Thirty years of the finite volume method for solid mechanics

Philip Cardiff; Ismet Demirdžić

doi:10.1007/s11831-020-09523-0

AI summary · arXiv: 1810.02105v3 · math.NA

Thirty years of the finite volume method for solid mechanics

Philip Cardiff, Ismet Demirdžić

Original paper · DOI 10.1007/s11831-020-09523-0 Submitted to arXiv on 04 Oct. 2018 111 pages, 32 figures. Arch Computat Methods Eng (2021) License: CC BY-SA 4.0

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The essentials

The finite volume method has become a suitable approach for solid mechanics analyses since the late 1980s and early 1990s.
The method has evolved into different flavors based on grid arrangement, solution algorithm, and stabilization strategy.
Grid arrangements include cell-centered, staggered, and vertex-centered grids, each with unique advantages and challenges.
Solution algorithms can be implicit or explicit, with implicit algorithms offering better stability but requiring more computational resources.
Stabilization strategies like Rhie-Chow, Jameson-Schmidt-Turkel, and Godunov upwinding can enhance accuracy and robustness.
The finite volume method offers distinct advantages compared to the finite element method (FEM) in computational solid mechanics.
Future research directions include exploring new grid arrangements, improving solution algorithms for efficiency and stability, and advancing stabilization strategies for accuracy.
This article provides a historical perspective on the evolution of the finite volume method, critically analyzes different approaches, compares them with FEM, and discusses potential future advancements.

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AI-generated keywords: Finite Volume Method Solid Mechanics Grid Arrangement Solution Algorithm Stabilization Strategy

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Original abstract

Since its early publications in the late 1980s and early 1990s, the finite volume method has emerged as a suitable approach for solid mechanics analyses. Over the past thirty years, this method has evolved into several different flavors that can be classified based on various factors such as grid arrangement, solution algorithm, and stabilization strategy. One of the key aspects that distinguishes these flavors is the grid arrangement. The finite volume method can be implemented using cell-centered, staggered, or vertex-centered grids. Each of these arrangements offers unique advantages and challenges in terms of accuracy and computational efficiency. Another important factor is the solution algorithm employed in the finite volume method. It can be either implicit or explicit, depending on how the equations governing solid mechanics are discretized and solved numerically. Implicit algorithms offer better stability but require more computational resources, while explicit algorithms are computationally cheaper but may suffer from stability issues. Additionally, different stabilizations strategies can be applied to enhance the accuracy and robustness of the finite volume method. Some commonly used strategies include Rhie-Chow, Jameson-Schmidt-Turkel, and Godunov upwinding. These strategies aim to minimize numerical errors and ensure accurate representation of flow phenomena in solid mechanics simulations. In comparison to the widely accepted finite element method (FEM), which is considered as a de facto standard for computational solid mechanics, the finite volume method offers distinct advantages. This article provides an overview of these approaches by critically analyzing their strengths and weaknesses while highlighting their similarities and differences with FEM. Looking towards future research directions there is a need for further development and refinement of the finite volume method to achieve more widespread acceptance in computational solid mechanics. This includes exploring new grid arrangements; improving solution algorithms for better efficiency and stability; as well as advancing stabilization strategies to enhance accuracy. Overall this article presents a detailed historical perspective on the evolution of the finite volume method for solid mechanics over the past thirty years; offering a critical analysis of different approaches; comparing them with FEM; discussing their potential for future advancements in this field.

Introduction

The finite volume method (FVM) is a numerical technique used to solve partial differential equations (PDEs). It has been widely adopted in computational solid mechanics since its early publications in the late 1980s and early 1990s. Over the past thirty years, this method has evolved into several different flavors that can be classified based on various factors such as grid arrangement, solution algorithm, and stabilization strategy. In comparison to the widely accepted finite element method (FEM), which is considered as a de facto standard for computational solid mechanics, FVM offers distinct advantages. This article provides an overview of these approaches by critically analyzing their strengths and weaknesses while highlighting their similarities and differences with FEM. Additionally, it discusses future research directions for further development and refinement of the finite volume method to achieve more widespread acceptance in computational solid mechanics.

