Mechanical MNIST Datasets
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Each dataset in the Mechanical MNIST collection contains the results of 70,000 (60,000 training examples + 10,000 test examples) finite element simulation of a heterogeneous material subject to large deformation. Mechanical MNIST is generated by first converting the MNIST bitmap images (http://www.pymvpa.org/datadb/mnist.html) to 2D heterogeneous blocks of material. Consistent with the MNIST bitmap ($28 \times 28$ pixels), the material domain is a $28 \times 28$ unit square. All simulations are conducted with the FEniCS computing platform (https://fenicsproject.org). The code to reproduce these simulations is hosted on GitHub (https://github.com/elejeune11/Mechanical-MNIST/tree/master/generate_dataset).
The paper "Mechanical MNIST: A benchmark dataset for mechanical metamodels" can be found at https://doi.org/10.1016/j.eml.2020.100659. All code necessary to reproduce the metamodels demonstrated in the manuscript is available on GitHub (https://github.com/elejeune11/Mechanical-MNIST). For questions, please contact Emma Lejeune (elejeune@bu.edu).
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Item Mechanical MNIST – Unsupervised Learning Dataset(2023) Nguyen, Quan; Lejeune, EmmaThe Mechanical MNIST dataset collection contains Finite Element simulations of heterogeneous materials undergoing applied displacement. Here, we introduce a new benchmark dataset designed specifically for assessing unsupervised learning methods where the goal is to discover patterns from unlabeled data. To obtain this dataset, we generate displacement fields from Finite Element simulations and uniformly sample approximately 1500 displacement markers on each domain of interest. Since unsupervised learning aims to identify patterns in labeled data, we provide a dataset where the primary objective is to explore unlabeled data, while simultaneously providing “ground truth” information to ultimately evaluate the efficacy of different unsupervised learning approaches. It is important to note however, that in the intended applications of these methods, ground truth information will likely be absent, particularly in experimental studies of intricate heterogeneous soft tissue. Broadly speaking, this computationally generated dataset mimics the behavior of soft materials, while simultaneously providing ground truth information for method evaluation. In total, the dataset contains the following combinations of conditions: 6 different heterogeneous material patterns, 2 constitutive models, 4 controlled boundary conditions, and 1 random boundary condition. Here, we include the tutorials for our dataset with the name “dataset_tutorials.pdf”. This document contains the information to understand the contents of our dataset, as well as the instructions on how to use the data. The many options from our dataset should enable researchers to explore unsupervised learning methods on soft materials.Item Mechanical MNIST – Distribution Shift(2022) Yuan, Lingxiao; Park, Harold S.; Lejeune, EmmaThe Mechanical MNIST – Distribution Shift dataset contains the results of finite element simulation of heterogeneous material subject to large deformation due to equibiaxial extension at a fixed boundary displacement of d = 7.0. The result provided in this dataset is the change in strain energy after this equibiaxial extension. The Mechanical MNIST dataset is generated by converting the MNIST bitmap images (28x28 pixels) with range 0 - 255 to 2D heterogeneous blocks of material (28x28 unit square) with varying modulus in range 1- s. The original bitmap images are sourced from the MNIST Digits dataset, (http://www.pymvpa.org/datadb/mnist.html) which corresponds to Mechanical MNIST – MNIST, and the EMNIST Letters dataset (https://www.nist.gov/itl/products-and-services/emnist-dataset) which correspond to Mechanical MNIST – EMNIST Letters. The Mechanical MNIST – Distribution Shift dataset is specifically designed to demonstrate three types of data distribution shift: (1) covariate shift, (2) mechanism shift, and (3) sampling bias, for all of which the training and testing environments are drawn from different distributions. For each type of data distribution shift, we have one dataset generated from the Mechanical MNIST bitmaps and one from the Mechanical MNIST – EMNIST Letters bitmaps. For the covariate shift dataset, the training dataset is collected from two environments (2500 samples from s = 100, and 2500 samples from s = 90), and the test data is collected from