Solving Non-linearity in Finite Element Analysis (FEA)

When you start out with a linear model to solve for a set of defined physics, solving one Newton-Raphson iteration is sufficient to compute the behavior of the system. But, in the world around us, there is a real dearth of linear systems. Non-linearity is in abundance and poses up challenges in Finite element analysis of systems with accuracy. In technical terms, these challenges arise primarily because the solver unit is set into repeated iterations and is unable to reach a conclusion, termed as failure in reaching convergence.

In one of the experiments, while trying to model the system (COMSOL Multiphysics 5.2a) with a high amount of surface charge and significantly high voltage input, i encountered the issue depicted on the left several times. Well, it can be explained since, the system became highly non-linear with those boundary conditions and took forever to compute thousands of nodes and respective iterations. In certain cases, i had to let the computation run for several hours only to encounter that the solver has failed after having computed for so long.

To solve such a highly complex system, one approach is to increase the error tolerance of the computing software. However, that should be done only if qualitative analysis is of interest. Most times, and as a researcher, studying the system to its closest possible ecosystem in real world is of interest to understand the underlying physics. In other words, i could not reduce my applied bias nor the surface charges but i needed to analyse the behavior. What do we do? Load ramping, comes to the rescue.

This concept breaks down the system into a set of several small Newton-raphson solutions where in, the solution of one set of equations under given constraints serves as an input for the following set. The approach is incremental and does not hit the system with large non-linearity at once. The following figure depicts a problem broken down to 2 simpler problems wherein the

To solve my system, i had to configure the surface charge values in increment of a fixed step sizes breaking the problem down to something simpler. The computation time reduced drastically to about 45 minutes for my problem and voila, i was analyzing the default plots generated pertaining to the configured physics.