# W24-09 Geometric Mechanics Formulations and Structure Preserving Discretizations: An Introductory Course

**Instructor(s):** Christopher Eldred, *Sandia National Laboratories; *Marc Gerritsma, *TU Deflt; *Artur Palha, *TU Delft*

**Description:**

Systems of partial differential equations (PDEs) underlie most of the models used in the study of continuum mechanics (CMMs), across a range of disciplines such as climate, aerospace, materials science, biomedicine, porous media, combustion and plasma dynamics. Since CMMs are generally intractable by hand, especially for complex multiphysics applications, they must be simulated using numerical methods instead. In doing so, stable and physically accurate simulations are needed. A powerful way to achieve this is through the use of geometric mechanics (GM) formulations (variational, Hamiltonian, metriplectic, etc.) combined with structure-preserving (SP) discretizations. In this approach, the equations of motion are first written in terms of a GM formulation, and then the GM formulation itself (rather than the equations of motion) is discretized using a SP discretization. By doing, so a discrete version of the GM formulation can be developed, with many of the same key properties as the continuous one. This includes conservation laws, involution constraints and freedom from spurious (computational) modes.

This course will serve as an introduction to these ideas, illustrating the entire GM/SP process for the shallow water equations, a prototypical example of a continuum mechanical system. It will consist of a combination of lectures and hands-on work, with the latter consisting of both coding and pen+paper derivations. Longer course notes will be made available online, that discuss the general theory in detail and provide many examples of continuum mechanics systems that can be treated using the GM/SP approach. A companion session (0306: Geometric mechanics formulations and structure-preserving discretizations for continuum mechanics and kinetic models) will happen during the main WCCM conference.

Prerequisites: vector calculus, basis knowledge of PDEs, basic knowledge of finite element methods

**Prelminary Schedule:**

**08:45 - 10:15 Intro / Motivating Example**

What are geometric mechanics formulations? What are structure-preserving discretizations? Why do they matter? How are they applied? In this lecture we will discuss answers to these questions, along with a motivating example: the shallow water equations.

**10:15 - 10:30 Coffee Break**

Drink a coffee, have a snack, contemplate the meaning of the universe...

**10:30 - 12:00 Geometric Mechanics Formulations**

In this lecture, we will give an introduction to Geometric Mechanics, focusing on basic geometric mechanics formulations (variational and Hamiltonian) for the shallow water equations. Additionally, there will be some brief discussion of more advanced topics such as formulations for (thermodynamically) irreversible processes (ex. metriplectic); and the semi-direct product theory and exterior calculus that underlies geometric mechanics formulations.

**12:00 - 13:00 Lunch Break**

Sample the delicious food available in Vancouver!

**13:00 - 14:30 Structure-Preserving Discretizations I**

In this lecture, we will cover the basic theory of structure-preserving discretiza- tions, concentrating on the core concept of the de Rham complex, and classifi- cation of fields that appear in physical field theories.

**14:30 - 14:45 Coffee Break**

Drink a coffee, have a snack, contemplate the meaning of the universe...

**14:45 - 16:15 Structure-Preserving Discretizations II**

In this lecture, we will continue the discussion of structure-preserving discretiza- tions, focusing on finite-element type methods (ex. finite element exterior cal- culus) in the context of the shallow water equations. Additionally, there will be some brief discussion of more advanced topics such as single & double de Rham complex methods, variational integrators and structure-preserving time discretization.

**16:15 - 16:30 Break**

Drink a coffee, have a snack, contemplate the meaning of the universe...

**16:30 - 17:00 Wrap Up**

We will wrap up the course with discussion about how things went, and point out additional more detailed material that is available online.