Normalization Method as a Potent Tool for Grasping Linear and Nonlinear Systems in Physics and Soil Mechanics

Abstract

To address physical problems that require solving differential equations, both linear and nonlinear analytical methods are preferred when possible, but numerical methods are utilized when necessary. In this study, the normalization technique is established, which is a simple mathematical approach that requires only basic manipulation of the governing equations to obtain valuable information about the solution. The methodology of this technique involves adopting appropriate references to obtain the dimensionless form of the governing equation, after which the terms of the equation are balanced, obtaining the dimensionless monomials governing the solution. Thorough knowledge of the physical processes involved is necessary to find the best references. The main advantages of this technique are the simplicity of the methodology, the acquisition of valuable information about the solution without the need for complex mathematical calculations, and its applicability to nonlinear problems. However, it is important to consider the difficulty in selecting appropriate references in more complex scenarios. This study applies this normalization methodology to different scenarios, showing how choosing appropriate references lead to the independent dimensionless monomials. Once obtained, it was possible to identify different situations concerning the value of monomials. It will be when they are close to unity, and therefore normalized, when they fundamentally affect the solution of the problem. Finally, we present two cases, one linear and one complex, about the application of normalization to the challenging problem of soil consolidation in ground engineering, illustrating how the technique was used to obtain the solution and its many advantages.

1. Introduction

During their engineering or science studies, graduates learn to derive independent dimensionless monomials governing a given problem formulated using a mathematical model [1,2,3,4]. To do so, they follow a procedure called nondimensionalization (a type of normalization of governing equations). By defining dimensionless independent and dependent variables, the governing equations are reduced to their dimensionless form [5,6,7,8], which, in turn, gives the dimensionless independent monomials, which are the ratios between the coefficients of the equations governing the problem [9,10]. This is expressed by the π theorem [9], which has been well known for more than a century in the scientific literature; that is, once the equations of the problem have been formulated and the appropriate references have been chosen, that will make the variables and subsequently the equations themselves dimensionless, the monomials, also without dimension, are obtained by balancing the terms of each of the equations. When the m ${\pi }_u$-groups contain different unknowns and the n ${\pi }_w$-groups are without unknowns, the solution for each m-group is an arbitrary function (Ψ) of the n-groups:

$${\pi }_{u,i\left(1\le i\le m\right)}=\Psi \left({\pi }_{w,1},{\pi }_{w,2},\dots {\pi }_{w,n}\right)$$

Finally, if the monomials are of the order of magnitude 1, which is why we use the concept normalization, it is evident that the arbitrary function also has this property. From these relationships, the order of magnitude of each unknown can be obtained.

However, students and even teachers reduce the application of this technique to certain types of linear problems, in which the number of variables and parameters involved is small. In other cases, due to an incorrect choice of references, the non-dimensionalization process [11] does not lead to the most accurate solution. For example, when one of the variables oscillates in an interval whose extremes are not zero, the reference to establish this variable as dimensionless cannot be any of the extremes, but the difference between them. This eliminates one of the independent dimensionless monomials. To avoid this type of error, it is necessary to thoroughly understand the phenomena involved.

The first step in performing the normalization process to an ordinary differential equation (ODE) or to a coupled system of such equations (CODE) is to define the independent and dependent variables in their dimensionless form using certain references [12,13] that forces the values of the new variables to be contained in the interval [0–1]. When this requirement cannot be satisfied, for example, in problems with an asymptotic solution, values closer to the asymptotic limit can be set without substantially modifying the results [14]. Thus, normalization is a special case of non-dimensionalization [15]. However, while the references are usually arbitrary in the nondimensionalization process, which causes the solution to differ from one process to another, in normalization, these references cannot be arbitrary. This is a type of discrimination [16,17,18,19] in which the minimum or maximum values of all dimensionless variables correspond to the same physical image of the problem. In this way, the range of the dimensionless variables is nothing more than the interval between the minimum and maximum values of these variables, whereas the dimensionless variables are established as the ratio between the deviation of the variable (with respect to the initial value) and the previous interval. The search for these references is perhaps the step that requires the most thorough consideration in the normalization protocol, as there is often no direct information in the formulation of the problem as to which is the best choice. In this case, the references are said to be “hidden” or not explicit.

It is tempting to ask why normalization, although it seeks general solutions, is not a widely applied technique in graduate studies or even in the scientific literature. Perhaps the answer is that mathematicians are interested in the entire solution and not just a part of it, whatever it is worth. The great advantage of normalization is that the effort needed to obtain valuable information in each problem requires only minor mathematical manipulations and deep physical knowledge of the problem.

When the problem is linear or pseudo-linear, the factors of each term of the dimensionless governing equations (made up of dimensionless variables and their derivatives or changes) can be assumed to be of the order of magnitude 1 and are therefore simplified in the equations. This changes the equations to a sum of the coefficients formed using the parameters and/or physical characteristics of the problem. This balance forces the coefficients to be of the same order of magnitude. The set of independent and complete relationships of these coefficients, now called dimensionless numbers, are monomials that govern the solution of the problem as a functional relationship between them [9,10]. By appropriately regrouping these numbers, the order of magnitude of the unknowns can be deduced. Dimensionless monomials obtained via normalization have two intrinsic properties: (i) their order of magnitude is 1, and (ii) they have a physical meaning in terms of balancing counteracting quantities in the problem.

To facilitate the understanding of this advantageous technique by students and graduates, normalization was applied to two illustrative examples related to dynamics and non-linear soil consolidation models. Their solutions were verified by solving them numerically using the network simulation method [20,21].

The work presented here has been structured, first explaining the accuracy of the method, as shown in Section 2 of the manuscript, and then using it in three engineering problems with a study of the true scope of this methodology and the great advantages it offers due to the simplicity of the calculations involved. This has been carried out in Section 3, in which the case of a forced oscillator, and one more complicated, on soil consolidation are studied thoroughly. Both of them contain nonlinearities, which, being real problems, are not too pronounced, giving the normalization methodology precise solutions. Finally, the discussion and conclusions are presented and Section 4 and Section 5 summarize the results and advantages of the methodology and future applications.

