Shapley values and machine learning to characterize metamaterials for seismic applications

08/01/2021
by   D. Oniz, et al.
University of Houston
0

Given the damages from earthquakes, seismic isolation of critical infrastructure is vital to mitigate losses due to seismic events. A promising approach for seismic isolation systems is metamaterials-based wave barriers. Metamaterials – engineered composites – manipulate the propagation and attenuation of seismic waves. Borrowing ideas from phononic and sonic crystals, the central goal of a metamaterials-based wave barrier is to create band gaps that cover the frequencies of seismic waves. The two quantities of interest (QoIs) that characterize band-gaps are the first-frequency cutoff and the band-gap's width. Researchers often use analytical (band-gap analysis), experimental (shake table tests), and statistical (global variance) approaches to tailor the QoIs. However, these approaches are expensive and compute-intensive. So, a pressing need exists for alternative easy-to-use methods to quantify the correlation between input (design) parameters and QoIs. To quantify such a correlation, in this paper, we will use Shapley values, a technique from the cooperative game theory. In addition, we will develop machine learning models that can predict the QoIs for a given set of input (material and geometrical) parameters.

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1. Introduction and Motivation

A crucial mission for urban planners and structural designers is to protect critical infrastructure (e.g., hospitals, bridges, power grids, to name a few) against earthquakes. It is also necessary to isolate sensitive equipment, such as MRI machines, in hospitals and laboratories from the ambient vibrations [Himmel, 2005]. The key strategy often employed to protect infrastructure and delicate instruments is through effective isolation systems that attenuate incoming disturbances, such as seismic waves and human-generated vibrations (e.g., from traffic).

An emerging approach for seismic isolation is the use of metamaterials as wave barriers [Miniaci et al., 2016; Xiang et al., 2012; Huang et al., 2021]. A metamaterial is an engineered composite that derives its functionalities through geometric layout of base materials [Cui et al., 2010]. The advancement of photonic crystals for the light reflection led to the development of seismic metamaterials that can achieve band gaps, which will limit the propagation of the elastic waves within these gaps [Khelif et al., 2006]. Initially, phononic crystals were introduced in 1995 for filtering the Rayleigh waves [Brûlé et al., 2020]. The structure of this material provides a full reflection of the elastic waves because of the periodically designed lattice [Olsson et al., 2011]. This periodic arrangement makes this metamaterial able to reflect the waves depending on whether its frequency creates constructive or destructive interference with the incoming waves. If the interference is destructive, the band-gap occurs. This phenomenon is commonly referred to as the Bragg scattering [Yi and Youn, 2016]. In 2000, Liu et al. [2000] introduced the local resonance mechanism, which is also known as sonic crystals. For these crystals, the waves can be prevented by a structure consisting of spheres coated with a hard matrix. This mechanism differs from Bragg scattering as it does not require periodicity [Ungureanu et al., 2015].

The basic idea of metamaterials-based wave barriers is to create band-gaps and ensure that these band gaps cover the frequencies of the incoming waves. It is essential to note two aspects: (a) Seismic waves have impulses in the low (5 – 50 Hz) and ultra-low (less than 5 Hz) frequency ranges. (b) The size of a unit cell is (inversely) proportional to the (frequency) wavelength. Thus, it will be practical to cover the seismic frequencies using lower band-gaps, preferably under the first band-gap, whereby maximizing the size of the wave barrier. This discussion implies that the first frequency cut-off (the lower boundary frequency between attenuation and propagation of the waves) and the width of the first band-gap (the distance between acoustic and optical branches of the waves) are the quantities of interest (QoIs).111Note that the first frequency cut-off is not the same as the first natural frequency of the structure. The scientific question of importance about metamaterials-based wave barriers is: how do the input parameters (material and geometric properties) affect these quantities of interest? Said differently, which input parameters should we vary to (i) lower the first frequency cut-off, and (ii) increase the width of the first band-gap.

Many prior studies related to metamaterials-based seismic isolation systems have used analytical, experimental, or statistical approaches. Under an analytical study, the focus is to obtain analytical expressions for the dispersion relation—a mathematical relationship between frequency () and wave number () [Kittel, 1996]. Solving the resulting dispersion relation provides the band diagram—a graph between and in the first Brillouin zone. The QoIs can then be determined from the band diagram. But except for canonical problems and simple layouts (e.g., layers), obtaining analytical dispersion relations is not possible; for example, metamaterials based on local resonance (i.e., sonic crystals) do not lend to analytical expressions.

Experimental studies on metamaterials have used either ultrasonic methods [Cheeke, 2017] or shake table testing [Xiang et al., 2012]. But both these approaches suffer from drawbacks. The former methods are unsuitable for seismic related studies, because of the limiting operating frequency ranges of the current ultrasonic instruments are covering the frequency ranges for ultrasonic waves only. There is no commercially available instrument capable of producing waves in the low and ultra-low frequency ranges. Next, a shake table test is expensive and labor intense. In addition, such a study needs special equipment (i.e., a shake table) and sophisticated instrumentation, available only in few universities and laboratories across the world (e.g., Pacific Earthquake Engineering Research, PEER, center [Ranf et al., 2009]).

Under statistical methods, researchers have used parameter studies and global sensitivity analysis; both these methods have drawbacks. The naïve parametric approach—varying one input variable while keeping others constant—can not capture the interactions among the input variables. In contrast, global variance-based methods can account for such interactions [Saltelli et al., 2008]. Recently, Witarto et al. [2019] have used Sobol analysis—a popular method from the global analysis of variance—for phononic metamaterials to rank the input parameters affecting the QoIs. Although this approach accurately ranked the dominant parameters, a significant drawback was the need to perform numerous realizations (100000) for the method to produce accurate results. So, this method is computationally prohibitive to carry out those many realizations, especially for metamaterials based on sonic crystals.

Thus, these drawbacks give rise to two specific needs:

  1. a simple procedure to quantify the effect of input parameters on the QoIs—the first frequency cut-off and the width of the first band-gap, and

  2. a robust method to predict QoIs in an economical and time-efficient way.

The aim of this paper is to address the said needs. Our innovation is to bring ideas from cooperative game theory and machine learning.

Our approach to address the first need is to pose the relationship between the input variables and QoIs as a cooperative (mathematical) game. We then use the Shapley values—a technique from game theory—to calculate each input parameter’s contribution towards each QoI, thereby ranking the importance of the input parameters. Unlike the Sobol analysis, this approach does not need a large number of realizations when we have small number of parameters [Narayanam and Narahari, 2010; Jia et al., 2019]. To address the second need, we utilize machine learning algorithms to create a regression model for the QoIs. Once trained with sufficient data, the model can predict the QoIs for a different input data-set with a fraction of the time compared to the original analysis model. Bereft of these methods and knowledge, designing effective and economic metamaterials-based seismic isolation systems will remain unattainable.

