Publications

Here you can find some of my publications with links to the source pages.

Pre-print versions of some of these publications can be found on my Research Gate page.

If you prefer, I can provide you with the published versions upon request.

Papers in international journals

Moldovan ID, Gomes Correia A – Optimisation of receiver’s location in bender element experiments using computational wave filtration, Soil Dynamics and Earthquake Engineering, 143, 106591, 2021

Bender elements are shear wave transducers, used for the computation of small strain shear moduli of geomaterials. However, the distortion of the output signal caused by residual compression waves may lead to important errors in the shear modulus estimates. We present a novel procedure for the optimisation of the location of the receiver bender element, to avoid regions where the distortion of the output signal is strong, without compromising the strength of the shear wave signal. The procedure is based on a computational technique naturally able to distinguish between the compression and shear waves present in the seismic response of geomaterials. This property enables the construction of compression and shear amplitude maps, that can be used to decide the best location for the receiver prior to running the experiment. The experimental validation of the procedure confirms that it leads to output signals which are easier to interpret than those obtained with the transmitter and receiver in the conventional, tip-to-tip configuration.

Borkowski M, Moldovan ID – Direct boundary method toolbox for some elliptic problems in FreeHyTE framework, Engineering Analysis with Boundary Elements, 125, 208-217, 2021

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.

Moldovan ID, Coutinho A, Cismasiu I – Hybrid-Trefftz finite elements for non-homogeneous parabolic problems using a novel dual reciprocity variant, Engineering Analysis with Boundary Elements, 106, 228-242, 2019

A new hybrid-Trefftz finite element for the solution of transient, non-homogeneous parabolic problems is formulated. The governing equations are first discretized in time and reduced to a series of non-homogeneous elliptic problems in space. The complementary and particular solutions of each elliptic problem are approximated independently. The complementary solution is expanded in Trefftz bases, designed to satisfy exactly the homogeneous form of the problem. Trefftz bases are regular, and defined independently for each finite element, using arbitrary orders. A novel dual reciprocity method is used for the approximation of the particular solution, to avoid domain integration. The same, regular basis is used for the expansions of the source function and particular solution, avoiding the cumbersome expressions of the latter that typify conventional dual reciprocity techniques. Moreover, the bases of the complementary and particular solutions are defined by the same expressions, with different wave numbers. The finite element formulation is obtained by enforcing weakly the domain equations and boundary conditions. To enhance the reproducibility of this work, the formulation is implemented in the computational platform FreeHyTE, where it takes advantage of the pre-programmed numerical procedures and graphical user interfaces. The resulting software is open-source, user-friendly and freely distributed to the scientific community through the FreeHyTE web page.

Borkowski M, Moldovan ID – On rank-deficiency in direct Trefftz method for 2D Laplace problems, Engineering Analysis with Boundary Elements, 106, 102-115, 2019

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.

Figueiredo EJF, Moldovan ID, Santos A, Campos P, Costa JCWA - Finite element-based machine learning approach to detect damage in bridges under operational and environmental variations, Journal of Bridge Engineering, 24(7), 04019061, 2019

In the last decades, the long-term structural health monitoring of civil structures has been mainly performed using two approaches: model based and data based. The former approach tries to identify damage by relating the monitoring data to the prediction of numerical (e.g., finite-element) models of the structure. The latter approach is data driven, where measured data from a given state condition are compared to the baseline or reference condition. A challenge in both approaches is to make the distinction between the changes of the structural response caused by damage and environmental or operational variability. This issue was tackled here using a hybrid technique that integrates model- and data-based approaches into structural health monitoring. Data recorded in situ under normal conditions were combined with data obtained from finite-element simulations of more extreme environmental and operational scenarios and input into the training process of machine-learning algorithms for damage detection. The addition of simulated data enabled a sharper classification of damage by avoiding false positives induced by wide environmental and operational variability. The procedure was applied to the Z-24 Bridge, for which 1 year of continuous monitoring data were available.

