The method of fundamental solutions for three-dimensional inverse geometric elasticity problems

Abstract

We investigate the numerical reconstruction of smooth star-shaped voids (rigid inclusions and cavities) which are compactly contained in a three-dimensional isotropic linear elastic medium from a single set of Cauchy data (i.e. nondestructive boundary displacement and traction measurements) on the accessible outer boundary. This inverse geometric problem in three-dimensional elasticity is approximated using the method of fundamental solutions (MFS). The parameters describing the boundary of the unknown void, its centre, and the contraction and dilation factors employed for selecting the fictitious surfaces where the MFS sources are to be positioned, are taken as unknowns of the problem. In this way, the original inverse geometric problem is reduced to finding the minimum of a nonlinear least-squares functional that measures the difference between the given and computed data, penalized with respect to both the MFS constants and the derivative of the radial coordinates describing the position of the star-shaped void. The interior source points are anchored and move with the void during the iterative reconstruction procedure. The feasibility of this new method is illustrated in several numerical examples.

The Method of Fundamental Solutions for Solving the Inverse Problem of Magma Source Characterization

“…However, the regularization term l 3 jCj is not included. In Karageorghis et al (2016), there were only a small number of components and numerical solutions with no implications for the instability of C. Initial coordinates of the magma source centre were set as X ¼0.1, Y ¼0.1, and the depth as Z ¼ À5.4 km. First, the problem was solved without considering the regularization parameters as shown in Figure 5(a).…”

“…Consequently, the results of the model do not conform to a sphere shape as expected from a magma source. In the second stage, l 1 >0; l 2 ¼ 0 as suggested in Karageorghis et al (2016) were considered. Using the Lcurve method, the regularization parameter was selected.…”

“…In comparison with Figure 5(a) where no regularization was employed, it can be seen that the modelled magma source is closer to a sphere shape. Third, the problem was solved using l 1 ¼ 0; l 2 >0, again as suggested in Karageorghis et al (2016).…”

Introductory Chapter: Analytical and Numerical Approaches in Engineering Elasticity

“…The first step was to check the basic differential equations in terms of satisfaction with the placement of the estimated displacement functions. The second step was to check the "initial values" or the "boundary conditions" of the problem [3][4][5]. Values were substituted into the differential equations in order to satisfy the conditions at these defined coordinates or at time domains.…”

“…In solid mechanics and elasticity theory, the governing partial differential equations, the constitutive and kinematics equations, and the initial and boundary conditions have been all defined. However, if at least one of the above conditions has remained partially or entirely unknown, then one has a so-called inverse problem (Figure 2) [5]. On the other hand, the elasticity "inverse problem" has been defined for the problems in which they consist of recovering the missing displacements to the solution space corresponding to the applied force data by using the iterative calculation steps.…”