2. Normalization Technique

In this section, we show the advantages of the method using three examples in which the normalization process has been carried out, paying attention to the specific weight of the monomials in the solution of the system of equations, which is observed when their value is around the order of magnitude 1. To understand this, we must bear in mind that each dimensionless monomial is the ratio between two terms of an equation and that if its value is around 1, it is because the two terms it balances are similar; otherwise, one of the two would be negligible and could be simplified from the equation. Thus, in the examples shown in this section, we first present an example in which all the monomials are of the order of magnitude 1; therefore, these monomials have the same weight in the solution of the system. The second case presents values for the monomials, both around unity and outside it, and it is interesting to study when they are close to 1. Once one of these moves away from this value, the other remains practically constant. Finally, an example is shown in which one monomial is practically negligible compared to the other, since it always presents a value close to unity, with this value being the one that controls the problem.

Example 1. 

In this first example, the equations governing the study of the Lotka–Volterra chemical oscillator are presented [22]. In this chemical oscillator, an initial concentration of species A, which is continuously replenished by an external source to keep it constant, reacts to become species x, which in turn is consumed by species y and finally transforms into species B. All this is represented by a system of equations as follows:

$$\frac{dA}{dt}=-k_{1}Ax$$

$$\frac{dx}{dt}=k_{1}Ax-k_{2}xy$$

$$\frac{dy}{dt}=k_{2}xy-k_{3}y$$

$$\frac{dB}{dt}=k_{3}y$$

In these equations, A, $k_1$, $k_2$, and $k_3$ are parameters that affect the rate at which the species are formed.

In that study [22], a series of reasoning is carried out, and the research is focused on an equilibrium process where the variation in the concentrations of species x and y is zero.

In this case, the stable values of x and y are:

$$x_{stable}=\frac{k_3 }{k_2 }{y}_{stable}=\frac{A{k}_1}{k_2}$$

$x_o$ and $y_o$ are defined from the initial value of the concentration of the species x, $\left(x_{stable}+x_0\right)$, which tends to decrease as the concentration of species y increases to a maximum, $\left(y_{stable}+y_0\right)$, and where $t_o$ is the time elapsed between the two extreme values of each species.

In the cases where the study is focused, the normalization process carried out with the system of equations presented results in four dimensionless monomials, with the expressions shown below. We focus on the values of these dimensionless monomials as they are normalized.

$${\pi }_1=\frac{-k_{1}A }{k_3 } {\pi }_2=\frac{-k_2 x_o}{k_3} {\pi }_3=\frac{-k_2{y}_o}{k_{1}A} {\pi }_4=\frac{1}{k_{1}A t_o}$$

Among these four monomials, the first two are formed only by the parameters of the system. The last two contain unknowns $y_o$ and $t_o$. Therefore, we focus on the values of the parameters that make ${\pi }_1$ and ${\pi }_2$, which have values of the order of magnitude 1, to observe what values will have the monomials ${\pi }_3$ and ${\pi }_4$, which are the ones that have the variables. By seeing the relationship between them and checking if they were also of the order of magnitude 1, they were normalized. The results are shown in

Table 1. Relationship between π₁, π₂, π₃, and π₄ for example 1.

${\pi }_1$	−1	−1	−1
${\pi }_2$	−1	−1	−1
${\pi }_3$	−0.998	−1	−0.998
${\pi }_4$	0.836	0.871	0.836

, verifying that they complied with this normalization.

The fact that, in this case, all monomials are of the order of magnitude 1 shows that they all have the same weight in the solution of the system of equations because they are balances of terms in which their ratio is close to 1.

Example 2. 

As a second example, we show the case studied in [23], of a chemical reaction of three species, A, B, and C, reacting to obtain products and occurring in a continuous-flow reactor. This reaction rate gives us a value for how fast the concentration of reactants decreases or the concentration of products increases. The equations governing these problems are coupled and nonlinear using the following expressions:

$$A+B\stackrel{a }{\to }C$$

$$\frac{dA}{dt}=-a{A}^{\alpha }{B}^{\beta }-b\left(A-A_o\right)$$

$$\frac{dB}{dt}=-a{A}^{\alpha }{B}^{\beta }-b\left(B-B_o\right)$$

$$\frac{dC}{dt}=a{A}^{\alpha }{B}^{\beta }-bC$$

With the following values of the coefficients in these equations:

$\alpha$ and $\beta$ are the partial orders of reaction for each reactant, a is the formation reaction constant, b is the flow rate, and A_o and B_o are the reactant concentrations at the input port.

Once the normalization process is applied, the dimensionless monomials achieved can be expressed as follows:

$${\pi }_{1,A}=t_{o}a{A}_o^{\alpha -1}{B}_o^{\beta } {\pi }_{2,A}=\frac{b}{a{A}_o^{\alpha -1}{B}_o^{\beta }}$$

$${\pi }_{1,B}=t_{o}a{A}_o^{\alpha }{B}_o^{\beta -1} {\pi }_{2,B}=\frac{b}{a{A}_o^{\alpha }{B}_o^{\beta -1}}$$

$${\pi }_{1,C}=\frac{t_{o}a{A}_o^{\alpha }{B}_o^{\beta }}{C_F} {\pi }_{2,C}=\frac{b{C}_F}{a{A}_o^{\alpha }{B}_o^{\beta }}$$

With $C_F$ being the product concentration at the output port and $t_o$ being the time at which the reaction rate was zero.

