Inverse problems in the presence of inclusions and unilateral constraints: a boundary element approach

Abstract

The authors present a boundary element method (BEM) numerical procedure for the solution of 2D nondestructive identi®cation problems in the presence of unilateral boundary conditions. Firstly, the position of a deformable inclusion in frictionless unilateral contact with the matrix is identi®ed on the basis of measurements surveyed at some sensor points on the external boundary where given static loads are applied (identi®cation problem). Then the procedure used is extended to the identi®cation of the position which a rigid/deformable inclusion must occupy within the matrix in order to maximise the structural stiffness of the matrix-inclusion system under prescribed external loads (optimisation problem). Matrix and deformable inclusion are both considered linear elastic. A minimisation problem is stated with design variables representing the size and the shape of the inclusion. The cost function is an error function that evaluates the difference between computed and observed displacements at the sensor points in the identi®cation problem and the strain energy accumulated by the matrixinclusion system in the optimisation problem. The minimisation is performed by using a ®rst-order nonlinear optimisation technique in which the cost function gradient is computed by implicit differentiation. Some simple but meaningful examples are presented and discussed in order to show the applicability of the proposed technique.

Material constants identification in anisotropic materials using boundary element techniques

“…First of all, it is obvious that Material constants identification in anisotropic materials 643 in regions of the domain where the stresses were zero, any distribution of elastic constants would produce the same distribution of displacements and tractions on the boundary. Moreover, if one component of the stress tensor is zero, say 12 , the corresponding elastic constant in the diagonal of matrix a, (a 66 in this case) will be undetermined. This problem will not appear as far as the region whose elastic constant we are trying to obtain is under a general state of stresses, which is not zero nor very small throughout the region of interest.…”

“…They make use of FEM and the Levenberg-Marquardt method to minimize the difference between the response of the system and the model. Mallardo and Alessandri [12] applied Boundary Element techniques to a very similar problem, but in this case the gradient of the functional was computed by implicit derivation, instead of the finite differences used in [11].…”

Manufacturing cost and reliability‐based shape optimization of plate structures

“…Previous work by the research community involving exact⧵implicit derivatives of BEM formulations have focused on 2D structures, 1,3,[21][22][23][24][25][26][27][28] with some work conducted on 3D structures, 13,29 and one work so far on plate structures. 20 One notable example is Huang et al in which the implicit derivatives for the 2D dual boundary element method (DBEM), a version of BEM effective at modeling cracks, were developed for the first time and used to estimate the reliability of 2D structures using the First-Order Reliability Method (FORM).…”

Shape Optimisation of Assembled Plate Structures with the Boundary Element Method

“…In addition, by exploiting the inherent advantages of the BEM mentioned earlier, it would also be more computationally efficient, since large parts of the BEM mesh would remain unchanged during the sensitivity analyses. Previous examples of work by the research community involving the derivation of the exact implicit derivatives of BEM formulations include [2,[20][21][22] for 2D structures, [23] for 3D structures, and [18,19] for simple single-plate structures. There have not yet been any works involving assembledplate structures.…”