I. Introduction

Every day, we are surrounded by beams. They are the fundamental structural elements that carry vertical loads. Though beams are traditionally used to describe building or civil engineering, beams can be found in all existing structures as structural elements, including machine frames, bones, carbon nanotubes, molecular chains, and other mechanical or structural systems. In these structures, the size-scale is paramount for a precise description of the mechanical properties of the beam [9].

In continuum mechanics, the analysis of movements and deformations is determined by the hypothesis that the medium is composed of matter in a homogeneous way. This theory ignores the existence of voids formed when molecules and atoms are not evenly distributed. The question that arises in micro- and nanoscale analysis is whether conventional models of continuum mechanics, such as Euler-Bernoulli and Timoshenko, may not be appropriate, given that such approaches do not take into account the scale factor in their models, see for example [9]. To address the scale issue, some non-classical continuum theories, as well as theories incorporating non-integer order derivatives in the [6, 7, 8, 9] models, have been investigated. Models with fractional dynamics have been shown to be more suitable for describing the properties of various real materials, e.g. [6], and thus have aroused the interest of engineering research.

In [7] an overview of the modeling of fractal media through the theory of continuous mechanics is presented using the ideas proposed in [8]. This theory consists of describing the laws of equilibrium for fractal media using fractional integrals. Using a map from the fractal to the continuous medium, those fractional integrals are rewritten as integrals in conventional Euclidean space. The interesting thing about this approach is that the essential condition of continuum mechanics, the separation of scales, can be replaced by the use of continuum field equations. In Section II, we present the deduction of the Euler-Bernoulli equation for beams in fractal media, using the techniques proposed in [8]. We also show that the analyzed model has a unique solution , which is known in the literature as the direct problem for the Euler-Bernoulli beam model in fractal media.

The main contribution of this work is the numerical investigation of an “inverse problem” for the Euler-Bernoulli beam model in fractal media. Indeed, the stable identification of the stiffness parameter associated with the Euler-Bernoulli equation in fractal media, as described in Section II, from indirect measurements of the nano (micro)-beam deflection of .

Given that measurements of the nano (micro)-beam deflection are subject to errors and that the inverse problems are generally ill-posed in the Hadamard sense [2, 3], the issue of instability in the identification of the stiffness parameter due to noise measurements of the beam deflection necessitates the use of some regularization strategy. In this contribution, we use the Landweber iterative method (see Section III), which will be used to numerically demonstrate the stability of the approximations for the identification coefficient in Section IV. In Section IV, we will present several numerical tests with varying levels of noise in the measurements. The presented scenarios demonstrated numerically that the Landweber iteration obtains stable approximate solutions for the coefficient under different fractal medium properties.

II. Euler-Bernoulli Equation in Fractal Media

In general, a fractal medium cannot be considered as a continuous medium, as there are points and domains that are not filled by particles of the medium. These domains can be called porous. Thus, the application of continuum theory to fractal media is not appropriate. To get around this difficulty, [8] proposed the use of fractional integrals to represent the mass of a region Importar tabla Importar tabla in three-dimensional Euclidean space E³ tabla as being:

with

where is the length scale measurement, D is the fractal dimension of mass, and is Euler's Gamma function. The coefficient establishes the transformation between fractal and continuous media. The term $d{V}_D$ tabla is the infinitesimal element of volume in fractal space, and $d{V}_3$ is the infinitesimal element of volume in $E^3$. However, the proposal of [8] implies that the measure of the fractal dimension in each of the directions of the medium must be the same. To contour this limitation, [8] proposed an approach in which the measure of length in relation to each coordinate of the medium is given by:

where represents mass density and a fractal dimension in the direction

Assuming that is given by the modified Riemann-Liouville integral, that is:

where is the total length along and is the characteristic length in the given direction, [8] showed that it is possible to reproduce almost all the known results of the mechanical theory of the continuum, in addition to allowing to represent more heterogeneous media.

A. One-dimensional fractal medium: the Euler-Bernoulli equation

Assume that we are in Euclidean dimension 1, in the direction. Let dimension be the dimension of the fractal structure in which we are immersed, in the direction . Then it follows from (3) that the element must be replaced by:

Rewriting the balance equations in variational form, with the measure given by (5) and using the generalized Green-Gauss Theorem, see [8], it follows that the Euler-Bernoulli equation in fractal means is given by:

wherein

is the bending moment with the stiffness coefficient. In (6) and (7), the operator is the Laplacian operator for fractal media, given by:

where is the transformation coefficient between the fractal medium and the continuous medium.

