Machine learning-based method for estimating mean pressure maps on the friction surface of brake pads during a braking event

Abstract

The present invention relates to a machine learning based method for estimating mean pressure maps on the friction surface of brake pads during a braking event, using as training data the results of numerical simulations based on the finite element method (FEM). Such a method allows accurately estimating the pressure distribution on pads with different sizes and shapes, which depend on the input parameters. To this end, different machine learning (ML) techniques are combined to create surrogate models of numerical simulations in which the domain depends on the input parameters and the discretization grid is not known a priori. Among the advantages of the invention are the highly reduced computation times as compared to those required for the numerical resolution of the equations of the model while maintaining a high accuracy of the estimate.

Description

[0001] Machine learning-based method for estimating mean pressure maps on the friction surface of brake pads during a braking event

[0002] The present invention relates to a machine learning- based method for estimating mean pressure maps on the friction surface of brake pads during a braking event.

[0003] Background art

[0004] WO2014170849A1 describes a method for estimating in real time the applied pressure and the noise in a braking element, in particular a brake pad, by processing the signal from a piezoceramic sensor placed between a block of friction material and a metal support element of the braking element, during a braking event. This is a temporal and not spatial distribution of the pressure on the points of the pad.

[0005] JP2004347095A describes a method for setting a specific spatial distribution of pressure on the pads.

[0006] The Inventors are not aware of methods for specifically estimating, when designing the brake pads, a spatial distribution of pressure on the brake pads.

[0007] However, finite element (FEM) simulations of mechanical components are known for the design thereof.

[0008] Applying such techniques to brake pads poses issues relating to complexity and slowness, which make them unusable in the specific application. "Surrogate-modeling", also referred to as "metamodeling" [1], which is an engineering method of using mathematical models to approximate a result of interest which cannot be measured or calculated easily, is also known.

[0009] Among the most important applications of surrogate models are optimization issues based on the outputs of numerical simulations [2]. In the field of engineering design, very often, in order to select the optimal geometry of a product, simulating the performance using mathematical models based on partial differential equations (PDE), which are solved numerically using the finite element method (FEM), is attempted. Typically, such simulations are computationally very intensive, therefore only a limited number of input configurations can be tested in a reasonable time. Surrogate models are used to address this issue, since they allow approximating the simulation output without solving the equations numerically.

[0010] Among the most used techniques for surrogate models are statistical interpolation methods [3] and, more recently, Machine Learning (ML) methods such as neural networks (NN) [4].

[0011] In the case of numerical simulations based on FEM methods, the domain of interest is discretized in a grid consisting of finite elements of encoded shape (typically, triangles in the case of 2D). The simulation result consists of a set of scalars or vectors which can be associated with each finite element. The size of the simulation output thus depends on the number of elements in the grid. In an optimization context, an objective function is typically defined, being the target to be maximized or minimized. Such an objective function returns a scalar value based on the simulation result. Many surrogate models simply estimate the value of the objective function based on the input parameters [4].

[0012] In the context of brake pad design, the objective function provides a reference scalar score which can be used in a "heuristic" manner to select the "best" pads; however, the final choice by designers can take into account other factors which are not easily enclosable into a single numerical value. For example, with the optimization process it is possible to identify some pad candidates based on the objective function; the final choice is then made with further analyses on the pressure distribution; therefore, hypothetically, it can be assumed that the most suitable geometry is not that with the most "optimal" value of the objective function. For this reason, it can be necessary to have an approximate value of the complete output, the size of which can however vary depending on the domain and the discrete grid.

[0013] This is particularly challenging for two reasons:

[0014] 1. Most machine learning models, including many neural network architectures, require a fixed number of inputs and a fixed number of outputs;

[0015] 2. When the size of the output greatly exceeds that of the inputs, issues such as underfitting can arise.

[0016] To overcome the first issue, some authors choose to develop specific surrogate models for a single domain with a fixed discretization. Others use approaches based on Graph Neural Networks (GNN) [5], or on "Point Cloud Neural Networks" [6] to cope with geometric variability. Both of these approaches require that the discrete grid be known a priori and that it is used as input to the model.

[0017] In the case of brake pad design, this is not possible precisely because there is no grid known a priori.

[0018] The need remains for a method of estimating the pressure distribution on brake pads in a braking event, in order to design brake pads.

[0019] Object and subject-matter of the invention

[0020] It is the object of present invention to provide a machine learning-based method for estimating mean pressure maps on the friction surface of brake pads during a braking event.

[0021] A method according to the appended claims is the subject-matter of the present invention.

