Ali Kadhum Idrees

Raphael Couturier
email: raphael.couturier@univ-fcomte.fr

Energy-saving Distributed Monitoring based Firefly Algorithm in Wireless Sensors Networks

Keywords: Wireless Sensor Networks, Scheduling, Firefly Optimization Algorithm, Coverage, Network lifetime, Distributed Computation

Coverage control represents an important research challenge during the design of Wireless Sensor Networks (WSNs) in an energy-efficient way. It is an indicator used to assess network services performance. In order to provide network services quality guarantee, it is essential to ensure the network coverage with a minimum consumed energy to extend the lifespan of the network. In this paper, an Energy-saving Distributed Monitoring based Firefly Algorithm (EDiMoFA) Protocol in Wireless Sensor Networks is proposed to ensure the coverage and to enhance the lifetime of WSNs. In the first phase, the sensing field is divided into smaller virtual regions using the concept divide-and conquer. The EDiMoFA protocol is distributed on every node in the resulted small regions in the second phase. The EDiMoFA protocol mixes three powerful approaches to work efficiently: virtual network division, dynamic distributed virtual region head selection in every region, wireless nodes scheduling based Firefly Algorithm (FA) is performed by every chosen head of the virtual region. The EDiMoFA protocol is periodic. Every period is composed of two different phases: a steady-state phase and monitoring one. The network information exchange, virtual region head selection, and a wireless sensors scheduling optimization-based FA are achieved in the steady-state phase. In the monitoring phase, the best sensor devices schedule produced by the FA will take the responsibility for monitoring the sensing field in every virtual region. The produced sensors schedule ensures coverage at a low consumed energy cost. Simulation results, which are obtained using the OMNeT++ network simulator, prove that the EDiMoFA protocol can increase the wireless sensors' lifetime and produces enhanced coverage control performances in comparison with some recent existing works in the literature. The EDiMoFA protocol has respectively prolonged the network lifetime from 3.2% up to 21.8%, from 10.4% up to 86.4%, from 35.2% up to 68.4%, and from 1.6% up to 6.7% in comparison with the DiLCO, DESK,

GAF, and PeCO protocols while maintaining the suitable level of coverage for the sensing field of interest.

Introduction

The advances in wireless networking and smart sensor technologies have led to the emergence of wireless sensor devices which constitute the Wireless Sensor Network (WSN) that represents the most implemented element in the Internet of Things (IoT) [START_REF] Alhussaini | Data transmission protocol for reducing the energy consumption in wireless sensor networks[END_REF]. These IoT smart sensors devices can be used in several applications such as danger alarm, vehicle tracking, battlefield surveillance, habitat monitoring, healthcare, military, etc [START_REF] Harb | Wireless sensor networks: A big data source in internet of things[END_REF][START_REF] Athraa | A comprehensive review for rpl routing protocol in low power and lossy networks[END_REF]. Wireless sensors devices can be used on a large scale of IoT applications. The smart devices are capable of sensing, communicating, and processing the collected sensed data before transmitting them across the Internet to the base station for further analysis [START_REF] Kadhum | Adaptive distributed energy-saving data gathering technique for wireless sensor networks[END_REF][START_REF] Kadhum Idrees | Distributed data aggregation based modified k-means technique for energy conservation in periodic wireless sensor networks[END_REF]. The lifetime coverage maximization problem has received a lot of attention, focusing in particular on how the physical space could be well monitored after the deployment. Coverage is one of the Quality of Service (QoS) parameters in WSNs, which is highly concerned with power depletion [START_REF] Li | Monitoring area coverage optimization algorithm based on nodes perceptual mathematical model in wireless sensor networks[END_REF]. In addition to the coverage, the energy consumption and the lifespan are QoS parameters. The lifetime coverage maximization refers to how well the sensing field of interest is monitored for a longer time by the deployed sensors devices with a minimum amount of energy whilst preserving a suitable level of coverage [START_REF] Kadhum Idrees | Multiround distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF]. The mission of preserving the desired level of coverage with a minimum number of sensor devices of the whole WSN is extremely important and represents a big challenge due to the limited resources of sensor devices like power, processing, memory, and bandwidth. Sensor devices have limited lifetime batteries which are often quite difficult to change, especially in a harsh environment such as underwater, deep forests, and so on. Therefore, the sensing field of interest must be deployed with a large number of sensor devices to exploit the redundant sensor devices and to improve the WSN lifespan. It is not essential to turn on the all sensor devices simultaneously in this high-density network. If all the sensor devices are turned on at the same time, the energy consumption would be quickly reduced which would thus decrease the lifespan of the network. Therefore, it is required to implement a scheduling technique to provide the schedule for the sensor devices by turning on some of them to provide the sensing services while turning off the other devices to save the energy thus extending the network lifetime of the WSN. The basic idea is to benefit from the overlapping sensing regions of some sensor nodes by putting some of them into a sleep mode during the sensing phase. The main advantages behind the scheduling technique are to turn on a minimum number of sensor devices, to maximize the lifespan of the sensor device battery, to reduce the collision on the channel, and to improve the lifespan of the WSNs [START_REF] Gupta | Analysis and comparison of sensor node scheduling heuristic for wsn and energy harvesting wsn[END_REF]. Several works have been proposed based on an integer program to optimize the coverage problem in WSNs. As the complexity of this problem increases, metaheuristics can be used to provide high-quality and fewer execution time solutions to difficult coverage optimization problems. This paper makes the following contributions: i) An Energy-saving Distributed Monitoring based Firefly Algorithm (EDiMoFA) Protocol in WSNs is proposed to ensure the coverage and to enhance the life-time of WSNs. The EDiMoFA protocol relies on the framework of the DiLCO protocol [START_REF] Kadhum Idrees | Distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF] to schedule sensor devices to be activated periodically so that the network lifespan is improved while maintaining a certain level of coverage. The sensing field is divided into smaller regions using the concept divide-and conquer. EDiMoFA protocol is operated inside every sensor device in every region and operates into periods. Every period is composed of two phases: a steady-state phase and a monitoring one. The network information exchange, region head selection, and wireless sensors scheduling optimization-based FA are achieved during the steady-state phase. In the monitoring phase, the best sensors schedule that will be produced by the FA will tackle the task of monitoring the sensing field in every region. The produced schedule of sensor devices ensures coverage at a low consumed energy cost. ii) The EDiMoFA protocol uses FA instead of GLPK optimization solver to solve the lifetime coverage optimization model to periodically produce the schedule of sensor devices inside the virtual region. The problem of lifetime coverage formulation is slightly adjusted to use sensor devices centers rather than the concept of primary points of sensor devices in the optimization model. Besides, the EDiMoFA protocol employs an efficient distributed region head election approach to select the head device in each region which is responsible for executing the FA to produce a better schedule of sensor devices sensing the monitoring phase of the current period. iii) The OMNeT++ network simulator has been used to conduct the results of several experiments to show the improved performance of the EDiMoFA. The achieved results of our EDiMoFA protocol are compared with some existing works in the literature such as PeCO, DiLCO, GAF, and DESK which are proposed in [START_REF] Kadhum Idrees | Perimeter-based coverage optimization to improve lifetime in wireless sensor networks[END_REF], [START_REF] Kadhum Idrees | Distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF], [START_REF] Xu | Geography-informed energy conservation for ad hoc routing[END_REF], and [START_REF] Vu | Distributed energy-efficient scheduling approach for k-coverage in wireless sensor networks[END_REF] respectively. EDiMoFA, PeCO, and DiLCO are based on the same network model and framework. PeCO and DiLCO are employed by GLPK optimization solver to solve the scheduling optimization model while the EDiMoFA protocol implements the firefly algorithm to produce the optimal or near optimal schedule of sensor devices per period.

