Computational Methods in Biomechanics and Physics

Computational Methods in Biomechanics and Physics

A Dissertation Presented to the Faculty of the Department of Mathematics University of Houston

In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

By Serguei Lapin May 2005

Computational Methods in Biomechanics and Physics

Serguei Lapin APPROVED:

Dr. Roland Glowinski, Chairman

Dr. Suncica Canic,

Dr. Tsorng -Whay Pan,

Dr. Doreen Rosenstrauch, Department of Cardiology, University of Texas Health Science Center

Dr. John L. Bear Dean, College of Natural Sciences and Mathematics

ii

ACKNOWLEDGMENTS I would like to express my deep gratitude and appreciation to my advisor, Dr. Roland Glowinski, for the significant impact he has had on me as an example of excellence in research and teaching. As my mentor, the support, guidance and patience he has provided for me during the last six years has enabled me to succeed in this venture where I might ˇ c otherwise have failed. I also would like to sincerely thank my co-advisor Dr. Suncica Cani´ for introducing me to the fascinating area of biomathematics. She has been a tremendous source of help during my time at the University of Houston. Her friendship and confidence in me has proved invaluable time and again. I have received an innumerable amount of sage advice and suggestions from Dr. T. -W. Pan and Dr. G. Guidoboni. I am indebted to them for their time and efforts on my behalf. I must also extend my thanks to Dr. Y. Kuznetsov for his advice and overall guidance throughout the course of my graduate studies. I am most thankful to my committee members for taking time from their busy schedules to be part of my dissertation defense. I would also like to thank all of my friends for their support and understanding during these few years as I have labored to finish this dissertation. I would extend a special thanks to Cynthia Chmielewski for her help in editing this manuscript. Finally, I am deeply indebted to and whole-heartedly thank, my family: my parents, brother and grandparents. I would have been lost without their unconditional love and support. If they had not had such confidence in me, I never would have gotten this far.

iii

Computational Methods in Biomechanics and Physics.

An Abstract of a Dissertation Presented to the Faculty of the Department of Mathematics University of Houston

In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

By Serguei Lapin May 2005

iv

ABSTRACT This dissertation considers the numerical solution of linear and non-linear wave propagation problems and numerical modeling of blood flow in compliant vessels. We consider the solution of the linear wave equation with discontinuous coefficients by a domain decomposition method. A time explicit finite difference scheme combined with a piecewise linear finite element method in space is used on semimatching grids. We apply a boundary supported Lagrange multiplier technique on the interface between the subdomains. We examine the numerical solution of the wave scattering problem via a fictitious domain method. We use a mixed variational formulation to construct a discrete problem, which is explicit in time along with a boundary supported Lagrange multipliers to enforce the boundary condition on the interface. The next problem deals with the solution of a vector-valued constrained nonlinear wave equation. We use a methodology based on the penalty treatment of the constraint. The numerical scheme combines the use of a continuous piecewise linear finite element approximation along with an implicit in time discretization. The stability and uniqueness results are provided for the forementioned problems . The goal of the next problem is to develop an efficient methodology for the numerical simulation of blood flow in bifurcated human arteries. The model is derived using an asymptotic reduction of the Navier-Stokes equations to model the blood, with Navier equation for the wall. We use Riemann invariants analysis in order to describe the continuity of pressure and momentum at the bifurcation point. We calculate shear stress rates for several different prostheses data and obtain results that indicate high shear stress rates in graft limbs where thrombosis is typically observed. Based on our findings, we suggest the optimal design of endovascular prostheses for endoluminal treatment of an aortic abdominal aneurysm. v

Contents 1 Introduction

1

1.1

Numerical methods for solving partial differentail equations . . . . . . . .

1

1.2

Fictitious domain methods . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Domain decomposition methods . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Absorbing boundary conditions . . . . . . . . . . . . . . . . . . . . . . . .

5

1.5

Fluid-structure interaction: hemodynamics applications . . . . . . . . . . .

6

1.6

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2 A Lagrange multiplier based domain decomposition method for wave propagation in heterogeneous media

10

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2

Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4

Fully discrete scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5

Energy inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6

Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Solution of a wave equation by a mixed finite element - fictitious domain method 37 3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2

Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 38

vi

3.3

Discretization of the problem. Energy inequality . . . . . . . . . . . . . . . 40

3.4

A fictitious domain method with boundary supported Lagrange multiplier . 44

3.5

Numerical experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 A penalty approach to the numerical simulation of a constrained wave motion 53 4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2

Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3

Discretization of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4

Solution method for the fully discrete problem . . . . . . . . . . . . . . . 65

4.5

Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5 Mathematical modeling of blood flow in compliant arteries 5.1

5.2

71

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.1.1

Background of the problem . . . . . . . . . . . . . . . . . . . . . 71

5.1.2

Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . 73

Effective model derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2.1

The 3-D fluid-structure interaction problem . . . . . . . . . . . . . 73

5.2.2

Weak formulation and a priori estimates . . . . . . . . . . . . . . . 76

5.2.3

Asymptotic reduction of the model . . . . . . . . . . . . . . . . . . 79

5.2.4

The reduced two-dimensional problem . . . . . . . . . . . . . . . 81

5.2.5

Derivation of the one-dimensional reduced model . . . . . . . . . . 82

5.3

Conditions at the bifurcation point . . . . . . . . . . . . . . . . . . . . . . 83

5.4

Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.5

Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Bibliography

100

vii

List of Figures 2.1

Computational domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2

Semimatching mesh on γ. . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3

Space Λ is the space of the piecewise constant functions defined on every union of half-edges with common vertex. . . . . . . . . . . . . . . . . . . 16

2.4

Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom) meters. Incident wave coming from the left. . . . . . . . . . . . . 23

2.5

Energy behavior versus time for L = 0.5 (top) and L = 0.25 (bottom).

. . 24

2.6

Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom) meters. Incident wave coming from the left. . . . . . . . . . . . . 25

2.7

Energy behavior versus time for L = 0.5 (top) and L = 0.25 (bottom).

. . 26

2.8

Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom) meters. Incident wave coming from the lower left corner with an angle of 45 degrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.9

Contour plot of the real part of the solution for L = 0.5 meters. . . . . . . . 29

2.10 Contour plot of the real part of the solution for L = 0.25 meters. . . . . . . 29 2.11 Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom). Incident wave coming from the left. . . . . . . . . . . . . . . . . 30 2.12 Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom). Incident wave coming from the lower left corner with an angle of 45 degrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 viii

2.13 Energy behavior for L = 0.5 meters. . . . . . . . . . . . . . . . . . . . . . 32 2.14 Energy behavior for L = 0.25 meters. . . . . . . . . . . . . . . . . . . . . 32 2.15 Obstacle in a form of an airfoil with a coating. . . . . . . . . . . . . . . . . 33 2.16 Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom). Incident wave coming from the left. . . . . . . . . . . . . . . . . 34 2.17 Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom). Incident wave coming from the left lower corner with a 45 degree angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.18 Energy behavior versus time for L = 0.5 (top) and L = 0.25 (bottom).

. . 36

3.1


3.2

Degrees of freedom for p . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3

Shaded rectangulars are boundary elements Tγ . . . . . . . . . . . . . . . . 45

3.4

Initial function u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5

Domain without an obstacle, time=0.5. . . . . . . . . . . . . . . . . . . . . 48

3.6

Domain without an obstacle, time=0.7. . . . . . . . . . . . . . . . . . . . . 49

3.7

Domain without an obstacle, time=1. . . . . . . . . . . . . . . . . . . . . . 49

3.8

Domain with a circular obstacle, time=0.2. . . . . . . . . . . . . . . . . . . 50

3.9

Domain with a circular obstacle, time=0.4. . . . . . . . . . . . . . . . . . . 50

3.10 Domain with a circular obstacle, time=0.6. . . . . . . . . . . . . . . . . . . 51 3.11 Domain with a circular obstacle, time=0.8. . . . . . . . . . . . . . . . . . . 51 3.12 Domain with a circular obstacle, time=1. . . . . . . . . . . . . . . . . . . . 52 3.13 Energy dissipation for ∆t = 1/250, h = 1/50 (dashed line) and ∆t = 1/500, h = 1/100 (solid line). . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1

Energy behavior (left) and max ||u|2 − 1| (right) for ε = 10−3 , h = x

1/50, ∆t = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

ix

4.2


1/50, ∆t = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3


1/50, ∆t = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.4

u3 component of u at t = 1/4 (left) and t = 1/2 (right) for ε = 10−3 , h = 1/50, ∆t = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5

u3 component of u at t = 3/4 (left) and t = 1 (right) for ε = 10−3 , h = 1/50, ∆t = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6

u3 component of u at t = 1/4 (left) and t = 1/2 (right) for ε = 10−5 , h = 1/50, ∆t = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.7

u3 component of u at t = 3/4 (left) and t = 1 (right) for ε = 10−5 , h = 1/50, ∆t = 1/100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.1

Abdominal aorta with and without aneurysm. . . . . . . . . . . . . . . . . 72

5.2


5.3

Blood flow through abdominal aorta (A.A) and iliac arteries (I.A.(1) and I.A.(2)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4

Aneurysmal abdominal aorta and iliac arteries with inserted bifurcated stent. 88

5.5

Oscillatory Shear Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.6

AneuRx bifurcated stent. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.7

Oscillatory Shear Index for AneuRx stent-graft with limb diameter equal to 12 mm and main body diameter 23 mm. . . . . . . . . . . . . . . . . . . 91

5.8

Oscillatory Shear Index for AneuRx stent-graft with limb diameter equal to 16 mm and main body diameter 28.5 mm. . . . . . . . . . . . . . . . . . 92

5.9

Oscillatory Shear Index for Wallstent stent with limb diameter equal to 14.5 mm and main body diameter 20 mm. . . . . . . . . . . . . . . . . . . . . . 93

x

5.10 Shear Stress Rates for AneuRx stent-graft with limb diameter equal to 12 mm and main body diameter 23 mm. . . . . . . . . . . . . . . . . . . . . . 94 5.11 Shear Stress Rates for AneuRx stent-graft with limb diameter equal to 16 mm and main body diameter 28.5 mm. . . . . . . . . . . . . . . . . . . . . 95 5.12 Oscillatory Shear Index for ”optimal” stent with limb diameter equal to 20 mm and main body diameter 28.5 mm. . . . . . . . . . . . . . . . . . . . . 96 5.13 Shear Stress Rates for the AneuRx stent-graft with a limb diameter equal to 14.5 mm and a main body diameter 28.5 mm. . . . . . . . . . . . . . . . 97 5.14 Shear Stress Rates for the AneuRx stent-graft with a limb diameter equal to 14.5 mm and a main body diameter 23 mm. . . . . . . . . . . . . . . . . 98 5.15 Shear Stress Rates for the Wallstent endoprosthesis with a limb diameter equal to 20 mm and a main body diameter of 28 mm. . . . . . . . . . . . . 98 5.16 Shear Stress Rates for an ”optimal” stent with a imb diameter equal to 20 mm and a main body diameter of 28.5 mm. . . . . . . . . . . . . . . . . . 99 5.17 Shear Stress Rates for the AneuRx stent-graft with a limb diameter equal to 20 mm and a main body diameter of 28.5 mm. . . . . . . . . . . . . . . 99

xi

Chapter 1 Introduction The numerical modeling of many physical phenomena leads to the solution of differential and integral equations. Using computer-implemented mathematical models it is possible to analyze complicated systems in science and engineering. An important aspect of many physical areas is the ability to simulate accurately wave phenomena in bounded and unbounded domains. For most applications closed form solutions of the underlying partial differential equations (PDEs) either do not exist or are intractable; therefore, numerical simulation is the only way to solve these PDEs. One of the applications where numerical mathematics has been focusing on recently is biomechanics. In particular, the modeling of circulatory system of human beings is of a great interest nowadays. The numerical modeling of blood flow leads to the so-called fluid structure interaction problems.

1.1 Numerical methods for solving partial differentail equations There are many different approaches for the numerical approximation of partial differential equations. Some of the most popular schemes are the finite difference based ones, and also 1

finite volume and finite element methods [1, 16, 93]. The theory of finite difference method can be traced back to the fundamental theoretical paper by Courant, Friedrichs and Lewy [15]. The intensive development of the finite difference methods began at the end of the 1940s and the beginning of the 1950s. It was stimulated by the necessity to deal with the complex problems of science and technology. Major works of that time belong to J. von Neumann [61] and F. John [40]. Some of the main contributors to the development of finite difference methods in 1950s and 1960s were J. Douglas [20], A. A. Samarskii [74, 73] and O. Widlund [91, 92]. At the end of this period the theory for initial boundary value problems in one space dimension was reasonably well developed and complete. On the contrary, the situation for multidimensional problems on general domains was much less satisfactory. One of the main reasons is the fact that finite difference methods rely on the values of the solution at the points of a uniform mesh, which do not necessarily fit the domain. In order to deal with this complication scientists developed methods based on variational formulations of the boundary value problems. One of these approaches, the Finite Element Method, is better suited for complex geometries. The first efforts to use piecewise continuous functions defined over triangular domains appear in applied mathematics literature with the work of Courant [16] in 1943. The actual solution of plane stress problems by means of triangular elements was first introduced by M. J. Turner and R. W. Clough in the presentation given at a meeting on Aeronautical Sciences in New York in 1953, this work was published in 1956 [83]. The current name Finite Element Method appeared only in 1960 in the paper by R. W. Clough [14]. Since its early development the Finite Element Method has become one of the most popular variational methods for solving partial differential equations. Many different variants and extensions of the classical Finite Element Method have been developed . Let us mention here, as examples, fundamental contributions in this area by Babuˇska and Strouboulis [2], Brezzi and Fortin [8] and Zienkiewicz [93]. 2

1.2 Fictitious domain methods Fictitious domain methods have a long history; to the best of our knowledge they were first introduced by Hyman [38]. Fictitious domain methods were then discussed in works by Saul’ev [75, 76] (who in fact coined the term ”fictitious domain”), and by Buzbee, Dorr, George and Golub [10]. The basic principle of these methods is the following: suppose that we have to solve a problem in a domain whose shape is complicated, so we are unable to use standard finite difference methods directly. One way to overcome this difficulty is to use finite element methods. Another way is to use a fictitious domain method which will allow us to enjoy the advantages of finite difference methods, e.g. simplicity of discretization and existence of fast numerical solvers. The idea is to extend a problem taking place on a geometrically complex domain to a larger domain of a simpler shape. Many different realizations of the fictitious domain concept exist. Fictitious domain methods for Dirichlet problems with boundary conditions enforced via Lagrange multiplier are discussed in detail in the work by Glowinski [27]. These methods use structured regular meshes over the extended domain. An example of a non-Lagrange multiplier based fictitious domain method can be found in the immersed boundary method by Peskin [65, 66, 67] and Peskin and McQueen [68] for the simulation of incompressible viscous flow in the presence of elastic moving boundaries. Auxiliary Domain Method is another example of the fictitious domain method; it uses a penalty as an alternative to Lagrange multipliers [48], [57], [31]. The works by Kuznetsov and Matsokin [45], Kuznetsov [44] and Finogenov and Kuznetsov [24] also contributed substantially to the field of fictitious domain methods.

3

1.3 Domain decomposition methods Among the methods providing a good and relatively easy way to implement numerical algorithms on complex domains while having a very rich mathematical foundation are Domain Decomposition Methods (DDM). The origin of the Domain Decomposition Methods extends back to the XIX century when H.A. Schwarz introduced the so-called alternating method bearing his name [77]. Nowadays DDM are quite popular. The main goal of DDM is to look for the solution of partial differential equations defined on domains with complicated geometries. DDM’s allow to decompose a given problem into subproblems with simpler and smaller geometries. This approach allows the use of different solvers on each subdomain along with some procedure (iterative in general) to match solutions at the interfaces. Some of the main reasons making DDM so popular are as follows: • DDM are well suited to parallel computations, • allow the use of the different numerical schemes for each subdomain, • have solid theoretical foundation, • they can be combined with other methods like local mesh refinement and multigrid methods. Conferences devoted to Domain Decomposition Methods are held regularly; they allow the presentation of new results and applications. The most recent conferences in this series took place in Berlin in 2003 [42] and in New York in 2005. While the first domain decomposition algorithms were mostly developed and analyzed for linear, self-adjoint, positive definite elliptic model problems with two subdomains, current applications of the DDM field concerns more complicated problems with more than two subdomain decompositions. 4

In his work H.A. Schwarz relied on the maximum principle to prove convergence results. In 1936 S. L. Sobolev gave a variational formulation based proof allowing the use of the Schwarz method without the maximum principle [81]. This approach was extended by P.L. Lions for the case of multiple subdomains. [50, 51, 52].

1.4 Absorbing boundary conditions The effective modeling of waves on unbounded domains by numerical methods depends on the particular boundary condition used to truncate the computational domain. One type of boundary condition used for this purpose is radiating or absorbing boundary conditions (ABC) [3, 22]. The quest for an ABC that produces negligible reflections has been, and continues to be, an active area of research. Most of the popular ABC’s can be classified as those that are derived from differential equations or those that employ an absorbing material. Differential-based ABC’s are generally obtained by factoring the wave equation and allowing a solution which permits only outgoing waves. Material-based ABC’s, on the other hand, are constructed so that fields are dampened as they propagate into an absorbing medium. Other techniques which may be used are exact formulations and superabsorption. Early techniques used to truncate the computational domain have included differentialbased ABC’s, such as those proposed by Merewether [56], Engquist and Majda [22], Lindman [47], and Mur [59]. These early techniques were considerably improved in the mid-1980’s by formulations proposed by Higdon [35, 36] and Keys [41]. Many other extensions of these differential-based ABC’s have been introduced since then. The idea of using a material-based absorbing boundary condition has existed for some time [37]. However, early material ABC’s did not provide a sufficiently low level of boundary reflection, because the characteristic impedance of the material boundary was matched to the characteristic impedance of free space only at normal incidence. In the early 1990’s, Rappaport et al. [70, 71] proposed a new ABC named the sawtooth anechoic chamber 5

ABC. This ABC employs pyramidally-shaped absorbing materials similar to those which are often found in anechoic chambers. The most recent advance in material-based ABC’s was put forward by Berenger [6]. His ABC, termed the perfectly matched layer (PML) absorbing boundary condition, appears to yield a major improvement in the reduction of boundary reflections.

