Claude Depollier

Analytical Calculation of the Magnetic Field Created by Permanent-Magnet Rings

We present analytical formulations, based on a coulombian approach, of the magnetic field created by permanent-magnet rings. For axially magnetized magnets, we establish the expressions for the three components. We also give the analytical 3-D formulation of the created magnetic field for radially magnetized rings. We compare the results determined by a 2-D analytical approximation to those for the 3-D analytical formulation, in order to determine the range of validity of the 2-D approximation.

Measuring the porosity and the tortuosity of porous materials via reflected waves at oblique incidence.

An ultrasonic reflectivity method is proposed for measuring porosity and tortuosity of porous materials having a rigid frame. Porosity is the relative fraction by volume of the air contained within a material. Tortuosity is a geometrical parameter which intervenes in the description of the inertial effects between the fluid filled the porous material and its structure at high frequency range. It is generally easy to evaluate the tortuosity from transmitted waves, this is not the case for porosity because of its weak sensitivity in transmitted mode. The proposed method is based on measurement of reflected wave by the first interface of a slab of rigid porous material. This method is obtained from a temporal model of the direct and inverse scattering problems for the propagation of transient ultrasonic waves in a homogeneous isotropic slab of porous material having a rigid frame [Z. E. A. Fellah, M. Fellah, W. Lauriks, and C. Depollier, J. Acoust. Soc. Am. 113, 61 (2003)]. Reflection and transmission scattering operators for a slab of porous material are derived from the responses of the medium to an incident acoustic pulse at oblique incidence. The porosity and tortuosity are determined simultaneously from the measurements of reflected waves at two oblique incidence angles. Experimental and numerical validation results of this method are presented.

An alternative Biot's displacement formulation for porous materials.

This paper proposes an alternative displacement formulation of Biots linear model for poroelastic materials. Its advantage is a simplification of the formalism without making any additional assumptions. The main difference between the method proposed in this paper and the original one is the choice of the generalized coordinates. In the present approach, the generalized coordinates are chosen in order to simplify the expression of the strain energy, which is expressed as the sum of two decoupled terms. Hence, new equations of motion are obtained whose elastic forces are decoupled. The simplification of the formalism is extended to Biot and Willis thought experiments, and simpler expressions of the parameters of the three Biot waves are also provided. A rigorous derivation of equivalent and limp models is then proposed. It is finally shown that, for the particular case of sound-absorbing materials, additional simplifications of the formalism can be obtained.

Inhomogeneous Biot waves in layered media

A description, using transfer matrices, is given of the propagation of the three inhomogeneous Biot waves in layered porous media. This is applied to predict the surface impedance of porous layered materials at oblique incidence, and an example is presented.

Acoustical properties of partially reticulated foams with high and medium flow resistance

A new method for measuring impedance has been used for evaluating the normal surface impedance of three foams in free field. The results have been interpreted using the Biot theory, the two dilatational waves being taken into account. It has been pointed out from the theory that, for high flow resistance, the ratio of the acoustical velocities of the frame and the air is close to 1 at the surface of the foam. This ratio decreases with flow resistance but is never negligible for the studied foams. For foams with high flow resistance, the contributions of the two waves must be taken into account when calculating the impedance, and a description with only one wave would not be realistic. For foams with medium flow resistance, the one‐wave approximation for calculating the surface impedance is a good approximation in the whole range of acoustical frequencies.

