In order to obtain an energypreserving scheme for the blood flow model, which satisfies a discrete counterpart of Theorem 1, we will introduce three variational formulations. The first one corresponds to the standard formulation that one obtains directly from System (1). This formulation has A and u as principal unknowns, where A is the crosssection of the aorta defined as \(A=Q/u\) and u is the blood velocity. Then, we introduce a second formulation that uses as a primary unknown the radius of the aorta \( R = \sqrt{A} / \sqrt{\pi } \) and u. This formulation is straightforwardly deduced from the first one and is a convenient intermediate step, since it introduces several simplifications. From these intermediate changes we deduce the last formulation that is written in the unknowns
$$\begin{aligned} v:= R \, u \quad \text{ and } \quad \varPhi := \varphi (R), \end{aligned}$$
where \( \varphi (R)\) is a smooth bijective function from \({\mathbb {R}}^+\) to \(I \subset {\mathbb {R}}\) that we define later. This change of variables has the main advantage to provide an “energycompliant” discretization, as it will be shown in the next chapter. As the reader will see, the energy is a quadratic functional of the new variables \(( \varPhi , v)\) .
Variational formulation in (A, u)
As a first step, we substitute Q with Au in (1) and assume that \( \Gamma = 0\), hence \( \psi _v = 0\). After some algebraic manipulations we obtain that (1) is equivalent to the system
$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}A+\partial _{s}(Au)={0}, \\ {\displaystyle \left( \frac{1}{2} \left( \partial _{t}A\right) u+A\partial _{t}u\right) } + {\displaystyle \left( \frac{1}{2} \, u^2 \, \partial _{s} A +\frac{3}{2} A \, u\, \partial _{s}u \right) }+ {\displaystyle \frac{A}{\rho }}\partial _{s}P \! \left( A\right) +K_{r}u={0}. \end{array}\right. } \end{aligned}$$
(21)
Note that we have rewritten System (1) in a specific form adapted to the derivation of the energy balance (in fact System (21) is obtained following the proof of Lemma 1 provided in [37]). Indeed, multiplying the second equation of System (21) by u, one can see that
$$\begin{aligned}&{\displaystyle \left( \frac{1}{2} \left( \partial _{t}A\right) u +A\partial _{t}u\right) } \, u \, = \frac{1}{2}\partial _t (A u^2), \nonumber \\&\quad \text {and } {\displaystyle \left( \frac{1}{2} \, u^2 \, \partial _{s} A +\frac{3}{2} A \, u\, \partial _{s}u\right) } \, u \, = \frac{1}{2} \partial _s (A u^3). \end{aligned}$$
(22)
These two equalities are, in fact, essential to prove the energy relation of Lemma 1. The objective is now to derive a weak formulation of System (21). Concerning the first equation, we multiply it by a spacedependent test function \({\tilde{\varPhi }}\) and we integrate in space, obtaining
$$\begin{aligned} (\partial _tA, {\tilde{\varPhi }}) + (\partial _s(Au), {\tilde{\varPhi }})=0, \end{aligned}$$
(23)
where \((\cdot ,\cdot )\) is the \(L^2\)scalar product in (0, L). We now focus on the second equation of System (21) and we repeat the procedure performed above, multiplying each term by a spacedependent test function \({\tilde{v}}\). After some manipulations, we obtain
$$\begin{aligned}&\Big (\frac{\rho }{2}(\partial _t A) u + \rho A\partial _t u,{\tilde{v}}\Big ) + a(u; {\tilde{v}}, A ) \nonumber \\&\quad  (\partial _s(A{\tilde{v}}), \psi (A) ) + \rho K_r (u,{\tilde{v}})= g({\tilde{v}}; A, u), \end{aligned}$$
(24)
where \(a( u ; \cdot ,\cdot )\) is bilinear in its two last arguments but nonlinear in u and is given by
$$\begin{aligned} a( u ; A, {\tilde{v}}) := \int _{0}^L \rho \left( \frac{1}{2} \, u^2 \, \partial _{s} A +\frac{3}{2} A \, u\, \partial _{s}u\right) \, {\tilde{v}} \, \mathrm {d}s \left. \frac{\rho }{2} A \, u^2 \, {\tilde{v}} \right _{0}^L, \end{aligned}$$
(25)
and the nonlinear functional g is defined as
$$\begin{aligned} g( A , u ; {\tilde{v}} ) := \left. \left( \frac{\rho }{2} A\, u^2 \,{\tilde{v}} + A \, {\tilde{v}} \, \psi (A) \right) \right _{0}^L =  A {\tilde{v}} \left( P_{\text {tot}}  P_{\text {ext}} \right) \Big _{0}^L . \end{aligned}$$
(26)
Observe that, by construction, g is linear in \( {\tilde{v}} \) and includes only boundary terms. Up to this point, the weak formulation of the problem described in System (21) is
