(v, p) (5.60) b(ū, q) = ρ [c(ū 1, ū, v) + c(ū, ū 1, v) + 1 ((ū 1 n )ū, v) Γ + 1 ((ū n )ū 1, v) Γ = 0 q L ( t n ), v H 1 D ( t n ), and the second variation ( η, η) is a solution o ρ s ( η, ξ) s + ν s d( η, ξ) s + λe( η, ξ) s = 0 ξ H 1 D (s ), ( η, γ) s ( η, γ) s = 0 γ H 1 D (s ). (5.61) Using Theorems and making the same assumptions, ũ 0, ũ 1, C 14 ( g n 0,ΓI + K 1 ), (5.6) 1 C14 ( g n 0,ΓI + K 1 ), (5.63) η 1, s C 15 ( g n 0,ΓI + K ), (5.64) ū i 0, ū i 1, 1 C16 ḡ i 0,ΓI or i = 1,, (5.65) C 16 ḡ i 0,ΓI or i = 1,, (5.66) η i 1, s C 17 ḡ i 0,ΓI or i = 1,, (5.67) and η η n 1 η + η n 1 0, s C 18 ( g n 0,ΓI + K 3 ). (5.68) 0, s For ease o notation, all trial and test unctions that ollow should be considered as 9

101 discrete inite element approximations. Use will be made o the inverse inequality, which requires a discrete setting. Using the trace theorem and (5.68), we can now bound D M(ũ n ( g n ), η n ( g n ), g n ) (ḡ 1, ḡ ) [ δ ḡ 1 0,ΓI ḡ 0,ΓI + C19 ū 1 1 ū 0, 1 1 ū 1, 1 ū 0, 1 1, and +C 19C 0 ū 1 1 ū 0, 1 1 1, [ + ū 1 ū 1 0, 1, η C19 ũ 1 ũ 1 0, 1, 1, s + C19C0 ū 1 0, ū 1 1, η + η n 1 +C 19C 1 1 0, s η 1 η η n 1 1, s + C 0 1 1, s η 1 η 1, s 1, s D M(ũ n ( g n ), η n ( g n ), g n ) (ḡ 1, ḡ 1) [ [ δ ḡ 1 0,Γ I ū 1 ū 1 0, 1, C19 ũ 1 0, ũ 1 1, η + η n 1 +C 19C 1 1 0, s η η n 1 1 1, s, where C 19, C 0, and C 1 are positive domain dependent constants which come rom the trace theorem and Poincaré Friedrich s inequality. ote that η 0,ΓI = 0 by considering (5.61) with ξ = η and γ = η; this is why these terms do not appear in the inequality. Substituting in (5.6) (5.68) and using that η n 1 1, s C, D M(ũ n ( g n ), η n ( g n ), g n ) (ḡ 1, ḡ ) [[ δ + C16C C16C17C19C0 + 1 C 17C0 ḡ 1 0,ΓI ḡ 0,ΓI + ū 1 ū 1 0, 1, [ 1 4 C14C 19( g n 0,ΓI + K 1) + 1 C19C 1 [C 18( g n 0,ΓI + K 3)(C 15( g n 1 0,ΓI + K ) + C ) (5.69) and [ [ D M(ũ n ( g n ), η n ( g n ), g n ) (ḡ 1, ḡ 1) δ ḡ 1 0,Γ I ū 1 ū 1 0, 1 1, 4 C14C19( g n 0,ΓI + K 1) + 1 C19C 1 [C 18( g n 0,ΓI + K 3)(C 15( g n 1 0,ΓI + K ) + C ), (5.70) where C 14,...,C 17 are positive constants described above. 93

102 ow, supposing that we can ind a bound or ū such that ū 1 ū 1 0, 1, < k C ḡ 1 0,ΓI ḡ 0,ΓI and ū 1 ū k 0, 1, < 1 C ḡ1 0,Γ I or k > 1, then we will have that D M(ũ n ( g n ), η n ( g n ), g n ) (ḡ 1, ḡ ) C ḡ 1 0,ΓI ḡ 0,ΓI and D M(ũ n ( g n ), η n ( g n ), g n ) (ḡ 1, ḡ 1 ) C ḡ 1 0,Γ I or any choice o g n in L (Γ I ) and suiciently small time step. Let us consider (5.60) with the choice o test unctions (v, q) = (ū, p) and immediately apply Young s inequality and use that z n W 1, : ū 0, [ρ ρ zn, + C 3 ν ū 1, + ρ [c(ū, ũ, ū) + 1 ((ū n )ũ, ū Γ + 1 ((ũ n )ū, ū) Γ = ρ [c(ū 1, ū, ū) + c(ū, ū 1, ū) + 1 ((ū 1 n )ū, ū) Γ + 1 ((ū n )ū 1, ū) Γ v H 1 D ( t n ). Here, C 3 is the constant based on Poincaré Friedrich s inequality which bounds C 3 ū 1, ū 0,. For all terms o the orm c(u, v, w), we can initially bound them rom above by c(u, v, w) C 4 [ u 1 u 1 0, v 1, 1, w 1 w 1 0, 1, + u 1 u 1 0, w 1, 1, v 1 v 1 0, 1,. (5.71) On the let hand side o the inequality, using Holder s inequality with p=4, q=4, and r=, the Sobolev imbedding o W 1, (Γ ) W0,4 (Γ ), and the trace theorem, we bound ((ū n )ũ, ū) Γ C 5 ū 1 ū 3 0, ũ 1, 1, (5.7) Additionally, noting that 94

