D. Polyzos

Modelling of pile wave barriers by effective trenches and their screening effectiveness

Abstract The three-dimensional problem of isolation of vibration by a row of piles is studied numerically on the basis of a model replacing the row of piles by an effective trench in order to reduce the modelling complexity. The analysis is accomplished with the aid of an advanced frequency domain boundary element method, which is used for both the infilled trench and the soil medium in conjunction with a coupling procedure based on enforcement of equilibrium and compatibility at the trench–soil interface. Linear elastic or viscoelastic material behaviour is assumed for both the piles and the soil. The piles can be tubular or solid and have circular or square cross-section. The vibration source is a vertical force, harmonically varying with time, and the row of piles acts as a passive wave barrier. The effective trench model is constructed by invoking well known homogenization techniques used in the mechanics of fibre-reinforced composite materials, and its accuracy is compared against a rigorous boundary element analysis modelling each pile separately in full contact with the soil medium. On the basis of the effective trench model, the screening effectiveness of a row of piles is studied through parametric studies.

Some studies on dual reciprocity BEM for elastodynamic analysis

The accuracy of the dual reciprocity boundary element method for two-dimensional elastodynamic interior problems is investigated. A general analytical method is described for the closed form determination of the displacement and traction tensor corresponding to radial basis functions and explicit expressions of these tensors are provided for a number of specific basis functions. For all these basis functions the accuracy of the dual reciprocity boundary element method is numerically assessed for three interior plane stress elastodynamic problems. The influence of internal points on the accuracy of the solution is also considered. Useful results concerning the suitability of the various basis functions for solving plane elastodynamic problems are obtained.

A boundary element method for solving 2-D and 3-D static gradient elastic problems: Part I: Integral formulation☆

Abstract A boundary element formulation is developed for the static analysis of two- and three-dimensional solids and structures characterized by a linear elastic material behavior taking into account microstructural effects. The simple gradient elastic theory of Aifantis expressed in the framework of Mindlin’s general theory is used to model this material behaviour. A variational statement is established to determine all possible classical and non-classical (due to gradient terms) boundary conditions of the general boundary value problem. The gradient elastic fundamental solution for both two- and three-dimensional cases is explicitly derived and used to construct the boundary integral representation of the solution with the aid of the reciprocal integral identity especially established for the gradient elasticity considered here. It is found that for a well-posed boundary value problem, in addition to a boundary integral representation for the displacement, a second boundary integral representation for its normal derivative is also necessary. Explicit expressions for interior displacements and stresses in integral form are also presented. All the kernels in the integral equations are explicitly provided.

The effect of boundary conditions on guided wave propagation in two-dimensional models of healing bone.

Guided wave propagation has recently drawn significant interest in the ultrasonic characterization of bone. In this work, we present a two-dimensional computational study of ultrasound propagation in healing bones aiming at monitoring the fracture healing process. In particular, we address the effect of fluid loading boundary conditions on the characteristics of guided wave propagation, using both time and time-frequency (t-f) signal analysis techniques, for three study cases. In the first case, the bone was assumed immersed in blood which occupied the semi-infinite spaces of the upper and lower surfaces of the plate. In the second case, the bone model was assumed to have the upper surface loaded by a 2mm thick layer of blood and the lower surface loaded by a semi-infinite fluid with properties close to those of bone marrow. The third case, involves a three-layer model in which the upper surface of the plate was again loaded by a layer of blood, whereas the lower surface was loaded by a 2mm layer of a fluid which simulated bone marrow. The callus tissue was modeled as an inhomogeneous material and fracture healing was simulated as a three-stage process. The results clearly indicate that the application of realistic boundary conditions has a significant effect on the dispersion of guided waves when compared to simplified models in which the bones surfaces are assumed free.

Free vibration analysis of non-axisymmetric and axisymmetric structures by the dual reciprocity BEM

The dual reciprocity boundary element method (DR/BEM) is employed for the free vibration analysis of three-dimensional non-axisymmetric and axisymmetric elastic solids. The method uses the elastostatic fundamental solution in the integral formulation of elastodynamics and as a result of that, an inertial volume integral is created in addition to the boundary ones. This volume integral is transformed into a surface integral by invoking the reciprocal theorem and expanding of the displacement field into a series involving seven different approximation functions. The approximation functions used are local radial basis functions (RBFs) and are applied in combination (or not) with global basis functions (augmentation). All these functions are compared in terms of the accuracy they provide. The axisymmetric case is efficiently treated with the aid of the fast Fourier transform (FFT) algorithm in order to provide even non-axisymmetric vibration modes. Two representative numerical examples involving the determination of natural frequencies and modal shapes of an elastic cube and an elastic cylinder serve to investigate in detail the potentiality of each of the seven approximation functions tested to provide results of high accuracy and to reach useful practical conclusions.