two additional environments (2000 samples from s = 75, and 2000 samples from s = 50). For the mechanism shift dataset, the training data is identical to the training data in the covariate shift dataset (i.e., 2500 samples from s = 100, and 2500 samples from s = 90), and the test datasets are from two additional environments (2000 samples from s = 25, and 2000 samples from s = 10). For the sampling bias dataset, datasets are collected such that each datapoint is selected from the broader MNIST and EMNIST inputs bitmap selection by a probability which is controlled by a parameter r. The training data is collected from two environments (9800 from r = 15, and 200 from r = -2), and the test data is collected from three different environments (2000 from r = -5, 2000 from r = -10, and 2000 from r = 1). Thus, in the end we have 6 benchmark datasets with multiple training and testing environments in each. The enclosed document “folder_description.pdf'” shows the organization of each zipped folder provided on this page. The code to reproduce these simulations is available on GitHub (https://github.com/elejeune11/Mechanical-MNIST/blob/master/generate_dataset/Equibiaxial_Extension_FEA_test_FEniCS.py).Item Mechanical MNIST - Cahn-Hilliard(2022) Kobeissi, Hiba; Lejeune, EmmaThe Mechanical MNIST Cahn-Hilliard dataset contains the results of 104,813 Finite Element simulations of a heterogeneous material domain subject to large equibiaxial extension deformation. The heterogeneous domain patterns are generated from a Finite Element implementation of the Cahn-Hilliard equation. Different stripe and circle patterns are obtained by varying four simulation parameters: the initial concentration, the grid size on which the concentration is initialized, parameter $\lambda$, and $b$, the peak-to-valley value of the symmetric double-well chemical free-energy function. Binary bitmap images of 400 x 400 pixels are converted into two-dimensional meshed domains of binary material using the OpenCV library, Pygmsh, and Gmsh 4.6.0. We also include in this dataset the 104,813 patterns (37,523 from case 1, 37,680 from case 2, and 29,610 from case 3) used in the Finite Element simulations stored as binary images in text files. After pattern generation, the material domain is modeled as a unit square of Neo-Hookean binary material (high concentration areas correspond to Young's Modulus 10, low concentration areas correspond to Young's Modulus 1). For equibiaxial extension, each of the four edges of the domain is displaced to 50% of the initial domain size in the direction of the outward normal to the surface with fixed displacements (d = [0.0,0.001,0.1,0.2,0.3,0.4,0.5]). Here we provide the simulation results consisting of the following: (1) change in strain energy reported at each level of applied displacement, (2) total reaction force at the four boundaries reported at each level of applied displacement, and (3) full field displacement reported at the final applied displacement d=0.5. All Finite Element simulations are conducted with the FEniCS computing platform (https://fenicsproject.org). The code to reproduce these simulations (both pattern generation simulations and equibiaxial extension simulations) is hosted on GitHub (https://github.com/elejeune11/Mechanical-MNIST-Cahn-Hilliard). The enclosed document “description.pdf'” contains additional details.Item Mechanical MNIST Crack Path(2021-07) Mohammadzadeh, Saeed; Lejeune, EmmaThe Mechanical MNIST Crack Path dataset contains Finite Element simulation results from phase-field models of quasi-static brittle fracture in heterogeneous material domains subjected to prescribed loading and boundary conditions. For all samples, the material domain is a square with a side length of 1. There is an initial crack of fixed length (0.25) on the left edge of each domain. The bottom edge of the domain is fixed in x (horizontal) and y (vertical), the right edge of the domain is fixed in x and free in y, and the left edge is free in both x and y. The top edge is free in x, and in y it is displaced such that, at each step, the displacement increases linearly from zero at the top right corner to the maximum displacement on the top left corner. Maximum displacement starts at 0.0 and increases to 0.02 by increments of 0.0001 (200 simulation steps in total). The heterogeneous material distribution is obtained by adding rigid circular inclusions to the domain using the Fashion MNIST bitmaps as the reference location for the center of the inclusions. Specifically, each center point location