The aim of this work is an analysis of some geometrical inverse problems related to the identification of cavities in linear elasticity framework. We rephrase the inverse problem into a shape optimization one using an energetic least-squares functional. The shape derivative of this cost functional is combined with the level set method in a steepest descent algorithm to solve the shape optimization problem. The efficiency of this approach is illustrated by several numerical results.
This work focuses on the sub-Cauchy problem for linear elasticity in two dimentional case. Solving such a problem may be formulated as follows: given the displacement and one component of the traction in a given part of the boundary of the elastic body, reconstruct the displacement field in all the domain. Author propose herein, an iterative method borrowed from the domain decomposition communauty to solve the sub-Cauchy problem. Numerical results highlight the efficiency of the proposed method.
This work is devoted to some geometric inverse problems in linear elasticity. The problem considered is the cavities identification in mechanical structures from the knowledge of partially overdetermined boundary data, namely the displacement field and the normal component of the normal stress. We state a uniqueness result from a single pair of data under some geometrical assumptions. We propose an iterative method based on the coupling of the data completion process through the Steklov-Poincaré operator to reconstruct the shear stress and of the shape gradient method combined with the level set method to identify cavities. Numerical simulations highlight the algorithm efficiency.
Highlights• A model for leak identification in pipes via the Cauchy problem for the heat equation is researched.• The model is reformulated to fit the application of a recently proposed regularising method.• Analyses of the regularizing method is presented.• The regularizing method is implemented using an open source Finite element code.• Conclusions and further research in this area are given.
AbstractThis work is an initial study of a numerical method for identifying multiple leak zones in saturated unsteady flow. Using the conventional saturated groundwater flow equation, the leak identification problem is modelled as a Cauchy problem for the heat equation and the aim is to find the regions on the boundary of the solution domain where the solution vanishes, since leak zones correspond to null pressure values. This problem is ill-posed and to reconstruct the solution in a stable way, we therefore modify and employ an iterative regularizing method proposed in [13,14]. In this method, mixed well-posed problems obtained by changing the boundary conditions are solved for the heat operator as well as for its adjoint, to get a sequence of approximations to the original Cauchy problem. The mixed problems are solved using a Finite element method (FEM), and the numerical results indicate that the leak zones can be identified with the proposed method.
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