Keywords

1 Introduction

The emergent applications of Artificial Intelligence algorithms in architectural and design practices opened a wide range of novel methods and processes to envision, create and optimize the design process, varying from the scale of urban design to the exploration of synthetic spaces. Naturally, data and the way how are organized plays a crucial role in the entire process, which can vary from datasets of images to numeric values. Nevertheless, applying AI-based algorithms to material computation requires a linear workflow to generate a specific dataset of values that can precisely represent a geometry. Requires that the design process be based on geometrical features that represent its physical characteristics.

This research project utilizes a regression Deep Neuronal Network to predict the distances of each cell's patterns from an auxetic surface that was previously actively bent, based on its Gaussian curvature, osculating point, pattern distance and the applied force parameters.

Auxetics are metamaterial structures with a negative Poisson’s ratio, in which the mechanical performance relies on the geometry rather than on the material itself. When stretched, they become thicker in the perpendicular direction to the applied force. This phenomenon occurs due to their internal structure and how this deforms when the sample is uniaxially loaded.

This makes them a material system with a wide range of applications on the architectural scale, for example, by reducing the amount of energy needed to create a three-dimensional shape [1], by distributing the internal mechanical forces of the system, achieving a relaxed and stable form.

Computing bent activated auxetic surfaces requires simulating each cell's behaviour that composes the surface and its global deformation, demanding a significant computational power. By creating and training a tailored regression DNN model, the simulation time and power can be reduced by just a fraction, offering the users an interactive tool to input the desired performance and receive a precise predicted pattern.

1.1 Auxetic Structures

Auxetic structures present the unique capacity of becoming wider when stretched and narrower when compressed [2]. The word auxetic comes from the Greek word αὐξητικός (auxetikos) which means “tends to increase”. Some of its edges and vertices work under compression to give a material the capacity to extend, reducing one axis its length, thus giving space for the other edges and vertices to elongate and, consequently, make the system extend. This relationship between compression and traction forces is defined as the Poisson ratio (v), which is the ratio of the transverse contraction of a material to the strain in the direction of the stretching force [3]. A negative Poisson ratio occurs when compression forces are applied, and, in contrast, the Poisson ratio is positive when there is tensile deformation (Fig. 1). The Poisson ratio values could have a wider range of values for anisotropic materials than isotropic materials.

Fig. 1.
figure 1

Source From the authors.

Left: non-auxetic material with a positive Poisson’s ratio under tensile stress. Right: auxetic material with a negative Poisson ratio under tensile stress.

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Bi-dimensional auxetic structures are results from the tessellation of a given plane with periodic regular polygons [4] working as an individual but concatenated cell, where its deformation magnitude and direction are directly related to the Poisson ratio. Because of this, auxetic structures materials should have a low density or be flexible. From this basic definition of the deformation ratio, auxetic surfaces have specific characteristics that contribute to a more refined understanding of their variabilities and geometric properties [5] as a material system:

  • Shear properties: shear modulus can be similar to the bulk module, meaning that the structure becomes hard to break but easy to deform.

  • Indentation resistance: Hardness can increase in an auxetic material due to its negative Poisson’s ratio.

  • Fracture toughness: because of the displacement for geometry, it possesses more crack resistance to fracture.

  • Synclastic curvature: will form a synclastic curvature geometry by a natural distribution of its internal forces.

  • Energy absorption: auxetic structures have a high capacity to absorb constant deformation loads in low frequencies.

  • Variable permeability: The pore-opening properties offer a high filtering capacity on the micro and macro scale.

1.2 Generative Particle-Spring System for Material Behaviour Simulation

A particle-spring system consists of particles given mass and position, which are connected by springs with stiffness and rest length [6]. By applying an external force over the network of particles and springs, each particle moves, affecting the others producing a concatenated deformation because of the springs, distributing the embedded energy to find its equilibrium state.

Physics simulators engines for generative design software such as Kangaroo Physics are powerful tools to simulate and fast preview the physical deformation of geometries submitted to external forces over a predefined geometrical system. Offering the capacity to compute the synclastic geometries generated by the application of tension in opposite axis directions on auxetic structures. Particle-Spring system simulations have been successfully applied to test different auxetic properties and the influence on the Poisson ratio when the internal angle of the hexagons of an auxetic cell is changed [7].

Despite the good performance of particle-spring physics simulators to quickly simulate geometric deformations, this process is based on iterative methods. It requires computing several variations of the same model demanding high computational power to simulate complex models.

1.3 Artificial Intelligence for Material Prediction

The application of machine learning algorithms to find a specific solution to a given problem is not new. In the early 90’s, shallow learning algorithms were used in the process of using inductive systems in knowledge acquisition [8] for the application of different civil engineering purposes, improving the understanding of a given domain through the systematic development of a system of decision rules governing that domain [9]. Problems that share the same domain are among the most common potential applications of trained Artificial Neuronal Network (ANN) models [10]. Offering one substantial difference from conventional iterative processes for searching for potential solutions. Because solutions emerge from local rules, exploring new outcomes relies on the generation of new global results [11].

These processes follow a common strategy: (1) the generation of a geometry-based dataset after optimization or physical simulation process; (2) architecture definition and training of an Artificial Neural Network (ANN); (3) prediction of the output value; (4) reconstruction of the global geometry.

Because Linear Regression (LR) models can only predict a specific value with a linear relationship between the features and the target, the ANN models require the target values to be continuous from an interval. Because Deep Learning algorithms work like the human brain neurons, it consists of an Input layer, Hidden layers and an Output layer, which can learn the complex relationship between the features and target due to the presence of activation function in each layer [12]. For this, a Forward propagation process for multiplying weights of each feature by adding them and a Backward propagation for updating the weights–requiring optimization and loss functions in the model- are enabled.