is generated randomly inside a square region defined by the corresponding Fashion MNIST pixel when the pixel has an intensity value higher than 10. In addition, a minimum center-to-center distance limit of 0.0525 is applied while generating these center points for each sample. The values of Young’s Modulus (E), Fracture Toughness (G_f), and Failure Strength (f_t) near each inclusion are increased with respect to the background domain by a variable rigidity ratio r. The background value for E is 210000, the background value for G_f is 2.7, and the background value for f_t is 2445.42. The rigidity ratio throughout the domain depends on position with respect to all inclusion centers such that the closer a point is to the inclusion center the higher the rigidity ratio will be. We note that the full algorithm for constructing the heterogeneous material property distribution is included in the simulations scripts shared on GitHub. The following information is included in our dataset: (1) A rigidity ratio array to capture heterogeneous material distribution reported over a uniform 64x64 grid, (2) the damage field at the final level of applied displacement reported over a uniform 256x256 grid, (3) the damage field at the final level of applied displacement reported over a uniform 64x64 grid, and (4) the force-displacement curves for each simulation. All simulations are conducted with the FEniCS computing platform (https://fenicsproject.org). The code to reproduce these simulations is hosted on GitHub (https://github.com/saeedmhz/phase-field).Item Mechanical MNIST - Fashion(2020) Lejeune, EmmaEach dataset in the Mechanical MNIST collection contains the results of 70,000 (60,000 training examples + 10,000 test examples) finite element simulation of a heterogeneous material subject to large deformation. Mechanical MNIST - Fashion is generated by first converting the fashion MNIST bitmap images (https://github.com/zalandoresearch/fashion-mnist) to 2D heterogeneous blocks of material. Consistent with the MNIST bitmap ($28 \times 28$ pixels), the material domain is a $28 \times 28$ unit square. In “Mechanical MNIST - Fashion,” the material is Neo-Hookean with a varying modulus. In the Uniaxial Extension (UE) case, the bottom of the domain is fixed (Dirichlet boundary condition), the left and right edges of the domain are free, and the top of the domain is fixed horizontally and moved vertically to a given fixed displacement (d). In the Equibiaxial Extension (EE) case, the top of the domain is free horizontally and moved vertically to a given fixed displacement (d), the right of the domain is free vertically and moved horizontally to a given fixed displacement (d), the bottom of the domain is free horizontally and moved vertically to a given fixed displacement (-d), and the left of the domain is free vertically and moved horizontally to a given fixed displacement (-d). The results of the simulations include: (1) change in strain energy at a perturbation level step (d=0.001), and at the final applied displacement (d=14 for UE, d=7 for EE) (2) total reaction force at a perturbation level step (d=0.001 for UE, d=.0005 for EE), and at the final applied displacement (d=14 for UE, d=7 for EE), (3) full field displacement at a perturbation level step (d=0.001), and at the final applied displacement (d=14 for UE, d=7 for EE), and (4) the components of the deformation gradient (F11, F12, F21, F22) at the final applied displacement (d=14 for UE, d=7 for EE). The x-reaction (first column) and y-reaction (second column) forces are given. For the UE case, this corresponds to the top boundary. For the EE case, this correspond to the left and top boundaries. All simulations are conducted with the FEniCS computing platform (https://fenicsproject.org). The code to reproduce these simulations and import these text files is hosted on GitHub (https://github.com/elejeune11/Mechanical-MNIST-fashion).Item Mechanical MNIST - Multi-Fidelity(2020-07) Lejeune, EmmaEach dataset in the Mechanical MNIST collection contains the results of 70,000 (60,000 training examples + 10,000 test examples) finite element simulation of a heterogeneous material subject to large deformation. Mechanical MNIST is generated by first converting the MNIST bitmap images (http://www.pymvpa.org/datadb/mnist.html) to 2D heterogeneous blocks of material. Consistent with the MNIST bitmap ($28 \times 28$ pixels), the material domain is a $28 \times 28$ unit square. The material is Neo-Hookean with a varying modulus dictated by the input bitmap. The simulation