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100 Chapter 5
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Figure 5.11
Velocity streamlines and the Q-criterion field at t ¼ 180 ms, horizontal long axis view.
Velocity streamlines and the Q-criterion field at t ¼ 380 ms, horizontal long axis view.
The computed velocity streamlines and the Q-criterion (Q-criterion is widely used in fluid
mechanics to identify flow regions with vortical structures [133]) fields at 180, 380, and
580 ms are shown in Figs. 5.11e5.13, respectively. These instances correspond to the
middle of the systole phase, beginning of the diastole phase, and beginning of the second
quarter of the diastole phase (cf. Fig. 5.10).
5.5 Integration in time and solving systems of algebraic equations
FE models as they were introduced in Sections 5.3.2 and 5.4.2 are not fully computable
yet. The approach we use follows the classical method of lines [190]: The application of
FEM in space (but not in time) reduces the mathematical model formulated in terms of
Figure 5.12
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Figure 5.13
Velocity streamlines and the Q-criterion field at t ¼ 580 ms, horizontal long axis view.
PDEs to a large system of ODEs. Next, we need to handle the time dependence. After
discretization in time is done, we obtain systems of algebraic equations, which are solved
numerically. We start discussion of time discretizations with the blood flow in the heart
model, which does not involve interaction with an elastic structure. For the considerations
in the following, the blood flow in vessel with nondeformable walls in Section 5.2.1 can
be seen as a particular case of time-independent physical domain and so the mapping x is
identity. We further proceed to the full FSI problem.
5.5.1 Coupled and splitting methods for the blood flow problem
Consider a time mesh on [0, T] with nodes tk¼ kDt; k ¼ 1; .; K, where Dt ¼ T=K is a
constant time step. We use the notation v
, x) and similar for p and other quantities of interest. For approximation of time
v(t
k
derivatives, it is convenient to use backward differentiation formulas. We shall need firstand second-order differences:
vf
vt
k
¼
fk f
Dt
k1
for a quantity f defined in time nodes.
Implicit discretizations are very common in computational hemodynamics due to their
superior numerical stability. Let v
0
h
operator. The fully implicit FE discretization of (5.28) reads: For k ¼ 1, 2,., find
k
to denote approximation of v at tk, vk(x) z
and
vf
vt
k
¼
3fk 4f
k1
2Dt
þ f
k2
;
¼ Ihðv0Þ, where Ihstands for FE interpolation
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k
k
; p
v
h
h
ℚhsatisfying v
˛ V
h
k
h
¼ I
k
x
h
t
on vbU
ns
, v
k
¼ I
h
k
v
h
D
on vbU
D
and the
following equations:
vv
k
2mJkD
^
U
qhJkF
^
U
vt
k
h
x
T
k
$Jhdx; þ
k
k
v
h
: Vv
: D
k
h
k
Jhdx
x
dx ¼
Z
J
Vv
k
^
U
Z
^
U
Z
Jkfk$Jhdx þ
^
U
k
h
k
p
JkF
h
1
F
k
T
k
k
w
$J
h
: VJhdx
dx
h
(5.29)
Z
Jkg Jhds
N
v^U
k
:¼v
h
k
k
x
and J
t
h
¼ det
k
for all J
ðF
Þ; Fk¼ Vxxk.
k
Z
J
^
U
Z
þ
Z
þ
0
˛ V
; qh˛ ℚh, with advection velocity w
h
h
On each time step, the method (Eq. 5.29) leads to a large system of nonlinear equations.
Newton or inexact Newton method is usually applied to its solution [129]. One may
minimize the computational cost of findingv
k
k
by making one step of the FEM more
; p
h
h
explicit. This should be done with certain care, since more explicit schemes may produce
numerical instabilities, unless the time step is sufficiently small (sometimes, it is too small
that reduces efficiency of simulation). Since the motions of the domain, J
, Fk, and xkare
k
given in Eq. (5.29), the only nonlinear term is the inertia term, the second one in Eq.
k
(5.29). This term is linearized by extrapolating advection velocity w
steps, e.g., w
k
h
¼2v
k1
h
v
k2
h
x
ð
Þ
t
for the linear (second order) extrapolation. With
k
t
from previous time
h
such modification of the inertia term, a linear algebraic system should be solved on each
time step. The resulting method is proved [174] to be unconditionally stable, which means
that numerical stability holds without a restriction on time step Dt. For the rigorous proof,
one needs to modify slightly the formulation by including the term:
Z
1
2
^
U
vJ
vt
k
þdivJkF
1
k
k
w
k
Jhdx;
v
h
h
which is consistent due to the identity (4.68). While computations show that in practice
this term can be skipped, analysis benefits from including it. In the analysis of FEM for
incompressible NaviereStokes equations, including this term corresponds to the Temam’s
[1] skew-symmetric form of the convective terms.
