Shape optimization of a thermoelastic body under thermal uncertainties
M. Dambrine, G. Gargantini, H. Harbrecht, V. Karnaev Shape optimization of a thermoelastic body under thermal uncertainties Journal of Computational Physics
Volume 527, 15 April 2025, 113794
Problem statement.
We consider a thermoelastic structure represented by a Lipschitz continuous bounded domain \(\Omega\).
According to the Duhamel-Neumann postulate, the stress tensor is
\[
\sigma(u, T) := \sigma_{el}(u) + \sigma_{th}(T),
\quad \sigma_{el}(u):=C:\varepsilon(u),
\quad \sigma_{th}(T):=(T-T_{in})B,
\]
where
\[
C:\varepsilon(u) = 2\mu\varepsilon(u) + \lambda\text{div}(u)I
\quad\text{and}\quad B = -\alpha(3\lambda + 2\mu)I.
\]
and \[
\mu=\frac{E}{2(1+\nu)}\quad\text{and}
\quad\lambda=\frac{E\nu}{(1+\nu)(1-2\nu)}.
\]
The thermal exchange with the environment is taken into
account through the Robin boundary conditions with external
temperature \( T_{ext} \in L^2((0,t_f); L^2(\Gamma_N\cup\Gamma_F))\).
Additionally, the body is influenced by a thermal body source \( Q\in L^2((0,t_f); L^2(D))\).
The temperature fields then solves
\[
\left\{\,
\begin{aligned}
&\rho\dfrac{\partial T}{\partial t} -\text{div}(k \nabla T) = Q \text{ in } (0,t_f)\times D, \\[1ex]
&(k \nabla T)\cdot n = -\beta (T-T_{ext}) \text{ on } (0,t_f)\times \Gamma_N\cup\Gamma_F, \\[1ex]
&T = T_{in} \text{ on } (0,t_f)\times \Gamma_D, \\[1ex]
&T = T_{in} \text{ in } \{t=0\}\times D.
\end{aligned}
\right.
\]
The thermoelastic body is subject to the body force
\(f \in L^2((0,t_f); L^2(D)^d)\) in the whole domain \(D\) and
to the surface force \(g \in L^2((0,t_f); L^2(\Gamma_N)^d)\) on the part
\(\Gamma_N\) of the boundary. The body is assumed to be fixed on the
part \(\Gamma_D\) of its boundary, while it is unconstrained
on the rest \(\Gamma_F\) of the boundary. The displacement \(u_D\) solves
\[
\left\{
\begin{aligned}
-&\text{div}(\sigma(u, T)) = f \text{ in } (0,t_f)\times D, \\[1ex]
&\sigma(u, T) n = g \text{ on } (0,t_f)\times \Gamma_N, \\[1ex]
&\sigma(u, T) n = 0 \text{ on } (0,t_f)\times \Gamma_F, \\[1ex]
&u = 0 \text{ on } (0,t_f)\times \Gamma_D, \\[1ex]
\end{aligned}
\right.
\]
We consider the thermoelastic
model in the case when the
external temperature in the Robin boundary condition
is random: the external temperature is
\[
T_{ext} \in L^2_\mathbb{P} [L^2((0,t_f); L^2(\Gamma_N\cup\Gamma_F))].
\]
Evolution of the exterior temperaure field at \( t=5,10,15\).
The shape optimization problem
We shall next introduce a first shape functional, which is the
spatial \(L^2\)-norm of the von Mises stresses
\[
\sigma_{\text{VM}}(D) = \sqrt{\frac{d}{2}\sigma_d(u_D):\sigma_d(u_D)},
\]
that is
\[
J_1(D,t) := \frac{2}{d}\int_D |\sigma_{\text{VM}}(u_D(t,x))|^2\,dx
= \int_D\sigma_d(u_D(t, x)): \sigma_d(u_D(t, x))\,dx.
\]
To address the time dimension, we introduce the
parameters \(\gamma,\,\delta\geq0\) and consider a weighted
combination of two options: \(J(D,t)\) averaged in time
and \( J(D,t)\) taken at the final time, i.e.,
\[
J_{time}(D) := \gamma\int_0^{t_f} J_1(D,t)\, d t + \delta\,J_1(D,t_f).
\]
and then the final criterion is its expectation:
\[
J(D) = \mathbb{E}[J_{time}(D)].
\]
Our contribution .
We demonstrate that the robust
constraint and its derivative are completely determined
by low order moments of the random input, thus computable
by means of low-rank approximation. The optimization is then tractable. See the paper for precise statements.
Numerical results: Optimal shape of a pilar.
Optimal design of a pilar. Left: deterministic case Right: robust case