The structure of the rest of this article is as follows. We will first provide preliminaries on phononic and sonic metamaterials and band-gap analysis of these metamaterials (§2). This discussion will be followed by the presentation of a sensitivity analysis framework using Shapley values (§3). The performance of the proposed framework will then be illustrated using numerical examples and exploratory data analysis for phononic crystals (§4) and sonic crystals (§5). After this, machine learning models will be presented to predict the QoIs (§6). Finally, we will draw conclusions along with a discussion on possible future research directions (§7).

2. Preliminaries: Metamaterials and Band-Gap Analysis

A metamaterial is a synthetic composite material system that is built to either imitate a response that is present in the nature or realize a new functionality [Engheta and Ziolkowski, 2006]. Metamaterials are often designed to either achieve a distinct material property or tailor the response of the material system to an external stimuli. To provide a few examples of the former case, researchers have developed optical materials with negative refractive index [Smith et al., 2004], and materials with negative Poisson’s ratio [Lakes, 1987; Babaee et al., 2013]. However, this paper addresses the other aspect of metamaterials—tailoring the response of a material system.

The central theme of this paper is manipulating the propagation of mechanical waves (such as seismic disturbances). Through an appropriate geometrical placement of the base materials, metamaterials can manipulate the propagation of waves through them. Depending on the material architecture, the waves can change direction, their intensity can diminish, or waves of certain frequencies will not be able to transmit at all: the resulting material system possesses band-gaps. For our purposes, a band-gap for a material system is a contiguous range of frequencies such that a wave within this frequency range will not be able to propagate through the system. In the parlance of condensed matter physics, it is the gap between acoustic and optical branches in the band diagram [Deymier, 2012].

There are two main mechanisms for creating band-gaps: back (or Bragg) scattering and local resonance. Metamaterials that utilize back scattering are referred to as phononic metamaterials, borrowing the name from phononic crystals [Huber, 2018]. Phononic crystals utilize periodic arrangement, as shown in Fig. 1. An incident wave upon such a material will undergo interference—a complex interaction among incident, transmitted and reflected components of the wave. This interference can be either destructive or constructive [Huber, 2018]. If the interference is destructive, the material can neutralize the amplitude of the wave, thereby creating frequency band-gaps [Deymier, 2012]. An attractive feature of a phononic metamaterial is its simple layout, which allows an analytical treatment of dispersion analysis. Using the transfer matrix method, Witarto et al. [2019] obtained the following dispersion equation—a mathematical relation or graph between the wave number and (angular) frequency —for a layered phononic metamaterial:

(2.1)

where represent the thickness of a unit cell, while , , and , respectively, denote the thickness, density, and wave speed in the th layer.

Figure 1. This figure shows the layered arrangement of base materials in a phononic metamaterial. The left figure shows a unit cell, which, in the case of a phononic crystal, is repeated infinite number of times.

On the other hand, sonic metamaterials utilize local resonance to achieve band-gaps. In a seminal paper entitled: “Locally Resonant Sonic Materials,” appeared in the journal of Science, the concept of local resonance is introduced as a way to realize band-gaps. Their key idea hinges on constructing a (sonic) crystal with a lattice constant equal to two order of magnitudes smaller than the wavelength [Liu et al., 2000]. The resulting sonic metamaterial comprised three parts. The core is made of lead which is covered with a thin layer of silicone rubber; this layer is commonly referred to as the resonator. The core and thin layer are put into a cube of epoxy, which has a stiffness between the lead and silicone rubber [Liu et al., 2000]. The layout of such a sonic crystal is shown in Fig. 2. When the resonance frequency of the resonator and the propagating seismic waves interact, a band-gap is created. The locations and widths off the frequency band-gap occurs are manipulated by tuning the parameters of the resonating (thin layer), elastic (epoxy) and inertial (core) components of the sonic metamaterial [Raghavan and Phani, 2013].

Figure 2. This figure shows the arrangement of base materials in a sonic crystal. The left depicts a unit cell, which is repeated to realize sonic crystal in two dimensions.

As mentioned above, we already know an analytical expression for dispersion relation for layered media. We therefore use this information to establish the utility and accuracy of Shapley value. After that, the proposed technique will be applied to the sonic metamaterial – for which an analytical dispersion relation does not exist.

3. A Sensitivity Analysis Framework Based on Shapley Values

Game theory is a mathematical approach often utilized in decision making, especially in situations involving social interactions. This approach is widely used in economics, politics, strategic game industry and also in fields such as engineering and computer sciences [Bhuiyan, 2018]. In the language of game theory, a situation is referred to as a ‘game’ and each party involved in the situation is referred to as a ‘player’ [Gilles, 2010]

. The study of game theory is broadly classified into cooperative and non-cooperative game theory. In a cooperative game, each player agrees to work together for achieving a desired result. On the other hand, non-cooperative game theory considers situations in which each player wants to achieve success, regardless of the other players decisions; that is, there can be competition among players. For further details on game theory, see

Osborne and Rubinstein [1994].

In a seminal paper, Shapley [1953] proposed an approach to evaluate a game by calculating the marginal contribution of each player; these marginal contributions are quantified by Shapley values [Hart, 1989]. Shapley value is helpful to determine the value of each player when there is cooperation among them [Castro et al., 2009]. When this idea is integrated into the field of metamaterials, a game—where the material properties are the players—can be constructed.

In this paper, cooperative game theory is utilized to characterize the mechanics of metamaterials, and the Shapley values will provide the relative importance of the input parameters over the design. Each of the two QoIs—decreasing the first-frequency cut-off, and increasing the first band-gap—is posed as a separate game. The input parameters are the players of the game. For each game, players’ pay-off will be calculated which provide the rankings of the parameters according to their Shapley values.

Since game theory is foreign to the field of mechanics, we will provide several examples to illustrate the main concepts and modifications used in this paper.

3.1. Illustrative examples

The first example provides an overview of cooperative game theory: introduces the notation such as a game and players. The second example demonstrates the Shapley value analysis, while the third shows the importance of satisfying the super-additive property. In mathematics, a function is said to satisfy the super-additive property if

(3.1)

where and are sets. In game theory,

denotes the characteristic function of the game. All the three games have a similar scenario but varies by minor explanatory differences.

