Dataset Open Access

# Zero Modes and Classification of Combinatorial Metamaterials

van Mastrigt, Ryan; Dijkstra, Marjolein; van Hecke, Martin; Coulais, Corentin

This dataset contains the simulation data of the combinatorial metamaterial as used for the paper 'Machine Learning of Implicit Combinatorial Rules in Mechanical Metamaterials', as published in Physical Review Letters.

In this paper, the data is used to classify each $$k \times k$$ unit cell design into one of two classes (C or I) based on the scaling (linear or constant) of the number of zero modes $$M_k(n)$$ for metamaterials consisting of an $$n\times n$$ tiling of the corresponding unit cell. Additionally, a random walk through the design space starting from class C unit cells was performed to characterize the boundary between class C and I in design space. A more detailed description of the contents of the dataset follows below.

Modescaling_raw_data.zip

This file contains uniformly sampled unit cell designs for metamaterial M2 and $$M_k(n)$$ for $$1\leq n\leq 4$$, which was used to classify the unit cell designs for the data set. There is a small subset of designs for $$k=\{3, 4, 5\}$$ that do not neatly fall into the class C and I classification, and instead require additional simulation for $$4 \leq n \leq 6$$ before either saturating to a constant number of zero modes (class I) or linearly increasing (class C). This file contains the simulation data of size $$3 \leq k \leq 8$$ unit cells. The data is organized as follows.

Simulation data for $$3 \leq k \leq 5$$ and $$1 \leq n \leq 4$$ is stored in numpy array format (.npy) and can be readily loaded in Python with the Numpy package using the numpy.load command. These files are named "data_new_rrQR_i_n_M_kxk_fixn4.npy", and contain a [Nsim, 1+k*k+4] sized array, where Nsim is the number of simulated unit cells. Each row corresponds to a unit cell. The columns are organized as follows:

• col 0: label number to keep track
• col 1 - k*k+1: flattened unit cell design, numpy.reshape should bring it back to its original $$k \times k$$ form.
• col k*k+1 - k*k+5: number of zero modes $$M_k(n)$$ in ascending order of $$n$$, so: $$\{M_k(1), M_k(2), M_k(3), M_k(4)\}$$.

Note: the unit cell design uses the numbers $$\{0, 1, 2, 3\}$$ to refer to each building block orientation. The building block orientations can be characterized through the orientation of the missing diagonal bar (see Fig. 2 in the paper), which can be Left Up (LU), Left Down (LD), Right Up (RU), or Right Down (RD). The numbers correspond to the building block orientation $$\{0, 1, 2, 3\} = \{\mathrm{LU, RU, RD, LD}\}$$.

Simulation data for $$3 \leq k \leq 5$$ and $$1 \leq n \leq 6$$ for unit cells that cannot be classified as class C or I for $$1 \leq n \leq 4$$ is stored in numpy array format (.npy) and can be readily loaded in Python with the Numpy package using the numpy.load command. These files are named "data_new_rrQR_i_n_M_kxk_fixn4_classX_extend.npy", and contain a [Nsim, 1+k*k+6] sized array, where Nsim is the number of simulated unit cells. Each row corresponds to a unit cell. The columns are organized as follows:

• col 0: label number to keep track
• col 1 - k*k+1: flattened unit cell design, numpy.reshape should bring it back to its original $$k \times k$$ form.
• col k*k+1 - k*k+5: number of zero modes $$M_k(n)$$ in ascending order of $$n$$, so: $$\{M_k(1), M_k(2), M_k(3), M_k(4), M_k(5), M_k(6)\}$$.

Simulation data for $$6 \leq k \leq 8$$  unit cells are stored in numpy array format (.npy) and can be readily loaded in Python with the Numpy package using the numpy.load command. Note that the number of modes is now calculated for $$n_x \times n_y$$ metamaterials, where we calculate $$(n_x, n_y) = \{(1,1), (2, 2), (3, 2), (4,2), (2, 3), (2, 4)\}$$ rather than $$n_x=n_y=n$$ to save computation time. These files are named "data_new_rrQR_i_n_Mx_My_n4_kxk(_extended).npy", and contain a [Nsim, 1+k*k+8] sized array, where Nsim is the number of simulated unit cells. Each row corresponds to a unit cell. The columns are organized as follows:

• col 0: label number to keep track
• col 1 - k*k+1: flattened unit cell design, numpy.reshape should bring it back to its original $$k \times k$$ form.
• col k*k+1 - k*k+9: number of zero modes $$M_k(n_x, n_y)$$ in order: $$\{M_k(1, 1), M_k(2, 2), M_k(3, 2), M_k(4, 2), M_k(1, 1), M_k(2, 2), M_k(2, 3), M_k(2, 4)\}$$.

Simulation data of metamaterial M1 for $$k_x \times k_y$$ metamaterials are stored in compressed numpy array format (.npz) and can be loaded in Python with the Numpy package using the numpy.load command. These files are named "smiley_cube_x_y_$$k_x$$x$$k_y$$.npz", which contain all possible metamaterial designs, and "smiley_cube_uniform_sample_x_y_$$k_x$$x$$k_y$$.npz", which contain uniformly sampled metamaterial designs. The configurations are accessed with the keyword argument 'configs'. The classification is accessed with the keyword argument 'compatible'. The configurations array is of shape [Nsim, $$k_x$$, $$k_y$$], the classification array is of shape [Nsim]. The building blocks in the configuration are denoted by 0 or 1, which correspond to the red/green and white/dashed building blocks respectively. Classification is 0 or 1, which corresponds to I and C respectively.