[0022] Detailed description of embodiments of the invention

[0023] List of Figures

[0024] The invention will now be described by way of non- limiting example, with particular reference to the figures in the accompanying drawings, in which:

[0025] - Figure 1 shows a typical geometry of the pad and pistons, according to the prior art;

[0026] — Figure 2 shows an example of a pressure map on the surface of a pad, output of the Abaqus numerical simulation model, according to the prior art;

[0027] - Figure 3 shows a method used to standardize the pressure maps (min-max scaling followed by 2D interpolation on a uniform grid), according to an aspect of the invention;

[0028] - Figure 4 shows how, from a standardized map, the original pressure distribution can be traced, since the geometry is known, according to an aspect of the invention. Note that the points where the pressure is reported are not the same since the positions of the nodes of the original grid are not known;

[0029] - Figure 5 shows a diagram which describes two flows: the encoding flow allows obtaining, from a pressure map, a "code vector" representing it. The reverse (decoding) flow allows tracing, from a "code vector", pressure maps, according to an aspect of the invention; and

[0030] - Figure 6 shows the flow of the surrogate model for predicting pressure maps from the input parameters, according to an aspect of the invention.

[0031] It is here specified that elements of different embodiments can be combined to provide further embodiments, without restrictions, while respecting the technical concept of the invention, as those ordinarily skilled in the art will effortlessly understand from the description.

[0032] The present description also makes reference to the prior art for the implementation thereof in relation to the detail features not described, such as elements of minor importance usually used in the prior art in solutions of the same type, for example.

[0033] When an element is introduced, it is always understood that there can be "at least one" or "one or more".

[0034] When a list of elements or features is given in this description, it is understood that the finding according to the invention "comprises" or alternatively "consists of" such elements.

[0035] When listing features within the same sentence or bulleted list, one or more of the individual features can be included in the invention without connection with the other features in the list.

[0036] Two or more of the parts (elements, devices, systems) described above can be freely associated and considered as a part kit according to the invention.

[0037] Embodiments

[0038] It is the object of the invention to develop a methodology for estimating pressure maps on the surface of the brake pads simulated during a braking event.

[0039] The need for this solution arises when designing a brake caliper; during this step, the hydraulics of the caliper, the effective radius and the area of the pad are defined based on system calculations. The number of pistons, the diameters thereof and the position thereof at the braking strip must then be defined.

[0040] These geometric features of the caliper affect various functional aspects, including the ability of the caliper to wear the brake pad uniformly. This is desired in order to take advantage of the entire thickness of the friction material and reduce the residual torque of the brake caliper during the life of the pad.

[0041] In order to ensure a uniform wear, a finite element simulation is performed, aimed at evaluating the distribution of the contact pressure present between the pad and the disc. The more uniform this pressure is, the more the wear of the pad is expected to be uniform.

[0042] Therefore, when designing the brake caliper, simulation activities are performed to estimate the best configuration of pistons in terms of number, diameters and position. This optimization process is limited by the need to perform a certain number of finite element method simulations which require significant preparation and computation times.

[0043] The possibility of performing the optimization via a surrogate model, such as that suggested by the present invention, allows exploring the space of possible solutions more effectively, and significantly reducing analysis time (furthermore, it does not require the use of commercial software subjected to the purchase of licenses).

[0044] In a modeling example, the geometry of the brake pads is described by 4 parameters: internal brake rotor radius Rinternai, external brake rotor radius Rexternal, length Rpad, and thickness (see Fig. 1). A fixed number of pistons (two), the shape and position of which vary as shown in Figure 1, is considered.

[0045] The input parameters of the simulation model are listed in the following table.

[0046] The data forming the target to be approximated are the pressure values calculated via a numerical simulation model at different points of the pad surface. The number of grid points and the position thereof depend on the geometry of the surface taken into consideration and are not known a priori, since they are determined by internal logics of the simulation software used (in a specific case, Abaqus®). An example of output of a simulation according to the prior art is shown in the following table and can be seen in Figure 2.

[0047] The output of the simulation model consists of pressure values (in MPa) associated with N points on the surface. The number and position of the points depend on the input parameters.

[0048] At this point, in order to build a surrogate model using machine learning, as in the prior art, it is necessary to have a training data set consisting of a series of input values and the associated pressure maps. "Classic" machine learning methods can be used to approximate functions where the size of the inputs and outputs are fixed (h and k, respectively).

[0049] In the context of brake pad design, the aim is to choose the best geometry for the brake pads by analyzing the pressure pattern on the contact surface with the disc. The FEM simulation software simulates the braking events and produces pressure maps. Since different geometries are to be tested, the shape and size of the pads change, and therefore also the discretization grid generated by the software and the number of nodes forming it.