$$\begin{aligned} {\left\{ \begin{array}{ll} (\partial _tA, {\tilde{\varPhi }}) + (\partial _s(Au), {\tilde{\varPhi }})=0, \\ \displaystyle \Big (\frac{\rho }{2}(\partial _t A) u + \rho A\partial _t u,{\tilde{v}}\Big ) + a( u; A, {\tilde{v}})  (\partial _s(A{\tilde{v}}), \psi (A) ) + \rho K_r (u,{\tilde{v}})= g( A, u; {\tilde{v}}). \end{array}\right. }\end{aligned}$$
(27)
Finally, observe that if we substitute
$$\begin{aligned} {\tilde{\varPhi }}=\psi (A), \quad {\tilde{v}}=u, \end{aligned}$$
in System (27), we can easily retrieve the energy relation presented in Lemma 1. Indeed, thanks to (22), one can see that
$$\begin{aligned} a(u; A, u) = 0. \end{aligned}$$
(28)
Moreover, we recover the energy relation
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} {\mathcal {E}}_{ar}(t) + {\mathcal {D}}_{ar}(t) = g( A, u; u) , \end{aligned}$$
(29)
with
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} {\mathcal {E}}_{ar}(t) = \Big (\frac{\rho }{2}(\partial _t A)u + \rho A \partial _t u, u\Big ) + (\partial _t A, \psi (A)) \quad \text{ and } \quad {\mathcal {D}}_{ar}(t)=\rho K_r (u,u). \end{aligned}$$
(30)
Here \({\mathcal {E}}_{ar}(t)\) is the total energy of the 1D model and \( {\mathcal {D}}_{ar}(t)\) represents the dissipative term with \(K_r\ge 0\).
An intermediate formulation in (R, u)
In order to obtain a formulation that leads to the achievement of the energy preservation at a discrete level, we construct an intermediate form of System (21). This formulation is obtained by replacing A with \(\pi R^2\), where R represents the radius of the lumen. The unknowns become u and R. The first equation of System (21) is now described as
$$\begin{aligned} 2 \pi \big ( R \, \partial _t R , {\tilde{\varPhi }} \big ) + \pi \big ( \partial _s (R^2 u), {\tilde{\varPhi }} \big ) = 0. \end{aligned}$$
(31)
Then, the first term in (24) can be rewritten, substituting \(\displaystyle {\tilde{v}} \leftarrow {{\tilde{v}}}/{R}\), as
$$\begin{aligned} \Big (\frac{\rho }{2}(\partial _t A) u + \rho A\partial _t u, \frac{{\tilde{v}}}{R}\Big ) =\pi \, {\rho } \, \big ( u \, \partial _t R + R \partial _t u,{\tilde{v}}\big ) =\pi \, {\rho } \, \big ( \partial _t (Ru),{\tilde{v}}\big ). \end{aligned}$$
Moreover, one can see that
$$\begin{aligned} a\Big ( u; A, \frac{{\tilde{v}}}{R}\Big ) = \displaystyle \pi \rho \, \int _{0}^L \left( \frac{ (Ru)^2 }{R^2} \, \partial _{s} R +\frac{3}{2} (Ru)\, \partial _{s}\frac{Ru}{R}\right) \, {\tilde{v}} \, \mathrm {d}s  \pi \, \left. \frac{\rho }{2} (R u)^2 \, \frac{{\tilde{v}}}{R} \right _{0}^L, \end{aligned}$$
and
$$\begin{aligned} \Big (\partial _s\Big ( A \frac{{\tilde{v}}}{R}\Big ), \psi (A) \Big ) = \pi ( \partial _{s} (R {\tilde{v}} ), \psi (\pi R^2 ) ) . \end{aligned}$$
Note that the substitution \(\displaystyle {\tilde{v}} \leftarrow {{\tilde{v}}}/{R}\) does not lead to any issue, since we consider solutions with \(R>0\) at any time and position. Finally, collecting the four expressions above, one can obtain a formulation with R and u as primary unknowns. It reads
$$\begin{aligned} {\left\{ \begin{array}{ll} 2 \pi \big ( R \, \partial _t R , {\tilde{\varPhi }} \big ) + \pi \big ( \partial _s (R^2 u), {\tilde{\varPhi }} \big ) = 0, \\ \displaystyle \pi \, {\rho } \, \big ( \partial _t (Ru),{\tilde{v}}\big ) + a\Big ( u; \pi \, R^2, \frac{{\tilde{v}}}{R}\Big ) \\ \qquad \displaystyle \, \pi ( \partial _{s} (R {\tilde{v}} ), \psi (\pi R^2 ) ) + \rho K_r \Big (u,\frac{{\tilde{v}}}{R}\Big )= g\Big ( \pi R^2, u; \frac{{\tilde{v}}}{R}\Big ). \end{array}\right. } \end{aligned}$$
(32)
It is worth noticing that the product Ru appears “almost” naturally and it is therefore tempting to define \(v := Ru \) as a new variable. It becomes even more obvious that this choice is suitable by looking at the energy density, defined as
$$\begin{aligned} e = \frac{\rho }{2}A{u}^2 + \Psi (A) = \frac{\pi \rho }{2} (R {u})^2 + \Psi (\pi R^2). \end{aligned}$$
(33)
This is precisely what motivates the introduction of the next formulation. Moreover, it is worth mentioning that now
$$\begin{aligned} \psi (\pi R^2) = \beta \frac{\sqrt{\pi }R  \sqrt{A_0}}{A_0}, \end{aligned}$$
(34)
so \(\psi (\pi R^2)\) is linear with respect to the unknown R and we will see in the next sections that \(\Psi (\pi R^2) \) is a thirdorder polynomial and this will simplify its analysis.