1.5 Fluid-structure interaction: hemodynamics applications The dynamic interaction between a fluid and a structure is a mechanism that is seen in many different physical phenomena: plane wings, suspended and cable-stayed bridges or tall buildings, vibrations in water pipes induced by a water hammer and, among others, blood flow in arteries. The common aspect in all these phenomena is the energy exchange between the fluid, be it air or water, and the structure; fluid and structure dynamics influence each other. The structure deforms under the load of a fluid, and the fluid follows the displacement of the structure. In the case where the deformation of the structure can be neglected, one can take a fixed domain and the presence of a structure can be treated by appropriate boundary conditions. In this case standard approaches and numerical approximation schemes can be employed. For those situations where the structure displacement is non neglegible, one has to solve the fluid equations in a moving domain. This is the case in hemodynamics, where the diameter of an artery may vary up to 10% of its unstressed state. Analytical solutions are available only to very simple fluid-structure interaction problems. Therefore, in general, different numerical approaches have to be employed in order to solve the problem. Let us mention here the Arbitrary Lagrangian Eulerian formulation [19], the space time approach [55], coupled finite element-boundary element technique [39, 23], the fictitious domain method [31, 32] and the immersed boundary method 6

[65, 68]. One of the fluid-structure interaction problems having received a lot of attention in recent years is the problem arising in hemodynamics applications. In large arteries, the mechanical interaction between blood and the arterial wall plays an important role in the control of blood flow in the circulatory system. In large arteries blood can be considered a homogeneous and incompressible fluid, and thus can be modeled using the Navier-Stokes equations. One approach to describe the behavior of the arterial wall is to use the Navier equation for the linearly elastic membrane [69, 60, 53]. This equation describes the ”effective response” of the arterial walls to the forces induced by the pulsatile blood flow. The mathematical and numerical study of the underlying fluid-structure interaction is a difficult task. Various numerical methods have been successfully proposed to study fluid-structure interactions arising in cardiovascular problems, see e.g. [27, 64, 69]. Nevertheless, they are rather complicated and time consuming whenever large three-dimensional sections of the cardiovascular system are simulated. This is why simplified, one dimensional effective models are introduced. These models have been used, for example, in the computation of blood flow through the aorta and coupled to 3D and lumped models to simulate the entire human vascular system [25]; using a ”structured tree” to model the ”arterial tree” [62]; assuming variable Young’s modulus this model has been used in [84, 86, 87] to study the properties of blood flow and the optimal design of stents (prostheses) in the endovascular treatment of an aortic abdominal aneurysm; in [4] a variable Young’s modulus with the one-dimensional equations have been implemented to study endovascular treatment of a stenosis.

1.6 Thesis outline Chapter 2 is dedicated to the solution of a linear wave equation with discontinuous coefficients by domain decomposition. We use an explicit in time finite difference scheme 7

combined with a piecewise linear finite element method in space on semimatching grids. We apply a boundary supported Lagrange multiplier technique on the interface between subdomains. The resulting ”saddle-point” system of linear equations is solved by a conjugate gradient method. In Chapter 3 we consider the numerical solution of the scattering problem via a fictitious domain method. The main advantage of this method is that a problem can be discretized on a uniform mesh, independent of the obstacle boundary. This approach also involves the use of boundary supported Lagrange multipliers to enforce the boundary condition on the boundary of the obstacle. We use a mixed variational formulation to construct a discrete problem, which is explicit in time. The resulting system of linear algebraic equations is of the ”saddle-point” type and is solved effectively using a conjugate gradient method. We also show the stability estimate and prove the uniqueness of the solution. In Chapter 4 we discuss the numerical solution of a vector-valued constrained nonlinear wave equation. The numerical methodology relies on the penalty treatment of the constraint and on an energy preserving time discretization leading to a scheme which is essentially unconditionally stable and second order accurate. We use a globally continuous piecewise linear finite element approximation for space discretization of the problem. At each time step of the time discretization we use a quasi-Newton method to solve the resulting nonlinear elliptic system. We also provide a proof of the existence and uniqueness of the solution. Numerical simulation of blood flow in medium-to-large arteries is considered in Chapter 5. As we already mentioned, the blood in large arteries is modeled by the Navier-Stokes equations and we use the Navier equation for the elastic membrane to model the arterial wall behavior. Since the numerical simulation of the resulting system is quite involved and costly we apply asymptotic reduction to obtain a set of reduced 1D equations. We employ Riemann invariants to derive the conditions at the bifurcation point of the artery. The second order Lax-Wendroff scheme is employed for the numerical solution of the resulting 8

model. Based on our numerical results we present suggestions for the optimal design of the bifurcated prosthesis. The results of Chapter 3 and Chapter 4 were published [30, 28, 29]. The main results of Chapter 5 were accepted for publication [87].

9

Chapter 2 A Lagrange multiplier based domain decomposition method for wave propagation in heterogeneous media 2.1 Introduction The main goal of this chapter is to address the numerical solution of a wave equation with discontinuous coefficients by a finite element method using domain decomposition and semimatching grids. A wave equation with absorbing boundary conditions is considered, the coefficients in the equation essentially differ in the subdomains. The problem is approximated by an explicit in time finite difference scheme combined with a piecewise linear finite element method in the space variables on a semimatching grid. The matching condition on the interface is taken into account by means of Lagrange multipliers. The resulting system of linear equations of the saddle-point form is solved by a conjugate gradient method.

10

Ω R γ

Ω2

Γext

Figure 2.1: Computational domain.

2.2 Problem formulation Let Ω ⊂ R2 be a rectangular domain with sides parallel to the coordinate axes and boundary Γext (see Fig. 2.1). Now let Ω2 ⊂ Ω be a proper subdomain of Ω with a curvilinear ¯ 2 . We consider the following linear wave problem: boundary and Ω1 = Ω \ Ω  2 ∂ u  −1  ε   ∂t2 − ∇ · (µ ∇u) = f in Ω × (0, T ),  p ∂u −1 ∂u −1 ε µ + µ = 0 on Γext × (0, T ),  ∂t ∂n     u(x, 0) = ∂u (x, 0) = 0. ∂t

(2.1)

∂u ∂u , ), n is the unit outward normal vector on Γext ; we suppose that ∂x1 ∂x2 ¯ i × [0, T ]). µi = µ|Ωi , εi = ε|Ωi are positive constants ∀i = 1, 2 and, fi = f |Ωi ∈ C(Ω Here ∇u = (

Let ε(x) = {ε1 if x ∈ Ω1 , ε2 if x ∈ Ω2 } and µ(x) = {µ1 if x ∈ Ω1 , µ2 if x ∈ Ω2 }. We define a weak solution of problem (2.1) as a function u such that u ∈ L∞ (0, T ; H 1 (Ω)),

∂u ∂u ∈ L∞ (0, T ; L2 (Ω)), ∈ L2 (0, T ; L2 (Γext )) ∂t ∂t 11

(2.2)

for a.a. t ∈ (0, T ) and for all w ∈ H 1 (Ω) satisfying the equation Z Z Z Z q ∂2u ∂u −1 −1 ε(x) 2 wdx + µ (x)∇u · ∇wdx + ε1 µ1 wdΓ = f wdx ∂t ∂t Ω

Γext

Ω

(2.3)

Ω

with the initial conditions u(x, 0) =

∂u (x, 0) = 0. ∂t

Note that the first term in (2.3) means the duality between (H 1 (Ω))∗ and H 1 (Ω). Now using a Faedo-Galerkin method (as in [17] ) one can prove the following Theorem 2.1. Under the assumptions (2.2) there exists a unique weak solution of problem (2.1). Let 1 E(t) = 2

Z Ω

2 Z ∂u 1 ε(x) dx + µ−1 (x)|∇u|2 dx ∂t 2 Ω

be the energy of the system. We take w = in (2.3) and obtain: Z Z q ∂u ∂u ∂u 2 dE(t) −1 + ε 1 µ1 ( ) dΓ = f dx ≤ kf kL2 (Ω) k kL2 (Ω) , dt ∂t ∂t ∂t ∂u ∂t

Γext

Ω

since E(0) = 0, the following stability inequality holds: E(t) ≤ constT kf kL2 (Ω×(0,T )) , ∀t ∈ (0, T ). In order to use a structured grid in a part of the domain Ω, we introduce a rectangular domain R with sides parallel to the coordinate axes, such that Ω2 ⊂ R ⊂ Ω with γ the boundary of R (Fig. 2.1). ˜ = Ω\R ¯ and let the subscript 1 of a function v1 mean that this function is Define Ω

˜ × [0, T ], while v2 is a function defined over R × [0, T ]. defined over Ω Now we formulate problem (2.3) variationally as follows: Let ˜ W1 = {v ∈ L∞ (0, T ; H 1 (Ω)),

∂v ∂v ˜ ∈ L∞ (0, T ; L2 (Ω)), ∈ L2 (0, T ; L2 (Γext ))}, ∂t ∂t 12

W2 = {v ∈ L∞ (0, T ; H 1 (R)),

∂v ∈ L∞ (0, T ; L2 (R)))}, ∂t

Find a pair (u1 , u2 ) ∈ W1 × W2 , such that u1 = u2 on γ × (0, T ) and for a.a. t ∈ (0, T )  Z Z Z ∂ 2 u1 ∂ 2 u2  −1  ε1 2 w1 dx + µ1 ∇u1 · ∇w1 dx + ε(x) 2 w2 dx    ∂t ∂t   ˜ ˜ R  Ω Ω Z Z Z q   ∂u1   + R µ−1 (x)∇u2 · ∇w2 dx + ε1 µ−1 w1 dΓ = f1 w1 dx + f2 w2 dx 1 ∂t (2.4) R  ˜ R Γext Ω     1 ˜ 1    for all (w1 , w2 ) ∈ H (Ω) × H (R) such that w1 = w2 on γ,     u(x, 0) = ∂u (x, 0) = 0. ∂t Now, introduce the interface supported Lagrange multiplier λ (a function defined over

γ × (0, T ) ), problem (2.4) can be written in the following way:

Find a triple (u1 , u2 , λ) ∈ W1 × W2 × L∞ (0, T ; H −1/2 (γ)), which for a.a. t ∈ (0, T ) satisfies Z

Z Z ∂ 2 u1 ∂ 2 u2 −1 ε1 2 w1 dx + µ1 ∇u1 · ∇w1 dx + ε(x) 2 w2 dx ∂t ∂t ˜ Z ˜ R Ω Ω Z Z q ∂u1 −1 −1 w1 dΓ + λ(w2 − w1 )dγ = + µ (x)∇u2 · ∇w2 dx + ε1 µ1 ∂t γ Γext ZR Z ˜ w2 ∈ H 1 (R); f1 w1 dx + f2 w2 dx for all w1 ∈ H 1 (Ω), ˜ Ω

(2.5)

R

Z

ζ(u2 − u1 )dγ = 0 for all ζ ∈ H −1/2 (γ),

(2.6)

γ

and the initial conditions from (2.1). Remark 2.1. We selected the time dependent approach to capture harmonic solutions since it substantially simplifies the linear algebra of the solution process. Furthermore, there exist various techniques to speed up the convergence of transient solutions to periodic ones (see, e.g. [9]).

13

2.3 Time discretization In order to construct a finite difference approximation in time of problem (2.5), (2.6) we partition the segment [0, T ] into N intervals using a uniform discretization step ∆t = T /N . Let uni ≈ ui (n ∆t) for i = 1, 2, λn ≈ λ(n ∆t). The explicit in time semidiscrete approximation to problem (2.5), (2.6) reads as follows: u0i = u1i = 0; ˜ un+1 ∈ H 1 (R) and λn+1 ∈ H −1/2 (γ) such for n = 1, 2, . . . , N − 1 find un+1 ∈ H 1 (Ω), 1 2 that

Z un+1 − 2un1 + u1n−1 1 n ε1 w1 dx + µ−1 1 ∇u1 · ∇w1 dx ∆t2 ˜ ˜ Ω Ω Z Z n+1 n−1 n u2 − 2u2 + u2 w2 dx + µ−1 (x)∇un2 · ∇w2 dx + ε(x) ∆t2 R Z n+1 ZR q n−1 u − u 1 1 w1 dΓ + λn+1 (w2 − w1 )dγ = + ε1 µ−1 1 2∆t γ Γext Z Z ˜ w2 ∈ H 1 (R); f1n w1 dx + f2n w2 dx for all w1 ∈ H 1 (Ω),

Z

˜ Ω

(2.7)

R

Z

ζ(un+1 − un+1 )dγ = 0 for all ζ ∈ H −1/2 (γ). 2 1

(2.8)

γ

Remark 2.2. The integral over γ is written formally; the exact formulation requires the use of the duality pairing h., .i between H −1/2 (γ) and H 1/2 (γ).

2.4 Fully discrete scheme To construct a fully discrete space-time approximation to problem (2.5), (2.6) we will use a lowest order finite element method on two grids semimatching on γ (Fig. 2.2) for the space ˜ and R, respectively. Further discretization. Namely, let T1h and T2h be triangulations of Ω r(e) ≤ q = const for all we suppose that both triangulations are regular in the sense that h(e) e ∈ T1h and e ∈ T2h , where q does not depend on e; r(e) is the radius of the circle inscribed in e, while h(e) is the diameter of e. 14

Ω γ R Figure 2.2: Semimatching mesh on γ. We denote by T1h a coarse triangulation, and by T2h a fine one. Every edge ∂e ⊂ γ of a triangle e ∈ T1h is supposed to consist of me edges of triangles from T2h , 1 ≤ me ≤ m for all e ∈ T1h Moreover, let a triangulation T2h be such that the curvilinear boundary ∂Ω2 is approximated by a polygonal line consisting of the edges of triangles from T 2h whose vertices belong to ∂Ω2 . Further, we say that a triangle e ∈ T2h lies in Ω2 if its larger part lies in Ω2 , i.e. meas(e ∩ Ω2 ) > meas(e ∩ (R \ Ω2 )), otherwise this triangle lies in R \ Ω2 .

˜ be the space of the functions globally continuous, and affine on each Let V1h ⊂ H 1 (Ω)

˜ : uh ∈ P1 (e) ∀e ∈ T1h }. Similarly, V2h ⊂ H 1 (R) is the e ∈ T1h , i.e. V1h = {uh ∈ H 1 (Ω) space of the functions globally continuous, and affine on each e ∈ T 2h . For approximating the Lagrange multipliers space Λ = H −1/2 (γ) we proceed as follows. Assume that on γ, T1h is two times coarser than T2h ; then let us divide every edge ∂e of a triangle e from the coarse grid T1h , which is located on γ ( ∂e ⊂ γ), into two parts using its midpoint. Now, we consider the space of the piecewise constant functions, which are constant on every union of half-edges with a common vertex (see Fig.2.3). Further, we use quadrature formulas for approximating the integrals over the triangles from T1h and T2h , as well as over Γext . For a triangle e we set Z e

3

X 1 φ(ai ) ≡ Se (φ) φ(x)dx ≈ meas(e) 3 i=1 15

Ω

γ R

Figure 2.3: Space Λ is the space of the piecewise constant functions defined on every union of half-edges with common vertex. where the ai ’s are the vertices of e and φ(x) is a continuous function on e. Similarly, 2

Z

∂e

X 1 φ(ai ) ≡ S∂e (φ), φ(x)dx ≈ meas(∂e) 2 i=1

where ai ’s are the endpoints of the segment ∂e and φ(x) is a continuous function on this segment. We use the notations: Si (φ) =

X

Se (φ), i = 1, 2, and SΓext (φ) =

e∈Tih

X

S∂e (φ).

∂e⊂Γext

Now, the fully discrete problem reads as follows: Let u0ih = u1ih = 0, i = 1, 2;

n+1 n+1 for n = 1, 2, . . . , N − 1 find (un+1 1h , u2h , λh ) ∈ V1h × V2h × Λh such that  ε 1 −1 n n n−1  S ((un+1  1h − 2u1h + u1h )w1h ) + S1 (µ1 ∇u1h · ∇w1h )  2 1  ∆t   1  n−1 n −1 n  S2 (ε(x)(un+1  2h − 2u2h + u2h )w2h ) + S2 (µ (x)∇u2h · ∇w2h )  +p ∆t2 Z ε1 µ−1 1 n−1 n+1  + SΓext ((u1h − u1h )w1h ) + λn+1  h (w2h − w1h )dγ =  2∆t    γ     S1 (f1n w1h ) + S2 (f2n w2h ) for all w1h ∈ V1h , w2h ∈ V2h ; Z n+1 ζh (un+1 2h − u1h )dγ = 0 for all ζh ∈ Λh . γ

16

(2.9)

(2.10)

n−1 n Note that in S2 (ε(x)(un+1 2h − 2u2h + u2h )w2h ) we take ε(x) = ε2 if a triangle e ∈ T2h

lies in Ω2 and ε(x) = ε1 if it lies in R \ Ω2 , and similarly for S2 (µ−1 (x)∇un2h ∇w2h ). Denote by u1 , u2 and λ the vectors of the nodal values of the corresponding functions and λn+1 for a fixed time tn+1 we have u1h , u2h and λh . Then in order to find un+1 , un+1 1 2 to solve a system of linear equations such as (2.11)

Au + B T λ = F ,

(2.12)

= 0,

Bu

where matrix A is diagonal, positive definite and defined by ε1 1 (Au, w) = S (u w ) + S2 (ε(x)u2h w2h ) + 1 1h 1h ∆t2 ∆t2

p −1 ε 1 µ1 SΓext (u1h w1h ), 2∆t

and where the rectangular matrix B is defined by (Bu, λ) =

Z

λh (u2h − u1h )dΓ,

γ

n−1 n−1 . and u2h and vector F depends on the nodal values of the known functions u n1h , un2h , u1h

Eliminating u from equation (2.11) we obtain BA−1 B T λ = BA−1 F ,

(2.13)

with a symmetric matrix C ≡ BA−1 B T . Let us prove that C is positive definite. Obviously, Ker C = Ker B T . Suppose, that B T λ = 0, then a function λh ∈ Λh corresponding to vector λ satisfies I≡

Z

λh uh dγ = 0

γ

for all uh ∈ V1h . Choose uh equal to λh in the nodes of T1h located on γ. Direct calculations give N

λ 1X hi + hi+1 2 (λi + λi+1 )2 I= [ λi + hi+1 ], 2 i=1 2 2

17

where Nλ is the number of edges of T1h on γ, hi is the length of i-th edge and hNλ +1 ≡ h1 , λNλ +1 ≡ λ1 . Thus, the equality I = 0 implies that λ = 0, i.e. Ker B T = {0}. As a consequence we have Theorem 2.2. Problem (2.9), (2.10) has a unique solution (u h , λh ). Remark 2.3. A closely related domain decomposition method applied to the solution of linear parabolic equations is discussed in [27].