The paper structure is arranged in the following way. The related works are reviewed in the next section. The scientific background of the Firefly Algorithm is presented in Section 3. The EDiMoFA protocol is explained in more details in Section 4. The simulation results are achieved using OMNeT++ [START_REF] Varga | Omnet++ discrete event simulation system[END_REF] and they are discussed and analyzed in Section 5. Section 6 presents the conclusion and the future works.

Related Works

Recently, The lifetime coverage maximization problem represents a challenging issue in the WSN [START_REF] Bagher Dowlatshahi | An energy aware grouping memetic algorithm to schedule the sensing activity in wsns-based iot for smart cities[END_REF]. Some relevant works concerning the problem of coverage are reviewed in this section and then the disadvantages of the presented works are described. The authors in [START_REF] Xu | Geography-informed energy conservation for ad hoc routing[END_REF] have proposed a technique named GAF (geographical adaptive fidelity) to save energy in the ad hoc networks. GAF divides the network into grids based on the position information. One sensor device is activated in each grid to perform the mission of sensing and routing. The other sensor devices in the same grid will be either in sleep or listening mode. DESK is a Distributed Energy-Efficient Scheduling Approach for K-Coverage in Sensor Networks that proposed by Vu et al. [START_REF] Vu | Distributed energy-efficient scheduling approach for k-coverage in wireless sensor networks[END_REF]. DESK works into rounds. It only needs the information of the one-hop neighbors. Each sensor device decides its status (Active or Sleep) based on the perimeter coverage model that was introduced by [START_REF] Huang | The coverage problem in a wireless sensor network[END_REF]. The work introduced in [START_REF] Yue | Improved whale optimization algorithm and its application in heterogeneous wireless sensor networks[END_REF] has proposed an enhanced whale coverage optimization system in WSNs. First, they build a mathematical model that considers energy saving, node utilization, and coverage. Then, to improve the whale optimization method, they use the sine and cosine algorithm. The work in [START_REF] Liang | Maximum target coverage problem in mobile wireless sensor networks[END_REF] proposed four algorithms to solve the problem of optimizing the coverage in WSN heuristically or approximately. This problem is defined as a set of targets that must be covered by a given set of mobile sensor nodes. The dispatch algorithm of a sensor is used to increase the covered targets and under the constraint of limiting the maximum moving distance for each node by threshold. The entire power minimization problem while the targets covering satisfies the requirement of the partial multi-cover is studied by [START_REF] Ran | Approximation algorithm for minimum power partial multi-coverage in wireless sensor networks[END_REF]. It is named MinPowerPMC (minimum power partial multi-cover problem) in WSNs. In this problem, a fixed number of targets such that each target should be covered by a given number of nodes (named covering requirement). In [START_REF] Manju | Target k-coverage problem in wireless sensor networks[END_REF], the authors proposed energy-aware heuristic to solve the k-coverage problem. The heuristic algorithm produces a set of various non-disjoint K-Covers to enhance the lifetime of WSNs. The work in [START_REF] Luo | Maximizing network lifetime using coverage sets scheduling in wireless sensor networks[END_REF] studies the Scheduling of Maximum Cover Sets in WSNs. The problem formulated as an integer program, and then a greedy approach is proposed to solve this problem. After that, the authors presented an approximation method to solve the formulated problem. The work in [START_REF] Li | Monitoring area coverage optimization algorithm based on nodes perceptual mathematical model in wireless sensor networks[END_REF] presented a mathematical model to solve the weak coverage problem. The authors formulated it as an optimization algorithm based on perceived probability around nodes. The proposed algorithm can decrease the distance of nodes moving and increase the time of service in the network.

The authors in [START_REF] Sahoo | Pre-scheduled and self organized sleepscheduling algorithms for efficient k-coverage in wireless sensor networks[END_REF] have proposed two algorithms to solve sleep-scheduling which are named PSKGS (Pre-Scheduling-based K-coverage Group Scheduling) and SKS (Self-Organized K-coverage Scheduling). PSKGS improves the quality lifetime detection and the SKS approach decreases the cost of processing and communication of devices thus enhancing the network lifespan. The authors in [START_REF] Lu | Square partition-based node scheduling algorithm for wireless passive sensor networks[END_REF] have proposed a clustering method based on the square division. The subdivision is based on the communication and sensing ranges as well as the secondary grouping. In every cluster, the scheduling algorithm is executed to ensure the coverage of the network. In [START_REF] Mishra | Design of probability density function targeting efficient coverage in wireless sensor networks[END_REF], the deployment field is partitioned into concentric circles where the region of every annulus is equivalent. In each annulus, the probability density function is planned according to the density of the node. An algorithm for node distribution based on the probability density function is proposed. This scheme of distribution ensures coverage, energy-saving, and lifetime improvement in the network. The authors in [START_REF] Mikitiuk | Maximization of the sensor network lifetime by activity schedule heuristic optimization[END_REF] exhibited three heuristic methods to schedule the nodes' activity: cellular automata, fine and random-tuning, and hypergraph methods. These methods can not be optimal solutions and the produced schedules can be used as input to a strategy based local search with neighborhood functions. In [START_REF] Tchuani Tchakonte | Lifetime optimization of wireless sensor networks with sleep mode energy consumption of sensor nodes[END_REF], the authors presented a greedy algorithm to schedule the sensor nodes into cover sets to maximize the WSN lifespan. This method gives the nodes with the lowest power a higher priority and limits the number of nodes monitoring critical targets. The coverage and connectivity problem is well studied by [START_REF] Yue | Hybrid artificial bee colony algorithm for improving the coverage and connectivity of wireless sensor networks[END_REF]. The researchers proposed a hybrid approach which consists of the bee colony method and the free search method with pheromone sensitivity to achieve suitable coverage and connectivity to optimize the lifetime of the WSNs. In [START_REF] Mishra | Design of probability density function targeting efficient coverage in wireless sensor networks[END_REF], The authors have studied the coverage problem with energy saving using Probability Density Function (PDF). They divided the field into circles where the annulus area is equal. The density of the nodes in each annulus is used to design the PDF.

Idrees et al. [START_REF] Kadhum Idrees | Distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF] introduced a protocol named Distributed Lifetime Coverage Optimization (DiLCO) to keep the coverage and enhance the network lifespan in WSNs. This protocol represents an ameliorated version of the protocol presented in [START_REF] Kadhum Idrees | Coverage and lifetime optimization in heterogeneous energy wireless sensor networks[END_REF]. The DiLCO protocol is based on partitioning the area of interest into subregions using a divide-and-conquer approach. After that, DiLCO is distributed on every sensor device in each subregion. The suggested DiLCO protocol is implemented in a periodic way, where each period consists of four phases: information exchange, leader election, decision, and sensing. The results of simulations show that DiLCO can improve the WSN lifetime and produces enhanced performances of coverage. In the PeCO protocol [START_REF] Kadhum Idrees | Perimeter-based coverage optimization to improve lifetime in wireless sensor networks[END_REF], a novel sensor devices perimeter-based optimization model is proposed. PeCO performs a centralized optimization to produce the sensor devices schedule inside the region while it is distributed on the sensor devices of the regions of the network. This schedule saves the energy while ensuring the required level of coverage for the region.

SHORTCOMINGS. Despite the introduction of various energy-saving protocols for coverage keeping and to enhance the lifespan of the network, no method could guarantee the coverage for the sensing field of interest with the optimal minimum number of sensor devices on the long-term. For instance, excellent solutions can be provided by the centralized approaches of optimization. Unfortunately this can lead to wasting large amount of energy due to the increased cost of the algorithm execution time and the sensor devices' communication in the network of the region. Hence, in dense networks, the centralized methods are not preferred. In the distributed approaches, the solutions are not optimal due to they depend on local information of the neighboring devices. These approaches can save energy because they decrease the communication and execution time costs in the sensor devices. Both distributed and centralized approaches would produce imperfect lifetime coverage in the WSNs.