Remark 8
The change of variable \( A =\pi R^2 \) is still meaningful even if the 1D hemodynamic model does not assume a perfect circle for the geometry of the crosssection. What matters here is that the new variable R depends on the square root of A. Obviously, the introduction of the factor \(\pi \) is natural to obtain a physical meaning for this new variable since, in practice, arterial crosssections are almost circular.
Variational formulation in \((\varPhi ,v)\)
A change of variables has to be made in order to demonstrate that the scheme is energypreserving after time discretization. More precisely, time discretization can easily deal with energies that involve quadratic terms of the unknowns. However, the energy density described in (33) is not a quadratic term of the unknowns (R, u) , but we can see that the first contribution is a quadratic term of
$$\begin{aligned} v := R u . \end{aligned}$$
Therefore, we propose to use v as a main unknown. A first naive choice is then to set \( \varphi (R) \) equal to \( \sqrt{\Psi ( \pi R^2 )} \), where \(\Psi (\pi R^2)\) is defined as
$$\begin{aligned} \Psi (\pi R^2)= \int _{A_0}^A \beta \frac{\sqrt{a}  \sqrt{A}_0}{A_0}da = \frac{\beta }{ A_0}\Big [\frac{2}{3}\pi ^{\frac{3}{2}}R^3  \sqrt{A}_0 \pi R^2 + \frac{1}{3} A_0^{\frac{3}{2}} \Big ], \end{aligned}$$
(35)
and set \(\varPhi \equiv \varphi (R)\) as the other main unknown. However, we show in “Variational formulation in \((\varPhi ,v)\)” section that this choice is not convenient, since \( \varphi (\cdot ) \) would not be a bijective function from \( {\mathbb {R}}^+\) to \( {\mathbb {R}}^+\). Instead, we define
$$\begin{aligned} \varphi (R) := \left\{ \begin{array}{ll} \sqrt{ \Psi (\pi R^2) } &{} \; R\ge R_0,\\  \sqrt{ \Psi (\pi R^2) } &{} \; 0\le R < R_0, \end{array} \right. \quad \text{ with } \quad R_0 = \frac{\sqrt{A_0}}{\sqrt{\pi }}. \end{aligned}$$
(36)
Before studying in more detail the impact of the choice described in System (36) (in particular the bijectivity of the function \(\varphi \)), we formally give the variational formulation associated with the new couple of unknowns \((v, \varPhi )\), where \( \varPhi := \varphi (R). \) Assuming for now that \( \varphi \) is bijective, we define the reciprocal function \( r( \varPhi ) := \varphi ^{1}( \varPhi ) \). Then, each term of the second equation of System (32) can be modified as follows:

i.
The term involving the time derivative reads
$$\begin{aligned} \pi \, {\rho } \, \big ( \partial _t (Ru),{\tilde{v}}\big ) = \pi \, {\rho } \, \big ( \partial _t v,{\tilde{v}}\big ). \end{aligned}$$
(37)

ii.
The nonlinear transport term reads
$$\begin{aligned} a\Big ( u; \pi \, R^2, \frac{{\tilde{v}}}{R}\Big )&= {\tilde{a}}\Big ( \frac{ v}{ r( \varPhi )}, v, {\tilde{v}}\Big ) \nonumber \\&:= \frac{\pi \rho }{2} \int _{0}^L \Big ( 2 \, {{\tilde{v}}} \, \frac{v}{ r( \varPhi )} \, \partial _s {v} + v \,{{\tilde{v}}} \, \partial _s \,\frac{v}{ r( \varPhi )} \Big ) \, \mathrm {d}s  \left. \frac{\pi \rho }{2} {{\tilde{v}}} \, v \, \frac{v}{ r( \varPhi )}\right _{0}^L, \end{aligned}$$
(38)
where \( {\tilde{a}} \) is now a trilinear form. Such reformulation will lead to the choice of an adapted space discretization that preserves, for all sufficiently smooth functions v and u, the property
$$\begin{aligned} {\tilde{a}}(u, v, v) = 0, \end{aligned}$$
(39)
and in particular for \(u = v/r( \varPhi )\).

