# Working/Study Groups

## Finite Element Study Group, Fall 2023

** Organizer:** Noel J. Walkington

This study group will meet weekly to learn about high level finite element software packages, and to compare and contrast their capabilities, strengths, and weaknesses. The first meeting will review the Fenics and Firedrake packages, and in subsequent meeting the NgSolve and Mfem finite element software will be considered.

Meeting will be in person 3:00pm – 4:00pm Tuesdays in Wean Hall 7218.

## Working Group, Fall 2022

** High-dimensional flows and approximation**

** Organizers:** Dejan Slepčev and Bob Pego

Participants will discuss recent works related to a variety of problems in data science and statistics that involve flows in high dimensions and their computation/approximation. Many standard methods for working with flows suffer from the `curse of dimensionality' in that the number of variables needed for good approximations explodes exponentially with the dimension. The working group will focus on a variety of new approaches that overcome the curse. Concepts that may be relevant for understanding better approaches to these problems include sliced Wasserstein distance, reproducing kernel Hilbert spaces, and the random batch method.

## Working Group, Spring 2022

**Continuum Mechanics of defects in solids (and associated mathematical questions)**

**Organizer:** Amit Acharya

We will discuss a continuum mechanics framework for 1 and 2 dimensional defects in elastic solids (e.g. dislocations, cracks, grain and phase boundaries, and g.disclinations – the ideas extend to corresponding defects in nematic and smectic liquid crystals as well). Such defects, if dealt within just (non)linear elasticity, often result in nasty singularities with non-integrable energies in finite bodies. Physically, one is also interested in the dissipative motion, and the mutual interaction of these defects, as well as with applied loads. The framework we will discuss covers a manageable setup to deal with such matters in a reasonable way. Results of finite element based computational proxy models of the nonlinear pde systems will be used to illustrate the ideas.

Time permitting, we will also discuss a mathematically formal idea for associating a variational principle with a general pde system, whose Euler-Lagrange equation is the pde system. The primary motivation for this arises from the above theory of defects which produces time–dependent systems of Hamilton-Jacobi equations (but covers systems like nonlinear elastostatics not arising from a stored energy function, and the Navier Stokes system). Such an ‘action’ principle is also a necessary condition for application of path integral based statistical analysis methods for such field theories.

Overall, the hope is that mathematicians may get interested and help out engineers in solving pressing problems of 21st century mechanics, with physical and mathematical origins in the early 20th century.

**The Working group discussion will be based on modern developments leading to continuing realization of a program originating in:**

- G. Weingarten. Sulle superficie di discontinuità nella teoria della elasticità dei corpi solidi. Rend. Reale Accad. dei Lincei, classe di sci., fis., mat., e nat., ser. 5, 10.1:57–60, 1901. English translation: D. H. Delphenich. On the surface of discontinuity in the theory of elasticity for solid bodies. http://www.neo-classical-physics.info/theoretical-mechanics.html.
- Vito Volterra. Sur l' équilibre des corps élastiques multiplement connexes. In Annales scientifiques de l' École normale supérieure, volume 24, pages 401–517, 1907. English translation: D. H. Delphenich. On the equilibrium of multiply-connected elastic bodies. http://www.neo-classical-physics.info/theoretical-mechanics.html.
- J.F. Nye Some geometrical relations in dislocated crystals Acta Metall., 1 (2) (1953), pp. 153-162.
- B.A. Bilby, R. Bullough, E. Smith Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry Proc. R. Soc. Lond. A, 231 (1185) (1955), pp. 263-273
- E. Kröner Continuum theory of defects Balian R., Kléman M., Poirier J.-P. (Eds.), Physics of Defects, Les Houches Summer School Proceedings, North-Holland, Amsterdam (1981), pp. 217-315
- T. Mura Continuous distribution of moving dislocations Philos. Mag., 8 (89) (1963), pp. 843-857
- N. Fox A continuum theory of dislocations for single crystals IMA J. Appl. Math., 2 (4) (1966), pp. 285-298
- J.R. Willis Second-order effects of dislocations in anisotropic crystals Int. J. Eng. Sci., 5 (2) (1967), pp. 171-190

