Numerical Modeling Of Wave Propagation Research Articles

Nonlinear wave model of towed system and numerical method for its calculation

Towed systems have found wide practical applications. Notable underwater systems for the long-distance extraction of minerals (concretions) from the ocean floor extend over 5-10 km. The existing mathematical models for solving dynamic problems of such systems in various environments are not entirely accurate regarding the diversity of wave processes. This necessitates the development of refined wave models. This article presents a new quasi-linear mathematical model describing the nonlinear four-mode dynamics of a towed system in a spatially inhomogeneous field of mass and surface forces. It is described by a nonlinear system of twelve first-order partial differential equations. The principles of boundedness and hyperbolicity are satisfied. The validation of the two-mode reduction of the model is based on the numerical solution of the problem of the propagation of two waves: longitudinal and configurational. Using a numerical algorithm and a program based on the finite difference method, a comparison of two difference schemes – Crank-Nicolson and Euler – was conducted. The main limitations for applying the finite difference method used for numerical modeling of wave propagation and reflection in a towed system are the peculiarities of the defining quasi-linear equations, which are related to the necessity of simultaneous computation of variables responsible for fast and slow processes. For such systems of equations, the term "singularly perturbed system of equations" is used. These perturbations result from the significant differences in the propagation speeds of longitudinal and configurational waves at the physical level. Consequently, it is necessary to employ special time-stepping regularization and filtering methods for numerical results. This imposes certain restrictions on the ability to model real processes and on the accuracy of the obtained results, necessitating the use of implicit difference schemes and high-frequency filtering.

2DH numerical model of nonlinear wave propagation based on a hybrid finite-volume and finite-difference scheme

A two-dimensional hybrid (2DH) numerical model suitable for nonlinear wave propagation from deep to shallow waters is developed based on a hybrid finite-volume and finite-difference scheme. To utilize this method conveniently, the selected governing equations are derived again in a conservative form. The fourth-order monotonic upstream-centered scheme for conservation laws–total variation diminishing scheme with a Riemann solver is employed to calculate the numerical flux. The finite-difference scheme is employed to discretize the other spatial derivatives, and the third-order strong stability-preserving (SSP) Runge–Kutta scheme is used for time-stepping. Wave breaking is simulated by locally switching the governing equations to nonlinear shallow-water equations when the Froude number exceeds a certain threshold. A comparison of the numerical results and analytical or experimental data showed that the numerical model performs well in simulating nearshore wave propagation.

A Digital Twin for Detecting Liquid-Liquid Interface in Containers

This study introduces a digital twin (DT) architecture for detecting liquid-liquid interfaces in containers, aimed at improving real-time monitoring and optimising the separation process in the petroleum industry. The research employs the DT concurrent to the physical twin to enhance the system performance by integrating response error measures with knowledge of industrial processes. The DT uses ultrasonic sensors to collect data and a high-fidelity numerical wave propagation model for an adaptive fitting algorithm to update the interface responses. Both experimental studies and numerical simulations confirm the ability of the DT to accurately determine liquid levels by analysing interfacial responses from ultrasonic signal features in transmission mode. Cross-validation and error metrics validate the DT’s adaptability and accuracy in interface detection, even with undersampled data.

A spectrum-of-spectrum filtering method to extract direct and multipath arrivals from simulations and measurements

In measurements and numerical modelling of wave propagation, undesired interference between direct and multipath arrivals can be reduced using Fourier-based signal processing methods. Existing methods, such as cepstral analysis and time-signal gating, are not applicable to all cases. Here, an alternative Fourier-based signal processing method is presented, called spectrum-of-spectrum (SoS) filtering. Its main advantage over existing methods is its ability to extract single direct or multipath arrivals for relatively short propagation distances even when subsequent arrivals do not become successively weaker. The method is based on the following steps:•Apply a lowpass filter to the real and imaginary parts of an input frequency spectrum individually, using a digital finite impulse response (FIR) filter in the frequency domain.•Recombine the filtered real and imaginary parts of the frequency spectrum to get the frequency spectrum of the direct arrival.•For extraction of the first multipath arrival, subtract the filtered frequency spectrum from the input frequency spectrum and repeat the previous steps. Repeat multiple times to extract subsequent multipath arrivals.

