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Finch, a domain specific language and code generation framework for partial differential equations (PDEs), is demonstrated here to solve two classical problems: steady-state advection diffusion equation (single PDE) and the phonon Boltzmann transport equation (coupled PDEs). Both finite volume and finite element methods are explored. In addition to work presented at the 2022 International Conference on Computational Science (Heisler et al., 2022), we include recent developments for solving nonlinear equations using both automatic and symbolic differentiation, and demonstrate the capability for the Bratu (nonlinear Poisson) equation. 

The phonon Boltzmann transport equation is a good model for heat transfer in nanometer scale structures such as semiconductor devices. Computational complexity is one of the main challenges in numerically solving this set of potentially thousands of nonlinearly coupled equations. Writing efficient code will involve careful optimization and choosing an effective parallelization strategy, requiring expertise in high performance computing, mathematical methods, and thermal physics. To address this challenge, we present the domain specific language and code generation software Finch. This language allows a domain scientist to enter the equations in a simple format, provide only basic mathematical functions used in the model, and generate efficient parallel code. Even very complex systems of equations such as phonon Boltzmann transport can be entered in a very simple, intuitive way. A feature of the framework is flexibility in numerical methods, computing environments, parallel strategies, and other aspects of the generated code. We demonstrate Finch on this problem using a variety of parallel strategies and model configurations to demonstrate the flexibility and ease of use. 

We introduce FINCH, a Julia-based domain specific language (DSL) for solving partial differential equations in a discretization agnostic way, currently including finite element and finite volume methods. A key focus is code generation for various internal or external software targets. Internal targets use a modular set of tools in Julia providing a direct solution within the framework. In contrast, external code generation produces a set of code files to be compiled and run with external libraries or frameworks. Examples include a matlab target, for smaller problems or prototyping, or C++/MPI based targets for larger problems needing scalability. This allows us to take advantage of their capabilities without needlessly duplicating them, and provides options tailored to the needs of the domain scientist. The modular design of FINCH allows ongoing development of these target modules resulting in a more extensible framework and a broader set of applications. The support for multiple discretizations, including finite element and finite volume methods, also contributes to this goal. Another focus of this project is complex systems containing a large set of coupled PDEs that could be challenging to efficiently code and optimize by hand, but that are relatively simple to specify using the DSL. In this paper we present the key features of FINCH that set it apart from many other DSL options, and demonstrate the basic usage and current capabilities through examples. 

Efficiently and accurately simulating partial differential equations (PDEs) in and around arbitrarily defined geometries, especially with high levels of adaptivity, has significant implications for different application domains. A key bottleneck in the above process is the fast construction of a ‘good’ adaptively-refined mesh. In this work, we present an efficient novel octree-based adaptive discretization approach capable of carving out arbitrarily shaped void regions from the parent domain: an essential requirement for fluid simulations around complex objects. Carving out objects produces an incomplete octree. We develop efficient top-down and bottom-up traversal methods to perform finite element computations on incomplete octrees. We validate the framework by (a) showing appropriate convergence analysis and (b) computing the drag coefficient for flow past a sphere for a wide range of Reynolds numbers (0(1-10 6 )) encompassing the drag crisis regime. Finally, we deploy the framework on a realistic geometry on a current project to evaluate COVID-19 transmission risk in classrooms.