OUR PROTOCOL. We propose an Energy-saving Distributed Monitoring based Firefly Algorithm (EDiMoFA) Protocol in WSNs to ensure a suitable level of coverage and to improve the lifetime of WSNs. The sensing field is divided into smaller regions using the concept divide-and-conquer in the first phase. In the second phase, the EDiMoFA protocol is distributed and executed on every node in each region. The EDiMoFA protocol works into periods. Each period is composed of two phases: steady-state and monitoring. The network information exchange, region head selection, and wireless sensors scheduling optimization-based FA are achieved in the steady-state phase. In the monitoring phase, the best sensors schedule that will be produced by the FA will take the responsibility of monitoring the sensing field in every virtual region. The produced sensors schedule ensures coverage at a low consumed energy cost and thus extends the network lifespan. The EDiMoFA protocol combines and employs the principal benefits of centralized and distributed strategies and gets rid of most of their disadvantages. In addition, the Firefly Algorithm is efficient, simple, light, and easy to implement and this is the main reason behind using the Firefly Algorithm by the proposed EDiMoFA protocol.

Firefly Algorithm

The Firefly algorithm is a nature-inspired metaheuristic optimization technique inspired by the real-world social life of the fireflies. The phrase Lightning bugs refers to the synonym of the Fireflies. In the world, the number of different firefly species is about 2000. The short and rhythmic flashes are produced by most of these firefly kinds. Each particular species has a personalized flashing pattern. The mating partners and potential preys are attracted by the signal of the flash of the firefly. Besides, the flashes can be used as an approach for secured warning. The firefly algorithm proposed by Yang can be explained as follows [START_REF] Yang | Nature-inspired metaheuristic algorithms[END_REF].

1. One firefly will be invited to join other fireflies without considering their sex where all fireflies are unisex. 2. Firefly Attractiveness is indicated by its light. The lower light firefly is moving toward the higher light one for any two flashing fireflies. The brightness is the measure of the attractiveness. When the distance of the of two fireflies grew, the attractiveness will be decreased. If the brightness of both fireflies is the same, one of them moves randomly. 3. The firefly brightness is calculated using an objective function. The brightness for the problem of the maximization is proportional to the objective function value. Algorithm 1 refers to the original firefly algorithm.

Algorithm 1 Firefly Technique 1: Goal function fun(y), y = (y 1 , ..., y d ) T 2: Create fireflies initial population y j , j = 1, ..., n 3: Intensity of light Intensity j at y j is identified by fun(y j ) 4: Set coefficient Υ (absorptionof light) 5: while stop condition true do 6:

for j ←-1 to n do 7:

for k ←-1 to n do 8:

if Intensity j < Intensity k then 9:

Shift firefly j in the direction of firefly k 10:

end if 11:

Modify attractiveness with distance dis by exp(-Υ * dis) 12:

Assess new solutions and modify the Intensity of light 13:

end for 14:

end for 15:

Order the fireflies and locate the current global best gbest 16: end while 17: Process the output of the algorithm and transform it into a visual form

Attractiveness

The firefly attractiveness is fixed via its intensity of light. The following equation gives the attractiveness

B(dis) = B 0 * e -Υ * dis 2 (1)
Where B 0 is the attractiveness at distance d = 0, Υ is the coefficient of absorption.

Distance

The Cartesian distance is used to calculate the distance between firefly j and firefly k, at Y j and Y k respectively, as follows

dis jk = |Y j -Y k | = d j=1 (Y j,i -Y k,i ) 2
(2)

Movement

The movement of a firefly j that is interested in another more winning firefly k is fixed via the following equation

Y j = Y j + B 0 * e -Υ * dis 2 j,k * (Y k -Y j ) + λ(rand -1/2) ( 3
)
Where λ is the randomization parameter and rand is the random generator function.

Discretization

If the firefly j shifts in the direction of firefly k, the firefly j position is adjusted from a value of the binary number to a real number value. Hence, it must replace this real value by a binary value. This will be done by using the following sigmoid function

S(Y j,k ) = 1 1 + exp(-Y j,k ) (4)
4 EDiMoFA protocol A randomly and uniformly deployed WSN composed of static homogeneous sensor devices is considered. The sensor devices have the same processing, communication, and sensing abilities. They have different initial energy levels. To guarantee initially a full coverage of the interesting sensing field, dense wireless sensor devices are deployed. Moreover, it is supposed that each sensor node knows its position by employing either a localization system or GPS. The sensing field is divided virtually using the concept divide and conquer concept, where each sensor device stores the information of the borders of each virtual region in the sensing field before the deployment process [START_REF] Kadhum Idrees | Distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF]. After the deployment, every sensor device will belong to a certain virtual region if its position is located inside the boundaries of that region. The sensors' devices are deployed uniformly over the virtual regions. Each sensor device includes the radio range Rs and the sensing range Rs. Therefore, if the distance length between any two devices less than or equal to 2Rs, they are considered as neighbors. The sensor device can sense any event that occurs in its Rs. The disk model is supposed for every device where its radius is Rs. Every device can communicate within the disk radius of Rc. The multi-hop communication is used during the information exchange among the sensor devices inside the virtual region as well as when the sensor devices transmit their measurements to the cluster head of the region. The EDiMoFA protocol is introduced in this section. Each sensor node in every virtual region of the sensing field includes the EDiMoFA protocol. Three energyefficient techniques are integrated to implement the EDiMoFA protocol. These techniques are virtual WSN division, region head selection, and sensors schedule production based on FA to ensure the coverage of the sensing field with a minimum consumed energy. The EDiMoFA protocol is periodic and each period consists of two phases: steady-state and monitoring. The former is composed of three steps: Information exchange, Region Head Selection, and Sensors Scheduling Production based FA. Figure 1 shows proposed EDiMoFA protocol. As exhibited in Figure 1, the suggested EDiMoFA protocol lifespan is broken into periods, where each period consists of two phases: steady-state and monitoring: in the former, information exchange, region head choice, and sensor activity scheduling based FA optimization. In the latter, the produced sensors schedule will monitor the sensing field during the current period. For every period, only one schedule of sensor devices is responsible for covering the sensing field of interest. The periodic implementation of the EDiMoFA protocol can enhance the robustness of the WSN against sensor failures [START_REF] Kadhum Idrees | Distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF]. Algorithm 2 refers to the EDi-MoFA protocol implemented in every sensor device in each region.

Algorithm 2 EDiMoFA(s j ) Exclude s j from joining in the current period 19: end if

1: if E R j ≥ E

Information exchange

As displayed in Algorithm 2, a sensor device s j tests whether it has adequate power. i.e, the residual power of s j ≥ E th . The E th represents the minimum power required by the sensor device to remain active throughout one period. If the condition is satisfied, sensor device exchanges INFO Messages with the all sensor devices belonging to the same region. It gathers from every sensor device its location coordinates, remaining power(E R j ), and sensor ID. These three parameters are exchanged only in the first period, while during the next periods, every sensor device will only send its E R j . If there is any change to its location, it can be sent with the remaining energy to update the information of the sensor devices in the region.

Region Head Selection

Once the first step in the steady-state phase is completed, each sensor device in the region implements the region head selection method to choose the region head of the current region. The region head selection method is executed in a distributed way in every sensor device. Each s j in the virtual region employing Eq. 5 and for every sensor device s, s ∈ S, where |S| is the number of sensor devices in the region. For each sensor device s in the region, the weight ws will be calculated. Hence, the head of the virtual region during the current period in the current virtual region is the sensor device s which has the maximum weight of ws. The same Eq. 5 will be applied by the all of the sensor devices in the same virtual region. Therefore, they will have the same maximum weight for the winning sensor device as the head of the virtual region. The parameter z in Eq. 5 is used to prevent the sensor devices from selecting more than one head of the virtual region at the same time in the case of equality between sensor devices.

ws =   σ * E R s Emax + ς *   i∈|S| 1 - dis(s, i) 2 * Rs     + z (5)
Where z = IDs |S| * δ, σ + ς = 1, s ∈ |S|, and δ = 0.005.

dis(s, i) = (xs -x i ) 2 + (ys -y i ) 2 (6)
The head of the virtual region is responsible for executing the Firefly Algorithm to provide the best sensor devices schedule for the current period (see Section 4.3.2. Every head of the virtual region will transmit an ActiveSleep message to the all sensor devices in the same region based on the produced schedule by the FA. This message will notify the sensor device either stay active or to go into sleep mode. Otherwise, if the sensor device is not the region head, it will listen for the ActiveSleep message to identify its status throughout the monitoring phase. The head of the virtual region will stay active during the current period and it will be responsible for receiving the measurements from the sensor devices inside the virtual cluster.