103 C 4 [ ū 0, ũ 1, ū 1, + ū 1 ũ 0, 1, ū 3 1, c(ū, ũ, ū) we can see that all trilinear terms on the let hand side o the inequality can be bounded above by, ρ [c(ū, ũ, ū) + 1 ((ū n )ũ, ū) Γ + 1 ((ũ n )ū, ū) Γ ρ [C 4 ū 0, ū 1, ũ 1, + (C 4 + C 5 ) ū 1 ū 3 0, ũ 1, 1, [ (ρ C 4 ) ũ C 3 ν 1, ū 0, + (ρ (C 4 + C 5 )) 4 4( 3 C ũ 4 3ν ) 3 1, ū 0, + C 3 ν ū 1, using Young s inequality with p=, q= and p=4, q= 4 3, respectively. For the second application o Young s inequality, this also requires multiplying by 1 in the orm o ( 3 C 3 ν ρ (C 4 +C 5 ) ( 3 C 3 ν ρ (C 4 +C 5 ) ext, we apply the inverse inequality and introduce the constant C 6 under the assumption that the mesh is quasi-uniorm, yielding ) 3 4 ) 3 4. ρ [c(ū, ũ, ū) + 1 ((ū n )ũ, ū) Γ [ C 6 h (ρ C 4 ) ũ C 3 ν 0, + 1 ((ũ n )ū, ū) Γ ū 0, + C4 6 h 4 (ρ (C 4 + C 5 )) 3 4( 3 C ũ 4 3ν ) 3 0, ū 0, + C 3 ν ū 1,. For terms on the right hand side o the inequality, again using Holder s inequality with p=4, q=4, and r=, the Sobolev imbedding o W 1, (Γ ) W0,4 (Γ ), and the trace theorem, ρ [ ((ū 1 n )ū, ū) Γ + ((ū n )ū 1, ū) Γ ρ C 5 ū 1 1, ū 1, ū 1 ū 1 0,. 1, 95

104 We apply Young s inequality with p = 4 3 and q = 4, ater multiplying by ( ) 1 C3 ν 4 C 5 ρ ( ) 1, to get C3 ν 4 C 5 ρ ρ [ ((ū 1 n )ū, ū) Γ + ((ū n )ū 1, ū) Γ C 3ν ū 4 1, + 3(C 5ρ ) 4 3 4(C 3 ν ) 1 3 ū ū 0, 4 3 ū 1, 3. 0, ow, we apply Young s inequality again with p = 3 and q = 3, and multiply by unity in the orm o 1 6, 1 6 ρ [ ((ū 1 n )ū, ū) Γ + 3(C 5ρ ) 4 3 4(C 3 ν ) ((ū n )ū 1, ū) Γ [ 1 ū 3 0, C 3ν ū 1, ū 1 1, ū 1,. ext we apply the inverse inequality, using ū 1 1, inverse inequality constant, to get C 7 h 1 ū 1 0,, where C 7 is the ρ [ ((ū 1 n )ū, ū) Γ + ((ū n )ū 1, ū) Γ C 3ν 4 + (C 5ρ ) 4 3 4(C 3 ν ) 1 3 ū 1, [ 1 ū 0, + C 7 h ū1 0, ū 1 1, ū 1,. Since t n R, as in (5.71), we have that ρ c(ū i, ū j, ū) ρ C 4 [ ū i 1 ū 0, i 1 ū 1, j 1, ū 1 0, ū 1 + ū i 1 ū 0, i 1 ū 1, 1, ū j 1 ū 0, j 1 1, 1,. ext, or the irst term we multiply by unity in the orm o inequality with p = 4 3 ( ) 1 C3 ν 4 C 4 ρ ( ) 1 C3 ν 4 C 4 ρ and apply Young s and q = 4, while or the second term we multiply by unity in the 96