A coupled acoustic/elastic discontinuous Galerkin finite element method: Application to ultrasonic imaging of 3D-printed synthetic materials

We present the derivation of upwind numerical fluxes for the space discontinuous Galerkin (dG) finite element method applied to the numerical modeling of wave propagation in multidimensional coupled acoustic/elastic media. The space dG method is formulated using the first-order velocity-pressure and velocity-stress systems for acoustic and elastic media, respectively. After eigenanalysis of the first-order hyperbolic systems highlighting the eigenmodes of wave propagation, the upwind numerical fluxes on the acoustic/acoustic and acoustic/elastic interface are obtained in terms of exact solutions of relevant Riemann problems. Thanks to the proposed approach, explicit closed-form expressions of the upwind numerical fluxes are obtained on the acoustic/elastic interface for the general case of multidimensional anisotropic heterogeneous solid media coupled with acoustic fluids. The developed numerical fluxes are validated by analytical/numerical comparison considering the example of an acoustic domain with a circular elastic inclusion. Finally, the coupled solver is used to perform a multiparametric study on the microstructure's echogenicity in a 3D-printed synthetic material under ultrasonic imaging.

Preliminary Numerical Analysis of Mechanical Wave Propagation Due to Elastograph Measuring Head Application in Non-Invasive Liver Condition Assessment

The authors of this paper focused their attention on developing numerical models of mechanical wave propagation along human tissue as a result of the application of the measuring head of the FibroScan® elastograph. The FibroScan® diagnostic device is used for diagnostic testing of liver fibrosis and steatosis. This examination is carried out using an in vivo method by directly applying the surface of the ultrasonic measuring probe to the patient’s skin at the site of the liver. The authors’ idea is to use this apparatus for non-invasive testing on the liver used for transplantation. In order to do this, the measuring head cap should be modified so that its application to the liver does not result in damage as a result of mechanical wave excitation. The purpose of the manuscript was to build numerical models of the liver and the tissues surrounding the liver. Then, the corresponding numerical simulations were carried out, the results of which corresponded to the mechanical–acoustic properties of the physical models of the tissues. The obtained results were validated on a set of commercial calibrated phantoms. High agreement of the numerical models was obtained.

Numerical model of nonlinear elastic bulk wave propagation in solids for non-destructive evaluation

Nonlinear ultrasonic techniques can be difficult and non-intuitive to understand due to the range of wave mixing combinations available between similar or different wave modes. To overcome this, a numerical model that uses a Finite Difference Time Domain (FDTD) scheme, without a staggered grid system, to solve the nonlinear elastic bulk wave equations in two dimensions is proposed in this paper, with the purpose of better understanding nonlinear ultrasonic techniques. Both material and geometrical nonlinearities are considered and a stress-type boundary condition is used to model the excitation. The energy transfer between frequency bands is shown and the amplitude trend of the harmonics are validated against available theories. It is then used to simulate the nonlinear ultrasonic field generated by a finite-size element of an array in a solid to better understand its behaviour. An array-element directivity at the second harmonic frequency is shown. Since the FDTD model does not account for attenuation, a correction using a ray-based approach is applied to the simulated result for direct comparison against experimental measurements. The field obtained from a typical array used in non-destructive testing is then simulated to characterise its behaviour. Experimental comparison of the nonlinear displacement fields for different focal positions showed good agreement, further validating the FDTD model. This FDTD model opens opportunities to virtually experiment and design appropriate nonlinear ultrasonic array inspections whilst isolating and understanding contribution from material nonlinearity.

Numerical Modeling of Elastic Wave Propagation in Porous Soils with Vertically Inhomogeneous Fluid Contents Due to Infiltration

The structure of soils is often heterogeneous with layered strata having distinct permeabilities. An advanced mathematical and numerical coupled model of elastic wave propagation in poroelastic multi-layered soils subjected to subsoil water infiltration is proposed in this study. The coupled model was based on the introduction of an inhomogeneous functionally graded fluid-saturation of the considered soil depending on the infiltration time, which was evaluated employing Richards’ equation. The time-harmonic solution was formulated in terms of the Fourier transform of Green’s matrix and the surface load that excites the vibration. The convergence and efficiency of the proposed approach are demonstrated. An example of dispersion curves for partially saturated porous strata made of loam, sand, and rock at different infiltration times is provided, and it is shown that the characteristics of the surface acoustic waves change with time, which can be further used for inverse problems’ solution.