2.5 Energy inequality Theorem 2.3. Let hmin denote the minimal diameter of the triangles from T1h ∪ T2h . There exists a positive number c such that the condition √ √ ∆t ≤ c min{ ε1 µ1 , ε2 µ2 } hmin

(2.14)

ensures the positive definiteness of the quadratic form 1 un+1 − un1h 2 1 un+1 − un2h 2 E n+1 = ε1 S1 (( 1h ) ) + S2 (ε( 2h ) )+ 2 ∆t 2 ∆t n n un+1 1 un+1 1 −1 1h + u1h 2 2h + u2h 2 S1 (µ−1 |∇( )| ) + S (µ |∇( )| ) 2 1 2 2 2 2 n n ∆t2 un+1 ∆t2 un+1 −1 1h − u1h 2 2h − u2h 2 − S1 (µ−1 |∇( )| ) − S (µ |∇( )| ), 2 1 8 ∆t 8 ∆t

(2.15)

which we call the discrete energy. System (2.9), (2.10) satisfies the energy identity p −1 ε 1 µ1 n−1 2 E n+1 − E n + SΓext ((un+1 1h − u1h ) ) = 4∆t n−1 n−1 1 1 n n+1 S (f n (un+1 1h − u1h )) + 2 S2 (f2 (u2h − u2h )) 2 1 1

(2.16)

and the numerical scheme is stable: there exists a positive number M = M (T ) such that n

E ≤ M ∆t

n−1 X k=1

(S1 ((f1k )2 ) + S2 ((f2k )2 )) ∀n 18

(2.17)

Proof. Let n ≥ 1; from equation (2.10) written for tn+1 and tn−1 we obtain Z n−1 n+1 n−1 ζh ((un+1 2h − u2h ) − (u1h − u1h ))dγ = 0 for all ζh ∈ Λh

(2.18)

γ

n−1 n−1 un+1 un+1 − u2h λn+1 1h − u1h , w2h = 2h in (2.9) and ζh = − h in (2.18) 2 2 2 we add these equalities. Using the identities

Choosing w1h =

n n−1 n+1 n−1 n+1 n 2 n n−1 2 (un+1 ih − 2uih + uih )(uih − uih ) = (uih − uih ) − (uih − uih )

and 1 n+1 n 2 n+1 n 2 unih un+1 ih = ((uih + uih ) − (uih − uih ) ) 4 after several technical transformations we obtain p −1 ε 1 µ1 n−1 2 SΓext ((un+1 E n+1 − E n + 1h − u1h ) ) = 4∆t n−1 n−1 1 1 n n+1 S (f n (un+1 1h − u1h )) + 2 S2 (f2 (u2h − u2h )). 2 1 1 Therefore n+1 n+1 1 − un1h 2 − un1h 2 1/2 1/2 u 1/2 u E n+1 ≤ E n + ∆tS1 ((f1n )2 )(S1 (( 1h ) ) + S1 (( 1h ) ))) 2 ∆t ∆t 1/2

1/2

+ 21 ∆tS2 ((f2n )2 )(S2 ((

n un+1 2h −u2h 2 ) ) ∆t

1/2

+ S2 ((

n un+1 2h −u2h 2 ) )). ∆t

(2.19)

Now, we will show that under condition (2.14) the quadratic form E n is positive definite; more precisely, that there exists a positive constant δ such that E n ≥ δ S1 ((

n un+1 un+1 − un2h 2 1h − u1h 2 ) ) + S2 (( 2h )) . ∆t ∆t

(2.20)

Obviously, it is sufficient to prove the inequality

4εe µe Se (vh2 ) ≥ ∆t2 Se (|∇vh |2 ) ∀e ∈ T1h ∪ T2h , ∀vh ∈ P1 (e),

(2.21)

where εe and µe are defined by εe = ε1 or εe = ε2 (respectively, µe = µ1 or µe = µ2 ). It is known that for a regular triangulation 2 Se (|∇vh |2 ) ≤ 1/c21 h−2 e Se (vh )

19

(2.22)

with a positive constant c1 , universal for all triangles e, where he is the minimal length of the sides of e. Combining (2.21) and (2.22) we observe that the time step ∆t should satisfy the inequality √ √ ∆t ≤ c εe µe he , (c = 2c1 ),

(2.23)

for all e ∈ T1h ∪ T2h . Evidently, (2.14) ensures the validity of (2.23). Further, using the relation (2.20), E 1 = 0 and summing the inequalities (2.19), one obtains the stability inequality (2.17):

E n ≤ M ∆t

n−1 X k=1

(S1 ((f1k )2 ) + S2 ((f2k )2 )), ∀n.

2.6 Numerical experiments In order to solve the system of linear equations (2.11)-(2.12) at each time step we use a Conjugate Gradient Algorithm in the form given by Glowinski and LeTallec [33]: 1. λ0 given, 2. Au0 = F − Bλ0 ; 3. g 0 = −B T u0 , if ||g 0 || ≤ ε0 take λ = λ0 , else 4. w0 = g 0 . For m ≥ 0, assuming that λm , g m , wm are known 5. A¯ um = Bw m . ¯ m. ¯m = BT u 6. g 20

7. ρm =

|g m |2 . ¯ m) (¯ g m ,w

8. λm+1 = λm − ρm wm . ¯m. 9. um+1 = um + ρm v 10. g m+1 = g m − ρm g¯ m , if

g m+1 ·g m+1 g 0 ·g 0

≤ ε then take λ = λm+1 ,

else 11. γm =

g m+1 ·g m+1 . g m ·g m

12. wm+1 = g m+1 + γm w m , do m = m + 1 and go to 5. We consider problem (2.9)-(2.10) with a source term given by the harmonic planar wave: uinc = −eik(t−a·x) ,

(2.24)

where {xj }2j=1 , {aj }2j=1 , k is the angular frequency and |a| = 1. For our numerical simulation we consider two cases: the first with the frequency of the incident wave f = 0.6 GHz and the second with f = 1.2 GHz, which gives us wavelengths L = 0.5 meters and L = 0.25 meters respectively. First, we consider the scattering by a perfectly reflecting obstacle. For the first experiment we have chosen Ω2 to be a perfectly reflecting disk with diameter d = 0.5 meters, and Ω is a 2 meter × 2 meter rectangle. Figure 2.4 shows the contour plot of the real part of the scattered solution for wavelength L = 0.5 meters and L = 0.25 meters. The second experiment shows the wave propagation through the domain Ω with an obstacle in form of an airfoil for wavelengths L = 0.5 meters and L = 0.25 meters. Figure 2.6 shows the contour plot for the case when the incident wave is coming from the left and Figure 2.8 shows the case when the incident wave is coming from the lower left corner 21

with an angle of 450 . For all the experiments we chose the time step to be ∆t = T /50, where T = 1/f = 1.66 × 10−9 sec is a time period corresponding to L = 0.5 meters and T = 1/f = 0.83 × 10−9 sec for L = 0.25 meters. We present the energy decay after the source term was set to zero in Figure 2.5 (for an obstacle in the form of a disk) and Figure 2.7 (for an obstacle in the form of an airfoil).

22

Figure 2.4: Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom) meters. Incident wave coming from the left. 23

2.5

2

Energy

1.5

1

0.5

0

0

0.2

0.4

0.6 0.8 time (sec)

1

3 time (sec)

5

1.2 −8

x 10

4.5 4 3.5

Energy

3 2.5 2 1.5 1 0.5 0

0

1

2

4

6 −9

x 10

Figure 2.5: Energy behavior versus time for L = 0.5 (top) and L = 0.25 (bottom).

24

Figure 2.6: Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom) meters. Incident wave coming from the left. 25

3

2.5

Energy

2

1.5

1

0.5

0

0

0.2

0.4

0.6 0.8 time (sec)

1

3 time (sec)

5

1.2 −8

x 10

4.5 4 3.5

Energy

3 2.5 2 1.5 1 0.5 0

0

1

2

4

6 −9

x 10


26

Figure 2.8: Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom) meters. Incident wave coming from the lower left corner with an angle of 45 degrees. 27

The next set of numerical experiments contains the simulations of wave propagation through a domain with an obstacle completely consisting of a coating material. We have taken the coating material coefficients to be ε2 = 1 and µ2 = 9, implying that the speed of propagation in the coating material is three times slower then in the air. As before Ω is a 2 meter × 2 meter rectangle and we ran our simulations for two different shapes of Ω 2 : 1. Ω2 is a disk with diameter d = 0.5 meters, and 2. Ω2 has the shape of an airfoil. For the solution of this problem for an incident frequency f = 0.6 GHz we have used a mesh with a total of 8435 nodes and 16228 elements. The time step was taken to be ∆t = T /50, where T = 1/f = 1.66 × 10−9 sec is a time period. We used a mesh consisting of 20258 nodes (39514 elements) for solving the problem for an incident wave with the frequency f = 1.2 GHz. The time step was equal to T /50, T = 1/f = 0.83×10 −9 sec. In Figures 2.9 and 2.10 we present the contour plot of the real part of the solution for the incident frequency L = 0.5 and L = 0.25 respectively. Figures 2.13 and 2.14 show the energy decay after four time periods when we set the source term equal to zero.

28

Figure 2.9: Contour plot of the real part of the solution for L = 0.5 meters.

Figure 2.10: Contour plot of the real part of the solution for L = 0.25 meters. 29

Figure 2.11: Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom). Incident wave coming from the left. 30

Figure 2.12: Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom). Incident wave coming from the lower left corner with an angle of 45 degrees. 31

−3

x 10

8 7

energy

6 5 4 3 2 1 0

0

0.2

0.4

0.6 0.8 time (sec)

1

1.2 −8

x 10

Figure 2.13: Energy behavior for L = 0.5 meters.

0.5 0.45 0.4 0.35

Energy

0.3 0.25 0.2 0.15 0.1 0.05 0

0

1

2

3 time (sec)

4

5

6 −9

x 10

Figure 2.14: Energy behavior for L = 0.25 meters.

32

Figure 2.15: Obstacle in a form of an airfoil with a coating. We also performed numerical computations for the case when the obstacle is an airfoil with a coating (Figure 2.15). The coating region is moon shaped and, as before, ε 2 = 1 and µ2 = 9. We show in Figure 2.16 the contour plot of the real part of the solution for the incident frequency L = 0.5 meters and L = 0.25 meters for the case when the incident wave is coming from the left. Figure 2.17 presents the contour plot of the real part of the solution for incident frequency, L = 0.5 meters and L = 0.25 meters for the case when incident wave is coming from the lower left corner with angle equal to 450 . One can observe the energy decay in Figure 2.18 when the source term has been set equal to zero. A crucial observation for all of the numerical experiments mentioned is that despite the fact that a mesh discontinuity takes place over γ together with a weak forcing of the matching conditions, we do not observe a discontinuity of the computed fields.

33

Figure 2.16: Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom). Incident wave coming from the left. 34

Figure 2.17: Contour plot of the real part of the solution for L = 0.5 (top) and L = 0.25 (bottom). Incident wave coming from the left lower corner with a 45 degree angle. 35

3

2.5

Energy

2

1.5

1

0.5

0

0

0.2

0.4

0.6 0.8 time (sec)

1

3 time (sec)

5

1.2 −8

x 10

4.5 4 3.5

Energy

3 2.5 2 1.5 1 0.5 0

0

1

2

4

6 −9

x 10


36

Chapter 3 Solution of a wave equation by a mixed finite element - fictitious domain method 3.1 Introduction In this chapter we discuss the numerical solution of a linear wave equation using a combined fictitious domain – mixed finite element methodology. We consider a linear wave equation with constant coefficient in a domain which is a rectangle containing a circular obstacle (Figure 3.1). Dirichlet boundary conditions are imposed on the obstacle boundary, while absorbing boundary conditions are prescribed on the external part of the boundary. We use a mixed variational formulation to construct a discrete problem. This discrete scheme is ”explicit” in time and is of the lowest order finite element approximation in space. To avoid difficulties concerning the implementation of a discrete scheme in a curvilinear domain, we construct a finite element approximation of the mixed problem in a rectangular domain containing the original domain (see e.g. [5]) . The interface conditions are treated by Lagrange multipliers. A conjugate gradient algorithm (discussed in [33], [18]) is used for solving the resulting system of linear algebraic equations (of the ”saddle-point” type). 37

Ω

γ

Γext


3.2 Formulation of the problem Let Ω be a bounded domain in R2 with boundary Γext . We consider the following wave problem: 1 ∂2u − ∆u = 0 c2 ∂t2

in

Ω × (0, T ),

(3.1)

Γext × (0, T ),

(3.2)

with the absorbing boundary condition 1 ∂u ∂u + =0 c ∂t ∂n

on

and the initial conditions u(0) = u0 ,

(3.3)

ut (0) = u1 .

Here n is the outward unit normal vector on Γext . We define a weak solution of problem (3.1) – (3.3) as a function u such that ∂u ∂u u ∈ L∞ (0, T ; H 1(Ω)), ∈ L∞ (0, T ; L2 (Ω)), ∈ L2 (0, T ; L2 (Γext )), ∂t ∂t and for a.a. t ∈ (0, T ) and for all w ∈ H 1 (Ω) satisfies the equation 1 c2

Z Ω

∂2u wdx + ∂t2

Z

1 ∇u · ∇wdx + c

Ω

Z

Γext

38

∂u wdΓ = 0 ∂t

(3.4)

and initial conditions (3.3). As in [17] one can prove the following Theorem 3.1. Let u0 ∈ H 1 (Ω), u1 ∈ L2 (Ω). Then there exists a unique weak solution of problem (3.1) – (3.3). Now, let us consider an equivalent approach based on a mixed formulation. We introduce the new variables: p = ∇u,

v=

∂u . ∂t

(3.5)

Clearly, p and v satisfy the following equations: ∂p/∂t = ∇v and

∂v/∂t = c2 ∇ · p in Ω × (0, T ),

v + cp · n = 0 on Γext × (0, T ), p(0) = ∇u0 ,

(3.6) (3.7) (3.8)

v(0) = u1 .

Multiplying the first equation in (3.6) by q ∈ H(Ω, div) = {q ∈ (L2 (Ω))2 : ∇ · q ∈ L2 (Ω)}, the second equation by w ∈ L2 (Ω), and then applying the divergence theorem we obtain the following variational formulation of problem (3.6)-(3.8): Z Ω

∂p · qdx + ∂t

Z

v∇ · qdx + c

Z

∂v wdx − c2 ∂t

Ω

Z

(p · n)(q · n)dΓ = 0,

Z

∇ · pwdx = 0,

∀q ∈ H(Ω, div),

(3.9)

Γext

∀w ∈ L2 (Ω),

(3.10)

Ω

Ω

p(0) = ∇u0 ,

v(0) = u1 .

Lemma 3.1. Problem (3.6)-(3.8) satisfies the energy dissipation property. Proof. Let us denote by 1 E(t) = 2

Z

1 |p(x, t)| dx + 2 2c 2

Ω

Z Ω

39

v 2 (x, t)dx

(3.11)

the energy of the system at time t > 0. We multiply the first equation in (3.6) by p and the second equation by v, add them and then integrate over Ω: Z Z Z Z ∂p ∂v 1 · pdx + 2 vdx = ∇v · pdx + ∇ · pvdx = 0. ∂t c ∂t Ω

Ω

Ω

Ω

Using the divergence theorem we have:   Z Z 1 2  1d  2 (p + 2 v )dx = vp · ndΓ 2 dt c Ω

Γ

Clearly dE/dt satisfies the inequality

dE(t) = −c dt

Z

(3.12)

(p · n)2 dΓ ≤ 0,

Γext

therefore we have a dissipation of the energy.

3.3 Discretization of the problem. Energy inequality In order to get a fully discrete problem which will be easy to implement, we first construct an ”explicit” in time semi-discrete problem. Let ∆t > 0, be the time step and let pn ≈ p(n∆t), v n+1/2 ≈ v((n + 1/2)∆t) for n = 0, 1, 2, ... approximate the corresponding functions p(t) and v(t). The time discretization of problem (3.9)-(3.11) reads as follows: For n ≥ 1, find (pn , v n+1/2 ) ∈ H(Ω, div) × L2 (Ω) such that Z n Z Z p − pn−1 pn + pn−1 n−1/2 · q dx + v ∇ · q dx + c · n)(q · n) dΓ = 0, (3.13) ( ∆t 2 Ω

Ω

Γext

∀q ∈ H(Ω, div), Z Z n+1/2 v − v n−1/2 2 w dx − c ∇ · pn w dx = 0, ∆t

∀w ∈ L2 (Ω),

(3.14)

Ω

Ω

p0 = ∇u0 ,

v 1/2 − v 0 = c 2 ∇ · p0 . ∆t/2 40

(3.15)

Now, we construct a fully discrete problem. For convenience, we set Ω to be a rectangular domain with boundaries parallel to the coordinate axes, namely Ω = (0, xL ) × (0, yL ). We define Rx : 0 = x0 < x1 < ... < xNx = xL and Ry : 0 = y0 < y1 < ... < yNy = yL to be partitions of [0, xL ] and [0, yL ], respectively, and denote by ∆i,j a mesh cell – rectangle [xi , xi+1 ] × [yj , yj+1 ]. We use T = {∆i,j } to denote the corresponding partition of Ω. Let Ph be a finite dimensional subspace of H(Ω, div) constructed via the lowest order Raviart-Thomas finite element method, namely, Ph = {ph ∈ H(Ω, div) : ph = (p1h , p2h ) ∈ P1,0 × P0,1 for each ∆i,j ∈ T }, where Pk,l is the space of polynomials of degree k in x and l in y. By Vh ⊂ L2 (Ω) we denote the space of piecewise constant functions, which are constant on every ∆ i,j . Further, let ∆i,j ∈ T , then we denote by 1 1 + a− i,j = (xi , (yj + yj+1 )), ai,j = (xi+1 , (yj + yj+1 )), 2 2 1 1 + b− i,j = ( (xi + xi+1 ), yj ), bi,j = ( (xi + xi+1 ), yj+1 ) 2 2 the mid–points of the corresponding edges (see Figure 3.2). The values of the fluxes, ph · n, at the mid–points of the finite elements ∆i,j ∈ T are taken as degrees of freedom − for ph ∈ Ph . Thus, for rectangular elements they are p1h (a− i,j ), p2h (bi,j ) etc. For vh ∈ Vh

the values at the center of the cells ∆i,j ∈ T are taken as the degrees of freedom. Further we denote by Ph and Vh the vectors whose components are the degrees of freedom for the corresponding functions ph ∈ Ph and vh ∈ Vh . Z We use the following quadrature formula to approximate ph · q h dx: Ω

1 − + + S∆ (ph · q h ) = meas ∆i,j (p1h (a− i,j )q1h (ai,j ) + p1h (ai,j )q1h (ai,j ) 4 + + − +p2h (b− i,j )q2h (bi,j ) + p2h (bi,j )q2h (bi,j )).

41

+

bi,j

−

a i,j

a

+ i,j

∆ i,j

−

bi,j

Figure 3.2: Degrees of freedom for p . The integral above is then approximated by X

SΩ (ph · q h ) =

∆i,j ∈T

S∆ (ph · q h ). n+1/2

The space discretized formulation of problem (3.13)-(3.15) involves the pair (p nh , vh Ph × Vh satisfying

pnh − phn−1 SΩ ( · qh) + ∆t

Z

n−1/2

vh

)∈

∇ · q h dx

Ω

Z

+c

(

pnh + phn−1 · n)(q h · n) dΓ = 0, 2

∀q h ∈ Ph ,

(3.16)

Γext

Z

n+1/2

vh

n−1/2

− vh ∆t

wh dx − c

2

Z

∇ · pnh wh dx = 0,

∀wh ∈ Vh ,

(3.17)

Ω

Ω

with initial conditions 1/2

p0h = (∇u0 )h ,

vh − vh0 = c2 ∇ · p0h , ∆t/2

where (∇u0 )h is an appropriate Ph -approximation of (∇u0 ). 42

(3.18)

n+1/2

Vector (pnh , vh

) can be found recurrently from a system of linear algebraic equa-

tions, namely Mp Ph = G 1 , M v Vh = G 2 , where Mp and Mv are diagonal matrices defined by pn (Mp Ph , Q) = SΩ ( h · q h ) + c ∆t

Z

(

pnh · n)(q h · n)dΓ 2

∀ph ∈ Ph , ∀q h ∈ Ph ,

Γext

1 (Mv Vh , W) = 2 c

n+1/2

vh wh dx ∆t

Z

∀vh ∈ Vh , ∀wh ∈ Vh .

Ω

Obviously, matrices Mp and Mv are positive definite; thus, there exists a unique solution n+1/2

(pnh , vh

) of problem (3.16), (3.17).