Static and dynamic boundary element analysis in incompressible linear elasticity

Abstract The boundary element formulation of incompressible, isotropic, linear elastostatics and frequency domain elastodynamics for the displacements and hydrostatic pressure is presented. Both two- and three-dimensional problems are considered. It is proven that the incompressible fundamental tensors can be obtained from the corresponding compressible ones by simply putting the Poisson ratio ν equal to 0.5 in elastostatics and the p -wavenumber k p equal to zero in elastodynamics. The various kernels employed in a complete static or dynamic boundary element analysis for the compressible case are written in a modified form that permits one to use already existing codes for compressible, incompressible or nearly incompressible cases without any problem. Numerical examples involving two- and three-dimensional problems under static and dynamic conditions are presented. These examples serve to illustrate the method and demonstrate its high accuracy and efficiency.

A boundary element method for solving 2-D and 3-D static gradient elastic problems. Part II: Numerical implementation

The boundary element formulation for the static analysis of two-dimensional (2-D) and three-dimensional (3-D) solids and structures characterized by a gradient elastic material behavior developed in the first part of this work, is treated numerically in this second part for the creation of a highly accurate and efficient boundary element solution tool. The discretization of the body is restricted only to its boundary and is accomplished by the use of quadratic isoparametric three-noded line and eight-noded quadrilateral boundary elements for the 2-D and 3-D cases, respectively. Advanced algorithms are presented for the accurate and efficient numerical computation of the singular integrals involved. Numerical examples involving a cylindrical bar in tension and a cylinder and a sphere in radial deformation are solved by the proposed boundary element method and the results are found in excellent agreement with the derived by the authors analytical solutions. The bar and sphere problems are solved in a 3-D context, while the cylinder problem is solved in a 2-D context (plane strain). Both the exterior and interior versions of the cylinder and sphere problems are considered.

An advanced boundary element method for axisymmetric elastodynamic analysis

An advanced boundary element method is appropriately combined with the fast Fourier transform (FFT) to analyze general axisymmetric problems in frequency domain elastodynamics. The problems are characterized by axisymmetric geometry and non-axisymmetric boundary conditions. Boundary quantities are expanded in complex Fourier series in the circumferential direction and the problem is efficiently decomposed into a series of problems, which are solved by the BEM for the Fourier coefficients of the boundary quantities, discretizing only the surface generator of the axisymmetric body. Quadratic boundary elements are used and BEM integrations are done by FFT algorithm in the circumferential direction and by Gauss quadrature in the generator direction. Singular integrals are evaluated directly in a highly accurate way. The Fourier transformed solution is then numerically inverted by the FFT to provide the final solution. The method combines high accuracy and efficiency and this is demonstrated by illustrative numerical examples.

An advanced BEM for electromagnetic wave scattering problems with axisymmetric dielectric particles

An advanced boundary element/fast Fourier transform (FFT) methodology for solving axisymmetric electromagnetic wave scattering problems with general, non-axisymmetric boundary conditions is presented. The incident field as well as the boundary quantities of the problem are expanded in complex Fourier series with respect to the circumferential direction. Each of the expanding coefficients satisfies a surface integral equation which, due to axisymmetry, is reduced to a line integral along the surface generator of the body and an integral over the angle of revolution. The first integral is evaluated by discretizing the meridional line of the body into isoparametric elements and employing Gauss quadrature. The integration over the angle of revolution is performed simultaneously for all the expanding coefficients through the FFT. The singular integrals are computed directly with high accuracy. Representative numerical examples demonstrate the accuracy of the proposed boundary element formulation.

Boundary Element Analysis of Gradient Elastic Problems

A boundary element method (BEM), suitable for solving two dimensional (2D) and three dimensional (3D) gradient elastic problems under static loading, is presented. The simplified Form-II gradient elastic theory (a simple version of Mindlin’s Form II general gradient elastic theory) is employed and the corresponding fundamental solution is exploited for the formulation of the integral representation of the problem. Three noded quadratic line and eight noded quadratic quadrilateral boundary elements are utilized and the discretization is restricted only to the boundary. The boundary element methodology is explained and presented. The importance of satisfying the correct boundary conditions, being compatible with Mindlin’s theory is demonstrated with a simple example. Three numerical examples are reported to illustrate the method and exhibit its merits.