Prediction of the internal structure of a lithium-ion battery using a single ultrasound wave response

This paper describes a means to predict the internal structure of a lithium-ion battery from the response of an ultrasonic pulse, using a genetic algorithm. Lithium-ion batteries are sealed components and the internal states of the cell such as charge, health, and presence of structural defects are difficult to measure. Ultrasonic inspection of lithium-ion batteries is a recent and growing area of research. Reflected and transmitted ultrasound pulses are proposed as a non-invasive means of gaining insights into the internal structure and changes within the closed body of a cell. However, the multiple layers present in a lithium-ion cell are problematic when attempting to interpret waveforms as many internal reflections superimpose. Attributing specific features of a cell to wave characteristics is challenging.In this work a genetic algorithm has been developed as a means to reverse engineer a single ultrasound wave response to predict the internal layered structure of a lithium-ion battery cell. A first randomised guess at the layered structure is made. A numerical wave propagation model is used to predict the ultrasound waveform associated with that structure. This waveform is then compared with a measured or reference waveform to establish its fitness. The layered structure is generationally mutated until the predicted waveforms converge on the reference signal. As this occurs the predicted layered body reveals insights into the cell structure under inspection.Initially, the algorithm was tested against an idealised model battery and its predicted waveform, giving a model-model verification. Further, experimental ultrasonic reflection signals were captured from small capacity lithium-ion cells. Estimations of layer structure predicted by the model were compared with CT-scans of the cells to assess performance. The genetic algorithm was found to be effective in converging the predicted wave response to the reference signal and creating accurate battery structures. It was shown that only part of the waveform was required to generate accurate predictions, which is helpful in avoiding parts of the signal contaminated by near field transducer effects.It was demonstrated that the genetic algorithm can predict material wave speed to 40–1100 m/s (3–29 %) accuracy when battery layer geometry is provided; and thicknesses to within approximately 0.2–7.5 μm (1–13 %) when material properties are provided. Providing the genetic algorithm with parameter constraints; either the layer topology and/or the material properties, substantially improved predictions to estimate the wave speeds on average to approximately ±50 m/s (3–4 %) and the layer thicknesses ±5 μm (7–8 %).This raises the possibility of the use of this approach to predict state of charge when the battery construction is known, or the presence of internal defects and damage to a known battery material composition.

How Ultrasonic Pulse-Echo Techniques and Numerical Simulations Can Work Together in the Evaluation of the Elastic Properties of Glasses

This paper presents the numerical simulation of the ultrasonic wave transmittance utilizing the elastodynamic finite integration technique (EFIT). With this methodology, it is possible to simulate the propagation of the ultrasound in a medium with a relatively low computational cost. The capability of this technique for determining the elastic properties of fluorophosphate and the aluminosilicate glasses is described in detail. The elastic constants of the glasses were calculated from the theoretically predicted longitudinal and transversal sound velocities and compared with the corresponding experimental data. Furthermore, the calculated and experimental elastic properties of the fluorophosphate and aluminosilicate glasses were correlated with the structural peculiarities of these glasses. This simulation technique is also suitable for unveiling the existence of possible defects in the glasses by comparing the experimental and simulation data. The EFIT technique is shown to be a very useful tool in order to provide fast and easy-to-acquire data regarding also the structural characteristics of various glassy systems. This can be used in conjunction with other spectroscopic techniques which can prove to be extremely useful for the non-destructive testing of vitreous materials. The latter can prove very important when vitreous materials used in optical or optoelectronic applications need continuous monitoring in order to ensure their optimum operation and functionality with limited intervention. The main contribution of this paper is the treatment of numerical time-domain modeling of 2D acoustic wave propagation in a viscoelastic medium by implementing the elastodynamic finite integration technique (EFIT).