105 orm o ( C3 ν 4C 4 ρ ) 1 ( C3 ν 4C 4 ρ ) 1 and then apply Young s inequality with p = and q = : ρ c(ū i, ū j, ū) [ 3(C 4 ρ ) 4 3 4( 1 C 3ν ) (C 4ρ ) C 3 ν ū i 3 ū 0, i 3 ū 1, j 4 3 ū 1, 3 + C 3ν ū 0, 8 1, ū i 0, ū i 1, ū j 0, ū j 1, + C 3ν ū 8 1,. Again we must apply Young s inequality with p = 3 and q = 3, multiplying by 1 in the orm o 6 1, to get 6 1 ρ c(ū i, ū j, ū) [ (C 4 ρ ) 4 3 4( 1 C 3ν ) (C 4ρ ) C 3 ν [ 1 4 ūi 0, ū i 1, ū j 1, + 1 ū 0, ū i 0, ū i 1, ū j 0, ū j 1, + C 3ν ū 4 1,. Substituting in the inequalities or terms on the let hand side, ū 0, [ρ ρ zn, C6 (ρ C 4 ) h C 3 ν h 4 C 4 6 (ρ (C 4 + C 5 )) 3 4( 3 C 3ν ) 3 ũ 4 0, ũ 0, + C 3 ν ū 1, ρ [c(ū 1, ū, ū) + c(ū, ū 1, ū) + 1 ((ū 1 n )ū, ū) Γ + 1 ((ū n )ū 1, ū) Γ. 97

106 ext, we make a substitution or terms on the right hand side, ū 0, [ρ ρ zn, C6 (ρ C 4 ) h ũ C 3 ν 0, C6 4 (ρ (C 4 + C 5 )) 3 h 4 4( 3 C ũ 4 3ν ) 3 0, 1 + C 3ν h + ū 1, C 7 (C 5 ρ ) 4 3 (C 3 ν ) 1 3 [ i,j=1,i j 5 4 (C 4 ρ ) 4 3 ( 1 C 3ν ) 1 3 ū 1 0, ū 1 1, ū 1, (C 4 ρ ) 4 3 ( 1 C 3ν ) (C 4ρ ) C 3 ν ū i 0, ū i 1, ū j 1, 1 (C 5 ρ ) 4 3 4(C 3 ν ) 1 3 ū i 0, ū i 1, ū j 0, ū j 1, Finally, substituting in (5.6) (5.63) and (5.65) (5.66), we have ū 0, [ ρ ρ zn, C6 (ρ C 4 ) h C C 3 ν 14( g n 0,ΓI + K 1 ) h 4 C 4 6 (ρ (C 4 + C 5 )) 3 4( 3 C 3ν ) 3 C 4 14( g n 0,ΓI + K 1 ) 4 1 (C 4 ρ ) 4 3 ( 1 C 1 (C 5 ρ ) 4 3 3ν ) 1 3 4(C 3 ν ) 1 3 C 4 16 ḡ 1 0,Γ I ḡ 0,Γ I [ 7 4 h C 7 (C 5 ρ ) 4 (C 3 ν ) C 3ν 4 (C 4 ρ ) 4 3 ( 1 C 3ν ) 1 3 ū 1, + 4(C 4ρ ) C 3 ν which yields the ollowing inequalities or a suiciently small : ū 1 0, ū 1 1, C 7 16 h 1 4 C 3 16 h 1 4 ( ḡ 1 0,ΓI ḡ 0,ΓI ) 1 (5.73) ( ḡ 1 0,ΓI ḡ 0,ΓI ) 1. (5.74) 98

107 Thereore, ū 1 ū 1 0, 1, 5 8 h 1 C (5.75) and we have continuity and coercivity o the second Fréchet derivative o the penalized unctional i < h 5 and is suiciently small. Let us make the assumption that the ball B rom which we make our initial guess has a radius o one. In the context o Theorem 5.3, our m a δ + C and m b δ C 1 8 δ Ch 1 8, based on (5.69), (5.70), and (5.75). Thereore, max{δ, C } m a, which means that (δ Ch 1 8 ) max{4δ,4c 4 } m b m a. We consider small values o δ, which in general is the case o interest. Thereore we concern ourselves only with the case δ C. In the steepest descent algorithm, ω n has the orm αm δ. According to Theorem 5.3, the algorithm is guaranteed to converge i αm δ (δ Ch 1 8 ), which implies that α 4C 4 m 4 δ Cδh 1 8 C suicient condition or convergence i h is small enough and the initial guess is close enough to the optimal solution. h 1 is a 5.6 umerical Results For an FSI problem which was presented by Astorino and Grandmont in [3, we will perorm computations and give rates o convergence to the true solution over a single time step. However, we use the avier Stokes operator or the luid rather than the Stokes operator. As in [3, we make the assumption o ininitesimal displacement o the structure and also o the luid domain, but with nonnegligible velocity o the interace. The densities o the luid and structure are similar, which adds the complication o having the added mass eect [1, 34. All terms including z will be dropped since z = 0 and z = 0 in the Eulerian ramework. This also means that the control will not be able to absorb the 1 (((un z n ) n ) u n, v) ΓI term and so the term 1 ((un n ) u n, v) ΓI will appear in the luid state equations as well as its respective derivatives in the adjoint equations. 99

