TY - JOUR

T1 - The neural particle method

T2 - An updated Lagrangian physics informed neural network for computational fluid dynamics

AU - Wessels, Henning

AU - Weißenfels, Christian

AU - Wriggers, Peter

PY - 2020/8/15

Y1 - 2020/8/15

N2 - Today numerical simulation is indispensable in industrial design processes. It can replace cost and time intensive experiments and even reduce the need for prototypes. While products designed with the aid of numerical simulation undergo continuous improvement, this must also be true for numerical simulation techniques themselves. Up to date, no general purpose numerical method is available which can accurately resolve a variety of physics ranging from fluid to solid mechanics including large deformations and free surface flow phenomena. These complex multi-physics problems occur for example in Additive Manufacturing processes. In this sense, the recent developments in Machine Learning display promise for numerical simulation. It has recently been shown that instead of solving a system of equations as in standard numerical methods, a neural network can be trained solely based on initial and boundary conditions. Neural networks are smooth, differentiable functions that can be used as a global ansatz for Partial Differential Equations (PDEs). While this idea dates back to more than 20 years (Lagaris et al., 1998), it is only recently that an approach for the solution of time dependent problems has been developed (Raissi et al., 2019). With the latter, implicit Runge–Kutta schemes with unprecedented high order have been constructed to solve scalar-valued PDEs. We build on the aforementioned work in order to develop an Updated Lagrangian method for the solution of incompressible free surface flow subject to the inviscid Euler equations. The method is straightforward to implement and does not require any specific algorithmic treatment which is usually necessary to accurately resolve the incompressibility constraint. Due to its meshfree character, we will name it the Neural Particle Method (NPM). It will be demonstrated that the NPM remains stable and accurate even if the location of discretization points is highly irregular.

AB - Today numerical simulation is indispensable in industrial design processes. It can replace cost and time intensive experiments and even reduce the need for prototypes. While products designed with the aid of numerical simulation undergo continuous improvement, this must also be true for numerical simulation techniques themselves. Up to date, no general purpose numerical method is available which can accurately resolve a variety of physics ranging from fluid to solid mechanics including large deformations and free surface flow phenomena. These complex multi-physics problems occur for example in Additive Manufacturing processes. In this sense, the recent developments in Machine Learning display promise for numerical simulation. It has recently been shown that instead of solving a system of equations as in standard numerical methods, a neural network can be trained solely based on initial and boundary conditions. Neural networks are smooth, differentiable functions that can be used as a global ansatz for Partial Differential Equations (PDEs). While this idea dates back to more than 20 years (Lagaris et al., 1998), it is only recently that an approach for the solution of time dependent problems has been developed (Raissi et al., 2019). With the latter, implicit Runge–Kutta schemes with unprecedented high order have been constructed to solve scalar-valued PDEs. We build on the aforementioned work in order to develop an Updated Lagrangian method for the solution of incompressible free surface flow subject to the inviscid Euler equations. The method is straightforward to implement and does not require any specific algorithmic treatment which is usually necessary to accurately resolve the incompressibility constraint. Due to its meshfree character, we will name it the Neural Particle Method (NPM). It will be demonstrated that the NPM remains stable and accurate even if the location of discretization points is highly irregular.

KW - Computational fluid dynamics

KW - Constraint problem

KW - Implicit Runge–Kutta

KW - Incompressibility

KW - Machine learning

KW - Physics-informed neural network

UR - http://www.scopus.com/inward/record.url?scp=85085728658&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2003.10208

DO - 10.48550/arXiv.2003.10208

M3 - Article

AN - SCOPUS:85085728658

VL - 368

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

M1 - 113127

ER -