One can further reduce the complexity of algebraic problems on each time step of the
numerical method for solving Eq. (5.27). One very common way of doing this is
decoupling the velocity and pressure update. This can be done by relaxing divergence-free
condition and treating pressure explicitly at predictor step and next correcting the velocity
field by projecting it back to the space of divergence-free functions. The new pressure or
pressure update appears as Lagrange multiplier on this projection step. This idea
is developed in the class of so-called projection or quasi-compressibility methods
(see Refs. [146,42] for a detailed exposition). An example of such projection is given in
3D vascular and heart hemodynamics 103
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the next section. Another splitting algorithm decouples nonlinearity and incompressibility
constraint through decomposition of Eq. (5.27) into the transport diffusion and the Stokes
problem on every time step (see, e.g., Ref. [145]).
5.5.2 Coupled and splitting methods for the fluidestructure interaction problem
In the same spirit of decoupling, one may integrate in time the FSI problem using the fluid
and structure solvers separately. This strategy, however, is well known to have pitfalls. In
particular, for certain range of physical parameters, such splitting schemes can be
numerically unstable for all values of the time step. This instability is related to so-called
“added mass effect” [121]: The presence of the fluid changes the natural vibration
frequencies of the elastic structure, which in a simplified setting can be interpreted as
adding extra “mass” to the structure. The effect is mostly pronounced if the density of the
elastic body is comparable with (or less than) the density of fluid (this is the case of blood
and blood vessel wall). Furthermore, geometric anisotropy also plays a role in amplifying
the added mass effect. It turns out that a segregate treatment of fluid and structure
problems often fails to address adequately this phenomenon and introduces spurious
frequencies independently on how small is the time step.
For the reasons outlined above, coupled (also known as monolithic) solvers have become
popular for the numerical integration of FSI problems (see, e.g.,
Refs. [154,155,177,185,199]). Of course, the superior numerical stability of monolithic
approaches comes with the expense of more computationally demanding algebraic systems
to be solved. This extra cost can be somewhat reduced by the linearization of the coupled
systems through time-lagging geometric terms. Moreover, nonlinear elasticity models can
be also linearized on each time step while keeping the overall stability of the algorithm
(we refer to Refs. [107,172,173] for further details).
We now consider a fully discrete method for the FSI FE problem (5.17)e(5.19). This
numerical strategy extends semi-implicit schemes for the blood flow problem from
Section 5.5.1. However, the FSI problem has stronger nonlinearity due to coupling
between the kinematic variables and the evolution of physical domain. In the ALE/
Lagrangian formulation (5.17) e(5.19), this evolution is accounted by Js, Fs,and
D
s, which depend on the solution. The fully implicit formulation is built by analog to
u
Eq. (5.29), when all solution-dependent quantities are computed on time step t
resulting method is computationally expensive. To linearize the formulation, we first make
all J, F,andD
change Fu
u
k
toeF
h
We next linearize the advection velocity in the inertia term similar to how we did it above
for Eq. (5.29), but now we also have to extrapolate the domain motion:
.The
k
terms explicit by extrapolating them from previous time steps, i.e., we
k
¼ F
k
w
h
k
e
u
h
¼2v
;eu
k
h
k1
h
¼eu
k1
h
vu
vt
eu
h
k2
; JktoeJk¼ det
h
k1
v
k2
h
e
F
,andD
k
vu
k2
h
vt
k
u
h
to D
k
~
u
h
:
.