Numerical analysis of Plastic Perforated Panel for Acoustic Protection

Acoustic protection is an important aspect in various industrial, commercial and residential applications. In order to reduce the transmission of noise, perforated panels are frequently used as a barrier. The present study aims to conduct a numerical analysis of plastic perforated panels for acoustic protection. The study employed a finite element method (FEM) approach and focused on the propagation of acoustic waves through perforations of varying diameters (30 mm, 40 mm, 50 mm, 60 mm, 70 mm and 90 mm) and at different frequencies (250 Hz, 500 Hz, 1000 Hz and 1500 Hz). The numerical analysis was conducted using the finite element software ANSYS. This work offers numerical analysis models of acoustic wave propagation, which can be used by those interested in similar problems, for different environments, in closed or open spaces. The results showed that the perforation diameter and frequency play a crucial role in the performance of the plastic perforated panels as an acoustic barrier. The results of the author�s research pointed out that the plastic materials can be used successfully in the construction of acoustic barriers. Next to it, the findings of this study can provide valuable insights for engineers and designers in the selection and optimization of plastic perforated panels for acoustic protection applications.

Numerical modeling of seismic wave propagation in loosely deposited partially saturated sands: an application to a mine dump monitoring case

Extensive mine dumps consisting of loosely deposited sands have been created as a result of open-pit lignite mining, with a risk of soil liquefaction under high water saturation and a corresponding initiating event. Soil compaction is one of the feasible methods for reducing the probability of liquefaction. For the monitoring of liquefaction events and the evaluation of compaction work, seismic survey methods with sensitivity to changes in soil saturation and structure may thus complement other methods. Compared to exploration methods for deep systems, the shallow subsurface presents some unique challenges. To this end, an open-source, customizable code based on Biot’s theory was developed in the FEniCS library, which takes into account partial saturation and porosity dependence of stiffness, permeability, and other quantities. Following code verification, a comprehensive investigation of parameter studies is conducted, from which the effects of different factors on wave propagation characteristics were obtained. The numerical model was applied to simulate the expected changes in seismic response following soil compaction. Furthermore, the position of the high saturation area could be detected from the reflection and refraction P waves. The goal of this work is to provide an analysis framework for the assessment of compaction works and monitoring liquefiable soils in mine dumps under conditions of variable saturation due to rising groundwater tables.

VERIFICATION OF THE WAVE FACTORIZATION METHOD FOR CALCULATION OF DISTRIBUTED SYSTEMS DURING TOWING IN A FLOW IN ACCELERATION MODE

For purposes of asymmetric confrontation with the Russian Navy, distributed systems (DS; cable, tethered, etc.) for unmanned underwater vehicles need further development and improvement. To date, specifics and methods of designing such DSs are insufficiently covered in scientific and technical literature. Distributed systems include towed systems of constant or variable length in a flow of liquid, underwater communication cables, pillars of offshore oil platforms, etc. This definition includes mechanical objects with one of the linear dimensions at least 10 – 20 times larger than the other two. The main limitations for application of the finite difference method (FDM) for numerical modeling of wave propagation and reflection in DS are the peculiarities of the defining quasi-linear equations. They are related to the need of simultaneous calculation of variables responsible for fast and slow wave processes. The term "singularly perturbed system of equations" is used for such systems of equations. These perturbations are the result of a significant difference in propagation velocities of longitudinal, configurational, bending, and torsional waves in DS at the physical level, etc. In this regard, it is necessary to apply the special step-by-step methods of regularization and filtering of numerical results. This imposes certain limitations on the possibility of simulating real processes and on the accuracy of obtained results, and forces the use of the implicit difference schemes and high-frequency filtering. The method of parallelization of the finite-difference operator and the program code according to the wave type is considered. The idea of parallelization by waves is based on the physical feature of propagation of different types of waves in distributed systems – a difference of 10 – 100 or more times between the propagation velocities of the longitudinal, transverse (configurational), bending and torsional waves in DS. The increase in productivity of the MPI version of a programming code when performing calculations on the SMP system is on average at least 30 – 100%, depending on the required accuracy of calculations and parallelization options. The version of the parallelized code, which uses the wave factorization method, is relevant when solving tasks of DS control, operational numerical analysis of transient modes of motion, etc., where performance is critically necessary. The comparative assessment of the accuracy of the experimental data, the original (non-parallelized) algorithm and the parallelized algorithm was carried out using the example of the numerical solution of the problem of the towing vessel’s movement in the acceleration mode when towing the DS.