Sensors Scheduling based Firefly Algorithm

The main objective of sensor scheduling after choosing the region head is to select the minimum number of sensor devices to enhance the lifetime of WSN while achieving a suitable rate of covering and monitoring of the region with a lower consumed energy. In each period and each region of the sensing field, one new sensor devices schedule will be provided to tackle the mission of sensing inside every region of the current period. The coverage problem represents an optimization problem and it can be solved using the FA technique.

Coverage Problem Formulation

The mathematical model of the coverage problem is formulated in this section as an integer linear program. This model is inspired by the model proposed in [START_REF] Pedraza | Efficient coverage algorithms for wireless sensor networks[END_REF] with some modifications and considers the center points of sensor nodes instead of the targets.

The problem of area coverage is declared as the monitor of a set of center points of the deployed sensors devices in the monitoring area. The set of center points of the deployed sensors devices is indicated by C and the set of alive sensors by S.

Definition 5 Inside Coverage Inc refers to the fact that if the center point c is covered (takes the value 0) or not (takes the value 1).

Inc = 1 if center point of sensor device is not covered, 0 otherwise.

Hence, the region coverage problem can then be mathematically formulated as follows

M inimize C c=1 (α Out • Outc + α In • Inc) (11) Subject to S s=1 (Φc,s • Ys) = 1 + Outc -Inc ∀c ∈ C (12)
Ys ∈ {0, 1}, ∀s ∈ S (13)

Outc ∈ N, ∀c ∈ C ( 14
)
Inc ∈ {0, 1}, ∀c ∈ C (15)
Furthermore, The constant parameters α Out and α In help to increase the importance for either Outside Coverage Outc or Inside Coverage Inc in Eq. 11.

Firefly Algorithm for Lifetime Coverage optimization

Nature-Inspired Metaheuristics are generic search approaches to investigate search spaces and to solve complex problems. Nature-Inspired Metaheuristic consists of two main elements: intensification (exploitation) and diversification (exploration) [START_REF] Yang | Nature-inspired metaheuristic algorithms[END_REF]. The latter consists in producing various solutions to investigate the search space on a global scale, whereas the former concentrates the search in a local search space to find the adequate solution by exploiting the information of this space. These approaches balance between the intensification of the aggregated search experience and the diversification of the search space. This balance can offer regions in the search space with excellent solutions, while avoiding to lose too much time via randomization in regions of the search space which have either been previously investigated or do not produce better quality solutions (i.e., increasing solutions diversity). Hence, Nature-Inspired Metaheuristic offers an adequate solution to any optimization problem, such as inadequate information or insufficient processing capability [START_REF] Absalom E Ezugwu | Metaheuristics: a comprehensive overview and classification along with bibliometric analysis[END_REF].

The Firefly Algorithm (FA) is one of the nature-inspired metaheuristic methods which copies the social behavior of fireflies. The FA is a highly effective method to locate global optima with increased success ratios [START_REF] Yang | Firefly algorithms for multimodal optimization[END_REF]. Xin-She Yang has proved that FA outperforms both PSO and GA in terms of success ratio and efficiency during the performance evaluation by simulation [START_REF] Yang | Firefly algorithms for multimodal optimization[END_REF]. The authors in [START_REF] Lukasik | Firefly algorithm for continuous constrained optimization tasks[END_REF] investigated the FA to solve optimization problems and they have indicated the effectiveness of FA. Since integer linear programming is NP-hard [START_REF] Korte | Combinatorial optimization: Theory and Algorithms[END_REF][START_REF] Bienstock | Integer programming and combinatorial optimization[END_REF] and many problem are complex, therefore the heuristic methods must be used instead. FA is a metaheuristic method suitable to solve NP-hard problems and produce approximate solutions. Compared to optimization solvers, FA gives a near-optimal solution with acceptable execution times. The FA also needs smaller amount of memory particularly for large size problems. Optimization solvers provide an optimal solution, but they need higher execution times and larger amounts of memory for large problems.

This section introduces a Nature-Inspired Metaheuristic based FA to solve the problem of coverage optimization. The suggested FA gives a near-optimal schedule of devices for sensing the interested monitoring field per period. The proposed FA depends on the model of optimization which is introduced in Section 4.3.1. The authors in [START_REF] K Chandrasekaran | Binary real coded firefly algorithm for solving unit commitment problem[END_REF] have explained that the Tanh function can improve the performance of the binary FA better than the Sigmoid function. Both Tanh and Sigmoid functions have scaled the values of the input vector in the [0,1] range. Therefore, the Tanh function is employed to enhance our binary firefly algorithm performances. Algorithm 3 exhibits the suggested FA to solve the coverage lifetime optimization problem.

The proposed protocol which is based on FA in the steady-state phase is named EDiMoFA. It aims to produce the optimal (or near-optimal) schedule of the sensor devices which takes the duty of monitoring the region of interest in the next phase. The individual (schedule) in FA is represented in binary values, where the 0 value refers to a sleep state and the 1 value refers to an active state. After generating the initial population in the FA, every individual is assessed and assigned a fitness value according to the fitness function illustrated in Eq. [START_REF] Liang | Maximum target coverage problem in mobile wireless sensor networks[END_REF]. It is computed based on both Eq. ( 11) and Eq.( 12). In the suggested FA, the optimal (or near-optimal) candidate solution, is the one with the minimum value for the fitness function.

Algorithm 3 Firefly Algorithm(S)

Require: all the parameters that are related to both coverage model and the FA algorithm Ensure: Y Best ( Sensors devices schedule) 1: Compute the parameter Φc,s using equation ( 7) 2: for i ←-1 to F iref ly N o do 3: Initialize Y i randomly by 1 if Rand() > 0.5 or 0 otherwise.

4:

Update Outside coverage Outc,Inside coverage Inc, and Penalty P enc 5:

Compute the fitness value F itness(Y i ) using equation ( 17) 6:

Intensity i ← F itness(Y i ) 7: end for 8: Initialize t ← 0 9: while (t M AXG) and (The M axM inF it condition is not met) do 10:

for i ←-1 to F iref ly N o do 11:

for j ←-1 to F iref ly N o do 12:

if Intensity i < Intensity j then 13:

dis i,j ← Distance(Y i , Y j ) 14:
Beta ← Attractiveness(Intensity 0 , Υ ; dis i,j ) 15: Compute the fitness value F itness(Y i ) using equation ( 17) 20:

Y i ← (1 -Beta) × Y i + Beta × Y j 16: Y i ← Y i + λ × (Rand() - 1
Intensity i ← F itness(Y i ) 21:
end for 22:

end for 23:

minF ← arg min i∈1,••• ,F iref ly N o F itness(Y i ) 24: for j ←-1 to S do 25: Y B j ← Y minF j + λ × (Rand() - 1 2 )
26: This function rewards the decrease in the sensor devices which covers the same center point and penalizes the decrease to zero in the sensor devices covering the center point.

