Publications

The small strain shear modulus (G0) is an important characteristic of geomaterials. Its laboratory measurements are very frequently based on bender element transducers installed in different geotechnical apparatuses. Despite their widespread use, there is room to improve the precision, reproducibility, and robustness of bender element measurements, which are currently hampered by an oversimplified interpretation model of the test. Aspects such as transducer directivity, apparatus-induced size effects, and boundary reflectivity are poorly understood and accounted for. This work presents a novel modified Rowe cell apparatus specifically designed to obtain experimental data relevant to those features. The modified Rowe cell uses transmitter and receiver bender elements on the same boundary of the sample, performing dynamic measurements in echo mode with variable initial specimen size. Additionally, the modified Rowe cell endorses the flexible installation of the receivers and enables the use of multiple receivers, with distinct orientations. Furthermore, the modified Rowe cell gives the option of using different boundary conditions at the reflecting end to test the dynamic properties of usual geotechnical testing apparatus interfaces such as (semi-) rigid enclosures and flexible membranes. A systematic reliability study of the bender element measurements on the modified Rowe cell is presented, along with some illustrative examples of the capabilities of the apparatus. 

Bridge authorities have been reticent to integrate structural health monitoring into their bridge management systems, as they do not have the financial and technical resources to collect long-term monitoring data from every bridge. As bridge authorities normally own huge amount of similar bridges, like the pedestrian ones, the ability to transfer knowledge from one or a small group of well-known bridges to help make more effective decisions in new bridges and environments has gained relevance. In that sense, transfer learning, a subfield of machine learning, offers a novel solution to periodically evaluate the structural condition of all pedestrian bridges using long-term monitoring data from one or more pedestrian bridges. In this paper, the applicability of unsupervised transfer learning is firstly shown on data from numerical models and then on data from two similar pedestrian prestressed concrete bridges. Two domain adaptation techniques are used for transfer learning, where a classifier has access to unlabeled training data (source domain) from a reference bridge (or a small set of reference bridges) and unlabeled monitoring test data (target domain) from another bridge, assuming that both domains are from similar but statistically different distributions. This type of mapping is expected to improve the classification accuracy for the target domain compared to a procedure that does not implement domain adaptation, as a result of reducing distributions mismatch between source and target domains. 

The effects of operational and environmental variability have been posed as one of the biggest challenges to transit structural health monitoring (SHM) from research to practice. To deal with that, machine learning algorithms have been proposed to learn from experience based on a reference data set. These machine learning algorithms work well if the operational and environmental conditions under which the bridge operates do not change over time. Meanwhile, climate change has been posed as one of the biggest concerns for the health of bridges. Although the uncertainty associated with the magnitude of the change is large, the fact that our climate is changing is unequivocal. Therefore, it is expected that climate change can be another source of environmental variability, especially the temperature. So, what happens if the mean temperature changes over time? Will it significantly affect the dynamics of bridges? Will the reference data set used for the training of algorithms become outdated? Are machine learning algorithms robust enough to deal with those changes? This is a pioneering work on the impact of climate change on the long-term damage detection in the context of bridge SHM. A classifier rooted in machine learning is trained using one-year data from the Z-24 Bridge in Switzerland and tested with current and future data. Three climate change scenarios are assumed, centered in three future periods, namely 2035, 2060, and 2085. This study concludes that climate change may be seen as another source of operational and environmental variability to be considered when using machine learning algorithms for long-term damage detection. 

