Introduction

The cohort participated in our 2024 CyberTraining workshop will be divided into 2 or 3 teams based on the mutual interests and complementary skills that each team possesses. Each team will be led by a PI and assisted by the TA to implement a POD project of their choice in C++. It is expected that each team is able to successfully complete the POD coding and perform the POD simulation of their selected project.

Learning Outcomes

Trainees are expected to demonstrate mastery of the POD method, ability to collect data using FEniCS, and implement POD using algorithms from PETSc and SLEPc in C++. More specifically, the projects can be divided into four milestones as follows:

1. Collect data by modifying the provided sample FEniCS code to fully describe the selected problem in the simulation domain with appropriate BCs and excitations

2 Prepare a code in C++ and use PETSc and SLEPc to perform the method of snapshots and solve a discrete eigenvalue problem to generate POD modes

3 Perform integrations of the POD modes and their gradients in C++ using tools PETSc and SLEPc

4 Solve ODEs to perform POD simulation in C++ using solvers in PETSc.

Project 1: 3D thermal simulation of a CPU

The desire for accurate and efficient spatiotemporal thermal predictions in microprocessors has been growing rapidly in recent years for thermal and power management due to high thermal gradients and serious hot spot formation in modern microprocessors. This project will implement the heat transfer equation in a multi-core CPU to study the dynamic thermal profile over the entire CPU heated by different dynamic power maps. The workshop participants will perform FEniCS simulation of a multi- core CPU to collect thermal data that will then be used to train the POD modes. The POD model constructed using the trained POD modes will be validated compared to FEniCS simulations in terms of accuracy and efficiency. The hot-spot distribution in the chip will be analyzed, compared to FEniCS simulation.

Project 2: Quantum eigenvalue problem of a 2D nanostructure

Quantum eigenvalue solution of the Schrӧdinger equation is needed for design and analysis of any nano- or atomic-scale electronic and photonic structures. A POD model for solving electron wave functions in a 2D quantum-dot structure influenced by potential variation will be constructed in the workshop. The participants will generate the training data of electron wave functions from a finite-difference Schrӧdinger solver. The method of snapshots is then applied to calculate the POD modes for the nanostructure to construct the physics-based quantum POD learning model. Wave functions determined by the POD simulation will be demonstrated against the numerical solution. The learning ability of the quantum learning model will be examined in cases of extrapolation where quantum POD simulations are performed beyond the training settings.

Project 3: Simulation of a 2D electromagnetic band gap structure

Electromagnetic band gap (EMBG) structures have numerous applications for designs of antennas, sensors, filters etc. for frequency ranging from microwave to optical spectrums. This project will develop a physics-based POD learning model for simulation of a 2D periodic electromagnetic structure to investigate electromagnetic fields in the band-gap structure. The training data of field solution will be collected from a finite difference electromagnetic eigenvalue solver, and the POD modes are generated from the method of snapshots. Comparison of the electric/magnetic fields and the dispersion relations between the POD and finite-different simulations will be performed. Variations of stop bands will be investigated in response to the permittivity variation in the structure.