108 Γ s D Γ s D s Γ s D Γ I0 Γ Γ Γ D Figure 5.1: Computational domain or a manuactured solution. Parameters or the problem are: ρ = 1.0 g/cm 3, ν = g/cm s, ρ s = 1.9 g/cm 3, ν s = 3 dyne/cm, and λ = 4.5 dyne/cm. Initial conditions, body orces, and boundary conditions are determined by the analytical solution according to the method o manuactured solutions: On t = = [0, 1 [0, 1 and s = [0, 1 [1, 1.5 (Fig. 5.1), u 1 = cos(x + t) sin(y + t) + sin(x + t) cos(y + t), u = sin(x + t) cos(y + t) cos(x + t) sin(y + t), p = ν (sin(x + t) sin(y + t) cos(x + t) cos(y + t)) + ν s cos(x + t) sin(y + t), η 1 = sin(x + t) sin(y + t), η = cos(x + t) cos(y + t). (5.76) We have used a uniorm mesh. We ix = 10 6 s, and begin two simulations; one at t 0 = 0.5 s and the other at t 0 = 0.8 s. We perorm the simulation over one time step in each case. The quadrilateral inite element pair, (Q, Q 1 ), were used or solutions on the luid domain, while Q inite elements were used to approximate the structure displacement and velocity. The FSI problem was repeatedly solved by Algorithm 4.11 using increasingly 100

109 ine spatial discretizations so as to see the rate at which the computed result converges upon the true solution (5.76). For Algorithm 4.11, δ = 10 4 and ɛ tol = All computations were made using the deal.ii inite element library [7, 8. Table 5.1: Fluid velocity and pressure convergence results or a single time step using the steepest descent algorithm at t = 0.5 s. h u n u true L Rate Rate Rate u n u true H 1 p n p true L 1/ e e e-01-1/ e e e / e e e / e e e / e e e / e e e Table 5.: Structure displacement and velocity convergence results or a single time step using the steepest descent algorithm at t = 0.5 s. h x h y η n η true L Rate η n η true H 1 Rate Rate η n η true L 1/9 1/ e e e-06-1/14 1/ e e e /1 1/ e e e /31 1/ e e e /46 1/ e e e /69 1/ e e e It is observed in Tables that we have ull theoretical spatial convergence or better upon the true solution, c. (5.53). Using (Q, Q 1 ) or the luid velocity and pressure, we expect second order convergence in the H 1 norm o the luid velocity and the L norm o the pressure. With Q elements used or the structure displacement and velocity, we expect second order convergence in the H 1 norm o the structure displacement and the L norm o the structure velocity. Because we are unable to get an H 1 bound on η n, we were orced to minimize the 101

110 Table 5.3: Fluid velocity and pressure convergence results or a single time step using the steepest descent algorithm at t = 0.8 s. h u n u true L Rate u n u true H 1 Rate p n p true L Rate 1/ e e e-01-1/ e e e / e e e / e e e / e e e / e e e Table 5.4: Structure displacement and velocity convergence results or a single time step using the steepest descent algorithm at t = 0.8 s. h x h y η n η true L Rate η n η true H 1 Rate η n η true L Rate 1/9 1/ e e e-06-1/14 1/ e e e /1 1/ e e e /31 1/ e e e /46 1/ e e e /69 1/ e e e unctional (5.1) rather than (5.). Despite not having the analytical support, in Tables we compute the convergence rate using the penalized unctional (5.) and optimizing over all time steps. Here we use the conjugate gradient algorithm, as given in Section 3.5, to ind the optimal solution at each time step. It is observed here that there is no loss o accuracy in space or time rom using our approach by optimization. 5.7 Conclusion Ater recasting the luid-structure interaction problem into an optimal control problem, the optimality system was derived. The optimality system was then rewritten in terms o a linear and nonlinear operator to which the BRR theory was applied [18. Finite element spaces were deined, and the existence o a solution to this system was shown along with 10

