CO-SIMULATION AND MULTIPHYSICS TECHNOLOGIES FOR …€¦ · CO-SIMULATION AND MULTIPHYSICS...

CO-SIMULATION AND MULTIPHYSICS TECHNOLOGIES FOR

COUPLED FLUID-STRUCTURE INTERACTION PROBLEMS

CO-SIMULATION AND MULTIPHYSICS TECHNOLOGIES FOR COUPLED FLUID-STRUCTURE INTERACTION PROBLEMS

Sharat Prasad, Ph.D.

Engineering Specialist*

Yun Yun Lu

Principal Development Engineer*

Shawn Harwood

Principal Development Engineer*

Karthik Mukundakrishnan

Development Engineer I*

Martin Sanchez Rocha

Development Engineer I*

*Dassault Systèmes Simulia Corp.

THEME

Multiphysics

KEYWORDS

Multiphysics, Fluid-Structure Interaction, Strong Physics Coupling,

Biomechanics.

SUMMARY

Fluid-structure interaction between incompressible fluids and compliant

structures pose a significant numerical challenge in various industrial

applications. One such application is in biomechanics where biofluids such as

blood interact with compliant biological structures such as arterial walls.

Understanding the hemodynamics in conjunction with the compliance of the

arteries has proven to be an important step towards understanding the effect of

mechanical factors in disease progression. The numerical challenges that are

faced in such problems range from non-linear and anisotropic behavior of

arterial tissues, non-Newtonian behavior of blood flow as well as strong

multiphysics coupling between structural and fluid dynamics due to near unity

fluid-solid density ratios with significantly flexible structures.



The current paper focuses on utilizing a scalable, parallel multiphysics

framework within Abaqus to solve complex and strongly coupled fluid

structure interaction problems. The framework is combined with an innovative

algorithm for ensuring stability in incompressible fluid-structure interaction

(FSI) problems that have previously required complex and time-consuming

iterative strategies. As an example to demonstrate this methodology and its

efficacy, we study the blood flow and its interaction with the compliant

abdominal aortic aneurysms (AAA) wall. The blood flow is modeled within

the multiphysics computational fluid dynamics capability in Abaqus/CFD. The

structural dynamics of aneurysm walls are modeled with Abaqus/Standard.

The time dependent fully-coupled fluid and structural coupling is enabled in

the background using the co-simulation engine (CSE) within Abaqus without

requiring any user intervention.

1: Introduction

Abdominal aortic aneurysm (AAA) is a prevalent disease that affects a

significant number of people. An aneurysm is a localized bulge of a blood

vessel. AAA occurs in the aortic segment located between the renal arteries

and iliac bifurcation. When the size of AAA increases, there is a significant

risk of rupture causing severe internal bleeding, other complications or even

death. The management and treatment of such conditions require an

understanding of the influence of mechanical factors on AAA growth and its

eventual rupture. It is well known that the arteries exist in a pre-stressed

configuration and the intraluminal pressure due to blood flow acts a normal

force affecting not only its structural behavior but also the biological behavior.

The wall stress is cited as primary mechanical factor affecting the arterial

properties as well as AAA growth and rupture. As a result, many

computational studies have been carried out to study blood flow, structural

dynamics of arterial walls and their coupled interaction with an aim to

understand and quantify the growth of AAA and predicting its rupture. Such a

computational study requires non-linear structural capabilities, computational

fluid dynamics capability as well as coupled multiphysics capabilities to study

the strongly coupled fluid-structure interaction effects.

In the current work, we utilize Abaqus/Standard as the structural solver and

Abaqus/CFD as the incompressible flow solver. Abaqus/CFD incompressible

flow solver uses a hybrid discretization built on the integral conservation

statements for an arbitrary deforming domain. For time-dependent problems,

an advanced second-order projection method is used with a node-centered

finite-element discretization for the pressure. This hybrid approach guarantees

accurate solutions and eliminates the possibility of spurious pressure modes

while retaining the local conservation properties associated with traditional

finite volume methods. The projection method enables legitimate segregation



of pressure and velocity fields for efficient solution of the incompressible

Navier-Stokes equations. The projection operators are constructed so that they

satisfy prescribed boundary conditions and are norm-reducing, resulting in a

robust solution algorithm for incompressible flows. An edge-based

implementation is used for all transport equations permitting a single

implementation that spans a broad variety of element topologies. The solution

methods in Abaqus/CFD use a linearly complete second-order accurate least-

squares gradient estimation. This permits accurate evaluation of dual-edge

fluxes for both advective and diffusive processes. All transport equations in

Abaqus/CFD make use of the second-order least-squares operators. The

implementation of advection in Abaqus/CFD is edge-based, monotonicity-

preserving, and preserves smooth variations to second order in space. The

advection relies on a least-squares gradient estimation with unstructured-grid

slope limiters that are topology independent.