Y B j ← exp(2 × |Y B j |) -1 exp(2 × |Y B j |) +
P enc = S S s=1 (Φc,s • Ys) = 0, 0 otherwise. ( 16
)
F i = C c=1 ((α Out • Outc + α In • Inc) + P enc) (17)
The proposed FA is terminated as soon as one of the two stopping criteria is achieved. The first criterion refers to the maximum number of iterations performed by the FA. The second criterion is called MaxMinFit. It is met when the maximum fitness value in the population is equal to the minimum fitness value. The best solution will be chosen as a schedule of sensor devices for the monitoring phase in the current period. The time complexity of the proposed Firefly Algorithm(S) 3 can be given as Θ(M AXG . F iref ly N o 2 ).

Performance Evaluation and Analysis

This section focuses on assessing the performance of the proposed protocol ED-iMoFA by conducting several simulation experiments using the OMNeT++ network simulator [START_REF] Varga | Omnet++ discrete event simulation system[END_REF]. Table 1 shows the parameters and their values which are used during the simulation. In this simulation, the sensor nodes in the network are deployed in a controlled way in order to guarantee the full coverage of the sensing field of interest of size (50 × 25) m 2 by the deployed sensor devices. They are deployed randomly only within every region of the sensing field. The conducted results are the average of 50 simulation runs. In our simulations, it is supposed that the concerned sensing field is partitioned into 16 regions in a similar way as explained in [START_REF] Kadhum Idrees | Distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF]. The consumed energy by the sensor devices is calculated according to the energy consumption model which is employed by [START_REF] Kadhum Idrees | Distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF]. This power model is described in Table 2 and the proposed protocol EDiMoFA takes into account all the sensor status when calculating the energy consumption inside each sensor device.

For simplicity's sake , the required amount of energy to change the sensor status from one state to another, to activate the radio, to start-up operation of the sensor device, etc are ignored. The proposed model of energy consumption in [START_REF] Kadhum Idrees | Distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF] is suggested to use 0.2575 mW as the cost of consumed energy for transmitting and receiving a one-bit message. Hence, the energy consumption for a message of length LM bits is equal to 0.2575 * LM .

Every sensor device initializes its energy randomly in the range [500 -700]. The minimum required energy inside every device to stay alive during one period is referred to as E th = 36 Joules, where E th = CE × T P . The EC is the consumed energy during active state (9.72 mW) and TP is the time of one period (3600s). Therefore, in our simulation, if the residual energy of the sensor device is less than E th , it cannot participate in the next period. The efficiency of EDiMoFA is evaluated using different performance metrics such as the coverage ratio, the active sensors ratio, the network lifespan, and the energy consumption. These metrics are used similarly in [START_REF] Kadhum Idrees | Distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF]. The execution time is used as an additional performance metric to evaluate the EDiMoFA protocol. In this simulation, the lifetime of the network is defined as the amount of time until the ratio of coverage is degraded below a certain threshold. The Lifetime95, Lifetime90, Lifetime85, and Lifetime50 are denoted by the amount of time in which the protocol can achieve a network coverage level greater than 95%, 90%, 85%, and 50% respectively. It is supposed that the network operates until the all sensor devices have died or have lost the connectivity of the network. The EDiMoFA protocol was simulated with a laptop (DELL) with an Intel Core i3 2370 M (1.8 GHz) processor (2 cores) whose MIPS (Million Instructions Per Second) rate is equal to 35330. In the simulation, every deployed sensor device uses Atmel's AVR ATmega103L microcontroller (6 MHz) having a MIPS rate equal to 6. Therefore, in order to be consistent with the use of this device, the original run time on the DELL laptop is multiplied by 2944.2 35330 2 × 1 6 . The effectiveness of EDiMoFA is tested and compared to four other techniques like DESK [START_REF] Vu | Distributed energy-efficient scheduling approach for k-coverage in wireless sensor networks[END_REF], GAF [START_REF] Xu | Geography-informed energy conservation for ad hoc routing[END_REF], DiLCO [START_REF] Kadhum Idrees | Distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF], PeCO [START_REF] Kadhum Idrees | Perimeter-based coverage optimization to improve lifetime in wireless sensor networks[END_REF]. The DESK is a distributed technique for solving coverage problems. The GAF technique partitions the sensing field into fixed cells. After that, only one sensor device will be activated inside each cell in the sensing field. The DiLCO protocol is the enhanced protocol of the protocol introduced in [START_REF] Kadhum Idrees | Distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF]. In the DiLCO protocol, the sensing field is virtually subdivided into small cells according to the divide and conquer method and then distributes the DiLCO protocol inside every small cell to optimize simultaneously both the coverage and lifespan in the cell. The PeCO protocol is also based on the virtual division of the sensing field. It performs the optimization inside each small cell in a distributed way based on the perimeter-coverage model and it also produces the optimal schedule of sensor devices inside each cell simultaneously.

In this paper, DiLCO, PeCO, and EDiMoFA protocols are based on the same framework of the network subdivision and they produce the optimal schedules of sensor devices based on solving the optimization model. The network subdivision consists of 16 subregions and this subdivision has given the best results as explained in [START_REF] Kadhum Idrees | Distributed lifetime coverage optimization protocol in wireless sensor networks[END_REF]. The size of the network and the size of the interested area can play an important role in adapting the number of regions. The three protocols mentioned above can be distinguished from each other by their mathematical formulations and the coverage models to introduce the schedule of sensor devices in each period. For example, the DiLCO protocol has achieved the optimization to cover a set of primary points. The PeCO protocol has performed the optimization to reach the best level of coverage for each perimeter of the sensor device. Both the DiLCO and PeCO protocols are employed by the integer program to produce the optimal schedule of sensor devices. This integer program is solved and executed by using a GLPK optimization solver. In the EDiMoFA protocol, the mathematical optimization model is implemented to achieve the coverage of the sensors centers points. EDiMoFA protocol employs the Firefly Algorithm to optimize both the network lifetime and the coverage.

Coverage Ratio

The trade-off between network energy-saving and a suitable level of coverage for the sensing field of interest is an essential challenge in the WSNs. The coverage ratio for 200 sensor devices and the five protocols are presented in Figure 2. In the first 33 rd periods, it can be seen that DiLCO, DESK, GAF can introduce a light better coverage ratio with 99.02%, 99.99%, and 99.91% respectively in comparison with 98.76% and 98% provided by PeCO and EDiMoFA respectively. On the one hand, EDiMoFA, DiLCO, and PeCO protocols turn off a larger number of redundant sensor devices that led to minimizing the coverage ratio slightly due to implementing the optimization. On the other hand, the other two protocols provide a higher coverage ratio due to activating a larger number of sensor devices because they did not apply the optimization. After the 33 rd period, it is clear that the proposed EDiMoFA protocol introduces a better coverage ratio than the PeCO protocol and it preserves the ratio of coverage better than other protocols for longer periods. For example, for network topology of 200 sensor devices, the ratio of the coverage is kept greater than 85% for 65, 70, and 72 periods in the DiLCO, PeCO, and EDiMoFA protocols respectively. The DiLCO, PeCO, and EDiMoFA protocols produced a coverage ratio greater than 95% for 60, 57, and 63 periods respectively. The energy saved by the EDiMoFA protocol in the early periods allows later for a substantial increase in the lifetime coverage performance.

Active Sensors Ratio

The energy of the sensor device represents the most critical factor that affects the performance of the network. Therefore, it is essential to cover the interesting sensing field by using the minimum number of sensor devices as possible to save energy and extend the lifetime of the network. The ratio of the active sensor devices for 200 sensor devices is illustrated in Figure 3. In the first 14 periods, DESK and GAF are activated closely 30.36 % and 34.96 % respectively of sensor devices, whilst DiLCO, PeCO, and EDiMoFA protocols are turned on 17.92%, 20.16%, and 19.44% of sensor devices during the same first fourteen periods. The reason behind activating a lower number of sensor devices with DiLCO, PeCO, and EDiMoFA protocols is that they have implemented an optimization model to produce the optimal (or near-optimal) schedule of sensor devices to optimize both the coverage and the lifespan of the network. DESK and GAF did not implement the optimization model to schedule the sensor devices in every period. As can be seen from Figure 2, the EDiMoFA protocol activated a large number of sensor devices after the 40 th period compared with other protocols to introduce a higher ratio of coverage for the sensing field.