3.1.1. Illustrating a cooperative game.

Three students—Alice (A), Bill (B) and Charlie (C)—have to write together a report of at least 70 pages within a week for a competition. Individually, these students can write 20, 27 and 35 pages within a week, respectively. If they work in groups of two: Alice and Bill can write 55 pages, Alice and Charlie can write 62 pages, and Bill and Charlie together can write 74 pages. All three can write together and accomplish 100 pages in a week. Then the decision to make is: what is the most economical way to meet the requirements for the competition—a report of at least 70 pages within a week.

Table 1 poses the mentioned scenario as a mathematical game of three players. Two combinations can meet the competition’s requirements—as given in bold: they all can work together to produce a 100-page report, or only Bill and Charlie can work together to produce a 74-page report. Thus, the economical option—to minimize labor—is for Bill and Charlie to work in a group of two.

This example perfectly illustrates the basic concept of cooperative game theory. In this example, number of written pages has no utility, where the only important thing is to qualify for the contest by writing at least 70 pages.

Combination A B C A&B A&C B&C A&B&C
Pages written 20 27 35 55 62 74 100
Table 1. A cooperative game of writing a report.

3.1.2. Illustrating Shapley value analysis.

An additional requirement for the competition was defined later as minimum three participants are required per group. Therefore, even though it is not the most economical combination, Alice, Bill and Charlie joined the competition all together. In the end, they won the competition and got the prize of 1000 dollars. They want to split the prize depending on the individual contribution of each. This contribution will be determined only by the number of pages written by that individual, not by the importance of what was written. The Shapley value analysis is shown in Table 2. In this table, the writing order A-B-C represents a scenario where Alice writes her share first (20 pages), followed by Bill until their combined page number (55 pages) is reached. Charlie writes last until 100-page report is done. After calculating the Shapley value—average marginal contribution—of each, multiplying the dominance percentage with the prize—1000 dollars—will give the prize each should take.

Writing order A B C
A-B-C 20 35 45
A-C-B 20 38 42
B-A-C 28 27 45
B-C-A 26 27 47
C-A-B 27 38 35
C-B-A 26 39 35
Total 147 204 249
Shapley value (in pages) 24.5 34 41.5
Dominance (in %) 24.5 34 41.5
Money award (in $) 245 340 415
Table 2. Shapley value analysis for the cooperative game of sharing the $1000 prize.

3.1.3. Illustrating super-additive property

This example considers human interactions. Just as in the original scenario, the students individually can write 20, 27, and 35 pages in a week, respectively. However, Alice does not work well in a group—she slows down the process by effecting others. When she pairs with Bill, both can write only write 10 pages in a week. Alice and Charlie together can write 17 in a week. On the other hand, Bill and Charlie work great together and can write up to 74 pages within a week. When all three students work together, they can complete 70 pages in a week. Table 3 summarizes this scenario.

In this scenario, when Alice is involved into a combination, it does not satisfy super-additive property.

Combination A B C A&B A&C B&C A&B&C
Pages written 20 27 35 10 17 74 70
Table 3. A cooperative game of writing a report without satisfying super-additive property.

For this competition, they have two ways to achieve their goal—working all together or working without Alice. If they decided to work all together and won the same prize, they again decide to split it according to their Shapley values. The Shapley analysis is given in Table 4.

Writing order A B C
A-B-C 20 -10 60
A-C-B 20 53 -3
B-A-C -17 27 60
B-C-A -4 27 47
C-A-B -18 53 35
C-B-A -4 39 35
Total -3 189 234
Shapley value (in pages) -0.5 31.5 39
Dominance (in %) -0.71 45 55.71
Money award (in $) -7.1 450 557.1
Table 4. Shapley value analysis for the cooperative game of sharing the $1000 prize without satisfying super-additive property.

As it can be seen from the Table 4, even if Alice contributed and they won, she has to give back 7.1 dollars to others and get no prize after all. In addition to this, the contribution of Alice is negative. Therefore, it makes it difficult to rank the importance of these students as they are effecting the result the opposite way. There is a chance that Alice may be effecting it in a more negative way than Charlie does in a positive way.

When super-additive property is not satisfied, data can be modified to get clearer results. Modifying the data-set can help students to determine who contributed the most and how they should split the prize. When combination of A and B is less than either A or B, the maximum value of these two should be taken as the new combination. Table 5 explains the modifying method, while Table 6 shows the Shapley value analysis with the modified data.

Combination A B C A&B A&C B&C A&B&C
Pages written 20 27 35 27 35 74 74
Table 5. A cooperative game of writing a report with modified data-set.
Writing order A B C
A-B-C 20 7 47
A-C-B 20 39 15
B-A-C 0 27 47
B-C-A 0 27 47
C-A-B 0 39 35
C-B-A 0 39 35
Total 40 178 226
Shapley value (in pages) 6.67 29.67 37.67
Dominance (in %) 9.01 40.09 50.91
Money award (in $) 90 401 509
Table 6. Shapley value analysis for the cooperative game of sharing the $1000 prize with modified data-set.

Modifying the calculations—mainly focused on Alice’s contribution to achieve the goal instead of her negative effect—resulted in her having a positive effect. Therefore, she is able to get some of the award for her work or at least making the team qualify for the contest. However, as the focus is on the positive effect for this game, when compared with others her contribution is considered as less valuable.

For the case where the qualification is changed to at least two people for the competition, Bill and Charlie chose to work together without Alice. By doing so they can share the award based on their contribution. Table 7 below explains the distribution of the award between Bill and Charlie. As it can be seen from the table, by neglecting Alice, Bill and Charlie can write more pages and gain much more money from the award. Therefore, if applicable, in some cases working with less parameters can lead to more economical results.

Writing order B C
B-C 27 47
C-B 39 35
Total 66 82
Shapley value (in pages) 33 41
Dominance (in %) 44.59 55.41
Money award (in $) 446 554
Table 7. Shapley value analysis for the cooperative game of sharing the $1000 prize without Alice.

In conclusion, when the super-additive property is not satisfied, there are two possible solutions available to overcome this problem, (i) modifying the data-set, (ii) neglecting the negative parameters. When a parameter has no positive effect both individually or within a combination, modification method used in this paper fully neglects that parameter.

3.2. Application on a simple continuous model

Till now we have applied the Shapley value analysis on discrete problems. We now consider a continuous model and demonstrate how to conduct Shapley value analysis on such a problem.

The mathematical model is given by Eq.  (3.2). The range for the two input parameters and is taken as . We layout a grid of . At each grid point, we perform a Shapley value analysis and quantify the relative importance of the input parameters – expressed in percentage.