The fluid-structure interaction (FSI) capability within Abaqus utilizes

Abaqus/Standard or Abaqus/Explicit as structural solver and Abaqus/CFD as

the fluid solver with a staggered solution methodology. The global elliptic

nature of the pressure in incompressible flows introduces numerical difficulties

for staggered coupling that are realized as instabilities with exponential growth

in the pressure. Incompressible FSI exhibits this behavior for near unity fluid-

solid density ratios with flexible structures (as commonly seen in

biomechanical problems). Abaqus/CFD deploys a stable, staggered, time-

accurate, incompressible FSI algorithm while taking into account added-mass

effect to solve strongly coupled FSI problems. For detailed information about

Abaqus/CFD solution methodology, we refer readers to [1].

We present the results for time-dependent response of blood flow through

AAA when the arterial walls are rigid (Abaqus/CFD analysis with appropriate

fluid boundary conditions). We then present the results for time-dependent

response of aneurysm walls subject to the time varying cardiac pressure pulse

without considering the effect of blood flow. We finally demonstrate a

coupled FSI methodology to investigate the strongly coupled interaction of

compliant arterial walls with blood flow. For each of these cases, we have also

presented reference results for a healthy artery of the same inlet diameter.

2: Methodology

In order to investigate blood flow, arterial dynamics and coupled FSI in AAA,

we utilize the aneurysm geometries proposed in [2]. These virtual aneurysm

models are used in absence of realistic aneurysm geometries. The CAD

models for aneurysms are created in Abaqus/CAE. The fluid domain has an

inlet and outlet diameter of d = 0.02 m and a maximum bulge of diameter D =

0.06 m at the midsection of the AAA sac. The arterial wall has a thickness of



W = 1.5x10-3

m. Aneurysm geometries in two different configurations are

studied. The asymmetry of the model is dictated by factor β which is the ratio

of radii r and R (see [2] and Figure 1(d)). A value of β = 1.0 represents an

axisymmetric aneurysm while β = 0.2 represents an asymmetric aneurysm

(Figure 1(b), 1(c)). We also study a healthy artery with the same uniform

diameter as the inlet and outlet diameter of the AAA (Figure 1(a)).

Figure 1: Aneurysm geometry, solid (artery) and fluid (blood) domains (a) healthy

artery (b) β β β β = 0.2 (asymmetric aneurysm) (c) β β β β = 1.0 (axisymmetric aneurysm)

(d) asymmetry ratio, ββββ

In the current work, blood is modeled as a Newtonian fluid with a density ρf =

1050 Kg/m3 and a dynamic viscosity µf

= 0.00385 Pa.s. Although blood is a

non-Newtonian fluid with shear thinning behavior, it is a reasonable

approximation to treat it as a Newtonian fluid in blood vessels of size greater

than 0.5 mm in diameters [3]. In vessels of larger diameters, the viscosity is

relatively constant due to high rates of shear.

The arterial walls typically exhibit non-linear, anisotropic hyperelastic

behavior. Several studies have indicated non-linear viscoelastic behavior as

well [4]. Hence, a wide range of material models have been proposed for

arterial walls. The approximations range from isotropic linear elastic behavior

[2] to isotropic non-linear hyperelastic behavior [5] to fully anisotropic non-

linear hyperelastic behavior [6]. While Abaqus supports different material

models, it is often difficult to obtain accurate material constants. In this current

work, for the sake of illustration, we have assumed an isotropic non-linear

elastic behavior with an elastic modulus E = 2.7 MPa and a Poisson’s ratio ν =

0.45. The arterial walls has a density of ρs = 1200 Kg/m

3.

r

R

(a) (b) (c) (d) Inlet

Outlet

Wall

(FSI region)



The boundary of the fluid domain is split into an inlet section where fluid

enters the solution domain; an outlet section where fluid leaves the solution

domain and a wall region which represents the interface with the artery (see

Figure 1(a)). The following boundary conditions are applied to the fluid

domain: (i) a time dependent fully developed velocity boundary condition on

the inlet section (Figure 2), (ii) a time dependent pressure boundary condition

on the outlet section (Figure 2) and (iii) a no-slip/no-penetration boundary

condition on the wall region.

For the laminar flow analysis in Abaqus/CFD with rigid walls, the no-slip/no-

penetration boundary condition dictates that fluid velocity at walls is zero. The

velocity and pressure waveforms conditions utilized here represent the normal

hemodynamic conditions in human abdominal aorta [2]. The Reynolds number

for the flow is 1640 based on the peak inlet velocity and 410 based on time

averaged inlet velocity.

Figure 2: Luminal pulsatile velocity and pressure waveform.

