Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods

International Journal of Computer Mathematics

Reeve

2011

International audienceWe investigate an application of the method of fundamental solutions (MFS) to heat conduction in two-dimensional bodies, where the thermal diffusivity is piecewise constant. We extend a recent MFS for one-dimensional heat conduction with the sources placed outside the space domain of interest (Engng. Anal. Boundary Elements 32, 697-703, 2008), to the two-dimensional setting. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be obtained efficiently with small computational cost

Section: Theorem 32 the Set Of Functions {Vmentioning

confidence: 99%

“…We have obtained the above equations by observing that the fundamental solution satisfies (10); the Heaviside function makes the fundamental solution equal to zero in equation (12), and (8) gives us (11). By the uniqueness Theorem 2.1, the only solution to the equations (10)- (12) [24].…”

Section: Denseness On the Lateral Surfacementioning

confidence: 99%

A method of fundamental solutions for two-dimensional heat conduction

Johansson

International Journal of Computer Mathematics

Reeve

2011

“…In the MFS, taking the pseudo-boundary similar to the boundary yields, in general, improved results as has been demonstrated by Gorzelańczyk and Kołodziej [15]. In (4), the singularities (ξ 2 n,m ) n=1,N ,m=1,M are not preassigned.…”

Section: The Methods Of Fundamental Solutions (Mfs)mentioning

confidence: 65%

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

Karageorghis

Marin

2016

Computers & Structures

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.

“…is a fixed constant (Gorzelańczyk and Ko lodziej, 2008). According to the notation used in Section 3, the corresponding MFS parameters have been set as follows: on Γ 1 , with ℓ ∈ { 4, 3, 2 } corresponding to cases (a)-(c), respectively.…”

Section: Numerical Resultsmentioning

confidence: 99%

Regularized MFS solution of inverse boundary value problems in three-dimensional steady-state linear thermoelasticity

Marin

Karageorghis

International Journal of Solids and Structures

2016

We investigate the numerical reconstruction of the missing thermal and mechanical boundary conditions on an inaccessible part of the boundary in the case of three-dimensional linear isotropic thermoelastic materials from the knowledge of over-prescribed noisy data on the remaining accessible boundary. We employ the method of fundamental solutions (MFS) and several singular value decomposition (SVD)-based regularization methods, e.g. the Tikhonov regularization method (Tikhonov and Arsenin, 1986), the damped SVD and the truncated SVD (Hansen, 1998), whilst the regularization parameter is selected according to the discrepancy principle (Morozov, 1966), generalized cross-validation criterion (Golub et al., 1979) and Hansen's L-curve method (Hansen and O'Leary, 1993).