Moldovan ID, Cismasiu I – FreeHyTE: a hybrid-Trefftz finite element platform, Advances in Engineering Software, 121, 98-119, 2018

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.

Cismasiu C, Pinho Ramos A, Moldovan ID, Ferreira DF, Filho JB - Applied element method simulation of experimental failure modes in RC shear walls, Computers and Concrete, 19(4), 365-374, 2017

With the continuous evolution of the numerical methods and the availability of advanced constitutive models, it became a common practice to use complex physical and geometrical nonlinear numerical analyses to estimate the structural behavior of reinforced concrete elements. Such simulations may yield the complete time history of the structural behavior, from the first moment the load is applied until the total collapse of the structure. However, the evolution of the cracking pattern in geometrical discontinuous zones of reinforced concrete elements and the associated failure modes are relatively complex phenomena and their numerical simulation is considerably challenging. The objective of the present paper is to assess the applicability of the Applied Element Method in simulating the development of distinct failure modes in reinforced concrete walls subjected to monotonic loading obtained in experimental tests. A pushover test was simulated numerically on three distinct RC shear walls, all presenting an opening that guarantee a geometrical discontinuity zone and, consequently, a relatively complex cracking pattern. The presence of different reinforcement solutions in each wall enables the assessment of the reliability of the computational model for distinct failure modes. Comparison with available experimental tests allows concluding on the advantages and the limitations of the Applied Element Method when used to estimate the behavior of reinforced concrete elements subjected to monotonic loading.

Moldovan ID, Gomes Correia A – Fixed point automatic interpretation of bender-based G0 measurements, Computers and Geotechnics, 89, 128-142, 2017

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.

Moldovan ID - A new approach to non-homogeneous hyperbolic boundary value problems using hybrid-Trefftz stress finite elements, Engineering Analysis with Boundary Elements, 69, 57-71, 2016

A new approach to the solution of non-homogeneous hyperbolic boundary value problems is casted here using the hybrid-Trefftz stress/flux elements. Similarly to the Dual Reciprocity Method, the technique adopted in this paper uses a Trefftz-compliant set of functions to approximate the complementary solution of the problem and an additional trial basis to approximate its particular solution. However, the particular and complementary solutions are obtained here in a single step, and not sequentially, as typical of the Dual Reciprocity Method. The trial functions used for both particular and complementary solutions are merged together in the same basis and offered full flexibility to combine so as to recover the enforced equations in the best possible way. This option enables Trefftz-compliant functions to compensate for deficiencies of the particular solution basis, meaning that accurate total solutions can be obtained with relatively poor particular solution approximations. The formulation preserves the Hermitian, sparse and localized structure that typifies the matrix of coefficients of hybrid-Trefftz finite elements and avoids the drawbacks of the collocation procedures that arise in the Dual Reciprocity Method. Moreover, all terms of the matrix of coefficients are reduced to boundary integral expressions provided the particular solution trial functions satisfy the static problem obtained after discarding both non-homogeneous and time derivative terms from the governing equation.

Moldovan ID, Gomes Correia A, Pereira C - Bender-based G0 measurements: a coupled numerical-experimental approach, Computers and Geotechnics, 73, 24-36, 2016

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.

Moldovan ID, Radu L - Trefftz-based Dual Reciprocity Method for hyperbolic boundary value problems, International Journal for Numerical Methods in Engineering, 106(13), 1043-1070, 2016

A new dual reciprocity-type approach to approximating the solution of non-homogeneous hyperbolic boundary value problems is presented in this paper. Typical variants of the dual reciprocity method obtain approximate particular solutions of boundary value problems in two steps. In the first step, the source function is approximated, typically using radial basis, trigonometric or polynomial functions. In the second step, the particular solution is obtained by analytically solving the non-homogeneous equation having the approximation of the source function as the non-homogeneous term. However, the particular solution trial functions obtained in this way typically have complicated expressions and, in the case of hyperbolic problems, points of singularity. Conversely, the method presented here uses the same trial functions for both source function and particular solution approximations. These functions have simple expressions and need not be singular, unless a singular particular solution is physically justified. The approximation is shown to be highly convergent and robust to mesh distortion.