Bridges are built to last more than 100 years, spanning many human generations. Throughout their lifetime, their service requirements may change, or they age and often suffer a material degradation process that can lead to the need of retrofitting. In bridge engineering, retrofitting refers to the strengthening of existing structures to make them more resistant and to increase the lifespan of bridges. Retrofitting normally increases the stiffness of bridge components, which can cause significant changes in the global modal properties. In the context of structural health monitoring, a classifier trained with datasets before retrofitting will most likely output many outliers after retrofitting, based on the premise that the new observations do not share the same underlying distribution. Therefore, how can long-term monitoring data from one bridge (labeled source domain) be reused to create a classifier that generalizes to the same bridge after retrofitting (unlabeled target domain)? This paper presents a novel approach based on transfer learning in the context of domain adaptation on datasets from two real bridges subjected to retrofit and under-monitoring programs. Based on the assumption that both bridges are undamaged before retrofitting, the results show that transfer learning can support the long-term damage detection process based on a classification using an outlier detection strategy. 

Classifiers based on machine learning algorithms trained through hybrid strategies have been proposed for structural health monitoring (SHM) of bridges. Hybrid strategies use numerical and monitoring data together to improve the learning process of the algorithms. The numerical models, such as finite-element (FE) models, are used for data augmentation based on the assumption of the existence of limited experimental data sets. However, a numerical model might fail in providing reliable data, as its parameters might not share the same underlying operating conditions observed in real situations. Meanwhile, the concept of transfer learning has evolved in SHM, in particular through domain adaptation techniques. The ability to adapt a classifier built on a well-known labeled data set to a new scenario with an unlabeled data set is an opportunity to transit bridge SHM from research to practice. Therefore, this paper proposes an unsupervised transfer learning approach for bridges with a domain adaptation technique, where classifiers are trained only with labeled data generated from FE models (source domain). Then, unlabeled monitoring data (target domain) are used to test the classification performance. As numerical and monitoring data are related to the same bridge, both domains are assumed to have similar statistical distributions, with slight differences caused by the uncertainties inherent to the FE models. The domain adaptation is performed using a transfer knowledge method called transfer component analysis, which transforms damage-sensitive features from the original space to a new one, called latent space, where the differences between feature distributions are reduced. This approach may increase the use of numerical modeling for long-term monitoring, as it overcomes some of the limitations imposed by the calibration process of FE models. The efficiency of this unsupervised approach is illustrated through the classification performance of classifiers built on source data with and without domain adaptation, and using the benchmark data sets from the Z-24 Bridge as the target data. 

The broad availability and low cost of smartphones have justified their use for structural health monitoring (SHM) of bridges. This paper presents a smartphone application called App4SHM, as a customized SHM process for damage detection. App4SHM interrogates the phone’s internal accelerometer to measure accelerations, estimates the natural frequencies, and compares them with a reference data set through a machine learning algorithm properly trained to detect damage in almost real time. The application is tested on data sets from a laboratory beam structure and two twin post-tensioned concrete bridges. The results show that App4SHM retrieves the natural frequencies with reliable precision and performs accurate damage detection, promising to be a low-cost solution for long-term SHM. It can also be used in the context of scheduled bridge inspections or to assess bridges’ condition after catastrophic events. 

In structural health monitoring of bridges, machine learning algorithms for damage detection are typically trained using an unsupervised learning strategy, with data gathered from monitoring systems, and assuming the structures are undamaged and functioning under normal operational conditions during a certain period of time. However, the scarcity of information regarding the structural response under seasonal environmental variations and less frequent operational conditions makes the distinction between these undamaged states and damaged ones very challenging and may cause damage detection algorithms to yield false indications. To overcome this limitation, hybrid approaches for the training of machine learning algorithms have recently been proposed. Rather than relying exclusively on monitoring data, hybrid approaches use finite element models of the structure to generate numerical data for less frequent undamaged scenarios. The numerical data are used for the training of machine learning algorithms together with the monitoring data. This paper addresses the reliability of numerical data for the training of machine learning algorithms by quantifying the damage detection performance of an algorithm trained with numerical data only. Monitoring data are used only for the initial calibration of the finite element model, which does not need to be exceedingly detailed, as the probabilistic variation of the uncertain parameters is considered. The damage detection performance is quantified both in terms of quality (number of ill-classified observations) and robustness to sub-optimal choices of the training data and algorithmic parameters. A general procedure for the generation of model-based data for the training of machine learning algorithms to detect damage is given and validated using the well-known Z-24 Bridge benchmark. 