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The resulting method takes the following form:
for all J
for all f
for all q
Z
r
Us
Z
þ
U
k
vv
h
vt
e
J
f
vv
k
vt
Jhdx þ
k
h
Jhdx þ
s
r
f
Z
e
k
D
Jhdx
J
k
f
e
u
h
Z
vu
h
vt
U
s
h
h
˛ V
˛ V
00
h
0
h
þ
,
, and
2m
U
s
Z
e
J
U
f
˛ ℚh. Note that although strong coupling (Eq. 5.20) is imposed on the interface,
h
k
fhdx
Vv
k
Z
e
Su
F
k
U
s
Z
r
J
f
U
f
Z
k
e
e
p
J
F
k
h
U
f
k
k
;eu
: VJ
e
F
: VJ
1
k
h
w
h
h
k
e
Vv
k
h
T
k
e
u
k
h
Z
k
v
fhdx ¼ 0 (5.31)
h
U
s
k
T
qhdx ¼0 (5.32)
: F
h
h
only a linear algebraic system should be solved on each time step.
dx
k
Jhdx
h
dx ¼ 0
h
(5.30)
The added mass phenomenon puts serious restriction on applicability of decoupling to the
FSI problem. Nevertheless, splitting schemes are not ruled out entirely. One attractive
splitting approach was suggested in Ref. [138] as a natural extension of the projection
method for fluid problems. Adopting this decoupling strategy to the ALE/Lagrangian FE
formulation with Saint VenanteKirchhoff material leads to the following three-step
algorithm:
Step 1. Given v
i
i
i
, u
, p
h
for i ¼ k 1, k 2, ., first find intermediate new velocitybv
h
h
k
h
satisfying
f
k
2m
Z
k
h
vbv
vt
f
U
bv
Dt
h
e
D
J
k
r
s
s
k
h
k
Jhdx þ
k
b
k
v
: D
~
u
h
h
k
h
k
vv
h
Jhdx þ
vt
dx
J
h
Z
r
U
f
k
Jhdx
~
u
h
, velocity v
Z
Z
p
U
f
1
k
Vbv
e
F
k
h
e
J
k
f
Z
k1
e
p
J
h
U
f
k
, and pressure p
h
k
e
Su
;eu
k
h
k1
e
J
h
k
h
U
F
s
p
k
w
Jhdx
h
(5.33)
k
e
F
k
h
T
k
: VJ
k
e
u
h
k
satisfying
h
h
: VJ
dx
dx ¼ 0
h
(5.34)
T
e
F
k
k
e
u
h
: VJ
k
dx ¼ 0
h
Z
e
r
J
f
U
f
Z
þ
U
for all J
Step 2. Find new deformation u
þ
h
Z
˛ V
r
U
f
0
.
h
v
e
J
k
f
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˛ V
h
˛ ℚh.
00
h
0
h
,
, and
Z
U
k
vu
h
fhdx
vt
s
Z
k
e
J
Vv
U
f
.
f
Z
k
v
fhdx ¼ 0 (5.35)
h
U
s
k
T
qhdx ¼ 0 (5.36)
: F
h
k
k
to the fluid domain, by solving, e.g.,
h
is solved, while on Step 2
f
for all Jh˛ V
for all f
for all q
Step 3. Extend continuously the deformation field u
h
auxiliary elasticity equation in U
We see that on Step 1 only the problem for velocity in U
velocity, pressure, and deformation serve as unknowns, but the system is greatly simplified
in U
. The stability of such splitting method was studied in Ref. [138].
f
5.5.3 Systems of linear algebraic equations
Algorithms for solving systems of linear algebraic equations are the cornerstone for the
numerical solution of many cardiovascular problems. Both implicit and semi-implicit
schemes typically lead to systems of algebraic equations with possibly millions of
unknowns. Depending on size, structure, and various algebraic properties of corresponding
matrices, direct or iterative methods are used to find (approximate) solution to these
systems. We will come across direct solvers later in this chapter. Since the systems of
algebraic equations resulting from Eqs. (5.29) or (5.30)e(5.32) are, in general,
nonsymmetric, generalized minimal residual method (GMRES) [186] is a suitable iterative
algorithm for solving them. For the system of equations:
GMRES takes initial guess x
i
; i ¼ 1; 2; . to the solution x. For given i, the iterate xisolves the minimization problem:
x
where K
is the so-called Krylov subspace, defined as
i
The computational complexity of passing from x
of matrixevector multiplication with the matrix A (at least for not too large is).