iii.
The coupling term \( ( \partial _{s} (R {\tilde{v}} ), \psi (\pi R^2 ) ) = ( \partial _{s} (r( \varPhi ) {\tilde{v}} ), \psi (\pi r( \varPhi )^2 ) ) \) is not modified.

iv.
The dissipation term reads
$$\begin{aligned} \Big (u,\frac{{\tilde{v}}}{R}\Big ) = \Big (\frac{ v}{r( \varPhi )},\frac{{\tilde{v}}}{r( \varPhi )}\Big ). \end{aligned}$$
(40)

v.
Finally, the boundary term g is given by
$$\begin{aligned} g\Big ( \pi R^2, u; \frac{{\tilde{v}}}{R}\Big ) = {\tilde{g}}( r( \varPhi ), {\tilde{v}} ) :=  \pi \, {\tilde{v}} \, r( \varPhi ) \big ( P_{\text {tot}}  P_{\text {ext}}\big ) \bigg _{0}^L , \end{aligned}$$
(41)
where, for simplicity, we assume that \( P_{\text {tot}} \) is given. Of course, if more general boundary conditions are considered, g must be modified accordingly.
Using all the expressions above, we obtain the following equation (corresponding to the second equation of System (32))
$$\begin{aligned} \pi \rho (\partial _t v, {\tilde{v}}) + {\tilde{a}}\Big ( \frac{v}{r},v, {\tilde{v}}\Big )  \pi \, (\partial _s (r{\tilde{v}} ) , \psi (\pi r^2)) + \rho K_r \Big ( \frac{v}{r},\frac{{\tilde{v}}}{r} \Big ) = {\tilde{g}}( r, {\tilde{v}}), \end{aligned}$$
(42)
where, for the sake of clarity, we have written \(r\) instead of \( r( \varPhi )\). The first term in (42) clearly shows how the introduction of v simplifies the dynamic behavior of the equation and it will help at the discrete level to demonstrate the energy preservation. Now we deal with the first equation of System (32) in which we use as a test function \(\displaystyle {\tilde{\varPhi }} \leftarrow \xi (R) {\tilde{\varPhi }} \), with
$$\begin{aligned} \xi (R):= \displaystyle \frac{\psi (\pi R^2)}{\varphi (R) }. \end{aligned}$$
(43)
We show in “Variational formulation in \((\varPhi ,v)\)” section that this function is smooth and positive. We obtain
$$\begin{aligned} 2 \pi \big (\xi (R) \, R \, \partial _t R , {\tilde{\varPhi }} \big ) + \pi \big ( \xi (R) \, \partial _s (R^2 u), {\tilde{\varPhi }} \big ) = 0. \end{aligned}$$
(44)
If we focus on the first term in (44), we can observe that
$$\begin{aligned} 2 \pi \, \xi (R) \, R \, \partial _t R = \pi \frac{\psi (\pi R^2)}{\varphi (R) } \partial _t R^2 = \frac{{\partial _t\Psi }(\pi R^2) }{\varphi (R) }. \end{aligned}$$
Now observe that, by definition, \( \varphi (R) = {\pm \sqrt{\Psi (\pi R^2)}}\). Thus, the term above can be rewritten as
$$\begin{aligned} 2 \pi \, \xi (R) \, R \, \partial _t R = 2 \, \partial _t\varphi (R). \end{aligned}$$
(45)
Since by definition we have \( \varPhi = \varphi (R) \) and \( R = r( \varPhi ) \), we can write
$$\begin{aligned} 2 \pi \big (\xi (R) \, R \, \partial _t R , {\tilde{\varPhi }} \big ) + \pi \big ( \xi (R) \, \partial _s (R^2 u), {\tilde{\varPhi }} \big ) = 2({\tilde{\varPhi }}, \partial _t \varPhi ) + \pi \, ( \partial _s (rv), \, {\tilde{\varPhi }} \, \xi (r)), \end{aligned}$$
(46)
where again we use the convention \(r\equiv r( \varPhi ). \) At this point, the formulation reads
$$\begin{aligned} \left\{ \begin{array}{l} 2({\tilde{\varPhi }}, \partial _t \varPhi ) + \pi \, ( \partial _s (R v), \, {\tilde{\varPhi }} \, \xi (R))= 0, \\ \displaystyle \pi \rho (\partial _t v, {\tilde{v}}) + {\tilde{a}}\Big ({\tilde{v}}, v,\frac{v}{R}\Big )  \pi (\partial _s (R {\tilde{v}} ) , \psi (\pi R^2))+ \rho K_r \Big ( \frac{v}{R },\frac{{\tilde{v}}}{R }\Big ) = {\tilde{g}}({\tilde{v}}, R),\\ R = r( \varPhi ). \end{array} \right. \end{aligned}$$
(47)
One can see in System (47) an apparent lack of symmetry. Indeed, one could expect the second term in the first equation to be equal to the third term in the second equation. This is true however, since we have, using (43),
$$\begin{aligned} (\partial _s (R {\tilde{v}} ) , \psi (\pi R^2)) = ( \partial _s (R {\tilde{v}}), \, \varphi (R) \, \xi (R)) = ( \partial _s (R {\tilde{v}}), \, \varPhi \, \xi (R)). \end{aligned}$$
