\documentstyle{SibMatJ} \topmatter \Author Vatulyan \Initial A. \Initial O. \Email aovatulyan\@sfedu.ru \ORCID 0000-0003-0444-4496 \AffilRef 1 \AffilRef 2 \endAuthor \Author Nesterov \Initial S. \Initial A. \Email 1079\@list.ru \AffilRef 2 \endAuthor \Affil 1 \Division Department of Theory of Elasticity \Organization Southern Federal University \City Rostov-on-Don \Country Russia \endAffil \Affil 2 \Division Department of Differential Equations \Organization Southern Mathematical Institute \City Vladikavkaz \Country Russia \endAffil \translator S.~G.~Pyatkov\endtranslator \iauthor Vatulyan~A.O. and Nesterov~S.A.\endiauthor \UDclass 539.3\endUDclass \Origin \Journal \VMZh \Year 2022 \CopyrightYear 2022 \Volume 24 \Issue 2 \Pages 75--84 \DOI 10.46698/v3482-0047-3223-o \endOrigin \pforename A.O.\endpforename \psurname Vatulyan\endpsurname \pemail aovatulyan\@sfedu.ru\endpemail \pforename S.A.\endpforename \psurname Nesterov\endpsurname \pemail 1079\@list.ru\endpemail \author A.~O.~Vatulyan and S.~A. Nesterov\endauthor \title Study of the Inverse Problems of Thermoelasticity for Inhomogeneous Materials \endtitle %\thanks \endthanks \datesubmitted October 26, 2021\enddatesubmitted \dateaccepted November 29, 2022\enddateaccepted %\? \address \endaddress \affil A.~O.~Vatulyan\\ Department of Theory of Elasticity\\ Southern Federal University, Rostov-on-Don, Russia\\ Department of Differential Equations\\ Southern Mathematical Institute, Vladikavkaz, Russia \endaffil \email aovatulyan\@sfedu.ru\endemail \orcid 0000-0003-0444-4496\endorcid \affil S.~A. Nesterov\\ Department of Differential Equations\\ Southern Mathematical Institute, Vladikavkaz, Russia \endaffil \email 1079\@list.ru\endemail \keywords inverse problem of thermoelasticity, functionally graded materials, operator equations, iterative process, algebraization method \endkeywords \abstract Under study is the coefficient inverse problem of thermoelasticity for finite inhomogeneous bodies. We obtain the operator equations of the first kind for the Laplace transform of a~solution for solving the nonlinear inverse problem by an iterative process. Solving the inverse problems of thermoelasticity in the originals relies on applying the inverse Laplace transform to these operator equations on using the theorems of operational calculus on the convolution and differentiation of the original. We provide some procedure for reconstructing the thermomechanical characteristics of a~rod, a~layer, and a~cylinder. Finding an initial approximation for the iterative process bases on two approaches. Then an initial approximation is found in the class of bounded positive linear functions, while the coefficients of the functions are determined as minimizers of the residual functional. The other approach to finding an initial approximation bases on algebraization. Numerical experiments were carried out to recover both monotone and nonmonotone functions. Only one characteristic is restored while the others are assumed known. Monotone functions are restored better than nonmonotone ones. While reconstructing the characteristics of stratified materials, the greatest error arises in a~neighborhood about the junction points. The reconstruction