(3.2)

Figure 3 shows the dominance of by plotting the parameter’s Shapley in percentage. The red color shows the regions for which the contribution of is greater than 50%, indicating that is the dominant parameter. While blue color denotes the regions for which parameter dominates, and the white color indicates the transition between the parameters. The predictions from the Shapley value analysis match with the analytical solution. However, the Shapley value analysis can be performed even for those problems without analytical solutions. This is often the case with material design such as creating seismic wave barriers using metamaterials. In such cases, graphs similar to Fig. 3 provide a lot of information to a design engineer in navigating through the design space.

For example, the Eq.  (3.2) represents a material design with two materials, where is density ratio and is thickness ratio between two materials. In a case where we have a particular type of steel and rubber available, changing density ratio will require purchase of new materials, which is not economical. If we have the information represented on Fig. 3, we can determine the ranges where the thickness ratio can govern the design. Let us say that the density ratio available is 4. Keeping the thickness ratio higher than 7.5 will give us the advantage of having the dominant parameter as thickness and alter the design by changing it as desired. To conclude, having the dominant parameter obtained from Shapley value helped us determine how to change the design characteristics as wanted in an economical and time-efficient way.

Figure 3. This figure presents the Shapley value analysis results for the mathematical model (3.2). The graph plots the dominance in percentage. Thus, in the regions with lower percentages [], the parameter is dominant.

4. Exploratory Data Analysis (Eda) for Bragg Scattering

4.1. Material properties and design of the phononic crystal

The material to be used in this paper is a 1D metamaterial with two materials. Figure 4 provides information about the arrangement of the phononic crystal. The effective parameters of design are taken as modulus of elasticity, density and thickness ratios. These ratios are taken as , while being the characteristic of that layer—Young’s modulus, density, and thickness. Poisson’s ratio is not varied as Witarto et al. [2019] showed that its effect on the design is negligible by using Sobol analysis.

Figure 4. This figure shows the arrangement of the base materials in the phononic metamaterial for the exploratory data analysis. The base material for layer 2 is rubber; only the geometrical properties of layer 2 are changed, while (dimensionless) unit cell width is fixed at 1. The properties of layer 1 are varied by altering the ratios. The changing parameters for the design are Young’s modulus ratio, density ratio, and thickness ratio. These ratios are calculated as for a given property .

For this metamaterial, properties of second material, presented by layer 2, are set same as the characteristics of rubber, where its Young’s modulus is equal to 3.49 MPa and its density is equal to 1100 . For the second material, only geometrical properties are changing during the analysis. However, the dimensionless thickness of unit cell, which is , is set to be 1. For the first material, represented by layer 1, properties are changed to see how they are affecting the design. The base value of all parameter ratios is taken as 0.1 for Shapley value analysis.

For the design, the main goals are to achieve a lower first frequency cut-off point to cover ultra-low frequency ranges and to achieve a larger band-gap width to cover more frequencies of seismic waves. Therefore, in this paper, if certain parameters are referred as negative, it means those parameters are causing to have higher first frequency cut-off point or smaller band-gap widths, which are not desired.

4.2. Data set

The used data is obtained by a Python code which was written for the analysis. This code first applies dispersion relation calculations to calculate the quantities of interest and then applies the Shapley value analysis to get the dominant parameter among the three input ratios. The data sets will be modified and discussed accordingly to determine the dominant parameters when decreasing first frequency cut-off point and when increasing the first band-gap width, separately. For this analysis, data set consist of the given ranges of the parameters, in Table 8.

Parameter Range
Young’s modulus ratio 0.1–50000
Density ratio 0.1–9.5
Thickness ratio 0.1–11
Table 8. Parameter ranges for Shapley value analysis of phononic crystals.

4.2.1. Visualizing labeled data for first frequency cut-off point.

The data set is modified for decreasing the first frequency cut-off point. The results of Shapley value analysis for decreasing the first frequency cut-off point is visualized by using 3D colormaps, as given in Fig. 5. Parameters are shown in terms of ratios.

As the data modification neglects a parameter when no positive effect of the parameter is present, the marginal contribution for Young’s modulus is obtained as zero for the given ranges. Therefore, there is no tertiary parameter for the design. In brief, increasing Young’s modulus, by itself or with other parameters, only increases the first frequency cut-off point instead of decreasing. The dominant and secondary parameters of the design can be seen in Fig. 5. The lack of data when both density and thickness ratios are equal to 1 is due to the lack of band-gap generation. In these graphs, the yellow and blue colors represent the thickness and the density ratios, respectively. For the lower ranges of thickness ratio [] and density ratio [], the thickness ratio appears as the dominant parameter to decrease the first frequency cut-off point. Above these ranges, the density ratio starts to govern the design while widening its dominance range as it increases.

Figure 5. Shapley value analysis for a phononic metamaterial. This figure presents the parameter ranking for the first QoI—decreasing the first frequency cut-off point. The dominant and secondary parameters are shown in subfigures (a) and (b), respectively. For this QoI, elasticity modulus has no effect over the design. For the lower ranges of thickness ratio [] and density ratio [], thickness ratio governs the design. Above this range, density ratio is the dominant parameter. However, this dominance zone of thickness ratio reduces as the density ratio increases.

4.2.2. Visualizing labeled data for width of first band-gap.

The data set is modified for increasing the first band-gap width. The results obtained for first band-gap width is visualized by using a colormap as shown in Fig. 6. For all the graphs, parameters are given in terms of ratios.

Figure 6. Shapley value analysis for a phononic metamaterial. This figure illustrates the ranking of parameters for the second QoI—increasing the band-gap width. For phononic crystal, no band-gap data is obtained when both density and thickness ratios are equal to 1. For the dominant parameter (subfigure a), for the lower ranges of thickness ratio [], thickness ratio governs the design. Above these ranges, thickness and Young’s modulus ratios have the same dominance over the design. For the secondary parameter (subfigure b), when thickness ratio is equal to 1, the other two parameters become ineffective. However, for lower ranges of thickness ratio [], density and elasticity modulus ratios have the same effect over the design. For intermediate ranges [2.5–10], elasticity modulus is the second dominant parameter. Above this point, density becomes the secondary parameter, as other two parameters are both dominant with same Shapley value.