Modescaling_classification_results.zip

This file contains the classification, slope, and offset of the scaling of the number of zero modes $$M_k(n)$$ for the unit cells of metamaterial M2 in Modescaling_raw_data.zip. The data is organized as follows.

The results for $$3 \leq k \leq 5$$ based on the $$1 \leq n \leq 4$$ mode scaling data is stored in "results_analysis_new_rrQR_i_Scen_slope_offset_M1k_kxk_fixn4.txt". The data can be loaded using ',' as delimiter. Every row corresponds to a unit cell design (see the label number to compare to the earlier data). The columns are organized as follows:

col 0: label number to keep track

col 1: the class, where 0 corresponds to class I, 1 to class C and 2 to class X (neither class I or C for $$1 \leq n \leq 4$$)

col 2: slope from $$n \geq 2$$ onward (undefined for class X)

col 3: the offset is defined as $$M_k(2) - 2 \cdot \mathrm{slope}$$

col 4: $$M_k(1)$$

The results for $$3 \leq k \leq 5$$ based on the extended $$1 \leq n \leq 6$$ mode scaling data is stored in "results_analysis_new_rrQR_i_Scen_slope_offset_M1k_kxk_fixn4_classC_extend.txt". The data can be loaded using ',' as delimiter. Every row corresponds to a unit cell design (see the label number to compare to the earlier data). The columns are organized as follows:

col 0: label number to keep track

col 1: the class, where 0 corresponds to class I, 1 to class C and 2 to class X (neither class I or C for $$1 \leq n \leq 6$$)

col 2: slope from $$n \geq 2$$ onward (undefined for class X)

col 3: the offset is defined as $$M_k(2) - 2 \cdot \mathrm{slope}$$

col 4: $$M_k(1)$$

The results for $$6 \leq k \leq 8$$ based on the $$1 \leq n \leq 4$$ mode scaling data is stored in "results_analysis_new_rrQR_i_Scenx_Sceny_slopex_slopey_offsetx_offsety_M1k_kxk(_extended).txt". The data can be loaded using ',' as delimiter. Every row corresponds to a unit cell design (see the label number to compare to the earlier data). The columns are organized as follows:

col 0: label number to keep track

col 1: the class_x based on $$M_k(n_x, 2)$$, where 0 corresponds to class I, 1 to class C and 2 to class X (neither class I or C for $$1 \leq n_x \leq 4$$)

col 2: the class_y based on $$M_k(2, n_y)$$, where 0 corresponds to class I, 1 to class C and 2 to class X (neither class I or C for $$1 \leq n_y \leq 4$$)

col 3: slope_x from $$n_x \geq 2$$ onward (undefined for class X)

col 4: slope_y from $$n_y \geq 2$$ onward (undefined for class X)

col 5: the offset_x is defined as $$M_k(2, 2) - 2 \cdot \mathrm{slope_x}$$

col 6: the offset_x is defined as $$M_k(2, 2) - 2 \cdot \mathrm{slope_y}$$

col 7: $$M_k(1, 1)$$

Additionally, results including classification for M2.ii can be found in the "results_analysis_unimodal_vs_oligomodal_vs_plurimodal_i_Scen_slope_M_M1k_kxk.txt and "results_analysis_unimodal_vs_oligomodal_vs_plurimodal_i_Scenx_Sceny_slopex_slopey_Mx_My_M1k_kxk.txt" files.

Random Walks Data

This file contains the random walks for $$3 \leq k \leq 8$$ unit cells of metamaterial M2. The random walk starts from a class C unit cell design (classification M2.ii), for each step $$s$$ a randomly picked unit cell is changed to a random new orientation for a total of $$s=k^2$$ steps. The data is organized as follows.

The configurations for each step are stored in the files named "configlist_test_i.npy", where i is a number and corresponds to a different starting unit cell. The stored array has the shape [k*k+1, 2*k+2, 2*k+2]. The first dimension denotes the step $$s$$, where $$s=0$$ is the initial configuration. The second and third dimension denote the unit cell configuration in the pixel representation (see paper) padded with a single pixel wide layer using periodic boundary conditions.

The class for each configuration are stored in "lmlist_test_i.npy", where i corresponds to the same number as for the configurations in the "configlist_test_i.npy" file. The stored array has the shape [k*k+1], where the index corresponds to the step $$s$$ and displays the class for the accompanying unit cell. The stored number corresponds to the class as $$\{0, 1\} = \{\mathrm{I}, \mathrm{C}\}$$

Files (8.0 GB)
Name Size
Modescaling_classification_results.zip
md5:79e649816a7473310f00d69b4053dae8
61.9 MB
Modescaling_raw_data.zip
md5:b498284b233f216f339b05c9c0438c8e
2.2 GB
Random_walks_data.zip
5.7 GB
• van Mastrigt, R., Dijkstra, M., van Hecke, M., & Coulais, C. (2022). Machine Learning of Implicit Combinatorial Rules in Mechanical Metamaterials. Physical Review Letters, 129(19), 198003.

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