Hybrid-Trefftz finite elements are well suited for modelling the response of materials under highly transient loading. Their approximation bases are built using functions that satisfy exactly the differential equations governing the problem. This option embeds relevant physical information into the approximation basis and removes the well-known sensitivity of the conventional finite elements to high solution gradients and short wavelength excitation. Despite such advantages, no public software using hybrid-Trefftz finite elements to model wave propagation through solid and porous media exists to date. This paper covers the formulation and implementation of hybrid-Trefftz finite elements for single-phase, biphasic and triphasic media, subjected to dynamic loads. The formulation is cast in a unified framework, valid for the three types of materials alike, and independent of the nature (harmonic, periodic or transient) of the applied load. Displacement, traction, elastic and absorbing boundary conditions are accommodated. The implementation is made in three novel, open-source and user-friendly computational modules which are freely distributed online.

FreeHyTE Direct Boundary Method Toolbox is a new computational framework for the solution of interior and exterior boundary value problems in two dimensions using three classes of direct methods: the Boundary Element Method, the Method of Fundamental Solutions and the Trefftz-Herrera Method. The toolbox, currently including solvers for Laplace and Helmholtz boundary value problems, is straightforward to use, featuring a simple graphical user interface and automatic mesh generators, and amenable to extension, as it provides modular computational procedures, directly applicable to other types of boundary elements and differential equations. The toolbox supports the definition of simply or multiply-connected domains, boundary elements of any order, complex wavenumbers, and Dirichlet, Neumann and Robin boundary conditions. FreeHyTE Direct Boundary Method Toolbox is freely distributed under the GNU General Public License and supported by manuals to quickly get new users started.

The application of the direct Trefftz method to the solution of Laplace equation defined on a 2D domain is frequently hindered by the numerical instability of the solving system, which may become ill-conditioned or even rank-deficient. Ill-conditioning is typically caused by a lack of domain scaling or by the oscillatory nature of the functions included in the weighting basis. Conversely, rank-deficiency may occur even for scaled domains and for low-order weighting bases. Its causes are related to the regularity properties of the weighting functions and to a lack of completeness of the weighting basis. The objective of this paper is to contribute to a better understanding of the mathematical grounds of rank-deficiency, and of its sensitivity to the definition of the referential and the (over-)determination of the basis. It shows that, while frequent, rank-deficiency can be avoided by slightly skewing the referential and by meshing the boundary such as to ensure that the basis is complete.

FreeHyTE is a public, open-source and user-friendly software for the solution of initial boundary value problems using hybrid-Trefftz finite elements. FreeHyTE is designed to be straightforward to use, even by analysts unacquainted to the Trefftz elements, and amenable to expansion by researchers willing to test their new ideas without having to code common procedures from scratch. To support users, FreeHyTE features intuitive graphical interfaces, automatic mesh generators and adaptive p-refinement procedures. To support developers, FreeHyTE approaches Trefftz elements through a unifying perspective, applicable to hyperbolic, parabolic and elliptic boundary value problems alike. It provides standardized procedures in all phases of the algorithm, including data input, construction and manipulation of the solving system, and post-processing of the results. Moreover, the modular structure of FreeHyTE enables the integration of existing procedures into new modules with minimal coding effort.

FreeHyTE’s distribution is free under the terms of the GNU General Public License and supported by theory, installation, user’s and developer’s manuals to quickly get new users and developers started.

We present a new model updating technique for the automatic calculation of the small strain shear modulus of geomaterials, based on bender element experiments. The technique searches for the shear modulus that maximizes the correlation between the output signal obtained from the experiment and the output signal acquired from its computational simulation. Instead of conventional extremum finding algorithms, a fixed point technique is used to update the shear modulus at each iteration. This option increases substantially the attraction basin of the absolute maximum correlation and improves the convergence of the algorithm.