This observation is fundamental to obtain an energy estimate. To summarize, we have deduced from the dynamics (32) the following formulation:
for all \( ({\tilde{\varPhi }}, {\tilde{v}})\) sufficiently smooth find, for all \(t >0\), \( (\varPhi (t), v(t))\) solution of
$$\begin{aligned} \left\{ \begin{array}{l} 2({\tilde{\varPhi }}, \partial _t \varPhi ) + \pi \, ( \partial _s (R v), \, {\tilde{\varPhi }} \, \xi (R))= 0, \\ \displaystyle \pi \rho (\partial _t v, {\tilde{v}}) + {\tilde{a}}\Big ({\tilde{v}}, v,\frac{v}{R}\Big )  \pi ( \partial _s (R {\tilde{v}}), \, \varPhi \, \xi (R)) + \rho K_r \Big ( \frac{v}{R },\frac{{\tilde{v}}}{R }\Big ) = {\tilde{g}}({\tilde{v}}, R),\\ R = r( \varPhi ), \end{array} \right. \end{aligned}$$
(48)
with the following initial data
$$\begin{aligned} \varPhi (0) = \varphi (R(0)) = \varphi (\sqrt{A}_0 / \sqrt{\pi }) = 0, \quad v = 0. \end{aligned}$$
(49)
This is what we call the energycompliant variational formulation. At the continuous level, the energy is easily obtained by choosing \( {\tilde{\varPhi }} = \varPhi \) and \( {\tilde{v}} = v \). This simple choice of test functions to deduce the energy relation at the continuous level will help in achieving the same energy relation property at a discrete level.
Remark 9
The formulation of System (48) can be obtained for other tube laws \(\psi (A)\). However, some properties should be satisfied by the function \( \psi \). In particular, \( \psi \) must be at least continuous and
$$\begin{aligned} \psi '(A) > 0, \quad \psi (A_0) = 0. \end{aligned}$$
Strong formulation
For the sake of completeness, we show the strong formulation of System (48). Choosing a smooth test function with compact support in [0, L], one can show, using integration by parts, that the following partial differential equations hold:
$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle 2\partial _t\varPhi + \pi \xi (R)\partial _s(Rv) = 0, \\ \displaystyle \pi \rho \partial _tv+\frac{\pi \rho }{2}\Big (2\frac{v}{R} \partial _sv + v\, \partial _s\frac{v}{R}\Big ) + \rho K_r\frac{v}{R^2} + \pi R\partial _s(\xi (R)\varPhi )= 0,\\ R = r( \varPhi ). \end{array} \right. \end{aligned}$$
(50)
Then, choosing a smooth test function in [0, L] vanishing at the boundaries in System (48) and using integration by parts for System (50), one can deduce the following boundary conditions:
$$\begin{aligned} \frac{\rho }{2} \frac{v^2}{R} + R \, \xi (R) \varPhi =  R \big ( P_{\text {tot}}  P_{\text {ext}}\big ) , \quad s \in \{0, L \}. \end{aligned}$$
Analysis of the function \(\varphi (R)\)
In this section we provide further details on the properties of the function \(\varphi (R)\). The definition in System (36) is motivated by the expression of \(\Psi =\Psi (\pi R^2)\) that is rewritten below:
$$\begin{aligned} \displaystyle \Psi (\pi R^2)=\frac{\beta }{A_0}\Big [\frac{2}{3}\pi ^{\frac{3}{2}}R^3  \sqrt{ A_0} \pi R^2 + \frac{1}{3}A_0^{\frac{3}{2}}\Big ]= \frac{\sqrt{\pi }\beta }{3R_0^2}(RR_0)^2 \, (2R+R_0), \end{aligned}$$
(51)
where—in this section—\(R_0 = \sqrt{A_0} / \sqrt{\pi } \) is the reference radius of the crosssection. The behavior of this function is shown in Fig. 3. It is straightforward to see that this function, as well as its square root, is not bijective. However, using System (36), the function \(\varphi (R)\) is then given by
$$\begin{aligned} \displaystyle \varphi (R)= \frac{(RR_0)}{R_0}\sqrt{\frac{\sqrt{\pi }\beta }{3}\Big (2R + R_0\Big )}. \end{aligned}$$
(52)
In Fig. 3 we can observe the comparison between \(\sqrt{\Psi (\pi R^2})\) and \(\varphi (R)\). For every \(R\ge R_0\) the two functions coincide, whereas for \(R<R_0\) they are opposite. However, we can also see that \(\varphi \) is bijective from \( {\mathbb {R}} \) to some interval I satisfying \({\mathbb {R}}^{+} \subset I \subset {\mathbb {R}} \). Moreover, it is easy to prove the following Property.