2 Research Objectives, Methods, and Processes

This research aims to build a clear workflow to predict an auxetic structure three-dimensional deformation pattern from a series of given properties as inputs to reconstruct the structure computationally. For this, a dataset generation is required from a base geometry with a defined set of rules after being subjected to a simulation of physical deformation activated with a vertical force in an active bending process. A Particle-Spring (PS) physics simulator is used to study the morphological modification of the deformed structure by measuring its geometrical features and then exported as a value dataset. Due to the high computational power required to simulate complex or large geometries, a trained ANN model with the dataset is used to predict, bypassing the physics simulator, and reducing computational power and time considerably. To achieve the research follows a workflow composed of five steps:

  • Computational simulation of the mechanical and physical behavior of an activated auxetic surface.

  • Parametric workflow for the generation of datasets of material features.

  • The construction and training of a regression Deep Neuronal Network (DNN).

  • Prediction of a specific feature of the geometry (output_Y) from given features values (input_X).

  • Reconstruction of the material system geometry of the predicted structure based on feature inputs.

2.1 Generative Physics Simulation

A ten-by-ten cell of 1 by 1 unit auxetic structure geometry was used as basic geometry. The edges of each cell work like pivots that moves towards its centre point at a relative distance between its centre point and the global force point. This movement gives the auxetic properties to the structure. If a cell centre point is closer to the vector force, the displacement is bigger, increasing its flexibility; on the contrary, the greater the distance, the displacement value is lower increasing its stiffness (Fig. 2).

Fig. 2.
figure 2

Source By the authors.

Auxetic structure, cell, force, and pivot displacement.

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The auxetic structure was input as a rigid body in a PS physics simulator. Each of the four vertices of the structure served as anchor points. The vector force was the force that randomly changed its position and amplitude, generating different outcomes from the simulator.

After applying a horizontal force to the global surface, the pivot point of each cell was deformed and displaced, generating a vector pointing to the interior of each cell. Finally, vector normals were reoriented to each cell, and its length was used to rebuild the surface in two dimensions. That length becomes extremely important in this process; the surface can be rebuilt in two dimensions to generate a specific three-dimensional deformation and simplify the potential manufacturing process.

2.2 Dataset Generation

The simulated geometry in the PS simulator was analyzed to extract the values of each geometric feature (Fig. 4), the six global features [13]: the osculating point and the Gaussian curvature of the global deformation, the start and endpoints from the U and V coordinates, and the displacement distance of the pivot point (Fig. 3). Also, the local cell data was extracted: the centroid position in x, y and z, U and V nodes coordinates, and each pivot point displacement distance.

Fig. 3.
figure 3

Source From the authors.

Auxetic deformation features analysis for the generation of the dataset: osculating Point U and V coordinate start and endpoints and displacement distance.

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Fig. 4.
figure 4

Source From the authors.

a A deformed shell structure with 25 cells. b deformation patterns from each cell. c Reconstruction of the 3D pattern cell of the original deformed mesh. d, e Global Features extraction: gaussian_curvature, osculating_point, u_node1, u_node2, v_node1, v_node2, extract global x bounding box dimension, extract global y bounding box dimension, extract global z bounding box dimension. f Local Features extraction of each cell: centroid x position, centroid y position, centroid z position, distance u_node1 to reference cell, distance u_node2 to reference cell, distance v_node1 to reference cell, distance v_node2 to reference cell, pattern distance 1, pattern distance 2, pattern distance 3, pattern distance 4.

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With this data a dataset of 3.000 values per feature was generated and to define the input_X–features to feed the network–, and output _Y–feature to predict-, which is associated with the pattern_distance.

2.3 Deep Neuronal Network Architecture Model

A Principal Components Analysis (PCA) was initiated to understand the correlation between the global features of the system and the local features of each cell, in order to detect and select the input features for training the model and to generate the output feature to predict (Table 1), the value from which the structure will be reconstructed.

Table 1. Data sample of each geometrical feature. The first ten features were used as inputs and the last one `pattern_distance´ as output to predict.
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After several iterations, an ANN Model was composed of six Dense Layers with a rectified linear activation function ReLU, and one Dense layer with a sigmoid activation to predict the value. The model was trained with the dataset produced under different geometrical deformations (Fig. 5), with 100 epochs and a validation split of 20%.

Fig. 5.
figure 5

Source From the authors.

Variations of the geometrical deformation of the system were used to train the regression Model.

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2.4 Geometry Reconstruction

The predicted values of the displacement were associated to each of the four pivots of each cell in x, y and z dimensions (Fig. 6). This allowed the rebuilding of the shell structure by reversing the parametric design process and bypassing the PS simulator (Fig. 7), at the same time, the final pattern was redrawn in two dimensions for a potential 3d printing manufacturing process.

Fig. 6.
figure 6

Source from the authors.

In blue is the reconstructed two-dimensional pattern using the predicted value of pattern_distance. In red is the original surface.

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Fig. 7.
figure 7

In blue is the reconstructed three-dimensional shell-like structure from the predicted value of pattern_distance. In green is the bi-dimensional pattern for 3d printing.

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3 Conclusions

This workflow offers a novel way to optimize and reduce the computational power needed to compute the three-dimensional physical deformation of structures and to invert the design process allowing the designer to input the desired parameter retrieving the optimal solution within the AI-based design model.

The material strength can be considerably optimized through geometry by applying an integrated material intelligence workflow to develop digital prototypes, decreasing the amount of embedded required energy during the design and manufacturing processes.

The next steps could be based on the comparison and analysis of the predicted values between this research workflow with Regression models plugins and Particle-Systems simulators in a generative software. Along with testing the process with other shell structures by increasing the heights and the tridimensionality of the structures.