procedure turned out resistant to noise in the input information. \endabstract \endtopmatter \head 1. Introduction \endhead To calculate the strength of structural elements in a~high-temperature environment, it is necessary to solve the problems related to recovering a~stress-strain state. Usually, these calculations are performed for homogeneous materials. However, inhomogeneous materials such as stratified composites and functionally graded materials (FGM) are applied widely as these days. The thermomechanical characteristics of inhomogeneous materials are functions of coordinates~[1] and thereby can be determined on using the apparatus of the coefficient inverse problems (CIPs). At present, many studies are devoted to solving CIPs of heat conductivity and elasticity~[7--9]. As a~rule, the study of inverse problems reduces to solving the corresponding extremal problems~[2, 3]. To this end, some residual functional is chosen that is minimized on a~finite-dimensional subspace by gradient methods~[2]. For some materials, we need to take into account the connectivity of fields and to solve the CIPs of thermoelasticity [10, 11] which are now poorly studied especially for inhomogeneous materials. Initially, the inverse problems of thermoelasticity were considered for stratified media. In [10] the thermomechanical characteristics and tree-layer plate thickness are recovered by reducing the inverse thermoelasticity problem to an extremal problem and applying the gradient method for minimizing the residual functional. Some new approach to solving the CIPs of mechanics of connected fields was proposed in~[12]. A~nonlinear inverse problem was solved by constructing the iteration process; at each step we look for a~solution to an operator equation of the first kind which is obtained on the base of a~weak statement, a~generalized reciprocity relation, and linearization. The present article is a~survey of the authors' earlier results on solving the CIPs of thermoelasticity for a~rod, a~layer, and a~cylinder by the developed iterative approach. \head 2. The General Statement of CIPs for a~Finite Body \endhead Consider the transient oscillations of a~finite thermoelastic body [12] which are described by the system $$ %\label{Vat1_eq1} \sigma _{ij,j} = \rho \ddot {u}_i, \tag1 $$ $$ %\label{Vat1_eq2} \sigma _{ij} = c_{ijkl} u_{k,l} - \gamma _{ij} \theta, \tag2 $$ $$ %\label{Vat1_eq3} (k_{ij}\theta_{,i})_{,j}=c\dot {\theta}+T_0\gamma_{ij}\dot {u}_{i,j}, \tag3 $$ $$ %\label{Vat1_eq4} \theta \vert _{S_T } = 0,\quad - k_{ij} \theta _{,i} n_j \vert _{S_q } = q, \tag4 $$ $$ %\label{Vat1_eq5} u_i \vert _{S_u} = 0,\quad \sigma _{ij} n_j \vert _{S_{\sigma} } = p_{i}, \tag5 $$ $$ %\label{Vat1_eq6} \theta (x,0) = u_i (x,0) = \dot {u}_i (x,0) = 0. \tag6 $$ Note that $\sigma _{ij} $ are the components of the stress tensor; $u_i $ are the components of the displacement vector; $\theta $ is the temperature increment from the natural state with a~temperature $T_0 $; $c_{ijkl} $ are the components of the elastic modulus tensor; $\rho$ is the density; $c_{\varepsilon}$ is the specific volumetric heat capacity at a~constant strain tensor; $k_{ij} $ are the components of the heat conductivity