Theorem 10
Assume \(R_0 > 0\), then
$$\begin{aligned} \begin{array}{lccc} \varphi : &{} [0, +\infty ) &{} \mapsto &{} \displaystyle {[} \varPhi _{min} , +\infty ) \\ &{} R &{} \mapsto &{} \varphi (R) \end{array} \qquad \text{ with } \qquad \varPhi _{min} = \sqrt{\frac{\beta \sqrt{\pi } R_0}{3}} \end{aligned}$$
is monotone increasing (hence bijective) and belongs to \(C^{\infty }([0, +\infty )).\)
Analysis of the function \( \xi (R)\)
We now focus on the property of the function \( \xi (R) \) that is defined by \( \displaystyle \xi (R) = {\psi (\pi R^2)}/{\varphi (R)}. \) In particular we want to check whether the function is smooth and bounded. This is not true because \( \varphi (R) \) vanishes and, as one can see in Fig. 3 and in (52), this happens at \(R=R_0\) where—in this section—\(R_0 = \sqrt{A_0} / \sqrt{\pi } \) is the reference radius of the crosssection. Using Eqs. (34) and (52) one can compute
$$\begin{aligned} \displaystyle \xi (R) = \frac{\beta \sqrt{\pi }}{ \pi R_0^2} \frac{ R  R_0 }{\varphi (R)}= \frac{\sqrt{3\beta }}{\pi ^{\frac{3}{4}}R_0\sqrt{(2R +R_0)}}. \end{aligned}$$
We see in Fig. 3 that \(\xi (R)\) has no singularity, it is smooth, strictly positive and monotonically decaying. This result is summarized in the following Property.
Theorem 11
Assume \(R_0 > 0\), then
$$\begin{aligned} \begin{array}{lccc} \xi : &{} [0, +\infty )\, &{} \longrightarrow &{} \displaystyle \Big (0, \sqrt{\frac{3\beta }{\pi ^{\frac{3}{2}}R_0^3}} \; \Big ] \\ &{} R &{} \mapsto &{} \xi (R) \end{array} \end{aligned}$$
is monotone decreasing, strictly positive, and belongs to \(C^{\infty }([0, +\infty )).\)
Extension of the model in a nonphysiological range
There is an equivalence between System (27) and System (48). More precisely, we can state the following Theorem.
Theorem 12
Let \(P_{\text {tot}}(t) \in C^0([0,T])\) be given. We have the following results:

Let \((A,u) \in C^1([0,T] \times [0,L])^2\) be solution of System (27). If \(A>0\) and if we define \( R = \sqrt{A}/ \sqrt{\pi } \), then \( (\varvec{\varPhi }(R), R \, u) \in C^1([0,T] \times [0,L])^2 \) is solution of System (48).

Reciprocally, if \( (\varPhi , v) \) is solution of System (48) and if
$$\begin{aligned} \varPhi > \varPhi _{min}, \end{aligned}$$
(53)
then \(( \pi \, r( \varPhi )^2 , v / r( \varPhi ) )\) is solution of System (27).
Although the bound defined in (53) is expected physiologically, after space discretization there is no guarantee that such property holds intrinsically at any time and any point. Therefore, we propose to modify System (48) for a nonphysiological range, e.g. close to \(R \simeq 0\), or equivalently, \( \varPhi \simeq \varPhi _{min} \). More precisely, \( r( \varPhi ) \) is not defined for \( \varPhi \) taking smaller values than \( \varPhi _{min}\). To circumvent this problem we introduce, for a given \(\epsilon > 0\)—a relaxation parameter—the function \({ r_\epsilon }\), defined by
$$\begin{aligned} r_\epsilon ( \varPhi ) := \left\{ \begin{array}{ll} r( \varPhi ) &{} \quad \displaystyle \varPhi \ge \varPhi _\epsilon , \\ \displaystyle a e^{ b \varPhi } &{} \quad \displaystyle \varPhi < \varPhi _\epsilon , \end{array} \right. \quad \text{ with } \quad \varPhi _\epsilon = \varPhi _{min} + \epsilon , \end{aligned}$$
(54)
where \((a,b) \in {\mathbb {R}}^2 \) are only defined by the constraint that \(r_\epsilon \in C^1({\mathbb {R}})\). In more detail, one needs to check that
$$\begin{aligned} r( \varPhi _\epsilon ) = a e^{ b \varPhi _\epsilon }, \quad ( r )' ( \varPhi _\epsilon ) =  a \, b \, e^{ b \varPhi _\epsilon }, \end{aligned}$$
hence one can compute that
$$\begin{aligned} b =  \frac{ ( r )' ( \varPhi _\epsilon ) }{r( \varPhi _\epsilon )} \quad \text{ and } \text{ then } \quad a = e^{ b \varPhi _\epsilon } \, r( \varPhi _\epsilon ). \end{aligned}$$
The main advantage of using \(r_\epsilon \) instead of r is that \(r_\epsilon \) is a bijective function from \( {\mathbb {R}} \) to \( {\mathbb {R}}^{+} \setminus \{0\} \). Hence, at the discrete level for any value of the unknown \( \varPhi \) we are able to compute a corresponding aortic radius R. In this process we introduce—mathematically speaking—a modeling error with respect to System (27). Nevertheless, we have the following straightforward result.