Energy Consumption

The energy consumption is an essential resource that must taken into account during designing energy-efficient WSN protocols due to the limited power of the sensor device batteries. Since replacing the sensor device battery is impractical in large networks or may even be impossible in hostile environments. Hence, it is essential to save the power of sensor batteries to improve the lifetime of the network. Energy consumption optimization has a direct impact on the network's lifetime. This experiment studies the impact of the consumed energy on the performance of the network. Different states of the energy consumption model implemented by the sensor devices are considered such as: active, sleep, transmitting, receiving, processing, and listening. The results of the five protocols are conducted and compared using different network densities. The energy consumption for different network densities and for both Lif etime 95 and Lif etime 50 is shown in conserves the energy of the sensor device from 9.4% up to 14.5%, from 41.8% up to 47.9%, from 25.9% up to 31.6%, and from 2.7% up to 10% in comparison with DiLCO, DESK, GAF, and PeCO respectively. It can be seen that the EDiMoFA protocol efficiently saves energy due to producing the most suitable set of active sensor devices per period using the Firefly optimization algorithm and puts the redundant sensor devices in sleep mode. The results have been shown the superiority of the EDiMoFA protocol from an energy-saving point of view. It consumes less energy than the other protocols which can lead to an extension of the network lifespan.

Network Lifespan

One of the most important factors that must be taken into account when designing the WSN is the network lifetime while preserving a suitable level of coverage for the sensing field. Figure 5 presents the network lifespan for different network sizes and for both (a) Lif etime 95 and (b) Lif etime 50 . The comparison results have proved that the EDiMoFA protocol has prolonged the network lifetime from 3.2% up to 21.8%, from 10.4% up to 86.4%, from 35.2% up to 68.4%, and from 1.6% up to 6.7% in comparison with the DiLCO, DESK, GAF, and PeCO protocols respectively. The results have shown the superiority of EDiMoFA protocol from the network lifetime point of view for Lif etime 95 , while the EDiMoFA and PeCO protocols have introduced the best network lifetime for Lif etime 50 . It is clear that the lifetime increases when the network density increases. The lifetime of the coverage for the EDiMoFA, DiLCO, and PeCO protocols are compared and shown in Figure 6. In this experiment, Protocol/85, Protocol/90, and Protocol/95 denote the amount of time during which the coverage protocol can respectively preserve the level of coverage ratio greater than 85%, 90%, and 95% . Hence, the term Protocol refers to EDiMoFA, DiLCO, or PeCO. It can be seen from the results that the EDiMoFA protocol outperforms PeCO and DiLCO for all coverage ratios. Moreover, the improvements in the network lifetime increase when the network size increases. Some WSNs applications do not need full coverage for the sensing field. For instance, forest fire applications should provide full coverage in the summer seasons, while in the rainy seasons, they only need 80% of coverage for the sensing field [START_REF] Li | Transforming complete coverage algorithms to partial coverage algorithms for wireless sensor networks[END_REF]. Fig. 6: Network lifetime for different coverage ratios.

Execution Time

In this experiment, the impact of the execution time on the performance of the network is investigated. Figure 7 presents the execution time in seconds which is required to solve the lifetime coverage optimization problem by EDiMoFA, PeCO, and DiLCO for different network densities. The PeCO and DiLCO protocols are employed in the optimization solver GLPK (GNU Linear Programming Kit) which applies the Branch-and-Bound technique to solve the lifetime coverage optimization problem. The EDiMoFA protocol applies the Firefly Algorithm to solve the same lifetime coverage optimization problem. As previously explained in this section, to be consistent with the use of a sensor device, the original execution time calculated on the DELL laptop is multiplied by 2944.2. As illustrated in the results, the Branch-and-Bound technique consumes a lot of time to solve the lifetime coverage optimization in the PeCO and DiLCO protocols. This has led to losing a large amount of energy by the region head to solve the lifetime coverage optimization problem. Notice that the EDiMoFA protocol always outperforms the PeCO and DiLCO protocols from the execution time point of view. This has a great impact on the increase of the network performance by reducing the energy consumption and thus improving the lifetime of the network while maintaining a suitable level of coverage for the sensing field.

The t-test metric

To show the significant difference between the results of the proposed EDiMoFA protocol with previous methods, the statistical analysis using t-test is used to explain that the conducted results of the proposed EDiMoFA protocol are significant. Hence, the t-test is implemented on the result of the comparison of energy consumption between EDiMoFA protocol and other methods like DiLCO, DESK, GAF, and PeCO for both Lif etime 95 and Lif etime 50 . For the case of Lif etime 95 , the t-test (with p-value) between EDiMoFA and DiLCO, EDiMoFA and DESK, EDiMoFA and GAF, EDiMoFA and PeCO are 1.54972E-05, 0.000649791, 3.76788E-05, and 0.001009918 respectively. While for the case of the Lif etime 50 , the ttest (with p-value) between EDiMoFA and DiLCO, EDiMoFA and DESK, EDi-MoFA and GAF, EDiMoFA and PeCO are 0.00204484, 0.00078877, 4.5961E-06, and 0.02568722 respectively. Hence, the t-test (with p-value < 0.05) shows that our results are significant in terms of energy consumption compared with other methods and the energy consumption is significantly reduced.

Conclusion and Perspectives

This paper has studied the lifetime coverage optimization problem in WSNs. In this paper, an Energy-saving Distributed Monitoring based Firefly Algorithm (ED-iMoFA) Protocol in WSNs is proposed to ensure the coverage and to enhance the lifetime of WSNs of IoTs. The main challenges consist in selecting the region head efficiently in the regions of the sensing field as well as optimizing the lifetime coverage to produce the best schedule of sensor devices and to extend the network lifetime while maintaining an acceptable coverage level. The sensing field is partitioned into smaller regions using the concept divide-and conquer. EDiMoFA protocol is distributed on every node in the resulted regions. EDiMoFA protocol operates into periods. Each period consists of two phases: steady-state and monitoring. The network information exchange, region head selection, and wireless sensors scheduling optimization-based FA are achieved in the steady-state phase.

In the monitoring phase, the best sensors schedule that will be produced by the FA will undertake to monitor every region in the sensing field. The produced sensors schedule ensures coverage at a low consumed energy cost. To evaluate the EDi-MoFA protocol performances, several simulations experiments are achieved using the OMNeT++ network simulator. The results prove that the EDiMoFA protocol outperforms some of the other existing protocols in terms of network lifetime, coverage ratio, active sensors ratio, processing time, and energy consumption. In the future, the proposed Firefly Algorithm will be extended to introduce schedules of sensor devices for several periods instead of executing it every period to give a new schedule. This can contribute to reducing energy consumption and maximize the lifetime of the network. Besides, the lifetime coverage optimization model will be improved to take into account the heterogeneous sensor devices.

Fig. 1 :

 1 Fig. 1: EDiMoFA Protocol.

Fig. 2 :

 2 Fig. 2: Coverage ratio for 200 deployed nodes.

Fig. 3 :

 3 Fig. 3: Active sensors ratio for 200 deployed nodes.

Figures 4 .

 4 As shown in both (a) and (b) of Figures 4, EDiMoFA

  protocol

  Lif etime 50 .

Fig. 4 :

 4 Fig. 4: Energy consumption per period.

  Lif etime 50 .

Fig. 5 :

 5 Fig. 5: Network Lifetime.

Fig. 7 :

 7 Fig. 7: Execution Time (s).

  Definition 1 Boolean Parameter Φc,s describes the actual relation between the center point c of sensor device and the sensor device s. Definition 3 Active node number is denoted as the number of sensor devices that covers the sensor device center point as s∈S Φc,s * Ys. Definition 4 Outside Coverage Outc refers to the number of sensor devices that monitor the center point c minus one.

