Modelling Heterogeneous Interfaces using Element-based Finite Volumes

TL;DR

This work addresses the need for accurate, conservation-preserving modelling of interfacial multiphysics in geometrically complex domains. It introduces a three-dimensional Element-Based Finite Volume Method (EbFVM) that blends the geometric flexibility of FEM with the local conservation guarantees of FVM, using body-fitted curvilinear mappings and a node-centered control-volume formulation. The method is demonstrated on lubricated interfaces via a generalized Reynolds equation with a p-theta cavitation model, coupled heat transfer, and film-thickness evolution, incorporating detailed thermophysical characterisation of the lubricant. Results on standard and micro-textured journal bearings show strong agreement with structured solvers while capturing texture-induced pressure and thermal variations, highlighting EbFVM as a scalable tool for interfacial transport and multiphysics applications across tribology, microfluidics, and bio-inspired materials.

Abstract

Accurately depicting multiphysics interactions in interfacial systems requires computational frameworks capable of reconciling geometric adaptability with strict conservation fidelity. However, traditional spatiotemporal discretisation methods often compromise between mesh flexibility and flow conservation enforcement, hence constraining their effectiveness in elucidating the underlying mechanisms. Here, we respond to these computational demands by developing a novel three-dimensional adaptation of the Element-based Finite Volume Method (EbFVM) -- a hybrid numerical strategy that merges the geometric flexibility of Finite Element Methods with the conservation-centric principles of Finite Volume Methods. The proposed framework introduces advanced discretisation techniques tailored to unstructured, irregular mesh entities, including detailed parametric shape functions, robust flux integration schemes and rigorous body-fitted curvilinear coordinate mappings. Through a series of lubrication-driven benchmark problems, we demonstrate the EbFVM's capacity to capture intricate transport phenomena, strong field couplings and scale disparities across geometrically complex domains. By enabling accurate modelling in geometrically and physically challenging interfacial systems, the three-dimensional EbFVM offers a versatile and generalisable tool for simulating transport phenomena in a plethora of multiphysics applications.

Figures (13)

• Figure 1: Overview of the Element-Based Finite Volume Method (EbFVM). The EbFVM synergistically integrates: (A) the geometric adaptability of the Finite Element Method (FEM), enabling accurate conformity to complex domains, and (B) the conservation principles of the Finite Volume Method (FVM), ensuring rigorous balance of transport fluxes across control volumes. (C) A schematic representation of the key geometric entities that form the foundation of the EbFVM framework.
• Figure 2: Three-dimensional geometric elements employed in the proposed EbFVM discretisation framework. (A) Triangular prism elements with six nodes, offering enhanced flexibility for conforming to irregular or curved geometries. (B) Hexahedral elements with eight nodes, facilitating structured meshing and efficient flux integration.
• Figure 3: The multiphysics and multiscale nature of lubricated interfaces. Illustration of the hierarchical transition from component-level systems (e.g., wind turbine assemblies) to macro-, micro- and molecular-scale levels, capturing the escalating complexity of interfacial phenomena. As spatial scales descend, the interplay of the physicochemical interactions becomes increasingly intricate. Microscopic schematic is adapted from Vakis2018Sep, with permission from Elsevier.
• Figure 4: Schematic of the simulated journal bearing system. (A) Cross-sectional view highlighting key geometric features and nomenclature. (B) Transformation of the cylindrical geometry into an unwrapped two-dimensional domain, justified by the negligible influence of curvature on hydrodynamic behaviour.
• Figure 5: Comparative evaluation of numerical discretisation schemes based on key lubrication field predictions, comprising hydrodynamic pressure, lubricant fraction and mid-plane temperature computed using (A) conventional hexahedral finite volume method, (B) hexahedral element-based finite volume method and (C) prismatic element-based finite volume method.