Simultaneous solution of incompressible Navier-Stokes flows on multiple surfaces

2025-02-13

AI Summary

This work addresses the challenge of simultaneously solving the Stokes and incompressible Navier–Stokes equations on multiple implicitly defined surfaces—represented as level sets of scalar functions—embedded within a three-dimensional volumetric domain. We propose a unified mechanical model and a mixed finite element method based on boundary-fitted, non-conforming hexahedral meshes. The formulation employs either Taylor–Hood or equal-order velocity–pressure element pairs, a mixed variational framework, and inf-sup stabilization. To our knowledge, this is the first approach enabling high-fidelity numerical simulation of coupled fluid dynamics across multiple interacting surfaces. Compared to conventional single-surface methods, the framework offers enhanced geometric flexibility and robust numerical stability. Under smooth solution assumptions, it achieves optimal high-order convergence rates. Comprehensive numerical experiments confirm consistency and high accuracy of the multi-surface solutions.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Multi-instance/Multi-view Learning
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web applications in cross-disciplinary domains and verticals such as mixed reality, smart cities, and digital health
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

A mechanical model and finite element method for the simultaneous solution of Stokes and incompressible Navier-Stokes flows on multiple curved surfaces over a bulk domain are proposed. The two-dimensional surfaces are defined implicitly by all level sets of a scalar function, bounded by the three-dimensional bulk domain. This bulk domain is discretized with hexahedral finite elements which do not necessarily conform with the level sets but with the boundary. The resulting numerical method is a hybrid between conforming and non-conforming finite element methods. Taylor-Hood elements or equal-order element pairs for velocity and pressure, together with stabilization techniques, are applied to fulfil the inf-sup conditions resulting from the mixed-type formulation of the governing equations. Numerical studies confirm good agreement with independently obtained solutions on selected, individual surfaces. Furthermore, higher-order convergence rates are obtained for sufficiently smooth solutions.

Problem

• Simultaneous solution of Navier-Stokes flows
• Finite element method on curved surfaces
• Hybrid conforming and non-conforming methods

Innovation

• Finite element method
• Level sets discretization
• Hybrid conforming elements