tensor; $\gamma _{ij}$ are the thermal stress tensor components; $n_j $ are the coordinates of the unit outward normal to~$S_{\sigma}$; $p_i $ are the components of the vector of active mechanical load applied to the body; $q$~is the heat flux density. To solve the direct thermoelasticity problem means to find the functions~$u_i (x,t)$ and $\theta (x,t)$ in~\Tag(1)--\Tag(6), where the thermomechanical characteristics $c_{ijkl} $, $\rho $, $c_{\varepsilon}$, and $k_{ij} $, $\gamma_{ij} $ are known. The inverse problem is to recover the laws of changing the thermomechanical characteristics $\rho $, $c_{ijkl} $, $c_{\varepsilon}$, $k_{ij} $, and $\gamma _{ij} $ from \Tag(1)--\Tag(6) on using the additional information on the components of the displacement vector measured of a~part of the boundary $S_\sigma$ or the information about the temperature increments measured on a~part of~$S_q$; i.e., we have either $$ %\label{Vat1_eq7} u_i \vert _{S_\sigma } = g_i , \quad t \in \left[ {T_1 ,T_2 } \right], \tag7 $$ or $$ %\label{Vat1_eq8} \theta \vert _{S_q } = f, \quad t \in \left[ {T_3,T_4 } \right]. \tag8 $$ Applying the Laplace transform in the variable $t$ to \Tag(1)--\Tag(3) and \Tag(4), \Tag(5) and taking \Tag(6) into account we derive that $$ %\label{Vat1_eq9} \widetilde{\sigma} _{ij,j} =p^2 \rho\widetilde{u}_i, \tag9 $$ $$ %\label{Vat1_eq10} \widetilde{\sigma} _{ij} = c_{ijkl} \widetilde{u}_{k,l} - \gamma _{ij} \widetilde{\theta}, \tag10 $$ $$ %\label{Vat1_eq11} (k_{ij}\widetilde{\theta}_{,i})_{,j}=pc\widetilde{\theta}+pT_0\gamma_{ij}\widetilde{u}_{i,j}, \tag11 $$ $$ %\label{Vat1_eq12} \widetilde{\theta} \vert _{S_T } = 0,\quad - k_{ij} \widetilde{\theta} _{,i} n_j \vert _{S_q } = \widetilde{q}, \tag12 $$ $$ %\label{Vat1_eq13} \widetilde{u}_i\vert _{S_u} = 0,\quad \widetilde{\sigma} _{ij} n_j \vert _{S_{\sigma} } = \widetilde{p}_{i}. \tag13 $$ The CIP of thermoelasticity is nonlinear. Its solution bases on constructing some iteration process whose operator equations in~[12] involve a~weak statement of the problem in terms the Laplace transform of a solution \Tag(9)--\Tag(13) and its linearization. The system of Fredholm integral equations of the first kind (FIE) for determining corrections for the thermomechanical characteristics $\delta \rho^{(n-1)}$, $\delta c_{ijkl}^{(n-1)} $, $\delta c_{\varepsilon}^{(n-1)}$, $\delta k_{ij}^{(n-1)}$, and $\delta \gamma _{ij}^{(n-1)}$ on the $(n-1)$th iteration is as follows: $$ \gather %\label{Vat1_eq14} \int\limits_{V} {\delta c_{ijkl}^{(n-1)} \widetilde{u}_{i,j}^{(n-1)}\widetilde{u}_{k,l}^{(n-1)}}\,dV + p^2\int\limits_{V} {\delta \rho^{(n - 1)}\left(\widetilde{u}_{i}^{(n - 1)}\right)^2}\,dV \\ - \int\limits_{V} {\delta\gamma _{ij}^{(n-1)}\widetilde{u}_{i,j}^{(n-1)}\widetilde{\theta}^{(n-1)}}\,dV = -\int\limits_{S_\sigma}{\widetilde{p}_{i}\left(\widetilde{g}_{i}-\widetilde{u}_{i}^{(n-1)}\right)}\,dS,\quad \tag14\endgather $$ $$ \gather %\label{Vat1_eq15} \int\limits_{V} {\delta k_{ij}^{(n-1)} \widetilde{\theta}_{i}^{(n-1)}\widetilde{\theta}_{j}^{(n-1)}}\,dV + p\int\limits_{V} {\delta c_{\varepsilon}^{(n-1)}\left(\widetilde{\theta}^{(n - 1)}\right)^2}\,dV \\ - pT_0\int\limits_{V} {\delta\gamma _{ij}^{(n-1)}\widetilde{u}_{i,j}^{(n-1)}\widetilde{\theta}^{(n-1)}}\,dV = \int\limits_{S_q}{\widetilde{q}\left(\widetilde{f}-\widetilde{\theta}^{(n-1)}\right)}\,dS. \tag15\endgather $$ \head 3. CIPs of Thermoelasticity for a~Rod \endhead As the first example, we consider the problem of longitudinal oscillations of an inhomogeneous thermoelastic rod of length $l$ fixed at the end $x = 0$ and we distinguish the thermal and mechanical methods of excitation of oscillations. The initial-boundary value problem in the former case is of the form [13] $$ %\label{Vat1_eq16} \frac{\partial \Omega }{\partial z} = \varepsilon^2 \bar {\rho }(z)\frac{\partial ^2U }{\partial \tau _1 ^2}, \tag16 $$ $$ %\label{Vat1_eq17} \Omega = \bar {E}(z)\frac{\partial U }{\partial z} - \bar {\gamma }(z)W, \tag17 $$ $$ %\label{Vat1_eq18} \frac{\partial }{\partial z}\left(\bar {k}(z)\frac{\partial W}{\partial z}\right) = \bar {c}(z)\frac{\partial W}{\partial \tau _1 } + \delta_0\bar {\gamma }(z)\frac{\partial ^2U}{\partial z\partial \tau _1 }, \tag18 $$ $$ %\label{Vat1_eq19} U (0,\tau _1 ) = W (0,\tau _1 ) = 0, \quad - \bar {k}(1)\frac{\partial W }{\partial z}(1,\tau _1 ) = \omega \phi (\tau _1 ), \quad \Omega (1,\tau _1 ) = 0, \tag19 $$ $$ %\label{Vat1_eq20} W (z,0) = U (z,0) = \frac{\partial U }{\partial \tau _1 }(z,0) = 0. \tag20 $$ Here $$z = \frac{x}{l}, \quad \bar {k}(z) = \frac{k(zl)}{k_0 }, \quad \bar {c}(z) =\frac{c_\varepsilon (zl)}{c_0 }, $$ $$ \bar {\rho }(z) = \frac{\rho (zl)}{\rho_0 }, \quad \bar {E}(z) = \frac{E(zl)}{E_0 }, \quad \bar {\gamma }(z) = \frac{\gamma (zl)}{\gamma _0 }, $$ $$ t_1 = \frac{l^2c_0}{k_{0 }}, \quad t_2 = l\sqrt {\frac{\rho _0 }{E_0 }}, \quad \tau _1 = \frac{t}{t_1 }, \quad \tau_2 = \frac{t}{t_2 }, $$ $$ W = \frac{\gamma _0 \theta }{E_0 }, \quad U =\frac{u}{l}, \quad \Omega = \frac{\sigma _x }{E_0 }, \quad \delta _0 =\frac{\gamma _0^2 T_0 }{c_0 E_0 }, \quad \omega = \frac{q_0 \gamma _0 l}{k_0 E_0}, $$ $$ \varepsilon = \frac{t_2 }{t_1 }, \quad k_0 = \max\limits_{x\in [0,l]} k(x), \quad c_0 = \max\limits_{x \in [0,l]} c_\varepsilon(x), $$ $$ E_0 = \max\limits_{x \in [0,l]} E(x), \quad \rho _0 = \max\limits_{x \in [0,l]} \rho (x), \quad \gamma _0 = \max\limits_{x \in [0,l]} \gamma (x), $$ $\delta _0 $~is a~dimensionless connectivity parameter, and~$\varepsilon$ is the ratio of characteristic times of sound and thermal perturbations. \goodbreak Direct problem \Tag(16)--\Tag(20) after applying the Laplace transform is solved by reducing the problem to a~system of Fredholm integral equations of the second kind and finding the originals by the Residue Theorem. In the inverse problem we need to recover one of the mechanical characteristics ($\bar {E}$, $\bar {\rho }$, and $\bar {\gamma })$ in \Tag(16)--\Tag(20) on using the information about the displacement at the end $z = 1$ of the rod which is written as $$ %\label{Vat1_eq21} U(1,\tau _2 ) = g(\tau _2 ), \quad \tau _2\in \left[ {c,d} \right], \tag21 $$ and one of the thermophysical characteristics ($\bar {c}$, $\bar {k}$, $\bar {\gamma }$) in \Tag(16)--\Tag(20) from information on the temperature increment at the end of the rod $$ %\label{Vat1_eq22} W(1,\tau _1 ) = f(\tau _1 ), \quad \tau _1\in \left[ {a,b} \right]. \tag22 $$ The operator equations for the Laplace