Theorem 13
Let \(P_{\text {tot}}(t) \in C^0([0,T])\) be given and \( A_{\epsilon } = \pi [{r_\epsilon ( \varPhi _\epsilon )}]^2\). We have the following results:

Let \((A,u) \in C^1([0,T] \times [0,L])^2\) be solution of System (27). If \(A\ge A_{\epsilon }\) and if we define \( R = \sqrt{A}/ \sqrt{\pi } \), then \( (\varvec{\varPhi }(R), R \, u) \in C^1([0,T] \times [0,L])^2 \) is solution of System (48) with \(r_\epsilon \) instead of r.

Reciprocally, if \( (\varPhi , v) \) is solution of System (48) with \(r_\epsilon \) instead of r and if
$$\begin{aligned} \varPhi \ge \varphi _{\epsilon }, \end{aligned}$$
(55)
then \(( \pi \, r_\epsilon ( \varPhi )^2 , v / \varphi _\epsilon ( \varPhi ) )\) is solution of System (27).
Note that we can choose \( \epsilon \) small enough so that the range of values \(A(x,t) \in (0, A_{\epsilon })\) for which the mathematical equivalence with System (27) is not satisfied can be set as desired. In particular, considering the application to hemodynamics, this interval can be chosen so that a solution of System (27) with values \( A < A_{\epsilon } \) is outside the validity of the the tube law described in (2).
Viscosity of the wall
In “Onedimensional blood flow model” section we introduced the third equation of System (1) that relates the pressure with the strain and strain rate of the wall. In particular, it takes into account the velocity of radial displacements [34] thanks to the term \(\psi _v\) that was assumed to vanish in “Variational formulation in (A, u)” section in order to derive the energycompliant variational formulation. In this section we address the treatment of this term, \(\psi _v\), through the change of variables introduced in “An energycompliant formulation for the blood flow model” section. Starting from System (21), we have
$$\begin{aligned} \frac{A}{\rho }\partial _sP(A) = \frac{A}{\rho }\partial _s(\psi _e(A)+\psi _v(A))= \frac{A}{\rho }\partial _s \Big (\frac{\beta }{A_0}(\sqrt{A}\sqrt{A_0})+\frac{\Gamma }{A_0\sqrt{A}}\partial _t A \Big ). \end{aligned}$$
(56)
Since we have already dealt in the previous section with the first term, related to \(\psi _e\), we focus now on the last one of the equation above, related to \(\psi _v\). Starting from (56) and using the first equation of System (21), we obtain
$$\begin{aligned} \frac{A}{\rho }\partial _s \Big ( \frac{\Gamma }{A_0\sqrt{A}}\partial _t A \Big ) = \frac{A}{\rho }\partial _s\Big (\frac{\Gamma }{A_0\sqrt{A}}\partial _sQ\Big ). \end{aligned}$$
(57)
This motivates the introduction of the nonlinear form \(c(\cdot ;\cdot ,\cdot )\) defined by
$$\begin{aligned} c(R; v, {\tilde{v}})= \frac{\sqrt{\pi }\, \Gamma }{R_0^2} \int _0^L\frac{1}{R} \, \partial _s(R{\tilde{v}}) \, \partial _s(Rv) \, \text{ d }s . \end{aligned}$$
(58)
Taking into account the manipulations performed in “An intermediate formulation in (R, u)” and “Variational formulation in \((\varPhi ,v)\)” sections, one can show that
$$\begin{aligned}  \Big ( \frac{A}{\rho }\partial _s\Big (\frac{\Gamma }{A_0\sqrt{A}}\partial _sQ\Big ) , \rho \frac{{\tilde{v}}}{R} \Big ) = c(R; v, {\tilde{v}}) + \pi \, R \, \psi _v(A) \, {\tilde{v}} \, \Big _{0}^L. \end{aligned}$$
(59)
Then, it can be shown that the second equation of System (48) can be replaced by
$$\begin{aligned} \pi \rho (\partial _t v, {\tilde{v}}) + {\tilde{a}}\Big ({\tilde{v}}, v,\frac{v}{R}\Big )  \pi ( \partial _s (R {\tilde{v}}), \, \varPhi \, \xi (R)) + c(R; v, {\tilde{v}}) + \rho K_r \Big ( \frac{v}{R },\frac{{\tilde{v}}}{R }\Big ) = {\tilde{g}}({\tilde{v}}, R). \end{aligned}$$
