Projects | Andrés Prieto

Numerical characterization of coastal seabed

Mon, 01 Jan 0001 00:00:00 +0000

Underwater hydroacoustic measurements are widely used in sonar applications not only for detecting target objects (such as vessels or fisheries) but also to characterize the nature of the seabed and quantify the presence of algae or bivalves. However, the coastal environment usually presents challenges due to varying sediment types and hydrodynamic conditions, which can affect the accuracy of acoustic measurements (see Figure 1). Poroelastic media has been used extensively for the mathematical modelling of seabed sediments and, more recently, including specific models for granular sediments. Most of these models are based on the Biot-Stoll theory, which incorporates various parameters that affect the acoustic properties of the seabed, such as porosity, permeability, and their elastic material coefficients, which encompass the vibro-acoustic behaviour of the seabed sediment (see Figure 1).

Figure 1: Coastal environment with the presence of bivalve shells in a sandy surface.

Under the time-harmonic assumption and since the transducer is insonifying the water column in a linear regime, the underwater coupled acoustic problem can be modelled using a linear model. This time-harmonic model involves a bilayer unbounded configuration consisting of seawater and granular seabed sediment, where the bivalves are buried. To address this complex scenario, a displacement-based Finite Element Method discretized by a Raviart-Thomas discretization has been considered. Since the original physical domain is unbounded, the computational domain must be truncated within the finite element framework. With this purpose, the Perfectly Matched Layer technique has been utilized to compute the scattering time-harmonic displacement field generated from a set of buried bivalves (see a more detailed description in [1]).

Figure 2: Frequency response function associated to different bivalve scenarios.

Those bivalves are assumed localized in a narrow spatial domain, which reduces the computational cost of the numerical simulations. Additionally, assuming that the coupling interface between the fluid and granular media is planar, the scattering field in the seawater has been evaluated at the far field using a Green's representation formula. This approach utilizes the explicit evaluation of the fundamental solution of the Helmholtz equation in bilayer media and it allows us to compute efficiently the frequency-response function (FRF) associated with the observed intensity field values measured at the echosounder location. Thanks to the proposed approach, a variety of scenarios can be simulated in silico and evaluate numerically their associated FRF to characterize numerically the acoustic signature of a specific configuration of bivalves buried in the seabed sediment (see Figure 2).

This research line has been supported by the NumSeaHy project TED2021-131660B-I00, which is supported by MICIU/AEI/10.13039/501100011033 and the European Union Next Generation EU/PRTR. This project is also an endorsed UNESCO project (no. 42.4) of the 2021-2031 United Nations Decade of Ocean Science for Sustainable Development.

References

[1] Pablo Rubial Yáñez
Vibroacoustic Quantification of a Coastal Seabed
Master thesis in Industrial Mathematics (2024). Advisor: A. Prieto.

Fluid-structure vibroacoustic problems

Mon, 01 Jan 0001 00:00:00 +0000

The present project focuses on a three-dimensional coupled problem involving a compressible fluid and a poroelastic material that is attached to an elastic plate. The Biot-Allard model describes the absorbing material's behaviour, while the elastic plate is modelled using the Naghdi model, which accounts for the transversal and in-plane structural displacements. The coupling between these three types of elements in the mechanical system increases the complexity of understanding the system's frequency response. Although a mixed pressure-displacement formulation could benefit the discretization of the poroelastic material, a displacement-displacement formulation is employed to achieve a natural coupling with respect to the degrees of freedom of the Naghdi plate displacements. Precisely, the main novelty of this work lies in addressing the coupling between the plate and the poroelastic medium using the Reissner-Mindlin formulation for the Naghdi plate model and the displacement-based formulation of the Biot-Allard model. In addition, an acoustic pressure-based model is used to describe the adiabatic compressional effects on the fluid cavity. The obtained numerical results have been compared with experimental data from two different types of absorbing materials (see figure on top).

Figure 1: Experimental validation setting with a detailed view of the interior with the frontal location of the loudspeaker in the bottom.