transforms of corrections of thermomechanical characteristics of a~rod for a~load $\phi (\tau _1 ) = H(\tau _1 )$ are as follows [13]: $$ \gather %\label{Vat1_eq23} \int\limits_0^1 {\delta \bar {E}^{(n - 1)}}\left(\frac{d\widetilde {U}^{(n - 1)}}{dz}\right)^2dz + p^2\int\limits_0^1 \delta \bar {\rho }^{(n - 1)}\left(\widetilde {U}^{(n - 1)} \right)^2\,dz \\ - \int\limits_0^1 {\delta \bar {\gamma }^{(n - 1)}} \frac{d\widetilde {U}^{(n - 1)}}{dz}\widetilde {W}^{(n - 1)}dz = -\frac{1}{p}\left(\widetilde {g}(p) - \widetilde {U}^{(n - 1)}(1,p)\right), \tag23\endgather $$ $$ \gather %\label{Vat1_eq24} \int\limits_0^1 {\delta \bar {k}^{(n - 1)}} \left(\frac{dW^{(n - 1)} }{dz}\right)^2\,dz + p \int\limits_0^1 \delta \bar {c}^{(n - 1)}\left(\widetilde {W}^{(n - 1)}\right)^2\,dz \\ + \delta _0 p\int\limits_0^1 {\delta \bar {\gamma }^{(n - 1)}} \frac{d\widetilde {U}^{(n - 1)}}{dz}\widetilde {W}^{(n - 1)}\,dz = \frac{\omega }{p}\left(\widetilde {f}(p) - \widetilde {W}^{(n - 1)}(1,p)\right). \tag24 \endgather $$ If we need to restore only one of the thermomechanical characteristics assuming that the remaining ones are known; then, assuming that all corrections for the remaining characteristics in~\Tag(23) and~\Tag(24) vanish, we arrive at a~FIE of the first kind with smooth kernels. So to find the corrections $\delta \bar {k}^{(n - 1)}(z)$ on the $(n - 1)$th iteration we have $$ %\label{Vat1_eq25} p\int\limits_0^1 {\delta \bar {k}^{(n - 1)}\left( {\frac{d\widetilde {W}^{(n - 1)}}{dz}} \right)} ^2\,dz = \omega \left(\widetilde {f}(p) - \widetilde {W}^{(n - 1)}(1,p)\right). \tag25 $$ Solving inverse thermoelasticity problems in originals bases on the inversion of the operator relations for the images. So, to find the corrections $\delta \bar {k}^{(n - 1)}(z)$ on the $(n - 1)$th iteration, we have the first kind~FIE $$ %\label{Vat1_eq26} \int\limits_0^1 {\delta \bar {k}^{(n - 1)}} R (z,\tau _1 )\,dz = \omega \left( {f(\tau _1 ) - W^{(n - 1)} (1,\tau _1 )} \right), \quad \tau _1 \in [a,b], \tag26 $$ \noindent where the kernel of equation \Tag(26) is of the form $$ R (z,\tau _1 ) = \int\limits_0^{\tau _1 } {\frac{\partial ^2W ^{(n - 1)}(z,\tau )}{\partial z\,\partial \tau }} \frac{\partial W ^{(n - 1)}(z,\tau _1 - \tau )}{\partial z}\,d\tau. $$ Solving \Tag(26) is an ill-posed problem. The regularization of such equations is realized by the Tikhonov method [14]. \head 4. CIPs of Thermoelasticity for a~Layer \endhead One more example is the problem of the transient oscillations of an isotropic thermoelastic layer inhomogeneous in the coordinate $x_3$ under the conditions of a~plane deformation. The lower face of a~layer is rigidly clamped and the mechanical and thermal loads are applied on the upper face. To solve the problem, the Fourier transform in the coordinate $x_1 $ is applied to the differential equations and boundary conditions in [12]. Equating the parameter of the Fourier transform to 0, we reduce the two-dimensional problem to two simpler one-dimensional problems. At the first step, solving the first inverse problem for a~known density we define the shear modulus. At the second step of identification, using the known density and the shear modulus, we find either one of the