Theorem 3.2. Let hmin denote the minimal mesh step size. Then the condition ∆t ≤

1 hmin 2c

(3.19)

ensures the positive definiteness of the quadratic form E

n+1

1 1 = SΩ ((pnh )2 ) + 2 2 2c

Z

n+1/2 2 ) dx (vh

∆t − 2

Ω

Z

n+1/2

∇ · pnh vh

dx,

(3.20)

Ω

which we call the discrete energy. System (3.16)-(3.18) satisfies the discrete energy identity E

n+1

n

− E = −c ∆t

Z

(

pnh + phn−1 · n)2 dΓ. 2

Γext

Thus, E n+1 ≤ E n , ∀n, and the discrete problem (3.16)-(3.18) is stable.

43

(3.21)

n+1/2

pn + phn−1 v Proof. Let us choose q = ∆t h in equation (3.16) and wh = ∆t h 2 in equation (3.17). By adding these equalities we obtain the identity (3.21).

n−1/2

+ vh 2 c2

Now, by direct calculations we obtain the inequality Z 16 |∇ · ph |2 dx ≤ 2 S∆i,j (|ph |2 ) hmin ∆i,j

for any ph ∈ Ph and any ∆i,j . From here we derive the estimate Z Z 4 n+1/2 n+1/2 2 n 2 1/2 ∇ · ph v dx ≤ (S∆i,j (|ph | )) ( |vh | dx)1/2 , h h min Ω

Ω

thus, under condition (3.19) we get the positive definiteness of E n for all n.

3.4 A fictitious domain method with boundary supported Lagrange multiplier Now we consider a domain ω ⊂ Ω with a piecewise smooth boundary γ. Let equation (3.1) and initial conditions (3.3) be valid in Ω \ ω ¯ and problem (3.1) - (3.3) to be completed by the following Dirichlet boundary condition on γ: u=0

γ × (0, T ).

on

(3.22)

As before, we introduce the new functions p and v, and get an initial boundary-value problem similar to (3.6) - (3.8) with the additional boundary condition v=0

γ × (0, T ).

on

(3.23)

Now, let the spaces Ph and Vh be defined as above and let Λη be a finite dimensional subspace of L2 (γ), which is defined below. First, let us define a family Tγ of the cells ∆ ∈ T , which we call ”boundary elements”. In Tγ the following cells ∆ are included: 44

γ

ω

Figure 3.3: Shaded rectangulars are boundary elements T γ . ◦

• ∆ such that γ∩ ∆6= ∅; ◦

• if γ∩ ∆= ∅, but γ ∩ ∂∆ 6= ∅, then we include ∆ in Tγ only if ∆ ∈ ω (Figure 3.3). Now, let γ = {δγi } be a partitioning of γ such that every interval δγi contains at least one interval γ ∩ ∆ of the intersection of γ with a cell ∆ from Tγ . Below we denote by Λη a space of piecewise constant functions which are constant on every δγ i . We also denote by Πγ wh a ”trace” of wh ∈ Vh on γ which is defined as a function piecewise constant on γ, which is equal to wh on each γ ∩ ∆ for ∆ ∈ Tγ . n+1/2

A fully discrete problem reads as follows: find a triple (pnh , vh

, λnη ) ∈ Ph × Vh × Λη

satisfying pn − phn−1 · qh) + SΩ ( h ∆t

Z

n−1/2

vh

∇ · q h dx

Ω

+c

Z

(

pnh + phn−1 · n)(q h · n) dΓ = 0, 2

∀q h ∈ Ph ,

(3.24)

Γext

Z Ω

n+1/2

vh

n−1/2

− vh ∆t

wh dx − c

2

Z

∇·

pnh wh

dx + c

2

Z γ

Ω

45

λnη Πγ wh dx = 0,

∀wh ∈ Vh , (3.25)

Z

n+1/2

γ

µη Π γ v h

dγ = 0,

∀µη ∈ Λη ,

(3.26)

1/2

p0h

= (∇u0 )h ,

vh − vh0 = c2 ∇ · p0h . ∆t/2

(3.27)

˜ h, V ˜ h and Λ ˜ h the vectors whose components are the degrees of freedom for Denote by P the corresponding functions ph ∈ Ph , vh ∈ Vh and λη ∈ Λη . Then for a fixed time tn = n∆t the first equation (3.24) is a system of linear algebraic ˜ h and Λ ˜ h we obtain a system equations with a mass matrix Mp defined above, while for V of linear algebraic equations:   Mv V ˜ h + BT Λ ˜ h = F,  ˜h = 0 BV

(3.28)

with Mv defined above and the rectangular matrix B defined by (BW, Λh ) =

Z

λη Πγ wh dx ∀λη ∈ Λη , wh ∈ Vh .

γ

˜ h, Λ ˜ h ). Theorem 3.3. System (3.28) has a unique solution (V Proof. Since Mv is a regular matrix it is enough to prove that KerBT = {0}. Suppose ˜ h = 0, i.e., BT Λ Z γ

λη Πγ wh dx = 0 ∀wh ∈ Vh .

Let us take wh ∈ Vh such that Πγ wh = 1 for one ∆i,j ∈ Tγ , while it equals zero in other cells ∆. Then λη = 0 for a δγk which contains γ ∩ ∆i,j . Because each δγk contains at least ˜ h ≡ 0. one interval γ ∩ ∆ with ∆ ∈ Tγ , then Λ Theorem 3.4. Let assumption (3.19) hold, then the system (3.24)-(3.26) satisfies the energy identity (3.21), where the discrete energy E n+1 is defined by (3.12). Therefore, problem 1 hmin . (3.24)-(3.26) is stable for ∆t ≤ 2c

46

Proof. We proceed as in the proof of Theorem 3.2. Namely, let us choose q = ∆t n+1/2

in equation (3.24), wh = ∆t Z

vh

γ

n−1/2

+ vh 2 c2

in equation (3.25) and µη = λnh in equation

n+1/2

µη (Πγ vh

pnh + pn−1 h 2

n−1/2

+ Π γ vh

)dγ = 0.

Adding these equalities we get identity (3.21). Now, the rest of the proof is the same as in Theorem 3.2. In order to solve the system of linear equations (3.28) at each time step we use the conjugate gradient algorithm in the form given in [33].

3.5 Numerical experiments. For our numerical experiments we have taken Ω = (0, 1) × (0, 1). The initial condition for the function u (Figure 3.4) has been set to   cos2 2πr, if r ≤ 1/4, u0 (x, y) =  0, otherwise, u1 = 0,

where r =

p

(x − 1/2)2 + (y − 1/2)2 . We have considered the following cases:

1. A wave problem in the domain Ω without an obstacle (Figures 3.5- 3.7). For {h, ∆t} we took hx = hy = 1/100 and ∆t = 1/500. 2. A wave problem in the domain Ω with ω the quarter of a disc in the right-lower corner. We have taken hx = hy = 1/100 and ∆t = 1/500. The radius of the disk has been set to R(ω) = 52 . We divided the boundary of ω into 25 intervals, i.e. δγ =

1 . 25

The fictitious domain method described in Section 4 has been used. Figures 3.8-3.12 show the wave propagation from the initial state (Figure 3.4) and its reflection from the obstacle. We observe a very good agreement of energy dissipation curves for different mesh and time step sizes in Figure 3.13. 47

1

0.5

0

−0.5

−1 1 0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

Figure 3.4: Initial function u .

Time=0.5

1

0.5

0

−0.5

−1 1 0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

Figure 3.5: Domain without an obstacle, time=0.5.

48

Time=0.7

1

0.5

0

−0.5

−1 1 0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

Figure 3.6: Domain without an obstacle, time=0.7.

Time=1

1

0.5

0

−0.5

−1 1 0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

Figure 3.7: Domain without an obstacle, time=1.

49

1

Time=0.2

1

0.5

0

−0.5

−1 1 0.8

1 0.6

0.8 0.6

0.4

0.4

0.2 0

0.2 0

Figure 3.8: Domain with a circular obstacle, time=0.2.

Time=0.4

1

0.5

0

−0.5

−1 1 0.8

1 0.6

0.8 0.6

0.4

0.4

0.2 0

0.2 0


50

Time=0.6

1

0.5

0

−0.5

−1 1 0.8

1 0.6

0.8 0.6

0.4

0.4

0.2 0

0.2 0


Time=0.8

1

0.5

0

−0.5

−1 1 0.8

1 0.6

0.8 0.6

0.4

0.4

0.2 0

0.2 0


51

Time=1

1

0.5

0

−0.5

−1 1 0.8

1 0.6

0.8 0.6

0.4

0.4

0.2 0

0.2 0

Figure 3.12: Domain with a circular obstacle, time=1.

2

1.8

1.6

Energy

1.4

1.2

1

0.8

0.6

0.4

0.2

0

0.1

0.2

0.3

0.4

0.5 Time

0.6

0.7

0.8

0.9

1

Figure 3.13: Energy dissipation for ∆t = 1/250, h = 1/50 (dashed line) and ∆t = 1/500, h = 1/100 (solid line). 52

Chapter 4 A penalty approach to the numerical simulation of a constrained wave motion 4.1 Introduction The main goal of this chapter is to discuss the solution of a constrained wave system which has applications in theoretical and applied physics. The numerical methodology relies on the penalty treatment of the constraint and an energy preserving time discretization leading to a scheme which is essentially unconditionally stable and second order accurate; both techniques are combined with a globally continuous piecewise linear finite element approximation. The results of numerical experiments illustrate the properties of the computational methods discussed here, particularly energy and length preservation.

4.2 Problem formulation Let Ω be a bounded domain in R2 with a piecewise smooth boundary ∂Ω and let u(x, t) : Ω × (0, T ) → R3 . We consider the following constrained wave problem ∂2u − ∆u + λu = 0, |u|2 = 1 in Ω × (0, T ) ∂t2 53

(4.1)

with Dirichlet boundary conditions u(x, t) = g(x, t) on ∂Ω × (0, T ),

(4.2)

and initial conditions u(x, 0) = u0 (x),

∂u (x, 0) = u1 . ∂t

Further we suppose that g = g(x) and   u0 (x) · u1 (x) = 0, |u0 (x)|2 = 1 for a.a. x ∈ Ω,  g(x) is the trace of u (x) on ∂Ω.

(4.3)

(4.4)

0

Remark 4.1. The function λ is clearly a Lagrange multiplier associated to the constraint |u|2 = 1. We approximate the constrained wave equation (4.1) by the following wave equation with a penalty term: 1 ∂2u − ∆u + (|u|2 − 1)u = 0, 2 ∂t ε

(4.5)

namely a kind of ”Ginzburg-Landau wave equation” (for more information on GinzburgLandau type models see, e.g. [7]). Theorem 4.1. For every fixed ε > 0 problem (4.5), (4.2), (4.3) has a unique solution ∂u such that u ∈ L∞ (0, T ; H 1 (Ω)3 ), ∈ L∞ (0, T ; L2 (Ω)3 ), provided that u0 ∈ H 1 (Ω)3 , ∂t u1 ∈ L2 (Ω)3 and the assumptions (4.4) are fulfilled. ∂u ∈ If, moreover, u0 ∈ H 2 (Ω)3 and u1 ∈ H01 (Ω)3 , then the solution u ∈ L∞ (0, T ; H 1(Ω)3 ), ∂t 2 ∂ u L∞ (0, T ; H01 (Ω)3 ) and 2 ∈ L∞ (0, T ; L2 (Ω)3 ). ∂t Proof. We give a proof which uses the Faedo-Galerkin method and is based on a proof given in [49] for a similar problem. 1) First, the initial conditions given in (4.3) make sense for a function u ∈ L ∞ (0, T ; H 1 (Ω)3 ), ∂u ∈ L∞ (0, T ; L2 (Ω)3 ). In fact, u ∈ C([0, T ]; L2 (Ω)3 ), i.e. it is continuous such that ∂t ∂2u from [0, T ] in L2 (Ω)3 . Further, from equation (4.5) it follows that 2 ∈ L∞ (0, T ; (H 1(Ω)∗ )3 ), ∂t 54

∂u ∂u ∈ L∞ (0, T ; L2 (Ω)3 ) it gives ∈ C([0, T ]; (H 1 (Ω)∗ )3 ) (here ∂t ∂t (H 1 (Ω)∗ )3 is dual to H 1 (Ω)3 ). and together with

2) Let V0 = H01 (Ω)3 and {w1 , . . . , wn , . . .} be a linear independent basis in V0 . We find an approximate solution in the form un (t) =

n X

gin (t)w i + u0 ,

i=1

where gin (t) satisfies the system of equations Z

00

un (t) · w j dx +

Ω

+

1 ε

Z

Z

∇un (t) : ∇w j dx

Ω

(4.6)

(|un |2 − 1)un · w j dx = 0, j = 1, . . . , n,

Ω

completed by initial conditions (4.7)

0

un (0) = u0 ; un (0) = u1n , 0

with u1n → u1 in L2 (Ω)3 as n → ∞. (Hereafter we use notation u =

∂u and similar for ∂t

second order derivatives). Owing to general results for non-linear system of ordinary differential equations problem (4.6), (4.7) has a solution in the interval (0, tn ) and due to a priori estimates obtained below tn = T . 3) To get the a priori estimates we multiply the j-th equation in (4.6) by g jn (t) and sum n X 0 0 these equations. Then taking into account the equality un (t) = gin (t)w i we obtain 0

i=1

1d 2 dt

Z Ω

|

∂un 2 1d | dx + ∂t 2 dt

Z

|∇un |2 dx +

Ω

1 d 4ε dt

Z

(|un |2 − 1)2 dx = 0.

(4.8)

Ω

It follows from (4.8) that the sequence {un } is bounded in L∞ (0, T ; H 1 (Ω)3 ) while {un } 0

is bounded in L∞ (0, T ; L2 (Ω)3 ) and as a consequence, the sequence {un } is bounded in H 1 (Ω × (0, T ))3 ). Recall also, that H 1 (Ω × (0, T ))3 ⊂ Lq (Ω × (0, T ))3 compactly for any 55

q < ∞ ( Ω ⊂ R2 ). This means that there exists a subsequence (we keep the same notation for it ) such that    un → u ∗ − weakly in L∞ (0, T ; H 1 (Ω)3 ),     0 0 un → u ∗ − weakly in L∞ (0, T ; L2 (Ω)3 ),      u → u strongly in L∞ (0, T ; Lq (Ω)3 )∀q < ∞ almost everywhere. n

(4.9)

(|un |2 − 1)un → (|u|2 − 1)u weakly in Lq (Ω × (0, T ))3 , ∀q < ∞.

(4.10)

Similar to [49] one can prove that

Using (4.9) and (4.10) we pass to the limit as n → ∞ in (4.6) for a fixed j and obtain Z Z Z d2 1 (|u|2 − 1)u · w j dx = 0 u(t) · w j dx + ∇u(t) : ∇w j dx + dt2 ε Ω

Ω

Ω

and then, in view of the density of the basis {w 1 , . . . , w n , . . .}, we get Z Z Z 1 d2 u(t) · wdx + ∇u(t) : ∇wdx + (|u|2 − 1)u · wdx = 0, ∀w ∈ V0 . dt2 ε Ω

Ω

Ω

The last equation means that the limit function u is a weak solution of (4.5). By standard methods one can prove that u (0) = u1 . 0

4) To prove the uniqueness we take two solutions u and v and let z = u − v. Then  2 ∂ z 1 1   − ∆z = (|v|2 − 1)v − (|u|2 − 1)u,   2  ε ε  ∂t 0 (4.11) z(0) = z (0) = 0,      0  z ∈ L∞ (0, T ; H01(Ω)3 ), z ∈ L∞ (0, T ; L2 (Ω)3 ).

Multiplying (formally) the first equation in expression (4.11) by z we get Z Z Z 1d 1d 1 0 0 2 2 |z (t)| dx + |∇z| dx = ((|v|2 − 1)v − (|u|2 − 1)u)z dx. 2 dt 2 dt ε 0

Ω

Ω

(4.12)

Ω

For the right-hand side we have the following estimate (H 1 (Ω)3 ⊂ Lq (Ω)3 for any q < ∞): Z Z 1 0 0 2 2 ((|v| − 1)v − (|u| − 1)u)z dx ≤ c(ε) (1 + |v|2 + |u|2 )|z||z |dx ε Ω

Ω

56

0

≤ c(ε)(1 + kv(t)kH 1 (Ω)3 + ku(t)kH 1 (Ω)3 )kz(t)kH 1 (Ω)3 kz (t)kL2 (Ω)3 0

(4.13)

≤ c(ε)kz(t)kH 1 (Ω)3 kz kL2 (Ω)3 . From (4.12) and (4.13) we have 0

kz (t)k2L2 (Ω)3 + kz(t)k2H 1 (Ω)3 ≤ 2c(ε)

Zt

0

(kz (σ)k2L2 (Ω)3 + kz(σ)k2H 1 (Ω)3 )dσ,

0

hence z = 0. The justification of this formal transformation (multiplication of equation (4.11) by z ) 0

can be found, for example, in [49]. 5) The last point is the proof of the regularity of the solution for more regular data. For this we use the traditional approach. Namely, we let {w 1 , . . . , wn , . . .} be a linearly independent basis in V0 ∩ H 2 (Ω)3 . Differentiating equation (4.6) by t and multiplying it by gjn (t) we get Z Z Z 00 1d ∂un 2 1 1d 0 2 0 00 0 00 | dx + | |∇un | dx = − ((|un |2 − 1)un un + 2(un un )(un un )dx. 2 dt ∂t 2 dt ε 00

Ω

Ω

Ω

The right-hand side is estimated above by Z 1 0 00 0 00 | ((|un |2 − 1)un un + 2(un un )(un un ))dx| ε Ω

00

0

00

0

≤ c(ε)kun kL2 (Ω)3 kun kH 1 (Ω)3 kun kH 1 (Ω)3 ≤ c(ε)kun kL2 (Ω)3 kun kH 1 (Ω)3 , thus, 1d 00 0 00 0 (kun k2L2 (Ω)3 + kun k2H 1 (Ω)3 ) ≤ c(ε)kun kL2 (Ω)3 kun kH 1 (Ω)3 , 2 dt From here we derive the estimate 00

0

kun (t)k2L2 (Ω)3 + kun (t)k2H 1 (Ω)3 ≤ c(ε)(1 +

Zt

00

0

(kun (σ)k2L2 (Ω)3 + kun (σ)k2H 1 (Ω)3 )dσ)

0

and the boundness {un } in L∞ (0, T ; H01 (Ω)3 ), {un } in L∞ (0, T ; L2 (Ω)3 ). 0

00

57

Thus, the limit function (solution to the problem) is such that u ∈ L∞ (0, T ; H01 (Ω)3 ) and 0

00

u ∈ L∞ (0, T ; L2 (Ω)3 ). Remark 4.2. Model (4.5), (4.2), (4.3) is energy preserving. Namely, by passing to a limit as n → ∞ in equation (4.8) we get Z Z Z 1d 1d ∂u 2 1 d 2 | | dx + |∇u| dx + (|u|2 − 1)2 dx = 0, 2 dt ∂t 2 dt 4ε dt Ω

Ω

Ω

which implies that the energy Z Z Z 1 1 ∂u(t) 2 1 2 | dx + Eε (t) = | |∇u(t)| dx + (|u(t)|2 − 1)2 dx 2 ∂t 2 4ε Ω

Ω

Ω

of the system is constant in time. Due to the assumption that |u0 (x)|2 = 1 the energy E(0) does not depend upon the penalty parameter ε and reduces to Z Z 1 2 E(0) = |u1 | dx + |∇u0 |2 dx . 2 Ω

Ω

This equality and the energy conservation imply the following estimate for any t > 0 Z Z Z 2 2 2 (|u| − 1) dx ≤ 2ε |u1 | dx + |∇u0 |2 dx . (4.14) Ω

Ω

Ω

4.3 Discretization of the problem First, we construct a semidiscrete problem by approximating the time derivative in equation (4.5). Let ∆t > 0 be a time step and set un ≈ u(n∆t) for n = 1, 2, . . .. Then, we consider the following time discretization scheme:

+

1 2ε

n+1 u + 2un + un−1 un+1 + un−1 − 2un −∆ ∆t2 4 ! n 2 2 n+1 n n−1 u + un+1 un + un−1 u + 2u + u + −2 = 0, 2 2 4

with corresponding boundary conditions. Scheme (4.15) is initialized with   u0 = u0 , u1 − u−1 = 2∆tu1 ,  u1 − 2u0 + u−1 = ∆t2 ∆u0 . 58

(4.15)

(4.16)

The semidiscrete problem keeps (in some sense) the energy preserving property of the initial problem. Approximate the energy Eε at t = (n + 1/2)∆t by 2 2 Z Z 1 un+1 − un 1 un+1 + un n+1/2 Eε = dx + 2 ∇ dx 2 ∆t 2 Ω

Ω

1 + 4ε

Z Ω

!2 n+1 n 2 u + u − 1 dx; 2

(4.17)

multiplying (4.15) by (un+1 − un−1 )/2 and integrating in x over Ω we obtain Eεn+1/2 = Eεn−1/2 ∀n.

