Nikolaos A. Kampanis

NUMERICAL SIMULATION OF LOW-FREQUENCY AEROACOUSTICS OVER IRREGULAR TERRAIN USING A FINITE ELEMENT DISCRETIZATION OF THE PARABOLIC EQUATION

Environmental noise raises serious concerns in modern industrial societies. As a result, the study of sound propagation in the atmosphere over irregular terrain is a subject of current interest in aeroacoustics. We use the standard parabolic approximation of the Helmholtz equation to simulate the far-field, low-frequency sound propagation in a refracting atmosphere, over terrains with mild range-varying topography. At an artificial upper boundary of the computational domain, described in range and height coordinates, a nonlocal boundary condition is used to model the effect of a homogeneous, semi-infinite atmosphere. We define a curvilinear coordinate system fitting the irregular topography. We discretize the transformed initial-boundary value problem with a finite element technique in height and a conservative Crank–Nicolson scheme for marching in range. The underlying transformation of coordinates allows the effective coupling with the nonlocal boundary condition. The resulting discretization method is accurate and efficient for the numerical prediction of noise levels in the atmosphere.

A FINITE ELEMENT CODE FOR THE NUMERICAL SOLUTION OF THE HELMHOLTZ EQUATION IN AXIALLY SYMMETRIC WAVEGUIDES WITH INTERFACES

We consider the Helmholtz equation in an axisymmetric cylindrical waveguide consisting of fluid layers overlying a rigid bottom. The medium may have range-dependent speed of sound and interface and bottom topography in the interior nonhomogeneous part of the waveguide, while in the far-field the interfaces and bottom are assumed to be horizontal and the problem separable. A nonlocal boundary condition based on the DtN map of the exterior problem is posed at the far-field artificial boundary. The problem is discretized by a standard Galerkin/finite element method and the resulting numerical scheme is implemented in a Fortran code that is interfaced with general mesh generation programs from the MODULEF finite element library and iterative linear solvers from QMRPACK. The code is tested on several small scale examples of acoustic propagation and scattering in the sea and its results are found to compare well with those of COUPLE.

Influence of Acute Anterior Cruciate Ligament Deficiency in Gait Variability

The objective of this study was to compare the gait variability of patients with isolated anterior cruciate ligament (ACL) deficiency (experimental group) with that of healthy individuals (control group). The hypothesis was that the gait variability of the experimental group would be higher than the control group. The experimental group consisted of 20 men with an ACL tear and the control group consisted of 20 healthy men without any neurological and/or musculoskeletal pathology or injury. The gait acceleration signal was analysed using the Gait Evaluation Differential Entropy Method (GEDEM). The GEDEM index of the experimental group in the medio-lateral axis was significantly higher than that of the control subjects. A receiver operating characteristic (ROC) analysis was used to assess the diagnostic value of the method and to determine a cut-off entropy value. The GEDEM cut-off value had a 95.6% probability of separating isolated ACL patients from healthy subjects.

Pre-operative versus post-operative gait variability in patients with acute anterior cruciate ligament deficiency.

Change in gait variability at least 6 months after surgical reconstruction of the anterior cruciate ligament (ACL) was assessed in 20 male patients with acute ACL deficiency and compared with pre-operative data and that from 20 healthy male controls. Gait was measured using a triaxial accelerometer and data were analysed by the Gait Evaluation Differential Entropy Method (GEDEM) to determine gait variability. Pain was assessed with a visual analogue scale and functional ability with the Oswestry Disability Index and the International Knee Documentation Committee score. Mean gait variability was significantly lower after than before surgery, with values for the anterior—posterior axis being in the normal range of controls after 6 months, whereas in the mediolateral axis mean gait variability remained significantly higher, indicating that some rotational instability remained in the time-frame of the study. Pain and functional ability scores improved after surgery compared with before surgery. The combination of accelerometry and GEDEM may be a useful orthopaedic tool for the post-operative evaluation of patients who have undergone ACL reconstruction.