thermophysical characteristics (the heat conductivity, the density, the thermal stress coefficient) or the Lam\'{e} coefficient. On using the thermoelasticity layer model, some numerical experiments were conducted to identify thermomechanical characteristics; i.e., (a)~the shear modulus of a~thermoelastic layer with destruction zones, (b) the heat conductivity coefficient of a~skin cover, and~(c)~the heat conductivity and Lam\'{e} coefficients of a~functional gradient cover. \head 5. CIPs of Thermoelasticity for a~Tube \endhead As in the third example, we consider the problem of radial oscillations of an inhomogeneous tube with some uniformly distributed thermal and mechanical loads applied on the outer boundary. The initial-boundary value problem for the mechanical excitation method of oscillations is of the form [15]: $$ %\label{Vat1_eq27} \frac{\partial \Omega _{rr} }{\partial \xi } + \frac{\Omega _{rr} - \Omega _{\phi \phi } }{\xi } = \bar {\rho }\,\frac{\partial ^2U}{\partial \tau _2^2}, \tag27 $$ $$ %\label{Vat1_eq28} \Omega _{rr} = \left(\bar {\lambda } + 2\bar {\mu }\right)\frac{\partial U}{\partial \xi } + \bar {\lambda }\frac{U}{\xi } - \bar {\gamma }W, \quad \Omega _{\phi \phi } = \bar {\lambda }\frac{\partial U}{\partial \xi } + \left(\bar {\lambda } + 2\bar {\mu }\right)\frac{U}{\xi } - \bar {\gamma }W, \tag28 $$ $$ %\label{Vat1_eq29} \frac{1}{\xi }\frac{\partial }{\partial \xi }\left(\bar {k}(\xi )\xi \frac{\partial W}{\partial \xi }\right) = \frac{1}{\varepsilon }\,\bar {c}(\xi )\frac{\partial W}{\partial \tau _2 } + \frac{\delta _0 }{\varepsilon }\,\bar {\gamma }(\xi )\left(\frac{\partial ^2U}{\partial \xi \partial \tau _2 } + \frac{1}{\xi }\frac{\partial U}{\partial \tau _2 }\right), \tag29 $$ $$ %\label{Vat1_eq30} \Omega _{rr} (\xi _0,\tau _2 ) = 0, \quad W(\xi _0,\tau _2 ) = 0, \tag30 $$ $$ %\label{Vat1_eq31} \Omega _{rr} (1,\tau _2 ) = \chi _0 \varphi (\tau_2), \quad W(1,\tau _2 ) = 0, \tag31 $$ $$ %\label{Vat1_eq32} W(\xi,0) = U(\xi,0) = \frac{\partial U}{\partial \tau _2 }(\xi ,0) = 0. \tag32 $$ Here $$ \xi = \frac{r}{r_2 }, \quad \xi_0 = \frac{r_1 }{r_2 }, \quad U = \frac{u_r }{r_2 }, \quad W = \frac{\gamma _0\theta }{\lambda _0 }, $$ $$ v = \sqrt {\frac{\lambda _0 + 2\mu _0 }{\rho _0 }}, \quad \tau _1 = \frac{t}{t_1 }, \quad \tau _2 = \frac{t}{t_2 }, \quad t_1 = \frac{r_2 c_0 }{k_0 }, $$ $$ t_2 = \frac{r_2 }{v}, \quad \delta _0 = \frac{\gamma _0^2 T_0 }{c_0 \lambda _0 }, \quad \varepsilon = \frac{t_2 }{t_1 }, \quad \Omega _{rr} = \frac{\sigma _{rr} }{\lambda _0 }, \quad \Omega _{\phi \phi } = \frac{\sigma _{\phi \phi } }{\lambda _0 }, $$ $$ \bar {\lambda } = \frac{\lambda }{\lambda _0}, \quad y_0 = \frac{\mu _0 }{\lambda _0 }, \quad \bar {\mu } = \frac{\mu }{\mu _0 }, \quad \bar {\rho } = \frac{\rho }{\rho _0 }, \quad \bar {\gamma } = \frac{\gamma }{\gamma _0 }, $$ $$ \bar {k} = \frac{k}{k_0 }, \quad \bar {c} = \frac{c}{c_0 }, \quad \chi _0 = \frac{p_0 }{\lambda _0 }, \quad \omega _0 = \frac{q_0 r_2 \gamma _0 }{k_0 \lambda _0 }. $$ After applying the Laplace transform, direct problem \Tag(27)--\Tag(32) is solved by the shooting method. In the inverse problem we need to recover one of the thermomechanical characteristics provided that the remaining