Aug 23, 2016 - Study theoretically/numerically the reconstruction of a damaged thermal insulating material in a copper converter using electrostatic/thermal ...
Introduction Some general inverse conductivity problems Conclusion
Reconstruction of interfaces using elliptic partial differential equations Application to the geometric inverse problem of corrosion detection in a copper converter
Jérémy Dalphin* - Axel Osses* - Matias Courdurier◦ *Centro de Modelamiento Matemático (CMM), Facultad de Ciencas Físicas y Matemáticas. ◦ Pontificia Universidad Católica de Chile (PUC), Facultad de Matemáticas.
Tuesday August 23th 2016
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Jérémy Dalphin - Reconstruction of interfaces using elliptic PDEs
August 23th 2016 - Instituto de Innovación en Minería y Metalurgia
Introduction Some general inverse conductivity problems Conclusion
General objectives associated with the project
General objectives associated with the project Study theoretically/numerically the reconstruction of a damaged thermal insulating material in a copper converter using electrostatic/thermal measurements. Give some quantitative/qualitative results concerning the unknown shape of the interface air/copper and the location of the purified copper, especially near the insulating layer.
Ωair
Ωcopper
Ωinsulation 2/6
Jérémy Dalphin - Reconstruction of interfaces using elliptic PDEs
August 23th 2016 - Instituto de Innovación en Minería y Metalurgia
Introduction Some general inverse conductivity problems Conclusion
Recovering the conductivity from boundary measurements A geometric inverse problem as a limiting case Another corrosion detection problem
Recovering the conductivity from boundary measurements The situation can be modelled by the following equation: div (γ∇u) = 0 in Ω := Ωair t Ωcopper t Ωinsulation u=ϕ on Γmeasure ∂n u = 0 on ∂Ω\Γmeasure . Goal: recover γ = γair 1Ωair + γcopper 1Ωcopper + γinsulating 1Ωinsulating from the knowledge of ∂n u|Γmeasure .
Input Ωair Γmeasure Ωcopper Output
Ωinsulation 3/6
Jérémy Dalphin - Reconstruction of interfaces using elliptic PDEs
August 23th 2016 - Instituto de Innovación en Minería y Metalurgia
Introduction Some general inverse conductivity problems Conclusion
Recovering the conductivity from boundary measurements A geometric inverse problem as a limiting case Another corrosion detection problem
A geometric inverse problem as a limiting case Since γair = 0 and ∆u = 0 u=ϕ u=0 ∂n u = 0
γcopper = +∞, we obtain the boundary value problem: in on on on
Ω Γmeasure Γcopper Γair and ∂Ω\ (Γmeasure t Γair t Γcopper ) .
Goal: recover Γair and Γcopper from the knowledge of ∂n u|Γmeasure , add some geometrical informations at the transition point (position, normal).
Γair
Γmeasure
Γcopper
Ω 4/6
Jérémy Dalphin - Reconstruction of interfaces using elliptic PDEs
August 23th 2016 - Instituto de Innovación en Minería y Metalurgia
Introduction Some general inverse conductivity problems Conclusion
Recovering the conductivity from boundary measurements A geometric inverse problem as a limiting case Another corrosion detection problem
Another corrosion detection problem The corrosion can also be modelled by the boundary value problem: in Ω ∆u = 0 u=ϕ on Γaccessible ∂n u + γu = 0 on Γinacessible , ∂n u = 0 on ∂Ω\ (Γaccessible t Γinaccessible ) Goal: recover γ from the knowledge of ∂n u|Γmeasure , locate the purified copper and the slag phase.
Γaccessible
Γinacessible
Ω 5/6
Jérémy Dalphin - Reconstruction of interfaces using elliptic PDEs
August 23th 2016 - Instituto de Innovación en Minería y Metalurgia
Introduction Some general inverse conductivity problems Conclusion
Some perspectives of research
Some perspectives of research Industrial issues: using asymptotic analysis and perturbations methods, give some informations about the transition air/copper, slag/purified phases, damage width of the insulating layer. Mathematical issues: wellposedness of the geometric inverse problems (uniqueness), good stability estimates (Lipschitz). Numerical issues: these problems can also be seen as shape optimization ones by considering for example the functional: Z inf |∂n uΩ − g|2 , Ω
Γmeasure
where uΩ is the solution of a PDE posed on a domain Ω with ϕ as a fixed input and g as the fixed output to reach. 6/6
Jérémy Dalphin - Reconstruction of interfaces using elliptic PDEs
August 23th 2016 - Instituto de Innovación en Minería y Metalurgia
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