Any boundary method can be used to approximate the complementary solution of the boundary value problem, once its particular solution is known. The option here is to use hybrid-Trefftz finite elements for this purpose. This option secures a domain integral-free formulation and endorses the use of super-sized finite elements as the (hierarchical) Trefftz bases contain relevant physical information on the modeled problem.

Moldovan ID - A new particular solution strategy for hyperbolic boundary value problems using hybrid-Trefftz displacement elements, International Journal for Numerical Methods in Engineering, 102, 1293-1315, 2015

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.

Arruda MRT, Moldovan ID - On a mixed time integration procedure for non-linear structural dynamics, Engineering Computations, 32(2), 329-369, 2015

The purpose of this paper is to report the implementation of an alternative time integration procedure for the dynamic non-linear analysis of structures. The time integration algorithm discussed in this work corresponds to a spectral decomposition technique implemented in the time domain. As in the case of the modal decomposition in space, the numerical efficiency of the resulting integration scheme depends on the possibility of uncoupling the equations of motion. This is achieved by solving an eigenvalue problem in the time domain that only depends on the approximation basis being implemented. Complete sets of orthogonal Legendre polynomials are used to define the time approximation basis required by the model. A classical example with known analytical solution is presented to validate the model, in linear and non-linear analysis. The efficiency of the numerical technique is assessed. Comparisons are made with the classical Newmark method applied to the solution of both linear and non-linear dynamics. The mixed time integration technique presents some interesting features making very attractive its application to the analysis of non-linear dynamic systems. It corresponds in essence to a modal decomposition technique implemented in the time domain. As in the case of the modal decomposition in space, the numerical efficiency of the resulting integration scheme depends on the possibility of uncoupling the equations of motion. One of the main advantages of this technique is the possibility of considering relatively large time step increments which enhances the computational efficiency of the numerical procedure. Due to its characteristics, this method is well suited to parallel processing, one of the features that have to be conveniently explored in the near future.

Moldovan ID, Cao DT, Freitas JAT - Elastic wave propagation in unsaturated porous media using hybrid-Trefftz stress element, International Journal for Numerical Methods in Engineering, 97(1), 32-67, 2014

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.

Moldovan ID, Cao DT, Freitas JAT - Hybrid-Trefftz displacement finite elements for elastic unsaturated soils, International Journal of Computational Methods, 11(2), 1342005, Special Issue on Computational Geomechanics, 2014

The displacement model of the hybrid-Trefftz finite element is formulated for elastodynamic problems defined on unsaturated soils. The mathematical formulation is based on the theory of mixtures with interfaces. The model considers the full coupling between the solid, fluid and gas phases, including the effects of relative (seepage) accelerations. The hyperbolic problem is integrated in time using a step-by-step implicit scheme that transforms it into a series of elliptic problems in space. The free-field solutions of these problems are derived in cylindrical coordinates and used to construct the domain approximation of the hybrid-Trefftz displacement element. This builds relevant physical information in the approximation basis, increasing the convergence of the elements under p-refinement and their robustness to wide variations of the frequency of the propagating wave.

Moldovan ID, Cao DT, Freitas JAT - Hybrid-Trefftz elements for biphasic elastostatics, Finite Elements in Analysis and Design, 66, 68-82, 2013

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.

Moldovan ID, Freitas JAT - Hybrid-Trefftz displacement and stress elements for bounded poroelasticity problems, Computers & Geotechnics, 42, 129-144, 2012

The hybrid-Trefftz displacement and stress elements for poroelasticity are applied here to the spectral analysis of bounded saturated porous media. The displacement model is derived from the direct approximation of the displacement and fluid seepage fields in the domain of the element and of the tractions and pore pressures in the solid and fluid phases, respectively, on the Dirichlet boundary of the element. Conversely, the stress model is obtained by the direct approximation of the total stress and pore pressure fields in the domain, while independent approximations of the displacement and fluid seepage fields are enacted on the Neumann boundaries. As typical of the Trefftz methods, for both models, the domain approximation bases are constrained to satisfy locally all field equations.