(4.18)

The basis of the previous formal calculations will be shown below, where the existence of a weak solution to problem (4.15), (4.16) and its corresponding regularity will be proved. We construct a finite dimensional approximation of the semidiscrete scheme (4.15) via the finite element method; the simplest quadrature formulas (based on the trapezoidal rule) are used to avoid non-diagonal mass matrices. Let Th be a regular triangulation of Ω, that we assume to be polygonal. For simplicity we denote by e the generic element of triangulation Th . By Vh ⊂ (H 1 (Ω))3 we denote the fi-

¯ vih |e ∈ P1 (e) for all i and for all e ∈ nite element space: {v h = (v1h , v2h , v3h ), vih ∈ C 0 (Ω), T Th }. For simplicity, let the function g(x) be continuous, i.e. g(x) ∈ (H 1/2 (∂Ω) C 0 (∂Ω))3 and let g h be it’s piecewise linear interpolate: g h (x) = g(x) at the boundary vertices.

We introduce the space Vhg = {v h ∈ Vh : v h = g h on ∂Ω} and a test function space T Vh0 = (H01 (Ω))3 Vh ). Further {ai }3i=1 will be the vertices of an element e ∈ Th of measure |e|. For any function φ(x) ∈ C 0 (¯ e), we use the quadrature formulas Z e

3

1 X φ(ai ). φ(x)dx ≈ Se (φ) ≡ |e| 3 i=1

¯ we use Sh (φ) = For a function φ ∈ C 0 (Ω)

59

X ¯ e∈Ω

Se (φ).

Now a discretization scheme for problem (4.5), (4.2), (4.3) can be written as follows: find uh ∈ Vhg , such that for all v h ∈ Vh0 and all n = 1, 2, . . . Sh

1 Sh 2ε

un+1 + uhn−1 − 2unh h · vh ∆t2

+

Z

∇

un+1 + 2unh + uhn−1 h : ∇v h dx+ 4

Ω

! ! n+1 n+1 n−1 n uh + unh 2 unh + uhn−1 2 u + 2u + u h h h + −2 · v h = 0, 2 2 4

(4.19)

with the discrete analogue of (4.16):    u0h = u0h ,    Sh (u1h · v h ) = Sh ((u0h + ∆tu1h ) · v h )   2 R   − ∆t2 ∇u0h : ∇v h dx ∀v h ∈ Vh0 . 

(4.20)

Ω

Here, u0h ∈ Vhg and u1h ∈ Vh0 approximate in some sense u0 and u1 , respectively. Remark 4.3. Higher order finite element methods can be also employed; the technique discussed below would still apply. The fully discrete scheme (4.19), (4.20) inherits the energy preserving property of the continuous and semidiscrete problems: if the discrete energy is defined by ! n+1 Z n 2 n 2 un+1 1 1 + u u − u n+1/2 h h h h + ∇ dx+ = Sh Eh 2 ∆t 2 2 Ω

!2  2 n+1 n u 1 + uh , Sh  h −1 4ε 2 

then

n+1/2

Eh

1/2

= Eh

(4.21)

(4.22)

1 − uhn−1 ) ∈ Vh0 in (4.19). for all n ≥ 1. To prove it, take v h = (un+1 2 h Theorem 4.2. 1) The fully discrete problem (4.19) has a solution for any h, ∆t and ε > 0. 60

√ 2) If u1 ∈ Lp (Ω)3 and ∆u0 ∈ Lp (Ω)3 for a p > 2, and ∆t ≤ 2 ε, then the semidisun+1 + 2un + un−1 crete problem (4.15) has a solution, such that un ∈ Lp (Ω)3 and ∈ 4 H 1 (Ω)3 for n = 1, 2, ... 3) If u1 ∈ H 1 (Ω)3 and ∆u0 ∈ H 1 (Ω)3 , then problem (4.15) has a solution, such that un ∈ H 1 (Ω)3 for all n = 1, 2, . . .. Proof. 1) To prove the existence of the solution for the fully discrete problem (4.19) we use the following variant of the Brouwer theorem [49]: Let V be a finite dimensional linear space endowed with an inner product (., .) and norm ||.||. Let P : V → V be a continuous mapping with the following property: there exists ρ > 0 such that (P ξ, ξ) ≥ 0 on the sphere ||ξ|| = ρ. Then there exists a vector ξ ∗ , ||ξ ∗ || ≤ ρ such that P ξ ∗ = 0. We equip the space Vh0 with the inner product (v h , wh )h = Sh (v h · w h ) and the corre1 1/2 0 − un−1 sponding norm ||v h ||h = Sh (v h · v h ). For a fixed n ≥ 1 let ξh = (un+1 h h ) ∈ Vh 2 and operator P : Vh0 → Vh0 be defined by the left-hand side of (4.19). In other words, we write (4.19) in the form (P ξ h , v h )h = 0 ∀vh ∈ Vh0 . The operator P is obviously continuous. Now, if we take v h = ξ h in (4.19), then n+1/2

(P ξh , ξ h )h = Eh

n−1/2

− Eh

.

(4.23)

are Let cn be a generic constant depending on unh and uhn−1 (we recall that unh and un−1 h supposed to be known). Then obviously, n−1/2

Eh

≤ cn

and n+1/2

Eh

≥

1 1 n 2 n+1 − u | ) − c ≥ S (|u Sh (ξ h · ξh ) − cn . n h h h 2(∆t)2 4(∆t)2

Hence, (P ξh , ξh )h ≥

1 Sh (ξ h · ξh ) − cn 4(∆t)2 61

and for ||ξh ||h large enough we have (P ξ h , ξ h )h ≥ 0, implying that problem (4.19) has a solution from the variant of the Brouwer theorem above. 2) Now we study the solvability of the semidiscrete problem (4.15), using the limit as h → 0 in the fully discrete problem. As it was mentioned in Remark 4.3, the previous existence result holds also for a fully discrete problem with high order finite element approximation in space variables (Lagrange or Hermit finite elements). Further we suppose in this the case, in particular, that ||∆u0h − ∆u0 ||Lp → 0. Let us introduce, for n ≥ 1 fixed, z n = 1/4(un+1 + 2un + un−1 ). We then have the following weak formulation of the problem (4.15): find un+1 ∈ Lp (Ω)3 with p > 2, such that z n ∈ H 1 (Ω)3 and 2 Z Z 4 n 1 un + un+1 n z η dx + ∇z ∇η dx + + ∆t2 2ε 2 Ω Ω !Ω n Z 2 u + un−1 4 n − 2 z n η dx = u η dx, ∀η ∈ H01 (Ω)3 . 2 ∆t2 Z

(4.24)

Ω

Under the assumptions un , un−1 ∈ Lp (Ω)3 , and for z n , η ∈ H 1 (Ω)3 , the integral ! Z n u + un+1 2 un + un−1 2 + zη dx 2 2 Ω

is well-defined, because of the Hölder inequality (

Z

uvw ≤ ||u||Lp1 ||v||Lp2 ||w||Lp3 , 1/p1 +

1/p2 + 1/p3 = 1) and the continuous embedding of H 1 (Ω) ⊂ Lq (Ω) for any q < ∞. From equation (4.16) we have that u1 = u0 + ∆tu1 + 1/2∆t2 ∆u0 ; if u1 ∈ Lp (Ω)3 and ∆u0 ∈ Lp (Ω)3 , then u0 , u1 ∈ Lp (Ω)3 . Let ||∆u0h − ∆u0 ||Lp → 0 and ||u1h − u1 ||Lp → 0 as h → 0. This ensures the uniform boundness in h in the Lp -norm of sequences {u1h }h and {∆u0h }h . Now, choose v h = z nh for (4.19) and we obtain 1 4 4 − Sh (|z nh |2 ) + ||∇z nh ||20 ≤ Sh (|unh ||z nh |). 2 2 ∆t ε ∆t 62

√ Under the induction assumption ||unh ||Lp ≤ c 6= c(h) (using the inequality ∆t ≤ 2 ε) we get the uniform estimate in h ||∇z nh ||0 ≤ c 6= c(h). Using this estimate and the induction assumptions ||uhn−1 ||Lp , ||unh ||Lp ≤ c 6= c(h) we can choose from the sequence {un+1 h }h a subsequence (we keep the same notation for it) such that z n+1 → z n+1 = 1/4(un+1 + 2un + un−1 ) h

weakly in H 1 (Ω)3 ,

un+1 → un+1 h

strongly in Lp (Ω)3 .

(4.25)

Now, passing to the limit as h → 0 in equation (4.19) with v h → v strongly in H 1 (Ω)3 , we derive the weak form of equation (4.15). 3) Under the assumptions of this section we prove that the sequences {||u0h ||1 }h and 1/2

{||u1h ||1 }h are uniformly bounded in h. So, Eh n+1/2

(4.22) Eh

≤ c 6= c(h) and as a consequence of

≤ c 6= c(h, n) for any n. This means that ||unh ||1 ≤ c 6= c(h, n) and we can

choose the subsequences of {unh }h , which converge weakly in H 1 (Ω)3 for any n. The rest of the proof is the same as in the previous case. Remark 4.4. When u1 ∈ H 1 (Ω)3 and ∆u0 ∈ H 1 (Ω)3 the energy E n in (4.17) is well defined and the energy preserving property (4.18) holds for any n. Direct calculations show that Eε1/2

1 = 2

Z 1 Z u − u0 2 dx + 1 ∆t 2 Ω

Z

Ω

Ω

|u1 |2 dx +

Z

u1 + u0 2 ∇ dx + 1 c1 ∆t4 = 2 ε

1 |∇u0 |2 + (c2 ∆t4 + c3 ∆t) ε Ω

(4.26)

with constants c1 , c2 and c3 depending on the H 1 (Ω)3 -norms of u1 and ∆u0 . As a consequence of (4.18) and (4.26) we have the following estimate  !2 n+1 Z Z 1 Z n 2 u u − u0 2 + u − 1 dx ≤ 2ε  dx + ∆t 2 Ω Ω Ω

63

 u1 + u 0 2 ∇ dx+O(∆t4 ) 2

= 2ε

Z

2

Ω

|u1 | dx +

Z

Ω

|∇u0 |2 dx + O(∆t) for all n ≥ 2.

(4.27)

Remark 4.5. The energy preserving properties (4.18) and (4.22) ensure the unconditional stability of schemes (4.15) and (4.19) respectively. It follows from the proof of Lemma 4.2 that a solution of the semidiscrete problem (4.15) is a weak limit when h → 0 of solutions of the fully discrete problem (4.19). When the assumptions u1 ∈ H 1 (Ω)3 and ∆u0 ∈ H 1 (Ω)3 hold, we get from (4.18) the un+1 − un ||0 } and uniform boundness in ∆t (or, equivalently, in n) of the sequences {|| ∆t un+1 + un {||∇ ||0 }. Using this result, we can prove that a solution of problem (3.7) – 2 (4.5) is a weak limit (in the corresponding norms) of a subsequence of solutions to problem (4.15). Theorem 4.3. Let u1 ∈ H 1 (Ω)3 and ∆u0 ∈ H 1 (Ω)3 . There exists a constant c˜, independent of ∆t and ε, such that for ∆t ≤ c˜ε problems (4.15) and (4.19) have unique solutions. Proof. We prove the uniqueness result only for problem (4.15), the proof for (4.19) is similar. Let un+1 and un+1 be two solutions of (4.15), and w = un+1 − un+1 ; assume that 1 2 1 2 uk1 = uk2 for all k ≤ n. From (4.15) we obtain 1 1 2 2 ||w||20 + ||w|| + ||∇w|| ≤ 0 0 2 ∆t ε c ε

Z

|w|2 |un+1 + un+1 + 2un1 ||un+1 + u2n−1 + 2un2 |dx 1 2 2

(4.28)

Ω

We estimate the right-hand side of (4.28) (denoted by S) by using the Hölder inequality, the continuous embedding H 1 (Ω) ⊂ Lq (Ω), ∀q < +∞, and the uniform boundness in n of the H 1 (Ω)3 -norms for the solutions of (4.15): 1 c S ≤ ||w||20 + max(||uk1 ||21 + ||uk2 ||21 )||w||0 ||w||1 ε ε k ≤

c ||w||20 + ||w||21 . ε2 64

Here c is a generic constant independent of n and ε. Now it is easy to see that for ∆t ≤ c˜ε with appropriate c˜, we have ||w||20 ≤ 0.

4.4

Solution method for the fully discrete problem

We solve equation (4.19) for n ≥ 1. Hereafter we drop the index h from the functions. Let us introduce at each time step the auxiliary variable z = (un+1 + 2un + un−1 )/4, then problem (4.19) can be written as follows (4z − 4un ) − ∆t2 ∆h z +

∆t2 (|2z − un−1/2 |2 + |un−1/2 |2 − 2)z = 0, 2ε

(4.29)

where un−1/2 = (un + un−1 )/2, and −∆h is the discrete Laplace operator. Equation (4.29) is completed by Dirichlet boundary conditions. Let us now denote the left-hand side of (4.29) by F (z). Obviously, function F (z) is differentiable and F 0 (z)w = 4w − ∆t2 ∆h w +

∆t2 (|2z − un−1/2 |2 + |un−1/2 |2 − 2)w 2ε 2 + ∆t2 (2z − un−1/2 , w)z, ε

where (.,.) is the inner product in R3 and matrix F 0 (z) has the form ∆t2 (|2z − un−1/2 |2 + |un−1/2 |2 − 2))I − ∆t2 ∆h (4.30) 2ε 4∆t2 T 2 2 zz − ∆t z(un−1/2 )T , + ε ε √ with the unit matrix I. Under the assumption ∆t ≤ 2 ε, the matrix F 0 (z) is positive (4 +

definite but not symmetric due to the term (2/ε)∆t2 z(un−1/2 )T . In order to solve the equation F (z) = 0 we use the following Quasi-Newton method F00 (z k )(z k+1 − z k ) + F (z k ) = 0, k = 0, 1, 2, ... where F00 (z) = (4 +

2∆t2 T ∆t2 (|2z − un−1/2 |2 + |un−1/2 |2 − 2))I − ∆t2 ∆h + zz , 2ε ε 65

(4.31)

√ which is obviously positive definite for ∆t ≤ 2 ε. Clearly, F00 (z) differs from F 0 (z) by the term 2ε ∆t2 z(z − un−1/2 )T which is equal to 1 3 un+1 + 2un + un−1 un+1 − un−1 T ∆t ε 4 2∆t at the exact solution and is expected to be ”small” enough when starting from a ”good” initial guess. To solve problem (4.31) at each time step we use a conjugate gradient algorithm.

4.5 Numerical results We have performed numerical experiments for problem (4.2)-(4.5) in Ω = (0, 1)×(0, 1) by using finite difference in space for (4.15) with a uniform mesh step size h. This difference scheme can be treated as a finite element scheme with P1 -interpolation and trapezoidal quadrature formulas to approximate the integrals. We considered different input data and different mesh and time step sizes h and ∆t, and penalty parameter ε > 0. We have found that our numerical results agree very well with the constrained verification estimate (4.27). Moreover, our calculations showed that the discrete scheme is stable and that the iterative method (4.31) is convergent with a high rate of convergence under much less restrictive conditions between h, ∆t and ε then those given by the developed theory. The following graphs show the energy and max ||u|2 − 1| behavior in time, the shape x

of the u3 component of u for different values of ε with the initial data defined as follows u01 =

x − x∗ y − y∗ cos(πx − πy) , u02 = , u03 = , r r r u1 = 0,

and boundary data given by: g1 = with r =

x − x∗ y − y∗ cos(πx − πy) , g2 = , g3 = , r r r

p ((x − x∗ )2 + (y − y ∗ )2 + cos2 (πx − πy)) and (x∗ , y ∗ ) = (1.1, 1.1). 66

10

0.5

9

0.4

8

0.3

7

0.2

6

0.1

5

0

4

−0.1

3

−0.2

2

−0.3

1

−0.4

0

0

10

20

30

40

50

60

70

80

90

100

−0.5

0

10

20

30

40

50

60

70

80

90

100

Figure 4.1: Energy behavior (left) and max ||u|2 − 1| (right) for ε = 10−3 , h = x

1/50, ∆t = 1/100. Remark 4.6. In both graphs in Figure 4.1 (which show the history of the discrete energy and deviation from the constraint |u|2 = 1), we observe oscillations taking place at the beginning of the evolution. These oscillations damped out as t increases. In order to reduce this unwanted phenomenon we have modified scheme (4.15), (4.16) as follows: • We started (4.15) from n = 0, instead of n = 1. • We used u0 = u0 and u1 − u−1 = 2∆tu1 as initial conditions. This modification slightly improves the numerical result (level of oscillations at the start is reduced by about 10%).

67

10

0.5

9

0.4

8

0.3

7

0.2

6

0.1

5

0

4

−0.1

3

−0.2

2

−0.3

1

−0.4

0

0

10

20

30

40

50

60

70

80

90

100

−0.5

0

10

20

30

40

50

60

70

80

90

100


1/50, ∆t = 1/100.

10

0.5

9

0.4

8

0.3

7

0.2

6

0.1

5

0

4

−0.1

3

−0.2

2

−0.3

1

−0.4

0

0

10

20

30

40

50

60

70

80

90

100

−0.5

0

10

20

30

40

50

60

70

80

90

100


1/50, ∆t = 1/100.

68

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 50

−1 50 40

40

50 30

50 30

40 30

20

20

10 0

40 30

20

20

10

10

0

0

10 0

Figure 4.4: u3 component of u at t = 1/4 (left) and t = 1/2 (right) for ε = 10−3 , h = 1/50, ∆t = 1/100.

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 50

−1 50 40

50 30 30

20 0

50 40 30

20

20

10

40 30

40

20

10

10

0

0

10 0

Figure 4.5: u3 component of u at t = 3/4 (left) and t = 1 (right) for ε = 10−3 , h = 1/50, ∆t = 1/100.