Propagation of stress waves in layered rock mass under the impact of high-pressure gas

High-pressure gas (HPG) blasting is safe and environmentally friendly, and has replaced traditional explosive blasting in some applications. The characteristics of HPG impact load and the propagation law of the stress wave in layered rock mass (LRM) under HPG blasting were investigated experimentally and numerically. Through experimental tests of hole wall pressure, a segmented exponential model of the hole wall pressure under the impact of HPG was developed; it accurately accounted for the time history characteristics of the hole wall load. Experimental tests of stress wave propagation were conducted; the soft rock–hard rock interface was found to have a significant effect on stress wave propagation in the LRM. Based on the experimental results, a specific strain model for 10 MPa HPG impact was fitted for a single-hole LRM. A polynomial exponential model and a specific strain model (20 MPa HPG impact) were found for the double-hole LRM. The stress wave attenuation laws for single-hole and double-hole LRM were characterised well. The propagation characteristics of stress waves in LRM under the impact of HPG were analysed and a numerical model of stress wave propagation was established. The numerical model adopted the Riedel–Hiermater–Thoma (RHT) material model for the test parameters. The proposed segmented exponential models of hole wall pressure were applied to the lower, middle and upper parts of the blasthole. With the feasibility of the numerical model analysed, the stress wave propagation characteristics were studied by numerical simulation of an underground pipe gallery. This study provides theoretical guidance and practical value for improving rock-breaking theory and optimising HPG blasting in LRM.

Comparative Study on Numerical Simulation of Wave-Current Nonlinear Interaction Based on Improved Mass Source Function

In coastal waters, wave propagation is often affected by rivers and tides. The wave current interaction increases the complexity of the wave propagation. In this study, we consider the Boussinesq type equation with an improved dispersion term as the governing equation and establish a numerical model of wave propagation in the coexistence of wave current environment. Firstly, we use the MIKE 21 BW model to simulate the propagation of dual-frequency waves. The Navier–Stokes equation wave model is used to verify the results and the Fourier transform is used to analyze and discuss the dual-frequency waves. Our findings show that the numerical model established by the Boussinesq equation can better describe the nonlinear interaction between waves more accurately at a much higher computational efficiency compared with the Navier–Stokes equation wave model. In addition, we set the constant current source point in the wave numerical model and conduct the numerical simulation of waves in the current environment, by improving the mass source wave generation method. The numerical simulation of wave-current interactions between uniform and variable water depths is performed, thus demonstrating its capability to describe accurately the influence of water flow on wave propagation.

Nearly perfectly matched layer implementation for time domain spectral element modelling of wave propagation in 3D heterogeneous and anisotropic porous media

Numerical modelling of wave propagation in the heterogeneous and anisotropic porous media is important for seismic exploration. When modelling time-domain poroelastic waves in an unbounded region, the implementation of perfectly matched layer (PML) is a common technique to absorb the outgoing waves. The PML in terms of first-order equations has been well proposed. However, the first-order PML needs fundamental and nontrivial reconstructions when applied to the second-order systems in displacements, such as finite element method (FEM) and spectral element method (SEM). It is well known SEM is an efficient algorithm for simulating seismic waves for its high accuracy and geometrical flexibility. However, at present, few literature studies the implementation of PML into time-domain SEM for simulating waves in the 3D unbounded heterogeneous porous media. In this work, combining with auxiliary ordinary differential equations (ODEs), we first systematically extend nearly PML (NPML) into time-domain SEM in terms of the second-order poroelastic system. The scheme preserves the original form of wave equations, leading to a simple and flexible incorporation into the existing algorithm. It avoids the temporary convolution through solving a set of first-order ODEs in time domain, resulting in high computational efficiency. Furthermore, to solve the coupling poroelastic/acoustic/elastic waves, we employ domain decomposition technique to handle the different regions independently, and apply the corresponding boundary conditions at the region interfaces to guarantee the coupling. Numerical examples demonstrate the accuracy and stability of the proposed method by comparing the numerical results with the analytical solutions and other independent numerical solutions. Finally, we apply the algorithm to solve wave propagation in a large-scale model involved a fluid-filled borehole embedded in heterogeneous porous formations.

Ultrasonic Guided Waves in Bone: A Decade of Advancement in Review.

