Self-learning locally-optimal hypertuning using maximum entropy and comparison of machine learning approaches for estimating fatigue life in composite materials

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Self-learning locally-optimal hypertuning using maximum

entropy, and comparison of machine learning approaches for

estimating fatigue life in composite materials

Miguel D´ıaz-Lagoa,†, Ismael Ben-Yeluna,†, Luis Saucedo-Moraa,b,c,∗, Miguel ´

Angel Sanza,

Ricardo Calladoa, Francisco Javier Mont´ansa,d

aE.T.S. de Ingenier´ıa Aeron´autica y del Espacio, Universidad Polit´ecnica de Madrid, Pza. Cardenal

Cisneros 3, 28040, Madrid, Spain

bDepartment of Materials, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK

cDepartment of Nuclear Science and Engineering, Massachusetts Institute of Technology, MA02139,

USA

dDepartment of Mechanical and Aerospace Engineering, Herbert Wertheim College of Engineering,

University of Florida, FL32611, USA

Abstract

Applications of Structural Health Monitoring (SHM) combined with Machine Learning

(ML) techniques enhance real-time performance tracking and increase structural integrity

awareness of civil, aerospace and automotive infrastructures. This SHM-ML synergy has

gained popularity in the last years thanks to the anticipation of maintenance provided by

arising ML algorithms and their ability of handling large quantities of data and considering

their inﬂuence in the problem.

In this paper we develop a novel ML nearest-neighbors-alike algorithm based on the

principle of maximum entropy to predict fatigue damage (Palmgren-Miner index) in

composite materials by processing the signals of Lamb Waves—a non-destructive SHM

technique—with other meaningful features such as layup parameters and stiﬀness matrices

calculated from the Classical Laminate Theory (CLT). The full data analysis cycle is ap-

plied to a dataset of delamination experiments in composites. The predictions achieve a

good level of accuracy, similar to other ML algorithms, e.g. Neural Networks or Gradient-

Boosted Trees, and computation times are of the same order of magnitude.

The key advantages of our proposal are: (1) The automatic determination of all the

parameters involved in the prediction, so no hyperparameters have to be set beforehand,

†M.D´ıaz-Lago and I. Ben-Yelun contributed equally to this work as ﬁrst authors.

∗Corresponding author

Email address:luis.saucedo@upm.es (Luis Saucedo-Mora).

Preprint submitted to Engineering Structures 21st October 2022

arXiv:2210.10783v1 [cs.LG] 19 Oct 2022

which saves time devoted to hypertuning the model and also represents an advantage

for autonomous, self-supervised SHM. (2) No training is required, which, in an online

learning context where streams of data are fed continuously to the model, avoids repeated

training—essential for reliable real-time, continuous monitoring.

Keywords: Maximum entropy, Machine Learning, Structural Health Monitoring,

Data-driven analysis

1. Introduction

Civil infrastructures, built since ages, are supposed to be robust and safe to provide

service to the daily life of millions of people during their operational lifespan. This concern

is also present in more recent ﬁelds, such as the aerospace or the automotive sector, where

the operation in highly aggressive dynamic environments and the necessary lightness and

tight safety factors may increase the possibility of damage and a lack of safety during their

service life. Therefore, it is fundamental to ensure the structural integrity of these systems,

which must be maintained over time. Structural health monitoring (SHM) techniques are

intended to undertake potential structural problems that may arise during the lifetime of

these structures, either to perform a corrective maintenance e.g. replace a certain damaged

part, or a predicted maintenance, trying to take advantage before the damage becomes

relevant and thus providing added safety and reducing maintenance costs to the given

infrastructure. A recent review of the many existing techniques in SHM has been done by

Rocha et al. [1], but in the last decades a large quantity of research has been performed

[2, 3, 4, 5, 6, 7, 8]. This monitoring essentially consists in information—captured by

sensors or similar devices—about the health of the component, that is, downstreaming

data that have to be interpreted by the user.

The real-time data generation which is constantly monitored in SHM is a key feature in

our digitalized world, where vast quantities of data are continuously shared, downloaded

and processed at the same time—according to a research conducted by Holst, 79·1021 bytes

of data were generate worldwide in 2021 [9]. Furthermore, this existing data ﬂow is taking

place not only in terms of the available amount of data but also in terms of the velocity

in which the information is generated, transferred, and demanded.

These two last claims have both enabled and likewise been enabled due to the use

of data driven techniques such as Machine Learning to process these gathered data and

make predictions of the health state of the structure. This eﬀective synergy has been

sought in the development of this work, which addresses damage mechanics monitored in

composite materials. Both topics introduce nonlinearities in terms of structural response,

so the models used to simulate these eﬀects become increasingly complex. Hence the

necessity of coming up with a surrogate, physics-based ML model that bypasses the need of

adjusting semi-analytical models for damage in composite materials, which has motivated

the present work. These latest advances in ML modeling have allowed the use of SHM

in other engineering areas like the one undertaken in this paper—a composite materials

dataset from the NASA. There is already related research work dealing with this dataset

which might be consulted [10, 11, 12, 13].

Regarding the physical-mathematical tools used to address this data-driven topic,

the statistical focus and derivatives has been utilized as common approach—see Ko and

Ni [4]—as well as autoreggresive (AR) models [14]. However, the latter method has the

limitations of lacking external variables, which is a critical issue when trying to forecast in

time series [15]. In another attempt to apply predictive maintenance in (infra)structures,

Gaussian Process (GP), probabilistic bayesian and transfer-bayesian models might be

found in Wan and Ni [16], Bull et al. [17] and Ierimonti et al. [18], with subsequent

improvements in predictions due to the ability of the models to handle non-linear input

data. Nevertheless, these models depend excessively on the train set i.e. the data used

for their ﬁtting, leading to non-unique solutions, and also present the main drawback of

computation time of GP—of order O(n3). Lastly, common ML techniques such as Neural

Networks (NN) and Support Vector Machine (SVM) have been recently used for SHM

applied to bridges [19], yet these algorithms require hypertuning, and more in particular,

the input data of the model proposed in [19] is purely autoreggresive, with the previously

mentioned disadvantages that this kind of models pose.