Bender elements are shear wave transducers, widely used for the experimental identification of the small-strain shear moduli of geomaterials. They offer good coupling with the sample and controllable loading signal and frequency, but some of their design and signal interpretation features are still under investigation. The research of these features has been approached mainly on experimental grounds and, in most cases, the conclusions were not consensual. This study aims at laying the foundations for a coupled, numerical–experimental approach to the problem. It uses hybrid-Trefftz finite elements to model the bender element test and the output signal obtained in the laboratory to validate the model.

Two main reasons justify the choice of the hybrid-Trefftz finite elements. First, hybrid-Trefftz elements are considerably less wavelength-sensitive than conforming displacement elements. This feature endorses the use of the same (coarse) mesh for modelling waves of very different types and frequencies, thus eliminating the need for time-consuming mesh refinements. Second, hybrid-Trefftz elements use physically meaningful approximation bases. This feature enables the numerical filtration of the spurious compression waves propagating laterally from the emitter and of their reflections from the envelope of the sample. Hybrid-Trefftz finite elements are shown to recover adequately the output signal obtained experimentally, for pulse excitations of various frequencies and two sets of boundary conditions. The decomposition of the output signal into its shear and compression components endorses the clear identification of the shear wave arrival and the amount of compression wave pollution.

A new indirect approach to the problem of approximating the particular solution of non-homogeneous hyperbolic boundary value problems is presented. Unlike the dual reciprocity method, which constructs approximate particular solutions using radial basis functions, polynomials or trigonometric functions, the method reported here uses the homogeneous solutions of the problem obtained by discarding all time-derivative terms from the governing equation. Nevertheless, what typifies the present approach from a conceptual standpoint is the option of not using these trial functions exclusively for the approximation of the particular solution but to fully integrate them with the (Trefftz-compliant) homogeneous solution basis. The particular solution trial basis is capable of significantly improving the Trefftz solution even when the original equation is genuinely homogeneous, an advantage that is lost if the basis is used exclusively for the recovery of the source terms. Similarly, a sufficiently refined Trefftz-compliant basis is able to compensate for possible weaknesses of the particular solution approximation. The method is implemented using the displacement model of the hybrid-Trefftz finite element method. The functions used in the particular solution basis reduce most terms of the matrix of coefficients to boundary integral expressions and preserve the Hermitian, sparse and localized structure of the solving system that typifies hybrid-Trefftz formulations. Even when domain integrals are present, they are generally easy to handle, because the integrand presents no singularity.

The stress model of the hybrid-Trefftz finite element is formulated for the analysis of elastodynamic problems defined on unsaturated porous media. The supporting mathematical model is the theory of mixtures with interfaces and considers the full coupling between the solid, fluid and gas phases, including the effect of seepage acceleration. Hybrid-Trefftz stress elements use the free-field regular solutions of the homogeneous Navier (or Beltrami) equation to construct the approximation of the generalized stresses in the domain of the element. The influence of non-homogeneous terms in the Navier equation is modelled using solutions of the corresponding static problem. The resulting elements are highly convergent under p-refinement and robust to both low and high excitation frequencies, as the trial functions embody relevant physical information on the modelled phenomenon.

This paper reports on the formulation and implementation of the displacement and stress models of the hybrid-Trefftz finite elements for elastostatic problems defined on saturated porous media. The supporting mathematical model is the (u–w) formulation of the Biot theory of porous media. The hybrid-Trefftz models are derived from the corresponding (pure) hybrid models by selecting the domain trial functions from the free-field solutions of the governing Navier equation. The resulting elements are highly robust and convergent, as they embody the physical characteristics of the modelled problem. Moreover, all coefficients present in the solving system are defined by boundary integral expressions.