For demonstration purposes, we will focus on the monomials that arise from the equations of species A, which are ${\pi }_{1,A}$ and ${\pi }_{2,A}$. From their study, as in [23], we can obtain the relation between as follows:

$${\pi }_1=2.3795{\pi }_2{}^{-0.7636}$$

From this relationship, we can tabulate the values of one monomial versus another. The monomial containing the unknown $t_o$ is ${\pi }_{1,A}$ and ${\pi }_{2,A}$ is a monomial containing only the parameters of the system, so we need to look at the values we obtain for ${\pi }_{1,A}$ when the monomial ${\pi }_{2,A}$ is of the order of magnitude 1.

It can be seen in

Table 2. Relationship between π_1,A and π_2,A for example 2.

π2,A	0.1	0.1	0.2	0.3	0.4	0.6	0.8	1.0	1.2	1.4	2.0	5.0	10.0	20.0	30.0
π1,A	15.0	13.8	8.1	6.0	4.8	3.5	2.8	2.4	2.1	1.8	1.4	0.7	0.4	0.2	0.2

that the values of the monomial ${\pi }_{1,A}$ are kept in the order of magnitude 1 when the same happens with the monomial ${\pi }_{2,A}$, when the values of this are in the range between 0.6 and 2.0. When the value of any of the monomials changes the order of magnitude, the value of the other monomial tends to be constant, becoming independent of each other. When both monomials are of the order of magnitude 1, they equally affect the solution of the problem. This solution is the value of t₀. In this case, $t_o$ will show dependence on the value of the two monomials; however, when this is not the case, $t_o$ will only show dependence on one of them. This is an important result that tells us which values the unknown depends on in the case of normalization.

Example 3. 

In this complex engineering problem, a reinforced concrete structure is immersed in salt water, and its pores are saturated with water [24]. Thus, the free chlorides from the salt water penetrate the reinforced concrete until they reach a stationary value throughout the structure. The mathematical model for the 1D problem is described by the following expressions, where the Langmuir isotherm was selected, although another expression could have been used:

$$\frac{\partial C_b}{\partial t}+C_f\frac{\partial {\varphi }_l}{\partial t}+{\varphi }_l\frac{\partial C_f}{\partial t}=D_c\frac{\partial {\varphi }_l}{\partial x}\frac{\partial C_f}{\partial x}+D_c{\varphi }_l\frac{{\partial }^2{C}_f}{\partial x^2}$$

$$C_b=\frac{C_{b,o}K{C}_f{\varphi }_l}{1+K{C}_f{\varphi }_l} \mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r} \mathrm{i}\mathrm{s}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m} \mathrm{o}\mathrm{f} \mathrm{L}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{m}\mathrm{u}\mathrm{i}\mathrm{r}$$

$$\varphi ={\varphi }_l={\varphi }_o-\frac{C_b}{{\rho }_F}$$

where D_c is the effective diffusivity (m²/s); C_f, is the free chloride concentration at the solution (kg/m³ solution); C_b, is the bound chloride concentration at the concrete (kg/m³ concrete); C_b,o, is the maximum bounded chloride ion concentration at the concrete (kg/m³ concrete); ϕ, the porosity (m³ of pores/m³ of concrete); ϕ_l, the pore solution content (m³ of solution/m³ of concrete); K, the equilibrium constant (m³ of concrete per kg of chloride); x, the location coordinate measured from the boundary (m); and finally, ρ_F, the density of Friedel’s salt (1892 kg/m³ of concrete).

By applying the normalization technique, knowing the concrete thickness (m), L, defining τ*, and defining the characteristic time (s) as the time at which the system reaches the steady state, as well as taking all of the above as reference values for the concentration of free chloride, the concentration of free chloride of salt water, and for the initial value of porosity of the pore solution content, the following monomials are obtained:

$${\pi }_1=\frac{D_c{\tau }^{*}}{L^2} {\pi }_2=\frac{C_{b,o}K}{{\left(1+K{\varphi }_{ref}{C}_f{}_{ref}\right)}^2}$$

Adjusting the relationship between both monomials by applying the π theorem

$${\pi }_1=4.495·{10}^{-5}{{\pi }_2}^2+1.590·{10}^{-3}{\pi }_2+1$$

If the relationship between the two monomials, as shown in

Table 3. Relationship between π₁ and π₂ for example 3.

${\pi }_2$	0.01	0.05	0.1	0.5	1	2	10	20	40	50
${\pi }_1$	1.000	1.000	1.000	1.001	1.002	1.003	1.020	1.050	1.136	1.192

, is studied, the ${\pi }_1$ monomial always takes a value close to unity, being therefore practically independent of the ${\pi }_2$ monomial, and the relationship between its variables controls the chloride diffusion problem. On the other hand, monomial ${\pi }_2$, which does not take values as high as those shown in Table 3, has the function of slightly delaying the entry of free chlorides into the concrete structure; that is, increasing the time in which the system reaches a steady state because very high values are required for ${\pi }_1$ to deviate significantly from unity.

In order to deepen in the concepts exposed in this section, we are going to show, in detail, the development of two cases in which the last two situations are observed for being the most interesting, because the first example is simply a particular situation of the second one. In the first, we will observe how, in general, one of the monomials presents a value close to unity when the other can take any value. In the second case, as a practical application to nonlinear soil consolidation, it will be observed how, for the cases around 90% degrees of settlement, near the end of the process, both monomials are important in the problem when they take values close to unity. When one of the monomials moves away from this value, the other tends to take a constant value; therefore, the influence on the problem of one that moves away from the unit value.

3. Illustrative Applications of the Normalization Technique

Two problems illustrate the application of the normalization procedure to linear and nonlinear ODE. The expressions of the unknowns deduced as functions of the rest of the problem parameters were verified via numerical simulations [21,25].

3.1. An Undamped Forced Oscillator

In this first application, we studied a linear system consisting of an undamped forced oscillator. The system consists of a mass attached to a spring under the influence of an external force. The physical setup of the problem is shown in Figure 1, and its model and solution are well known.