111 Table 5.5: Fluid velocity and pressure convergence results using the conjugate gradient algorithm rom t = 0.5 to t = 1.0 s. h u n u true L (L ) Rate un u true L (H 1 ) Rate pn p true L Rate 1/6 1/ e e e-03-1/9 1/7.6513e e e /14 1/ e e e /1 1/ e e e /31 1/ e e e /46 1/ e e e Table 5.6: Structure displacement and velocity convergence results using the conjugate gradient algorithm rom t = 0.5 to t = 1.0 s. h x h y η n η true L (L ) Rate ηn η true L (H 1 ) Rate η n η true L (L ) Rate 1/4 1/16 1/ e e e-03-1/6 1/4 1/8.0519e e e /9 1/36 1/ e e e /14 1/54 1/ e e e /1 1/81 1/ e e e /31 1/1 1/ e e e spatial approximation estimates to the discretized in time, continuous in space, solution. ext, an algorithm or optimization by steepest descent was outlined along with a proo o convergence o the algorithm. A numerical study was made based on a known analytical solution. It assumed ininitesimal displacement o the luid domain, with nonnegligible velocity on the interace. Full spatial convergence was observed, demonstrating that there was no spatial degradation o the solution over a single time step. Additional numerical results included show that using a unctional lacking proos o existence or an optimal solution and Lagrange multipliers, ull order convergence was observable in both space and time over all simulations or the problem with a known analytical solution. Chapter 6 extends this work by consider luid-structure interaction in the case o a ewtonian luid and a nonlinear elastic solid. This approach by optimization shows great promise o eiciently decoupling highly nonlinear FSI problems. 103

112 Part II avier-stokes / onlinear St. Venant-Kirchho Elasticity 104

113 Chapter 6 APPLICATIO TO OLIEAR ELASTICITY 6.1 Introduction An investigation is made o employing the optimization-based algorithm developed in this thesis to decouple and solve nonlinear nonsteady luid-structure interaction. The constitutive equation or elasticity coupled with a ewtonian luid in luid-structure interaction is modeled here by the St. Venant Kirchho hyperelastic model. Motivation is irst given or the advantages o modeling FSI using a nonlinear elastic material. A derivation is next presented or relating the St. Venant Kirchho constitutive equations to the amiliar gradient o the elastic displacement ield. The equations are transormed to the reerence coniguration, where appropriate. A linearization o the state equations are given which deine the inner optimization problem to be solved as part o the Gauss ewton iterations presented in Chapter 3. Several numerical studies have been perormed comparing the simulation results using the optimization-based algorithm against an implicit partitioned accelerated Aitken s approach. The irst problem this has been applied to is a case o two dimensional FSI with nonlinear elasticity [54. The second results are rom a three dimensional pulsatile low 105

114 through a straight cylinder [19 and they include a record o optimization iterations needed as the cylinder is reined. Using the number o Gauss ewton and state solves, a study is made o the computational workload relative to a orward solve with a known traction orce. 6. onlinear Elasticity Remaining consistent with previous notation, let us denote the displacement o the elastic material as η, and the rest coniguration as X. Thereore the physical coniguration is x = X + η. The St. Venant Kirchho model is an improvement over the linear elasticity model or modeling blood low. The artery and vessel walls are sot tissue, which generally undergo large displacements. Large displacements obviously violate the small displacement assumption made in deriving the linear elasticity model. St. Venant Kirchho is the most basic nonlinear elastic model, but is oten used in numerical simulations using inite elements because o its ease o implementation relative to other nonlinear elastic models. In posing the linear elasticity equations, one makes the assumption that η T η 0 in the strain tensor, X η x η, and that J = det(i + x η T ) 1. The St. Venant Kirchho model makes none o these assumptions. It is oten reerred to as the large displacement-small strain model. Only partial existence results exist [3, p. 99 or this model primarily since nothing prevents det(f) rom becoming zero or even negative. The Jacobian that results rom the displacement, generally denoted F, is deined as: F i,j = x i X j = 1[i = j + η i X j = I i,j + η j,i = [I + η T i,j or F = I + η T. The deormation tensor locally characterizes the dierence between current and reerence 106

115 coniguration. Strain should not take into account rotations and translations, i.e., rigidbody motions. George Green deined what has come to be known as the right Cauchy Green deormation tensor: C i,j = k F k,i F k,j or equivalently, C = F T F. This measure o strain is symmetric and positive deinite and is invariant to rigidbody motions. The Green St. Venant strain tensor is deined as: E = 1 [C I, where C is the right Cauchy Green deormation tensor, and is a measure dierence between the deormed material modulo rigid-body motions and the reerence coniguration. This strain tensor is also symmetric and positive deinite. The strain-energy density unction describes the energy per unit o volume stored by the elastic structure due to its deormation. For the St. Venant Kirchho equations, this unction is W (E) = λ T r(e) + ν s T r(e ), (6.1) where T r( ) is the trace operator, λ is Lamé s irst parameter, and ν s is the shear modulus. Dierentiating this strain-energy density unction (6.1) with respect to the strain tensor E gives the second Piola Kirchho stress, also called the material stress: Σ = W (E) E = λ T r(e)i + ν s E The second Piola Kirchho stress is related to the Cauchy stress, also called the true stress, through the stress-strain relation σ = 1 J FΣFT. (6.) With deormation, strain, and stress tensors deined, we are now prepared to present the 107