While other studies focused on: (a) comparing numerical results with experimental outcomes for the same coupled fluid-porous-plate problem using three types of porous materials modeled through the wall impedance model, empirical model, and equivalent fluid model; (b) they have introduced the mobile wall impedance model concept. In contrast, this project addresses a direct modeling of the porous medium through Biot's approach, experimentally validating the proposed approach (see Figure 1). Furthermore, the numerical frequency responses obtained with the Biot-Allard model and the fluid-equivalent Allard-Champoux model have also been analyzed when the acoustic source is placed in a frontal or lateral location with respect to the position of the absorbing poroelastic layer. (see Chapter 3 in [1] for a detailed description of the project).

References

[1] Xabier Sagartzazu
Contributions in the acoustic study of fluid-structure interaction problems with porous materials and thin structures
Phd thesis in mathematical modelling and numerical simulation in engineering and applied science, Universidade da Coruña, 2022. Advisor: L. M. Hervella, A. Prieto.

Characterization of hydroacoustic materials

Mon, 01 Jan 0001 00:00:00 +0000

The material characterization is not an easy task for engineering materials such as polymers (see figure on top) or composite materials. In some cases a suitable choice of the viscoelastic model is fundamental: the more appropriate the model is, the more accurate its mechanical response will be in comparison with the experimental data. Well-known viscoelastic material models such as Maxwell, Zener, and Kelvin-Voigt models, or the more recent fractional derivative viscoelasticity models are common choices for modeling linear wave propagation in viscoelastic materials. Usually, to estimate the unknown parameters, the constitutive laws are first fixed, and then the available experimental data are fitted with the response of the mathematical model.

Figure 1: Experimental hydroacoustic setting to measure the mechanical properties of a layer of viscoelastic material.

However, in this project, a data-driven approach is considered. This methodology avoids the need of choosing a constitutive law for fitting. Instead of this, the fitting problem consists of minimizing the distance between a set of experimental data and the computed values. Therefore, the choice of the viscoelastic model is based only on the experimental ultrasound measurements, and not on imposing any functional dependence of the parameters in terms of frequency. In this project (see Chapter 3 in [1]), a viscoelastic material has been characterized by using a data-driven approach instead of a classical parametric model. This material is part of a coupled problem formed by the material surrounded by water (see Figure 1).

References

[1] Laura del Río Martín
Numerical characterization of complex materials and vibro-acoustic systems
Phd thesis in mathematical modelling and numerical simulation in engineering and applied science,Universidade da Coruña, 2020. Advisor: A. Prieto.

Data-driven porous models

Mon, 01 Jan 0001 00:00:00 +0000

The acoustic characterization of porous materials with rigid solid frame plays a key role in the prediction of the acoustic behavior of any dynamic system that incorporates them. In order to obtain an accurate prediction of its frequency-dependent response, a suitable choice of the parametric models for each material is essential. However, such models could be inadequate for a given material or only valid in a specific frequency range.

Figure 1: Wavenumber characterization of a composed porous material.

In this project (see [1]), a novel non-parametric methodology is proposed for the characterization of the acoustic properties of rigid porous materials. Unlike most widespread methodologies, this technique is based on the solution of a sequence of frequency-by-frequency well-posed inverse problems (see Figure 1), thus increasing the characterization accuracy. Once a reduced number of experimental measurements is available, the proposed method avoids the a priori choice of a parametric model.

References

[1] J. Carbajo, A. Prieto, J. Ramis, L. Río-Martín.
A non-parametric fluid-equivalent approach for the acoustic characterization of rigid porous materials.
Applied Mathematical Modelling, 76 (2019), 8, 330-347. (pre-print) (dataset)

Dynamic stiffness modelling

Mon, 01 Jan 0001 00:00:00 +0000

The standard ISO 9052-1 is used to determine the dynamic stiffness of elastic materials used under floating floors (see the experimental setting in the figure on top), which is one of the parameters used to determine the acoustic insulation of these floors. However, such value is not directly related to the elastic coefficients typically used in the Hooke's linear model, such as the Young modulus, the Poisson coeffcient, or the loss factor. Hence, an additional numerical or experimental procedure is required to determine from a quantitative point of view those material coefficients.

Figure 1: Impulse response of the acceleration in an impact experiment.

In this project, a numerical methodology based on a hierarchical modeling approach is proposed, using only those experimental data obtained from the standard ISO 9052-1 framework (see a detailed discussion in Chapter 6 in [1]). Consequently, the purpose of this chapter is focused on the computation of some elastic coeffcients of viscoelastic and poroelastic materials by using a hierarchy of models. This project is part of a joint collaboration with Jesús Carbajo, Pedro Poveda, and Jaime Ramis from the Department of Physics, System Engineering and Signal Theory of the University of Alicante, and the available experimental data (see Figure 1) they have provided.