An undamped forced oscillator is a mechanical system that continues to oscillate at a specific frequency when subjected to periodic external forces [26]. In any forced oscillatory motion, we can distinguish two stages or regimes: an initial transient non-harmonic one, where the mechanical energy is not constant, and a stationary regime where the mechanical energy is constant. In this study, we focused on the stationary regime [27].

Energy transferred to the system accumulates and is released in the form of oscillations. The differential equation of the undamped forced oscillator with a mass attached to a spring can be expressed as follows:

$$m\frac{d^{2}x}{d{t}^2}+kx-Bcos\left(\omega t\right)=0$$

(1)

where $m$ represents the mass of the object, $k$ is the spring constant, $x\left(t\right)$ is the position of the object as a function of time, and $\omega$ is the angular frequency of the external force.

The well-known general solution of this differential equation can be written as the sum of the complementary solution and a particular solution [28]:

$$x\left(t\right)=Acos\left({\omega }_{0}t-\phi \right)+\frac{B}{m\left({\omega }_0^2-{\omega }^2\right)}cos\left(\omega t-\delta \right)$$

(2)

where $A$ and $\phi$ are the integration constants determined via the initial conditions of the system, ${\omega }_0$ is the natural frequency of the system $\left(\sqrt{k/m}\right),$ and δ is the integration constant determined via the initial conditions of the system and the complementary solution.

In this section, we demonstrate how the same solutions can be obtained via the technique of normalization using a simple mathematical manipulation of the equation that governs the physical situation of the undamped forced oscillator. This requires prior knowledge of the important references for each variable involved to transform the equation into its dimensionless version while also confining these variables to the range [0,1], thus normalizing their values. Once the dimensionless monomials that govern the complete solution of the problem are obtained, the expression for the solution is given by applying the π theorem. Therefore, the most important aspect is the appropriate choice of the references, where $x_0$ is the position of the maximum amplitude of the oscillator and $t_0$ is the time required by the mass to reach the position x₀ from the start point, which is a quarter of the period of the movement. With these references, the dimensionless variables are $x^{\prime }=x/x_0$ and $t^{\prime }=t/t_0$.

The next step is to introduce in Equation (1) the dimensionless variables, and using the series development of the cosine function as follows,

$$Cos\left(\omega t\right)=1-\frac{{\left(\omega t\right)}^2}{2!}+\frac{{\left(\omega t\right)}^4}{4!}-\dots$$

(3)

Keeping the first two terms of the series, since the use of more terms does not lead to different results, the governing equation can be written in the following dimensionless normalized form,

$$\frac{m{x}_0}{t_{0_{}}^2}\frac{d^2{x}^{\prime }}{d{t^{\prime }}^2}+k{x}_0{x}^{\prime }-B+\frac{B{\omega }^2{t}_0^2{t^{\prime }}^2}{2}=0$$

(4)

The coefficients and independent terms of this equation are,

$$C_1=\frac{m{x}_0}{t_{0_{}}^2}, C_2=k{x}_0, C_3=B, C_4=\frac{B{\omega }^2{t}_0^2{t^{\prime }}^2}{2}$$

Considering that B is included in the fourth coefficient, from their quotient, the following dimensionless monomials are obtained,

$${\pi }_I=\frac{t_0^{2}k}{m}$$

$${\pi }_{II}=Bm{x}_0{\omega }^2$$

(5)

According to the π theorem [9], the monomial ${\pi }_I$ (which contains the unknown t₀) is a function of the rest of the dimensionless monomials of the problem (which do not contain unknowns); that is, ${\pi }_{II}$:

$${\pi }_I=\Psi ({\pi }_{II})$$

(6)

where $\Psi$ is an unknown function a priori. Thus, the order of magnitude of the unknown $t_0$ is

$$t_0=\sqrt{\frac{m}{k}}\Psi \left(Bm{x}_0{\omega }^2\right)$$

(7)

To test this solution obtained via the normalization process, some validations were performed in two different scenarios around the resonance frequency by varying the value of the oscillator parameters. The first is when the frequency of the external force, $\omega$, is lower than the natural frequency of the oscillator, ${\omega }_0=\sqrt{k/m}$, which is tested in validations 1 and 2. In the second scenario concerning validations 3 to 6, the parameters were varied using values for the frequencies above the resonance frequency. It should be noted that in the case of resonance, which occurs when the frequency of the external force, $\omega$, is equal to the natural frequency of the system, the system oscillates but without the characteristic amplitude because it tends to infinity in time.

Thus,

Table 4. Validation of the normalization process for an undamped forced oscillator.

	Initial Parameters					Simulation Results		Normalization
Validation	$\mathit{m}$	$\mathit{k}$	$\mathit{B}$	$\mathit{\omega }$	${\mathit{x}}_0$	${\mathit{T}}_{\mathit{s}\mathit{i}\mathit{m}}$	${\mathit{t}}_{0,\mathit{s}\mathit{i}\mathit{m}}$	$\sqrt{\mathit{m}/\mathit{k}}$	$\mathit{B}\mathit{m}{\mathit{x}}_0{\mathit{\omega }}^2$	$\mathit{\Psi }$
1	1	1	0.1	0.6	0.29	7.37	1.8425	1	0.01044	1.8425
2	1	1	0.1	0.9	1.04	6.58	1.6450	1	0.08424	1.6450
3	2	2	1	1.1	4.74	5.77	1.4425	1	11.4708	1.4425
4	2	2	3	1.5	2.45	3.34	0.8350	1	33.075	0.8350
5	2	1	2	0.9	6.42	7.14	1.7850	1.414	20.8008	1.2622
6	2	1	3	0.9	5.99	5.46	1.3650	1.414	29.1114	0.9652

presents the values of the period ($T_{sim}$) obtained by simulating the analytical solution, Equation (2). In particular, the period was calculated as the average of the distances between two consecutive maxima on the time axis, as shown in the representations of the solutions from the simulation in Figure 2, Figure 3 and Figure 4.

