Volume 4 - Year 2017 - Pages 1-9

DOI: 10.11159/jffhmt.2017.001

### Application of Variational Principle to Form Reduced Fluid-Structure Interaction Models in Bifurcated Networks

A.S. Liberson, Y. Seyed Vahedein, D. A. Borkholder

Rochester Institute of Technology

James E. Gleason Building, 76 Lomb Memorial Drive

Rochester, NY 14623-5604, United States

Asleme@rit.edu; Yashar.seyed.vahedein@rit.edu; David.borkholder@rit.edu

**Abstract** - *Reduced fluid-structure interaction models have received a considerable attention in recent years being the key component of hemodynamic modeling. A variety of models applying to specific physiological components such as arterial, venous and cerebrospinal fluid (CSF) circulatory systems have been developed based on different approaches. The purpose of this paper is to apply the general approach based on Hamilton’s variational principle to create a model for a viscous Newtonian Fluid - Structure Interaction (FSI) in a compliant bifurcated network. This approach provides the background for a correct formulation of reduced FSI models with an account for arbitrary nonlinear visco-elastic properties of compliant boundaries. The correct boundary conditions are specified at junctions, including the interface between 3D and 1D models. The hyperbolic properties of the derived mathematical model are analyzed and used, constructing a monotone finite volume numerical scheme, second-order accuracy in time and space. The computational algorithm is validated by comparison of numerical solutions with the exact manufactured solutions for an isolated compliant segment and a bifurcated structure. The accuracy of applied total variation diminishing (TVD) and Lax-Wendroff schemes are analyzed by comparison of numerical results to the available analytical smooth and discontinuous solutions, demonstrating a superior performance from the TVD algorithm.*

** Keywords:** Hamilton’s variational principle, reduced fluid-structure interaction (FSI), bifurcated arterial networks, multiscale 3D/1D approach, total variation diminishing method (TVD), Lax-Wendroff method, manufactured test, break-down solution.

© Copyright 2017 Authors - This is an Open Access article published under the Creative Commons Attribution License terms Creative Commons Attribution License terms. Unrestricted use, distribution, and reproduction in any medium are permitted, provided the original work is properly cited.

Date Received: 2017-04-24

Date Accepted: 2017-08-08

Date Published: 2017-08-18

#### Nomenclature

*PWV*: Pulse wave velocity (m/s)*FSI*: Fluid structure interaction- : Volume (m
^{3}) - : Variation, Del operator
- : Coefficients replacing the derivative of the function distributions
- : Flow behavior index
- : Axial normal and shear viscous stress (Pa)
- : Kinematic viscosity
- η: Nondimentional wall displacement
- : Density of incompressible fluid (kg/m
^{3}) - : Function distribution in radial Direction for displacement and Velocity
*A*: Cross sectional area (m^{2})*C*: Moens – Korteweg speed of Propagation (m/s)*H, h*: Thickness of the wall at a zero stress and loaded conditions respectively (m)- : Action component for fluid and solid respectively
*K*: Kinetic energy of the vessel wall (J)*L, L*: Lagrangian density functions for solids and solids, respectively._{f}*P*: Pressure (Pa)*R, r*: Internal wall radii in zero stress & loaded conditions respectively (m)*S*: Entropy (J/K)*t , T*: Time (s), Temperature (K)*u*: Displacement vector (m)*U, U*: Internal, elastic and dissipative Energy (J)_{el},U_{d}*W*: Work of external load (J)_{p}*V*: Velocity (m/s)