The Three Exact Components of the Magnetic Field Created by a Radially Magnetized Tile Permanent Magnet

This paper presents the exact analytical formulation of the three components of the magnetic field created by a radially magnetized tile permanent magnet. These expressions take both the magnetic pole surface densities and the magnetic pole volume density into account. So, this means that the tile magnet curvature is completely taken into account. Moreover, the magnetic field can be calculated exactly in any point of the space, should it be outside the tile magnet or inside it. Consequently, we have obtained an accurate 3D magnetic field as no simplifying assumptions have been used for calculating these three magnetic components. Thus, this result is really interesting. Furthermore, the azimuthal component of the field can be determined without any special functions. In consequence, its computational cost is very low which is useful for optimization purposes. Besides, all the other expressions obtained are based on elliptic functions or special functions whose numerical calculation is fast and robust and this allows us to realize parametric studies easily. Eventually, we show the interest of this formulation by applying it to one example: the calculation and the optimization of alternate magnetization magnet devices. Such devices are commonly used in various application fields: sensors, motors, couplings, etc. The point is that the total field is calculated by using the superposition theorem and summing the contribution to the field of each tile magnet in any point of the space. This approach is a good alternative to a finite element method because the calculation of the magnetic field is done without any simplifying assumption.

Application of the Biot model to ultrasound in bone: Inverse problem

This paper concerns the ultrasonic characterization of human cancellous bone samples by solving the inverse problem using experimentally measured signals. The inverse problem is solved numerically by the least squares method. Five parameters are inverted: porosity, tortuosity, viscous characteristic length, Young modulus, and Poisson ratio of the skeletal frame. The minimization of the discrepancy between experiment and theory is made in the time domain. The ultrasonic propagation in cancellous bone is modelled using the Biot theory modified by the Johnson-Koplik-Dashen model for viscous exchange between fluid and structure. The sensitivity of the Young modulus and the Poisson ratio of the skeletal frame is studied showing their effect on the fast and slow waveforms. The inverse problem is shown to be well posed, and its solution to be unique. Experimental results for slow and fast waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.

Permanent Magnet Couplings: Field and Torque Three-Dimensional Expressions Based on the Coulombian Model

This paper presents three-dimensional expressions for the optimization of permanent-magnet couplings. First, we give a fully analytical expression of the azimuthal field created by one arc-shaped permanent magnet radially polarized which takes into account its magnetic pole volume density. Such an expression has a very low computational cost and is exact for all points in space. Then, we propose two semianalytical expressions of the azimuthal force and the torque exerted between two arc-shaped permanent magnets. These expressions are valid for thick or thin arc-shaped permanent magnets. Furthermore, this approach allows us to realize easily parametric studies and optimizations. The analytical approach taken in this paper, based on the Coulombian model, is a good alternative compared to the finite element method generally used to study such configurations.

Determination of transport parameters in air-saturated porous materials via reflected ultrasonic waves

An ultrasonic reflectivity method of evaluating the acoustic parameters of porous materials saturated by air (or any other gas) is discussed. The method is based on experimental detection of waves reflected at normal incidence by the first and second interface of the material. This method is based on a temporal model of direct and inverse scattering problems for the propagation of transient ultrasonic waves in a homogeneous isotropic slab of porous material with a rigid frame [Fellah et al., J. Acoust. Soc. Am. 113, 61-73 (2003)]. Generally, the conventional ultrasonic approach can be used to determine tortuosity, and viscous and thermal characteristic lengths via transmitted waves. Porosity cannot be estimated in transmitted mode because of its very weak sensitivity. First interface use of the reflected wave at oblique incidence leads to the determination of porosity and tortuosity [Fellah et al., J. Acoust. Soc. Am. 113, 2424-2433 (2003)] but this is not possible at normal incidence. Using experimental data of reflected waves by the first and second interface at normal incidence simultaneously leads to the determination of porosity, tortuosity, viscous and thermal characteristic lengths. As with the classic ultrasonic approach for characterizing porous material saturated with one gas, both characteristic lengths are estimated individually by assuming a given ratio between them. Tests are performed using weakly resistive industrial plastic foams. Experimental and numerical results, and prospects are discussed.

Modelization of layered sound absorbing materials with transfer matrices

A method is presented that allows the prediction of the acoustical properties of layered porous or fibrous materials in normal incidence from simple measurements. Transfer matrices obtained from the Biot theory are used to describe homogeneous layers and screens constituting layered materials. The description of an absorbing material with the method is given.