116 equations or nonlinear elasticity having derivatives in the deormed coniguration and use appropriate substitutions to return them to the reerence domain. The equations or elasticity are ρ d η dt (X) J x σ = J s in s or equivalently, ρ d η dt J x σ = J s in s (6.3) η dη dt = 0 in s (6.4) The Piola transorm [3, pp allows us to pull back the divergence operator or a tensor or vector as J x v = X (JF 1 v) and J x σ = X (JσF T ). Using the Piola transorm, we can rewrite the elasticity equations (6.3) (6.4) as ρ d η dt X (JσF T ) = J s in s η dη dt = 0 in s. ext, we apply the relationship between the Cauchy and Piola Kirchho stress tensors (6.) to get ρ d η dt X (FΣ) = J s in s (6.5) η dη dt = 0 in s, (6.6) and FΣ is the irst Piola Kirchho stress tensor. In order to make connection with the Lagrangian mapping clear, here substitutions 108

117 are made into the stress tensor to the granularity o η. FΣ = F[λ T r(e)i + ν s E [ ( ) 1 1 = F λ T r [C I I + ν s [ ( ) 1 = F λ T r [FT F I [ = [I + X η T λ T r [C I I + ν s 1 [FT F I ( 1 [[I + X ηt T [I + X η T I ) I + ν s 1 [[I + X ηt T [I + X η T I (6.7) Temporal discretization o the luid subsystem (.34) (.35) by implicit Euler and o the variational ormulation o the o the strong orm o the structure subsystem (6.5) (6.6) by a second order midpoint scheme yields: ind (u n, p n, η n, η n ) H 1 D ( t n ) L ( t n ) H 1 D (s ) L ( s ) such that ρ [(u n, v) (u n 1, V(v)) t n 1 [ + ρ ((u n z n ) u n, v) (( z n )u n, v) [ + ν a(u n, v) + b(v, p n ) b(u n, q) (ν D(u n ) n p n n, v) ΓIt [ = ( n, v) + (u n, v) Γ v H 1 D ( t n ), (6.8) = 0 q L ( t n ), (6.9) and ( ρ s η n η n 1, ξ ) + s = ( F n Σ n + F n 1 Σ n 1, X ξ ) ([ F n Σ n + F n 1 Σ n 1 n s, ξ ) Γ It0 [ ( n s + s n 1, ξ ) + ( η n s + η n 1, ξ) ξ H 1 Γ s D (s ), (6.10) ( η n + η n 1, γ ) = ( η n η n 1, γ ) γ L ( s ), (6.11) s s 109

118 and satisying the interace continuity equations u n Ψ = η n on Γ It0 (6.1) and [[ν D(u n ) p n I n Ψ = FΣ n s on Γ It0. (6.13) 6.3 Description o the Optimization Problem Since [[ν D(u n ) p n I n Ψ = FΣ n s on Γ It0, these terms can be substituted rom the variational ormulation o the FSI problem in (6.8) (6.11) with a control g n deined on the reerence domain Γ It0, ensuring the continuity o traction orce or any choice o the control. In order to ind a choice o control in L ( t 0 ) that enorces continuity o velocity, we seek to minimize the penalized unctional J n (u n, η n, g n ) = 1 u n V( η n ) dγ + ɛ g n dγ, (6.14) Γ I Γ I subject to ρ [(u n, v) b(u n, q) (u n 1, V(v)) t n 1 [ + = ν a(u n, v) [ + ρ ((u n z n ) u n, v) (( z n )u n, v) + b(v, p n ) [ (V(g n ), v) ΓI + ( n, v) + (u n, v) Γ v H 1 D ( t n ), (6.15) = 0 q L ( t n ), (6.16) 110

119 and ( ρ s η n η n 1, ξ ) + s = ( F n Σ n + F n 1 Σ n 1, X ) [ ( g n + g n 1, ξ ) Γ It0 + ( n s + n 1 s + ( η n + η n 1, ξ) Γ s, ξ ) s ξ H 1 D (s ), (6.17) ( η n + η n 1, γ ) s = ( η n η n 1, γ ) s γ L ( s ). (6.18) 6.4 Linearization o the First Piola Kirchho Stress Tensor We now linearize the structure equations with respect to η in order to get each ewton iteration update to solve the nonlinear PDE. Solving this problem will have the orm K (η) (η (k) η (k 1) ) = K(η (k 1) ), where K is the elasticity operator (6.5) (6.6). Focus is irst placed on linearizing the irst Piola Kirchho stress tensor (6.7), denoted S, since all other terms in the structure equations are linear. S (η) (φ) = (FΣ) η (φ) [ λ = [I + X η T (k 1) ( ) T r X φ[i + X η T (k 1) + [I + X ηt (k 1) T X φ T I [ +ν s X φ[i + X η T (k 1) + [I + X ηt (k 1) T X φ T ( ) 1 + X φ [λ T T r [[I + X ηt (k 1) T [I + X η T (k 1) I I 1 + ν s [[I + X ηt (k 1) T [I + X η T (k 1) I (6.19) Using this deinition or the linearized irst Piola Kirchho stress tensor, we can substitute η (k) η (k 1) in place o φ to get the stress tensor s contribution or a ewton iteration. 111

