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Aerospace Engineering

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#### Use of the conventional and tangent derivative boundary integral equations for the solution of problems in linear elasticity

Regularized forms of the traction and tangent derivative boundary integral equations of elasticity are derived for the case of closed regions. The hypersingular and strongly singular integrals of the displacement gradient representation are regularized independently, through identities of the fundamental solution and its various derivatives, before the boundary integral equations are formed. Besides the displacements and the tractions, only the tangential derivatives of the displacements evaluated at the singular point appear in the regularized equations making them well suited for numerical treatment. The regularization of the hypersingular integrals demands that the displacement components have Holder continuous first derivatives at the singular point. Consistent with this requirement, the regularization of the strongly singular integrals is effective if the tractions and the unit vectors normal and tangent to the surface are continuous at that location.;Higher order elements for two and three dimensional elastostatic problems are implemented through the coincident collocation of regularised forms of the displacement and the tangent derivative equations. The nodal values of the displacements, the fractions and their tangential derivatives are used as the degrees of freedom associated with the functional representation of the boundary variables. The tangential derivatives of the displacements and the tractions at the functional nodes are directly recovered from the boundary solution with comparable accuracy as the primative variables. Hence, the nodal values of the stress components are directly obtained through Hooke's law and need not be determined in a post processing manner. Several numerical examples demonstrate the advantages of the higher order elements versus the conventional ones. In two dimensions, four degrees of freedom per node Hermitian elements are used for functional interpolation only on those portions of the boundary where the gradients are high and quadratic Lagrangian elements are employed for the remaining parts of the modelled region. In three dimensions, nine degrees of freedom per node, incomplete quartic elements are employed for the approximation of the displacements and the tractions. Finally, the methodology presented here is general and can be extended to other problems amenable to a boundary integral formulation.

#### Application of displacement and traction boundary integral equations for fracture mechanics analysis

A general formulation by dual boundary integral equations and a computational solution algorithm for the general mixed-mode crack in a linearly elastic, isotropic medium is presented. Traction boundary integral equations are collocated at the points on one crack surface while displacement boundary integral equations are collocated at the opposite points on the other surface to ensure a unique solution. The hypersingular and strongly singular integrals in the traction and displacement boundary integral equations are regularized before the numerical implementation. The singular integration elements on both crack surfaces are replaced by smooth curved auxiliary surfaces to avoid the direct integration over the singular elements. Usage of these detoured auxiliary contours is justified by certain identities of the fundamental solutions. Convergence tests for integration order and subdivision of the auxiliary surface elements were performed. To demonstrate the accuracy and efficiency of the present technique, the deformation and the stress intensity factors for two- and three-dimensional embedded and edge crack problems are given.

#### Efficient inversion of Fourier and Laplace domain boundary element solutions for elastodynamic scattering

In ultrasonic nondestructive testing, the actual time domain signature of a scattered wave due to a pulsed input is the common observable (via oscilloscope) quantity. Numerically simulated solutions are thus desirable in the time domain. These can be obtained by working directly in the time domain or by inversions of integral transform solutions in the frequency (Fourier) or Laplace domains. Direct time solutions are suitable for short times but generally deteriorate for longer times. Alternately, the transform techniques generally require solutions over a wide spectrum in the transform variable to provide accurate inversions back to the time domain. However, without special provisions, numerical implementations like boundary or finite elements are limited to lower frequencies or Laplace parameter values. Hence, ways to minimize the number of transform variable solutions for canonical problems are important. In this dissertation the Fourier and Laplace transform methods are compared and investigations are made into ways of minimizing the solutions in the spectrum space in a boundary element setting. Three common scattering models--a spherical void and two spherical inclusions--are used in the analysis and the numerical results are compared to analytical solutions and to experimental results.

#### Boundary element solutions to wave scattering by surface anomalies on a fluid-solid interface

The direct multi-region boundary element method (BEM) is developed to simulate the interaction of ultrasonic waves with submerged finite elastic bodies. This technique is further specialized to model fluid-solid half-space problems with fluid-filled indentations and air bubbles on the solid surface. These problems usually occur in ultrasonic immersion testing and are of particular interest to the NDE community. The problems are formulated in the Fourier (frequency) domain and are described by the three dimensional acoustic and elastodynamic boundary integral equations (BIE) with pressure and displacement serving as primary variables. The techniques developed are general and may be used with any kind of incident wave, although plane waves are used in all numerical experiments. The equations governing the acoustic region are first converted mathematically to equations like those of an elastodynamic region and the resulting two elastodynamic regions are coupled and solved for the interface displacements. Tractions, pressures, and pressure gradients are subsequently computed using these displacements and the interface conditions. Numerical results are obtained for some typical problems and justified using reciprocity relations and by comparison with solutions available for the half-space elastodynamic problem.