Boundary Element Regularized Algorithm for Boundary Condition Inverse Problems in Elasticity and Its Comparative Analysis
|School||Hefei University of Technology|
|Keywords||Inverse problems Boundary element method Regularization methods Boundary conditions Nearly singular integrals|
The boundary element method is developed to analyze the Cauchy boundarycondition inverse problems in2-D isotropic elasticity. Truncated singular valuedecomposition (TSVD) technique is applied to solving the ill-posed problem. L-Curvemethod is proposed to select the regularization parameter.i.e. the optimal truncationnumber, and then the solution of this problem can be obtained. The comparison ofnumerical and analytical solutions in numerical examples shows that the TSVDalgorithm is effective and stable. The regularization errors are also analyzed. Theaccuracy of the solution can be improved with respect to reducing the amount of noiseadded into the known data and refining the boundary element mesh size.Displacement or traction boundary conditions are given on a part of boundary,whilst all of displacement and traction vectors are unknown on the rest of the boundary.All the unknown boundary conditions are to be determined with some measurabledisplacement information at some interior points. The preconditioned conjugate gradientmethod (PCGM) based on implicit transformation and the conjugate gradient method(CGM) are employed to regularize the ill-posed problem, and the Morozov’s discrepancyprinciple is used to select the regularization parameter. In the case of boundary conditioninverse problems relating to interior point information, there exist the difficulties toaccurately calculate nearly singular integrals (NSI) when the interior points are veryclose to the boundary, due to the failure of the conventional Gauss numerical quadrature.Analytical integral algorithms are applied to evaluating the nearly singular integrals first,and then the ill-posed inverse problems are regularized. The accuracy of the solution canbe improved by reducing the random noise added into the known data and increasingmore interior points. The numerical solutions of the proposed iterative methods convergeto analytical values with respect to increasing the number of boundary elements, at thesame time keeping a fixed number of interior points. Both of the two iterative methodsare not sensitive to the position of interior points when the interior points distribute alongthe whole boundary. The computation time of the CGM is a bit less than the computationtime of the PCGM owing to less iteration steps. Numerical examples show theeffectiveness and stability of these two iterative methods.