120 S (η) (η (k) η (k 1) ) = [I + X η T (k 1) [ λ T r ([ X η T (k) X ηt (k 1) T [I + X η T (k 1) +[I + X η T (k 1) T [ X η T (k) X ηt (k 1) ) I +ν s [[ X η T (k) X ηt (k 1) T [I + X η T (k 1) +[I + X η T (k 1) T [ X η T (k) X ηt (k 1) ( ) 1 + [ X η T (k) X ηt (k 1) [λ T r [[I + X ηt (k 1) T [I + X η T (k 1) I I 1 + ν s [[I + X ηt (k 1) T [I + X η T (k 1) I Adding the contribution o (FΣ)(η (k 1) ) rom the right hand side to the let hand side gives that (FΣ) (η) (η (k) η (k 1) ) = (FΣ)(η (k 1) ) is equivalent to [I + X η T (k 1) [ λ T r ([ X η T (k) X ηt (k 1) T [I + X η T (k 1) +[I + X η T (k 1) T [ X η T (k) X ηt (k 1) ) I +ν s [[ X η T (k) X ηt (k 1) T [I + X η T (k 1) +[I + X η T (k 1) T [ X η T (k) X ηt (k 1) ( ) 1 + [I + X η T (k) [λ T r [[I + X ηt (k 1) T [I + X η T (k 1) I I + ν s 1 [[I + X ηt (k 1) T [I + X η T (k 1) I = 0. (6.0) Substituting the let hand side o (6.0) into the structure equation (6.17) in place o FΣ results in the ewton iteration operator or the structure subproblem. For the linearized problem which is used or a linear least squares solve in the Gauss ewton algorithm, the operator or (6.17) (6.18) is linearized in the orm o K (η) (φ), 11

121 resulting in solving: ind (w, r, η, ϕ) H 1 D ( t n ) L ( t n ) H 1 D (s ) L ( s ) such that ρ (w, v) + ρ [c(u n, w, v) + c(w, u n, v) + 1 ((un n )w, v) Γ b(w, q) + 1 ((w n )u n, v) Γ 1 (( zn )w, v) c(z n, w, v) + ν a(w, v) + b(v, r) = (h, v) ΓI v H 1 D ( t n ), (6.1) = 0 q L ( t n ), (6.) ρ s (ϕ, ξ) s + (S (η) (θ), x ξ) s = (h, ξ) Γ It0 ξ H 1 D (s ), (6.3) (ϕ, γ) s (θ, γ) s = 0 γ L ( s ). (6.4) where h is a unction determined by the optimization routine selected, and S (η) (θ) is deined in (6.19). 6.5 umerical Results Haemodynamic Experiment The irst numerical experiment we study plots the vertical displacement o the structure at three locations on the interace. We revisit the problem that was described in Section 4.10, but this time using the St. Venant Kirchho equations as the constitutive equation or the elastic structure. A comparison is made between a sequentially staggered Dirichlet eumann approach augmented by Aitken s relaxation and our optimization-based approach. For more details on sequentially staggered approaches, see [30. The plots in Figure 6. demonstrate the contrast between using linear and the St. Venant Kirchho constitutive equation or the elastic structure, in order to emphasize the signiicant dierence in response between a linear and nonlinear elastic as well as the agreement between the optimization 113

122 approach and the relaxed sequentially staggered method. η D = 0 η = 0 s = [0, 6 [1, 1.1 η D = 0 u = b(t) 0 = [0, 6 [0, 1 Γ I0 u = 0 u D = 0 Figure 6.1: Domain and boundary conditions or numerical experiment A orce b(t) is applied to the let luid boundary (Fig. 6.1) at t s where ( 10 3 (1 cos πt.05 b(t) = ), 0) dyne/cm, t 0.05 (0, 0), 0.05 < t < T. The unction b(t) deines the stress on the inlet denoted by u in (.4). The volume orce or the luid and structure are (t) = (0, 0) dyne/cm. The other boundary conditions on the domain coniguration are homogeneous Dirichlet or eumann (Fig. 6.1), and the simulation begins at rest. The reerence domain or the luid subsystem has height 1 cm and length 6 cm. The density o the luid, ρ, is 1 g/cm 3 and the viscosity o the luid, ν, is g/cm s. The structure domain has height 0.1 cm and length 6 cm. The density o the structure, ρ s, is 1.1 g/cm 3. The Young s Modulus o the structure, E, is dyne/cm and its Poisson ratio, ν, is 0.3. Spatial discretization in the x direction is h x = 0. cm and in the y direction is h y = 0.1 cm or both luid and structure domains on a uniorm quadrilateral mesh. The simulation was perormed with = 10 4 s rom T = 0 s to T = 0.1 s. Computations were perormed in deal.ii [7,8 using the tensor product (Q, Q 1 ) inite element pair or the luid, and tensor product Q elements or the structure. The stopping criteria used or Aitken s ( ) 1 relaxation was (η n ΓIt0 (k) ηn (k 1) ) dγ < 10 14, while δ = 10 4 and ɛ tol = or the Gauss-ewton Algorithm 3.3 adapted to the state and linearized equations (6.15) (6.18) and (6.1) (6.4). In Figure 6.1, close agreement can be observed between solutions computed using the 114