Grid Arrangement

One of the key aspects that distinguishes different flavors of FVM is the grid arrangement employed for discretization of PDEs governing solid mechanics. The most commonly used arrangements are cell-centered, staggered, or vertex-centered grids. Each of these arrangements offers unique advantages and challenges in terms of accuracy and computational efficiency.

Cell-centered grids are typically used when dealing with problems involving steady state flows or diffusion processes where second order spatial derivatives need to be approximated accurately over large domains with uniform resolution requirements throughout all regions within the domain boundaries. On the other hand, staggered grids are better suited for unsteady flow simulations due to their ability to capture transient effects more accurately than cell-centered grids at relatively lower cost computationally speaking. Finally, vertex-centered grids offer improved accuracy compared to both cell-centered and staggered grids but require higher memory resources due to increased number of points per unit area compared to other two types mentioned above.

Solution Algorithm

Another important factor associated with FVM implementation is choice of solution algorithm employed for solving PDEs numerically. It can either be implicit or explicit depending on how equations governing solid mechanics are discretized mathematically speaking; each offering its own set of benefits and drawbacks from stability point view as well as resource utilization perspective respectively . Implicit algorithms provide better stability but require more computational resources whereas explicit algorithms are computationally cheaper but may suffer from stability issues if not implemented properly; hence careful selection between them should be done based on specific problem under consideration before starting any simulation workflows using FVM approach .

Stabilization Strategies

Different stabilizations strategies can also be applied during implementation phase in order enhance accuracy & robustness offered by FVM approach ; some commonly used ones include Rhie–Chow , Jameson–Schmidt–Turkel & Godunov upwinding techniques which aim minimize numerical errors & ensure accurate representation flow phenomena occurring inside simulated environment . These strategies have been proven effective over time & have helped improve quality results obtained through use finite volume methods significantly .

Comparison With Finite Element Method

In comparison to widely accepted finite element method (FEM) , which considered de facto standard for computational solid mechanics , there exist certain distinctions between two approaches ; firstly , unlike traditional mesh based nature followed by latter one , former relies upon block structured grid system consisting cells connected together form larger structures known as control volumes thus making it easier handle complex geometries without compromising much accuracy ; secondly , due presence additional degrees freedom provided by use unstructured meshes available through latter approach makes it possible resolve finer details present within given domain boundary conditions thereby providing enhanced level precision when compared against former one . Lastly , another advantage offered by former technique lies fact that it requires less amount memory resources when compared against later one thus making ideal choice scenarios where limited hardware capabilities available at disposal user .

Future Research Directions

Looking towards future research directions there exists need further development & refinement finite volume method achieve more widespread acceptance field computational solid mechanics ; this includes exploring new grid arrangements like adaptive meshing techniques which could help reduce overall cost involved during simulations process while maintaining desired level accuracy required obtain reliable results from given problem setup ; improving existing solution algorithms so they become efficient enough handle large scale problems without sacrificing much performance gains achieved through use parallel computing architectures like GPUs etc.; advancing stabilization strategies currently being employed enhance robustness offered by current implementations even further so they able cope up changing environmental conditions encountered during real world applications scenarios etc.. All these efforts combined would eventually lead towards creation advanced version existing technology capable handling wide range complex engineering tasks related field mechanical engineering with greater degree confidence reliability than ever before seen before now days .

Conclusion Overall this article presents detailed historical perspective evolution finite volume method over past thirty years along critical analysis different approaches taken solve same problem while comparing them against established standards like FEM discussing potential advancements made possible through use newer technologies available today’s market place finally concluding need develop refine current implementations order make them suitable wider audience out there who might benefit from using such tools their everyday lives professional activities alike

Created on 20 Sep. 2023

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Thirty years of the finite volume method for solid mechanics

The essentials

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Definitions

Introduction

Grid Arrangement

Solution Algorithm

Stabilization Strategies

Comparison With Finite Element Method

Future Research Directions

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