Page 527 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 527
NONSMOOTH VARIATIONAL METHODS FOR PDES AND APPLICATIONS IN
MECHANICS (MS-8)
[5] S. Giuffrè, Lagrange multipliers and non-constant gradient constrained problem, Journal
of Differential Equations 269 (1), 542-562, 2020.
[6] H. Ishii, S. Koike, Boundary regularity and uniqueness for an elliptic equation with gradi-
ent constraint, Comm. in Partial Diff. Eq., 8(4) (1983), 317- 346.
[7] P.L. Lions, Generalized Solutions for Hamilton-Jacobi Equations, Pitman Advanced Pub-
lishing Program, Boston-London-Melbourne, 1982.
[8] M. Wiegner, The C1,1-character of solutions of second order elliptic equations with gradi-
ent constraint, Comm.Part. Diff. Eq., 6, 361-371 (1981).
On an inverse crack problem in a linearized elasticity by the enclosure
method
Hiromichi Itou, h-itou@rs.tus.ac.jp
Tokyo University of Science, Japan
In this talk, we discuss a reconstruction problem for several linear cracks located on a line be-
tween two linearized elastic plates from measured data which are a loading surface traction and
the resulted displacement field on the boundary of the joined plates. This is a typical problem
from the nondestructive testing of materials. For this problem, we introduce an extraction for-
mula of the cracks from a single set of the data by means of the enclosure method. In the case
of a single linear crack, the extraction formula of the location and shape of an unknown crack
is established by using the enclosure method [2]. However, this result cannot be extended to
several cracks case directly because the original enclosure method can give an extraction for-
mula of the convex hull of cracks. As one of ways to overcome the difficulty, we apply the
Kelvin transform to the indicator function of the classical enclosure method. In [1, 3], by virtue
of this transform we derived extraction procedure of information about the location of tips of
several cracks located on a line between two electric conductive plates from a single set of an
electric current density and the corresponding voltage potential on the boundary of the material
formed by the plates. In the present talk, I will consider further extension of the result [1] to the
linearized elastic case.
This research is based on a joint work with Masaru Ikehata (Hiroshima University) and is
partially supported by Grant-in-Aid for Scientific Research (C)(No. 18K03380) and (B)(No.
17H02857) of Japan Society for the Promotion of Science and JSPS and RFBR under the Japan
- Russia Research Cooperative Program (project No. J19-721).
References
[1] A. Hauptmann, M. Ikehata, H. Itou and S. Siltanen, Revealing cracks inside conductive
bodies by electric surface measurements, Inverse Problems, 35, 025004 (24pp) (2019).
[2] M. Ikehata and H. Itou, Reconstruction of a linear crack in an isotropic elastic body from
a single set of measured data, Inverse Problems, 23, 589-607 (2007).
[3] M. Ikehata, H. Itou and A. Sasamoto, A., The enclosure method for an inverse problem
arising from a spot welding, Math. Methods Appl. Sci., 39, 3565-3575 (2016).
525
MECHANICS (MS-8)
[5] S. Giuffrè, Lagrange multipliers and non-constant gradient constrained problem, Journal
of Differential Equations 269 (1), 542-562, 2020.
[6] H. Ishii, S. Koike, Boundary regularity and uniqueness for an elliptic equation with gradi-
ent constraint, Comm. in Partial Diff. Eq., 8(4) (1983), 317- 346.
[7] P.L. Lions, Generalized Solutions for Hamilton-Jacobi Equations, Pitman Advanced Pub-
lishing Program, Boston-London-Melbourne, 1982.
[8] M. Wiegner, The C1,1-character of solutions of second order elliptic equations with gradi-
ent constraint, Comm.Part. Diff. Eq., 6, 361-371 (1981).
On an inverse crack problem in a linearized elasticity by the enclosure
method
Hiromichi Itou, h-itou@rs.tus.ac.jp
Tokyo University of Science, Japan
In this talk, we discuss a reconstruction problem for several linear cracks located on a line be-
tween two linearized elastic plates from measured data which are a loading surface traction and
the resulted displacement field on the boundary of the joined plates. This is a typical problem
from the nondestructive testing of materials. For this problem, we introduce an extraction for-
mula of the cracks from a single set of the data by means of the enclosure method. In the case
of a single linear crack, the extraction formula of the location and shape of an unknown crack
is established by using the enclosure method [2]. However, this result cannot be extended to
several cracks case directly because the original enclosure method can give an extraction for-
mula of the convex hull of cracks. As one of ways to overcome the difficulty, we apply the
Kelvin transform to the indicator function of the classical enclosure method. In [1, 3], by virtue
of this transform we derived extraction procedure of information about the location of tips of
several cracks located on a line between two electric conductive plates from a single set of an
electric current density and the corresponding voltage potential on the boundary of the material
formed by the plates. In the present talk, I will consider further extension of the result [1] to the
linearized elastic case.
This research is based on a joint work with Masaru Ikehata (Hiroshima University) and is
partially supported by Grant-in-Aid for Scientific Research (C)(No. 18K03380) and (B)(No.
17H02857) of Japan Society for the Promotion of Science and JSPS and RFBR under the Japan
- Russia Research Cooperative Program (project No. J19-721).
References
[1] A. Hauptmann, M. Ikehata, H. Itou and S. Siltanen, Revealing cracks inside conductive
bodies by electric surface measurements, Inverse Problems, 35, 025004 (24pp) (2019).
[2] M. Ikehata and H. Itou, Reconstruction of a linear crack in an isotropic elastic body from
a single set of measured data, Inverse Problems, 23, 589-607 (2007).
[3] M. Ikehata, H. Itou and A. Sasamoto, A., The enclosure method for an inverse problem
arising from a spot welding, Math. Methods Appl. Sci., 39, 3565-3575 (2016).
525
