Application of Calderón’s inverse problem in civil engineering. (English) Zbl 1513.65338

Havelka, Jan; Sýkora, Jan

Appl. Math., Praha 63, No. 6, 687-712 (2018).

In specific fields of research such as preservation of historical buildings, medical imaging, geophysics and others, it is of particular interest to perform only non-intrusive boundary measurements. The idea is to obtain comprehensive information about the material properties inside the considered domain while keeping the test sample intact. The authors put emphasis on the obstacles arising in civil engineering problems such as a limited number of measurement nodes on the boundary, reduced number of heaters, discontinuous or non-differentiable fields, i.e., synthesize the physical model of interest with the method of boundary reciprocals. The authors intend to apply identical principles used in the Calderón problem for the time-dependent model \[ \begin{aligned} g_s(x)c_p(x)\frac{\partial u}{\partial t}(x,t)-\nabla\left (\lambda_s(x)\nabla u(x,t)\right)=0,\quad &x,t\in\Omega\times(0,t_s)\\ \lambda_s(s)\frac{\partial u}{\partial n}(x,t)=f_N(x,t),\quad &x\in\partial\Omega_N,\\ \alpha\left (u(x,t)-u_0(x,t)\right) =\lambda_s(x)\frac{\partial u}{\partial n},\quad &x\in\partial\Omega_T,\\ u(x,0)=0,\quad &x\in\partial\Omega=\partial\Omega_N\cup\partial\Omega_N, \end{aligned} \] where \(g_s(x)\) is the volumetric mass density, \(c_p(x)\) is the specific heat capacity, \(t_c\) is the final time of simulation, \(\Omega\subset \mathbb R^2\) is a bounded domain with a piecewise smooth boundary \(\partial\Omega\); \(\partial\Omega_N\), \(\partial\Omega_T\) are non intersecting subsets of the boundary \(\partial\Omega\) with corresponding environmental factors \(u_0(x,t)\), \(\alpha\) and \(f_0(x,t)\). The model is represented here by the time-dependent heat equation with transport parameters that are subsequently identified using a modified Calderón problem which is numerically solved by a regularized Gauss-Newton method. The proposed model setup is computationally verified for various domains, loading conditions and material distributions.

Reviewer: Elena V. Tabarintseva (Chelyabinsk)

MSC:

65M32	Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35K05	Heat equation
80A23	Inverse problems in thermodynamics and heat transfer
80M50	Optimization problems in thermodynamics and heat transfer
65K10	Numerical optimization and variational techniques
35R30	Inverse problems for PDEs
35Q79	PDEs in connection with classical thermodynamics and heat transfer

Keywords:

Calderón problem; finite element method; diffusion equation; boundary inverse value method; Neumann-to-Dirichlet map