For the colormaps for parameter rankings, each color represents a different parameter. While dark blue means no parameter is dominant for the range, modulus of elasticity ratio, density ratio and thickness ratio are blue, light blue and green, respectively. For the ranges there are two parameters having the same Shapley value, yellow color is used. As it can be seen from the Fig. 6a, thickness ratio is the dominant parameter that governs the design for lower ranges of the thickness ratio []. Above this range, both elasticity modulus and thickness ratios have the same marginal contribution to the design. The lack of dominant parameter, when density and thickness ratios are both equal to 1, is due to the lack of band-gap generation. As given in Fig. 6b, there is no secondary parameter, both for when thickness ratio is equal or less than 1 and for lower ranges of density ratio []. Above these ranges, both density and elasticity modulus ratios have the same effect over the design, until thickness ratio reaches 3. Above this point, Young’s modulus starts to govern (blue in Fig, 6b), until it becomes the dominant parameter along with thickness ratio (yellow in Fig. 6a).

4.3. Discussion on EDA for Bragg Scattering.

In this section, dispersion relation and Shapley value analysis were used to understand the behavior of the phononic metamaterial. Firstly, band-gap characteristics of the crystal were found using the dispersion relation for various ranges and combinations of the design parameters. After obtaining these QoIs, Shapley value analysis is applied to determine the dominant parameter that could alter the design in the desired way, time-efficiently and economically.

For the Shapley value analysis, the modification changes for the QoIs, as lower first frequency cut-off point and larger band gap width are the ultimate goals. Therefore, Shapley value analysis is applied twice; one for increasing the QoIs and one for decreasing them. As a result, the dominant parameters are found separately for each case.

Main findings of this exploratory data analysis on Bragg scattering data set are as follows. To decrease the first frequency cut-off point, Young’s modulus ratio has no effect over the data set. For lower ranges of density [] and thickness ratios [], the dominant parameter is thickness ratio. However, as these parameters get higher, density ratio starts to govern the design. Therefore, to achieve a lower first frequency cut-off point: (i) Young’s modulus ratio should be kept same or in lower ranges as much as the design allows, (ii) When designing with lower density and lower thickness ratios, increasing thickness ratio can yield a lower first frequency cut-off point, (iii) For the higher ranges of both thickness and density ratios, increasing density ratio will give better results.   To increase the width of first band-gap, thickness ratio is the only effective parameter for low ranges of both density [] and thickness ratios []. For intermediate ranges [], the dominant parameter is thickness ratio. However, as thickness ratio increases [], both Young’s modulus and thickness ratios have the same dominance over the design. Therefore, to achieve a larger band-gap: (i) When working with low ranges of thickness [] and density ratios [], larger band-gap can be achieved, if only thickness of layer 1 is increased. Increasing other two parameters, within these ranges, will only decrease the width, (ii) Increasing the thickness ratio will also increase the band-gap for higher ranges up to the point thickness ratio is higher than 9.5. After this point, Young’s modulus ratio can also be increased.

4.3.1. Comparison of results with Sobol analysis

In prior studies, Witarto et al. [2019] discussed the dominant and effective parameters on the design of phononic crystals. Sobol analysis was used to obtain information on the crystal behavior between the given ranges in Table 9. In this table Poisson’s ratio range in the prior study is not included as it was not taken into account in this paper. The parameter ranges used for Sobol analysis falls into our observation range. However, the range used for Shapley value analysis uses a larger end-value for Young’s modulus and lower end-value for density ratio. Additionally, one of the important differences between Sobol and Shapley value analysis is that Sobol analysis also captures the interaction between the parameters, which was not captured by Shapley value analysis.

Parameter Range
Young’s modulus ratio 10–10000
Density ratio 1–1000
Thickness ratio 0.11–9
Table 9. Parameter ranges for Sobol analysis from Witarto et al. [2019].

After obtaining results with Sobol analysis, Witarto et al. [2019] presents the following. For the first frequency cut-off point, the predominant parameter is found to be density ratio, followed by density & thickness ratio interaction and thickness ratio, respectively. Additionally for P-waves, Poisson’s ratio and its interaction with other parameters were also found to be effective on the design.   For the width of first band-gap, thickness ratio is found to be the dominant parameter over the design. Secondary parameter is the interaction between Young’s modulus and thickness ratio. The tertiary parameter is the interaction between density and thickness ratios and it is followed by the Young’s modulus ratio and density ratio, separately. Additionally for P-waves, Poisson’s ratio was found to be effective on the design.

The comparison between these result from Sobol analysis and our results on Shapley value can be done as follows. For the first frequency cut-off point: (i) From the Shapley results, density ratio is obtained as the dominant parameter for higher ranges. As the Sobol analysis goes beyond our range for the density ratio values, it is possible that density continues to govern for most of the values within the range. Therefore, density being the dominant parameter for Sobol analysis can be explained, as the sensitivity analysis does not determine if there is a transition between dominant parameters. (ii) Shapley value also determines that Young’s modulus ratio has no effect over lowering the first frequency cut-off point, which also does not appear as an effective parameter with the Sobol analysis.   For the width of first band-gap, (i) Thickness ratio is obtained as the dominant ratio with Sobol analysis. When the observation range of the Sobol analysis is considered, it directly fits into the ranges that the thickness ratio governs the design by itself in the Shapley value results. (ii) When interaction is not considered, secondary and tertiary parameters predicted by Sobol analysis align perfectly with Shapley value analysis results.

5. Exploratory Data Analysis (Eda) for Local Resonance

5.1. Material properties and design of the sonic crystal

The sonic crystal built for this part is a 2D model with three materials. Figure 7 shows the arrangement of the base materials in the metamaterial.

Figure 7. This figure shows the arrangement of base materials for During the Shapley value analysis, only properties of layer 1 are varied. The design parameters are taken as , where is the property of material—such as density, Young’s modulus and thickness.

For the sonic crystal model, second materials properties are set to be same as rubber. The values of rubber are given in §4. For layer 2, only the geometrical properties are changing along with thickness ratio. For the third layer, the properties of epoxy are used. For epoxy, Young’s modulus is taken as 3 GPa, while density is taken as 1200 . For the unit cell, the dimensionless thickness of epoxy layer is equal to 1. For the layer 1, which is the inner core of the sonic crystal, the properties will be changed to see how it affects the design. The effective parameters for sonic crystal is taken as Young’s modulus ratio, density ratio and thickness ratio. The ratios are taken in terms of , where is the referred characteristic. The changes of layer 3 properties are not involved in Shapley value analysis.

For the design, the main goal is same as Bragg scattering model, achieving lower first frequency cut-off point and larger first band-gap. For the data set, the base value is taken as when all parameters are equal to 1. Therefore, at the base, there is no band-gap generation.

5.2. Data set

The data is obtained by eigen-frequency study via COMSOL models. After obtaining the corresponding QoIs for the parameters, the Python code for Shapley value analysis is used. This Python code also modifies the data either for decreasing first frequency cut-off point or increasing first band-gap width, separately. For this analysis, parameters are examined within the given ranges in Table 10. For the elasticity modulus ratio axis, all graphs are plotted in logarithmic scale.

