How Accurate are Reduced Models?
Hello. In our last article (Speed-up your simulation using reduced-order modeling) we introduced you to the concept of reduced order models. This is a hot topic in computational engineering. There is a strong reason for this. Reduced-order models, shortly ROMs, lie on the cross-section of machine learning, data-driven engineering and classical scientific computing. They use the most important concepts from those fields, and leverage them to speed up the design process, prototyping and evaluation of your products. Today, I want to uncover more details on the accuracy of this methods.
I structured the article in three sections. In the first one, I discuss the idea of solution space and its relevant subspace that we are interested in. Next, we go on to do some numerical experiment and compare the reduced results with full finite element solution. In the last section, I discuss various options how this process can be extended, and different types of modes that we can use. I hope you will have a chance to learn something new, and get more interested in this broad and exciting topic!
Currently, finite element method is the workhorse of thermal analysis. In the classic FEA we approximate the solution with piece-wise polynomial functions. Those functions are defined by the mesh. It describes a space of solutions. However, for the majority of problems, the resulting solution space is too big. A lot of solutions possible to represent are not physical and out of our interest. To overcome this problem we look at different methods to represent the functions. The classic methods to solve an unsteady partial differential equations is through variable splitting. Assume that our solution is a product of space and time solution. In the end, we obtain a Fourier series. This is the foundation for reduced order model. Mathematically, we can write it using a simple matrix formula:
Columns of L matrix encode our new, reduced basis. As we will see below, size of this basis is closely related to errors.
In the simple example we look at a metallic rod, that is heated from one side, and cooled from the other. This is naturally a one dimensional example, but provides a lot of intuition on the behavior of ROMs. I tested the model with different number of modes, and compared with full solution.
The average error over time falls down as we increase the number of modes. Only 30 modes allows to drop the error below 0,1%. At the same time, size of our numerical model is orders of magnitude smaller than the original model. Note, that staircase effect as well. It is caused by symmetric and asymmetric trigonometric modes.
The reduction comes with an additional benefit. I managed to reduce the simulation time by an order of magnitude. On this small example, the benefit is not critical. However, this speed up does not scale linearly. Let’s see how it performs on larger example.
In the second example I run the simulation of multi-component battery system with electric heating. I tested it for different number of modes in cell models.
The above plot shows a zoomed in mean temperature history. As we can see, the differences are below 0,1 Celsius degrees. The simulation took 60 seconds to run 60 time steps on my laptop. The full model consists of over 600 thousands of nodes, and could take hours to run and debug. Changing number of modes in cells leads to indistinguishable differences. Such level of accuracy and speed of execution of full wold not be possible on a regular computer.
My goal in this short analysis was to show you that errors involved in model reduction can be controlled, and kept very small. This is the key requirement for successful adoption of this technique. In the small example I showed that we can obtain 99% accuracy with the simplest reduction possible.
This was only achieved with Fourier expansion. Broad range of other functions are available. New possibilities open if we pick special functions to augment effects of boundary conditions. Another challenge is the treatment of nonlinear terms. If you got interested in this topic, I highly recommend you to check out the following sources:
- Data Driven Science and Engineering, 2019, Brunton and Kutz – Excellent book on reduced models, data science methods, and different ways to apply them to engineering
- Certified Reduced Basis Methods for Parametrized Partial Differential Equations, 2017, Hesthaven, Rozza and Stamm – Another great book on the topic. Includes a lot of mathematical insight.
- Quickersim Q-Bat Theory Guide – Theory guide and training material for our reduced modelling software. See the link in the first comment.
I hope that you found this post interesting. Do you use reduced models in your research?