References

Windshields for PU probes

Mon, 01 Jan 0001 00:00:00 +0000

The industrial partner Microflown Technologies designs and produces PU probes, which are able to measure particle velocity and acoustic pressure fields simultaneously, are sensitive to the effect of wind, since they are based on thermal transducers and hence highly dependent on the variations of thermal flow velocity. Objectives of this research project are the mathematical modelling and numerical simulation of thermo-acoustic coupled Systems (involving PU probes, the compressible fluid in the presence of flow, and the multilayer windscreen). The numerical results will play a key role in the design of novel windscreens to mitigate the flow effects on the measures of acoustic probes.

Figure 1: Scattering field of a plane wave impiging the PU device with a porous wind shield.

The present project is part of the European project ROMSOC (an MSCA European Industrial Doctorate programme). More precisely, since the wind-shield enclosures generate complex acoustics fields inside (see [1] for more details), three different approaches will be analysed by solving numerically the following coupled problems:
(A) Modelling and numerical simulation of the three dimensional coupled acoustic behaviour of a multilayer system consisting in the combination of waterproof fabrics with open porous foams. Novel effective structural-acoustic models will be developed for the structural elements and the sponge layer (see [2])
(B) Modelling and numerical simulation of the three dimensional coupled thermal-acoustic behaviour of a multilayer structure composed of metallic micro-perforated screens surrounding the USP probe. The characterization of the mechanical impedance of this protected screen will be computed in the presence of flow.
(C) Modelling and numerical simulation of the three dimensional coupled thermal-acoustic behaviour of micro-machined wind-shields consisting in a twolayer structure (a nano-perforated thin foil glued to a silicon substrate where the device wires are lying). The turbulence flow around the micro-machined transducer will be characterized. In addition, since the shape of the structure around the sensor can improve the behaviour under flow noise and weak signals, different shapes around the sensor will be tested and compared in different SNR scenarios.

References

[1] Ashwin S. Nayak
Mathematical Modelling and Numerical Simulation of Coupled Acoustic Multi-layer Systems for Enabilng Particle Velocity Measurements in the Presence of Airflow
Phd thesis in mathematical modelling and numerical simulation in engineering and applied science, Universidade da Coruña, 2021. Advisors: D. Fernández Comesaña (Microflown Technologies), A. Prieto.
[2] A. Nayak, A. Prieto, D. Fernández-Comesaña
Model coupling for acoustic sensors in layered media
14th World Congress on Computational Mechanics (WCCM) ECCOMAS Congress 2020,11–15 January 2021.

Crack detection

Mon, 01 Jan 0001 00:00:00 +0000

Health monitoring techniques are crucial in detecting defects early in industrial pipelines to avoid those problems. Among others, ultrasound acoustic measurements can help check the pipe status and locate potential cracks on its structure. In this project (see [1]), we have focused on the numerical simulation of wave propagation of Loves waves in bi-layered materials using a modal-based #PUFEM technique (a joint-project with my colleagues Philippe Destuynder, Luis María Hervella Nieto, Paula María López Pérez, and José María Orellana)

Figure 1: Lamb mode computed in a bi-layered structure.

The time-harmonic propagation of elastic waves in layered media is simulated numerically by means of a modal-based Partition of Unity Finite Element Method (PUFEM). Instead of using the standard plane waves or the Bessel solutions of the Helmholtz equation to design the discretization basis, the proposed modal-based PUFEM explicitly uses the tensor-product expressions of the eigenmodes, the so-called Love and interior modes (see Figure 1), of a spectral elastic transverse problem, which fulfil the coupling conditions among layers. This modal-based PUFEM approach does not introduce quadrature errors since the coefficients of the discrete matrices are computed in closed-form. A preliminary analysis of the high condition number suffered by the proposed method is also analyzed in terms of the mesh size and the number of eigenmodes involved in the discretization. The numerical methodology is validated through a number of test scenarios, where the reliability of the proposed PUFEM method is discussed by considering different modal basis and source terms. Finally, some indicators are introduced to select a convenient discrete PUFEM basis taking into account the observability of cracks located on a coupling boundary between two adjacent layers.