The central objective of the paper is to present a consistent set of tests designed to evaluate the ability of the proposed models to accurately predict the response of saturated porous media subjected to harmonic excitation and to assess the corresponding convergence patterns. Emphasis is placed on some key advantages of the presented formulation, namely the insensitivity of the results to gross mesh distortion, to near-incompressibility of the medium and to the wavelength content of the propagating wave, which enables the use of frequency-independent finite element meshes.

Freitas JAT, Moldovan ID - Hybrid-Trefftz stress element for bounded and unbounded poroelastic media, International Journal for Numerical Methods in Engineering, 85(10), 1280-1305, 2011

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.

Freitas JAT, Moldovan ID, Cismasiu C - Hybrid-Trefftz displacement element for poroelastic media, Computational Mechanics, 48, 659-673, 2011

The elastodynamic response of saturated poroelastic media is modelled approximating independently the solid and seepage displacements in the domain and the force and pressure components on the boundary of the element. The domain and boundary approximation bases are used to enforce on average the dynamic equilibrium and the displacement continuity conditions, respectively. The resulting solving system is Hermitian, except for the damping term, and its coefficients are defined by boundary integral expressions as a Trefftz basis is used to set up the domain approximation. This basis is taken from the solution set of the governing differential equation and models the free-field elastodynamic response of the medium. This option justifies the relatively high levels of performance that are illustrated with the time domain analysis of unbounded domains.

Freitas JAT, Moldovan ID, Toma M - Mixed and hybrid stress elements for biphasic media, Computers & Structures, 88(23), 1286–1299, 2010

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.

Moldovan ID, Freitas JAT - Hybrid-Trefftz stress and displacement elements for dynamic analysis of bounded and unbounded saturated porous media (invited article), CAMES, 15, 289–303, 2008

The displacement and stress models of the hybrid-Trefftz finite element formulation are applied to the dynamic analysis of two-dimensional bounded and unbounded saturated porous media problems. The formulation develops from the classical separation of variables in time and space. A finite element approach is used for the discretization in time of the governing differential equations. It leads to a series of uncoupled problems in the space dimension, each of which is subsequently discretized using either the displacement or the stress model of the hybrid-Trefftz finite element formulation. As typical of the Trefftz methods, the domain approximation bases are constrained to satisfy locally all domain equations. An absorbing boundary element is adopted in the extension to the analysis of unbounded media. The paper closes with the illustration of the application of the alternative hybrid-Trefftz stress and displacement elements to the solution of bounded and unbounded consolidation problems.

Books and theses

Moldovan ID, Cismasiu I, Teixeira de Freitas JA - Unified Hybrid-Trefftz Finite Element Formulation for Dynamic Problems. In Alves C, Karageorghis A, Leitão V, Valtchev S (eds.) “Advances in Trefftz Methods and Their Applications”. Springer, 2020. ISBN: 978-3-030-52804-1

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.

Figueiredo EJF, Moldovan ID, Marques MJMB (eds.) - Condition Assessment of Bridges: Past, Present and Future. Católica Editora, Lisbon, 2013. ISBN: 978-972-54-0401-0

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.

Moldovan ID – Hybrid-Trefftz Finite Elements for Elastodynamic Analysis of Saturated Porous Media, PhD Thesis, Technical University of Lisbon, (2008)

The displacement and stress models of the hybrid-Trefftz finite element formulation are applied to the dynamic analysis of two-dimensional bounded and unbounded saturated porous media problems. The formulation develops from the classical separation of variables in time and space. Periodic problems are discretized in time using Fourier analysis. Conversely, a finite element approach is used for the discretization in time of transient problems. Either strategy leads to a series of uncoupled problems in the space dimension, which are subsequently discretized using either the displacement or the stress model of the hybrid-Trefftz finite element formulation. As is typical of Trefftz methods, for both models, the approximation bases are constrained to satisfy locally all domain equations. The displacement model is based on the direct approximation of the solid displacement and fluid seepage fields in the domain of the element and of the forces and pore pressures on the solid and fluid phases, respectively, on the boundary of the element. Conversely, to derive the stress model, the stress and pore pressure fields are approximated in the domain, while independent approximations of the displacement and fluid seepage fields are assumed on the boundary of the element.

