Compatible Finite Element Methods for Geophysical Flows

Abstract

Union's Horizon 2020 research and innovation programme (grant agreement no 741112). vii 1 Geophysical fluid dynamics and simulation The hierarchy of models we will build up in this chapter is shown in Figure 1.1. This is just one choice of hierarchy, since for example one can apply the hydrostatic approximation directly to the compressible Euler equations without making Boussinesq /anelastic approximations. Similarly, we do not consider quasi-geostrophic approximations which are valid for rapidly rotating systems; these approximations can be made at any step of our hierarchy. Governing equations: Full, three-dimensional compressible Euler system. Boussinesq/anelastic: Assumption: small variations in density. Impact: no acoustic waves. Hydrostatic: Assumption: fluid is in a thin layer. Impact: No vertical acceleration term, vertical velocity becomes diagnostic from the continuity equation. Shallow water system Assumption: columnar motion (horizontal velocity is independent of height). Impact: equations become two-dimensional, prognostic variables are horizontal velocity and layer height. 1.1.1 The compressible Euler system We start by presenting the compressible Euler equations, which have the fewest approximations amongst our hierarchy. We assume that the air is dry (no moisture), inviscid (no viscous forces), and adiabatic (no sources or diffusion of temperature). The governing equations for a dry, inviscid, adiabatic, compressible fluid in a rotating reference frame with angular velocity Ω may be written in the form

Automating Finite Element Methods for Geodynamics via Firedrake

“…Firedrake is a next-generation automated system for solving variational problems using the finite element method (e.g. Rathgeber et al, 2016;Gibson et al, 2019). It has a number of features that are ideally suited to simulating geophysical fluid dynamics problems, as exemplified by its use in application areas such as coastal ocean modelling (Kärnä et al, 2018), numerical weather prediction (Shipton et al, 2018), and glacier flow modelling (Shapero et al, 2021).…”

“…Equations (8-10) are a more general, mathematically rigorous representation of the continuous PDEs in strong form (Equations 6, 2 and 7), provided suitable function spaces with sufficient regularity are chosen (see, for example Zienkiewicz and Taylor, 1991;Elman et al, 2005). Galerkin finite element discretisation proceeds by restricting these function spaces to We note that there are many other choices of finite element function spaces available in Firedrake, although they are not considered herein (see Gibson et al, 2019, for an overview). All that is required for their implementation is that a basis can be found for the function space such that each solution can be written as a linear combination of basis functions.…”

A Comparative Study of the Application of Different Commercial Software for Topology Optimization

“…In this table, only the software platforms used in the comparative study presented later in this paper are included, which are SolidWorks, ANSYS Mechanical, and ABAQUS. Interested readers can access and edit the library online at the following links: https://docs.google.com/document/d/1CnQOA492EkOdX1gi59EWaPnrJBL-fOQs/edit?usp=sharing&ouid=110678641076107916949&rtpof=true&sd=true (accessed on 15 December 2021 ) or Supplementary Material [13,15,16,[22][23][24][26][27][28]. The authors' intention was to create an updated database of the existing TO software where every CAD designer could add new or missing TO software.…”

“…In addition, there are some scripts that implement different optimization methods, such as the CalculiX [17], which uses the bi-directional evolutionary structural optimization (BESO) [18], or the Allaire_Scilab [19], which applies the level set method [20]. Furthermore, there are a few open source platforms that offer a toolbox of TO scripts, such as the FreeFem [21] and the Firedrake [22]. On the other hand, the commercial CAD/FEM platforms have integrated TO over the last years due to the lack of computational power.…”

A compatible finite‐element discretisation for the moist compressible Euler equations

“…Further information on hybridisation techniques and their implementation can be found in Gibson et al . (2019; 2020). More on this particular linear solver and its performance will be detailed in future work.…”