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Particle Image Velocimetry Investigation of Hemodynamics via Aortic Phantom

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Physics-driven proper orthogonal decomposition: A simulation methodology for partial differential equations.

Alessandro Pulimeno^1,2, Graham Coates-Farley³, Martin Veresko³

¹Department of Mechanical and Aerospace Engineering, Clarkson University, Potsdam, NY 13699, USA.

|July 10, 2023

Summary

A new physics-driven Proper Orthogonal Decomposition (POD) simulation method uses learning algorithms to solve partial differential equations (PDEs). This approach significantly reduces computational effort for complex physical problems while maintaining high accuracy.

Keywords:

Galerkin projection Heat transfer Machine learning Partial differential equations Physics-driven simulation methodology based on Proper Orthogonal Decomposition enabled by Galerkin projection (POD-GP)Quantum nanostructures

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Area of Science:

Computational physics
Scientific computing
Numerical analysis

Background:

Solving partial differential equations (PDEs) often requires significant computational resources.
Direct Numerical Simulations (DNS) provide accurate solutions but are computationally expensive.
Model reduction techniques are sought to accelerate simulations.

Purpose of the Study:

To present a novel simulation methodology based on Proper Orthogonal Decomposition (POD) and Galerkin projection.
To develop a physics-driven approach for solving PDEs efficiently.
To demonstrate the methodology's effectiveness on complex physical problems.

Main Methods:

A learning algorithm based on Proper Orthogonal Decomposition (POD) is employed.
Solution data from DNSs are used to train POD modes.
Galerkin projection is applied to the PDE within the POD space.
The methodology involves data collection, POD mode calculation, and model derivation.

Main Results:

The POD-Galerkin methodology enables a significant reduction in degrees of freedom (DoF).
High accuracy is maintained despite the reduction in DoF.
Computational effort is drastically decreased compared to DNS.
Successful simulations were performed for dynamic thermal analysis of microprocessors and the Schrödinger equation.

Conclusions:

The physics-driven POD-Galerkin method offers a computationally efficient alternative to DNS.
This approach effectively solves complex PDEs by reducing model complexity.
The methodology shows promise for accelerating simulations in various physical domains.