Ferrite.jl - Finite Element Toolbox - Summer of Code

Ferrite.jl is a Julia package providing the basic building blocks to develop finite element simulations of partial differential equations. The package provides extensive examples to start from and is designed as a compromise between simplicity and generality, trying to map finite element concepts 1:1 with the code in a low-level . Ferrite is actively used in teaching finite element to students at several universities across different countries (e.g. Ruhr-University Bochum and Chalmers University of Technology). Further infrastructure is provided in the form of different mesh parsers and a Julia based visualizer called FerriteViz.jl.

Below we provide a four of potential project ideas in Ferrite.jl. However, interested students should feel free to explore ideas they are interested in. Please contact any of the mentors listed below, or join the `#ferrite-fem` channel on the Julia slack to discuss. Projects in finite element visualization are also possible with FerriteViz.jl.

Difficulty: Medium-Hard

Project size: 300-350 hours

Problem: While Ferrite.jl provides infrastructure for the most common tasks in finite element assembly, it lacks an adaptive mesh infrastructure for adaptive finite element technology. A preliminary implementation of a p4est type adaptive mesh refinement has been started, but not finalized yet. Information about the existing implementation is summarized here.

Minimum goal: Finalize the basic p4est implementation as described in the original paper [1], either starting from the existing branch (recommended) or from scratch together with a set of tests.

Extended goal: Interesting extensions might be to implement the optimizations proposed by Tobin Isaac [2], anisotropic refinement as in p6est [3] or generalizations to other geometries as in t8code [4].

Recommended skills:

• Basic knowledge the finite element method

• Basic knowledge about mesh geometries

Mentors: Maximilian Köhler and Dennis Ogiermann

References

1. Burstedde, C., Wilcox, L. C., & Ghattas, O. (2011). p4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM Journal on Scientific Computing, 33(3), 1103-1133.

2. Isaac, T., Burstedde, C., Wilcox, L. C., & Ghattas, O. (2015). Recursive algorithms for distributed forests of octrees. SIAM Journal on Scientific Computing, 37(5), C497-C531.

3. Isaac, T., Stadler, G., & Ghattas, O. (2015). Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics. SIAM Journal on Scientific Computing, 37(6), B804-B833.

4. Holke, J. (2019). t8code-Extreme Scale Adaptive Mesh Refinement with Arbitrary Elements.

Discontinuous Galerkin Infrastructure

Difficulty: Medium-Hard

Project size: 300-350 hours

Problem: Discontinuous Galerkin methods combine different advantages of finite element and finite volume schemes at the cost of having more degrees of freedom than classical continuous Galerkin methods. Currently Ferrite.jl can not handle such problems, because there is no H(div) conforming (i.e. Raviart-Thomas) element implemented.

Minimum goal: A minimum goal is to implement a H(div) conforming element together with a mixed Poisson example and introduce corresponding test coverage.

Extended goal: If the time allows then it might be interesting to move one step further and explore Runge-Kutta Discontinuous Galerkin (RKDG) schemes for a convection-dominated problem.

Recommended skills:

• Knowledge the finite element method

Mentors: Dennis Ogiermann and Fredrik Ekre

Fluid-Structure Interaction Example

Difficulty: Easy-Medium (depending on your specific background)

Project size: 150-300 hours

Problem: Ferrite.jl is designed with the possibility to define partial differential equations on subdomains. This makes it well-suited for interface-coupled multi-physics problems, as for example fluid-structure interaction problems. However, we currently do not have an example showing this capability in our documentation. We also do not provide all necessary utilities for interface-coupled problems.

Minimum goal: The minimal goal of this project is to create a functional and documented linear fluid-structure interaction example coupling linear elasticity with a stokes flow in a simple setup. The code should come with proper test coverage.

Extended goal: With this minimally functional example it is possible to extend the project into different directions, e.g. optimized solvers or nonlinear fluid-structure interaction.

Recommended skills:

• Basic knowledge the finite element method

• Basic knowledge about solids or fluids

• The ability (or eagerness to learn) to write fast code

Mentors: Dennis Ogiermann and Fredrik Ekre

Investigation of Performant Assembly Strategies

Difficulty: Medium

Project size: 250-350 hours

Problem: Ferrite.jl has an outstanding performance in single-threaded finite element simulations due to elaborate elimination of redundant workloads. However, we recently identified that the way the single-threaded assembly works makes parallel assembly memory bound, rendering the implementation for "cheap" assembly loops not scalable on a wide range of systems. This problem will also translate to high-order schemes, where the single-threaded strategy as is prevents certain common optimization strategies (e.g. sum factorization).

Minimum goal: As a first step towards better parallel assembly performance it is the investion of different assembly strategies. Local and global matrix-free schemes are a possibility to explore here. The code has to be properly benchmarked and tested to identify different performance problems.

Extended goal: With this minimally functional example it is possible to extend the project into different directions, e.g. optimized matrix-free solvers or GPU assembly.

Recommended skills:

• Basic knowledge the finite element method