ABSTRACT. Maxwell’s equations describe the movement of waves through space and are well-known throughout the scientific world. We seek to solve the exterior scattering problem in 3 dimensions by truncating the domain and numerically solving over a finite domain. Usually, obtaining a numerical solution is achieved using transmission boundary conditions, PML, or the DtN map, but we instead apply an exact but nonlocal boundary condition that has been proven to work for the Helmholtz equation. Since this involves using layer potentials and the dyadic Green’s function, we computationally utilize both Pytential and Firedrake to obtain accurate numerical results, converging at the expected rate as our mesh is refined. Proving the quasi-optimal convergence of our solution requires using the theory of collective compactness after converting our variational formulation to a second-kind integral equation.

ABSTRACT. The modeling of many processes, such as chemical reactions and vegetation dynamics, involve systems with complex long-range interactions or fast-diffusion processes that cannot always be describe accurately with local diffusion operator. This leads us to introduce and analyze a nonlocal version of the diffusion-reaction Gray-Scott model for chemical reaction where the diffusion process is represented by an integral operator (i.e. convolution or fractional Laplacian operator) that is better suited to represent the above phenomena.

For this talk, we will first introduce a nonlocal Gray-Scott model for one dimensional domain. We will summarize briefly our analysis the well-posedness of the resulting model that concludes on the existence of small-time weak solution for problems with either nonlocal Dirichlet or Neumann boundary constraints. Then, we will present a finite element discretization of our nonlocal Gray-Scott model. The resulting algorithm is validated using manufactured solutions before we study the generation of pulse solution. Numerical illustrations will be provided to show the impact of the type of diffusion (local vs nonlocal) and also on the impact of nonlocal boundary constraints (Dirichlet vs Neumann) on the generated pulse solutions.

ABSTRACT. In this talk, we outline a structure preserving discretization for the resistive magnetohydrodynamic equations in a tokamak domain, as commonly considered for magnetic confinement fusion simulations. The discretization is based on compatible finite elements, and ensures a discrete form of the magnetic field's zero divergence condition. In the course of the presentation, we highlight various aspects of the Firedrake-based code implementation, including a structure preserving interior penalty-based transport stabilization method, a way of establishing subdomains to differentiate between plasma and wall regions, and initialization/postprocessing considerations.

ABSTRACT. Our work focuses on the development of high-resolution Digital Light Processing Stereolithographic (DLP-SLA) 3D printing techniques for fabricating microfluidic lab-on-a-chip devices. These devices prioritize the creation of negative features, which serve as the channels for fluid processing and manipulation. To meet our stringent resolution demands, we have engineered custom-built 3D printers and created associated resin formulations, as well as specialized control software to optimize the fabrication process. Our capabilities enable the production of microfluidic structures with unprecedented precision, including channel cross-sections as small as 1.9 microns by 2.0 microns, incorporating both passive and active elements such as valves, pumps, and mixers.

As we push the boundaries of resolution, modeling the photopolymerization process has become important for understanding fundamental limitations and enhancing performance. To address this, we employ a finite element method (FEM) framework using Firedrake to solve a system of five coupled nonlinear partial differential equations (PDEs). These equations incorporate nonlinearities in species concentration, reaction kinetics, and diffusivity terms.

In this presentation, I will outline our numerical approach, present key findings, and introduce a layer-by-layer simulation strategy that iteratively models the photopolymerization of successive resin layers. This methodology allows for seamless integration of previously cured layers onto a newly initialized mesh, facilitating the simulation of multi-layered structures typical of our 3D printing approach. We are currently looking at open-sourcing our code and have therefore created a number of ancillary software affordances to make its use more convenient.

ABSTRACT. In this presentation we will take a detailed look into the recent developments of Automatic Differentiation in MFEM (https://mfem.org) which leverages modern compiler framework techniques of the LLVM project. We're going to establish a connection between the recently added firedrake ExternalOperator and the related functionality used in MFEM by introducing the Finite Element Operator Decomposition, one of the core concepts of MFEM. The presentation is concluded by a live demonstration of the framework.

ABSTRACT. pyop3 is a new domain-specific language (DSL) for expressing the application of computational kernels over an unstructured mesh. It was created to address shortcomings with PyOP2, a core component of the Firedrake framework, and is slated to replace it sometime within the next 3 months.

Compared with PyOP2, the pyop3 DSL is much more composable; enabling users to express – and hence generate code for - a wider selection of problems that aren’t necessarily FEM specific. A much wider range of possible data layouts are also supported, enabling users to experiment to achieve the best performance.

In this talk we will introduce some of the foundational concepts behind pyop3 before giving a few examples. Emphasis will be placed on how the changes to pyop3 will appear to a typical Firedrake user, and the sorts of novel features that its integration should make possible.

ABSTRACT. We discuss bringing Python natively to the CUDA ecosystem. From low level bindings to domain specific applications, CUDA is supporting Python standards and ecosystem. New libraries include nvmath-python for managing optimized mathematics libraries, cccl-python for cooperative threading and device parallelism, cuda-core for managing the complete CUDA toolstack from Python with no need for C++, and finally numba-cuda for generating device side kernels with integration of C++ device libraries and LTO IR.