In this paper we propose a novel ML nearest-neighbors-alike algorithm which maxim-

izes the entropy (i.e. amount of information) of the surrounding data points and makes

a prediction through a convex combination of these selected neighbors—the set of points

that maximizes the entropy. This algorithm is applied to a real dataset of delamination

tests of composite layups to predict their fatigue life, and compared to other commonly

used Machine Learning algorithms such as k-Nearest Neighbors, gradient boosting trees

and Neural Networks. There are two key advantages with respect to the other algorithms:

The ﬁrst one is that all the neighbors parameters (i.e. the radii) are optimized during

the prediction, thus saving the need of ﬁnding a suitable hyperparameter for the model.

The second one is the accomplishment of a real-time ML model, able to handle with

new, continuously-generated data streams. The latter feature not only avoids repeated

training of the model but also represents a supply of new data points that enriches the ac-

curacy for further predictions—something which is essential for real-time and continuous

monitoring.

This paper is organized as follows. Firstly, a theoretical review of composites and struc-

tural health monitoring (SHM) techniques is performed in Section 2. Then, the maximum

entropy algorithm is introduced in Section 3, explaining the steps of the algorithm itself

and showing two examples as a proof of its functionality. Then, in Section 4 the proposed

methodology to predict fatigue-life in composites is addressed. The full data analysis

cycle is applied to the experiments conducted at Stanford Structures and Composites

Laboratory (SACL) in collaboration with the Prognostic Center of Excellence (PCoE)

of NASA Ames Research Center [20]. Finally, the results are detailed in Section 5, and

several concluding remarks are outlined in Section 6.

2. Theoretical Framework

2.1. Classical Laminate Theory

Using some assumptions, the Classical Laminate Theory (CLT) [21] simpliﬁes the

complex nature of laminates in terms of mechanical properties of continua. Each ply has

an orthotropic behaviour, from the notable diﬀerence in Young’s moduli between the stiﬀ

ﬁber direction and its perpendicular plane, being the latter only determined by the resin

properties. The CLT provides a method to calculate the continuum stiﬀness matrix of

any laminate, which relates the forces and moments applied to a laminate plate to the

in-plane strains εxx, εyy, γxy and curvatures κxx, κyy, κxy. The stress tensor is obtained

from the constitutive equations. The terms of the stiﬀness matrix will be later used as

input features of the model, to feed it with elastic, i.e. physics-based, information.

The constitutive equations of each ply expressed in the main ply axes, [Q]12, assuming

stresses remain in-plane are

[ε]12 = [Q]12[σ]12 ⇒





ε1

ε2

γ12







=





1/E1−ν12/E10

−ν12/E11/E20

0 0 1/G12





12







σ1

σ2

τ12







,(1)

where E1,E2,ν12,G12 are the Young’s moduli, Poisson’s ratio and shear modulus of the

ply. Then, to use a common reference frame for all plies, we convert them to the laminate

axes, [Q]xy, through suitable rotation matrices. After that, applying the Kirchhoﬀ hypo-

theses for plates and integrating across the z−coordinate of the laminate (with respect to

the middle plane), the stiﬀness matrix of the laminate can be obtained:







Nxy

Mxy













A11 A12 A16 B11 B12 B16

A21 A22 A26 B21 B22 B26

A61 A62 A66 B61 B61 B66

B11 B12 B16 D11 D12 D16

B21 B22 B26 D21 D22 D26

B61 B62 B66 D61 D62 D66













εxx

εyy

γxy

κxx

κyy

κxy







,(2)

where the terms of the three distinctive matrix boxes are

Aij =

k=1

ij (zk−zk−1), Bij =1

k=1

ij (z2

k−z2

k−1), Dij =1

k=1

ij (z3

k−z3

k−1),(3)

being [Q]kthe constitutive matrices of each ply kin laminate axes, and zkare the locations

of the plies relatives to the middle plane. This is the stiﬀness matrix of the laminate.

Submatrix A is called extensional stiﬀness matrix and relates extensional and shear forces

and strains. Submatrix B is the coupling stiﬀness matrix, since it relates bending strains

with extensional and shear forces and vice-versa. Lastly, submatrix D is called the bending

stiﬀness matrix, for it relates the curvature with the bending moments. For symmetric

laminates, there is an extension-bending decoupling, i.e. Bij = 0. For balanced laminates

extension and shear forces and strains are decoupled, A16 ≈A26 ≈0.

2.2. Structural Health Monitoring and Lamb Waves

Structural Health Monitoring (SHM) is deﬁned as the process of implementing a dam-

age detection strategy for aerospace, civil and mechanical engineering infrastructures [22].

Damage in laminate components like delaminations, cracks, or porosity, can be produced

during manufacturing or during service life, and grow damaging the mechanical properties

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Self-learninglocally-optimalhypertuningusingmaximumentropy,andcomparisonofmachinelearningapproachesforestimatingfatiguelifeincompositematerialsMiguelDaz-Lagoa;,IsmaelBen-Yeluna;y,LuisSaucedo-Moraa;b;c;*,MiguelAngelSanza,RicardoCalladoa,FranciscoJavierMontansa;daE.T.S.deIngenieraAeronauticayd...

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Self-learning locally-optimal hypertuning using maximum entropy and comparison of machine learning approaches for estimating fatigue life in composite materials

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