In the construction field, ultra-high performance concrete (UHPC) has drawn remarkable attention owing to its outstanding mechanical properties and durability, with increasingly extensive applications in flat components. Moreover, due to its dense microstructure, UHPC is highly susceptible to explosive spalling at elevated temperatures. In this paper, we comprehensively review the application status of UHPC in this domain, encompassing aspects such as bridge deck overlays, composite slabs, steel-concrete composite systems, joint connections, rehabilitation and strengthening, and thin-walled members. We deeply analyzed the spalling mechanisms of UHPC at high temperatures, mostly including thermal stress and vapor pressure mechanisms, and thoroughly investigated influencing factors such as permeability, heating rate, fiber and aggregate types, specimen size, cooling method, external load, and restraint. Additionally, we summarize effective methods to mitigate fire-induced spalling, such as the application of fire insulation, optimization of curing processes, incorporation of fibers or aggregates, and the utilization of thermal spalling-resistant admixtures. Despite the significant potential of UHPC in flat component applications, numerous challenges persist, including further validation of application feasibility, optimization and improvement of interface performance, in-depth elucidation of spalling mechanisms, research and exploration of new fiber materials, full consideration of the scale effect, and exploration and exploitation of innovative improvement solutions for fire resistance. Researchers should concentrate on addressing these issues to promote the broader and more efficient application of UHPC in the construction field.
Citation: Xiaodong Cheng, Jun Xia, Theofanis Krevaikas, Luigi Di Sarno. A review of the applications of ultra-high performance concrete in flat components and the associated fire-induced spalling risk[J]. AIMS Materials Science, 2025, 12(1): 165-202. doi: 10.3934/matersci.2025010
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In the construction field, ultra-high performance concrete (UHPC) has drawn remarkable attention owing to its outstanding mechanical properties and durability, with increasingly extensive applications in flat components. Moreover, due to its dense microstructure, UHPC is highly susceptible to explosive spalling at elevated temperatures. In this paper, we comprehensively review the application status of UHPC in this domain, encompassing aspects such as bridge deck overlays, composite slabs, steel-concrete composite systems, joint connections, rehabilitation and strengthening, and thin-walled members. We deeply analyzed the spalling mechanisms of UHPC at high temperatures, mostly including thermal stress and vapor pressure mechanisms, and thoroughly investigated influencing factors such as permeability, heating rate, fiber and aggregate types, specimen size, cooling method, external load, and restraint. Additionally, we summarize effective methods to mitigate fire-induced spalling, such as the application of fire insulation, optimization of curing processes, incorporation of fibers or aggregates, and the utilization of thermal spalling-resistant admixtures. Despite the significant potential of UHPC in flat component applications, numerous challenges persist, including further validation of application feasibility, optimization and improvement of interface performance, in-depth elucidation of spalling mechanisms, research and exploration of new fiber materials, full consideration of the scale effect, and exploration and exploitation of innovative improvement solutions for fire resistance. Researchers should concentrate on addressing these issues to promote the broader and more efficient application of UHPC in the construction field.
Scientists have been always fascinated by the functioning of the human brain and always attempted to understand its complexity.
Only in 1899 Santiago Ramon y Cajal, exploiting the experimental techniques developed by Camillo Golgi, was able to discover that the nervous system is made by individual cells, later called neurons [31]. His studies laid the foundations for the so-called ''Neuron Doctrine'' and gave him the Nobel Prize in Physiology and Medicine in 1906 [11]. Although some scientists suggest to rethink the Neuron Doctrine [6], it remains the pillar of modern neuroscience.
It is worth observing that in each individual there is not an only type of neuron, but several ones. Nevertheless, they share many common properties. From the cell body (called soma) starts a number of ramifying branches called dendrites. These structures constitute the input pole of a neuron. From the soma originates also a long fiber called the axon. It is considered as the output line of a neuron since through the axons terminals, called synapses, the exchange of information with other neurons takes place [33,35].