Note that the value of $t_{0,sim}$ shown in the table is a quarter of $T_{0,sim}$ extracted from the simulation of the analytical solution.

The values of the $\Psi$ function are on the order of magnitude 1. First, we can observe in validations 1 and 2, where the ratios between the values of $m$, $k,$ and $B$ are maintained; thus, the value of the function $\Psi$ is of the order of magnitude 1, as expected, with the solution that reached with the normalization being very close to the simulation value. For illustrative purposes, validations 2, 4, and 6 are simulated and plotted. The simulation of validation 2 is shown in Figure 2.

As can also be observed, in validation 4, which is simulated in Figure 3, and validation 6, simulated in Figure 4, the values extracted for the function Ψ remains in the values of the order of magnitude 1, even when in these validations, the relationships between the problem parameters have been varied, with the intention of addressing all the possible scenarios, demonstrating the advantages of the use of the normalization process for this general case of the undamped forced oscillator.

To further investigate these results, and in the line of research followed in this study,

Table 5. Relationship between π_I and π_II for all validations.

Validation	1	2	3	5	6	4
${\pi }_I$	3.395	2.706	2.081	1.593	0.932	0.697
${\pi }_{II}$	0.010	0.084	11.471	20.801	29.111	33.075

shows the values of the monomials ${\pi }_I$ and ${\pi }_{II}$ for the cases studied, listed in decreasing order for the values of ${\pi }_I$.

It is noteworthy that the monomial ${\pi }_I$, which contains the unknown $t_0$, remains perfectly within the order of magnitude 1 for a wide range of values that can take the monomial ${\pi }_{II}$. As already mentioned, it is well known that dimensionless monomials are formed using the balances between the terms of the equations; thus, with the monomial ${\pi }_I$ always being quite close to unity, it is this one that governs the solution searched. This solution is $t_0,$ corresponding to the fourth part of the period of the motion, which is the main period that can be seen in the figures in this section, while the monomial ${\pi }_{II}$ is related to the amplitude modulation becuase the term $x_0$ is contained in it with a slight damping of the period with respect to the natural period of the oscillator.

3.2. Non-Linear Soil Consolidation

The following example in which we apply the normalization technique is a clear example of a nonlinear system. It is a soil consolidation model in the field of geotechnical engineering, for which there are, in principle, quite a few bibliographical references that have been solved via numerical methods before [29], although it has recently been studied using the dimensioning technique [30]. We will see that the application of normalization, a special case of dimensioning, extracts the information to reach the solutions from the dimensionless monomials with which the solution is constructed more precisely and intuitively.

For the study of the nonlinear consolidation of soils, in recent years, there have been studies on different types of dependence between the void ratio (related to the porosity) and effective stress on the one hand, and hydraulic conductivity and effective stress (or void ratio), on the other. Thus, we highlight the models of Davis and Raymond [31], Juárez-Badillo [32], and Cornetti and Battaglio [33], the latter of which was reformulated by Arnod et al. [34]. Among the assumptions made using these models are the incompressibility of the fluid and soil skeleton, a constant volume (1 + e) in the soil compression term in the equilibrium equation and the lack of consideration of creep effects. The most widely used model in the literature, that of Cornetti and Battaglio [33], assumes logarithmic dependencies between both the void ratio $\left(e\right)$ and effective stress $\left({\sigma }^{\prime }\right)$, and between the void ratio $\left(e\right)$ and hydraulic conductivity $\left(k\right)$. These dependencies are commonly known as soil-constitutive relationships.

Recently, the dimensionless monomials that characterize the model have been derived from their governing equations, as well as their extensions to more general and precise formulations, where the assumption of $1+e$ and $dz$ both as non-constants [30] allows the proposal of a mathematical model free of restrictions (which, far from simplifying the problem from the point of view of nondimensionalization, complicates it and makes it somewhat incongruous). The resulting model, called the Cornetti and Battaglio model, extended for the non-constant $1+e$ and $dz$, as shown in Figure 5, is governed by the following differential Equation (8) in terms of the dependent variable $\zeta =e-e_f$, which is an expression of the settlement in terms of the instant void ratio and the final void ratio (once the consolidation process has concluded):

$$\frac{\partial \zeta }{\partial t}=\frac{{\left(1+e\right)}^{2}Ln\left(10\right){\sigma }_o^{\prime }{k}_o}{I_c{\gamma }_w}{\left(\frac{{\sigma }^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }\left[-\lambda \frac{Ln\left(10\right)}{I_c}{\left(\frac{\partial \zeta }{\partial z}\right)}^2+\left(\frac{{\partial }^2\zeta }{\partial z^2}\right)\right]{\left(\frac{1+e_0}{1+e}\right)}^2$$

(8)

Using the reference quantities $H_0$ (initial soil thickness), $\mathsf{\Delta }\zeta ={\zeta }_f-{\zeta }_0$ (variation interval of the variable $\zeta$) and ${\tau }_{0,s}$ (characteristic time at which a certain fraction or percentage of the final settlement is reached), the dimensionless variables can be defined as $z^{\prime }=z/H_0$, ${\zeta }^{\prime }=\left(\zeta -{\zeta }_0\right)/\left({\zeta }_f-{\zeta }_0\right)$ and $t^{\prime }=t/{\tau }_{0,s}$. Note that variables $z^{\prime }$ and ${\zeta }^{\prime }$ are normalized in the interval [0–1] because variable $z$ varies between 0 and $H_0$, while variable $\zeta$ does so between ${\zeta }_0$ and ${\zeta }_f$. The variable $t^{\prime }$ is also normalized in the interval [0–1], but not until the end of the consolidation process, which is asymptotic in nature, but rather for the percentage of settlement that is taken as reference.