Parameter Range
Young’s modulus ratio 1–100000
Density ratio 1–10
Thickness ratio 1–10
Table 10. Parameter ranges for Shapley value analysis of sonic crystals.

5.2.1. Visualizing the labeled data for decreasing first frequency cut-off point.

For this data set, the Shapley data is modified for decreasing the first frequency cut-off point. For this data set, only effective parameter is found to be density ratio, in Fig. 8. The density ratio has dominance percentage of 100%. In brief, increasing the density of layer 1 is the only effective parameter to decrease the first frequency cut-off point. Increasing Young’s modulus and thickness ratios will increase the first frequency cut-off point.

Figure 8. Shapley value analysis for a sonic metamaterial. This figure presents the parameter ranking for the first QoI—decreasing the first frequency cut-off point. Over a wide range for the input parameters, the dominant parameter is the density ratio. Other two parameters are not found to be effective.

5.2.2. Visualizing the labeled data for increasing the first band-gap width.

For this data set, the Shapley value analysis is done by modifying the data for increasing first band-gap width. The results are visualized by 3D heat maps. For all ranges, the dominant parameter for increasing the first frequency band-gap width is the Young’s modulus. The blue color represents the Young’s modulus in Fig. 9a. For the secondary parameter, in Fig. 9b, the yellow points represents the thickness ratio dominance, while dark blue is density ratio.

Figure 9. Shapley value analysis for a sonic metamaterial. This figure shows the ranking of effective parameters for the second QoI—the first band-gap width. As shown in (a), the dominant parameter is the Young’s modulus ratio for all ranges. For secondary parameter, given in (b), thickness ratio governs the design for most of the ranges. This dominance is replaced by density ratio, when density ratio is higher [] and thickness ratio is equal to 2.

5.3. Discussion on EDA for local resonance

In this section, COMSOL software models were used, as dispersion relationship is not known for sonic crystals, followed by Shapley value analysis. Firstly, the COMSOL models were built and band structure of the sonic crystal is obtained for various design parameters. When the data set is complete, Shapley value analysis was applied to determine the dominant parameter of the design.

For the sonic crystals, Shapley value analysis is applied twice as it was done for Bragg scattering. Main results obtained from visualization and Shapley value analysis are as follows. To decrease the first frequency cut-off point, density ratio is the only effective parameter for the design. Therefore to achieve a lower first frequency cut-off point, increasing density ratio is the only effective approach as increasing Young’s modulus and thickness ratios tend to increase the first frequency cut-off point. These two ratios should be kept at minimum as much as permitted by design. To increase the width of band-gap, Young’s modulus is the dominant parameter for the design for all ranges followed by thickness and density ratios for different ranges. Density ratio only governs when density ratio is in higher ranges and thickness ratio is in lower ranges. To achieve a larger band-gap, changing any parameter would result in a larger band-gap, even though the most effective approach is to make the inner core stiffer.

6. Machine Learning and Deep Learning Models

Supervised learning is training of the model where it becomes able to map the outputs for a newly introduced data. This approach is highly used for regression and classification problems, where the outcome or class of the data is originally known and desired to be found for a new data point. However, clustering problems can be solved by unsupervised learning where model seeks for a similarity between points and divides into groups [Cunningham et al., 2008]. In this paper, as the outputs are known, supervised learning will be used as the common approach for machine learning algorithms.

To train the models with supervised learning, several machine and deep learning algorithms are widely used. Machine learning models consist of one layer of multiple neurons while deep learning models consist of multiple layers each with multiple neurons. Due to its complex and bigger modelling nature, the neural networks can provide accurate results without being time-consuming while using on large and complex data sets

[Bonaccorso, 2017].

In this paper, several models are built for the prediction of QoI values both for phononic and sonic crystals. For the regression models, the used modeling approaches are: (a) neural network, (b) linear regression, and (c) random forest regressor.

For regression, the neural network is built for comparison of deep learning models with machine learning models. Linear regression is a simple and common approach for approximation of output values by introducing a function that could represent the behavior of the input data [Vasilev et al., 2019]. This function could be linear, quadratic or even from an higher degree. Finally, the random forest regression algorithm constructs ndecision trees with bootstrap samples and takes average of the results of each to predict the new data [Segal, 2004].

In the following subsections, these models will be constructed for Bragg scattering and local resonance data sets. To evaluate the final models that were built for same material, both root mean-squared error (RMSE) and scores are calculated for each model. The root mean-squared error is a widely used evaluation method for regression models. It can be calculated as in Eq. 6.1 [Chai and Draxler, 2014]. This error indicates how closely spread the predicted values are compared to the ground truth. score, also referred as coefficient of determination, is also a common and effective approach for the evaluation of the regression models. The Eq. 6.2 shows its mathematical expression [Renaud and Victoria-Feser, 2010]. While having an score of 0 means that the model is same as the mean model, where the mean of the data set is predicted for all, having scores closer to 1 indicates a better fit. The results for each model will be provided and the hyper-parameter tuning will be done for the best model.

(6.1)

where y is the ground truth and is the predicted value. n is the number of samples.

(6.2)

where is the mean of ground truth.

6.1. Bragg Scattering

The data used for this project is consists of one set containing the information on the following Table 11, which will be used for regression models. It has the parameter ratios as input, while the output is the quantities of interest obtained from the dispersion relation. This data set consists of 2269 data points.

Parameter Quantity of Interest
E ratio ratio h ratio
First frequency
cut-off point
First band-gap
0.1–50000 0.1–9.5 0.1–11
From
dispersion
relation
From
dispersion
relation
Table 11. Data set for Bragg scattering models.

For the neural networks, the input values within the data set—parameter ratios—are scaled using Scikit-learn for each application [Buitinck et al., 2013]. Since the random forest regressor is a tree-based algorithm, there is no need to normalize the data. This aspect was evident even in our numerical simulations—the results were the same with and without normalized data for the random forest regressor. We also observed that the linear regression gave the same results with and without normalization of the data, although linear regression is not graph-based. For all the models, the data set is divided into train/validation/test subsets. For the reproducibility of the results, the randomness is fixed for all the models. The same seed number is used for the random forest regressor and linear regression when dividing the data sets. For the neural network, the optimum seed number is used.

6.1.1. Regression Models

As the data generation by building a model is not time-efficient for metamaterials that lack the dispersion relationship, there is need for a machine learning model that can predict the corresponding quantities of interest. To be able to see the efficiency of possible models with different type of data set, it is also used for Bragg scattering. Models that are built to act as a regressor are discussed with their efficiencies, based on the RMSE value and score.