Indeed, the basic elements of the communication among the neurons are pulsed electric signals called action potentials or spikes. In fact, the neuronal cell is surrounded by a membrane, across with there is a difference in electrical charge (called also potential), that depends on the different concentrations of ions, especially Sodium (Na
Moreover, a neuron can exhibit rich dynamical behaviors, such as resting, excitable, periodic spiking, and bursting activities. In particular, the ability of periodic firing has been recorded in isolated neurons since the 1930s [37] and Hodgkin [17,19] was the first to propose a classification of neural excitability, depending on the frequency of the action potentials generated by applying external currents.
In literature several nonlinear dynamical systems have been proposed to suitably model the dynamics of the electrical activity observed in a single neuron.
However, the paper by Hodgkin and Huxley [18] on the physiology of the squid giant axon remains a milestone in the science of nervous system, and the model proposed therein has been extensively studied, due to its richness.
It is known [13,15,32] that, depending on the value of the external current stimuli
In general, it is not an easy task to detect a periodic solution of a nonlinear dynamical system. Several methods are exploited to predict the existence of limit cycles and to study their stability, both in time and frequency domain [2,4,5,23,25,27,28,30]. In particular, for the HH model, this is even more difficult due to the high nonlinear structure of the system.
Our aim is to propose a technique based on the harmonic balance method [2,27] in order to characterize the periodic solutions exhibited by a nonlinear dynamical system. In particular, for the HH model we will show how this technique is more efficient than the previous approaches based either on finite differences, collocation or shooting methods [9,13,14,15,32]. Indeed, by applying the harmonic balance method we obtain an approximated but analytical expression of the periodic solutions, therefore we get a good approximation of the monodromy matrix and the Floquet analysis is straightforward. Moreover, it is worth noting that the harmonic balance method has an exponential convergence, while the collocation methods exploited in the previous papers [9,13,14,15,32], even with a mesh adaptation, have only a polynomial one. Finally, only from 2 to 50 harmonics are needed to correctly approximate the periodic solutions, so the linear system to solve is low-dimensional.
In the end, the application of the harmonic balance method to the Hodgkin-Huxley model displays several advantages: a straightforward implementation without mesh adaptation, exponential convergence rate, an analytical representation of the stable and unstable periodic solutions. Moreover, this method can be coupled with a simple continuation method based on a selective parameterization instead of the exploitation of the Keller arc-length parameterization [7].
The paper is structured as follows: in Section 2 the basics of collocation and harmonic balance methods are briefly recalled and our technique, that we call multi-base harmonic balance method is introduced. In Section 3 firstly we present the structure of the HH model and its main characteristics. We show how it is possible to obtain the bifurcation diagram of the HH model by mainly exploiting the multi-base harmonic balance method. Furthermore, all the bifurcations are analyzed via the harmonic balance method and Floquet analysis. Finally, concluding remarks are offered in Section 4.
There are several methods that are exploited to predict the existence of limit cycles in nonlinear systems and to study their stability. Classical methods, based on integration schemes, such as the simple shooting method, are generally sensitive to the stability properties of solutions, so they cannot be exploited to detect unstable periodic solutions. In this section, first of all, we review two methods that overcome this problem and belong to a class of spectral methods. Indeed, we present collocation and harmonic balance methods, whose main idea is the approximation of the exact solution by the projection on a finite-dimensional subspace. It has been shown that the harmonic balance method is more efficient than the collocation method [20], but is computationally onerous when the periodic solution contains high harmonics. Therefore, in this paper we propose a new technique, a multi-base harmonic balance method, that permits to obtain a good approximation of a periodic function even if the nonlinear dynamical system under study is highly nonlinear.
Let us consider an autonomous dynamical system
˙x=f(x) | (1) |
where
It is interesting to notice that searching a periodic solution of an ODE is equivalent to the resolution of a boundary value problem (BVP). In fact, if
{dxdt=f(x)x(0)=X(0)=X(T). | (2) |
System (2) belongs to the general class of nonlinear boundary value problems and several methods have been proposed in literature to solve it [2]. The more intuitive one is probably the simple shooting method [24]. The idea is to find an initial condition
G(X0,T)=φ(X0,T)−X0, | (3) |
where
G(X0,T)=φ(X0,T)−X0=0, | (4) |
has
On the contrary, the collocation method is independent on the stability of the periodic solutions under consideration. We briefly recall the main properties of this method [34].