In this work, we will consider (7) with the following boundary conditions:

corresponding to a cantilever beam.

B. The inverse problem as an equation of operators

To formulate the inverse problem that we are interested in this work, we first need to formulate some hypotheses for which problem (7) with boundary conditions (9), has a single solution, that is, the direct problem is well posed.

A1: The stiffness coefficient and the transformation coefficient are measurable functions on that satisfy the condition and for known constants . The set of coefficients satisfying the hypothesis A1 will be denoted by Ad in this manuscript and referred to as the admissible set

A1’: The stiffness coefficient and the transformation coefficient besides satisfying A1 have uniformly bounded that is, the coefficients belong to the set where ||.|| denotes the norm with and

A2: The bending moment

Consider the vector space

measurable, such that

with the inner product below

It follows from the Assumption A1 that the space of functions with the norm induced by the inner product (11) is a space of Hilbert. Furthermore, the space will denotate the Sobolev space of all functions in , with the derivatives in the weak sence, also belongs to , and satisfies . See, for example [1].

In order to prove existence and uniqueness for a solution of the problem (7) with conditions in (9), we use the theory of weak solution as follows. First, we consider as a weak solution to problem (7) any function such that

for any test function

Lemma 1: Assume that the assumptions A1 and A2 are satisfied. If there is satisfying the problem (7) and (9), then is a weak solution to the problem (12). Conversely, if is a weak solution to (12), then satisfies (7) almost always.

Proof: It follows from Hypothesis A1 and A2 that $\textcolor{red}{}c.M/a\in L^2[0,L]$. Therefore, from (7) and conditions (9), and satisfy (12) (see the Green’s identities in [1]).

Reciprocally, if is a weak solution to (7), it is, satisfies (12), then boundary conditions (9) for the problem in (7) are satisfied. Furthermore, as satisfied (12) by assumption, we have, after an integration by parts, that

Therefore, it follows from the density of in that , almost everywhere. Hence, by the Hahn-Banach theorem (see [1]) the result for (13) can be extended to .

Given the Lemma 1, it is possible to prove through the Lax Milgram’s Theorem [1] the existence of a unique solution that satisfies (12). Indeed, notice that is a weak solution of (7) if and only if satisfies

where $A:H_0^1[0,L]\times H_0^1[0,L]\to ℝ$ is the bilinear form defined as

and $l:H_0^1[0,L]\to ℝ$ is the linear functional given by

Theorem 1: Assuming that the hypotheses A1 and A2 are satisfied, there is a unique solution to (12). As a consequence of the Lemma 1, there is a unique weak solution to (7).

Sketch of Proof: Following the same ideas in [4], it is possible to prove that the linear functional defined in (16) is continuous and furthermore that the bilinear form defined in (15) is continuous and coercive in . Therefore, it follows from the Lax-Milgram Theorem [1, Corollary 5.8] the existence of a unique function satisfying (12).

As a result of Theorem 1, it follows that, for any given function , satisfying Assumption A1, the operato

where is the unique solution of (7), is well defined.

$F_{c\left(x\right)}$ is called the forward operator in the theory of inverse problems, see for example [2, 3].

C. The inverse problem

Assume that the functions and are known. The inverse problem that we are interested in this work deals with the identification of the stiffness coefficient from measurements , with noise level , satisfying:

of the deflection solution of (7) with boundary conditions (9). Equivalently, determine in the operator equation (17), from the measures satisfying (18).

Inverse problems, in general, do not have the property of continuous dependence of the measures . This implies that small perturbations of magnitude in the measurements can generate large perturbations in obtaining the solution of the inverse problem of interest, e.g., [2, 3]. As a result, obtaining stable and convergent solutions with respect to the noise level requires the use of regularization methods. See for example [2, 3, 4, 5].