Two alternative approaches are used to extend the proposed formulations to semi-infinite (half-space) media, namely a finite element with absorbing boundary and an infinite element. For the displacement (stress) model, the tractions (displacements) are approximated independently on the absorbing boundary and used to enforce on average the asymptotic approximation of the Sommerfeld radiation condition. The domain approximations used in the derivation of infinite elements satisfy explicitly the Sommerfeld radiation condition, thus eliminating the uncertainties regarding possible spurious reflections in the vicinity of the absorbing boundary.

Invited Talks

Harvesting the numbers. (Trefftz) finite elements and their applications, Engineering School, University of Bristol, 2020

Structural Health Monitoring: State of the Art, Research and Transition to Industry, Technical University of Cluj-Napoca, 2018

Trefftz finite elements and their applications, Rzeszów University of Technology, 2018

The meaning in the numbers, Technical University of Cluj-Napoca, 2017

Merging of model updating and machine learning algorithms for Structural Health Monitoring, Catholic University of Leuven, 2017

Trefftz Dynamic Analysis of Multi-Phase Soils, Engineering School, Minho University, 2014

Hybrid-Trefftz Dynamic Analysis of Biphasic Media, University of Edinburgh, 2011

Soils and cartilages through the finite element method perspective, Faculty of Engineering, Catholic University of Portugal, 2008

Conference papers

Gomes Correia A, Moldovan ID – Optimisation of receiver's location in bender element tests using advanced computational techniques, Invited lecture, Workshop on Transportation Geotechnics GDRI-CSU (Changsa, Hunan, China, 2021)


Moldovan ID, Gomes Correia A – Coupling experimental and numerical techniques for improving the reliability of bender element experiments, Proceedings of the 17th European Conference on Soil Mechanics and Geotechnical Engineering, (Reykjavik, Iceland, 2019)


Moldovan ID, Cismasiu I, Freitas JAT – A unified hybrid-Trefftz formulation and its applications, The Joint International Conference on Trefftz Method IX and Method of Fundamental Solutions V, (Lisbon, Portugal, 2019)


Coutinho A, Moldovan ID, Cismasiu I – Hybrid-Trefftz finite elements for transient parabolic problems, The Joint International Conference on Trefftz Method IX and Method of Fundamental Solutions V, (Lisbon, Portugal, 2019)


Bud MA, Nedelcu M, Radu L, Moldovan ID, Figueiredo EJF – On the reliability of finite element models for training machine learning algorithms for damage detection in bridges, Proceedings of the 12th International Workshop on Structural Health Monitoring, (Stanford, the USA, 2019)


Moldovan ID, Gomes Correia A – Automatic interpretation of G0 measurements using bender elements, Proceedings of the 3rd International Conference on Information Technology in Geo-Engineering, (Guimarães, Portugal, 2019)


Souza L, Lopes RAC, Figueiredo EJ, Moldovan ID, Souza JAM, Rabelo JJ, Sousa RFM, Prazeres PGC, Costa JC – Ensaios dinâmicos e modelação da ponte sobre o Rio Itacaiúnas, Proceedings of Betão Estrutural - BE2018, (Lisbon, Portugal, 2018)


Santos A, Figueiredo EJ, Campos P, Moldovan ID, Costa JC - A generalized approach to integrate machine learning, finite element modeling and monitoring data for bridges, Proceedings of the 11th International Workshop on Structural Health Monitoring, (Stanford, the USA, 2017)