Substituting these dimensionless variables in Equation (8) yields

$$\frac{-\partial \zeta ’{\zeta }_o}{\partial t’{\tau }_{o,s}}=\frac{{\left(1+e_o\right)}^{2}Ln\left(10\right){\sigma }_o^{\prime }{k}_o}{I_c{\gamma }_w}{\left(\frac{{\sigma }^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }\left[-\lambda \frac{Ln\left(10\right)}{I_c}\frac{\partial \zeta {’}^2{\zeta }_o{}^2}{\partial z{’}^2{H}_o{}^2}-\frac{{\partial }^2\zeta ’{\zeta }_o}{\partial z{’}^2{H}_o{}^2}\right]$$

(9)

Assuming that dimensionless variables and their changes are of the order of magnitude 1, the coefficients $C_1$, $C_2,$ and $C_3$ of Equation (9)

$$C_1=\frac{1}{{\tau }_{o,s}},C_2=\frac{{\left(1+e_o\right)}^{2}Ln\left(10\right){\sigma }_o^{\prime }{k}_o}{I_c{\gamma }_w}{\left(\frac{{\sigma }_m^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }\lambda \frac{Ln\left(10\right){\zeta }_o}{I_c{H}_o{}^2},C_3=\frac{{\left(1+e_o\right)}^{2}Ln\left(10\right){\sigma }_o^{\prime }{k}_o}{I_c{\gamma }_w}{\left(\frac{{\sigma }_m^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }\frac{1}{H_o{}^2}$$

are of the same order of magnitude. For its part, the dependent variable’s effective stress, ${\sigma }^{\prime }$, will intervene in the coefficients $C_2$ and $C_3$, taking a value $\sigma {\prime }_m$ between $\sigma {\prime }_0$ and $\sigma {\prime }_f$ (values of effective soil stress at the beginning and end of the process consolidation, respectively).

The ratios $C_1/C_3$ and $C_2/C_3$ give rise to two dimensionless monomials.

$${\pi }_I=\frac{C_1}{C_3}=\frac{I_c{\gamma }_w{H}_o^2}{{\tau }_{o,s}{\left(1+e_o\right)}^{2}Ln\left(10\right){\sigma }_o^{\prime }{k}_o}{\left(\frac{{\sigma }_m^{\prime }}{{\sigma }_o^{\prime }}\right)}^{-\lambda }$$

$${\pi }_{II}=\frac{C_2}{C_3}=\lambda \frac{Ln\left(10\right){\zeta }_o}{I_c}= \lambda Ln\left(10\right)lo{g}_{10}\left(\frac{{\sigma }_f^{\prime }}{{\sigma }_o^{\prime }}\right)=Ln\left(10\right)lo{g}_{10}{\left(\frac{{\sigma }_f^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }$$

(10)

For the expression of ${\pi }_{II}$, it was considered that ${\zeta }_o=e_o-e_f=I_{c}lo{g}_{10}\left(\frac{{\sigma }_f^{\prime }}{{\sigma }_o^{\prime }}\right)$.

As for expression (16) of the monomials ${\pi }_I$ and ${\pi }_{II}$, since they are related by function $\Psi$, they can be simplified by eliminating the constant $Ln\left(10\right)$ so that they can be rewritten as:

$${\pi }_I=\frac{I_c{\gamma }_w{H}_o^2}{{\tau }_{o,s}{\left(1+e_o\right)}^2{\sigma }_o^{\prime }{k}_o}{\left(\frac{{\sigma }_m^{\prime }}{{\sigma }_o^{\prime }}\right)}^{-\lambda }$$

$${\pi }_{II}=lo{g}_{10}{\left(\frac{{\sigma }_f^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }$$

(11)

However, because of the previous reasoning, it is also possible to simplify the monomial ${\pi }_{II}$ again because the function $\Psi$ can collect the effect of the logarithm in base 10. This is:

$${\pi }_I=\frac{I_c{\gamma }_w{H}_o^2}{{\tau }_{o,s}{\left(1+e_o\right)}^2{\sigma }_o^{\prime }{k}_o}{\left(\frac{{\sigma }_m^{\prime }}{{\sigma }_o^{\prime }}\right)}^{-\lambda }$$

$${\pi }_{II}={\left(\frac{{\sigma }_f^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }$$

(12)

since the coefficients $C_1$, $C_2,$ and $C_3$ are of the order of magnitude 1, so are the monomials; therefore, their inverse expressions are equally valid.

$${\pi }_I=\frac{{\tau }_{o,s}{\left(1+e_o\right)}^2{\sigma }_o^{\prime }{k}_o}{I_c{\gamma }_w{H}_o^2}{\left(\frac{{\sigma }_m^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }$$

$${\pi }_{II}={\left(\frac{{\sigma }_f^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }$$

(13)

Based on these last expressions, and considering that the quotient $\sigma {\prime }_m/\sigma {\prime }_0$ is, in essence, a function of $\sigma {\prime }_f/\sigma {\prime }_0$ (remember that $\sigma {\prime }_f$ is the sum of $\sigma {\prime }_0$ and the increment of the load applied to the soil), the effect of the factor ${(\sigma {\prime }_m/\sigma {\prime }_0)}^{\lambda }$ is present in the monomial ${\pi }_I$ and can also be included within the function $\Psi =\Psi {(\sigma {\prime }_m/\sigma {\prime }_0)}^{\lambda }$, further simplifying the expressions of the monomials:

$${\pi }_I=\frac{{\tau }_{o,s}{\left(1+e_o\right)}^2{\sigma }_o^{\prime }{k}_o}{I_c{\gamma }_w{H}_o^2}$$

$${\pi }_{II}={\left(\frac{{\sigma }_f^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }$$

(14)

From the expression ${\pi }_I$, Equation (14), the order of magnitude of the unknown ${\tau }_{0,s}$ can be derived as a function of the soil’s geotechnical parameters:

$${\tau }_{o,s} \approx \frac{I_c{\gamma }_w{H}_o^2}{{\left(1+e_o\right)}^2{\sigma }_o^{\prime }{k}_o}$$

But also, from Equation (6), using the expressions of Equation (14), we have,

$$\frac{{\tau }_{o,s}{\left(1+e_o\right)}^2{\sigma }_o^{\prime }{k}_o}{I_c{\gamma }_w{H}_o^2}={\Psi \left(\frac{{\sigma }_f^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }$$

This allows us to write the following.

$${\tau }_{o,s}=\frac{I_c{\gamma }_w{H}_o^2}{{\left(1+e_o\right)}^2{\sigma }_o^{\prime }{k}_o}{\Psi \left(\frac{{\sigma }_f^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }$$

(15)

To determine the function $\Psi$, it is necessary to model and solve the governing Equation (8), simulating a vast variety of scenarios in which the soil’s geotechnical parameters are modified to cover the interval of real values that monomial ${\pi }_{II}$ can take in practice. A network method was used as a numerical tool [29].