The problem of identifying the stiffness coefficient in a beam does not have the property of continuous dependence of the measures , as demonstrated in the case of in [4, 5]. As a result, the stable identification of the stiffness coefficient , requires some regularization methods [2, 3]. In this contribution, we will use an iterative regularization method called the Landweber method [2] to recover the parameter in a stable and convergent manner with respect to the noise level . In other words, we will show numerically that the Landweber iteration (see equation (19)) together with a stop criterion, called the discrepancy principle (see equation (20)), generates approximate stiffness coefficients , for , with values that are stable and convergent to , as a function of the noise level in the data . The iterative algorithm is presented in Section III, while the numerically simulated scenarios for the recovery of the stiffness coefficient is presented in Section IV.

III. Landweber’s Iterative Method

The Landweber iteration (Landweber's iterative method) for the identification of the coefficient in (7), is given by

where $\gamma$ is a relaxation parameter. $F_{c\left(x\right)}^{\text{'}}\left(a_k^{\delta }\right)*$ denotes the adjunct of the Fréchet derivative of the parameter-to-measurement operator , defined in (7). is the initial guess of the iteration (14), that shall be chosen properly.

Because the data contains noise, the iterative method must be combined with a stop rule, as mentioned in [2, 3, 4, 5]. In this work, we use the discrepancy principle's stopping criterion, which states that (19) must be stopped at the first step that satisfies

for some . Thus, the number of iterations determines the stopping rule of the method.

The numerical implementation to obtain the coefficient iteratively according to (19) is given by the following algorithm:

(1) Choose an initial value for and satisfying Assumptions A1 and A1', respectively. Choose also the parameter values

(2) Add the uniformly distributed random variable $z\left(x\right)$ to [0,1] the solution of the forward problem to generate the noise data , satisfying $\left|\right|u\left(x\right)-u^{\delta }\left|\right|\le \delta$

3) As long as the iteration (19) is such that $a_k^{\delta }$ , for$k<k^{*}$, where $k^{*}$ denotes the iteration index that satisfies the discrepancy principle, do the following steps:

(4) Solve the problem with the initial conditions

with the initial conditions

(5) Evaluate the residue

where in $F_{c\left(x\right)}\left(a_k^{\delta }\right)$ is solution of differential equation calculated in Step (4).

(6) To calculate $F_{c\left(x\right)}^{\text{'}}\left(a_k^{\delta }\right)*r_k$ , firstly,

(6.1) Solve the differential equation

with finals conditions

(6.2) Then, find the adjunct

solving

wherein ${{\nu }_k}_{}$ is solution of equation obtained in Step (6.1).

(7) Update $a_{k+1}^{\delta }$ wherein $a_{k+1}^{\delta }=a_k^{\delta }-\gamma A{D}_k$

(8) Go back to Step (3) while the discrepancy principle given by (20) is not reached.

(9) Otherwise, the regularized solution is , where is deteremined by the discrepancy principle (20).

It is important to mention that, for the calculations of in Step (6) in the algorithm, it is necessary to define an auxiliary operator given by

for . It is straightforward to show that the operator defined in (21) is linear, bounded and bijective. Therefore, it also has a linear and bounded inverse . The adjunct operator of is such that , corresponding the Step (6.1) in the algorithm. Because it is also linear and bounded, its inverse is given by

where is the unique solution of the

The equations obtained in Step (6) of the above algorithm are calculated by taking the Fréchet derivative of the operator $F_{c\left(x\right)}(·)$ defined in (12) and integrating by parts with respect to the inner product given by (11). Therefore, we obtain

with $A^{*}{\left(a\right))}^{-1}$ as (22) applied to residue $r=u^{\delta }-F_{c\left(x\right)}\left(a^{\delta }\right)$

Hence, the Steps (6.1) and (6.2) of the algorithm are equivalent to (23).

IV. Numerical Examples

In this section, we use the Landweber regularization method given by (19) to identify the beam stiffness coefficient , in the fractal media Euler-Bernoulli equation modeled by (7) and (9). In all the simulations presented below, we use , and the bending moment . Also, we use as the initial guess of the Landweber iteration method (19). The finite difference method was used to obtain the numerical solution for (7) at points where for points for the simulated scenario of Example 1 and points for the other simulated scenarios. In all examples, the noisy data is generated by adding a random variable , evenly distributed, to the solution of (7), such that , where is the noise level.