69

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 50

−1 50 40

40

50 30

50 30

40 30

20

20

10 0

40 30

20

20

10

10

0

0

10 0

Figure 4.6: u3 component of u at t = 1/4 (left) and t = 1/2 (right) for ε = 10−5 , h = 1/50, ∆t = 1/100.

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 50

−1 50 40

50 30 30

20 0

50 40 30

20

20

10

40 30

40

20

10

10

0

0

10 0

Figure 4.7: u3 component of u at t = 3/4 (left) and t = 1 (right) for ε = 10−5 , h = 1/50, ∆t = 1/100.

70

Chapter 5 Mathematical modeling of blood flow in compliant arteries 5.1 Introduction 5.1.1 Background of the problem In this chapter we consider a problem of modeling blood flow in the human arteries. The focus is on numerical simulation of the flow through branching arteries with an inserted prostheses called stents and stent-grafts. Both the arteries and the stents are assumed to be elastic. The Young’s modulus of elasticity for arteries was obtained from [60]. The Young’s modulus of elasticity of stents was obtained from the measurements by K. RaviChandar and R. Wang [88, 89]. The underlying problem consists of modeling the flow through the human aorta branching into iliac arteries (see Figure 5.1), that suffers from an aortic abdominal aneurysm (AAA). An aneurysm can be described as a bulging of the human aorta (Figure 5.1). There is a 90% mortality rate associated with AAA rupture. Treatment of AAA consists of either surgical interventions or nonsurgical AAA repair for high-risk patients. Non-surgical repair 71

Figure 5.1: Abdominal aorta with and without aneurysm. entails inserting a prosthesis (either a stent or a stent-graft) inside the diseased aorta to redirect the flow of blood and lower the pressure to the aneurysmal walls, thereby lowering the probability of rupture. Stents and stent-grafts are tube-like devices made of stainless steal or nitinol with a mesh (stent) that can be covered with polyester or Dacron grafts. In a bifurcated aneurysm, bifurcated stent-grafts are typically used for AAA-repair. There are various complications associated with this procedure, they include stent-graft migration, occlusion of the graft limbs and formation of new aneurysms near the anchoring sites [82]. The purpose of the study presented here is to use mathematical modeling and numerical simulations to understand some of the hemodynamic factors that might be responsible for the complications listed above, and suggest an improved prosthesis design that might minimize some of the problems. More details will be presented in section 5.5.

72

Σε R+ηε

R x

Ωε


5.1.2 Mathematical model We use a reduced, one dimensional model, together with the special conditions derived via Riemann invariants, to simulate the flow through the branching arteries. The model is derived using an asymptotic reduction of the Navier-Stokes equations modeling blood with the Navier equation for the wall. The resulting set of equations is used to study the optimal prostheses design by investigating the impact of shear stress rates on the initiation of focal atherogenesis.

5.2

Effective model derivation

In this section we follow the derivation of effective equations given in detail in [85].

5.2.1 The 3-D fluid-structure interaction problem We consider the unsteady axisymmetric flow of a Newtonian incompressible fluid in a thin elastic cylinder whose radius is small with respect to its length. We define the ratio ε = R/L, where R is the radius and L is the length of the cylinder. Now, for every fixed ε > 0 we introduce the computational domain (Figure 5.2) Ωε (t) = {x ∈ R3 ; x = (rcosΘ, rsinΘ, x), r < R + η ε (x, t), 0 < x < L} 73

This domain is filled with fluid modeled by the incompressible Navier-Stokes equations. We assume that the angular velocity is zero, then the problem in an Eulerian framework in cylindrical coordinates in Ωε (t) × R+ reads as follows 2 ε ε ε ε ∂vr ∂ vr 1 ∂vrε ∂ 2 vrε vrε ∂pε ε ∂vr ε ∂vr + + vr + vx + + − = 0, ρ −µ ∂t ∂r ∂x ∂r 2 r ∂r ∂x2 r2 ∂r 2 ε ε ε ∂vxε ∂pε ∂ vx 1 ∂vxε ∂ 2 vxε ε ∂vx ε ∂vx ρ + vr + vx + + + = 0, −µ ∂t ∂r ∂x ∂r 2 r ∂r ∂x2 ∂x

∂vrε ∂vxε vrε + + = 0. ∂r ∂x r

(5.1)

(5.2) (5.3)

Further we assume that the lateral wall of the cylinder Σε (t) = {r = R + η ε (x, t)} × (0, L) is elastic and allows only radial displacement. We can describe its motion (in Lagrangian coordinates) using the Navier equations for a linearly elastic membrane. The radial contact force is given by

Fr = −

∂2ηε ∂2ηε h(ε)E(ε) η ε + h(ε)G(ε)k(ε) − ρ h(ε) , w 1 − σ 2 R2 ∂x2 ∂t2

(5.4)

where Fr is the radial component of external forces (coming from the stresses induced by the fluid), η ε is the radial displacement from the reference state Σ0ε := Σε (0), h = h(ε) is the membrane thickness, ρw is the wall volumetric mass, E = E(ε) is the Young’s modulus, 0 < σ ≤ 0.5 is the Poisson ratio, G = G(ε) is the shear modulus and k = k(ε) is the Timoshenko shear correction factor ([69, 72]). The fluid equations are coupled with the membrane equation through the lateral boundary conditions requiring the continuity of velocity and balance of forces. We require that the fluid velocity evaluated at the deformed interface (R + η ε , x, t) equals the Lagrangian velocity of the membrane

vrε (R

∂η ε (x, t) on (0, L) × R+ + η , x, t) = ∂t ε

vxε (R + η ε , x, t) = 0 on (0, L) × R+ 74

(5.5) (5.6)

Now we consider the balance of forces, namely we set the radial contact force equal to the radial component of the force exerted by the fluid. The fluid contact force is given in Eulerian coordinates as, Ff = ((pε − pref )I − 2µD(v ε))ner , where D(v ε )) is the symmetrized gradient of velocity defined by, 1 D(v ε ) = (∇v ε + (∇v ε )t ). 2 We need the Jacobian of the transformation from Eulerian to Lagrangian coordinates in order to perform the coupling. We consider Borel subsets B of Σ0ε and require that r Z Z ∂η ε 2 ε ε ε ((p − pref )I − 2µD(v ))ner (R + η (x, t)) 1 + ( ) dx = −Fr R dx, ∂x B

(5.7)

B

for all B ∈ Σ0ε , where J =

q ε 1 + ( ∂η )2 is the Jacobian determinant of the transformation ∂x

from dx to dΣ0ε /(2πR). Pointwise we have

ηε −Fr = ((pε − pref )I − 2µD(v ε ))ner (1 + ) R

r

1+(

∂η ε 2 ) on Σ0ε × R+ . ∂x

(5.8)

The initial data are given by, ηε =

∂η ε = 0 and v ε = 0 on Σε (0) × {0}. ∂t

(5.9)

We assume that the end-points of the tube are fixed and that the problem is driven by a time-dependent pressure drop between the inlet and outlet boundary. Therefore, we have the following inlet-outlet boundary conditions: vrε = 0, pε + ρ(vxε )2 /2 = P1 (t) + pref on (∂Ωε (t) ∩ {x = 0}) × R+ ,

(5.10)

vrε = 0, pε + ρ(vxε )2 /2 = P2 (t) + pref on (∂Ωε (t) ∩ {x = L}) × R+ ,

(5.11)

η ε = 0 for x = 0, η ε = 0 for x = L and ∀t ∈ R+ . We will assume that the pressure drop, A(t) = P1 (t) − P2 (t) ∈ C0∞ (0, +∞). 75

(5.12)

Parameters

Values

ε

0.002-0.06

Characteristic radius: R0

0.0025-0.012 m

Characteristic length: L

0.065-0.2 m

Young’s modulus: E

105 − 106 Pa [60]

Wall thickness: h

1 − 2 × 10−3 m

Blood density: ρ

1050 kg/m3

Reference pressure: p0

13000 Pa = 97.5 mmHg

Table 5.1: Parameter values

5.2.2 Weak formulation and a priori estimates In this section we will introduce the variational formulation of the problem (5.1)-(5.3), (5.5)-(5.12) and a priori solution estimates, which are necessary for the reduced model derivation. We proceed, first, by introducing the following norms

||P12 (q, T )||2ν = max{||P12 ||∞ , ||P22 ||∞ } Z T 0 0 2 21 +4q T max{P12 (qτ ), P22 (q, τ )} dτ, T 0 ||A(q, T )||2aver

1 = T

Z

(5.13)

T 0

|A(qτ )|2 dτ,

P ≡ ||P12 (q, T )||2ν + 24π 2 T 2 ||A(q, T )||2aver ,

(5.14)

(5.15)

where q is the frequency of oscillations. Further, we have the following bounds for the velocities v ε and radial displacements η ε , 76

sup 0≤t≤T

2µ π

Z

h(ερw R) ∂η ε h(ε)E(ε) ε 2 ||η (t)|| + || (t)||2L2 (0,L) 2 L (0,L) R(1 − σ 2 ) 2 ∂t h(ε)G(ε)R ∂η ε R3 L(1 − σ 2 ) 2 2 + || (x)||L2 (0,L) ≤ 2 P , 2 ∂t h(ε)E(ε)

T 0

||D(v ε )(τ )||2L2 (Ωε (τ )) dτ +

(5.16)

ρ R3 L(1 − σ 2 ) 2 sup ||v ε )(τ )||2L2 (Ωε (τ )) ≤ 2 P . (5.17) 2π 0≤t≤T h(ε)E(ε)

See [85] for more details. Now, we are ready to introduce the solution and tests spaces. Let the space Γ consists of all the functions γ ∈ L∞ (0, T ; H 1 (0, L))∩W 1,∞ (0, T ; L2 (0, L)) such that γ(t, 0) = γ(t, L) = 0 and such that the bound (5.16) is satisfied. The space U consists of all the functions u = (ur , ux ) ∈ L2 (0, T ; H 1/2−δ (ΩRm ax )) × H 1 (ΩRmax ) ∩ L∞ (0, T ; L2 (ΩRmax )2 ) for some δ > 0, such that ∇ · u = 0 in ΩRmax × R+ , ur = 0 for x = 0, L and the bound (5.17) is satisfied. Let Ωγ (t) = {(r, x)|0 < r < R + γ(t, x), x ∈ (0, L)} and Σγ (t) = {r = R + γ(t, x)} × (0, L). The set of solution candidates K consists of all the functions (γ, u), where the u’s are axially symmetric, such that ∂γ for R + γ(t, x) ≤ r ≤ Rmax , ∂t ur ∈ H 1 (Ωγ (t))ux (r, x, t) = 0 for R + γ(t, x) ≤ r ≤ Rmax }.

K = {(γ, u) ∈ Γ × U |ur (r, x, t) =

(5.18)

Let us now define the space

V (Ωγ (t)) = {ϕ = ϕr er + ϕx ex ∈ H 1 (Ωγ (t))|ϕr (r, 0) = ϕr (r, L) = 0, ϕx (R + γ(x, t), x) = 0 and ∇ · ϕ = 0 in Ωγ (t)a.e.}, 77

(5.19)

then the test space is the space H 1 (0, T ; V (Ωγ (t))) The weak formulation of the problem (5.1)-(5.3) and (5.5)-(5.12) is the following

For a given (γ, u) ∈ K find (η ε , vrε , vxε ) ∈ K such that ∀ϕ ∈ H 1 (0, T ; V (Ωγ (t))) we have

2µ

R

D(v ε ) : D(ϕ)r drdx + ρ

Ωγ (t)

R

ε

Ωγ (t)

{∂ v∂t + (u(t)∇)v ε }ϕ r drdx

RL 2 ε ∂ +R {h(ε)G(ε)k(ε) ∂∂xη2 ∂x ϕr (R + γ(x, t), x, t) + 0

+Rρw h

RL ∂ 2 ηε 0

∂t2

h(ε)E(ε) η ε ϕ (R 1−σ 2 R2 r

+ γ(x, t), x, t)} dx

RR ϕr (R + γ(x, t), x, t)} dx = − {P2 (qt) − ρ2 (ux vxε )|x=L }ϕx |x=L r dr 0

RR

+ {P1 (qt) − 2ρ (ux vxε )|x=L }ϕx |x=0 r dr,

(5.20)

∂η ε = 0 on (0, L) × {0} and v ε (r, x, 0) = 0. ∂t

(5.21)

0

and

ηε =

Remark 5.1. Integrals involving scalar products with

∂η ε ∂t

and

∂ 2 ηε ∂t2

in fact are duality pair-

ings. Now, we present a priori solution estimates for the problem (5.20)-(5.21). The a priori estimates are important for determining an asymptotically reduced model (for the detailed derivation of these estimates see [85]). Theorem 5.1. The following are the a priori solution estimates for the fluid-structure interaction problem defined above R4 (1 − σ 2 )2 2 1 ε ||η (t)||2L2 (O,L) ≤ 4 P , L h(ε)2 E(ε)2

||v ε (τ )||2L2 (Ωε (τ )) ≤ 4π 78

R3 L(1 − σ 2 ) 2 P , ρh(ε)E(ε)

(5.22)

(5.23)

Z t 0

Z t 0

vrε 2 ∂vxε 2 ∂vrε 2 + || ||L2 (Ωε (τ )) + || || 2 || 2 dτ || ∂r L (Ωε (τ )) r ∂x L (Ωε (τ )) s R(1 − σ 2 ) 2 R2 P , ≤ 2µ ρh(ε)E(ε)ρ

s R(1 − σ 2 ) 2 ∂vxε 2 ∂vrε 2 R2 || ||L2 (Ωε (τ )) + || ||L2 (Ωε (τ )) dτ ≤ P . ∂r ∂x 2µ ρh(ε)E(ε)ρ

(5.24)

(5.25)

5.2.3 Asymptotic reduction of the model In order to represent the equations (5.1)-(5.3) in nondimensional form we introduce the independent variables r˜, x˜ and t˜ • r = R˜ r, • x = L˜ x, • t=

1 ˜ t, ωε

where ω ε =

1 L

q

hE . Rρ(1−σ 2 )

Based on the a priori estimates (5.22) - (5.25) we introduce the following asymptotic expansions v ε = V {˜ v 0 + ε˜ v 1 + ...}, with V =

η ε = Ξ{˜ η 0 + ε˜ η 1 + ...}, with Ξ =

s

R(1 − σ 2 ) P, ρhE

R2 (1 − σ 2 ) P, hE

pε = ρV 2 {˜ p0 + ε˜ p1 + ...}.

(5.26)

(5.27)

(5.28)

After ignoring the terms of order ε2 and smaller the momentum equations and the incompressibility condition become

79

where Sh =

∂˜ vxε ∂vzε vxε vxε 1 1 ∂ ε ∂˜ ε ∂˜ Sh r˜ = 0, + v˜x + v˜r − ∂t ∂ x˜ ∂ r˜ Re r˜ ∂ r˜ ∂ r˜

(5.29)

∂ p˜ = 0, ∂ r˜

(5.30)

∂ ∂ (˜ rv˜r ) + (˜ rv˜x ) = 0, ∂ r˜ ∂ x˜

(5.31)

Lω ε V

and Re =

ρV R2 . µL

The leading order Navier equations read G(ε)k(1 − σ 2 )ε2 ∂ 2 η˜ + O(ε2 ). −Fr = P η˜ − E(ε) ∂ x˜2

(5.32)

Since G = E/(2(1 + σ)) and k = O(1), the second term on the right-hand side can also be neglected. The asymptotic form of the contact force derived from (5.4) becomes Ξ ε ε 2 ε 2 ((p − pref )I − 2µD(v ))ner = ρV (p − pref + O(ε )) 1 + η˜ . R

Therefore, we obtain the following leading order relationship between the pressure and

the radial displacement P P Ξ Ξ η˜ (˜ p − p˜ref ) = = η˜ 1 − η˜ + ... = ρV 2 1 + RΞ η˜ ρV 2 Ξ R R 2 ! Ξ Ξ PR η˜ − η˜ ... . 2 ρV Ξ R R ε

(5.33)

We assume that Ξ/R is of order ε so the last term on right-hand side of (5.33) can be ignored and we obtain

p˜ε − p˜ref =

PR η˜, ρV 2

(5.34)

which in dimensional variables gives us the Law of Laplace,

pε − pref =

Eh η . 2 (1 − σ )R R 80

(5.35)

5.2.4 The reduced two-dimensional problem Now, having the results of the previous section in hand, we proceed by introducing the reduced two-dimensional model written in non-dimensional variables. We define the domain (scaled) Ξ ˜ t˜) = {(˜ x, t˜), 0 < x˜ < 1}, Ω( x, r˜) ∈ R2 |˜ r < 1 + η(˜ R Ξ ˜ t˜) = {˜ and the lateral boundary Σ( r =1+ R η(˜ x, t˜)} × (0, 1).

Now we formulate the two-dimensional problem in the following way,

Find a triple (˜ vx , v˜r , η˜) satisfying vxε vxε ∂ p˜ 1 1 ∂ ∂˜ vxε ∂vzε ε ∂˜ ε ∂˜ + v˜x + v˜r + = Sh r˜ , ∂ x˜ ∂ r˜ ∂ x˜ Re r˜ ∂ r˜ ∂ r˜ ∂ t˜

(5.36)

∂ ∂ (˜ rv˜r ) + (˜ rv˜x ) = 0, ∂ r˜ ∂ x˜

(5.37)

(˜ pε − p˜ref ) =

v˜r (˜ x, 1 +

PR η˜, ρV 2

∂ η˜ Ξ η˜(x, t), t˜) = , v˜x = 0, R ∂ t˜

(5.38)

(5.39)

with the initial and boundary conditions given by η˜ =

∂ η˜ = 0 at t˜ = 0, ∂ t˜

(5.40)

v˜rε = 0, p˜ = (P1 (t˜) + pref )/(ρV 2 ) on (∂Ωε (t) ∩ {˜ x = 0}) × R+ ,

(5.41)

v˜rε = 0, p˜ = (P2 (t˜) + pref )/(ρV 2 ) on (∂Ωε (t) ∩ {˜ x = 1}) × R+ ,

(5.42)

η˜ = 0 for x˜ = 0, η˜ = 0 for x˜ = 1 and ∀t˜ ∈ R+ .

(5.43)

81

The model defined above is a free-boundary degenerate hyperbolic system with parabolic regularization. This system, though already simplified, is still quite complex. The theoretical analysis and numerical simulation of (5.36)-(5.43) is a difficult task; that is why we will proceed further in order to obtain a simplified one-dimensional reduced model.

5.2.5 Derivation of the one-dimensional reduced model In order to derive a simplified one-dimensional model we express the two-dimensional equations in terms of the averaged quantities across the cross-sectional area. Ξ ˜ where η˜)2 and m ˜ = A˜U Let us introduce A˜ = (1 + R

Z 1+ Ξ η˜ R 2 ˜ = U v˜x r˜d˜ r, A˜ 0 Z 1+ Ξ η˜ R 2 2v˜x 2 r˜d˜ r. α ˜ = 2 ˜ ˜ AU 0

(5.44) (5.45)

Now we integrate the equations (5.36)-(5.37) from r˜ = 0 to r˜ = 1 +

Ξ η˜ R

and then

express them in terms of the averaged quantities. Using the no-slip condition at the lateral boundary we obtain ∂ A˜ Ξ ∂ m ˜ = 0, + ∂ t˜ R ∂ x˜ (5.46) ∂m ˜ ∂ + ˜ ∂ x˜ ∂t

2

p ∂˜ ∂ p ˜ 2 v x + A˜ . = A˜ ∂ x˜ Re ∂ r˜ Σ˜

α ˜m ˜ A˜ The next step is to specify the axial velocity profile v˜x in terms of the averaged quantiSh

ties. The typical approximation (corresponding to Poiseuille flow [90]) is " γ+2˜ v˜x = U 1− γ

r˜ Ξ 1+ R η˜

!γ #

.

