An energy approach for a Cauchy problem in elasticity

We are interested in the problem of data completion, which consists in recovering the data on an inaccessible part of the boundary of a solid using overspecified data measured on another part of it. This is an old problem mathematically known as the Cauchy problem. This kind of problem arises in many industrial, engineering or biomedical applications under various forms: identification of boundary conditions or expansion of measured surface fields inside a body. But also it can be the first step in general parameters identification problems and damage detection where only partial boundary data are under control. Hence, robust and efficient data completion method is an essential and basic tool in structural identification. In this paper we present a method for data completion based on the minimization of an energy-like error functional which depends on lacking data. We consider homogeneous elastic solid Ω with smooth boundaries, where Γc is the part of the boundary where the data Uc and Fc, respectively the displacement and the pressure fields, are known, and Γi is the part of the boundary where the data have to be recovered. Following, (σ, ε and u) will denotes the stress, strain and displacement fields. The problem can be stated as follows, find (Ui, Fi) on Γi such that:

1

$$ \left\{ \begin{gathered} \begin{array}{*{20}c} {div\left( {\sigma \left( u \right)} \right) = 0} & {in \Omega ,} & {\sigma \left( u \right).n = F_c ,u = U_c on \Gamma _c } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\sigma \left( u \right) = \mathbb{C}:\varepsilon \left( u \right)} & {in \Omega ,} & {\sigma \left( u \right).n = F_i ,u = U_i on \Gamma _i } \\ \end{array} \hfill \\ \end{gathered} \right. $$

where ℂ is the fourth-order elasticity tensor. In the approach presented here, we consider, for a given pair (

f, d

), the following two mixed and well-posed problems, whose solutions are denoted by

u

1

and

u

2

:

$$ \begin{array}{*{20}c} {\left\{ {\begin{array}{*{20}c} \begin{gathered} div\left( {\sigma \left( {u_1 } \right)} \right) = 0 \hfill \\ \sigma \left( {u_1 } \right) = \mathbb{C}:\varepsilon \left( {u_1 } \right) \hfill \\ \end{gathered} & \begin{gathered} in \Omega ,u_1 = U_c \hfill \\ in \Omega ,\sigma \left( {u_1 } \right).n = f \hfill \\ \end{gathered} & \begin{gathered} on \Gamma _c \hfill \\ on \Gamma _i \hfill \\ \end{gathered} \\ \end{array} } \right.} & {\left\{ {\begin{array}{*{20}c} \begin{gathered} div\left( {\sigma \left( {u_2 } \right)} \right) = 0 \hfill \\ \sigma \left( {u_2 } \right) = \mathbb{C}:\varepsilon \left( {u_2 } \right) \hfill \\ \end{gathered} & \begin{gathered} in \Omega , \sigma \left( {u_2 } \right).n = F_c \hfill \\ in \Omega , u_2 = d \hfill \\ \end{gathered} & \begin{gathered} on \Gamma _c \hfill \\ on \Gamma _i \hfill \\ \end{gathered} \\ \end{array} } \right.} \\ \end{array} $$

The displacements fields

u

1

and

u

2

are equal when the pair (

f, d

) meets the real data (

U

i

F

i

) on the boundary Γi. Hence, we propose to solve the data completion problem via the minimization of the energy functional:

$$ \left( {F_i ,U_i } \right) = \mathop {\arg \min }\limits_{\left( {f,d} \right)} \int\limits_\Omega {\left( {\sigma \left( {u_1 } \right) - \sigma \left( {u_2 } \right)} \right):\left( {\varepsilon \left( {u_1 } \right) - \varepsilon \left( {u_2 } \right)} \right)} $$

To explore the efficiency of this method, several numerical examples are presented for 2D and 3D situations. The results are in good agreement with the actual ones. The method turns out to be very efficient with respect to the precision of the solution but also with respect to the amount of computation needed. The formulation is very general and can be used with heterogeneous materials and other physical phenomena. Some variant of the problem (1) will also be addressed illustrating the flexibility of the approach and potential applications in experimental methods.