123 Displacement (cm) (1.5,1.0) 0.1 (1) 0.05 () (3) 0 (4) (1) () (3) (4) (3.0,1.0) Time (seconds) (1) () (3) (4) (4.5,1.0) Figure 6.: Vertical displacement at three points on the interace using (1) optimization and () Aitken s relaxation with the St. Venant Kirchho constitutive equation and (3) optimization and (4) Aitken s relaxation with the linear elastic constitutive equation. optimization-based approach presented in this thesis and the Aitken relaxed sequentially staggered Dirichlet eumann coupling approach. The dierence in the response o the vertical displacement o the structure to the linear or nonlinear elastic physics is pronounced D Pulsatile Flow Through a Cylinder Figure 6.3: Computational domain or 3D pulsatile low through a cylinder. In three dimensions, a pressure driven low through a cylinder is commonly simulated numerically [19, 9, 34, 49. In this setting, a luid modeled by the avier Stokes equations is in contact with an elastic solid modeled by the St. Venant Kirchho equations. The luid has parameters µ = poise, ρ = 1 g/cm 3, in an initially straight vessel o radius

124 cm and length 5 cm. The structure has parameters ρ s = 1. g/cm 3, E = 3.0e+6 dynes/cm, and ν = 0.3, with a surrounding structure thickness o 0.1 cm. There is an overpressure on the inlet boundary o 1.333e+4 dynes/cm or s. The inlet and outlet boundaries are clamped, i.e., no displacement. The simulation step size is = 10 4 s. All parameters exactly match those used in [19. In Figure 6.4, snapshots are taken o the luid pressure on the moving luid domain with the domain deormation scaled by a actor o 10. The enveloping elastic structure is clipped and the deormation is scaled by the same actor. Snapshots are included or our time steps using the tensor product, LBB deicient (Q 1, Q DC 0 ) inite element pair or the luid velocity and pressure, and using tensor product Q 1 elements or the ALE mesh and structure velocity and displacement. A quantity o interest when applying optimization to a problem is how the number o optimization iterations needed or convergence respond to increasingly reining the computational domain. For this sequence o computations perormed on increasingly reined meshes, the tensor product Q 1 elements are used or the luid velocity, luid pressure, mesh update, structure displacement, and structure velocity in the deal.ii inite element library. This luid velocity and pressure pair do not satisy the Ladyzhenskaya Babuska Brezzi (LBB) condition, and so they are stabilized by a stabilization method or low-order mixed inite elements [13. Work Factor = Fluid Solves (Total) + Gauss ewton Iterations Fluid Solves (Stress Determined) (6.5) In the case o CPU wall times, it is possible that certain accelerations may be added to a code and omitted rom another, intentionally or unintentionally, so that when compared one will outperorm the other. In order to prepare as air o an estimate as possible or the computational complexity o our optimization-based algorithm, a metric is proposed which will compare the computational eort as a multiplier o the eort to perorm a orward 116

125 (a) t=0.005 s (b) t= s (c) t= s (d) t= s Figure 6.4: Snapshots o luid pressure and scaled solid deormation (by a actor o 10) using (Q 1, Q DC 0 ) elements or the luid pressure and velocity, Q 1 elements or structure and mesh updates. solve, had the correct boundary conditions on the interace been known a priori. In Table 6.1, the average number o Gauss ewton iterations per time step are listed along with the average number o GMRES solves needed per Gauss ewton iteration. The work actor is determined using deinition (6.5). The rationale or this metric is based on the act that in every Gauss ewton iteration, a sequence o GMRES iterations must be perormed, but it is only the irst iteration that requires matrix assembly and actorization. Ater this irst iteration, the actorization can be cheaply reused and so the primary eort lies in the initial actorization which is approximately the same as one nonlinear state solve iteration. For this reason, the computational eort or the GMRES solves are represented in the Work Factor ormula by two times the number o Gauss ewton iterations. 117