Mixed Virtual Element Approximation for Steady Boussinesq Problem with Temperature-Dependent Parameters

Konsep Inti

The author develops a new mixed-VEM for the steady two-dimensional Boussinesq equation with temperature-dependent parameters, utilizing a fixed-point strategy and convergence analysis.

Abstrak

The content discusses the development of a mixed virtual element method for the Boussinesq problem with temperature-dependent parameters. It includes theoretical analysis, discretization techniques, convergence analysis, and numerical examples to illustrate the proposed method's performance.
Free convection processes are modeled using the Boussinesq approximation, considering fluid properties affected by temperature. Various numerical approximations, including finite element methods and virtual element methods, have been explored in previous studies. The authors propose a fully mixed virtual element method based on pseudostress-velocity formulation for solving the Boussinesq problem with temperature-dependent parameters. The continuous formulation is presented, followed by discretization using virtual element subspaces. Solvability and convergence analyses are conducted to establish optimal error estimates.
Researchers have extended virtual element methods to solve linear and nonlinear problems in fluid mechanics. The proposed method aims to broaden the applicability of mixed-VEM to nonlinear models featuring variable coefficients. By extending previous works on VEMs for fluid mechanics models, the authors introduce a fully mixed VEM for the Boussinesq problem with temperature-dependent parameters.

Statistik

A priori convergence analysis shows an optimal rate of convergence.
Banach spaces-based approach extended to VEM framework.
Bilinear forms exhibit ellipticity properties.
Constants αS and αT ensure stability conditions are met.
Inf-sup conditions guarantee well-defined operators L_S and L_T.
Estimates show unique solutions exist for uncoupled problems (3.4) and (3.9).

Kutipan

"Various types of free convection are present in nature and in industry."
"In recent years, researchers have focused on extending virtual element methods to solve linear and nonlinear problems in fluid mechanics."
"The proposed scheme is rewritten as an equivalent fixed point operator equation."

Wawasan Utama Disaring Dari

Mixed virtual element approximation for the five-field formulation of the steady Boussinesq problem with temperature-dependent parameters

by Zeinab Ghari... pada arxiv.org 03-13-2024

https://arxiv.org/pdf/2403.07173.pdf

Pertanyaan yang Lebih Dalam

How does the proposed mixed-VEM approach compare to traditional finite element methods

The proposed mixed-VEM approach offers several advantages over traditional finite element methods. One key difference is the flexibility it provides in handling complex fluid dynamics problems with temperature-dependent parameters. By incorporating pseudostress, vorticity, velocity, pseudoheat vector, and temperature fields as additional variables in the formulation, the mixed-VEM method can capture more intricate physical phenomena accurately. This leads to improved accuracy and reliability in simulating fluid flow behavior under varying conditions.
Furthermore, the use of virtual element techniques allows for greater adaptability to irregular geometries and mesh structures compared to traditional finite element methods. The ability to work with H(div6/5) and L6(Ω) spaces enables a more efficient representation of the solution space while maintaining stability and convergence properties.
Overall, the mixed-VEM approach presents a modern and versatile framework for solving complex fluid dynamics problems that goes beyond the limitations of traditional finite element methods.

What computational challenges may arise when implementing fully mixed VEMs

Implementing fully mixed VEMs may pose certain computational challenges due to their increased complexity compared to standard finite element methods. Some potential challenges include:

Increased Computational Cost: Fully mixed VEMs involve solving coupled systems of equations for multiple unknowns simultaneously (e.g., pseudostress, vorticity, velocity), which can lead to higher computational costs compared to single-field formulations.

Nonlinear Iterative Solvers: The presence of nonlinear terms in the variational formulations requires iterative solvers capable of handling nonlinearity efficiently. Convergence issues may arise if not properly addressed.

Mesh Generation: Generating suitable meshes that conform to both H(div6/5) and L6(Ω) spaces can be challenging for complex geometries or unstructured domains.

Stability Analysis: Ensuring stability and well-posedness of fully mixed VEM formulations often requires rigorous mathematical analysis due to the interplay between different fields involved.

Addressing these challenges typically involves a combination of advanced numerical techniques, careful algorithm design, adaptive mesh refinement strategies, and thorough validation against analytical solutions or experimental data.

How can the concept of pseudostress be applied in other fluid dynamics simulations

The concept of pseudostress plays a crucial role in enhancing our understanding of fluid dynamics by capturing additional physics beyond conventional stress tensors like Cauchy stress tensor or viscous stress tensor.
In other fluid dynamics simulations:

In viscoelastic fluids modeling: Pseudostress helps account for elastic effects alongside viscous effects present in viscoelastic materials.
In multiphase flows: Pseudostress aids in characterizing interface behavior between different phases by accounting for surface tension forces.
In non-Newtonian fluids: Pseudostress contributes towards modeling shear-thinning or shear-thickening behaviors observed in non-Newtonian fluids based on their rheological properties.
By incorporating pseudostress into various fluid dynamics simulations across different applications, researchers can achieve more comprehensive models that better represent real-world phenomena with enhanced predictive capabilities.