Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 113

discretized problem mimics the energy of the continuous problem. As we shall see
later, this guarantees stability of the scheme.

2.5.1 The Energy of the Coupled Problem

We present here a general approach to deriving an energy estimate of the coupled
FSI benchmark problem, described in Sect. 2.3 , for the class of problems in which
the Koiter shell is linear. Thus, we consider a clampedlinearly(visco)elasticKoiter
shell ( 2.58 ), coupled with the equations of linear elasticity ( 2.59 ), and the flow of an
incompressible, viscous fluid modeled by the Navier-Stokes equations ( 2.60 ). The
inlet and outlet data are given by the dynamicpressure data,specifiedin( 2.64 ).
We first recall from Sect. 2.2.1 that the linear operators

L el and L vis are defined

as follows:

Z Z

G ./ W G . / p a C

h 3
48

R ./ W R . / p a; 8 2 C c :
(2.70)

h
2

h L el ; i WD

! A ! A

Z Z

h
2

G .P / W G . / p a C

h 3
48

R .P / W R . / p a; 8 2 C c ;
(2.71)

h L el P ; i WD

! B ! B

where G and R are the change of metric, and change of curvature tensors,
respectively, and are the elasticity tensor and the viscoelasticity tensor,

respectively, defined in Example 4 of Sect. 2.2.1 . This will be used to obtain the
following energy estimate for the coupled problem:

A and B

Proposition 2.1. The coupled FSI benchmark problem ( 2.58 ) - ( 2.69 ) with dynamic
inlet and outlet pressure data satisfies the following energy estimate:

d
dt .E kin .t/ C E el .t// C D.t/ C.P in .t/;P out .t//;

(2.72)

where

2 F k u k L 2 . S / ;

1

2 2 2

E kin .t/ WD

L 2 . F .t// C K hk@ t k L 2 ./ C S k@ t d k

(2.73)

2 E el ./ C 2k D . d /k L 2 . S / ;

1

2 2

E el .t/ WD

L 2 . S / C kr d k

denote the kinetic and elastic (internal) energy of the coupled problem, respectively,
and the term D.t/ captures viscous dissipation:

2
L 2 . F .t// :

D.t/ WD E vis .@ t / C F k D . u /k (2.74)