The equations that govern the dynamic response of saturated porous media are first discretized in time to define the boundary value problem that supports the formulation of the hybrid-Trefftz stress element. The (total) stress and pore pressure fields are directly approximated under the condition of locally satisfying the domain conditions of the problem. The solid displacement and the outward normal component of the seepage displacement are approximated independently on the boundary of the element. Unbounded domains are modelled using either unbounded elements that locally satisfy the Sommerfeld condition or absorbing boundary elements that enforce that condition in weak form. As the finite element equations are derived from first-principles, the associated energy statements are recovered and the sufficient conditions for the existence and uniqueness of the solutions are stated. The performance of the element is illustrated with the time domain response of a biphasic unbounded domain to show the quality of the modelling that can be attained for the stress, pressure, displacement and seepage fields using a high-order, wavelet-based time integration procedure.

The hybrid–mixed stress element for transient analysis of compressible and incompressible biphasic media is based on the approximation of the stress and pressure fields in the domain of the element and of the displacements in the solid and fluid phases independently in the domain and on the boundary. The hybrid and hybrid-Trefftz variants are derived constraining the domain approximation to satisfy locally the equilibrium condition and all domain conditions, respectively. As it is typical of stress elements, in these alternative formulations the inter-element and boundary continuity conditions are enforced on the surface forces acting on the solid and fluid phases of the medium. The hybrid-Trefftz stress element is applied to the transient analysis of two-dimensional and axisymmetric biphasic media. The performance of the element is illustrated in terms of sensitivity to distortion and to wavelength, full incompressibility and spectral and transient modelling of both bounded and unbounded domains.

A teoria da elasticidade é uma disciplina basilar dos cursos de licenciatura e/ou mestrado da maior parte das áreas da engenharia, como, por exemplo, aeroespacial, ambiente, biomédica, civil, física, de materiais, ou mecânica. A sua vasta aplicabilidade vem do facto de que, em muitas situações práticas, nos interessa compreender o comportamento de porções macroscópicas de matéria quando sujeitas a esforços ou variações de temperatura.

Este livro apresenta o essencial da teoria da elasticidade através da obtenção das equações diferenciais que governam a resposta dos materiais às solicitações externas a partir das leis físicas fundamentais, da descrição das respetivas condições de fronteira, e da exploração dos métodos matemáticos disponíveis para a sua resolução. Pressupõe conhecimentos de álgebra linear, cálculo infinitesimal e mecânica newtoniana, mas não de cálculo tensorial ou de mecânica analítica, de modo que, sem prejuízo do rigor, seja acessível a um vasto leque de leitores com formação matemática porventura menos extensa.

Os autores não se limitam a deduzir as equações básicas do problema da elasticidade linear, mostrando como as mesmas podem ser resolvidas, e quais são os benefícios e as limitações dos métodos de resolução analíticos, semi-analíticos e numéricos. Esta estratégia conduz naturalmente ao Método dos Elementos Finitos, de grande utilidade prática, e raras vezes abordado em obras desta natureza publicadas em português.

A obra contém problemas resolvidos ("Exemplos") para facilitar a assimilação da matéria e ilustrar as suas aplicações. Além disso, no final de cada capítulo é proposto um conjunto de exercícios, dos quais se fornecem as soluções (e, no caso de exercícios mais elaborados, um esboço da resolução) no final do livro. 

Hybrid-Trefftz finite elements combine favourable features of the Finite and Boundary Element methods. The domain of the problem is divided into finite elements, where the unknown quantities are approximated using bases that satisfy exactly the homogeneous form of the governing differential equation. The enforcement of the governing equations leads to sparse and Hermitian solving systems (as typical to Finite Element Method), with coefficients defined by boundary integrals (as typical to Boundary Element Method). Moreover, the physical information contained in the approximation bases renders hybrid-Trefftz elements insensitive to gross mesh distortion, nearly-incompressible materials, high frequency oscillations and large solution gradients. A unified formulation of hybrid-Trefftz finite elements for dynamic problems is presented in this chapter. The formulation reduces all types of dynamic problems to formally identical series of spectral equations, regardless of their nature (parabolic or hyperbolic) and method of time discretization (Fourier series, time-stepping procedures, or weighted residual methods). For non-homogeneous problems, two novel methods for approximating the particular solution are presented. The unified formulation supports the implementation of hybrid-Trefftz finite elements for a wide range of physical applications in the same computational framework.