	Φc,s =	1 if center point c is covered by sensor device s; ∀c ∈ C, ∀s ∈ S 0 otherwise.	(7)
	Definition 2 Boolean Parameter Ys Indicating the activation (1 if yes) or not (0
	if not) of the sensor s.
		Ys =	1 if sensor device s is Active, 0 otherwise.	(8)
	Outc =	0 (	if center point of sensor device is not covered,

s∈S Φc,s * Ys) -1 otherwise.

Table 1 :

 1 simulation parameters and their values.

	Parameter	Value
	Network Field	(50 × 25) m 2
	No. of Devices	100, 150, 200, 250 and 300 nodes
	Primary Energy	500-700 joules
	Period of monitoring	60 Minutes
	Rs	5 m
	E thr	36 Joules
	Rc	10 m
	M axIter	1000
	λ	0.5
	Υ	1.0
	Spop	50
	α Out	1
	α In	|C 2 |
	σ	0.9
	ς	0.1
	B 0	1.0

Table 2 :

 2 Power consumption values

	Sensor status	MCU Radio	Sensing	Power (mW)
	COMPUTATION	ON	ON	ON	26.83
	ACTIVE	ON	OFF	ON	9.72
	LISTENING	ON	ON	ON	20.05
	SLEEP	OFF	OFF	OFF	0.02
	The required energy to transmit or receive a 2-bit message	0.515

Acknowledgement

This paper is funded by the EIPHI Graduate School (contract "ANR-17-EURE-0002").


Gianni Comandini
email: gianni.comandini@bristol.ac.uk

Morvan Ouisse

Fabrizio Valeska P Ting

Fabrizio Scarpa

Acoustic Transmission Loss in Hilbert Fractal Metamaterials

come

Acoustic transmission loss in Hilbert fractal metamaterials

Introduction

Excessive noise can significantly impact human health and lead to serious pathology [START_REF] Zhou | Environmental noise exposure degrades normal listening processes[END_REF][START_REF] Basner | Auditory and non-auditory effects of noise on health[END_REF] . Traditional noise transmission mitigation strategies [START_REF] Liu | On the definition of noise[END_REF][START_REF] Tao | Recent progress in acoustic materials and noise control strategies-a review[END_REF] have followed mass law principles [START_REF] Adkins | Transformer noise reduction witk a close-fitting mass-law enclosure[END_REF] , with a resulting drawback in terms of high mass and thickness of the materials used [START_REF] May | The optimum weight of highway noise barriers[END_REF] . Mass-spring-mass models exploit other mechanisms, with the development of designs based on one or multiple coupled degrees of freedom and combinations of materials [START_REF] De Melo Filho | Metamaterial foam core sandwich panel designed to attenuate the mass-spring-mass resonance sound transmission loss dip[END_REF] . Spring-mass models are typically used in acoustic architecture to separate environments with rational use of the space available. Other more recent but consolidated technologies exploit the use of porous materials as perforated [START_REF] Starkey | Experimental characterisation of the bound acoustic surface modes supported by honeycomb and hexagonal hole arrays[END_REF][START_REF] D'aguanno | Broadband metamaterial for nonresonant matching of acoustic waves[END_REF][START_REF] Magliacano | Computation of dispersion diagrams for periodic porous materials modeled as equivalent fluids[END_REF] and/or microperforated panels [START_REF] Bravo | Sound absorption and transmission through flexible micro-perforated panels backed by an air layer and a thin plate[END_REF] , or through thermo-viscous acoustic energy dissipation in the internal porous skeletons of foams with various internal geometry and materials compositions [START_REF] Kühl | Absorption of sound by porous materials[END_REF][START_REF] Kuczmarski | Acoustic absorption in porous materials[END_REF][START_REF] Ba | Soft porous silicone rubbers with ultra-low sound speeds in acoustic metamaterials[END_REF][START_REF] Caniato | A sustainable acoustic customization of open porous materials using recycled plastics[END_REF][START_REF] Jia | Highly compressible and anisotropic lamellar ceramic sponges with superior thermal insulation and acoustic absorption performances[END_REF] . Thermoviscous energy dissipation technologies also theoretically allow acoustic dampening of any frequency, with the only limitation represented by the available room for the porous liners. Because of space constraint limitations, porous materials are generally adopted to mitigate noise within the middle and high-frequency ranges. Another technique to obtain a desired transmission loss (TL) for a predetermined interval of frequencies is by using Helmholtz resonators [START_REF] Romero-García | Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators[END_REF][START_REF] Huang | Sound trapping in an open resonator[END_REF][START_REF] Long | Reconfigurable sound anomalous absorptions in transparent waveguide with modularized multi-order helmholtz resonator[END_REF][START_REF] Lan | Manipulation of acoustic wavefront by gradient metasurface based on helmholtz resonators[END_REF] . However, resonators are only efficient for narrow sets of frequency ranges and may involve the use of considerable space for the spring part of the Helmholtz resonator for low frequencies applications [START_REF] Tang | Theory of a generalized helmholtz resonator[END_REF][START_REF] Abbad | Numerical and experimental investigations on the acoustic performances of membraned helmholtz resonators embedded in a porous matrix[END_REF] . More recent techniques for noise management are promising because of their lightweight characteristics and their use of different mechanisms to achieve sound absorption and transmission loss [START_REF] Jia | Highly compressible and anisotropic lamellar ceramic sponges with superior thermal insulation and acoustic absorption performances[END_REF][START_REF] Cummer | Controlling sound with acoustic metamaterials[END_REF][START_REF] Zhu | Implementation of dispersion-free slow acoustic wave propagation and phase engineering with helicalstructured metamaterials[END_REF][START_REF] Kadic | 3d metamaterials[END_REF][START_REF] Assouar | Acoustic metasurfaces[END_REF][START_REF] Zhao | A scalable high-porosity wood for sound absorption and thermal insulation[END_REF][START_REF] Jiang | Ultralight metal foams[END_REF][START_REF] Qiu | Biomimetic superelastic graphene-based cellular monoliths[END_REF][START_REF] Si | Ultralight nanofibre-assembled cellular aerogels with superelasticity and multifunctionality[END_REF] . Metamaterials for noise control are moving from conceptual design [START_REF] Xue | Topological acoustics[END_REF][START_REF] Yin | Band transition and topological interface modes in 1d elastic phononic crystals[END_REF][START_REF] Beli | Wave attenuation and trapping in 3d printed cantilever-in-mass metamaterials with spatially correlated variability[END_REF][START_REF] Hu | Experimental demonstration of a three-dimensional omnidirectional and broadband acoustic concentrator using an anisotropic metamaterial[END_REF] to practical implementation [START_REF] Morandi | Standardised acoustic characterisation of sonic crystals noise barriers: Sound insulation and reflection properties[END_REF][START_REF] Xie | Wavefront modulation and subwavelength diffractive acoustics with an acoustic metasurface[END_REF][START_REF] Singh | Hybrid fractal acoustic metamaterials for low-frequency sound absorber based on cross mixed micro-perforated panel mounted over the fractals structure cavity[END_REF][START_REF] Gupta | Metamaterial foundation for seismic wave attenuation for low and wide frequency band[END_REF][START_REF] Tang | Hybrid acoustic metamaterial as super absorber for broadband low-frequency sound[END_REF][START_REF] Xie | Acoustic imaging with metamaterial luneburg lenses[END_REF][START_REF] Song | Sound pressure level gain in an acoustic metamaterial cavity[END_REF][START_REF] D'alessandro | Low frequency 3d ultra-wide vibration attenuation via elastic metamaterial[END_REF] . Fractal geometries have been used to design electromagnetic and mechanical metamaterials [START_REF] Park | Unusually high ratio of shear modulus to young's modulus in a nano-structured gyroid metamaterial[END_REF][START_REF] Jiang | 3d printed auxetic mechanical metamaterial with chiral cells and re-entrant cores[END_REF][START_REF] Lv | Origami based mechanical metamaterials[END_REF][START_REF] Gatt | Hierarchical auxetic mechanical metamaterials[END_REF] . Moreover, fractal mechanical metamaterials possess strength, shape stability under large deformations, and significant damage tolerance [START_REF] Zhang | Harnessing fractal cuts to design robust lattice metamaterials for energy dissipation[END_REF] . The Hilbert lattice [START_REF] Hilbert | Über die stetige abbildung einer linie auf ein flächenstück[END_REF] is one of the most heavily explored fractals in metamaterial design, especially for electronics/electromagnetic applications. Fractal antennas have shown enhanced performance compared to traditional designs since they can operate at multiple frequencies without load and are more compact than those featuring other geometries [START_REF] Thekkekara | Bioinspired fractal electrodes for solar energy storages[END_REF][START_REF] Maragkou | Fractal future[END_REF][START_REF] Fan | Fractal design concepts for stretchable electronics[END_REF][START_REF] Elwi | Printed microwave metamaterial-antenna circuitries on nickel oxide polymerized palm fiber substrates[END_REF] . Recent studies in the field of acoustics have improved the understanding of the Hilbert space-filling curve pattern in metamaterials, providing numerous benefits ranging from high porosity, multiple resonances, multi-modes, and sub-wavelength scales [START_REF] Comandini | Sound absorption in hilbert fractal and coiled acoustic metamaterials[END_REF][START_REF] Song | Broadband focusing acoustic lens based on fractal metamaterials[END_REF] . This work describes, for the first time, a numerical and experimental evaluation of the acoustic transmission loss capability of Hilbert fractal metamaterials.