First of all, since
of the time scale, it is possible to write (2) on the interval
{dudτ=Tf(u)u(0)=u(1), | (5) |
where
The idea of the collocation methods for BVPs is to approximate the analytical solution by a piecewise polynomial vectorial function
Let us consider the partition
P(t)/[ti,ti+1]=Pi(t),i=0,…,N−1, |
where
tij=ti+ρj(ti+1−ti), i=0, …, N−1, j=1, …, m, |
where
Then, the request that
1T˙P(tij)=f(P(tij)),i=0,…,N−1,j=1,…,m, | (6) |
with the boundary conditions
P(0)=P(1). | (7) |
Thus, the state vector is given by
U=(P(0),P(tij)0≤i≤N−1,1≤j≤m,P(1),T)∈Rq, |
where
Moreover, a further condition is necessary to determine the unknown parameter
˙u1(0)=0, |
where
Several solvers using collocation methods have been proposed in the literature. For example, COLSYS/COLNEW [1,3] and AUTO [8] which use collocation methods with Gaussian points, or bvp4c, bvp5c and bvp6c [22,34] that are routines with Lobatto points.
In particular, the bvp4c method exploits 3 Lobatto points on each subinterval. In this case
The bvp4c methods controls the error
˙P(t)=F(P(t))+r(t), H(P(0),P(1))=δ, |
where
The basic idea of this technique is to minimize the residue over each sub-interval
It is worth noting that any periodic smooth function of period
X(t)=A0+∞∑k=1(Akcos(k2πTt)+Bksin(k2πTt)), | (8) |
where
A0=1T∫T0X(t)dt,Ak=2T∫T0X(t)cos(k2πTt)dt, k=1,...,∞.Bk=2T∫T0X(t)sin(k2πTt)dt, | (9) |
The idea of the harmonic balance method [27] is to search for an approximation of the solution of (1) as a truncated series
XK(t)=A0+K∑k=1(Akcos(k2πTt)+Bksin(k2πTt)), | (10) |
where
Ak=(−1)(n−1)rn2T∫T0X(n)(t)sin(rt)dt |
and
Bk=(−1)(n)rn2T∫T0X(n)(t)cos(rt)dt, |
with
Analogously, the function
f(XK(t))=N0+K∑k=1(Mkcos(k2πTt)+Nksin(k2πTt)), | (11) |
wherein each coefficient
N0=1T∫T0f(XK(t))dt,Mk=2T∫T0f(XK(t))cos(k2πTt)dt,Nk=2T∫T0f(XK(t))sin(k2πTt)dt. | (12) |
We note
ej(t)={1 if j=0,cos(j2πTt) if j=2k, k=1,...K.sin(j2πTt) if j=2k+1, |
the Fourier base. Introducing (10) and (12) in (1), and taking into account the orthogonality of the elements of the Fourier base, we obtain a nonlinear algebraic system in the unknowns
Unfortunately, if the function
Remark 1. It is worth recalling the drawbacks of the methods presented before:
In the following we present a technique mixing Fourier series and trigonometric Lagrange interpolation in the harmonic balance method that permits to avoid such inconveniences.