characteristics in \Tag(27)--\Tag(32) are known on using some additional information about the temperature or displacements measured on the outer surface of a~tube. The operator equations for the Laplace transform of corrections of the thermomechanical characteristics of a~tube under the load $\psi (\tau _2 ) = H(\tau _2 )$ are of the form [15]: $$ %\label{Vat1_eq33} \gather \int\limits_{\xi _0 }^1 {\delta \bar {\lambda }^{(n - 1)}\left( {\frac{d\widetilde {U}^{(n - 1)}}{d\xi } + \frac{\widetilde {U}^{(n - 1)}}{\xi }} \right)^2\xi} \,d\xi \\ +y_0 \int\limits_{\xi _0 }^1 {\delta \bar {\mu }^{(n - 1)}\left( {\left( {\frac{d\widetilde {U}^{(n - 1)}}{d\xi }} \right)^2 + \left( {\frac{\widetilde {U}^{(n - 1)}}{\xi }} \right)^2} \right)\xi}\, d\xi + p^2\int\limits_{\xi _0 }^1 {\delta \bar {\rho }^{(n - 1)}\left( \widetilde {U}^{(n - 1)}\right)^2\xi}\,d\xi \\ - \delta _0 \int\limits_{\xi _0 }^1 {\delta \bar {\gamma }^{(n - 1)}\left( {\frac{d\widetilde {U}^{(n - 1)}}d\xi +\frac{\widetilde {U}^{(n - 1)}}{\xi }} \right)\widetilde {W}^{(n - 1)}\xi}\,d\xi = -\frac{\chi _0 }{p}\left(\widetilde {g}(p) - \widetilde {U}^{(n - 1)}(1,p)\right), \tag33 \endgather $$ $$ \gather \int\limits_{\xi _0} {\delta \bar {k}^{(n - 1)}\left( {\frac{d\widetilde {W}^{(n - 1)}}{d\xi }} \right)} ^2\xi\, d\xi + p\int\limits_{\xi _0 }^1 {\delta \bar {c}_\varepsilon ^{(n - 1)} \left(\widetilde {W}^{(n - 1)}\right)^2\xi}\, d\xi \\ + \,\delta _0 p\int\limits_{\xi _0 }^1 {\delta \bar {\gamma }^{(n - 1)}\left( {\frac{d\widetilde {U}^{(n - 1)}}{d\xi } + \frac{\widetilde {U}^{(n - 1)}}{\xi }} \right)\widetilde {W}^{(n - 1)}\xi}\, d\xi = \frac{\omega _0 }{p}\left(\widetilde {f}(p) - \widetilde {W}^{(n - 1)}(1,p)\right). \tag34\endgather $$ \head 6. Peculiarities of Determining the Initial Approximation \endhead We propose two approaches to determining the initial approximation of the iteration process of reconstructing thermomechanical characteristics. In the first approach, we look for an initial approximation in the set of bounded positive linear functions of the form $C_1 + C_2 z$. The constants $C_1 $ and $C_2 $ are determined on using a~priori information about the range of thermomechanical characteristics. Relying on a~priori information, we can construct the limits of these constants which form the range of the coefficients $C_1 $ and~$C_2 $ which is a~compact subset of~$R^2$. Introducing a~grid in this domain and minimizing the corresponding functional on this compact set, we determine an appropriate pair $(C_1,C_2 )$. For a~thermal load of a~rod, the residual functional is as follows: $$ %\label{Vat1_eq35} J_1=\int\limits_a^b \left(f(\tau_1)-W^{(n-1)}(1,\tau_1)\right)^2\,d{\tau_1}. \tag35 $$ For a~mechanical load of a~rod, the residual functional is written as $$ %\label{Vat1_eq36} J_2=\int\limits_c^d \left(g(\tau_2)-U^{(n-1)}(1,\tau_2)\right)^2\,d{\tau_2}. \tag36 $$ In the second approach, we seek an approximate solution to the inverse problem in the form of the expansion $$ %\label{Vat1_eq37} \bar {c}(z) = \sum\limits_{j = 1}^m {s_j } z^{j - 1}, \quad j = 1,\dots,m. \tag37 $$ Represent the images of the displacements and the temperature under the Laplace transform as expansions (similar to that