#### 1. Introduction

Extensive work has been done for developing different models, applied to specific
components of hemodynamic pulsating flow, such as arterial, venous and CSF
circulations [1–4]. Functions of
each component of the cardiovascular system and the relation of blood pressure
to arterial disease predictions have been reviewed by Vosse and Stergiopulos –
2011 [1] and Quarteroni *et al. – 2017* [5]. Numerical
models are able to predict blood flow rate and pressure non-invasively and in a
patient-specific setting, thus enabling non-invasive and less expensive disease
diagnostics compared with clinical tests. Parker – 2009 [6], presents a
historical review of arterial fluid mechanics models. Detailed derivation of
simplified reduced Fluid – Structure Interaction (FSI) models for a linear
elastic arterial system with account of visco-elasticity and inertia of the
wall can be found in Peiro and Veneziani – 2009 [7]. Physical
nonlinearity of thin and thick arterial walls coupled with large deformations
have been introduced in FSI dynamics by Liberson *et al.* – 2016 [8] and Lillie *et
al*. – 2016 [9]. An analytical
solution for the pulse wave velocity (PWV) of a nonlinear FSI model was
presented in Liberson *et al*. – 2014 [10]. The
variational approach, yielding governing equations of physical phenomena,
serves as an indispensable tool, when the interaction of the system components are
non-trivial, for instance, when their interaction contain, strong
nonlinearities, kinematic constraints, or high derivatives. The monumental book
of Berdichevsky – 2010 [11,12] presents a
variety of variational principles applied separately to fluids and solids. Kock
and Olson – 1991 [13] developed a
variational approach for FSI system, restricting analysis by a linear elastic thin-walled
cylinder and an inviscid, irrotational and isentropic fluid flow. Lagrangian multipliers
are used to enforce continuity equation constraint on boundary conditions.
Multiple references can be found in this paper relating to the applications of
the variational approach to the analysis of small vibrations of elastic bodies
in a potential fluid.

We demonstrate the effectiveness of Hamiltonian variational principle in analyzing FSI, without any limitations on dissipative fluid dynamics or physical properties of an adjacent flow path wall. We do not use the Lagrangian multiplier, accounting for the continuity equation explicitly, which simplifies the entire procedure. Internal boundary conditions are specified at junctions, including matching points in a combined 3D/1D approach, following from the Euler-Lagrange conditions.

Numerical effectiveness in a simulation of a pulsating flow is characterized by its ability to track a propagating wave for a few time periods, without suffering from numerical dissipation (errors in amplitude) or numerical dispersion (artificial oscillations). The most popular numerical methods in this area are the Lax-Wendroff finite volume method, its Taylor-Galerkin finite element counterpart, and discontinuous Galerkin spectral finite element method (As mentioned in [14]). Here, we also demonstrate the superior accuracy of the second order Total Variational Diminishing (TVD) approximation [15-17], compared to the more common Lax-Wendroff method. This higher accuracy can be essential when simulating a model with discontinuity in the load or material properties.

#### 2. The Variational Principle for Fluid–Structure Interaction Problems

Hamilton’s variational principle is enunciated as a universal principle of nature unifying mechanical, thermodynamic, electromagnetic and other fields in a single least action functional, subject to extremization for a true process [11]. According to the mentioned principle, the variation of the action functional being applied to the FSI problem can be determined as:

Here are variations of action components across fluid and solid volumes , ; t – time, -density of the fluid, - the Lagrangian density functions for fluid and solids, respectively.

#### 2.1. Fluid Domain

As mentioned by Berdichevsky - 2010 [11,12], variation of the Lagrangian density function in Eulerian coordinates can be written as follows:

Where – is a velocity vector, – is internal energy as a function of density, *S*
– entropy, – a distortion tensor (gradient of a displacement
vector ), and T – temperature. Velocity, density and the displacement
vector are not subject to independent variations. To avoid the use of
Lagrangian multipliers, variation of density is directly extracted from the
mass conservation law:

Presenting variation of a velocity as a substantial derivative of a variation of displacement vector, we have:

Equations (3) and (4) reduce the equation (2) to only the independent variables - displacement and entropy. Substituting (3) and (4) into equation (2) gives:

In which, according to Maxwell’s thermodynamic identity, pressure , and the stress tensor is introduced as .