[0050] Since the outputs of the brake pad pressure map simulations do not have a fixed size, the first step of the methodology according to the invention consists in standardizing the pressure maps (Figure 3(a)) so that they are all characterized by a constant number of points, which can be associated with topologically equivalent regions. To do this, first a min-max scaling (Figure 3(b)), or more generally a feature scaling of the coordinate values of the points, is performed, so that the maps are included in the square [0,1]X [0,1]; a new uniform grid is then defined with a predetermined number of points (see Figure 3(b)).

[0051] The pressure maps must now be projected, according to the invention, onto the new grid (Figure 3(c)).

[0052] In order to project the pressure maps onto the new grid, a 2D interpolation was chosen using the technique known as Kriging. Note that, since the perimeter of the pad surface is known a priori, from a standardized (Figure 3(c)) pressure map, it is possible to trace an approximation of the original pressure map through a back-transformation (see Figure 4, in which (a) shows the result of the back-transformation and (b) shows the original pressure map). It is apparent that the approximation is good: with this procedure it is not possible to exactly reconstruct the original map, also because the information of the original grid is lost, however, there is an approximation which is very close, if it is not essential to maintain the same optimization grid, and the loss of information / the introduced error are limited.

[0053] The uniform grid on the square [0,1]X [0,1] can be more or less dense depending on the level of detail to be obtained. Assuming a grid of 40x40 points, each pressure map is identifiable with a vector of size 1600, resulting from the flattening of the pressure values on the grid points. Such a vector would be the output of the surrogate model to be build. In principle, it would be possible to use a neural network the final output of which is a vector of size 1600; however, when the size of the output is significantly larger than that of the input data, issues can arise, such as underfitting. Furthermore, training a network with an output of such a size would be computationally expensive. To address this issue, the invention uses an auto-encoding technique, for example the one known as Principal Component Analysis (PCA). PCA (or another equivalent technique) allows compressing the standardized pressure maps (in the example, having a size of 1600) into "code vectors" with very small size with a negligible loss of information .

[0054] In brief, each pressure map is standardized by sequentially applying min-max scaling of the coordinates and a 2D interpolation on a predefined grid of N points. A PCA or equivalent is then fitted on the standardized maps and is applied to each map to produce code-vectors of a sufficiently small size (M << N), without however causing too significant a loss of information. Finally, a machine learning model is used to approximate the function RMwhich associates the code-vector (of size M) associated with the corresponding output pressure map of the simulation with a set of input parameters (of size H).

[0055] In an embodiment, a Multi-Layer Perceptron (MLP), i.e., a "fully connected" neural network with 2 layers, was chosen, but other neural networks can also be used. Starting from a code-vector generated by the MLP it is possible to obtain a pressure map by sequentially applying reverse PCA transformation (or equivalent), and then back-transformation which, from a standardized map, produces a pressure distribution on the original domain. The encoding-decoding flow of the pressure maps is described in Figure 5, while the flow of the surrogate model is described in Figure 6 (the pre-processing of the inputs can be a Standard scaling (e.g., x' = (x - mean (x)) / std (x)).

[0056] Auto-encoding thus allows switching from a sample of regionalized values representative of a 2D scalar field on domains of variable geometry, to a vector of a sufficiently small size representing it. The exemplary choice of the encoding method was made considering several factors, including the amount of data needed, the training time and the simplicity of implementation. In the end, a three-step process was chosen: rescaling, interpolation and auto-encoding (PCA). This was possible since, in the case of brake pad pressure maps, the geometry of the domain can be obtained from the input parameters; therefore, it is possible to obtain a back- transformation .

[0057] Finally, it was decided to use a uniform grid on which to interpolate the values of the 2D field; such a grid can be adapted to the specific geometry of each domain as long as the number of points remains constant.

[0058] The approach of the present invention has several advantages, including conceptual simplicity and highly fast training and prediction times. Although the effectiveness of the methodology of the invention has been proven when solving the problem of estimating pressure maps on the surface of simulated brake pads, such an inventive approach is valid for any parametric map on two-dimensional geometric domains of any physical product, where the parametric map is a map of a physical parameter associated with the points of a two-dimensional geometric domain of a product, the physical parameter being any simulatable physical quantity of the product.

[0059] References

[0060] [1] Saouma, V. & Hariri-Ardebili, Mohammad Amin. (2021). Metamodeling and Machine Learning. 10.1007 / 978-3-030- 57434-5_20 .