References

[1] P. Destuynder, L. Hervella-Nieto, P.M. López-Pérez, J. Orellana, A. Prieto.
A modal-based Partition of Unity Finite Element Method for elastic wave propagation problems in layered media.
Computer and Structures, 265 (2022), 106759. (pre-print)

Kelvin wakes

Mon, 01 Jan 0001 00:00:00 +0000

A novel linear potential model is presented to compute free surface flows of incompressible fluids produced by the motion of a floating rigid body in the presence of an underlying non-uniform flow (see figure on top). In particular, the proposed model enables the accurate numerical simulation of the Kelvin wake pattern in a computational domain of reduced size. The governing equations are obtained by using an Arbitrary Lagrangian Eulerian (ALE) formulation, which involves the underlying velocity of the fluid flow around the floating body, and an adimensional analysis to derive the novel linear system of equations for free surface flows.

Figure 1: Kelvin wake pattern generated by a double twin hull.

The discretization of the proposed model is made by a standard Galerkin finite element method, where a SUPG-inspired upwinding strategy has been used in combination with a Perfectly Matched Layer technique, which allows to truncate the original unbounded fluid domain without introducing spurious reflections in the Kelvin wake pattern. The numerical simulations computed with the proposed approach are compared with respect to those other results obtained by the classical linear potential model with uniform underlying flow and the full incompressible Navier-Stokes equations equipped with the k-$\omega$ SST turbulent model. This numerical comparison is discussed in terms of a classical hydrodynamic floating body benchmark involving the Wigley hull (see [1] for more details).

References

[1] A. Bermúdez, O. Crego, A. Prieto.
Upwind finite element-PML approximation of a novel linear potential model for free surface flows produced by a floating rigid body.
Applied Mathematical Modelling, 103 (2022), 2, 824-853. (pre-print)

Micro-perforated panels

Mon, 01 Jan 0001 00:00:00 +0000

The goal of this case study is focus on the numerical simulation of the dynamical behavior of a Micro-Perforated Plate (MPP), which plays the role of an structural actuator, and its effects on the pressure waves propagated along a duct. Since the objective of the actuator unit consists in minimizing the amplitude of such waves, the optimal adaptive strategy for the actuators depends on a reliable simulation of the propagation phenomena involving: an accurate simulation of the thermal and viscous boundary layer present of the compressible fluid around the MPP and the elastic structural behavior of the MPP material (see reference [1]).

Once these three models are coupled, a mixed Finite Element formulation in terms of the fluid temperature, the displacement vector field of the MPP structure and the pressure and velocity fields in the fluid is used to compute the numerical approximation of the pressure and velocity differences between in-front and back regions separated by the MPP.

Figure on the top shows a vertical cut on a section of the plane perpendicular to the MPP, where the pressure field and the velocity streamlines are plotted. In this numerical simulation, the MPP (depicted in dark yellow) has been designed with micro-slits. As in this proposed collaboration project, the simulation code has been developed specifically to handle accurately the thermal-structural-acoustic interaction of the MPP with the duct fluid.

References

[1] A. Bermúdez, J.L. Ferrín, A. Prieto.
Numerical characterization of the acoustic impedance for micro-perforated plates (MPP).
Book of Abstracts at Euronoise 2012. Prague, 10-13 June (2012)

Seabed classification

Mon, 01 Jan 0001 00:00:00 +0000

The main goal of SIMNUMAR project is focused on the development and application of novel techniques of numerical simulation to acoustic wave propagation phenomena at high-frequency regime in highly heterogeneous and complex coastal environments.

The researchers involved in the SIMNUMAR project are A. Prieto, L. Hervella-Nieto and J. Tarrío from the University of A Coruña, N. Sánchez-Carnero (see [1]) from the University of Vigo, and D. Santamarina from the University of Santiago de Compostela. SIMNUMAR project is funded currently by the Xunta de Galicia grant “emerging projects” (EM2013/052).

Taking into account the specific goals of this project, the following tasks will be addressed:

Mathematical modelling of the coastal fluid media (highly heterogeneous and depending on several state variables) and the application of novel structural models to govern the seabed as a poroelastic medium.
Application and numerical analysis of techniques based on absorbing layers to truncate unbounded domains in presence of interfaces between heterogeneous fluid and poroelastic media.
Review and development of efficient and robust numerical procedures to approximate accurately the harmonic motion in hydro-acoustic problems at the high-frequency regime, involving measurements of side-scan sonars in a range from 1kHz to 200kHz.

