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On the use of quasi-equidistant source points over the sphere surface for the method of fundamental solutions

Publication . Araújo, António; Serranho, Pedro

The method of fundamental solutions is broadly used in science and engineering to numerically solve the direct time-harmonic scattering problem. In 2D the choice of source points is usually made by considering an inner pseudo-boundary over which equidistant source points are placed. In 3D, however, this problem is much more challenging, since, in general, equidistant points over a closed surface do not exist. In this paper we discuss a method to obtain a quasi-equidistant point distribution over the unit sphere surface, giving rise to a Delaunay triangulation that might also be used for other boundary element methods. We give theoretical estimates for the expected distance between points and the expect area of each triangle. We illustrate the feasibility of the proposed method in terms of the comparison with the expected values for distance and area. We also provide numerical evidence that this point distribution leads to a good conditioning of the linear system associated with the direct scattering problem, being therefore an adequated choice of source points for the method of fundamental solutions.

2019Journal article

Open access

On the numerical solution of the inverse elastography problem for time-harmonic excitation

Publication . Serranho, Pedro; Barbeiro, Sílvia; Henriques, Rafael; Batista, Ana; Santos, Mário; Correia, Carlos; Domingues, José Paulo; Loureiro, Custódio; Cardoso, João M. R.; Bernardes, Rui; Morgado, António Miguel

In this paper we address the numerical solution of the inverse elastography problem, from the knowledge of the excitation field on the boundary and the displacement field in a grid of points within the domain. We suggest using a representation of the solution by the method of fundamental solutions and using a Newtontype method to iteratively approximate the Lam´e coefficients of the medium from elastography displacement measurements. We consider a toy model to illustrate the performance of the method.

2022Conference object

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The method of fundamental solutions for the direct elastography problem in the human retina

Publication . Barbeiro, Sílvia; Serranho, Pedro

This paper addresses the numerical simulation of the mechanical waves propagation and induced displacements in the human retina, for the elastography imaging modality. In this way, we use a model for the human eye and numerically approximate the propagation of time-harmonic acoustic waves through the different media of the eye and the respective elastic excitation in the retina, through a layered representation approach based on the method of fundamental solutions. We present numerical results showing the feasibility of the method.

2020Book part

Restricted access

Time-dependent elastic numerical model for Optical Coherence Elastography of the murine retina

Publication . Correia, Carlos; Batista, Ana; Barbeiro, Sílvia; Cardoso, João; Domingues, José Paulo; Henriques, Rafael; Loureiro, Custódio; Santos, Mário J.; Serranho, Pedro; Bernardes, Rui; Morgado, Miguel

We present the initial stages of development of a Finite Element Method-based time-dependent elastic numerical model which seeks to support the employment of our Optical Coherence Elastography system for assessing murine retinal elasticity. The current model is able to reconstruct displacement maps in both homogeneous and heterogeneous domains with errors up to a few hundredths relatively to a known exact displacement map, within 1 millisecond. The results demonstrate the robustness of the numerical algorithm under different elastic domains, and model parametrization with real Optical Coherence Elastography data is already in progress.

2023Conference object

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