If
Ln(X(t),t)=n∑j=0X(tj)lj(t), |
where
lj(t)=sin((n+12)(tT2π−tj))sin(12(tT2π−tj)),tj=jT2n+1,j=0,1,...,2n. |
Then, the functions
Let us suppose
(YL(¯XK))j=f(XK(tj)). |
Moreover, it is possible to deduce
YF(¯XK)=PΓ−1YL(¯XK), |
where
P=(I2K+10…0)∈R2K+1,2n+1, |
Γ−1=(1cos(1×t0)sin(1×t0)⋯⋯⋯cos(n×t0)sin(n×t0)⋮⋮⋮⋯⋯⋯⋮⋮⋮⋮⋮⋯⋯⋯⋮⋮1cos(1×t2n)sin(1×t2n)⋯⋯⋯cos(n×t2n)sin(n×t2n)) |
It is worth observing that the choice of
As far as we know, this joint exploitation of Fourier and Lagrange basis in the harmonic balance method is an original approach. It is worth noting that the change of basis between the Fourier and the Lagrange ones permits to avoid to find directly the Fourier coefficients of
Finally, from (10) we obtain
˜D¯XK=YF(¯XK), | (13) |
where
D=(0⋯⋯⋯⋯0⋮D1⋱⋱⋱⋮⋮⋱⋱⋱⋱⋮⋮⋱⋱Dk⋱⋮⋮⋱⋱⋱⋱⋮0⋯⋯⋯⋯DK) |
and
Dk=(0k−k0). |
Remark 2. Generally, the choice of
||XK−XK+1||≤ε, | (14) |
for a given
XK=XK(t)=AK0+K∑k=1(AKkcos(k2πTt)+BKksin(k2πTt))XK+1(t)=AK+10+K+1∑k=1(AK+1kcos(k2πTt)+BK+1ksin(k2πTt)), |
then, in general, we have
In this section we will show how the computation of the Floquet multipliers, and thus the stability analysis of the limit cycles under consideration, is straightforward when the limit cycles have been detected via the harmonic balance method [10].
Let
˙Z=Dxf(x(t))Z, Z(0)=I, | (15) |
Since, in general, it is not easy to solve (15), the monodromy matrix
Ah(t)=Ah,k, t∈[tk, tk+1),(k=0,…,n−1), |
where
Thus, it is easy to see that the solution
˙Zh=Ah(t)Zh, Zh(0)=I |
is given by
Zh(t)=exp((t−tk)Ahk)exp(hAh,k−1)⋯exp(hAh0),t∈[tk, tk+1], |
so, the following approximation of the monodromy matrix
Mh=Zh(T)=∏k∈{0,..N−1}exp(hAh,k). |
For more details, see [10].
We remark that when the periodic solution is found via the harmonic balance method, we have an approximated but analytical solution in the form (10). Therefore, the matrix
In this section we show how our technique based on the HB method permits to efficiently characterize the periodic solutions of the Hodgkin-Huxley (HH) model. First of all, the structure of the HH model and its main characteristics are briefly presented.
The Hodgkin-Huxley model for a neuron consists in a set of four nonlinear ordinary differential equations in the four variables
{CdVdt=−I−[(V−ENa)¯gNam3h+(V−EK)¯gKn4+(V−EL)¯gL],dndt=αn(V)(1−n)−βn(V)n,dhdt=αh(V)(1−h)−βh(V)h,dmdt=αm(V)(1−m)−βm(V)m, | (16) |
where
αn(V)=0.1expc(0.1(10+V)),βn(V)=exp(V/80)/8,αh(V)=0.07exp(V/20),βh(V)=1/(1+exp(0.1(30+V))),αm(V)=expc(0.1(25+V)),βm(V)=4exp(V/18), |
and
expc(x)={xexp(x)−1ifx≠01ifx=0. |
Finally, the typical values for the other parameters are [13,15]:
EK=12mV, ENa=−115mV, EL=−10.599mV |
¯gK=36mS/cm2,¯gNa=120mS/cm2,¯gL=0.3mS/cm2. |
For small values of the current stimulus
At present, few works about the detection of the periodic solutions of the HH model and their related bifurcations exist in literature, for example [9,13,14,15,32], because of the high dimension of the system and its high nonlinearity. Furthermore, the existing works approach the problem by exploiting different methods (finite differences, collocation or shooting methods), that are not so simple to handle with.
We show that through our technique based on the harmonic balance method [2,27] it is possible to efficiently characterize and numerically approximate the periodic solutions exhibited by the HH model (16) and the related bifurcations, depending on the intensity of the external current stimuli
In particular, we point out how the harmonic balance method can be efficiently exploited in the route-to-chaos region found by [13], that is in the region of the Hodgkin-Huxley bifurcation diagram where the dynamics is more awkward and thus more interesting. This method permits to obtain analytically the stable and unstable periodic solutions of the Hodgkin-Huxley model, therefore we get a better characterization of the unstable chaos and of the Hopf bifurcations. In fact, the Floquet multipliers can be easily computed and the flip bifurcation effortlessly detected. In particular, we show that in the regions close to the Hopf bifurcations we are able to obtain the unstable periodic solution with just two harmonics (thus 12 unknowns variables).