of~[12]) in a~system of basis functions, i.e., $$ %\label{Vat1_eq38} \widetilde {W}_1 (z,p) = \varphi _0 (z,p) + \sum\limits_{i = 1}^n \widetilde {b}_i (p)\,\varphi _i (z), \quad \widetilde {U}_1 (z,p) = \phi _0 (z,p) + \sum\limits_{i = 1}^n \widetilde {a}_i (p)\,\phi _i (z) . \tag38 $$ Here $\phi _0 (z,p)$ % $\varphi _0 (z,p)$ is a~function satisfying inhomogeneous boundary conditions, while $\phi _i (z)$ and $\varphi _i (z)$, with $i=1,\dots,n$, are orthogonal functions satisfying the homogeneous boundary conditions. Insert \Tag(37) and \Tag(38) in \Tag(16)--\Tag(20). Multiply~\Tag(16) by $\phi_i (z)$, and~\Tag(18) by $\varphi _i (z)$ and integrate with respect to $z$ from~0 to~1. We obtain some system of algebraic equations relatively the coefficients~$\widetilde {a}_i (p)$ and $\widetilde {b}_i (p)$ of the expansions. An analytic expression for the determinant of the system found in Maple is a~polynomial in~$p$ and the unknowns $c_1,\dots,c_m $. To find the unknown coefficients $c_1,\dots,c_m $ of the expansion, we use the additional information of~\Tag(21) and~\Tag(22). We approximate the additional information given tabular on an informative time segment is approximated by the span of exponential functions. The exponents of these functions are determined by the Prony algorithm as in~[12]. Given a~collection $c_1,\dots,c_m $, the unknown function $\bar {c}(z)$ is recovered in accord with ~\Tag(37). \head 7. Results of Numerical Experiments \endhead Numerical experiments on recovering thermomechanical characteristics of a~rod, a~layer, and a~cylinder were conducted. One of the characteristics is recovered and the remaining ones are assumed known. The iteration process stops when functional~\Tag(35) or~\Tag(36) achieves its limit value $10^{-6}$. The hypothetic laws of inhomogeneity in the sets of power, exponential, and trigonometric functions as well as the laws modelling the material characteristic of real FGM, on the base of $Ni-TiC$, for example, are recovered. The results of numerical experiments are as follows: (1) Choosing constants in the initial approximation in order to achieve the limit value in \Tag(35) or \Tag(36) requires more iterations than that in the choice among linear functions; (2) Monotone functions are recovered better than nonmonotone; (3) The maximal error when restoring the heat conductivity, the density, the thermal stress coefficient takes place in a~neighborhood about the end $z=0$ which is connected with the kernel peculiarities of the corresponding integral equations; (4)~In the case of recovering the characteristics of stratified materials, the maximal error arises in a~neighborhood about the junction points; (5)~The reconstruction procedure gives satisfactory results even for noisy input information; (6)~The algebraization method allows us to find an initial approximation in shorter runtime than in minimizing the residual functional, but the method yields instable solutions under a~noisy input information. \enddemo \Refs \ref\no 1 \by Wetherhold~R.C., Seelman~S., and Wang~S. \paper The use of functionally graded materials to eliminated or control thermal deformation \jour Compos. 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