Adaptive Finite Element Simulation of Nonlinear Coupled Flow-Temperature Model with Temperature-Dependent Density and Viscosity

Core Concepts

This work develops adaptive finite element schemes using goal-oriented error control for a highly nonlinear flow-temperature model with temperature-dependent density and viscosity. The dual-weighted residual method is employed to compute error indicators that steer mesh refinement and solver control.

Abstract

The authors present a mathematical model for the Navier-Stokes equations coupled with a heat equation, where the density and viscosity depend on the temperature. This results in a highly nonlinear coupled PDE system.
The numerical solution algorithms are based on monolithic formulations, where the entire system is solved all-at-once using a Newton solver. The Newton tolerances are chosen according to the current accuracy of the quantities of interest, which is achieved by a multigoal-oriented a posteriori error estimation with adjoint problems, using the dual-weighted residual (DWR) method.
The error estimators are localized using a partition-of-unity technique, which enables adaptive mesh refinement. Several numerical examples in 2D are presented, including comparisons between the new model with temperature-dependent density and viscosity, and a simpler Boussinesq model. The results show robust and efficient error reduction, with effectivity indices close to one.

Stats

The authors use the following parameter values in their numerical experiments:

ρ0 = 998.21
ν0 = 2.216065960663198 × 10−6
EA = 14906.585117275014
k = 0.5918

Quotes

"The resulting coupled problem is highly nonlinear."
"The adjoint needs to be explicitly derived for our coupled problem."
"Our numerical examples are inspired from applications in laser material processing."

Key Insights Distilled From

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

by Sven Beuchle... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.01823.pdf

Deeper Inquiries

How would the results change if the boundary conditions were modified, e.g., using Neumann or Robin conditions instead of Dirichlet

Modifying the boundary conditions from Dirichlet to Neumann or Robin conditions would have a significant impact on the results of the numerical simulations.

Neumann Conditions: Neumann boundary conditions specify the derivative of the solution at the boundary instead of the value itself. This would affect the behavior of the solution near the boundaries, especially in terms of flux and how it interacts with the domain. The flow patterns and temperature distribution could be altered, leading to different vortices and temperature gradients compared to Dirichlet conditions.

Robin Conditions: Robin boundary conditions are a combination of Dirichlet and Neumann conditions, incorporating both the value of the solution and its derivative. By introducing Robin conditions, the balance between the prescribed values and the flux at the boundaries would change, potentially influencing the overall flow and temperature profiles within the domain.
In summary, changing the boundary conditions to Neumann or Robin conditions would introduce new constraints and dynamics to the system, resulting in variations in the flow patterns, temperature distribution, and overall behavior of the model.

What are the potential challenges in extending this framework to 3D problems, and how would the computational cost and performance scale

Extending the framework to three-dimensional (3D) problems poses several challenges and considerations:

Increased Complexity: Moving from 2D to 3D introduces a higher level of complexity in terms of geometry, mesh generation, and numerical computations. The additional dimensionality requires more computational resources and memory to handle the increased data and calculations.

Mesh Generation: Generating and refining 3D meshes is more challenging and computationally intensive compared to 2D meshes. The mesh quality and resolution play a crucial role in accurately capturing the behavior of the flow and temperature fields in 3D space.

Computational Cost: The computational cost of solving 3D problems is significantly higher than 2D problems due to the increased number of degrees of freedom, larger matrices to solve, and more complex numerical algorithms required for accurate simulations.

Scalability: The performance of the framework in 3D would depend on its scalability with respect to the problem size, mesh refinement, and computational resources. Efficient parallelization and optimization techniques would be essential for handling the increased computational demands of 3D simulations.
In conclusion, extending the framework to 3D problems would require addressing these challenges to ensure accurate and efficient numerical simulations.

Can the proposed approach be adapted to handle time-dependent problems, and how would the error estimation and adaptivity strategies need to be modified

Adapting the proposed approach to handle time-dependent problems would involve modifications to the error estimation and adaptivity strategies to account for the temporal dimension. Here are some key considerations:

Time-Dependent Error Estimation: Error estimation techniques would need to incorporate the temporal error in addition to spatial errors. This could involve tracking the evolution of errors over time and adjusting the mesh adaptivity based on both spatial and temporal accuracy requirements.

Temporal Adaptivity Strategies: Adaptive mesh refinement and time-stepping strategies would need to be coordinated to ensure efficient and accurate simulations. This could involve dynamically adjusting the time step size based on the error estimates and adapting the mesh in both space and time.

Multi-Time-Step Methods: Implementing multi-time-step methods could be beneficial for time-dependent problems, allowing for adaptive time integration schemes that adjust the time step size locally based on the error estimates.

Transient Boundary Conditions: Time-dependent problems may require transient boundary conditions that evolve over time. Adapting the boundary conditions dynamically could be necessary for accurately capturing the behavior of the system.
In summary, handling time-dependent problems would require a comprehensive approach that integrates temporal considerations into the error estimation, adaptivity strategies, and numerical algorithms to ensure robust and efficient simulations.