Figure 1: Sonar time-dependent signals obtained from different seabeds: rocky, sand, and sand with vegetation.

During the SIMNUMAR project, both supervised and non-supervised classification methods have been compared and applied to a large dataser of hydroacoustic signals (mono-beam echosound) depicted in Figure 1 (see [2]). Seabed characterization in coastal environments is usually based on acoustic techniques since direct measurements are very costly and time-consuming. The echosounders (from singlebeam to multibeam ones) insonify the sea bottom, sending an acoustic signal from the boat (or towfish), and record the backscatter as a continuous signal, the echo. The standard methodology for seabed classification uses features extracted from corrected (or uncorrected) echoes and applies a classical multivariate approach (PCA and cluster analysis) to group them into acoustic classes. Although this approach is widely used and provides satisfactory results, selecting only a number of features means to discard a large amount of information that could help to discriminate more seabed details. Moreover, traditional approaches require pulse-length correction to avoid depth dependent results, waiving resolution in the data. The present work introduces alternative automatic and statistical methodologies based on either time series cluster methods or non--hierarchical cluster techniques for functional data analysis (FDA), that allow to work with the whole signal without information reduction. More precisely, unsupervised methods such as the FDA $k$-means method, the multivariate medoids clustering, and time series cluster techniques have been applied. The supervised FDA techniques such as functional generalized linear models (FGLM) and functional generalized spectral additive models (FGSAM) have been also considered. Performance of these techniques has been illustrated using acoustic data acquired in a controlled setup over three different bottom types. Both FDA $k$-means and time series clustering approaches have provided seabed classification with accuracies over $90\%$. Moreover, these methodologies were used to analyze the classification power of the echoes identifying the topmost significant portions of the signals. Furthermore, this could point to new relevant features that would improve the results of traditional multivariante approaches.

References

[1] N. Sánchez Carnero
Técnicas acústicas y software libre : aplicaciones en la gestión costera.
PhD Thesis. Departamento de Bioloxía Animal, Bioloxía Vexetal e Ecoloxía. Universidade da Coruña (2012).

[2] J. Tarrío-Saavedra, N. Sánchez-Carnero, A. Prieto.
Comparative Study of FDA and Time Series Approaches for Seabed Classification from Acoustic Curves.
Mathematical Geosciences, 52 (2020) 669-692. (R-code) (R-notebooks) (dataset)

Trimmed multilayered patches

Mon, 01 Jan 0001 00:00:00 +0000

The aim of this joint project with a Spanish bus manufacturer consists in the development of novel green technologies to increase the acoustical comfort of the public transportation passengers (see reference [2]). More precisely, the optimal design of new coating materials for the interior vehicle have been suggested by means of their numerical characterization. Since most of the industrial suppliers for absorbing trimmed patches characterized their acoustic behavior by using international standards, for instance, ISO-10534, and such standards involves experimental measurements using a Kundt's tube, which can be mathematically modeled by a normal plane-wave analysis, the experimental data has been used to fit the physical coefficients in the non-linear frequency response mathematical models.

In fact, to determine the mechanical impedance associated to these patches or to deduce the frequency behavior of new multilayered patches with arbitrary thickness, it has been assumed: a) patches are local reacting panels and b) their time-harmonic motion of the foam layers are governed by the Allard-Champoux rigid-frame fibrous model and those micro-perforated layers are described by the Maa's impedance formula (see reference [1] for further details). In this model, the dynamic mass density and the dynamic bulk modulus depends on the frequency and the flow resistivity of the material. The computation of the last coefficient is obtained by solving an optimization problem to fit the experimental data in a wide frequency range (see Figure 1).

Figure 1: Experimental data and fit values of the imaginary part of the mechanical impedance associated to a coating patch.

Since the novel multilayered absorbing patches are composed by combining micro-perforated plates, porous layers and inner air cavities, these new configurations must be optimized by analyzing the plane-wave absorbing coefficients optimizing its geometrical configuration and the material composition in each layer. Once, the final optimal design for the multilayered coating is computed, the values of the mechanical impedances can be included in a standard finite element formulation of the three-dimensional structural-acoustic problem. This methodology is illustrated in the numerical simulation of a passive control noise implemented in a bus (see Figure on the top of the page).