The differential equations corresponding to Step (6) of the algorithm were solved using backward Euler's method to account for the final conditions. The steps of the algorithm resulting from the Landweber method, presented in Section III, were implemented in Python (version 3.8.5).

Example 1: The simulated scenario corresponding to this first example consists in identifying the stiffness coefficient in the fractal medium Euler-Bernoulli beam where the fractal medium transformation coefficient is .

Fig. 1 compares the coefficients and recovered by the Landweber method given by (14) for noise levels and , respectively. The simulations were performed on a mesh with points. The numerical results shown in Fig. 1 demonstrate that Landweber's iterative method produces stably approximate solutions for the stiffness coefficient , as a function of the noise level .

In the simulations presented in Fig. 1, the reconstructed coefficient is obtained by using the discrepancy principle, for with the iteration is stopped after iterations for the noise level . While, the iteration is stopped after and , for the noise level of and , respectively.

In the following simulated scenarios, we will consider the identification of a non-constant stiffness coefficient , as a way of evaluating the performance of Landweber's iterative method in more unfavorable scenarios

Example 2: The simulated scenario of this example corresponds to the identification of the stiffness coefficient $a^{*}\left(x\right)=\frac{1}{2-x}$ , from noise measurements of the fractal Euler-Bernoulli beam equation (6), where the fractionality of the medium is given by .

The results for noise levels of and are shown in Fig. 2. The mesh was chosen uniformly with points. This figure show that the Landweber iteration with the stopping criterion given by the discrepancy principle in (20) produced stable and satisfactory approximations for the non-constant coefficient .

The principle of discrepancy given in (20) is reached for the scenarios of this example with , and , for the simulated noise levels for , and , respectively.

In the simulated scenarios that follow, we present approximate solutions for recovering the coefficient as in Examples 1 and 2, where the fractionality of the medium is distinct.

Example 3: The simulated scenario of this example corresponds to the Euler-Bernoulli fractional media beam, where the coefficient of fractionality is given by . The simulations for a noise level of and , are presented in order to recover the coefficient , in Fig. 3.

Fig. 3 shows the coefficient recovered by the Landweber method given in (19) for different noise levels and . The simulation was performed on a mesh with points.

In Fig. 3, the reconstructed coefficients for different noise levels and satisfied the discrepancy principle, respectively, with and .

Example 4: The simulated scenario of this example corresponds to the Euler-Bernoulli fractional media beam, where the coefficient of fractionality is given by . The simulations for noise level and are presented in Fig. 4 in order to recover the coefficient $a^{*}\left(x\right)=\frac{1}{2-x}$

Fig. 4 shows the coefficient recovered by the Landweber method given in (19) for different noise levels and respectively. The simulation was performed on a mesh with points.

The principle of discrepancy given in (20) is reached of this example for the simulated noise levels, where for for and for .

The numerical results shown in Examples 1, 2, 3, and 4 demonstrate that Landweber's iterative method terminated with the discrepancy principle produces stably approximate solutions $a_k^{\delta }·\left(x\right)$ for the simulated scenarios with the stiffness coefficient $a\left(x\right)=1$ and $a^{*}\left(x\right)=\frac{1}{2-x}$ for different proposed noise levels with distinct fractionality transformation coefficient .

It is worth noting that the discrepancy principle in Examples 3 and 4 is stretched after more iterations than previous examples, which is possible due to the polynomial degree of the coefficient .

It will be investigated in future contributions.

V. Conclusions

In this paper, we present a fractal mechanics-based version of the Euler-Bernoulli equation for beams at micro- and nanoscales, as well as the inclusion of the parameter responsible for characterization of the fractionality of scales. We investigated the inverse problem of identifying the Euler-Bernoulli equation coefficient from measures of noisy data corresponding to the bending of a fractal media beam. As this problem is ill-posed in the Hadamard sense, we numerically analyze the Landweber iteration method as a regularization, in order to obtain stable and convergent solutions for the parameter of interest in terms of the noise level. We present some numerical examples for different noise levels in the recovery of constant and non-constant stiffness coefficients . In addition, we performed tests with two different functions for the fractal parameter , evaluating the performance of the method in these cases as well. In fact, the numerical results presented showed that the proposed iterative method satisfactorily recovered the stiffness coefficient, reaching the stopping criterion with a similar number of iterations in the different tests, even when simulated for different parameters .

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