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MS31 2: Inverse Problems in Elastic Media
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Presentations | ||
An inverse problem for the porous medium equation 1National Yang Ming Chiao Tung University, Taiwan; 2National Institute of Science Education and Research, India; 3Inha University, Korea
The porous medium equation is a degenerate parabolic type quasilinear equation that models, for example, the flow of a gas through a porous medium. In this talk I will present recent results on uniqueness in the inverse boundary value problem for this equation. These are the first such results to be obtained for a degenerate parabolic equation.
Comparison of variational formulations for the direct solution of an inverse problem in linear elasticity 1Boston University, United States of America; 2Rochester Institute of Technology, United States of America
Given one or more observations of a displacement field within a linear elastic, isotropic, incompressible object, we seek to identify the material property distribution within that object. This is a mildly ill-posed inverse problem in linear elasticity. While most common approaches to solving this inverse problem use forward iteration, several variational formulations have been proposed that allow its direct solution. We review five such direct variational formulations for this inverse problem: Least Squares, Adjoint Weighted Equation, Virtual Fields, Inverse Least Squares, Direct Error in Constitutive Eqn. [1, 2, 3, 4, 5]. We briefly review their derivations, their mathematical properties, and their compatibility with Galerkin discretization and numerical solution. We demonstrate these properties through numerical examples.
[1] P. B. Bochev, M. D. Gunzburger. Finite element methods of least-squares type, SIAM Review, 40(4): 789--837, 1998.
[2] P. E. Barbone, C. E. Rivas, I. Harari, U. Albocher, A. A. Oberai, Y. Zhang. Adjoint-weighted variational formulation for the direct solution of inverse problems of general linear elasticity with full interior data, International Journal for Numerical Methods in Engineering 81(13): 1713--1736, 2010.
[3] F. Pierron, M. Grédiac. The Virtual Fields Method: Extracting Constitutive Mechanical
Parameters from Full-field Deformation Measurements, Springer Science & Business Media, 2012.
[4] G. Bal, C. Bellis, S. Imperiale, F. Monard. Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements, Inverse Problems 30(12): 125004, 2014.
[5] O. A. Babaniyi, A. A. Oberai, P. E. Barbone. Direct error in constitutive equation formulation for plane stress inverse elasticity problem, Computer Methods in Applied Mechanics and Engineering 314: 3--18, 2017.
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