Nuno F. M. Martins

Recovering the source term in a linear diffusion problem by the method of fundamental solutions

This work considers the detection of the spatial source term distribution in a multidimensional linear diffusion problem with constant (and known) thermal conductivity. This work can be physically associated with the detection of non-homogeneities in a material that are inclusion sources in a heat conduction problem. The uniqueness of the inverse problem is discussed in terms of classes of identifiable sources. Numerically, we propose to solve these inverse source problems using fundamental solution-based methods, namely an extension of the method of fundamental solutions to domain problems. Several examples are presented and the numerical reconstructions are discussed.

The direct method of fundamental solutions and the inverse Kirsch- Kress method for the reconstruction of elastic inclusions or cavities

In this work we consider the inverse problem of detecting inclusions or cavities in an elastic body, using a single boundary measurement on an external boundary. We discuss the identifiability questions on shape reconstruction, presenting counterexamples for Robin boundary conditions or with additional unknown Lame parameters. Using the method of fundamental solutions (MFS) we adapt a method introduced twenty years ago by Andreas Kirsch and Rainer Kress [17] (in the context of an exterior problem in acoustic scattering) to this inverse problem in a bounded domain. We prove density results that justify the reconstruction of the solution from the Cauchy data using the MFS. We also establish some connections between this linear part of the KirschKress method and the direct MFS, through matrices of boundary layer integrals. Several numerical examples are presented, showing that with noisy data we were able to retrieve a fairly good reconstruction of the shape (or of its convex hull) with this MFS version of the Kirsch-Kress method.

A meshfree method for elasticity problems with interfaces

In this work, we develop a meshfree method based on fundamental solutions basis functions for a transmission problem in linear elasticity. The addressed problem consists in, given the displacement field on the boundary, compute the corresponding displacement field of an elastic object (which has piecewise constant Lame coefficients). The Lame coefficients are assumed to be constant in non overlapping subdomains and, on the corresponding interface (interior boundaries), non homogeneous jump conditions on the displacement and on the traction vectors are considered. The main properties of the method are analyzed and illustrated with several numerical simulations in 2D domains.

An iterative shape reconstruction of source functions in a potential problem using the MFS

In this work, we address the reconstruction of characteristic source functions in a potential problem, from the knowledge of full and partial boundary data. The inverse problem is formulated as an inverse obstacle problem and two iterative methods are applied. A decomposition method based on the Kirsch–Kress method (requires Cauchy data reconstruction) and a Newton-type method based on the domain derivative (requires the resolution of direct transmission problems) has been applied. For the reconstruction of Cauchy data we use the method of fundamental solutions (MFS) and we show that, for partial data, we can consider only one exterior artificial boundary. We test the domain derivative method using the MFS (for transmission problems) and present theoretical results that justifiy this numerical approximation. The feasibility of these methods will be illustrated by numerical simulations for both full and partial data.

Identification and reconstruction of elastic body forces

We address the identification and reconstruction of 2D elastostatic and elastodynamic body forces from pairs of displacement and traction boundary data. As in the scalar acoustic case, an elastic body force cannot be fully identified from a single boundary measurement. We present some partial identification results for a single and multiple boundary measurements. We show that full identification can be obtained by considering boundary measurements along a full set of frequencies. We present and test two numerical methods for retrieving the body force.

The Method of Fundamental Solutions applied to a heat conduction inverse problem

The Method of Fundamental Solutions (MFS) is a meshless method that is used to solve an inverse heat conduction problem. The inverse problem consists in finding the shape of a cavity inside a domain R2 with prescribed temperature on the boundary of the cavity. On the exterior boundary of the domain the temperature is imposed and we measure the induced heat flux. We prove that this problem has a unique solution, under certain hypothesis and use MFS to solve it numerically. Density results for the MFS justify the adequacy of this method as a direct solver and suggest its use as an inverse method. However the straightforward application of the MFS as an inverse method lead to poor numerical reconstructions. Therefore we also introduce an alternative use of the MFS with a minimization Quasi-Newton method that uses the MFS as forward solver. Using this procedure, the numerical reconstructions are much better and stable to the introduction of random noise in the measured data. Recovery of the cavities with partial access for measurements is also considered (see Fig. 1). Open image in new window Figure 1 Left plot - Accessible parts of the exterior boundary. Right plot - Results obtained with the iterative scheme. Full line: starting curve; dashed lines: intermediate curves; dotted line: final curve; full bold line: cavity shape.

On Inverse Problems for Characteristic Sources in Helmholtz Equations

We consider the inverse problem that consists in the determination of characteristic sources, in the modified and classical Helmholtz equations, based on external boundary measurements. We identify the location of the barycenter establishing a simple formula for symmetric shapes, which also holds for the determination of a single source point. We use this for the reconstruction of the characteristic source, based on the Method of Fundamental Solutions (MFS). The MFS is also applied as a solver for the direct problem, using an equivalent formulation as a jump or transmission problem. As a solver for the inverse problem, we may apply minimization using an equivalent reciprocity functional formulation. Numerical experiments with the barycenter and the boundary reconstructions are presented.

Identification results for inverse source problems in unsteady Stokes flows

We establish identification results in an inverse source problem for the two-dimensional Brinkman equations. This identification problem is cast in the context of a nondestructive evaluation problem that consists in retrieving a pair of forces, namely a body force and a divergence force, from the corresponding Cauchy data. Results are established for data obtained from a single measurement and from several measurements. Extension to three-dimensional problems is also discussed.

Meshfree methods for nonhomogeneous Brinkman flows

Abstract We present a meshfree method for the numerical solution of non homogeneous Brinkman systems with Dirichlet and traction boundary conditions. The aim of the paper is to propose a representation for the fluid flow as a linear superposition of unsteady Stokeslets and elastodynamic P and S waves. A theoretically study for the proposed choice of basis functions is given and several numerical examples are presented in order to illustrate and discuss the feasibility of the method.

A stokeslets approach for the numerical solution of Brinkman systems

In this work we present some properties of unsteady hydrodynamic potentials in order to justify a meshfree numerical method for Brinkman systems with Dirichlet and Neumann boundary conditions. The problems here addressed concerns interior 2D problems and can be easily extended for exterior and 3D problems. We will illustrate the feasibility of the methods with some numerical simulations.