Also historically important:

- K. Kondo On the analytical and physical foundations of the theory of dislocations and yielding by the differential geometry of continua Int. J. Eng. Sci., 2 (3) (1964), pp. 219-251

### References:

- Amit Acharya, Robin J. Knops, Jeyabal Sivaloganathan (2019) On the structure of linear dislocation field theory,
*Journal of the Mechanics and Physics of Solids*, 130, 216-244. - Rajat Arora, Amit Acharya (2020) A unification of finite deformation J2 Von-Mises plasticity and quantitative dislocation mechanics,
*Journal of the Mechanics and Physics of Solids*, 143, 104050. - Chiqun Zhang and Amit Acharya (2018) On the relevance of generalized disclinations in defect mechanics,
*Journal of the Mechanics and Physics of Solids*, 119, 188-223. - Amit Acharya, Variational principles for nonlinear pde systems via duality, 2021
- Amit Acharya, An action for nonlinear dislocation dynamics, 2021.
- Xiaohan Zhang, Amit Acharya, Noel J. Walkington, Jacobo Bielak, (2015) A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations,
*Journal of the Mechanics and Physics of Solids*, 84, 145-195. - Chiqun Zhang, Amit Acharya, Alan C. Newell, Shankar C. Venkataramani (2021) Computing with non-orientable defects: nematics, smectics and natural patterns,
*Physica D: Nonlinear Phenomena*, 417, 132828.

Weingarten and Volterra's 1901 and 1907 papers may be difficult to parse; a revisit that may help (with extensions to finite deformations) is in

- Amit Acharya (2019) On Weingarten-Volterra defects,
*Journal of Elasticity*, 34, 79-101.

## Working Group, Fall 2021

**Preconditioners**

**Organizers:** Noel Walkington, Yangwen Zhang and Franziska Weber

We plan to study mainly multigrid and algebraic preconditioners from a mathematical point of view and try to find connections between those.

**Topics to be addressed:** V-cycle method, BPX, applications to linear and nonlinear PDEs, the Schwarz method and algebraic preconditioners using graph theoretical methods.

## Working Group, Fall 2020

**Nonlocal PDE and variational problems for data**

**Organizers:** Dejan Slepčev and Raghav Venkatraman

In the first few weeks we will discuss variational problems on random graphs and their consistency. In particular the recent results on graph Laplacian. The second part of the semester will be devoted to Stein Variational Descent and the relevant related geometries. In particular the novel transportation based geometries with nonlocal norms on the velocity and nonlocal continuity equation. Other topics may be discussed depending on the interests of participants.

## Working Group, Fall 2019

### Progress in nonlinear optimization and gradient descent

**Organizers:** Bob Pego, Hayden Schaeffer and Noel Walkington

Participants in this working group will discuss recent works that concern novel and interesting methods for determining optimizers and understanding gradient-like dynamics in nonlinear systems. Topics of interest include, e.g.: relaxed variational principles, acceleration techniques, stochastic approaches to minimization and approximation, and use of analytic structure.

## Working Group, Spring 2019

### Calculus of Variations

**Organizers:** Irene Fonseca and Giovanni Leoni

**Topics to be Addressed:** Rigidity estimates; Minimizing movements; BV spaces, fracture, crystal defects; Convex integration. If time remains: Free boundary problems; Topological methods.

**Fenics Study Group, Fall 2018**

**Author:** Noel Walkington

This study group will meet weekly learn about the FEniCS finite element software package and related issues. This will include an introduction the Python programing language which is required to utilize this software.

**Syllabus:** Python Basics (I); Python Basics (II); Fenics and Paraview demonstration (Neilan); Fenics Basics; Introcution to numpy (Weber); FireDrake vs Fenics; Discontinuous Spaces - Convection Problem; DG Schemes - Convection Diffusion Problem; Nonlinear Problems.