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Mathematical modeling and numerical multigoal-oriented a posteriori error control and adaptivity for a stationary, nonlinear, coupled flow temperature model with temperature dependent density

Sven Beuchler, Ayhan Demircan, Bernhard Endtmayer, Uwe Morgner, Thomas Wick

TL;DR

The authors address a nonlinear stationary flow problem coupled with heat transfer where density and viscosity depend on temperature, formulating a monolithic Navier–Stokes–Fourier system with ρ(θ)=ρ_0 e^{−∫ α(θ) dθ} and ν(θ)=ν_0 e^{E_A/(R θ)}. They develop a multigoal dual‑weighted residual framework, employing enriched discrete adjoints and partition‑of‑unity localization to obtain reliable and efficient a posteriori error estimators for multiple quantities of interest, and they implement adaptive mesh refinement and solver control based on these estimators. The work demonstrates the method on laser‑driven test cases, examining prerefinement near heat sources, temperature‑dependent expansion, and comparisons between density‑based and Boussinesq models, reporting favorable error reductions and robust effectivity indices across scenarios. The contributions provide a practical, principled approach to QoI accuracy in multiphysics flows with temperature‑dependent material properties, enabling targeted refinement and automatic solver balancing in complex simulations.

Abstract

In this work, we develop adaptive schemes using goal-oriented error control for a highly nonlinear flow temperature model with temperature dependent density. The dual-weighted residual method for computing error indicators to steer mesh refinement and solver control is employed. The error indicators are used to employ adaptive algorithms, which are substantiated with several numerical tests. Therein, error reductions and effectivity indices are consulted to establish the robustness and efficiency of our framework.

Mathematical modeling and numerical multigoal-oriented a posteriori error control and adaptivity for a stationary, nonlinear, coupled flow temperature model with temperature dependent density

TL;DR

The authors address a nonlinear stationary flow problem coupled with heat transfer where density and viscosity depend on temperature, formulating a monolithic Navier–Stokes–Fourier system with ρ(θ)=ρ_0 e^{−∫ α(θ) dθ} and ν(θ)=ν_0 e^{E_A/(R θ)}. They develop a multigoal dual‑weighted residual framework, employing enriched discrete adjoints and partition‑of‑unity localization to obtain reliable and efficient a posteriori error estimators for multiple quantities of interest, and they implement adaptive mesh refinement and solver control based on these estimators. The work demonstrates the method on laser‑driven test cases, examining prerefinement near heat sources, temperature‑dependent expansion, and comparisons between density‑based and Boussinesq models, reporting favorable error reductions and robust effectivity indices across scenarios. The contributions provide a practical, principled approach to QoI accuracy in multiphysics flows with temperature‑dependent material properties, enabling targeted refinement and automatic solver balancing in complex simulations.

Abstract

In this work, we develop adaptive schemes using goal-oriented error control for a highly nonlinear flow temperature model with temperature dependent density. The dual-weighted residual method for computing error indicators to steer mesh refinement and solver control is employed. The error indicators are used to employ adaptive algorithms, which are substantiated with several numerical tests. Therein, error reductions and effectivity indices are consulted to establish the robustness and efficiency of our framework.
Paper Structure (20 sections, 4 theorems, 41 equations, 29 figures, 4 tables, 3 algorithms)

This paper contains 20 sections, 4 theorems, 41 equations, 29 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The model problem and discretization
  3. The model problem
  4. Discretization and numerical solution
  5. Goal oriented error estimation with the dual-weighted residual method
  6. Single goal
  7. The classical approach
  8. An approach on discrete spaces
  9. Multiple goals
  10. Combined Functional
  11. Algorithms
  12. Numerical experiments
  13. Prerefinement around laser
  14. Temperature dependent thermal expansion coefficient
  15. Flow in a square with two lasers
  16. ...and 5 more sections

Key Result

Theorem 3.1

Let $\tilde{Z}\in V_h$ be an approximation to $Z_h$ and $\tilde{U} \in U^D+V_h$ be an approximation to $U_h$. Furthermore let $U$ be the solution to the model problem weak form and $Z$ be the solution to the adjoint problem Goee: cont. adjoint. Additionally let $A\in \mathcal{C}^3$ and $J \in \mathc with and $\mathcal{R}^{(3)}$ as higher order term. For more information on $\mathcal{R}^{(3)}$, we

Figures (29)

  • Figure 1: $f_{E,y}(x)$ with $y=\frac{1}{2}(1,1)$ and $E=1$ on the domains $[0,1]^2$ for $\sigma=10^{-1}$ (left), $[0.45,0.55]^2$ for $\sigma=10^{-2}$ (center) and $[0.495,0.505]^2$ for $\sigma=10^{-3}$ (right).
  • Figure 2: Example 1+2: The domain $\Omega$$\Gamma_{\text{no-slip}}$ and $x_1$ and $x_2$ (left) and the initial mesh (right)
  • Figure 3: Example 1: Magnitude of velocity(left), pressure and adaptively refined mesh (center) and streamlines and temperature (right)
  • Figure 4: Example 1: Effectivity index.
  • Figure 5: Example 1: Relative errors for $J_1$, $J_2$, $J_3$ and absolute error for $J_c$.
  • ...and 24 more figures

Theorems & Definitions (18)

  • Theorem 3.1: Error representation(see RanVi2013EnLaWi18EnLaNeWiWo20)
  • proof
  • Corollary 3.2
  • proof
  • Remark 3.3
  • Definition 3.4: Efficient and reliable error estimator for $J$
  • Definition 3.5
  • Corollary 3.6: Efficiency and reliability for $\eta$
  • proof
  • Corollary 3.7: see EndtLaRiSchafWi24_book_chapter
  • ...and 8 more