S. Nintcheu Fata

ELASTIC SCATTERER RECONSTRUCTION VIA THE ADJOINT SAMPLING METHOD

An inverse problem dealing with the reconstruction of voids in a uniform semi-infinite solid from near-field elastodynamic waveforms is investigated via the linear sampling method. To cater to active imaging applications that are characterized by a limited density of illuminating sources, existing formulation of the linear sampling method is advanced in terms of its adjoint statement that features integration over the receiver surface rather than its source counterpart. To deal with an ill-posedness of the integral equation that is used to reconstruct the obstacle, the problem is solved by alternative means of Tikhonov regularization and a preconditioned conjugate gradient method. Through a set of numerical examples, it is shown (i) that the adjoint statement elevates the performance of the linear sampling method when dealing with scarce illuminating sources, and (ii) that a combined use of the existing formulation together with its adjoint counterpart represents an effective tool for exposing an undersam...

Semi-analytic integration of hypersingular Galerkin BIEs for three-dimensional potential problems

An accurate and efficient semi-analytic integration technique is developed for three-dimensional hypersingular boundary integral equations of potential theory. Investigated in the context of a Galerkin approach, surface integrals are defined as limits to the boundary and linear surface elements are employed to approximate the geometry and field variables on the boundary. In the inner integration procedure, all singular and non-singular integrals over a triangular boundary element are expressed exactly as analytic formulae over the edges of the integration triangle. In the outer integration scheme, closed-form expressions are obtained for the coincident case, wherein the divergent terms are identified explicitly and are shown to cancel with corresponding terms from the edge-adjacent case. The remaining surface integrals, containing only weak singularities, are carried out successfully by use of standard numerical cubatures. Sample problems are included to illustrate the performance and validity of the proposed algorithm.

Explicit expressions for three-dimensional boundary integrals in linear elasticity

On employing isoparametric, piecewise linear shape functions over a flat triangle, exact formulae are derived for all surface potentials involved in the numerical treatment of three-dimensional singular and hyper-singular boundary integral equations in linear elasticity. These formulae are valid for an arbitrary source point in space and are represented as analytical expressions along the edges of the integration triangle. They can be employed to solve integral equations defined on triangulated surfaces via a collocation method or may be utilized as analytical expressions for the inner integrals in a Galerkin technique. A numerical example involving a unit triangle and a source point located at various distances above it, as well as sample problems solved by a collocation boundary element method for the Lame equation are included to validate the proposed formulae.

Boundary integral approximation of volume potentials in three-dimensional linear elasticity

A systematic treatment of volume potentials appearing in a boundary integral equation formulation of the three-dimensional Lame equation is rigorously investigated and its usefulness demonstrated in the context of a collocation boundary element method. Developed to effectively deal with volume potentials without a volume-fitted mesh, the proposed approach initially converts elastic volume potentials, defined in the form of domain integrals featuring a non-trivial body force, into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. Details of these domain-to-boundary integral transformations are provided along with some examples to show the correctness of the calculation of the elastic Newton potential. Moreover, with the aid of an analytic integration technique developed to accurately compute singular surface integrals in linear elasticity, numerical examples dealing with mixed boundary-value problems for the three-dimensional Lame equation are included to validate the proposed approach.

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.