The boundary conditions for the structural model for artery include a zero

displacement condition on the ends of the artery. While more realistic

boundary conditions can be applied including the effect of interaction with

surrounding tissues, a rather simplistic approach is chosen due to lack of data

on such interactions. For studying the dynamics of AAA without including the

effect of fluid flow, we will load the AAA internally with a time dependent

luminal pressure shown in Figure 2. This approach incorporates the effect of

normal forces on the arterial walls but excludes the shear forces exerted by the

blood flow.

For fully coupled FSI simulation, we will computationally solve the laminar

blood flow in Abaqus/CFD and couple it with structural response of AAA in

Abaqus/Standard. Such a simulation will enable inclusion of shear forces on

the arterial walls and it will also enable incorporating the effect of arterial

deformations on blood flow. In addition to the boundary conditions mentioned

above, such an analysis would also require additional constraints at the FSI

interface namely (i) continuity of displacements at blood/AAA interface (us =

uf), (ii) continuity of velocity at blood/AAA interface, (vs = vf, no-slip/no-



penetration condition) and (iii) equilibrium of tractions at the blood/AAA

interface (σσσσs .n = σσσσf .n). The interfacial constraints are automatically honored

by Abaqus when a coupled fluid-structure interaction is defined at the FSI

interface in structural and fluid models.

3: Computational Mesh

The computational meshes utilized in the simulations for solid and fluid

domains are depicted in Figure 3. The meshes for both solid as well fluid

models are created in Abaqus/CAE.

Figure 3: Computational mesh for fluid and solid domains

For the element choice for the artery, we tested fully integrated (C3D8),

reduced integrated (C3D8R) (with enhanced hourglass control) and

incompatible mode (C3D8I) elements in Abaqus/Standard. Various element

choices yielded nearly identical results. Hence, we have used fully integrated

solid elements for both structural dynamics of arteries and fully coupled fluid-

structure interaction study. 18228 solid elements were used in the structural

analysis of the artery.

The fluid domain is meshed with FC3D8 elements in Abaqus/CFD. Four

different meshes ranging from 30480 to 121920 fluid elements were analyzed

to arrive at a mesh convergent solution. A change of less than 5% in velocities

at the centre of fluid domain is used as an acceptable criterion for mesh

convergence.

For FSI analysis, we chose the coarsest fluid mesh (30480 elements) for sake

of computational efficiency and acceptable solution times.

4: CFD Simulation of Blood Flow in AAA



Blood flow is simulated in AAA with boundary conditions mentioned in

section 2. A transient Navier-Stokes solver in Abaqus/CFD is used to obtain

solution for 8 cycles until a time-periodic solution is obtained. Figure 4 depicts

the velocities at five interesting times of the cardiac cycle for β = 0.2.

Figure 4: Velocity contours in AAA, β β β β = 0.2

Figure 5 depicts the mass flow rate at the inlet and outlet and also the error in

mass conservation for AAA with β = 0.2. It can be seen that Abaqus/CFD

obtains an excellent mass conservation with a maximum error of 0.1%.

Figure 5: Mass flow rate and error in mass conservation, β β β β = 0.2

Figure 6 depicts the velocities for axisymmetric aneurysm. The corresponding

values for the healthy artery are depicted in Figure 7.

t = 0.2 s t = 0.3 s t = 0.52 s t = 1.0 s t = 0.4 s



Figure 6: Velocity contours in AAA, β β β β = 1.0

Figure 7: Velocity contours in healthy artery

The solution time to simulate eight cardiac cycles for the finest CFD mesh

(121920 elements) was 14 minutes on 16 cores of a Quad Core 2.27 GHz

machine.

At the beginning of the cardiac cycle, a mildly recirculating flow pattern is

seen at the walls for the healthy artery. For AAA, a similar mild recirculation

is seen which extends all the way along the walls of the aneurysm sac. At peak

inlet flow (t = 0.3 s), the effect of strong systolic acceleration results in a

completely attached flow in the healthy artery as well aneurysm models. The

flow starts recirculating again in the aneurysm sac at its intersection with the

inlet section around t = 0.4 s when the cardiac pressure pulse is at its peak. The

recirculation strength (as seen by reversed x velocity component) reaches its

maximum at t = 0.52 s when the inlet flow has also reversed. The recirculation

adheres to the walls of the aneurysm sac extending all the way to its

intersection with the outlet section.