Two neural networks

both with two hidden layers are built for the regression, using ReLU as the activation function. First neural network predicts the first frequency cut-off point, while second predicts the width of first band-gap. These models are separated in order to perform hyper-parameter tuning and accuracy comparison with other models. For these neural networks, optimizer is selected as Adam while loss function is selected as mean squared error loss. For both models, the training/test subsets are divided as 0.8/0.2 of the whole batch. The validation set is determined by further dividing of the training set by 0.2. For the

first frequency cut-off point prediction

model, the learning rate is selected as 0.0025, with number of epochs set to 10000. Additionally, weight decay is introduced as 0.3. The loss history for the training and validation subsets can be seen in

Fig. 10a. Loss history graph of the neural network indicates over-fitting. This losses are given in terms of the MSE loss. The final testing RMSE value is obtained as 10.49, while the score is 0.9971. For the width of the band-gap prediction model, the second neural network, the learning rate is set to be 0.0008, while number of epochs is 25000. For this model, weight decay is set to be 0.5. The loss history of training and validation sets is shown in Fig. 10b, which shows over-fitting. For the testing subset, the RMSE and scores are obtained as 14.93 and 0.9993, respectively.

Figure 10. Phononic metamaterial. This figure shows the training and validation loss history. The loss is given in terms of MSE loss. Both graphs indicate over-fitting. Due to the large MSE losses for lower epochs, both graphs are plotted for lower ranges of the loss.

For the linear regression models, two models were built separately for QoIs. The models are trained by adding polynomial features. For the first frequency cut-off point prediction model, the optimum model is obtained with third degree polynomials and their interactions being introduced. The training and validation losses are 81.21 and 81,66, respectively, in terms of RMSE loss. The testing subset has its RMSE value as 76.45, while the score is 0.8318. The model does not fit the data properly. For the width of band-gap prediction model, the polynomial degree is selected as four. The training loss is 131.49, while the validation loss is 135.42. This indicates over-fitting within the model. The RMSE of the testing subset is 129.42 and its score is 0.9550. This model fits the data set better than the first frequency cut-off point. However, when compared to other models, it provides poor results.

For the random forest regressor

models, hyper-parameter tuning for both models were done by cross validation. For both QoIs, models were built separately and tuned for maximum tree depth and number of estimators. For the

first frequency cut-off point prediction model, the changes in loss is plotted for the tuned hyper-parameters in Fig. 11a-b. The testing RMSE vlaue is obtained as 4.16 with a score of 0.9996. These values are obtained with the optimum model, where maximum tree depth is taken as 10 and number of estimators is selected as 800. For the width of band-gap prediction model, the hyper-parameter tuning is also done with the same parameters. The graphs for the tuning are given in Fig. 11c-d. The testing subset yields the RMSE as 12.75 and score as 0.9996. For the optimum model, maximum tree depth is taken as 10 and number of estimators is selected to be 1100.

Figure 11. Phononic metamaterial. This figure shows the hyper-parameter tuning of the random forest regressor algorithm. For the first frequency cut-off point, the optimum maximum tree depth is 10, whereas the optimum number of trees is 800; see subfigures (a) and (b). For the first band-gap width, the optimal maximum tree depth and number of trees are 10 and 1100, respectively; see subfigures (c) and (d).

6.2. Local resonance

The data set for local resonance is constructed by COMSOL models. The information on this data set are given in the Table 12. For this data sets, the Young’s modulus is plotted on logarithmic scale as the parameter value altered as product of ten between 1000 and 100000. The regression data set has the inputs as the parameter values, while output is the QoI values.

E ratio ratio h ratio
First frequency
cut-off point
First band-gap
1–100000 1–10 1–10
Obtained from
COMSOL model
Obtained from
COMSOL model
Table 12. Data set for local resonance models.

For the neural networks, only the input values from the data set—the parameter ratios but not the outputs (QoI values)—are scaled with the same procedure as the one used for Bragg scattering. As indicated for the Bragg scattering models, the random forest regressor and linear regression models used unnormalized data set. For all the models, the data set is divided into train/validation/test subsets. The randomness in the models for the local resonance models is fixed following the procedure used for models under the Bragg scattering.

6.2.1. Regression Models

Since the dispersion relation is not known for materials such as sonic crystals, computational models should be built to achieve it. This approach is time-consuming as the design requires many realizations to achieve the optimum result. To make this process time-efficient, the need for a faster and accurate model can be addressed by machine learning approaches. Various regression models are built for this purpose and explained with their efficiencies depending on their RMSE and score.

Two neural networks were built for prediction of QoIs, separately. For both models, ReLU is used as the activation function. For both the loss function is selected as MSE loss and the final loss is presented as the square root of MSE loss. This loss is calculated for each—both for training, validation and testing subsets. For both models, training/testing sizes are 0.8/0.2 and validation is built by dividing training again by 0.8/0.2. Both neural networks are built with two hidden layer structure. For first frequency cut-off point prediction model, number of epochs is selected as 12500. With Adam optimizer, learning rate of 0.0002 yields the optimum results, when combined with the use of whole training batch during training of the model. The training and validation loss history can be seen in Fig. 12a. The testing RMSE and score are found to be 4.57 and 0.8395, respectively. For the width of the band-gap prediction

model, number of epochs is set to be 1500. The optimizer is selected as stochastic gradient descent (SGD) with a learning rate of 0.0002. For this model, mini-batches of 64 points is used during training. The loss history of training and validation subsets can be seen in Fig. 

12b. This graph indicates over-fitting. The testing RMSE is obtained to be 17.11, while the score of the model is 0.3663. This model does not provide a good fit.

Figure 12. Sonic metamaterial. This figure shows the loss history of training and validation subsets for neural network models. In these graphs, the loss is plotted in terms of MSE loss. The graph for the width of the first band-gap (right) indicates over-fitting.

The linear regression models built separately for QoIs. Both models are trained by adding third degree polynomial features, as yielded the optimum results. The losses for subsets are calculated as square root of mean squared error. The first frequency cut-off point prediction model yields the training RMSE loss as 3.71, where validation RMSE is 3.21. For the testing subset, the RMSE value and score are 3.76 and 0.8996, respectively. For the band-gap width prediction model, the training RMSE loss is 9.57, followed by 10.2 as validation loss. The testing RMSE is obtained as 8.57. score of the model is 0.8893. Unlike the Bragg scattering model, these models does not indicate over-fitting or poor accuracy.

