We consider the design of finite element methods for inverse problems with
full-field data governed by elliptic forward operators. Such problems arise in
applications in inverse heat conduction, in mechanical property
characterization, and in medical imaging. For this class of problems, novel
finite element methods have been proposed (Barbone et al., 2010) that
give good performance, provided the solutions are in the H^1(Ω) function
space. The material property distributions being estimated can be discontinuous,
however, and therefore it is desirable to have formulations that can
accommodate discontinuities in both data and solution. Toward this end, we
present a mixed variational formulation for this class of problems that handles
discontinuities well. We motivate the mixed formulation by examining the
possibility of discretizing using a discontinuous discretization in an irreducible finite
element method, and discuss the limitations of that approach. We then derive a
new mixed formulation based on a least-square error in the constitutive
equation. We prove that the continuous variational formulations are well-posed
for applications in both inverse heat conduction and plane stress elasticity. We
derive a priori error bounds for discretization error, valid in the limit
of mesh refinement. We demonstrate convergence of the method with mesh
refinement in cases with both continuous and discontinuous solutions. Finally we
apply the formulation to measured data to estimate the elastic shear modulus
distributions in both tissue mimicking phantoms and in breast masses from data
collected in vivo.
| Identifer | oai:union.ndltd.org:bu.edu/oai:open.bu.edu:2144/14640 |
| Date | 17 February 2016 |
| Creators | Babaniyi, Olalekan |
| Source Sets | Boston University |
| Language | en_US |
| Detected Language | English |
| Type | Thesis/Dissertation |
| Rights | Attribution 4.0 International, http://creativecommons.org/licenses/by/4.0/ |
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