Thus, Figure 6 shows the relationship between the monomials ${\pi }_{II}$ and ${\pi }_I$ for the characteristic times of consolidation at which the settlement grades of 30%, 50%, 70%, 90%, 95%, 99%, and 99.9% are reached and of the total settlement grade, ${\stackrel{-}{\mathrm{U}}}_s$.

At this point, it is interesting to recall the definition of the average degree of settlement, ${\stackrel{-}{\mathrm{U}}}_s$.

$${\stackrel{-}{\mathrm{U}}}_s=1-\frac{H-H_f}{H_o-H_f}$$

(16)

Alternatively, it can be expressed as a percentage.

$${\stackrel{-}{\mathrm{U}}}_s(\%)=\left(1-\frac{H-H_f}{H_o-H_f}\right)·100$$

(17)

Thus, in Figure 6, each of the seven curves depicted provides the (characteristic) time for which a given soil (whose properties reside in the expressions of the monomials ${\pi }_I$ and ${\pi }_{II}$) reaches a certain degree of settlement. In other words, ${\tau }_{0,s}{}_{70}$ is the time it takes for the soil to reach an average settlement degree of 70% (${\stackrel{-}{\mathrm{U}}}_s=0.7$).

Thus, a universal solution to the problem is provided, covering the evolution of the consolidation process from the early stages of settlement to the practical completion of the transitory phenomenon.

As is logical, the greater the percentage of settlement considered, the greater the characteristic consolidation time, and therefore, the greater the value of ${\pi }_I$. However, it can be observed that the greatest influence of one monomial, with respect to the other, occurs in the interval [0–1]. In other words, as ${\pi }_{II}$ increases (outside this interval), the value of ${\pi }_I$ (and therefore $\mathsf{\tau }$_o,s) is less affected.

In conclusion, the fact that the monomials ${\pi }_I$ and ${\pi }_I$ exhibit this behavior (strong dependence within the interval [0–1], with little or no influence outside it) is due to the correct application of the non-dimensionalization technique in combination with the normalization of variables.

As an illustration, we present an application of a real nonlinear consolidation problem governed by the Cornetti and Battaglio model and extended for the non-constant $1+e$ and $dz$, as shown in Equation (8).

The initial ground state parameters, before applying the load increase $\Delta {\sigma }^{\prime }$, are

$$e_o=2$$

$${\sigma }_o^{\prime }=30000 \mathrm{N}/{\mathrm{m}}^2$$

$$k_o=0.02 \mathrm{m}/\mathrm{year}$$

$$H_o=1 \mathrm{m}$$

On the other hand, the constitutive parameters of the material have the following values:

$$I_c=1.5$$

$$\lambda =-0.5$$

The specific weight of water is:

$${\gamma }_w=9800 \mathrm{N}/{\mathrm{m}}^3$$

Under these conditions, a load increase was instantaneously applied to the ground surface as follows:

$$\Delta {\sigma }^{\prime }=16780 \mathrm{N}/{\mathrm{m}}^2$$

Thus, the final effective stress can be determined as

$${\sigma }_f^{\prime }={\sigma }_o^{\prime }+\Delta {\sigma }^{\prime }=46780 \mathrm{N}/{\mathrm{m}}^2$$

Knowing these parameters, we obtain the values that the monomials ${\pi }_I$ and ${\pi }_{II}$ in Equation (14) take for our problem:

$${\pi }_I=\frac{{\tau }_{o,s}{\left(1+e_o\right)}^2{\sigma }_o^{\prime }{k}_o}{I_c{\gamma }_w{H}_o^2}=0.367{\tau }_{o,s}$$

$${\pi }_{II}={\left(\frac{{\sigma }_f^{\prime }}{{\sigma }_o^{\prime }}\right)}^{\lambda }=0.8$$

In this way, and since the monomial ${\pi }_{II}$ is completely known, from Figure 6, we can obtain the value of the characteristic consolidation time ${\tau }_{0,s}$, corresponding to the required average degree of settlement, ${\stackrel{-}{\mathrm{U}}}_s$, i.e., 30%, 50%, 70%, 90%, 95%, 99%, or 99.9%. However, for a more accurate reading of ${\tau }_{0,s}$, Figure 7 shows a zoomed in Figure 6 for the low values of ${\pi }_I$ and ${\pi }_{II}$ (where the dependence between these monomials is higher).