Considering
2D axisymmetric flow in a long compliant tube, (*i.e.* long wave
approximation [7]), we neglect
variability of the radial velocity component and the pressure in radial
direction. Thus, Equation (5) is transformed to the following form:

Here, – is an axial velocity, – is an axial component of displacement, – axial normal and shear viscous stress
components, - internal radius of a tube as a function of axial
coordinate and time. The reduced models are based on assumptions regarding the
radial profiles, *i.e*.

With the aim of application to the incompressible flow, density is assumed constant. Integrating the functional (7) over the cross section with the following integration by parts, arrive at the reduced momentum equation

Where the coefficients are:

;

In case of Newtonian fluid (), generalized Hagen-Poiselle profile and a constant profile for the function distribution in radial direction, equation (8) takes the form presented by San and Staples – 2012 [18].

Besides equation (8), Hamilton’s equation in a form of yields natural boundary conditions. In case of a multiscale model, matching section of a coupled 3D and 1D require continuity following from natural boundary conditions

It should be noted that we neglect the effect of dissipation on boundary conditions.

#### 2.2. Solid Domain

Consider a circular thin-wall cylinder relating to the polar system of coordinates. Let be the radius of the wall under the load, – radius in a load free state, - the wall thickness, – circumferential stretch ratio, – nondimensionalized wall normal displacement. Introducing wall kinetic energy , elastic energy and a dissipative energy and work of external load , the Hamiltonian functional relating to the solid domain can be presented as

The normal velocity of the moving wall, defines kinetic energy per unit length

Internal elastic energy is composed of hyperelastic exponential strain energy (Fung, 1990 [19]) and an energy, accumulated by a longitudinal pre-stress force N, per unit area

Where , and are material constants from Fung *et al*.
anisotropic model [19]. Assuming the wall model is a system of independent
nonlinearly elastic rings (Perktold and Rapitsch [20]), and simplifying the
equation (14) by only keeping the principle quadratic terms (the forth power
for and quadratic
terms for the slope), arrive at

Elementary work produced by the viscous component of circumferential stress, relating to the Voight type material and external pressure load are presented as

Substituting (13)-(16) into equation (12), and equating to zero, we obtain the equation of motion of an axisymmetric cylinder in the explicit form with respect to pressure

Momentum equation (8), equation of a boundary wall motion (17) and the continuity equation (3), averaged over the cross-section

Create a closed-form reduced mathematical model for fluid-structure interaction in a compliant channel.

#### 3. Numerical Simulation

Numerical
solvers utilized in a computational hemodynamics are typically based on the
Lax-Wendroff scheme, its Taylor-Galerkin finite element counterpart, or a
discontinuous Galerkin spectral finite element method [14]. In this article,
the second-order accuracy monotonicity preserving TVD scheme is presented,
which demonstrates its superiority for the problems dealing with a
discontinuity of a load function, including discontinuity of its derivative, or
material properties. This scheme can be applied to the linear scalar wave
equation ( – is the unknown scalar function, *a* –
coefficient)

The following numerical scheme (interested reader is referred to the book by Leveque [15]) has been applied

Here, the upper and lower indices correspond to the discrete time and a space coordinate, , is the limiter to maintain monotonicity ( for ).

The details of TVD methods applied to the arbitrary system of hyperbolic equations can be found in [15]. The following testing cases compare TVD and LW schemes.

#### 3.1. Break-down Solution

Numerical solution of the break-down problem (Riemann problem [14]) for the acoustics model, Equation (22), is simulated with the TVD (Figure 1) and the Lax-Wendroff (Figure 2) methods.

;

The solution was initialized with the piece-wise constant initial pressure and
velocity data (shown by the black line in Figures 1 & 2). *P *and *V*
values at the boundary were set to their constant initial values. The model
runtime was set to stop before the wave front reach the boundaries. The same
values of , , Courant number and cell count of N was used for both schemes. Discontinuity
resolved by the TVD method, practically reproduces the exact solution (Figure
1), whereas discontinuity simulated by the Lax-Wendroff method gives rise to
artificial oscillations (Figure 2). The Lax-Wendroff method is clearly
dispersive, and does not perform well around discontinuities.

