Speed-up your simulation using reduced-order modeling
In our last article (How battery thermal design can be accelerated) we have mentioned reduced-order models as a way to speed up simulations. In this article, we will describe the model reduction method which is used in Q-Bat simulation software. We will also present a simple example that will help to understand how this method works. Finally, we will present our model reduction method in action on the battery pack consisting of multiple cells and cooling plates.
Reduced-order models (ROMs) are computationally inexpensive mathematical representations, which offer the potential for near real-time analysis. Thanks to that, ROMs have a lot of applications such as embedded controllers, system-level simulations, and even real-time simulators.
One can distinguish two stages in the model reduction:
- Offline stage, during which a reduced model is created. This stage is carried out only once. It should contain all computationally intensive operations.
- Online stage, which should be as efficient as possible, because it will be executed many times, for multiple operating points.
There are two main types of the reduction model methods:
- Data-driven This method requires a large quantity of previously measured samples to abstract a simulation model. This approach could incorporate response surfaces, neural networks and other models suited for regression tasks. All the models created with this approach are able only to interpolate the underlying function. If a new sample will be located outside of the learned distribution, the model will not be able to predict proper response.
- Physics-based This method is built on the high-fidelity models based on the finite element and finite volume discretization. In this article, we will focus on the reduced basis method, which mathematically transforms the model into a new vector space. Physics-based models do not require any training data and are able to work properly for all ranges of the input parameters.
Reduced Basis method
The main idea of the Reduced Basis method (RB) is to reduce the complexity of the system without any significant loss of information or accuracy of the solution. It is possible because we know that the solution field will be smooth, so we can accurately approximate it with a set of properly constructed functions. To do so, we start by assembling matrices describing our governing equation. Below we present governing equation of the unsteady heat transfer problem.
Where M is the mass matrix, K is diffusion matrix and F is the source term and T stands for temperature field.
Next, we can create the reduced basis L, in which we will approximate the solution. In standard CFD or FEM codes, the solution representation basis consists of solution values at all nodes (usually a couple of million nodes). For reduced models, L can consist of a few hundred vectors that represent smooth, physically feasible solution shapes. The procedure of reduced basis generation is a key component of the whole method because it affects the solution accuracy. In Q-Bat we generate the reduced basis L including eigenvectors, trigonometric functions, and polynomial functions. Below we present the formula for the approximated solution, in this notation, “r” in the lower index stands for reduced representation.
After substitution of the approximated solution to the governing equations and left multiplication by the transpose of L, we obtain the following reduced system of equations.
Where the reduced matrices and source term with “r” index are defined by:
Let’s consider the steady-state heat transfer problem in a rod with assigned values of temperature at both ends. Assuming that the values at both ends are different, the expected temperature distribution in the rod is linear, as presented in the picture below.
In this case, we imposed Dirichlet boundary conditions at both ends of the rod with values 2.5 and 6 deg C. In this case, the reduced basis generated by Q-Bat consists of two vectors, which represent constant and linear temperature distribution. The distribution fields represented by these vectors are presented below.
The solution of this reduced system of equations yields a reduced temperature vector consisting of two values: 2.5 for constant distribution and 3.5 for linear distribution. We can reconstruct the full solution by multiplying each value of the reduced solution with a corresponding vector from the reduced basis and adding them together. The obtained solution is the same as the expected one.
In this example, we will perform model reduction on a complicated battery pack with 4 battery modules and 4 cooling plates connected hydraulically in series. Each battery module represents 20 prismatic cells in 1P20S configuration (20 cells connected in series). We will simulate ten minutes of battery usage with the current load varying in range from 0C up to 5C. The geometry of the battery pack is shown in the picture below.
During the reduction process, a new reduced basis is generated for each part of the battery assembly. During the generation of each basis Q-Bat takes into account boundary conditions, thermal contact between bodies, and in the case of the cooling plates, heat transfer between the pipe and the coolant. Some of the generated reduced basis vectors are presented in the picture below.
Thanks to the model reduction we have managed to reduce the number of unknowns in our model from 650990 to only 704. By doing so, we were able to reduce the computational time from several hours to only a few seconds. Despite the significant speed-up of calculations, we maintain maximum relative error between full and reduced models below 1%.
In the next article, we will show how the number of vectors, used to generate the reduced basis, affects the accuracy of the solution.
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