Meanwhile, from the numerical solution of the governing Equation (8) of the problem, the curve for the average degree of settlement, as shown in Figure 8, of the proposed real case was obtained. For this purpose, because the average degree of settlement ${\stackrel{-}{\mathrm{U}}}_s=1-\left(H-H_f\right)/\left(H_o-H_f\right)$ depends on the instantaneous and final thicknesses ($H$ and $H_f$), it is necessary to obtain these parameters during the simulation of the problem, which are directly related to the void ratio ($e$) by means of the following expression:

$$\frac{1+e}{1+e_o}=\frac{H}{H_o}$$

(18)

The void ratio ($e$) is related to the effective stress (${\sigma }^{\prime }$) by the following constitutive relation:

$$e=e_o-I_{c}lo{g}_{10}\left(\frac{{\sigma }^{\prime }}{{\sigma }_o^{\prime }}\right)$$

(19)

Thus, the instantaneous values of ($e$) and ($H$) are obtained during the calculation process from the instantaneous values of the effective stress (${\sigma }^{\prime }$). On completion, these parameters have the following values.

$$e_f=e\left({\sigma }_f^{\prime }\right)=1.709$$

$$H_f=H\left(e_f\right)=0.906$$

From Figure 8, it is possible to know the time required to reach a given average settlement degree, ${\stackrel{-}{\mathrm{U}}}_s$. To compare these results with those obtained from Figure 7, we plotted the times at which the degrees of settlement were: 30%, 50%, 70%, 90%, 95%, 99%, and 99.9%. For example, in Figure 8, $t_{70}$ is the time it takes for the soil to reach an average degree of settlement of 70% (${\stackrel{-}{\mathrm{U}}}_s=0.7$).

Table 6. ${\tau }_{0,\stackrel{-}{\mathrm{U}}s}$ (estimated) and $t_{\stackrel{-}{\mathrm{U}}s}$ (simulated) values for different average degrees of settlement ${\stackrel{-}{\mathrm{U}}}_s$ (%).

${\mathit{Ū}}_{\mathit{s}}$ (%)	30	50	70	90	95	99	99.9
${\pi }_{I,\tau }{}_{0,\stackrel{-}{\mathrm{U}}s}$ for ${\pi }_{II}$ = 0.8 (Figure 7)	0.04	0.11	0.22	0.46	0.61	0.96	1.47
${\tau }_{0,\stackrel{-}{\mathrm{U}}s}$ (yr) estimated (Figure 7)	0.1063	0.2997	0.5995	1.2534	1.6621	2.6158	4.0054
$t_{\stackrel{-}{\mathrm{U}}s}$ (yr) Simulated (Figure 8)	0.1053	0.2976	0.5931	1.2478	1.6620	2.6249	4.0006
rel. error (%) for $t_{\stackrel{-}{\mathrm{U}}s}$	0.92	0.71	1.07	0.45	0.01	0.35	0.12

shows both the estimated values for the characteristic time ${\tau }_{0,s}$ (or ${\tau }_{0,\stackrel{-}{\mathrm{U}}s}$) from the ${\pi }_I$-${\pi }_{II}$ dependence (Figure 7) and the times $t_{\stackrel{-}{\mathrm{U}}s}$ obtained via the simulation (Figure 8) for the different average degrees of settlement ${\stackrel{-}{\mathrm{U}}}_s$. Finally, the table includes the relative error calculated as follows:

$$rel. error (\%)=\frac{\left|{\tau }_{0,\stackrel{-}{\mathrm{U}}s}-t_{\stackrel{-}{\mathrm{U}}s}\right|}{t_{\stackrel{-}{\mathrm{U}}s}}·100$$

(20)

As can be seen, the differences between the estimated values (from the ${\pi }_I$-${\pi }_{II}$ dependence) and the simulated ones are practically negligible (relative error never greater than 1%), which shows the high reliability of these curves (Figure 6 and Figure 7) provided to determine the characteristic consolidation time ${\tau }_{0,s}$ without having to simulate the problem.

4. Discussion

The normalization of an ODE or CODE system is a reliable and fast procedure that, without complex analytical or numerical efforts, allows the determination of dimensionless numbers governing the process and, from them, the order of magnitude of some unknowns of interest. It is necessary to understand the importance of the order of magnitude of the dimensionless monomials obtained, as they will significantly affect the solution of the problem as long as they are normalized to unity. This is because these monomials are formed by balancing two terms of the equation, and if they are not of the order of unity, it is because one of the terms is negligible compared to the other, and therefore, can be removed from the equation. This information is fundamental for PhD students, graduates, and researchers in the design and solution of different engineering problems.

One possible application of this normalization technique is in the mineral processing industry. The large number of independent variables and their heterogeneity imply high uncertainty and the impossibility of performing accurate comparative analyses with respect to the same operating variable [35]. An example of this condition is the thickening process, which consists of separating the phases that make up the slurries generated during flotation [36,37]. The main operational problems of thickeners are related to the complexity of the system of tailings and pulp concentrates because of the heterogeneity of their mineralogical composition and size distribution in the feed pulps, which affect the operational parameters [38]. This is a multifactorial problem that affects process efficiency and needs to be analyzed. The normalization and nondimensionalization of the variables that compose the process implies the generation of models, allowing for the comparative analysis of the variables that affect the unitary operations in a more precise manner to collaborate in optimization processes. This will be a future research direction for using this technique.

5. Conclusions

When the normalization technique is applied, the mathematical manipulations necessary to define the governing equations in their dimensionless form are simple, but thorough knowledge of the physical phenomena involved is required to choose the appropriate references that make the independent and dependent variables of the equations dimensionless. Often, these appropriate references are not explicit in the problem formulation, despite the large number of parameters whose values are explicit, and the researcher must explore other hidden references. This occurs in many mechanical (dynamic) problems, in which each variable has an initial and final value that can be chosen as a reference. Usually, these are unknowns whose order of magnitude are given via normalization.

Both the proposed dynamic application as a linear system and the nonlinear soil consolidation model demonstrated that the order of magnitude of the appropriate references for each situation is derived from the normalization process.

The normalization of variables has also been successfully applied to the nonlinear soil consolidation problem, a model that considers logarithmic constitutive relationships and is free of restriction because both $1+e$ and $dz$ are not constants. The solutions, expressed in terms of dimensionless monomials that govern the problem, provide the evolution of the consolidation phenomenon from the early stages of settlement to the end of the process. In addition, the influence of these monomials is practically bounded to the interval [0,1], which is proof of the successful application of the normalization and nondimensionalization technique while allowing us to understand in a much more precise and intuitive way the influence of the different soil’s geotechnical parameters in the solution pattern of the problem.