Moldovan ID, Gomes Correia A - A coupled numerical–experimental approach for bender-based G0 measurements in geomaterials, Proceedings of the 19th International Conference on Soil Mechanics and Geotechnical Engineering, (Seoul, South Korea, 2017)


Moldovan ID, Cismasiu I - FreeHyTE: a hybrid-Trefftz finite element platform, The Joint International Conference on Trefftz Method VIII and Method of Fundamental Solutions IV, (Poznan, Poland, 2017)


Moldovan ID, Radu L - Trefftz-based Dual Reciprocity Method for non-homogeneous hyperbolic boundary value problems, VII European Congress on Computational Methods in Applied Sciences and Engineering, (Crete, Greece, 2016)


Moldovan ID, Freitas JAT, Cao DT - Hybrid-Trefftz Finite Elements for Multiphase Soils, 6th International Conference on Computational Methods for Coupled Problems in Science and Engineering, (Venice, 2015)


Moldovan ID, Freitas JAT - A new particular solution strategy for hyperbolic problems using hybrid-Trefftz finite elements, 11th World Congress on Computational Mechanics, (Barcelona, 2014)


Moldovan ID, Cao DT, Freitas JAT - Dynamic analysis of unsaturated porous media using hybrid-Trefftz finite elements, 11th International Conference on Vibration Problems, (Lisbon, 2013)


Rodrigues LFG, Cismasiu I, Moldovan ID - A p-adaptive algorithm for hybrid-Treffz stress element for elastodynamic analysis, 11th International Conference on Vibration Problems, (Lisbon, 2013)


Moldovan ID, Cao DT, Freitas JAT – Hybrid-Trefftz displacement and stress elements for biphasic elastostatics, Computer Methods in Mechanics, (Warsaw, 2011)


Moldovan ID, Cao DT, Freitas JAT – Hybrid-Trefftz stress element for biphasic elastostatics, Congresso de Métodos Numéricos em Engenharia, (Coimbra, 2011)


Moldovan ID, Freitas JAT, Cao DT - Sensitivity Assessment of Hybrid-Trefftz Stress and Displacement Elements for Poroelasticity, European Conference on Computational Mechanics, (Paris, 2010)


Moldovan ID, Freitas JAT - Hybrid-Trefftz stress and displacement elements for dynamic analysis of bounded and unbounded saturated porous media, LSAME 2008 Trefftz Conference, (Leuven, 2008)


Freitas JAT, Moldovan ID, Toma M – Mixed and hybrid stress elements for biphasic media, ACME07 (invited lecture), (Glasgow, 2007)


Moldovan ID, Freitas JAT - Hybrid-Trefftz Finite element models for bounded and unbounded elastodynamic problems, Third European Conference on Computational Mechanics, (Lisbon, 2006)


Freitas JAT, Moldovan ID, Toma M - Trefftz spectral analysis of biphasic media, 6th World Congress on Computational Mechanics (invited lecture), (Beijing, 2004)


Pop I, Moldovan ID – Tuned Liquid Dampers for Mitigating the Response of Structures Subjected to Dynamic Excitations – a state-of-the-art, Constructions 50 Conference, (Cluj-Napoca, 2003)


Souza LSH, Lopes RAC, Figueiredo EJF, Moldovan ID, Prazeres PGC, Costa JCWA, Souza JAM, Rabelo JJ, Sousa RFM – Análise dinâmica da ponte sobre o rio Itacaiúnas, 60º Congresso Brasileiro do Concreto, 2018 (Foz do Iguaçú, Brasil, in portuguese)


Braz Fernandes FM, Fava Gaspar CS, Mahesh KK, Moldovan ID - Caracterização estrutural, térmica e mecânica de ligas Ni-Ti com memória de forma para aplicação em Engenharia Civil, REPAR 2010 (LNEC, Lisboa, 2010, in portuguese)

Moldovan ID, Climent N, Bendea ED, Cismasiu I, Gomes Correia A – A hybrid-Trefftz finite element platform for solid and porous elastodynamics, Engineering Analysis with Boundary Elements, 124, 155-173, 2021

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.