[0061] [2] Forrester, A.I., & Keane, A.J. (2009). Recent advances in surrogate-based optimization. Progress in Aerospace Sciences, 45, 50-79.

[0062] [3] Le Gratiet, Loic. (2012). Recursive co-kriging model for Design of Computer experiments with multiple levels of fidelity. International Journal for Uncertainty Quantification. 4. 10.1615 / Int.J .UncertaintyQuantification.2014006914.

[0063] [4] White, Daniel & Arrighi, William & Kudo, Jun & Watts, Seth. (2018). Multiscale topology optimization using neural network surrogate models. Computer Methods in Applied Mechanics and Engineering. 346. 10.1016 / j.cma .2018.09.007. [5] Franco, Nicola Rares et al. (2023). Deep Learning- based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks.

[0064] [6] Kashefi, Ali & Rempe, Davis & Guibas, Leonidas J. A point-cloud deep learning framework for prediction of fluid flow fields on irregular geometries. Physics of Fluids 1 February 2021; 33 (2): 027104.

[0065] [7] Kriging - Wikipedia

[0066] [8] Principal component analysis - Wikipedia

[0067] Preferred embodiments have been described above and variations of the present invention have been suggested, but it should be understood that those skilled in the art may make modifications and changes without departing from the related scope of protection, as defined by the appended claims.

Claims

CLAIMS1. A computer-implemented method for simulating parametric maps, a parametric map being a map of a physical parameter associated with the points of a two- dimensional geometric domain of a product, the method comprising the use of a trained neural network that takes as input input parameters associated with the two- dimensional geometric domain of the product, the neural network outputting a simulated parametric map, wherein in the training step of the neural network the following steps are performed:T1. obtaining training parametric maps calculated via a numerical simulation model based on corresponding sets of training input parameters, each training parametric map being characterized by a corresponding grid of points;T2. standardizing the training parametric maps so that they are all characterized by a uniform grid with a constant number of points, by carrying out the following sub-steps:T2.

1. performing a feature scaling of the coordinate values of the points, so that the training parametric maps are included in the square [0,1]x [0,1],T2.

2. defining a uniform grid with a predetermined number of points;T2.

3. projecting the training parametric maps of step T1 onto the uniform grid via interpolation, thus obtaining uniform training parametric maps;T3. compressing the uniform training parametric maps via auto-encoding, obtaining compressed uniform training parametric maps;T4. training the neural network with the compressed uniform training parametric maps as outputs and the corresponding sets of training input parameters, thus obtaining a trained neural network; and wherein a simulation step S is carried out in which the following sub-steps are performed:S1. providing a set of simulation input parameters relating to a two-dimensional geometric domain of a product to be simulated with a predefined grid of points;S2. performing the operations of steps T2 and T3 on said two-dimensional geometric domain of the product obtaining a compressed uniform parametric map of the two-dimensional geometric domain for the product to be simulated;S3. using the trained neural network with said set of simulation input parameters to obtain a compressed uniform simulated parametric map;S4. decompressing the compressed uniform simulated parametric map via reverse auto-encoding as compared to that of step T3, obtaining a simulated uniform parametric map; andS5. back-transforming the simulated uniform parametric map of step S4 onto said predefined grid of points via feature scaling, thus obtaining a simulated parametric map.

2. A method according to claim 1, wherein the neural network is a Multi-Layer Perceptron (MLP).

3. A method according to claim 1 or 2, wherein after step S5 the following step is performed:P. producing said product based on the simulated parametric map.

4. A method according to one of claims 1 to 3, wherein said interpolation of step T2.3 is carried out by Kriging.

5. A method according to one of claims 1 to 4, wherein said auto-encoding of step T3 is carried out via Principal Component Analysis.

6. A method according to one of claims 1 to 5, wherein said reverse auto-encoding of step S4 is carried out via Principal Component Analysis.

7. A method according to one of claims 1 to 6, wherein said feature scaling of steps T2 and S5 is carried out via min-max scaling.

8. A method according to one of claims 1 to 7, wherein the parametric maps on two-dimensional geometric domains are brake pad pressure maps, i.e., maps of pressure on the working surface of a brake pad, wherein the input parameters relating to the two-dimensional geometric domain of the product are: length of the pad, externalradius of the brake rotor, internal radius of the brake rotor, two-dimensional coordinates of the center of two or more brake pistons, radii of said two or more pistons of the brake, and thickness of the pad.

9. A computer program comprising instructions which, when the program is executed by a computer, cause the computer to carry out the steps of the method according to any one claim 1 to 8.

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