The neural particle method – An updated Lagrangian physics informed neural network for computational fluid dynamics

• Published in: Computer methods in applied mechanics and engineering Vol. 368; p. 113127
• Main Authors: Wessels, Henning, Weißenfels, Christian, Wriggers, Peter
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.08.2020; Elsevier BV
• Subjects: Boundary conditions; Computational fluid dynamics; Computer simulation; Constraint problem; Continuous improvement; Design engineering; Euler-Lagrange equation; Fluid flow; Free surfaces; Implicit Runge–Kutta; Incompressibility; Incompressible flow; Machine learning; Mathematical analysis; Meshless methods; Neural networks; Numerical analysis; Numerical methods; Partial differential equations; Physics; Physics-informed neural network; Runge-Kutta method; Simulation; Solid mechanics; Time dependence; Constraint problem; Physics-informed neural network; Computational fluid dynamics; Incompressibility; Implicit Runge–Kutta; Machine learning
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2020.113127

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. •A feed-forward neural network is used to construct a global geometric ansatz function.•No special treatment of the incompressibility constraint is necessary.•High order implicit Runge–Kutta time integration is employed.•Excellent conservation properties are demonstrated in numerical examples.•The computations remain stable even for irregularly distributed discretization points.