(5.47)

We assume that γ = 9, which corresponds to the non-Newtonian flow of blood [78, 79]. We calculate α and obtain α = 1.1.

82

After plugging (5.47) into (5.43) the right hand side becomes −2(γ + 2) m ˜ . Re A˜ Using A0 to denote the non-stressed area R2 we obtain a one-dimensional system written in dimensional variables

∂ ∂m + ∂t ∂x

αm2 A

p := p − pref =

+

∂A ∂m + = 0, ∂t ∂x

(5.48)

A ∂p 2µ m = − (γ + 2) , ρ ∂x ρ A

(5.49)

!!

(5.50)

Eh (1 − σ 2 )R

r

A −1 A0

We use σ = 0.5, so equation (5.50) reduces to ! r A p = G0 −1 , A0 with G0 =

.

(5.51)

4Eh . 3R

5.3 Conditions at the bifurcation point We use the system of equations (5.48)-(5.49) and (5.51) to model blood flow through the abdominal aorta branching into the iliac arteries (Fig.5.3). One of the important questions here is how to model the flow at the branching point. We use the continuity of pressure and conservation of mass condition to couple the flow exiting the abdominal aorta with the flow entering the iliac arteries [62]. It was shown in [54] that the 1-D approximation to the flow at the branching point for the fixed geometry provides ε1/2 accuracy to the 3-D flow for small to moderate Reynolds numbers. The analysis demonstrating the accuracy of the 1-D approximation on compliant branching points remains an open problem. 83

I.A.(1) A.A. I.A.(2)

Figure 5.3: Blood flow through abdominal aorta (A.A) and iliac arteries (I.A.(1) and I.A.(2)). We calculate the continuity of pressure and conservation of mass condition at the branching point using the concept of Riemann invariants for a hyperbolic system. More precisely, let us write the system (5.48)-(5.49) and (5.51) in quasilinear form:        0 A 0 1 A  x  =    t + α 1 2m m m2 0 −2 α−1 ν A mx − A2 + ρ Ap (A) A mt

(5.52)

The eigenvalues of (5.52) are m λ= − A

s

s

1/2

m 1 G0 , µ= + A 2 ρ q −1/2 Let us further use the following notation: c˜ = 12 Gρ0 A0 . 1 G0 2 ρ

A A0

A A0

1/2

.

(5.53)

One can show that the right eigenvectors corresponding to λ and µ are given by     1 1 rλ =   , rµ =   . (5.54) λ µ

Consider Riemann invariants w and z corresponding to the eigenvalues λ and µ respectively. Then w and z satisfy the following equations [80]: ∇w · rλ = 0, 84

∇z · rµ = 0,

(5.55)

Direct calculations give us m w= +4 A

m −4 z= A

s

s

1 G0 2 ρ

A A0

1/2

1 G0 2 ρ

A A0

1/2

,

(5.56)

.

(5.57)

Now, we assume that the bifurcation occurs at a point and that there is no leakage at the bifurcation. Then the outflow from the abdominal aorta (”parent” vessel) should be balanced by the inflow into the iliac arteries (”daughter” vessels) [62]:

mp = md1 + md2 ,

(5.58)

which describes conservation of mass. We also assume the continuity of pressure pp = pd1 = pd2 ,

(5.59)

here subscript p corresponds to the ”parent” vessel and d1, d2 correspond to the ”daughter” vessels. The Riemann invariants for the ”parent” and the two ”daughter” vessels read as follows: mp + c˜p 4A1/4 p , Ap

(5.60)

zd1 =

md1 1/4 − c˜d1 4Ad1 , Ad1

(5.61)

zd2 =

md2 1/4 − c˜d2 4Ad2 . Ad2

(5.62)

wp =

Recalling the pressure-area relationship (5.51) and assuming that G 0 at the branching point is the same for the ”parent” vessel and the ”daughter” vessels we can rewrite (5.59)

85

in terms of the cross-sectional areas: Ad1 Ap = , A0p A0d1 (5.63) Ap Ad2 = . A0p A0d2 Adding (5.61) and (5.62) and using (5.58) we have 5/4

5/4

mp = md1 + md2 = Ad1 zd1 + c˜d1 4Ad1 + Ad2 zd2 + c˜d2 4Ad2 . Further, the use of equations (5.60) and (5.63) gives us Ap wp −

c˜p 4A5/4 p

5/4 Ap A0d1 Ap A0d1 = zd1 + c˜d1 4 A0p A0p

5/4 Ap A0d1 Ap A0d2 zd2 + c˜d2 4 + . A0p A0p Let us introduce auxiliary variables b1 =

A0d1 A0p

and b2 =

A0d2 . A0p

Then we have the following

nonlinear equation for Ap : 5/4

5/4

(wp − b1 zd1 − b2 zd2 )Ap − 4(˜ cp + c˜d1 b1 + c˜d2 b2 )A5/4 = 0. p Due to physical reasons we assume that Ap 6= 0, hence Ap can be determined through A1/4 = p

w − b1 zd1 − b2 zd2 p . 5/4 5/4 4 c˜p + c˜d1 b1 + c˜d2 b2

(5.64)

Now using equations (5.63) and (5.61)-(5.62) it is easy to evaluate A d1 , Ad2 , mp , md1 and md2 . Thus, using Riemann invariants and conditions (5.58) and (5.59) we can calculate A p , Ad1 , mp , md1 and md2 in terms of the Riemann invariants wp , zd1 and zd2 . The ”forward” Riemann invariant wp and the ”backward” Riemann invariants zd1 and zd2 are known from the initial conditions and the inlet and outlet boundary data. 86

5.4 Numerical method In order to solve equations (5.48)-(5.49) and (5.51) we rewrite them first in conservation form. We take into account that A0 can depend on x and write the equations in conservation form as follows ∂ ∂ U+ F = S, ∂t ∂x

(5.65)

where 

U =

and

A m



, 

S(U ) = 



F (U ) = 

αm2 A

+

G0 3ρ

m 3/2 A A0

0 α m −2 α−1 A

+

G0 3ρ

A A0

3/2

A00

A0



(5.66)

,



(5.67)

.

For the numerical solution of the problem stated above we apply the two-step LaxWendroff method [46]. We assume here that the grid is uniform, with ∆x denoting the n mesh step size and ∆t the time step, then we define Um to be the approximation of the

solution at (m∆x, n∆t). The method takes the following form n+1 n Um = Um −

∆t ∆t n+1/2 n+1/2 n+1/2 n+1/2 (F (Um+1/2 ) − F (Um−1/2 )) + (S(Um+1/2 ) + S(Um−1/2 )) ∆x 2

where n+1/2 Uj

=

n n Uj+1/2 + Uj−1/2

2 ∆t + 2

−

n n F (Uj+1/2 ) − F (Uj−1/2 )

∆x

+

n n S(Uj+1/2 ) + S(Uj−1/2 )

2

!

for j = m + 1/2 and j = m − 1/2. The method is stable if the CFL condition s 1/2 2 ∆t G0 1 A ∆t m αm + max |λ, µ| = max ± α(α − 1) ∆x < 1, ∆x A A 2ρ A 0

is satisfied.

87

!!!!!!!!!!!!!!! !!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Figure 5.4: Aneurysmal abdominal aorta and iliac arteries with inserted bifurcated stent.

5.5 Numerical experiments We use the numerical method described above to study blood flow through the aneurysmal prostheses (see Figure 5.4). We considered two kinds of prostheses: • Fabric covered ”rigid” stent-grafts, such as AneuRx TM . • Highly compliant bare stents, such as Wallstent. Presently, bare stents are no longer in use. We considered stents in this study because they have drastically different elastic properties than stent-grafts such as AneuRx TM . One of the goals of this study was to understand the influence of the elastic properties of the prosthesis on the stresses exerted by the prostheses to the walls of the native aorta. In particular we were interested in understanding what hemodynamic factors are related to the occlusion of graft limbs, observed in patients and reported in [13, 12, 63, 82]. Many researchers have shown that the wall shear stress has a significant influence on blood coagulation and thrombosis, endothelial cell structure and function. For instance, C.G. Caro et al. [11] stated that low wall shear stress could produce an area of low mass transfer. On the contrary, D. L. Fry [26] showed that wall shear stress in excess of 400 dyn/cm2 can damage endothelial cell layer. C.K. Zarins et al.([43],[58]) proposed a method called Oscillatory Shear Index (OSI) to quantify the degree of oscillation in shear direction. OSI can be calculated using the 88

τ

$#"$#" $#"$#" $#"$#" $#"$#" $#"$#" A$#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $"$" $"#$"#$#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $"$" $"#$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $"$" $"#$"#$#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $"$" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $"$" $"#$"#$#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $#"$#" $"$" pos

%#%#%#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %#%#&& A%#%#&& %#%#&& %#%#&& %%&& %#%#%#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %%&& %#%#%#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %#%#&& %%&& pos

T

A neg

Figure 5.5: Oscillatory Shear Index. following formula: OSI =

|Aneg | , |Apos | + |Aneg |

where Aneg and Apos are the areas under the shear stress vs. time when the shear stress is negative and positive respectively (see Figure 5.5). M. Haidekker, C. White and J.A. Frangos suggested that the spatial and temporal gradients of the shear stress must be separated in order to identify which one is the primary cause of atherosclerotic plaque. In their paper [34], they presented a study implicating high shear stress rate (or the temporal gradient of shear stress) as the main hemodynamic factor responsible for the initiation of focal atherogenesis. We studied the Oscillatory Shear Index and the shear stress rate to detect possible causes for graft limb occlusion. The following two parameters were varied in our study: • prostheses flexibility (Young’s modulus), • prostheses diameter. 89

Figure 5.6: AneuRx bifurcated stent. We have used the following data in our computations: • Young’s modulus Ec = 8700P a and Er = 217500P a for compliant and rigid stents respectively • Young’s modulus of a human aorta Ea = 105 P a First, we present the results of the numerically computed OSI for two different prostheses flexibility parameters and different diameters of ”parent”(main body) and ”daughter” (limb) components. On each of the Figures 5.7-5.9 the solid curve shows the Oscillatory Shear Index for the main body component from the anchoring site to the bifurcation point and the ”solid-star” curve presents the OSI corresponding to the limb component from the bifurcation point downstream to the distal anchoring site. The Young’s modulus of the prostheses shown in Figures 5.7- 5.8 corresponds to the one of AneuRx stent-graft, the main body dimeters are 23 mm and 28.5 mm and the limb 90

Oscillatory Shear Index (OSI) Dmb=23mm Dl=12mm

0.145 0.14 0.135 0.13 0.125 0.12 0.115 0.11 0.105 0.1

5

10

15

20

25

30

35

40

45

50

55

60

Figure 5.7: Oscillatory Shear Index for AneuRx stent-graft with limb diameter equal to 12 mm and main body diameter 23 mm. diameters are 12 mm and 16 mm. We observed that the smaller the diameter of the prosthesis the higher the OSI. We also tested a flexible prosthesis with a Young’s modulus corresponding to a Wallstent endoprosthesis. The diameter of the main body for this case was taken to be 20 mm and the limb diameter is 14.5 mm. The amplitude of the Oscillatory Shear Index drastically increased for this case, indicating a much higher probability for occlusion. Although the OSI showed that the smaller the diameter of the limbs the higher the OSI, it did not capture the patient-observed prosthesis characteristics that show higher probability for thrombosis at the distal site and not typically at the proximal site of the prosthesis. In contrast, the shear stress rate of the bifurcated prostheses did show to be higher at the distal anchoring site thereby conforming better to the complications observed in patients. Figures 5.10-5.16 present the temporal gradient of shear stress at the systolic peak for different prostheses data. Each figure has two curves: 91

Oscillatory Shear Index (OSI) Dmb=28.5mm Dl=16mm

0.145 0.14 0.135 0.13 0.125 0.12 0.115 0.11 0.105 0.1

5

10

15

20

25

30

35

40

45

50

55

60

Figure 5.8: Oscillatory Shear Index for AneuRx stent-graft with limb diameter equal to 16 mm and main body diameter 28.5 mm. • one curve corresponding to the shear stress rate along the ”parent” prosthesis part (main modular element) from the proximal anchoring site to the bifurcation point. • the second curve corresponding to the shear stress rate along the graft limb from the bifurcation point downstream to the distal anchoring site. The Young’s modulus of the AneuRx stent-graft shown in Figure 5.6 was used in the numerical experiments to describe the flexibility of the prosthesis. Several different stentgraft designs were tested for shear stress rates. The parent (main body) diameter ranged from 23 mm to 28.5 mm and the limbs ranged in diameter from 12 mm to 14.5 mm for smaller prostheses, and from 12 mm to 16 mm for larger prostheses. Figures 5.10 and 5.11 show the shear stress rates corresponding to the two extreme cases: the prosthesis with the smaller limb diameter (12 mm) and the prosthesis with the largest limb diameter (16mm), respectively. We observe that the smaller the diameter of the limbs the larger the shear stress rate, indicating a higher chance for occlusion. The 92

Oscillatory Shear Index (OSI) Dmb=23mm 0.5

Dl=14.5mm

0.4

0.3

0.2

0.1

0

5

10

15

20

25

30

35

40

45

50

55

60

Figure 5.9: Oscillatory Shear Index for Wallstent stent with limb diameter equal to 14.5 mm and main body diameter 20 mm. worst performance was obtained for the smallest prostheses (Figure 5.10) and the best performance was for prostheses with a limb diameter equal to 16 mm (Figure 5.11). We also observed that the size of the diameter of the main body influences the magnitude of the shear stress rate. Figures 5.13 and 5.14 show the comparison between two prostheses with the same limb diameters, but with different sizes for the main body: one with a main body diameter of 23 mm and the second one with a main body diameter of 28.5 mm. We can see an improvement in the shear stress rates for the larger main body prosthesis. We calculated the shear stress rates for a bifurcated prosthesis with a compliancy corresponding to the Wallstent endoprosthesis. The Young’s modulus of the Wallstent was measured [88, 89, 87] and is equal to 8700N/m2 , which is about ten times more elastic then the wall of the human aorta [87]. As in the previous experiments for the ”stiff” stent, we assumed that the diameter of the main modular component ranged from 23 mm to 28.5 93

Time derivative of Shear Stress (main body diameter=23mm, limbs diameters=12mm)

80

Dmb=23mm Dl=12mm

70 60 50

Pa/s

40 30 20 10 0 −10 −20

5

10

15

20

25

30 35 mesh points

40

45

50

55

60

Figure 5.10: Shear Stress Rates for AneuRx stent-graft with limb diameter equal to 12 mm and main body diameter 23 mm. mm, and that the limbs diameter is betwen 12 mm and 16 mm. Figure 5.15 presents the results of the numerical computations which indicate that the shear stress rates are drastically higher then for the case of stiff prosthesis. The results presented above indicate that the magnitude of the shear stress rate is affected mainly by the size of the limbs and by the prostheses elasticity. In an attempt to obtain an optimal design minimizing shear stress rates we tested different prosthesis structures varying the limb diameter and elasticity parameters. We have found that the shear stress rate is minimized for the prosthesis with the following two characteristic: • The diameter of the prosthesis limbs should be around

√ 2/2 of the diameter of the

main body component: if Dp denotes the diameter of the parent (main body) component and Dd denotes the diameter of the daughter (limb) component, then we obtained numerically that the lowest shear stress rates in the limbs are observed when Dd =

√ 2 Dp . 2

This relationship is a consequence of the conservation of mass princi94

Time derivative of Shear Stress (main body diameter=28.5mm, limbs diameters=16mm)

80

Dmb=28.5mm Dl=16mm

70 60 50

Pa/s

40 30 20 10 0 −10 −20

5

10

15

20

25

30 35 mesh points

40

45

50

55

60

Figure 5.11: Shear Stress Rates for AneuRx stent-graft with limb diameter equal to 16 mm and main body diameter 28.5 mm. ple. • Variable elasticity: The prosthesis which is stiffer in the central section, where there is no support to the prosthesis walls by the walls of native aorta, and softer at the overlap with the iliac arteries, showed best performance. In this test we used the fact that the stiffness of the composit prosthesis/vessel structure is equal to the combined stiffness of each structure, [21]. Hence, the smaller the stiffness of the prosthesis in the overlap anchoring region, the smaller the difference between the stiffness of the native vessel and prosthesis. Figure 5.16 shows the behavior of the shear stress rates for the ”optimal” prosthesis with characteristics obtained from the suggestions above: • Young’s modulus in the overlap region Eoverlap = 8700P a (corresponds to Wallstent elasticity), 95

Oscillatory Shear Index (OSI) Dmb=28.5mm Dl=20mm

0.145 0.14 0.135 0.13 0.125 0.12 0.115 0.11 0.105 0.1

5

10

15

20

25

30

35

40

45

50

55

60

Figure 5.12: Oscillatory Shear Index for ”optimal” stent with limb diameter equal to 20 mm and main body diameter 28.5 mm. • Young’s modulus of the prosthesis in the aneurysm sac region is equal to 100000P a, i.e. equal to AneuRx Young’s modulus, • diameter of the main body component is Dmb = 28.5mm, • diameter of the limbs is Dl = 20mm (which is about

√ 2 Dmb ). 2

We observed a drastic improvement in the limb shear stress rates with respect to the previous prosthesis structures. We also calculated the Oscillatory Shear Index for the ”optimal prosthesis” described above (see Figure (5.12)). The results also showed a much smaller amplitude of OSI compared to the cases given in Figures (5.7)-(5.9). For a comparison, we tested the shear stress rates for a bifurcated AneuRx stent-graft prosthesis with uniform stiffness, with a main body diameter of 28.5 mm and a limb diameter of 20 mm. The results are presented in Figure 5.17. We can see that this prosthesis 96

Time derivative of Shear Stress (main body diameter=28.5mm, limbs diameters=14.5mm)

80

Dmb=28.5mm Dl=14.5mm

70 60 50

Pa/s

40 30 20 10 0 −10 −20

5

10

15

20

25

30 35 mesh points

40

45

50

55

60

Figure 5.13: Shear Stress Rates for the AneuRx stent-graft with a limb diameter equal to 14.5 mm and a main body diameter 28.5 mm. design produces shear stress rates that are close to optimal.

