MAC-MIGS Deep Dive on Finite Element Software – 12th/13th November
Various powerful finite element packages have been developed in recent years, such as FEniCS, Firedrake, and NGSolve, were developed. These packages provide a high-level Python interface to their underlying efficient C++ implementations. Users may specify the weak form and finite element spaces similarly to writing down the mathematical formulas.This enables the incorporation of up-to-date results from mathematical research on finite element methods. These finite element packages have achieved significant impact in the engineering community for real-world applications from solid mechanics and other areas, and have opened new research opportunities.This Deep Dive brings in leading experts to provide students with introduction to the state-of-the-art analysis and applications of finite element methods and software. The Deep Dive will conclude with an ACM research seminar.
Details and timetable are below. Note different locations.
Tuesday 12th November
| 10.00-11.00 (ICMS 5.10) | Lucy Weggler (TU Berlin) | Boundary Element Methods in Practice: Solving 3D Problems with NGSolve and NGBem |
| 11.00-12.00 (ICMS 5.10) | Cécile Daversin-Catty (Simula) | Mixed-dimensional coupled finite elements in FEniCS |
| 12:00-14:00 (Bayes Centre 5th Floor) | Lunch for registered participants | |
| 14.00-16.00 (Lister Learning and Teaching Centre 2.14) | Patrick Farrell (Oxford) | Solving PDEs with Firedrake (Part 1) |
Wednesday 13th November
| 9:45-10:45 (40 George Sq. (Lecture Theatre B)) | Patrick Farrell (Oxford) | Solving PDEs with Firedrake (Part 2) |
| 11:00-12:00 (Newhaven Lecture Theatre (South College St.)) | Patrick Farrell (Oxford) | Solving PDEs with Firedrake (Part 2) |
| 15.00-16.00 (Appleton Tower 2.14) | Patrick Farrell (Oxford) | Designing conservative and accurately dissipative numerical integrators in time |
| 16.00 – (Bayes centre 5th floor) | Coffee for all seminar participants at Bayes, 5th Floor. |
Organisers: Lehel Banjai, Manolis Georgoulis, Kaibo Hu, John Pearson
Abstracts
Lectures
Title: Boundary Element Methods in Practice: Solving 3D Problems with NGSolve and NGBem
Lecturer: Lucy Weggler
In this talk, we will introduce the fundamentals of Boundary Element Methods (BEM) and provide a practical demonstration of the NGSolve add-on, NGBem, which facilitates the solution of several classical problems using BEM, including:
- The Laplace Equation in 3D (Electrostatics)
- The Helmholtz Equation in 3D (Acoustics)
- Harmonic Maxwell Equations in 3D (Electrodynamics)
Although BEM is less widely known compared to the Finite Element Method (FEM), it offers unique advantages, particularly for problems involving infinite domains. The relative obscurity of BEM likely stems from the increased complexity of both its mathematical foundation and implementation.
Unlike FEM, BEM does not rely on a variational formulation of the differential equation itself, but instead on a variational formulation based on an ansatz for the solution of the boundary value problem. Given the central importance of this solution ansatz in BEM, the first part of the talk will focus on its development. Starting from a simple 1D example, we will derive a representation formula involving a single-layer potential, a double-layer potential, and a Newton potential. With this foundational understanding, we will then address a more complex 3D problem. From the representation formula, we will derive the corresponding variational formulation and discuss the numerical discretization in detail.
In the second part of the talk, we will turn to the practical application of BEM using NGBem. This tool integrates four decades of academic research, offering high accuracy through the use of high-order boundary elements and efficient performance via advanced matrix assembly techniques. NGBem is an add-on to NGSolve, a robust software suite for solving similar problems using FEM. The combination of FEM and BEM allows engineers to leverage the strengths of both methods, providing flexibility and improved results in various engineering applications. We will conclude with an overview of the current capabilities of the software and demonstrate additional example problems.
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Title: Mixed-dimensional coupled finite elements in FEniCS
Lecturer: Cécile Daversin-Catty
Mixed-dimensional partial differential equations (PDEs) are equations coupling fields defined
over distinct domains, which may differ in topological dimension. These PDEs naturally arise in
a wide range of fields (e.g. geology or bio-medicine), and are also employed to impose
non-standard conditions through Lagrange multipliers. Only a few open-source software
packages include the abstraction and features necessary to solve such problems, involving
nested meshes of heterogeneous topological dimension, and a non-standard assembly process.
The FEniCS project aims at automating the numerical solution of PDE-based models using finite
element methods. A core feature is a high-level domain-specific language for finite element
spaces and variational forms, close to mathematical syntax. Lately, FEniCS gave way to its
successor FEniCSx, including major improvements over the legacy library. A dedicated
mixed-dimensional framework was developed in core FEniCS legacy libraries. The specific
software abstractions and algorithms involved were recently ported to FEniCSx, taking
advantage of the underlying upgrades in the library features and design.
This talk gives an overview of the abstractions and algorithms involved, and their
implementation in the FEniCS project core libraries. The introduced features are illustrated by
concrete applications in biomedicine.
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Title: Solving PDEs with Firedrake (Short Course)
Lecturer: Patrick Farrell
The lectures cover a range of topics, including
– solving Poisson with optimal complexity p-multigrid at high polynomial degree;
– constructive solid geometry for building complicated domains;
– adaptivity on the L-shaped domain;
– time discretisation with implicit Runge-Kutta schemes;
– and if time permits adaptive discretisation of the Poisson eigenproblem.
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ACM Seminar
Title: Designing conservative and accurately dissipative numerical integrators in time
Lecturer: Patrick Farrell
Numerical methods for the simulation of transient systems with structure-preserving properties are known to exhibit greater accuracy and physical reliability, in particular over long durations. These schemes are often built on powerful geometric ideas for broad classes of problems, such as Hamiltonian or reversible systems. However, there remain difficulties in devising higher-order-in-time structure-preserving discretizations for nonlinear problems, and in conserving non-polynomial invariants.
In this work we propose a new, general framework for the construction of structure-preserving timesteppers via finite elements in time and the systematic introduction of auxiliary variables. The framework reduces to Gauss methods where those are structure-preserving, but extends to generate arbitrary-order structure-preserving schemes for nonlinear problems, and allows for the construction of schemes that conserve multiple higher-order invariants. We demonstrate the ideas by devising novel schemes that exactly conserve all known invariants of the Kepler and Kovalevskaya problems, arbitrary-order schemes for the compressible Navier-Stokes equations that conserve mass, momentum, and energy, and provably dissipate entropy, and energy-conservative schemes for the Benjamin-Bona-Mahony equation.