A FINITE ELEMENT METHOD FOR THE PARABOLIC EQUATION IN AEROACOUSTICS COUPLED WITH A NONLOCAL BOUNDARY CONDITION FOR AN INHOMOGENEOUS ATMOSPHERE

The standard parabolic equation is used to simulate the far-field, low-frequency sound propagation over ground with mild range-varying topography. The atmosphere has a lower layer with a general, variable index of refraction. An unbounded upper layer with a squared refractive index varying linearly with height is considered and modeled by the nonlocal boundary condition of Dawson, Brooke and Thomson.1 A finite element/transformation of coordinates method is used to transform the initial-boundary value problem to one with a rectangular computational domain and then discretize it. The solution is marched in range by the Crank–Nicolson scheme. A discrete form of the nonlocal boundary condition, which is left unaffected by the transformation of coordinates, is employed in the finite element method. The fidelity of the overall method is shown in the numerical simulations performed for various cases of sound propagation in an inhomogeneous atmosphere over a ground with irregular topography.

Accelerometry for Evaluation of Gait Pattern in Healthy Soccer Athletes

An accelerometer system was used to measure the characteristics of the motion of 133 healthy male soccer athletes in a 30-s walking test and the data obtained were analysed using the gait evaluation differential entropy method (GEDEM). GEDEM processes gait acceleration data and calculates an index that provides a quantitative evaluation of a subjects gait, at low cost and with negligible effect on the subject. The GEDEM index was not significantly correlated with age, body weight, body mass index, or the number of years of active training. The GEDEM value for the anterior-posterior axis showed a small negative statistically significant correlation with height and the vertical axis was moderately and statistically significantly positively correlated with the time spent training per week. The triaxial accelerometry system described here is easy for subjects and testers to use, and enables measurements to be made on the sports field to evaluate an athletes musculoskeletal condition with respect to gait stability.

ON GALERKIN METHODS FOR THE WIDE-ANGLE PARABOLIC EQUATION

We consider the third-order, wide-angle parabolic approximation of underwater acoustics in a medium with depth- and range-dependent speed of sound in the presence of dissipation and horizontal interfaces. We first discuss the theory of the existence and uniqueness of solutions to the problem and derive an energy estimate. We then discretize the problem in the depth variable using two types of Galerkin/finite element formulations that take into account the interface conditions, and in the range variable by the Crank–Nicolson and also a fourth-order accurate, implicit Runge–Kutta method. The resulting high-order numerical schemes are stable and convergent and are also shown to compare favorably with classical, implicit finite difference schemes in terms of computational effectiveness when applied to standard benchmark problems.

HIGH-ORDER ACCURATE NUMERICAL SCHEMES FOR THE PARABOLIC EQUATION

Efficient, high-order accurate methods for the numerical solution of the standard (narrow-angle) parabolic equation for underwater sound propagation are developed. Explicit and implicit numerical schemes, which are second- or higher-order accurate in time-like marching and fourth-order accurate in the space-like direction are presented. The explicit schemes have severe stability limitations and some of the proposed high-order accurate implicit methods were found conditionally stable. The efficiency and accuracy of various numerical methods are evaluated for Cartesian-type meshes. The standard parabolic equation is transformed to body fitted curvilinear coordinates. An unconditionally stable, implicit finite-difference scheme is used for numerical solutions in complex domains with deformed meshes. Simple boundary conditions are used and the accuracy of the numerical solutions is evaluated by comparing with an exact solution. Numerical solutions in complex domains obtained with a finite element method show excellent agreement with results obtained with the proposed finite difference methods.

Error estimates for finite element methods for a wide-angle parabolic equation

Abstract We consider a model initial-and boundary-value problem for the third-order wide-angle parabolic approximation of underwater acoustics with depth- and range-dependent coefficients. We discritize the problem in the depth variable by the standard Galerkin finite element method and prove optimal-order L2-error estimates for the ensuing continuous-in-range semidiscrete approximation. The associated ODE systems are then discretized in range, first by a second-order accurate Crank-Nicolson-type method, and then by the fourth-order, two-stage Gauss-Legendre, implicit Runge-Kutta scheme. We show that both these fully discrete methods are unconditionally stable and possess L2-error estimates of optimal rates.

Fast and Accurate Finite Element Methods for the Numerical Prediction of the Acoustic Field

The standard parabolic approximation of the reduced wave equation, used to describe sound propagation in a multi—layered, range—dependent ocean, is solved numerically using finite element methods to discretize in depth and an implicit range—stepping method of Runge—Kutta type. Efficiency tests performed ensure that in terms of computational effectiveness (achieved accuracy vs. computational work), the resulting higher—order accurate, numerical methods compare favorably with classical, second—order accurate, implicit finite difference schemes. They are also faster in obtaining equally accurate results when applied to standard benchmark problems of underwater acoustics.