Note that the boundary terms in (59) are indeed taken into account, since \({\tilde{g}}\)—defined in (41)—involves the total pressure that is given by (6) and now reads
$$\begin{aligned} P_{\text {tot}} = P_{\text {ext}} + \psi _e\left( A\right) + \psi _v\left( A\right) + \frac{\rho }{2} \, {u}^{2}. \end{aligned}$$
Outflow conditions, inflow conditions and energy relation
In order to complete the weak formulation of the problem given in System (48), the outflow and inflow conditions need to be specified. This is done by expanding the term \({\tilde{g}}\) using the coupling condition described in (7) at the outlet, whereas at the inlet we use
$$\begin{aligned} P_{\text {ar}}(t) = P_{\text {tot}}(0,t)  P_{\text {ext}}, \quad \pi \, R(0,t) \, v(0,t) = Q_{\text {ar}}(t), \end{aligned}$$
where \( P_{\text {ar}}(t) \) and \( Q_{\text {ar}}(t) \) are the arterial pressure and the arterial flow, respectively. We obtain the following system of equations:
$$\begin{aligned} {\left\{ \begin{array}{ll} &{}2( \partial _t \varPhi , {\tilde{\varPhi }}) +\pi (\partial _s(R \, v), {\tilde{\varPhi }}, \xi (R)) = 0, \\ &{}\displaystyle \pi \rho (\partial _t v, {\tilde{v}}) + {\tilde{a}}\Big ({\tilde{v}}, v,\frac{v}{R}\Big )  \pi ( \partial _s (R {\tilde{v}}), \, \varPhi \, \xi (R)) + c(R; v, {\tilde{v}}) + \rho K_r \Big ( \frac{v}{R },\frac{{\tilde{v}}}{R }\Big )\\ &{}\qquad \qquad = \pi {\tilde{v}}(L) R(L)\big (P_c + R_c \, \pi \, R(L) \, v(L)\big ) +\pi \, {\tilde{v}}(0) \, R(0) \, P_{\text {ar}} ,\\ &{}R = r( \varPhi ), \\ &{} \pi \, R(0) \, v(0) = Q_{\text {ar}},\\ &{} \displaystyle C_c \frac{\mathrm {d}}{\mathrm {d}t} P_c + \frac{P_c}{R_{\text {per}}} = {\pi \, R(L)\, v(L)} . \end{array}\right. }\end{aligned}$$
(60)
Note that a similar energy identity to the one given in Theorem 2 can be derived for this system, as we state below.
Theorem 14
Any smooth solution of System (60) satisfies the conservation property
$$\begin{aligned} \, \frac{\mathrm {d}}{\mathrm {d}t}\Big ( {{\mathcal {E}}_{\text {ar}}} + {{\mathcal {E}}_{\text {w}}} \Big ) + {\mathcal {D}}_{\text {ar}} + {\mathcal {D}}_{\text {w}} = P_{{\text {ar}} } \, Q_{\text {ar}} , \end{aligned}$$
(61)
where,
$$\begin{aligned} {\mathcal {E}}_{\text {ar}} = \int _{0}^L \varPhi ^2 \, \mathrm {d}s + \frac{ \pi \, \rho }{2}\int _{0}^L v^2 \, \mathrm {d}s, \quad {\mathcal {E}}_{\text {w}} = \frac{C_c}{2} P_c^2, \end{aligned}$$
and,
$$\begin{aligned} {\mathcal {D}}_{\text {ar}} = \rho \, K_r \int _{0}^L \frac{v^2}{R^2} \, \mathrm {d}s\, + \frac{\sqrt{\pi }\, \Gamma }{R_0^2} \int _0^L\frac{1}{R} \, (\partial _s(Rv))^2 \, \, \mathrm {d}s , \quad {\mathcal {D}}_{\text {w}} = \frac{P_c^2}{R_{\text {per}}} + R_c ( {\pi \, R(L)\, v(L)} )^2. \end{aligned}$$
Note that System (14) can be easily used with or without coupling with the reduced heart model. Hence, we consider two cases:

Case 1: Imposed inlet flux;
In this case the arterial pressure \( P_{{\text {ar}} } \) is considered as a new unknown, namely a Lagrange multiplier for the constraint \( \pi \, R(0) \, v(0) = Q_{\text {ar}}. \)

Case 2: Coupling with the reduced heart model.
System (60) should then be completed with Eqs. (10), (14) and (17), that describe the reducedorder cardiac mechanics, the microscopic actinmyosin binding model and the valve model, respectively. Note that in this model \( Q_{\text {ar}} \) is an unknown that can be straightforwardly substituted in (17) using the relation \( Q_{\text {ar}} = \pi \, R(0) \, v(0).\)