Therefore, the approach is particularly attractive for solving systems of linear algebraic
equations resulting from FEMs, since the corresponding matrices are sparse, i.e., they have
very few nonzero entries compared with the total number of entries. The sparsity property
Ax ¼b; A ˛ R
0
˛ RNand builds a sequence of approximations (iterates)
i
b
¼ min
Ax
¼spanr0; Ar0; .; Air
K
i
y ˛ K
NN
; x; b ˛ RN; (5.37)
0
Ax
i
þy b
0
; r
i1
to xiis largely defined by complexity
;
0
¼ b Ax0:
106 Chapter 5
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makes the matrixevector product operation computationally cheap and highly suitable for
parallel computing. Another popular choice of the iterative solver for nonsymmetric
nondefinite systems is the BiCGstab method [204], which needs less computer memory
than GMRES but is less robust: its convergence can be nonmonotone and, in general, is
not guaranteed.
Although, in the exact machine arithmetic, the GMRES method would give the exact
solution to Eq. (5.37) for some i N, the method is rarely used as direct (exact) solver. It
is much more common to terminate the process for smaller values of i, when the iterate x
is close enough to the true solution x. The convergence of the method (which measures
how fast x
i
approaches to x for increasing i), however, strongly depends on algebraic
properties of the matrix A, for example, the bounds on pseudospectrum or field of values
for the matrix. Iterative methods based on Krylov subspaces, including their convergence
properties, have been extensively addressed in the literature (we refer to the monographs
[169,179,187] for further details).
In general, algebraic properties of the matrix A from Eq. (5.37) should be improved by
preconditioning for GMRES, BiCGstab, or other iterative method to be efficient. One may
think of preconditioning as looking for a matrix P ˛ R
NN
, which is called preconditioner,
such that
i
• the equation Py ¼ b is “easily” solved with any given right-hand side vector b and
• the product matrix P
1
A or AP1is more suitable for applying the iterative method.
Different approaches for building preconditioners have been proposed [206]; some of them
explicitly exploit information and properties of the underlying PDE system and FE
method, whereas others are entirely based on entries of matrix A. Both approaches are
used for FE discretizations of hemodynamics problems. In particular, for the blood flow
problems in nondeformable volumes, it is common to design P by exploiting the block
structure of the matrix A. This structure is well seen after regrouping velocity unknowns
first and pressure unknowns next:
AB
B 0
T
¼
v
p
b
v
b
5 Ax ¼ b; (5.38)
p
v and p are the vectors of nodal values of FE velocity and pressure functions. Then one
may construct P (or directly P
A and for the pressure Schur complement matrix BA
1
) through defining preconditioners for the velocity block
1BT
, which arise if one applies the
blockwise Gauss elimination procedure to the system (5.38). In turn, a preconditioner for
A can be built upon a PDE-based technique such as multigrid or domain-decomposition
algebraic solver [116,179], and for BA
1BT
, less standard approachers should be used.
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Preconditioners exploiting the block structure of system (5.38) in this or other ways are
well studied in the literature (see, e.g. Refs. [109,135,179] and references therein).
An alternative to block preconditioners is based on elementwise incomplete factorizations
of the 2 2 block matrix from Eq. (5.38). Among various factorizations, the ILU
factorization is the most commonly considered for the purpose of preconditioning. In this
approach, one looks for a left triangle matrix L and a right triangle matrix R such that
A ¼LR þ E (5.39)
where E is the error matrix of the decomposition. The factors L and R are easily
“invertible” (assuming all diagonal elements are nonzero); hence, LR can be a good
preconditioner if E has a small norm compared with A or/and E has a small rank [187].
Preconditioners based on incomplete LU factorization are less studied for the discrete
fluid system than various block preconditioners. Several approaches to building ILU
preconditioners for nonsymmetric saddle-point systems that arise from the FE
discretization of the incompressible NaviereStokes equations have been suggested in
Refs. [126,161,162,205]. Numerical results in Section 5.2.4 were computed using the
second-order threshold ILU preconditioners [161,162]. In particular, the method was
successfully applied to simulate a flow in a reconstruction of the right coronary artery of
a real patient. The assessment of block preconditioners in the hemodynamics context can
be found in Ref. [131]. Comparison studies of some block and ILU preconditioners were
published in Refs. [205,192]. Finally, we note that applying numerical stabilization such
as SUPG method discussed in Section 5.2.3 may perturb the matrix structure in Eq.
(5.38), leading to (1,2)-block not equal to the transpose of (2,1)-block; also (2,2) may
become nonzero. These changes m ay affect algebraic properties of the matrix making
them more or less favorable, depending on the stabilizatio n method applied and algebraic
solver to be used.