results included here are the change in strain energy at a fixed level of applied displacement. The cases considered are as follows: *UE: uniaxial extension, full fidelity dataset (fully refined mesh, quadratic triangular elements, applied displacement is $1/2$ of a side length); *EE: equibiaxial extension, full fidelity dataset; *3D: uniaxial extension and out of plane twist, full fidelity three dimensional dataset (fully refined mesh, quadratic tetrahedral elements, applied displacement is $1/7$ of a side length, twist is $\pi/8$ radians, block thickness is $1/7$ of a side length); *UE-CM-28: uniaxial extension, $28 \times 28 \times 2$ linear triangular elements; *UE-CM-14: uniaxial extension, $14 \times 14 \times 2$ linear triangular elements; *UE-CM-7: uniaxial extension, $7 \times 7 \times 2$ linear triangular elements; *UE-CM-7-quad: uniaxial extension, $7 \times 7 \times 2$ quadratic triangular elements; *UE-CM-4: uniaxial extension, $4 \times 4 \times 2$ linear triangular elements; *UE-CM-4-quad: uniaxial extension, $4 \times 4 \times 2$ quadratic triangular elements; *UE-perturb: uniaxial extension, applied displacement is a perturbation (.001 units); *UE-CM-28-perturb: uniaxial extension, $28 \times 28 \times 2$ linear triangular elements, applied displacement is a perturbation (.001 units). All simulations are conducted with the FEniCS computing platform (https://fenicsproject.org). The code to reproduce these simulations is hosted on GitHub (https://github.com/elejeune11/Mechanical-MNIST-Transfer-Learning).Item Mechanical MNIST: A benchmark dataset for mechanical metamodels(Elsevier BV, 2020-04) Lejeune, EmmaMetamodels, or models of models, map defined model inputs to defined model outputs. Typically, metamodels are constructed by generating a dataset through sampling a direct model and training a machine learning algorithm to predict a limited number of model outputs from varying model inputs. When metamodels are constructed to be computationally cheap, they are an invaluable tool for applications ranging from topology optimization, to uncertainty quantification, to multi-scale simulation. By nature, a given metamodel will be tailored to a specific dataset. However, the most pragmatic metamodel type and structure will often be general to larger classes of problems. At present, the most pragmatic metamodel selection for dealing with mechanical data has not been thoroughly explored. Drawing inspiration from the benchmark datasets available to the computer vision research community, we introduce a benchmark data set (Mechanical MNIST) for constructing metamodels of heterogeneous material undergoing large deformation. We then show examples of how our benchmark dataset can be used, and establish baseline metamodel performance. Because our dataset is readily available, it will enable the direct quantitative comparison between different metamodeling approaches in a pragmatic manner. We anticipate that it will enable the broader community of researchers to develop improved metamodeling techniques for mechanical data that will surpass the baseline performance that we show here.Item Mechanical MNIST - Shear(2020-02) Lejeune, EmmaEach dataset in the Mechanical MNIST collection contains the results of 70,000 (60,000 training examples + 10,000 test examples) finite element simulation of a heterogeneous material subject to large deformation. Mechanical MNIST is generated by first converting the MNIST bitmap images (http://www.pymvpa.org/datadb/mnist.html) to 2D heterogeneous blocks of material. Consistent with the MNIST bitmap ($28 \times 28$ pixels), the material domain is a $28 \times 28$ unit square. In "Mechanical MNIST - Shear," the material is Neo-Hookean with a varying modulus. The bottom of the domain is fixed (Dirichlet boundary condition), the left and right edges of the domain are free, and the top of the domain is fixed vertically and moved horizontally to a set of given fixed displacements (d = [0.0, 0.001, 0.01, 0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5]). The results of the simulations include: (1) change in strain energy at each step, (2) total reaction force at the top boundary at each step, and (3) full field displacement at each step. All simulations are conducted with the FEniCS computing platform (https://fenicsproject.org). The code to reproduce these simulations is hosted on GitHub (https://github.com/elejeune11/Mechanical-MNIST/tree/master/generate_dataset).Item Mechanical MNIST - Equibiaxial Extension(2020-02) Lejeune, EmmaEach dataset in the