#### 3.2. Single Segment Nonlinear FSI Problem

The nonlinear mathematical model used for this problem is based on the momentum equation (10) (α=1), continuity equation (18) and a linear constituent equation in the form of (17), accounting for the linear viscoelastic behavior of the wall. To generate this testing case, the method of manufactured solutions (MMS) [21,22] is used. We start by manufacturing an explicit expression for the solution as a superposition of Fourier harmonics, satisfying the corresponding linear equations. Then, we substitute the solution to the equations (10), (17) and (18), evaluating the source terms. Given the source terms, inlet velocity, exit pressure and initial conditions found from the exact solution, we use the simulation tool to obtain a numerical solution and compare it to the originally assumed solution with which we started. Results, presented in Figure 3 show that numerical and exact solutions of the nonlinear FSI problem, cannot be distinguished from one another in the pressure and velocity plots as a function of time. For smooth solutions (without the discontinuities), the Lax-Wendroff approach and TVD solutions give practically the same answer.

#### 3.3. Bifurcated Elements Nonlinear FSI Problem

Schematic of a symmetric bifurcated structure with a single parent vessel and two identical daughter vessels (twins) are presented in Figure 4. We have manufactured an explicit expression for the solution as a superposition of three Fourier harmonics, satisfying a priori flow and pressure continuity at the matching sections.

The flow for the daughter and parent vessels are created such that the continuity condition at matching sections is satisfied automatically. , and – is the local coordinate at each vessel, varying from 0 to in daughter segment, and from 0 to in a parent vessel.

The following equations are used to specify flow at each interface (i.e. for parent or daughter vessels)

The pressure distributions for the parent, P_p (x,t), and the daughter, P_d (x,t), vessels are manufactured providing automatic match P_p (x=L_3,t)=P_d (x=0,t) at junctions (e.g. the gray colored area in Figure 4).

As mentioned before, we substitute the solution to the equations (10), (17) and (18), evaluating the source terms. Given the source terms, boundary and initial conditions just obtained, we use the simulation tool to obtain a numerical solution and compare it to the originally assumed solution with which we started. As presented in Figure 5, resulted P and V plots, cannot be distinguished from one another in case of numerical (straight line) or exact solution (square-dotted line). For smooth solutions, the Lax-Wendroff approach and TVD solutions give practically the same answer.

#### 3.4. Sin-Wave Introduced in the Middle of the Parent Vessel in a Bifurcation

Lastly, the effectiveness of TVD scheme in preserving the shape of a propagating wave is evaluated by introducing a single Sin shape wave in the middle of the parent vessel. Figure 6 shows the propagation of this wave in a TVD based scheme compared to the Lax-Wendroff method. The flow properties similar to section 3.1, ρ=0.25 kg/m^3 , c=2 m/s, CFL=0.25 are used for both cases.

It can be seen that small oscillations exist in the left side of the sin wave in case of the Lax-Wendroff approximation. These oscillations (dispersion) increases as the sin wave propagates forward.

#### 4. Conclusions

A general approach to derive the fluid-structure interaction problem have been applied based on Hamilton’s variational principle. Fluid is assumed viscous, and a boundary wall – nonlinear viscoelastic. Internal boundary conditions at the matching sections of a multiscale 3D-1D approach are derived. Numerical results based on a TVD approach compared to the solutions provided by the Lax-Wendroff. It is proved that the Lax-Wendroff method is clearly dispersive, providing artificially oscillations simulating physical problems with discontinuity. These oscillations are not present in TVD method, making it the optimum choice in solving 1D FSI problem. Derived internal boundary conditions enable the coupling of 1D FSI model to a local 3D FSI model of the arteries.

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