Thermal Analyses#

This page presents the thermal analyses currently supported by OpenPisco and their corresponding mathematical formulations.

Overview#

OpenPisco provides thermal and thermo-mechanical analyses used in topology optimization workflows.

Supported Analyses#

  • Thermal conduction

  • Thermal conduction based on eigenvalues

  • Thermoelasticity

Thermal Conduction#

We are looking for the temperature \(T\), solution of the following problem

\[\begin{split}\begin{equation} \left\{ \begin{aligned} - \rho C_p \frac{\partial T}{\partial t} - \text {div} (\lambda \nabla T) &= S \qquad &\text{ in } \Omega \times \mathbb{R}^{+},\\ \lambda \nabla T \cdot n &= -h (T-T_{ext}) \qquad &\text{ on } \Gamma_N \times \mathbb{R}^{+}, \\ \lambda \nabla T \cdot n &= 0 \qquad &\text{ on } \Gamma \times \mathbb{R}^{+}, \\ T &= T_d\qquad &\text{ on } \Gamma_D \times \mathbb{R}^{+},\\ T(t=0) &= T_{init}\qquad &\text{ in } \Omega.\\ \end{aligned} \right. \end{equation}\end{split}\]

where:

  • \(S\) is the thermal source

  • \(T_{ext}\) is the emperature of the fluid in contact with the structure

  • \(T_d\) is the prescribed temperature

  • \(h\) is the convection coefficient

  • \(\rho\) is the material density

  • \(C_p\) is the thermal capacity

  • \(\lambda_c\) is the thermal conductivity

  • \(T_{init}\) is the initial temperature

  • the temperature is prescribed at \(T_d\) on \(\Gamma_D\)

Thermal Conduction Based on Eigenvalues#

The unknown temperature \(T\) of following thermal problem

\[\begin{split}\begin{equation} \left\{ \begin{aligned} - \rho C_p \frac{\partial T}{\partial t} - \text {div} (\lambda \nabla T) &= S \qquad &\text{ in } \Omega \times \mathbb{R}^{+},\\ T &= T_d\qquad &\text{ on } \partial \Omega \times \mathbb{R}^{+},\\ T(t=0) &= T_{init}\qquad &\text{ in } \Omega.\\ \end{aligned} \right. \end{equation}\end{split}\]

can be computed using a spectral decomposition involving the following eigenvalue problem

\[\begin{split}\begin{equation} \left\{ \begin{aligned} \text {div} (\lambda_c \nabla T_i) &= \mu_i T_i \qquad &\text{ in } \Omega \times \mathbb{R}^{+},\\ T &= T_d\qquad &\text{ on } \partial \Omega \times \mathbb{R}^{+},\\ \end{aligned} \right. \end{equation}\end{split}\]

where:

  • \(\mu_i\) is the i-th eigenvalue

  • \(T_i\) is the i-th eigenmode

  • \(T_d\) is the fixed temperature

  • \(\rho\) is the material density

  • \(C_p\) is the thermal capacity

  • \(\lambda_c\) is the thermal conductivity

Associated Implementations#

  • OpenPisco.PhysicalSolvers.AsterThermalEigenValues

Thermoelasticity#

We investigate a weakly coupled thermoelastic problem. As such, the first step is to perform a classic thermal analysis to find the temperature \(T\) (see thermal analysis for more details). The second step is, for a given temperature field, to find the displacement governed by the following problem

\[\begin{split}\begin{equation} \left\{ \begin{aligned} \text {div} (\sigma(u,\Delta T)) + f &= 0 \qquad &\text{ in } \Omega,\\ \sigma(u,\Delta T) &= {\cal A} e(u) + {\cal A} e^{ther}(\Delta T) \qquad &\text{ in } \Omega,\\ \sigma(u) \cdot n &= g \qquad &\text{ on } \Gamma_N, \\ \sigma(u) \cdot n &= 0 \qquad &\text{ on } \Gamma, \\ u &= 0\qquad &\text{ on } \Gamma_D.\\ \end{aligned} \right. \end{equation}\end{split}\]

where:

  • \(e(u)\) is the strain tensor

  • \(e^{ther}\) is the thermal strain equal to

\[\begin{equation} e^{ther}(\Delta T) = -\alpha {\cal A} (\Delta T I). \end{equation}\]

  • \({\cal A}\) is the fourth order elasticity tensor equal to

\[\begin{equation} {\cal A}_{ijkh}=\lambda\,\delta_{ij}\delta_{kh}+\mu(\delta_{ik}\delta_{jh}+\delta_{ih}\delta_{jk}) \end{equation}\]

with \(\delta_{ij}\) the kronecker symbol.

  • \(\lambda\) and \(\mu\) are the Lamé coefficients

  • \(g\) is the density of surfaces forces

  • \(f\) is the density of volumic forces

  • the body is clamped on \(\Gamma_D\)