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Crack identification in laminated composites based on modal responses using metaheuristics, artificial neural networks and response surface method: a comparative study

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Abstract

Structures that are difficult to maintain and access need to have an efficient and robust process for continuous monitoring. Such monitoring through damage detection and identification studies is present in several engineering applications as it allows that corrective measures be applied in order to guarantee the structural safety of a given structure, machine or equipment. In particular, laminated composite materials, often used in aeronautical structures, have a complex failure mechanism where delamination or cracks in these materials are often not visible on the surface. Thus, the use of optimization methods for the characterization of damages in these materials becomes relevant. In this study, both the metaheuristic sunflower optimization, the artificial neural networks and the response surface method were used to solve an inverse crack identification problem. The crack was modeled as a thin elliptical hole in a rectangular laminated plate numerically modeled using the finite element method. As a result of the methods used, different approaches to the problem were obtained that present reliable shape, size and position identification of a crack sized between 3 and 30 mm. The results showed substantial and promising results in the uses of both metaheuristic techniques and artificial neural networks. However, neural networks have a certain competitive advantage over optimization techniques as long as the data that feeds the model present a certain level of quantity and quality. Results obtained were able to identify all damage parameters (location, extension and orientation), with errors less than 1%.

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Abbreviations

K :

Stiffness matrix

\(\omega _{j}\) :

jth natural frequency of interest

M :

Mass matrix

\(\phi _{{\left( {i,j} \right)}}\) :

Modal shift of the ith node in the jth mode of interest

\(\hat{\phi }_{{\left( {i,j} \right)}}\) :

Modal shift of the ith node in the jth normalized mode of interest

\(Q_{i}\) :

Amount of heat received by the ith plant

P :

Source power

\(r_{i}\) :

Distance between the sun and the ith plant

\(\vec{s}_{i}\) :

Plant direction toward the sun

X* :

Best individual parameter vector

\(X_{i}\) :

Parameter vector of the ith plant

\(d_{i}\) :

Step of the ith plant

λ :

Constant value that defines an “inertial” displacement of the plants

\(P_{i}\) :

Probability of reproduction of the ith plant

\(X_{{\max }}\) :

Upper limit of the parameter vector

\(X_{{\min }}\) :

Lower limit of the parameter vector

\(N_{{{\text{pop}}}}\) :

Number of plants

\(\vec{X}_{{\left( {i,j} \right)}}\) :

ITh plant in the jth generation

\(v_{k}\) :

K Neuron linear combination factor

\(w_{{kj}}\) :

Synapse weight between k and j neurons

ϕ :

Activation function

\(b_{k}\) :

Neuron k trend

\(Y_{i}\) :

ITh response predicted by the artificial neural network

\(\hat{Y}_{i}\) :

ITh response expected by the artificial neural network

γ :

Regularization factor

\(E_{x}\) :

Young's modulus in the x direction

\(E_{y}\) :

Young's modulus in the y direction

\(\nu _{{xy}}\) :

Poisson's ratio in the xy plane

\(G_{{xy}}\) :

Shear modulus in the xy plane

\(x_{{\text{o}}}\) :

Position of the center of the ellipse at x

\(y_{{\text{o}}}\) :

Y Ellipse center position

a :

Larger radius of the ellipse

b :

Minor radius of the ellipse

θ :

Inclination angle of the ellipse's largest radius in relation to the x axis

\(f_{i}\) :

Ith natural frequency of interest (in Hz)

t :

Parameter independent of the parametric equation of the ellipse

\(x_{{\text{e}}}\) :

X rectangular enclosure size

\(y_{{\text{e}}}\) :

Y rectangular enclosure size

\(\phi _{{{\text{h}}(i,j)}}\) :

Modal shift of the ith plate node without damage in the jth mode

\(\phi _{{{\text{d}}(i,j)}}\) :

Modal shift of the ith node of the damaged plate in the jth mode

\(\Delta \hat{\phi }_{{\left( {i,j} \right)}}\) :

Difference between modal shift of the ith node of the real structure in the jth mode of interest and the undamaged structure

\(\Delta \phi _{{\left( {i,j} \right)}}\) :

Difference between modal shift of the ith node in the jth mode

ϕ h(i,j) :

Modal shift of the ith plate node without damage in the jth mode

ϕ d(i, j):

Modal shift of the ith plate node with damage to the jth mode of interest

J :

Objective function

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Acknowledgements

The authors would like to acknowledge the financial support from the Brazilian agency CNPq and FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais – Grant APQ-00385-18).

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Authors and Affiliations

  1. Mechanical Engineering Institute, Federal University of Itajubá (UNIFEI), Itajubá, Brazil

    Felipe Mouallem de Assis & Guilherme Ferreira Gomes

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  1. Felipe Mouallem de Assis
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Correspondence to Guilherme Ferreira Gomes.

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This manuscript is self-contained, in that it contains all necessary theory to reproduce the results, including the preliminaries, i.e., the inverse crack identification in laminate composite plates. The formulation of the inverse problem is described in detail, and all parameters for the numerical examples are provided.

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de Assis, F.M., Gomes, G.F. Crack identification in laminated composites based on modal responses using metaheuristics, artificial neural networks and response surface method: a comparative study. Arch Appl Mech 91, 4389–4408 (2021). https://doi.org/10.1007/s00419-021-02015-y

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