In this work, we are interested in detecting periodic solutions of HH system, depending on the values of the external current
Therefore, the numerical computation of a periodic solution is based on the resolution of the nonlinear algebraic system
F(X,T,I)=0, | (17) |
issue of one of the methods presented above (mainly HB method), where
The continuation method implementation exhibits two main difficulties: on the one hand, the construction of a right initial solution, and on the other hand the progression of the algorithm for critical values of parameter
Let us remark that, in our case, the branches are locally linear, so, for suitable choices of
Our codes are available at: http://lmah.univ-lehavre.fr/codes/codes.html
Both collocation and harmonic balance methods are appropriate for detecting all the periodic solutions, both the stable and unstable ones. The multi-base harmonic balance method uses Fourier series, therefore we have an analytical expression of the solution and the convergence rate is of exponential order. In our case, in general 50 harmonics are enough to get the desired accuracy (see Remark 2), so we have at most only 405 unknowns variables.
In Fig. 1 the bifurcation diagram for the HH model has been obtained by jointly and optimally exploiting the three methods presented above, that is shooting, collocation and multi-base HB methods. The stability analysis of the detected limit cycles has been carried out by the calculation of the Floquet multipliers, by applying the numerical algorithm proposed in [10] to the approximated solution.
It is possible to see that the dynamical behavior of HH system can be decomposed in two main regions, depending on the value of the external current
For
Therefore, we can conclude that multi-base HB method works very well in the region between
In the following, we analyze more accurately the various limit cycles bifurcations that take place.
Hopf bifurcations. In this paragraph, we are interested in Hopf bifurcations, that take place at
It is worth noting that in both cases over a large interval of
Saddle node of cycles bifurcations. In our case, there are two types of saddle node of cycles bifurcation: for
The Floquet multipliers for these three cases are represented in Fig. 9. It is possible to see that a multiplier leaves or enters in the unit circle through
Period doubling bifurcation. Finally, in this section, we consider the period-doubling bifurcation. By exploiting the Floquet analysis, we can easily detect this bifurcation since in this case a Floquet multiplier crosses the unit circle through
I | ||||
7.92197799 | 1.000 | 0.000 | -2940.687 | -1.041 |
7.92197793 | 1.000 | -0.000 | -2964.042 | -1.033 |
7.92197787 | 1.000 | 0.000 | -2987.386 | -1.025 |
7.92197781 | 1.000 | 0.000 | -3010.719 | -1.017 |
7.92197775 | 1.000 | -0.000 | -3034.042 | -1.009 |
7.92197768 | 1.000 | 0.000 | -3057.354 | -1.001 |
7.92197762 | 1.000 | -0.000 | -3080.655 | -0.993 |
7.92197756 | 1.000 | 0.000 | -3103.946 | -0.986 |
7.92197750 | 1.000 | 0.000 | -3127.225 | -0.978 |
7.92197743 | 1.000 | 0.000 | -3150.494 | -0.9713 |
In 1952 Hodgkin and Huxley developed the pioneer and still up-to-date mathematical model for describing the activity of the squid giant axon. Depending on the value of the external current stimuli, this fourth-order nonlinear dynamical system exhibits many complex behaviors, such as multiple periodic solutions (both stable and unstable) and chaos.
Previous works have treated this problem by using several numerical methods, such as shooting and finite difference methods, that are not so simple to handle with. In this paper, we propose a multi-base HB method, a technique based on the harmonic balance method, permitting to detect the stable and unstable periodic solutions and the associated bifurcations of a nonlinear dynamical system. In particular, we have shown how our multi-base harmonic balance method is extremely handy, permits to obtain the bifurcation diagram of the HH model, and works very well in the most complex part of such diagram. Furthermore, harmonic balance and Floquet analysis have permitted to suitably detect the period-doubling bifurcation that entails a route-to-chaos in the HH model.
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