# Analysis of heterogeneous computing approaches to simulating heat transfer in heterogeneous material

## TL;DR

This paper presents a parallelizable, assembly-free finite element method for simulating heat transfer in heterogeneous materials, optimized for GPU computing, and demonstrates its efficiency on large-scale, real-world problems.

## Contribution

The paper introduces a novel assembly-free FEM approach adapted for GPU architectures, enabling efficient simulation of heat transfer in complex heterogeneous materials.

## Key findings

- GPU implementation outperforms CPU sparse matrix methods
- Dual GPU setup accelerates simulations significantly
- Method effectively handles large-scale, real-world inverse problems

## Abstract

The simulation of heat flow through heterogeneous material is important for the design of structural and electronic components. Classical analytical solutions to the heat equation PDE are not known for many such domains, even those having simple geometries. The finite element method can provide approximations to a weak form continuum solution, with increasing accuracy as the number of degrees of freedom in the model increases. This comes at a cost of increased memory usage and computation time; even when taking advantage of sparse matrix techniques for the finite element system matrix. We summarize recent approaches in solving problems in structural mechanics and steady state heat conduction which do not require the explicit assembly of any system matrices, and adapt them to a method for solving the time-depended flow of heat. These approaches are highly parallelizable, and can be performed on graphical processing units (GPUs). Furthermore, they lend themselves to the simulation of heterogeneous material, with a minimum of added complexity. We present the mathematical framework of assembly-free FEM approaches, through which we summarize the benefits of GPU computation. We discuss our implementation using the OpenCL computing framework, and show how it is further adapted for use on multiple GPUs. We compare the performance of single and dual GPUs implementations of our method with previous GPU computing strategies from the literature and a CPU sparse matrix approach. The utility of the novel method is demonstrated through the solution of a real-world coefficient inverse problem that requires thousands of transient heat flow simulations, each of which involves solving a 1 million degree of freedom linear system over hundreds of time steps.