Arif Masud: Advanced Computational Methods and Non-Newtonian Blood Models for Blood-artery ...

33 Pages·2016·9.95 MB·English

by  Steve Kleinvehn

Advanced Computational Methods
and Non-Newtonian Models for Blood-artery
Interactions in Patient Specific Geometries
Arif Masud
JaeHyuk Kwack
Soonpil Kang
Blue Waters Symposium, June 13, 2016
Research Highlights
Developing advanced numerical methods for cardiovascular biofluid dynamics
• 
New Generalized Oldroyd-B models for viscoelastic fluids"
• 
New Class of Stabilized FE methods for Non-Newtonian fluids"
• 
Application to CF-VAD assisted flows in diseased Carotid artery to assess progression of
disease and risk of stroke"
• 
Blood-artery interaction models (hypertension)"
• 
Intracranial hemodynamics during high levels of +Gz accelerations"
Developing tools for Automated Geometry Reconstruction
• 
Java & Python codes to extract geometry from MRI, CT-Scans"
• 
Codes for processing patient models for 3D printing"
Collaborators:
• 
Prof. Bill Gropp, University of Illinois"
• 
Mark Vanmoer, NCSA"
• 
Dr. G. Bhatt, Heart Institute, Advocate Christ Medical Center, Oak Lawn, IL, USA!
2
"
Shear-rate Dependent Constitutive Models

Viscosity vs. shear-rate Shear-stress vs. shear-rate
Constitutive relations for viscosity:
(a) Newtonian model: η(γ):= η
(b) Power Law model: η(γ):=µγ n−1
(n−1)
(c) Carreau-Yasuda model: η(γ):+=− η +(η η )(1 (λγ)a) a
∞ 0 ∞
3
Incompressible Shear-rate Dependent Fluids

Governing Equations
ρv +⋅ρ∇v −∇v ⋅ =∇+σ (v) Ω p× ρf in ]0,T[
,t v
∇⋅v= 0 inΩ × ]0,T[
v =Γg × on ]0,T[
g
σ ⋅=n (σ−(v) ⋅=pI) n h Γ × on ]0,T[
v h
v(x,0) =Ωv × on {0}
0
• σ = 2η(γ)ε(v) := viscous stress tensor
v
•γ = 2ε(v): ε(v) := shear rate
The Standard Weak Form
v v
Find v ∈= Ω(H1( ))nsd a∈nd= pΩ P∩ CΩ0( ) L∀2( ),∈ s.t. w ∈ and q P
0
ρ(w∇,v )++⋅ρ∇(w−,v∇=⋅ v) ( w,+σ (v)) ( w, p) ρ(w, f ) (w,h)
,t v
Γ
h
(q,∇⋅v)= 0 4
The Variational Multiscale Method

v(x,t) = v (x,t) + vʹ(x,t); w(x) = w(x) + wʹ(x)
1 2 3 1 2 3 { {
coarse scale fine scale coarse scale fine scale
•  vʹ
: Piecewise polynomials of sufficiently high order, continuous in space
but discontinuous in time
Modified variational form
ρ(w + wʹ,v )+ρ(w + w+ʹ+,(∇⋅v∇+v+ʹ) +(v vʹ)) ( (w wʹ),σv( v ʹ))
,t v
−(∇⋅(w+ wʹ), p)= ρ(w+ wʹ, f +) (w+ wʹ,h)
Γ
h
(q,∇⋅(v + vʹ))= 0
•σ (v + vʹ) = 2ηεv( +v ʹ) ε•v (v +εvʹ) =εv( )+ ( ʹ)
v
•η is a function of the shear-rate from the coarse-scale velocity field
5
(Hughes CMAME 1995); (Kwack & Masud Comp Mech. 2014)
Nonlinear Stabilized Form

Embed fine-scale solution in the coarse problem:
ρ(w∇,v )++⋅ρ∇(w,−v∇∇⋅+v⋅) ( w,2ηεv( )) ( w , p) (q, v)
,t
⎛⎛ρ(−∇v ⋅w +(∇⋅v )w + v ⋅∇w)⎞ ⎞
⎜⎜ ⎟ ⎛−ρv −ρv∇+⋅∇⋅v 2η εv( )⎞⎟
−+⎜∇⎜ η∇(⋅Δ+( w) w) ⎟,τ ⎜ ,t ⎟⎟ = ρ(w, f )+(w,h)
⎟ + ∇ ⋅ −∇ ⎜+⎜ ⎟ ⎜ 2 η εv( ) p ρf ⎟ Γh
⎝ ⎠
⎜⎜+∇η⋅((∇⋅w)1+∇w)+∇q ⎟ ⎟
⎝⎝ ⎠ ⎠
• 
Smooth variation of shear rate over element interiors ð non-zero
gradients of the viscosity field
• 
Additional contributions to the stabilization terms
• 
These terms are important for obtaining optimal convergence for
higher-order elements
6
Implementation Strategy
and Performance of the Code
-  Mesh partition employing a “ghost” node
approach
- METIS library to partition the mesh
- Balanced work load
- Unbalanced communication pattern

- PETSc to solve the system of equations

- Implicit Method with Newton-type strategy

- Generalized alpha method for time integ.

- Most part of the computations are done
at the element level
- Degradation of scalability due to
- Communication unbalance
- “Ghost” nodes
Blood Flow in Carotid Artery

Inflow:
Common Caro,d Artery
8
Blood Flow in Carotid Artery

D = 7.38 mm
D = 4.10 mm
D = 7.97 mm
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Blood Flow in Carotid Artery

D = 7.38 mm
D = 4.10 mm
D = 7.97 mm
10