The maximum wall shear stress in healthy artery and AAA is observed at the

peak flow (t = 0.3 s). The maximum wall shear stress seen in AAA is about

12.4% higher than the healthy artery (see Figure 8). The AAA asymmetry

does not have any remarkable effect in the peak wall shear stress.

t = 0.2 s t = 0.3 s t = 0.52 s t = 1.0 s t = 0.4 s

t = 0.2 s t = 0.3 s t = 0.52 s t = 1.0 s t = 0.4 s



Figure 8: Maximum wall shear stress (t = 0.3 s) in asymmetric and axisymmetric

AAA

5: Structural Dynamics of Aortic Aneurysm

We now present the result for structural dynamics simulation (without

modeling blood flow) of an aortic aneurysm subjected to a time dependent

intraluminal pressure. Figure 9 depicts the Von Mises stresses and

displacements in the AAA for β = 0.2 and β = 1.0 at minimum and maximum

luminal pressure. Corresponding values for the healthy artery are also

depicted.

As can be seen from the results, the maximum Von Mises stress seen in the

artery is 156% more in an asymmetric aneurysm and 135% more in an

axisymmetric compared to the healthy case at maximum luminal pressure (t =

0.4 s). The presence of aneurysm invokes higher stresses on arterial walls

making it prone to rupture. The effect of asymmetry in aneurysm geometry is

to increase the stresses even further.

Healthy

β β β β = 0.2



Figure 9: Von Mises stresses and displacements in the artery at minimum (t = 0.2

s) and maximum luminal pressure (t = 0.4 s) for healthy artery and AAA with β β β β =

0.2, β β β β = 1.0

6: Coupled Blood Flow/Arterial Wall Dynamics Using FSI Approach

We now demonstrate the coupled FSI analysis for the AAA. Figure 10 shows

the Von Mises stresses in the AAA at peak systolic acceleration and peak

cardiac pulse. The corresponding values for the healthy artery are also shown.

The FSI analysis incorporates the effect of both the normal as well as shear

forces due to blood flow. During the cardiac cycle, the time-dependent fluid

forces will deform the inner wall which in turn will affect the fluid velocity

field. The sudden application of fluid forces at t = 0 can cause convergence

issues in the FSI analysis since a sudden pressure pulse causes large

accelerations in the structure. In order to circumvent this problem, an equal

and opposite pressure load is applied to the structure at t = 0. This dummy load

is ramped down over a 0.1s period to load the structure more smoothly. The

coupled FSI simulation is run for six cycles to obtain a time-periodic solution.

t = 0.2 s t = 0.4 s

β β β β = 1.0



Figure 10: Von Mises stresses in the artery at peak systolic acceleration (t = 0.3 s)

and peak cardiac pulse (t = 0.4 s) in the FSI analysis

As can be seen from the results, the coupled simulation is still dominated by

the normal forces exerted by the cardiac pressure pulse.

7: Conclusion

In this study, the methodology for investigating the coupled fluid-structure

interaction of pulsatile blood flow through a compliant artery (with and without

an aneurysm sac) was demonstrated. The staggered FSI solution approach was

implemented using Abaqus/Standard as the structural solver and Abaqus/CFD

as the incompressible Navier-Stokes solver. The case study successfully

demonstrates the efficacy of multiphysics capabilities within Abaqus in

studying structural, fluid and strongly coupled dynamics of blood flow and

compliant arteries with nearly identical fluid-solid density ratios. The stability

of the strongly coupled FSI problem is ensured with a single step algorithm

within a staggered solution paradigm without resorting to iterative strategies.

Future work will involve incorporating the effect of material properties such as

hyperelasticity and viscoelasticity of the artery and the aneurysm sac in a

coupled FSI simulation.

t = 0.3 s t = 0.4 s



REFERENCES

[1] Abaqus version 6.11 Analysis User’s Manual, Dassault Systemes Simulia

Corp, 2011

[2] Scotti, C., Shkolnik, A., Muluk, S. and Finol, E. - Fluid-structure

interaction in abdominal aortic aneurysms: effects of asymmetry and wall

thickness, BioMedical Engineering OnLine, Vol 4, pp. 64, 2005.

[3] McDonald, D. – Blood flow in arteries, Wilkins & Wilkins, 1960

[4] Kuchařová, M., Ďoubal, S., Klemera, P., Rejchrt, P. and Navrátil, M. - Viscoelasticity of biological materials – Measurement and practical impact on

biomedicine, Physiol. Res. 56 (Suppl. 1), pp. S33, 2007

[5] Raghavan, M. L. and Vorp, David A. – Towards a biomechanical tool to

evaluate rupture potential of abdominal aortic aneurysm: identification of a

finite strain constitutive model and evaluation of its applicability, Journal of

Biomechnics, Vol 33, pp. 475, 2000

[6] Vorp, David A. – Biomechanics of abdominal aortic aneurysm, Journal of

Biomechanics, Vol 40, pp. 1887, 2007.

[7] Scotti, C. and Finol, E. – Compliant biomechanics of abdominal aortic

aneurysm: A fluid-structure interaction study, Computers and Structures, Vol

85, pp. 1097, 2007.

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