References

[1] A. Bermúdez, P. Gamallo, L. Hervella-Nieto, A. Prieto.
Numerical simulation of active-passive cells with microperforated plates or porous veils.
Journal of Sound and Vibrations, 329 (2010), 3233-3246.

[2] A. Bermúdez, F. J. González Diéguez, A. Prieto.
Characterization and optimization of trimmed models in passive control noise problems.
Poster at Mathematics for Innovation: Large and Complex Systems, Tokyo, 28 February-4 March (2012).

Noise cancellation

Mon, 01 Jan 0001 00:00:00 +0000

In this project, the numerical study is focused on the comparative study of three different strategies with the objective of reducing the broadband noise in an interior room, by means of passive-active cells supported on the ceiling. Each actuation unit is composed by a box with lateral rigid walls, a porous veil in the front face (in contact with the air room cavity), and it contains a combination of sensors (microphones) and actuators (local loudspeakers) in the cell interior, which allows to implement different feedback strategies of active control (see reference [1]).

More precisely, the so-called pressure release and the impedance matching s trategies have been tested in an 5x6 array configuration in two actuation scenarios: a) the entire cell system is turn on, and b) only five of the cells are acting on the noise room. In both cases, the root mean square of the pressure field (measured in decibels) has been compared for a low frequency range where classical passive solutions, based on absorbing materials, do not work efficiently.

Figure 1: Root-mean-square of the pressure field (in decibels (dB)) computed in the interior room with three different passive-active strategies.

Figure on the top shows three two-dimensional slices of the pressure field in the interior of a room. In this numerical simulation, the noise source is supposed to come from the corridor joint to a door (see left front wall) and a window (see right front wall) has been assumed to behaves as an absorbing surface. If the entire 5x6 array is adaptively acting to minimize the root-mean-square of the pressure measured by the sensors, both control active strategies has been compared at different frequencies (see Figure 1), showing different response in the frequency band from 10Hz to 275Hz. As in this proposed collaboration project, the simulation code has been developed specifically to handle accurately not only the acoustical modeling of the room but also to deal with the optimization of the adaptive control strategies of the unit cells.

References

Optical leaky modes

Mon, 01 Jan 0001 00:00:00 +0000

During the last decade, several authors have addressed that the PML technique [1] can be used not only for the computation of the near-field in time-dependent and time-harmonic scattering problems, but also to compute numerically the resonances in open cavities. Despite such complex resonances are not natural eigen-frequencies of the physical system, the numerical determination of this kind of eigenvalues provides information about the model, what can be used in further applications. For instance, the computation of leaky modes in an open waveguide is used as the spectral basis for a modal decomposition technique [2] of complex scattering patterns (see Figure on the top).

Figure 1: Pseudospectra of the discrete non-normal operator associated to a wave scattering problem truncated with a Perfectly Matched Layer.

In the framework of the PML technique, to ensure that the complex eigenvalues are independent of the particular choices of the so-called PML absorption coefficient, the theoretical background is related with the complex scaling technique developed in the seventies by Aguilar and Combes. However, the theoretical results do not guarantee any convergence result on the PML error, which comes from the spurious reflections originated by the truncation of the PML layer to a finite thickness.

Moreover, the numerical computation of these complex resonances have been revealed highly unstable for standard finite element methods (FEM). The numerical results depend strongly not only on the thickness of the PML layer but also on the position of its inner PML boundary with respect to physical domain of the problem. All these numerical difficulties can be interpreted by using the concept of pseudospetrum (see Figure 1) associated to the PML problem and its non-normal behavior (see, for details [3]) .

References

[1] J. P. Bérenger.
A perfectly matched layer for the absorption of electromagnetic waves.
Journal of Computational Physics, 114 (1994) 185-200.

[2] A.-S. Bonnet-BenDhia, B. Goursaud, c. Hazard, A. Prieto.
Finite element computation of leaky modes in stratified waveguides.
In: Ultrasonic Wave Propagation in Non Homogeneous Media, Chapter 7 (A. Leger, M. Deschamps eds.), Springer (2009), 73-86.

[3] L. N. Trefethen.
Spectra and pseudospectra.
Princeton University Press (2005).