97

Time derivative of Shear Stress (main body diameter=23mm, limbs diameters=14.5mm)

80

Dmb=23mm Dl=14.5mm

70 60 50

Pa/s

40 30 20 10 0 −10 −20

5

10

15

20

25

30 35 mesh points

40

45

50

55

60

Figure 5.14: Shear Stress Rates for the AneuRx stent-graft with a limb diameter equal to 14.5 mm and a main body diameter 23 mm. Time derivative of Shear Stress (main body diameter=23mm, limbs diameters=14.5mm)

80

Dmb=23mm Dl=14.5mm

70 60 50

Pa/s

40 30 20 10 0 −10 −20

5

10

15

20

25

30 35 mesh points

40

45

50

55

60

Figure 5.15: Shear Stress Rates for the Wallstent endoprosthesis with a limb diameter equal to 20 mm and a main body diameter of 28 mm. 98

Time derivative of Shear Stress (main body diameter=28.5mm, limbs diameters=20mm)

80

Dmb=28.5mm Dl=20mm

70 60 50

Pa/s

40 30 20 10 0 −10 −20

5

10

15

20

25

30 35 mesh points

40

45

50

55

60

Figure 5.16: Shear Stress Rates for an ”optimal” stent with a imb diameter equal to 20 mm and a main body diameter of 28.5 mm. Time derivative of Shear Stress (main body diameter=28.5mm, limbs diameters=20mm)

80

Dmb=28.5mm Dl=20mm

70 60 50

Pa/s

40 30 20 10 0 −10 −20

5

10

15

20

25

30 35 mesh points

40

45

50

55

60

Figure 5.17: Shear Stress Rates for the AneuRx stent-graft with a limb diameter equal to 20 mm and a main body diameter of 28.5 mm. 99

Bibliography [1] O. Axelsson and V. Baker. Finite element solution of boundary value problems. Theory and computation. Academic Press, 1984. [2] I. Babuˇska and T. Strouboulis. The Finite Element Method and Its Reliability. Oxford University Press, 2001. [3] J. Balschak and G. Kriegsmann. A comparative study of absorbing boundary conditions. J.Comput. Phys., 77:109–139, 1988. [4] A. Barnard, W. Hunt, W. Timlake, and E. Varley. A theory of fluid flow in compliant tubes. Biophysical Journal, 6:717–724, 1966. [5] E. Becache, P.Joly, and C.Tsogka. Fictitious domains, mixed finite elements and perfectly matched layers for 2d elastic wave propagation. INRIA, Rapport de Recherche, 3889, 2000. [6] J. P. Berenger. A perfectly matched layer for the absorbption of electromagnetic waves. J. Comput. Phys., 114:185–200, 1994. [7] F. Bethuel, H. Brezis, and F. Helein. Ginzburg Landau Vortices. Birkhäuser, 1994. [8] F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer-Verlag, 1991.

100

[9] M. O. Bristau, E. Dean, R. Glowinski, V. Kwok, and J. Periaux. Exact controllability and domain decomposition methods with non-matching grids for the computation of scattering waves. In R. Glowinski, J. Periaux, and Z. Shi, editors, Domain Decomposition Methods in Sciences and Engineering, pages 291–307. John Wiley and Sons, 1997. [10] B. L. Buzbee, F. W. Dorr, J. A. George, and G. H. Golub. The direct numerical solution of the discrete Poisson equation on irregular regions. SIAM J. Numer. Analysis, 8:722–736, 1971. [11] C. G. Caro, J. M. Fitz-Gerald, and R. C. Schroter. Atheroma and arterial wall shear, observations, correlation and proposal of a shear dependant mass transfer mechanism for atherogenesis. Procedings Royal Society London B, 177:109–133, 1971. [12] J. P. Carpenter, D. G. Neschis, R. M. Fairman, C. F. Baker, M. A. Golden, O. C. Velazquez, M. E. Mitchell, and R. A. Baum. Failure of endovascular abdominal aortic aneurysm graft limbs. J. Vasc. Surgery, 33(2):296–302, 2001. [13] A. Carrocio, P. L. Faries, N. J. Morissey, V. Teodorescu, J. A. Burks, E. C. Gravereaux, L. H. Hollier, and M. L. Marin. Predicting iliac limb occlusions after bifurcated aortic stent grafting: anatomic and device-related causes. J. Vasc. Surgery, 36(4):679–684, 2002. [14] R. W. Clough. The finite element method in plane stress analysis. In Proceedings of 2nd ASCE conference on electronic computation. 1960. [15] R. Courant, K. Friedrichs, and H. Lewy. Uber die differenzengleichungen der mathematisschen physik. Math. Ann, 100:32–72, 1928. [16] R. L. Courant. Variational methods for the solution of problems of equilibrium and vibration. Bull. Amer. Math. Soc., 5:1–23, 1943. 101

[17] R. Dautray and J.-L. Lions. Mathematical analysis and numerical methods for science and technology, volume 5. Springer-Verlag, 1992. [18] E. Dean and R. Glowinski. Domain decomposition methods for mixed finite element approximations of wave problems. Computers and mathematics with applications, pages 207–214, 1999. [19] J. Donea. Arbitrary Lagrangian Eulerian methods. In T. Belytschko and T. J. R. Hughes, editors, Computational methods for transient analysis, pages 474–516. North Holland, Amsterdam, 1983. [20] J. Douglas. The solution of the diffusion equation by a high order correct difference equation. J. Math. Phys., 35:145–151, 1956. [21] J. F. Dyet, W. G. Watts, D. E. Ettles, and A. A. Nicholson. Mechanical properties of metallic stents: How do these properties influence the choice of stent for specific leisons? Cardiovasc. Interv. Radiology, 23:47–54, 2000. [22] B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comput., 31:629–651, 1977. [23] G. C. Everstine and F. M. Henderson. Coupled finite element boundary element approach for fluid structure interaction. Journal of Acoustical Society of America, 87(5):1938–1947, 1990. [24] S. A. Finogenov and Y. A. Kuznetsov. Two-stage ficitious domain component methods for the Dirichlet boundary value problem. Soviet J. Numer. Methods Math. Modeling, 3:301–324, 1988. [25] L. Formaggia, F. Nobile, A. Quarteroni, and A.Veneziani. Multiscale modeling of the circulatory system: a preliminary analysis. Computing and Visualization in Science, 2:75–83, 1999. 102

[26] D. L. Fry. Acute vascular endothelial changes associated with increased blood velocity gradients. Circ. Res., 22:165–197, 1968. [27] R. Glowinski. Finite Element Methods for Incompressible Viscous Flow , Vol. IX of Handbook of Numerical Analysis, Eds. P.G. Ciarlet and J.L. Lions. North-Holland, Amsterdam, 2003. [28] R. Glowinski, A. Lapin, and S. Lapin. On the numerical simulation of the constrained wave motion: a penalty approach. In G. C. Cohen, E. Heikkola, P. Joly, and P. Neittaanmaki, editors, Proceedings of The Six International Conference on Mathematical and Numerical Aspects of Wave Propagation Held at Jyvaskyla, Finland. SpringerVerlag, 2003. [29] R. Glowinski, A. Lapin, and S. Lapin. A penalty approach to the numerical simulation of the constrained wave motion. Journal of Numerical Mathematics, 11(4):289–300, 2003. [30] R. Glowinski and S. Lapin. Solution of a wave equation by a mixed finite elementfictitious domain method. Computational Methods in Applied Mathematics, 4:431– 444, 2004. [31] R. Glowinski, T. Pan, and J. Periaux. A ficitious domain method for Dirichlet problem and applications. Comp. Meth. Appl. Mech. Engrg., 111(3-4):283–303, 1994. [32] R. Glowinski, T. Pan, and J. Periaux. A ficitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations. Comp. Meth. Appl. Mech. Engrg., 111(1-4):133–148, 1994. [33] R. Glowinski and P.LeTallec. Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics. SIAM, Phildelphia, PA, 1989.

103

[34] M. Haidekker, C. White, and J.A.Frangos. Analysis of temporal shear stress gradients during the onset phase of flow over a backward-facing step. J. Biomech. Eng., 123:455–463, 2001. [35] R. L. Higdon. Absorbing boundary conidtions for difference approximations to the multi-dimensional wave equations. Math. Comput., 47:437–459, 1986. [36] R. L. Higdon. Numerical absorbing boundary conditions for the wave equation. Math. Comput., 49:65–90, 1987. [37] R. Holland and J. W. Williams. Total-field versus scattered-field finite-difference codes: A comparative assessment. IEEE Trans. Nucl. Sci., 30:4583–4588, 1983. [38] M. A. Hyman. Non-iterative numerical solution of boundary value problems. Appl. Sci. Res. Sec. B2, pages 325–351, 1952. [39] R. A. James and I. C. Mathews. Solution of fluid-structure interaction problems using coupled finite element and variational boundary element technique. Journal of Acoustical Society of America, 88(5):2459–2466, 1990. [40] F. John. On the integration of parabolic equations by difference methods, I: Linear and quisilinear equations for the infinite interval. Comm. Pure Appl. Math., 5:155– 211, 1952. [41] R. G. Keys. Absorbing boundary conditions for acoustic media. Geophysics, 50:892– 902, 1985. [42] R. Kornhuber, R. Hoppe, J. Periaux, O. Pironneau, O. Widlund, and J. Xu, editors. Domain Decomposition Methods in Science and Engineering (Lecture Notes in Computational Science and Engineering). Springer, 2004.

104

[43] D. N. Ku, D. Giddens, C. K. Zarins, and S. Glagov. Pulsatile flow and atherosclerosis in the human carotid bifurcation: positive correlation between plaque location and low and oscillating shear stress. Atherosclerosis, 5:293–302, 1985. [44] Y. A. Kuznetsov, G. I. Marchuk, and A. M. Matsokin. Fictitious domain and domain decomposition methods. Soviet. J. Numer. Methods Math. Modeling, 1:3–35, 1986. [45] Y. A. Kuznetsov and A. M. Matsokin. Numerical solution of the Helmholtz equation by the ficitious domain method (in russian). Publication of the Computer center, pages 127–144, 1988. [46] R. Leveque. Numerical Methods for Conservation Laws. Birkhäuser, Basel;Boston, 1992. [47] E. L. Lindman. Free-space boundary conditions for the time dependent wave equation. J. Comput. Phys., 18:67–78, 1975. [48] J.-L. Lions. Une remarque sur les problèmes d’évolution non linéaires dans des domaines non-cylindriques. Rev. Roumaine Math. Pures Appl., 9:11–18, 1964. [49] J.-L. Lions. Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires. Dunod, Paris, 1969. [50] P.-L. Lions. On the Schwarz alternating method I: a variant for nonoverlaping subdomains. In R. Glowinski, G. H. Golub, G. A. Meurant, and J. Periaux, editors, Proc. First Conference on Domain Decomposition Methods, pages 1–42. SIAM, Philadelphia, PA, 1988. [51] P.-L. Lions. On the Schwarz alternating method II: a variant for nonoverlaping subdomains. In T. F. Chan, R. Glowinski, J. Periaux, and O. B. Widlund, editors, Proc. Second Conference on Domain Decomposition Methods, pages 47–70. SIAM, Philadelphia, PA, 1989. 105

[52] P.-L. Lions. On the Schwarz alternating method III: a variant for nonoverlaping subdomains. In T. F. Chan, R. Glowinski, J. Periaux, and O. B. Widlund, editors, Proc. Third Conference on Domain Decomposition Methods, pages 202–223. SIAM, Philadelphia, PA, 1990. [53] X. Ma, G. Lee, and N. Tseng. Nonlinear elastic analysis of blood vessels. J.Biomech. Engrg., 106:376–383, 1984. [54] E. Marusic-Paloka. Fluid flow through a network of thin pipes. Comptes Rendus de l’Academie des Sciences Serie II Fascicule B - Mecanique, 329(2):103 – 108, 2001. [55] A. Masud and T. J. R. Hughes. A space-time Galerkin/least squares finite element formulation of the Navier-Stokes equations for moving domain problems. Comput. Methods Appl. Mech. Engrg, 146:91–126, 1997. [56] D. E. Merewether. Transient currents on a body of revolution by an electromagnetic pulse. IEEE Trans. Electromagn. Compat., 13:41–44, 1971. [57] A. L. Mignot. Métodes d’approximation de solutions de certains problèmes aux limites linéaires. Rend. Sem. Mat. Univ. Padova, 40:1–138, 1968. [58] J. E. Moore, C. Xu, S. Glagov, C. K. Zarins, and D. N. Ku. Fluid wall shear stress measurements in a model of the human abdominal aorta: oscillatory behavior and relashionship to atherosclerosis. Atherosclerosis, 110:225–240, 1994. [59] G. Mur. Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations. IEEE Trans. Electromagn. Compat., 23:377–382, 1981. [60] W. W. Nichols and M. F. O’Rourke. McDonald’s Blood Flow in Arteries: Theoretical, experimental and clinical principles. Arnold and Oxford University Press Inc., New York, London, Sydney, Auckland, 4th edition, 1998. 106

[61] G. G. O’Brien, M. A. Hyman, and S. Kaplan. A study of the numerical solution of partial differential equations. J. Math. Phys., 29:223–251, 1951. [62] M. Olufsen, C.Peskin, W.Kim, E.Pedersen, A.Nadim, and J.Larsen. Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Annals of Biomedical Engineering, 28:1281–1299, 2000. [63] F. N. Parent, V. Godziachvilli, G. H. Meier, F. M. Parker, K. Carter, R. G. Gayle, R. J. Demassi, and R. T. Gregory. Endograft limb occlusion and stenosis after ANCURE endovascular abdominal aneurysm repair. J. of Vasc. Surg., 35(4):686–690, 2002. [64] K. Perktold and G. Rappitsch. Mathematical modeling of local arterial flow and vessel mechanics. In J. Crolet and R. Ohayon, editors, Computational Methods for Fluid Structure Interaction. Pitman Research Notes in Mathematics, volume 306, pages 230–245. Addison Wesley, 1994. [65] C. S. Peskin. Numerical analysis of blood flow in the heart. J. Comput. Phys., 25:220– 252, 1977. [66] C. S. Peskin. Lectures on mathematical aspects of physiology. In F. C. Hoppensteadt, editor, Lectures in Applied Mathematics, volume 19, pages 69–107. 1981. [67] C. S. Peskin. The immersed boundary method. Acta Numerica, pages 1–39, 2002. [68] C. S. Peskin and D. M. McQueen. Modeling prosthetic heart valves for numerical analysis of blood flow in the heart. J. Comput. Phys., 37:113–132, 1980. [69] A. Quarteroni, M. Tuveri, and A. Veneziani. Computational vascular fluid dynamics: problems, models and methods. survey article. Comput. Visual. Sci., 2:163–197, 2000. [70] C. Rappaport and L. Bahrmasel. An absorbing boundary condition based on anechoic absorber for EM scattering computation. J. Electromagnetic Waves and Applications, 6:1621–1634, 1992. 107

[71] C. M. Rappaport and T. Gürel. Reducing the computational domain for FDTD scattering simulation using the sawtooth anechoic chamber abc. IEEE Trans. Magnetics, 31:1546–1549, 1995. [72] H. Reismann. Elastic plates: theory and applications. Wiley, New York, 1988. [73] A. A. Samarskii. A priori estimates for difference equations (in russian). U.S.S.R. J. Comp. Math. and Math. Phys., 1:1138–1167, 1961. [74] A. A. Samarskii. A priori estimates for the solution of the difference analogue of a parabolic differential equation (in russian). U.S.S.R. J. Comp. Math. and Math. Phys., 1:487–512, 1961. [75] V. K. Saul’ev. On a method for automatization of solution of boundary value problems on high perfomance computers. Dokl. Acad. Sci. USSR, 144:497–500, 1962. [76] V. K. Saul’ev. On the solution of some boundary value problems on high perfomance computers by fictitious domain methods. Siberian J. Math, 4:912–925, 1963. [77] H. A. Schwarz. Gesammelte Mathematische Abhandlungen. 2:133–143, 1890. [78] N. Smith, A. Pullan, and P. Hunter. The generation of an anatomically accurate geometric coronary model. Annals of Biomedical Engineering, 28:1:14–25, 2000. [79] N. Smith, A. Pullan, and P. Hunter. An anatomically based model of transient coronary blood flow in the heart. SIAM Journal on Applied Mathematics, 62:990–1018, 2002. [80] J. Smoller. Shock waves and reaction-diffusion equations. Springer-Verlag, New York, 2nd edition, 1994. [81] S. L. Sobolev. Schwarz algortihm in elasticity theory (in russian). Dokl. Acad. Sci. USSR, IV((XIII) 6):243–246, 1936. 108

[82] T.Umscheid and W. Stelter. Time-related alterations in shape, position, and structure of self-expanding, modular aortic stent-grafts. J. Endovasc. Surg., 6:17–32, 1999. [83] M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp. Stiffness and defection analysis of complex structures. J. Aero. Sci., 23:805–824, 1956. ˇ c. Blood flow through compliant vessels after endovascular repair: wall defor[84] S. Cani´ mations induced by the discontinuous wall properties. Computing and Visualization in Science, 4(3):147–155, 2002. ˇ c, S. A. Mikelic, D. Lamponi, and J. Tambaˇca. Self-consistent effective equa[85] S. Cani´ tions modeling blood flow in medium-to-large compliant arteries. Accepted to SIAM J. Multiscale Analysis and Simulation. ˇ c and D. Mirkovic. A hyperbolic system of conservation laws in modeling [86] S. Cani´ endovascular treatment of abdoninal aortic aneurysm. Hyperbolic Problems: Theory, Numerics, Applications, 141(1):227–236, 2000. ˇ c, K. Ravi-Chandar, Z. Krajcer, D. Mirkovic, and S. Lapin. A comparison [87] S. Cani´ between the dynamic responses of bare-metal wallstent endoprosthesis and aneurx stent-graft: A mathematical model analysis. Accepted to THIJ. [88] R. Wang and K. Ravi-Chandar. Mechanical response of a metallic aortic stent-I. Pressure-diameter relashionship. Submitted to Journal of Applied Mechanics. [89] R. Wang and K. Ravi-Chandar. Mechanical response of a metallic aortic stent-II. A beam on elastic foundation model. Submitted to Journal of Applied Mechanics. [90] F. M. White. Viscous Fluid Flow. McGraw-Hill, New York, 1974. [91] O. B. Widlund. On the stability of parabolic difference schemes. Math. Comp., 19:1– 13, 1965. 109

[92] O. B. Widlund. Stability of parabolic difference schemes in the maximum norm. Numer. Math., 8:186–202, 1966. [93] O. C. Zienkiewicz. The Finite Element Method in engineering science. McGraw-Hill, London, 1971.

110

Computational Methods in Biomechanics and Physics

Computational Methods in Biomechanics and Physics

Suggest Documents

Computer Methods in Biomechanics and

Research Methods in Biomechanics

Virtual Biomechanics and Its Physics

Computer Methods in Biomechanics and Biomedical Engineering

Computer Methods in Biomechanics and Biomedical Engineering

Computer Methods in Biomechanics and Biomedical ...

Computer Methods in Biomechanics and Biomedical

Computer Methods in Biomechanics and Biomedical ...

Computer Methods in Biomechanics and ... - Semantic Scholar

Computer Methods in Biomechanics and Biomedical ...

Computer Methods in Biomechanics and Biomedical ...

Computer Methods in Biomechanics and Biomedical ...

Computer Methods in Biomechanics and Biomedical ...

Computer Methods in Biomechanics and Biomedical

Computational Biomechanics in Thoracic Aortic ...

Computational Biomechanics of the Musculoskeletal

Communications in Computational Physics

SPRING 2012 Computational Methods in Physics - Eastern Illinois ...

Computational Physics

ANALYTICAL AND COMPUTATIONAL METHODS

computational physics 330 - Physics and Astronomy - Brigham ...

Research Methods in Biomechanics-2nd Edition Information

computer methods biomechanics & biomedical ...

computer methods biomechanics & biomedical ...