Vianey Villamizar

On the multi-frequency inverse source problem in heterogeneous media

The inverse source problem where an unknown source is to be identified from knowledge of its radiated wave is studied. The focus is placed on the effect that multi-frequency data have on establishing uniqueness. In particular, it is shown that data obtained from finitely many frequencies are not sufficient. On the other hand, if the frequency varies within a set with an accumulation point, then the source is determined uniquely, even in the presence of highly heterogeneous media. In addition, an algorithm for the reconstruction of the source using multi-frequency data is proposed. The algorithm, based on a subspace projection method, approximates the minimum-norm solution given the available multi-frequency measurements. A few numerical examples are presented.

Coupling of Dirichlet-to-Neumann boundary condition and finite difference methods in curvilinear coordinates for multiple scattering

The applicability of the Dirichlet-to-Neumann technique coupled with finite difference methods is enhanced by extending it to multiple scattering from obstacles of arbitrary shape. The original boundary value problem (BVP) for the multiple scattering problem is reformulated as an interface BVP. A heterogenous medium with variable physical properties in the vicinity of the obstacles is considered. A rigorous proof of the equivalence between these two problems for smooth interfaces in two and three dimensions for any finite number of obstacles is given. The problem is written in terms of generalized curvilinear coordinates inside the computational region. Then, novel elliptic grids conforming to complex geometrical configurations of several two-dimensional obstacles are constructed and approximations of the scattered field supported by them are obtained. The numerical method developed is validated by comparing the approximate and exact far-field patterns for the scattering from two circular obstacles. In this case, for a second order finite difference scheme, a second order convergence of the numerical solution to the exact solution is easily verified.

Generation of smooth grids with line control for scattering from multiple obstacles

A new approach to generate structured grids for two-dimensional multiply connected regions with several holes is proposed. The bounding curves may include corners or cusps. The new algorithm constitutes an extension of the Branch Cut Grid Line Control (BCGC) technique introduced byVillamizar et al. [V. Villamizar, O. Rojas, J. Mabey, Generation of curvilinear coordinates on multiply connected regions with boundary-singularities, J. Comput. Phys. 223 (2007) 571-588] to domains with a finite number of holes. Regions with multiple holes are reduced to several contiguous single hole subregions. Then, the BCGC algorithm is applied to each single hole subregion producing a smooth grid with line control. Finally, the subregions with their respective grids are joined and their interfaces are smoothed resulting a globally smooth grid. The advantages of the novel grids are revealed by employing them to numerically solve acoustic scattering problems in the presence of multiple complexly shaped obstacles.

The DtN nonreflecting boundary condition for multiple scattering problems in the half-plane

Abstract The multiple-Dirichlet-to-Neumann (multiple-DtN) non-reflecting boundary condition is adapted to acoustic scattering from obstacles embedded in the half-plane. The multiple-DtN map is coupled with the method of images as an alternative model for multiple acoustic scattering in the presence of acoustically soft and hard plane boundaries. As opposed to the current practice of enclosing all obstacles with a large semicircular artificial boundary that contains portion of the plane boundary, the proposed technique uses small artificial circular boundaries that only enclose the immediate vicinity of each obstacle in the half-plane. The adapted multiple-DtN condition is simultaneously imposed in each of the artificial circular boundaries. As a result the computational effort is significantly reduced. A computationally advantageous boundary value problem is numerically solved with a finite difference method supported on boundary-fitted grids. Approximate solutions to problems involving two scatterers of arbitrary geometry are presented. The proposed numerical method is validated by comparing the approximate and exact far-field patterns for the scattering from a single and from two circular obstacles in the half-plane.

A least squares finite element method with high degree element shape functions for one-dimensional Helmholtz equation

An application of least squares finite element method (LSFEM) to wave scattering problems governed by the one-dimensional Helmholtz equation is presented. Boundary conditions are included in the variational formulation following Cadenas and Villamizars previous paper in Cadenas and Villamizar [C. Cadenas, V. Villamizar, Comparison of least squares FEM, mixed galerkin FEM and an implicit FDM applied to acoustic scattering, Appl. Numer. Anal. Comput. Math. 1 (2004) 128-139]. Basis functions consisting of high degree Lagrangian element shape functions are employed. By increasing the degree of the element shape functions, numerical solutions for high frequency problems can be easily obtained at low computational cost. Computational results show that the order of convergence agrees with well known a priori error estimates. The results compare favorably with those obtained from the application of a mixed Galerkin finite element method (MGFEM).

High order local absorbing boundary conditions for acoustic waves in terms of farfield expansions

We devise a new high order local absorbing boundary condition (ABC) for radiating problems and scattering of time-harmonic acoustic waves from obstacles of arbitrary shape. By introducing an artificial boundary S enclosing the scatterer, the original unbounded domain is decomposed into a bounded computational domain and an exterior unbounded domain +. Then, we define interface conditions at the artificial boundary S, from truncated versions of the well-known Wilcox and Karp farfield expansion representations of the exact solution in the exterior region +. As a result, we obtain a new local absorbing boundary condition (ABC) for a bounded problem on , which effectively accounts for the outgoing behavior of the scattered field. Contrary to the low order absorbing conditions previously defined, the error at the artificial boundary induced by this novel ABC can be easily reduced to reach any accuracy within the limits of the computational resources. We accomplish this by simply adding as many terms as needed to the truncated farfield expansions of Wilcox or Karp. The convergence of these expansions guarantees that the order of approximation of the new ABC can be increased arbitrarily without having to enlarge the radius of the artificial boundary. We include numerical results in two and three dimensions which demonstrate the improved accuracy and simplicity of this new formulation when compared to other absorbing boundary conditions.