The algebraic system becomes even more complicated for the problem where
blood flow interacts with the elastic walls of the vessel. For the semi-implicit method
(Eqs. (5.30)e(5.32)), the algebraic system can be written in the block form:
0
Ef00I
B
0 A
B
B
B
B
B
@
f
0 B 000
00 0 M
v
0 J
sf
fs
BT0 J
sCs
p
J
EsA
sf
1
0
0
1
u
B
B
B
B
@
f
C
v
f
C
C
¼r.h.s. vector; (5.40)
p
C
A
u
s
v
s
C
fs
C
C
C
A
s
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vf; p are the vectors of nodal values of FE fluid (blood) velocity and pressure functions, u
is ALE displacement field in the fluid domain, and usand vsare the displacement and
velocity of the elastic body.
From Eq. (5.40), we see that u
variables v
, p, us, and vs, however, cannot be decoupled. To solve for them numerically,
f
can be found separately after usbecomes available. Other
f
either a preconditioner for GMRES/BiCGstab iterations has to be designed or one can use
a direct solve based on exact factorization of the matrix. The latter is a feasible option
only if the number of degrees of freedom is not too large. For numerical examples in
Sections 5.3.3 and 5.4.4, we use multifrontal sparse direct solver MUMPS [106]. The
acceptable accuracy with moderate number of degrees of freedom (not significantly
exceeding 10
5
d.o.f.) is achieved, thanks to the higher-order FE combined with a mesh
adaptation.
There exists an extensive literature on building suitable preconditioners for linear systems
resulting from monolithic FSI FE formulations like Eqs. (5.30)e(5.32). Most of them
exploit the block structure of the matrix in Eq. (5.40). One natural strategy would be to
build a preconditioner by removing some coupling off-diagonal blocks in Eq. (5.40) so
that fluid and structure problems can be solved independently, when the preconditioner is
applied. Such decoupling may inherit a deficiency related to the added mass effect
(see Section 5.5.2) in the sense that the convergence of the preconditioned iterations may
deteriorate for the same values of physical parameters, when the splitting time-stepping
method exhibits numerical instability. In Ref. [107], it was noted that the stable splitting
Eqs. (5.33)e(5.36) can be translated into more robust preconditioning of the coupled
system (5.40). More on preconditioning of FSI linear algebraic systems can be found, for
example, in Refs. [119,125,130,160].
f
CHAPTER 6
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Lumped parameter models
6.1 Introduction
6.1.1 General idea of the lumped models
Depending on a biomedical application, a different level of details is required from the
mathematical model of the cardiovascular system. If only basic (averaged) information is
required, then a common approach is to apply spatial reduction techniques. The extreme
spatial reduction produces lumped parameter models (also known as zero-dimension or 0D
models). In a lumped parameter model, the region of interest (whole organism or its local
part) is virtually represented by a set of compartments connected to each other. Each
compartment may represent an organ, a portion of a tissue, or a part of the body. For
example, the 0D model presented in Ref. [211] relates average values of the flow rate,
velocity, and pressure in the compartments without resolving any spatial details,
whereas the 0D model from Ref. [212] takes into account some spatial elements like
heart valves.
Lumped parameter models usually need a few parameters that can be measured easily for
a specific patient. The 0D models have low computational cost. They allow fast
hemodynamic simulations over dozens of heart cycles. The lumped parameter
hemodynamic models can be classified into heart function models, vascular compartment
models, and terminal resistance models. Examples of such models are the fourcompartment dynamical model of the heart from Ref. [213], leading to a stiff system of
ordinary differential equations (ODEs), or the model of cerebral hemodynamics from
Ref. [214], resulting in the nonlinear ODE of Van der PoleDuffing oscillator.
In this chapter, we focus on a model of the heart function. The model is simple enough
and shares a lot of features with many lumped parameter models. We note that a detailed
mathematical model of the heart function should account the action potential propagation,
myocardium tissue mechanics, hydrodynamics of the blood in the chambers, and coronary
flow with myocardium perfusion, although modeling some of these phenomena separately
may have practical sense as well. Closed models of the cardiovascular system, studied in
Refs. [213,216e222], describe the heart function in terms of the pressureevolume
relationship and/or the cardiac output and the ejection fraction. Both statistics depend on
the venous inflow to the auricles (preload conditions) and arterial outflow from the
Personalized Computational Hemodynamics. https://doi.org/10.1016/B978-0-12-815653-7.00006-3
Copyright © 2020 Elsevier Inc. All rights reserved.
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