The use of guided wave ultrasonography as a means to assess cortical bone quality has been a significant practice in bone quantitative ultrasound for more than 20 years. In this article, the key developments within the technology of ultrasonic guided waves (UGW) in long bones during the past decade are documented. The covered topics include data acquisition configurations available for measuring bone guided waveforms, signal processing techniques applied to bone UGW, numerical modeling of ultrasonic wave propagation in cortical long bones, formulation of inverse approaches to extract bone properties from observed ultrasonic signals, and clinical studies to establish the technology's application and efficacy. The review concludes by highlighting specific challenging problems and future research directions. In general, the primary purpose of this work is to provide a comprehensive overview of bone guided-wave ultrasound, especially for newcomers to this scientific field.

Time domain numerical modeling of wave propagation in poroelastic media with Chebyshev rational approximation of the fractional attenuation

In this work, we investigate the poroelastic waves by solving the time‐domain Biot‐JKD equation with an efficient numerical method. The viscous dissipation occurring in the pores depends on the square root of the frequency and is described by the Johnson‐Koplik‐Dashen (JKD) dynamic tortuosity/permeability model. The temporal convolutions of order 1/2 shifted fractional derivatives are involved in the time‐domain Biot‐JKD model, causing the problem to be stiff and challenging to be implemented numerically. Based on the best relative Chebyshev approximation of the square‐root function, we design an efficient algorithm to approximate and localize the convolution kernel by introducing a finite number of auxiliary variables that satisfy a local system of ordinary differential equations. The imperfect hydraulic contact condition is used to describe the interface boundary conditions and the Runge‐Kutta discontinuous Galerkin (RKDG) method together with the splitting method is applied to compute the numerical solutions. Several numerical examples are presented to show the accuracy and efficiency of our approach.

The calculation of loads on buildings and structures caused by outdoor explosions of the fuel-air mixture

Introduction. An important engineering task, to be solved in the process of designing buildings and structures for hazardous industrial facilities, is to determine values of loads caused by outdoor explosions of the fuel-air mixture. Nowadays software packages, that use the computational fluid dynamics (CFD) approach, are widely applied in the design practice to assess various effects on building structures. In this regard, it is necessary to develop a load calculation method, that employs numerical simulation, and verify it in comparison with the experimental data.Goals and objectives. The purpose of this work is to use the method of computational fluid dynamics to analyze external sympathetic detonation loads on various types of buildings and structures.The body of the article. The article addresses the “compressed balloon” method used to analyze loads, caused by outdoor explosions of gas. Dependencies, proposed in the article, are needed to set the input data and make numerical calculations using the computational fluid dynamics (CFD) technique. The numerical modeling of various experiments in the ANSYS Fluent software package was conducted. The authors compared the results of numerical modeling and standard engineering methods with various experiments to assess the accuracy of the “compressed balloon” method used to analyze an outdoor detonation explosion.Conclusions. The authors have proven the qualitative and quantitative convergence of the numerical model of blast wave propagation and the experimental data. This calculation method allows to accurately apply the pressure profile to any surface of a building or structure in the course of an outdoor detonation explosion and estimate the bearing capacity of building structures. The proposed method can be used in the design of buildings or structures that feature various configurations.

Spatial Response Identification Enables Robust Experimental Ultrasound Computed Tomography.

Ultrasound computed tomography techniques have the potential to provide clinicians with 3-D, quantitative and high-resolution information of both soft and hard tissues such as the breast or the adult human brain. Their practical application requires accurate modeling of the acquisition setup: the spatial location, orientation, and impulse response (IR) of each ultrasound transducer. However, the existing calibration methods fail to accurately characterize these transducers unless their size can be considered negligible when compared with the dominant wavelength, which reduces signal-to-noise ratios below usable levels in the presence of high-contrast tissues such as the skull. In this article, we introduce a methodology that can simultaneously estimate the location, orientation, and IR of the ultrasound transducers in a single calibration. We do this by extending spatial response identification (SRI), an algorithm that we have recently proposed to estimate transducer IRs. Our proposed methodology replaces the transducers in the acquisition device with a surrogate model whose effective response matches the experimental data by fitting a numerical model of wave propagation. This results in a flexible and robust calibration procedure that can accurately predict the behavior of the ultrasound acquisition device without ever having to know where the real transducers are or their individual IR. Experimental results using a ring acquisition system show that SRI produces calibrations of significantly higher quality than standard methodologies across all transducers, both in transmission and in reception. Experimental full-waveform inversion (FWI) reconstructions of a tissue-mimicking phantom demonstrate that SRI generates more accurate reconstructions than those produced with standard calibration techniques.