Mechanical MNIST collection contains the results of 70,000 (60,000 training examples + 10,000 test examples) finite element simulation of a heterogeneous material subject to large deformation. Mechanical MNIST is generated by first converting the MNIST bitmap images (http://www.pymvpa.org/datadb/mnist.html) to 2D heterogeneous blocks of material. Consistent with the MNIST bitmap ($28 \times 28$ pixels), the material domain is a $28 \times 28$ unit square. In "Mechanical MNIST - Equibiaxial Extension," the material is Neo-Hookean with a varying modulus. The top of the domain is free horizontally and moved vertically to a set of given fixed displacements (d = [0.0, 0.0005, 0.005, 0.05, 0.25, 0.5, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0 ]), the right of the domain is free vertically and moved horizontally to a set of given fixed displacements (d = [0.0, 0.0005, 0.005, 0.05, 0.25, 0.5, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0 ]), the bottom of the domain is free horizontally and moved vertically to a set of given fixed displacements (d = [-0.0, -0.0005, -0.005, -0.05, -0.25, -0.5, -1.0, -2.0, -3.0, -4.0, -5.0, -6.0, -7.0 ]), and the left of the domain is free vertically and moved horizontally to a set of given fixed displacements (d = [-0.0, -0.0005, -0.005, -0.05, -0.25, -0.5, -1.0, -2.0, -3.0, -4.0, -5.0, -6.0, -7.0 ]). The results of the simulations include: (1) change in strain energy at each step, (2) total reaction force at the top and right boundaries at each step, and (3) full field displacement at each step. All simulations are conducted with the FEniCS computing platform (https://fenicsproject.org). The code to reproduce these simulations is hosted on GitHub (https://github.com/elejeune11/Mechanical-MNIST/tree/master/generate_dataset).Item Mechanical MNIST - Confined Compression(2020-02) Lejeune, EmmaEach dataset in the Mechanical MNIST collection contains the results of 70,000 (60,000 training examples + 10,000 test examples) finite element simulation of a heterogeneous material subject to large deformation. Mechanical MNIST is generated by first converting the MNIST bitmap images (http://www.pymvpa.org/datadb/mnist.html) to 2D heterogeneous blocks of material. Consistent with the MNIST bitmap ($28 \times 28$ pixels), the material domain is a $28 \times 28$ unit square. In "Mechanical MNIST - Equibiaxial Extension," the material is Neo-Hookean with a varying modulus. The top of the domain is free horizontally and moved vertically to a set of given fixed displacements (d = [-0.0, -0.001, -0.01, -0.1, -0.5, -1.0, -1.5, -2.0, -2.5, -3.0, -3.5] ), the right and left sides of the domain are free vertically and fixed horizontally, and the bottom of the domain is free horizontally and fixed vertically. The results of the simulations include: (1) change in strain energy at each step, (2) total reaction force at the top and right boundaries at each step, and (3) full field displacement at each step. All simulations are conducted with the FEniCS computing platform (https://fenicsproject.org). The code to reproduce these simulations is hosted on GitHub (https://github.com/elejeune11/Mechanical-MNIST/tree/master/generate_dataset).Item Mechanical MNIST - Uniaxial Extension(2019-12) Lejeune, EmmaEach dataset in the Mechanical MNIST collection contains the results of 70,000 (60,000 training examples + 10,000 test examples) finite element simulation of a heterogeneous material subject to large deformation. Mechanical MNIST is generated by first converting the MNIST bitmap images (http://www.pymvpa.org/datadb/mnist.html) to 2D heterogeneous blocks of material. Consistent with the MNIST bitmap ($28 \times 28$ pixels), the material domain is a $28 \times 28$ unit square. In “Mechanical MNIST - Uniaxial Extension,” the material is Neo-Hookean with a varying modulus. The bottom of the domain is fixed (Dirichlet boundary condition), the left and right edges of the domain are free, and the top of the domain is fixed horizontally and moved vertically to a set of given fixed displacements (d = [0.0, 0.001, 0.01, 0.1, 0.5, 1.0, 2.0, 4.0, 6.0, 8.0, 10.0, 12.0, 14.0 ]). The results of the simulations include: (1) change in strain energy at each step, (2) total reaction force at the top boundary at each step, and (3) full field displacement at each step. All simulations are conducted with the FEniCS computing platform (https://fenicsproject.org). The code to reproduce these simulations is hosted on GitHub (https://github.com/elejeune11/Mechanical-MNIST/tree/master/generate_dataset).