Improved and more continuous condition assessment of bridges has been demanded by our society to better face the challenges presented by aging civil infrastructure. Indeed, the recent collapses of the Hintze Ribeiro Bridge that killed 59 people, in Portugal, and the I-35W Bridge in the United States, that killed 13 people, pointed out the need for new and more reliable tools to prevent such catastrophic events. Besides those events, the financial implications and potential impact through optimal bridge management are vast. For instance, facing an ageing infrastructure, the United Kingdom Government’s 2010 Infrastructure Plan signaled the need for enormous investments in infrastructures, equivalent to £200 billion over the next five years. On the other hand, the American Society of Civil Engineers reports the cost of eliminating all existing US bridge deficiencies at $850 billion. These values clearly show that planned bridge maintenance can lead to considerable savings. In the last two decades, bridge condition assessment techniques have been developed independently based on two complementary approaches: Structural Health Monitoring (SHM) and Bridge Management Systems (BMSs). The SHM refers to the process of implementing monitoring systems to measure in real time the structural responses, in order to detect anomalies and/or damage at early stages. On the other hand, BMS is a visual inspection-based decision-support tool developed to analyze engineering and economic factors and to assist the authorities in determining how and when to make decisions regarding maintenance, repair, and rehabilitation of structures. While the BMS has already been accepted by the bridge owners around the world, even though with inherent limitations posed by the visual inspections, the SHM is becoming increasingly appealing due to its potential ability to detect damage at early stages, with the consequent life-safety and economical benefits. Recent research suggests that, in an effort to create more robust bridge management, the SHM should be integrated into the BMS in a systematic way. Nowadays, there is a generalized consensus about this integration, but few real applications have been accomplished, mainly because of the lack of interaction between all the participants involved in the bridge management field. 

Hybrid-Trefftz finite elements are well suited for modeling the response of materials under highly transient loading. Their approximation bases are built using functions that satisfy exactly the differential equations governing the problem. This option embeds relevant physical information into the approximation basis and removes the well-known sensitivity of the conventional finite elements to high solution gradients and short wavelength excitations. Despite such advantages, no public software using hybrid-Trefftz finite elements to model wave propagation through solid and porous media exists to date. This paper covers the formulation and implementation of hybrid-Trefftz finite elements for single-phase, biphasic and triphasic media, subjected to dynamic loads. The formulation is cast in a unified framework, valid for the three types of materials alike, and independent of the nature (harmonic, periodic or transient) of the applied load. Displacement, traction, elastic and absorbing boundary conditions are accommodated. The implementation is made in three novel, open-source and user-friendly computational modules which are freely distributed online.

The hybrid-Trefftz displacement element formulation for poroelastodyamics in three-dimensional saturated media is presented in this article. The governing equations are integrated in time using a wavelet decomposition which expands the time-dependent problem into several spectral problems. Displacements are discretized in each finite element and tractions are approximated on each essential element boundary (displacement model). The displacement approximation functions satisfy the governing equations of the media (Trefftz constraint) and are not linked to the nodes. Trefftz bases are derived solving the governing equations and are composed of four types of modes: two compression waves and two shear waves. The weak enforcement of the Navier equation and boundary conditions yields a solving system where the unknown are the weights of the approximation bases. Numerical tests have been carried out to assess the convergence of the models. The results of the simulations are satisfactory: the numerical tests converge to known analytical solutions and the simulation of a propagation of a vertical shock wave is consistent to the one obtained with a 2D plane simulation under the same conditions. The results of a bender element test simulation in three dimensions are also presented.