Two random forest regressor models were built by cross validation and hyper-parameter tuning. The hyper-parameter tuning is done for maximum tree depth and number of estimators. The changes in loss with hyper-parameter tuning are given in Fig. 13 for both models. The loss is presented in terms of the square root of MSE loss. For the first frequency cut-off point prediction model, for a model with 8 as maximum tree depth and 800 as number of estimators, testing subset yields the RMSE value as 1.65 and score as 0.9825. For the band-gap width prediction model, for a model with 12 as maximum tree depth and 1500 as number of estimators, the testing loss is found to be 3.55, while the score is 0.9827.

Figure 13. Sonic metamaterial. This figure shows the hyper-parameter tuning of the random forest regressor model. For the first frequency cut-off point, the optimum maximum tree depth is 8 whereas the optimum number of trees is 800; see subfigures (a) and (b). However, the number of trees has no significant effect on the mean squared error (MSE) loss. For the width of the first band-gap, the optimal maximum tree depth and number of trees are 12 and 1500, respectively; see subfigures (c) and (d).

6.3. Discussion on machine learning models

The main finding in this section are given separately for phononic and sonic crystals. For the regression problem, used algorithms are: (i) neural network, (ii) linear regression, and (iii) random forest regressor. The results will be provided in terms of the algorithms and also how these served on various data sets. The obtained RMSE values and scores are given in the Table 13.

Mechanism Quantity of Interest Evaluation metric Neural network Linear regression Random Forest Regressor
Bragg scattering First frequency cut-off point
RMSE
10.49 76.45 4.19
score
0.9971 0.8318 0.9996
First band-gap width
RMSE
14.93 129.42 12.75
score
0.9993 0.9550 0.9996
Local resonance First frequency cut-off point
RMSE
4.57 3.76 1.65
score
0.8395 0.8996 0.9825
First band-gap width
RMSE
17.11 8.57 3.55
score
0.3663 0.8893 0.9827
Table 13. Testing root-mean-square error (RMSE) values and scores of the regression models.

The main findings of the machine learning algorithms for layered phononic metamaterials are as follows. For the first frequency cut-off point prediction, while both neural network and random forest regressor algorithms working well on the data set, random forest yields a lower RMSE value and a faster model. On the other hand, linear regression does not provide a good predictions for this data. For the band-gap width prediction, random forest regressor and neural network both provides good accuracy, with random forest providing the best fit within a better prediction time. Linear regression provides the poorest fit.

The main findings of the machine learning algorithms for sonic metamaterials are as follows. While predicting the first frequency cut-off point, all three models yield low losses. However, based on their scores, random forest regressor provides the best model. On the other hand, linear regression and neural network models fit the data around same efficiency, with linear regression resulting in a better fit. While predicting the band-gap width, best working model is random forest regressor, and it is followed by linear regression. The score and RMSE value of neural network indicates that it provides a poor fit.

7. Summary and Concluding Remarks

This paper addresses the effect of material and geometrical properties of a metamaterial on band structure characteristics under Bragg scattering and local resonance. Given our interest in seismic wave barriers, we have considered the first frequency cut-off and first band-gap as the two quantities of interest (QoIs). The contributions of our paper are twofold. First, we have developed a sensitivity analysis method using Shapley values—a technique from cooperative game theory—for quantifying the dominance of input parameters on the QoIs. Second, we have presented machine learning tools for obtaining regression models for the QoIs.

7.1. Shapley value analysis

The main advantages of Shapley value analysis are it: (i) requires lesser amount of data to rank the input parameters compared to Sobol analysis, (ii) demarcates the transition regions between dominant parameters, and (iii) it gives information on how to alter the parameters, whether to increase or decrease. On the other hand, the main disadvantages are: (i) when compared to Sobol analysis, Shapley value analysis does not rank the interactions among the individual parameters. (ii) It does not offer regression expressions—or the so-called reduced-order models—for predicting the QoIs.

Based on a Shapley value analysis, the main findings for phononic metamaterials are as follows. To decrease the first frequency cut-off point, Young’s modulus ratio does not contribute much; increasing this ratio only causes the first frequency cut-off point to increase. Coming to the other two parameters—thickness and density ratios— in the ranges of [0.1–2] for the density ratio and [0.1–9.5] for the thickness ratio, the thickness ratio is dominant. Above these ranges, density ratio governs the design and increases its dominance range. To increase the band-gap width, for small ranges of thickness ratio [0.1–1.5], both Young’s modulus and density ratios are not effective, meaning that increasing these parameters by a small amount would decrease the width of first band-gap. For these parameters to be effective, the increase should be by a larger amount. For larger thickness ratio values, Young’s modulus starts to have the same dominance as thickness ratio. In cases where the large increase of Young’s modulus is not viable, this material property can be kept the same while introducing only geometrical changes.

The corresponding conclusions for the sonic metamaterials are as follows. To decrease the first frequency cut-off point, the only effective parameter is the density ratio with the inner core being denser; increasing the ratios for the two parameters—Young’s modulus and thickness—has a negative effect on the QoI. To increase the width of first band-gap, contrast in Young’s modulus (with the inner core being the stiffer material) is the dominant parameter for all ranges. However, on contrary to a phononic metamaterial, increasing thickness ratio () or density ratio () will also increase this QoI, but these effects are secondary compared to the Young’s modulus ratio.

These findings are valuable as they provide information on how the design parameters can be altered economically. For example, when two materials are already available to use, the design engineer can arrange the base materials within the ranges so that the geometrical properties govern the design. Thus, the approach presented herein provides a viable route to manipulate QoIs with the existing materials.

7.2. Machine learning

Three regression models—based on neural network, linear regression and random forest regressor algorithms—were built for each QoI separately for Bragg scattering and local resonance. For Bragg scattering, the best working model is the random forest regressor for both models, while linear regression yields the poorest fit. For local resonance, the best evaluation values are obtained using the random forest regressor, although all models yielded low RMSE values. For this model, neural network yielded the lowest scores for both QoIs. The chosen machine learning algorithms, except linear regression, behaved similarly on the data sets for phononic and sonic crystals. As the field of deep learning is evolving rapidly as we speak, the use of newer algorithms and computer architectures can further minimize the computational time. In future, these deep-learning-based models can serve as an economical alternative to the commonly employed numerical formulations.

A plausible future work can be towards developing a framework using deep learning tools to address the inverse problem—the so-called band engineering. Said differently, what we need is a deep learning framework for designing metamaterials (i.e., selection of base materials and geometrical properties) for a desired band structure (e.g., the first frequency cut-off and first band-gap).

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