A point of particular novelty here is in the improved understanding of the physics behind the frequencies corresponding to transmission loss peaks. As will be demonstrated, those frequencies correspond to the phenomenon of impedance mismatch, with an analytical formulation that describes how to calculate those frequencies also being presented. Furthermore, we demonstrate that the use of the Hilbert fractal pattern to design passive tools for acoustic control relies on the capability of the metamaterials geometries to provide tunable transmission loss in the lower part of the acoustic spectrum. This is an advantage when compared with commonly used acoustic porous materials having the same thickness [START_REF] Droz | Improving sound transmission loss at ring frequency of a curved panel using tunable 3d-printed small-scale resonators[END_REF][START_REF] Mulholland | The effect of sound-absorbing materials on the sound insulation of single panels[END_REF] or -more significantly -to the narrow-band effect of Helmholtz resonators [START_REF] Abbad | Adaptive helmholtz resonator based on electroactive polymers: modeling, characterization, and control[END_REF][START_REF] Fahy | Foundations of engineering acoustics[END_REF] or straight slits with similar or narrower widths (see the Supplementary Information). The fractal topologies covered here range from the 0 th (Fig. 1a) to the 4 th order (Fig. 1e). We have used additive manufacturing to generate samples of fractal metamaterials made using polylactic acid (PLA) because of its recyclability and good printability. One of the design features of the fractal specimens is the presence of inlets and outlets for the impinging and exiting acoustic waves (Fig. 1ae). The geometry of the fractal patterns is here defined in nondimensional terms, using the lateral dimension of the metamaterial sample n (50.8 mm in our case), the gap width w (see Fig. 1g), and the fractal order ξ . The parameter η = w/n represents the ratio between gap width (w) and side dimension of the specimen (n); meanwhile, the porosity is defined as ϕ = η(2 ξ ). An impedance tube with four microphones was used to obtain the experimental measurements of the TL (Fig. 1i). Numerical Finite Element simulations using COMSOL Multiphysics [START_REF] User's | version 5.6, comsol[END_REF] featuring different models representing the air inside the cavities of the fractal metamaterials, and analytical models were also performed. The models were used here to benchmark the results and understand some of the physical mechanisms underpinning the amplitude and frequency of the TL peaks.

Results and Discussion

We performed a comparison between experimental and finite element-simulated transmission losses for the Hilbert fractal metamaterials with a gap width of 1.0 mm (i.e., corresponding to a value of η equal to 2 %). The size of the gap is the same 1a). (c) First (Fig. 1b). (d) Second (Fig. 1c). (e) Third (Fig. 1d). (f) Fourth (Fig. 1e) orders.

2/10 (a) (b) (c) (d) (e) (f)

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on both sides of the sample. The finite element results here are related to narrow-acoustics models of the cavity [START_REF] User's | version 5.6, comsol[END_REF] inside the fractal metamaterials. There is a good agreement between experiments and simulations, as can be seen in Fig. 2. As also shown in the Supplementary Information (Fig. S2), Hilbert geometries from the second order with w = 1 mm can provide a greater transmission loss compared to a block of bulk PLA with the same dimension n. The mechanism that generates the peaks of the transmission loss is an impedance mismatch between the air inside the tube and the metamaterial.

The metamaterial provides an increasing number of minimum transmission coefficient (TC) values in the frequency spectrum investigated according to the fractal order, whereas the peaks of the transmission loss correspond to frequencies related to the minimum TC values. The transmission loss has a behaviour similar to an inverse pass filter, for which the transmission loss is maximum and therefore the TC is at its minimum. Moreover, as shown in Fig. S4, the high values of the peak of the reflection coefficient highlight the multi-resonance behaviour due to the Fabry-Pérot resonance. The sample made of bulk PLA has its own Fabry-Pérot resonant frequencies. Those however lie outside the spectrum of frequencies investigated in this work. At the same time, the metamaterial can minimise and maximise the mismatch at multiple frequencies, as shown in Fig. S4. A dip in the reflection coefficient corresponds to a drop in the transmission loss output. The fractal acoustic metamaterial can therefore provide multiple maximisations and minimisations of the impedance mismatch by a single fractal geometry cavity. A description of the model, and a comparison between lossless, viscous and experimental results related to configurations from the 0 th to the 4 th fractal order are presented in the Supplementary Information. Both viscous-based and lossless formulations for the fluid are able to capture the overall transmission loss properties inside the fractal cavities, and the narrow-acoustics simplification is adequate to follow the experimental results; moreover, as described by the work of Ward, et al 60 , the particle velocity field inside the fractal channels is lower when thermoviscous losses are considered in the FEM model, as shown in the Supplementary Information Fig. S5. The lossless model however provides large dips of transmission loss at frequencies slightly larger than those predicted by the viscous models, and provided by the experimental results. It is interesting to note the behaviour of the TL for the various configurations. The order zero (i.e., the topology with the cavity extending to the full length of the cubic sample -Fig. 1a). has a peak of the transmission loss equal to 37 dB at 1546 Hz (Fig. 2b). A peak of 37 dB is however observed at 785 Hz in the case of the first fractal order (Fig. 2c). The second Hilbert fractal order possesses two peaks at 390 Hz and 1212 Hz, both with TL values of 37 dB and 38 dB, respectively (Fig. 2d). The third order shows the presence of four transmission loss peaks at 195 Hz, 611 Hz, 1033 Hz, and 1455 Hz, all with a TL value of 38 dB (Fig. 2e). Finally, the fourth fractal order exhibits seven TL peaks at 310 Hz, 526 Hz, 743 Hz, 961 Hz, 1180 Hz, 1399 Hz, and 1618 Hz, with TL values of 38 dB, 38 dB, 39 dB, 39 dB, 40 dB, 40 dB, and 41 dB, respectively (Fig. 2 f ). We also include the results related to the transmission losses of the empty holder (Fig. 1 f ); as expected, the TL values here were the lowest amongst all the configurations considered (Fig. 2a). In summary, both experiments and simulations show the presence of multiple peaks and minima of the transmission loss; the number of those peaks and dips depends on the fractal order of the pattern of the metamaterial considered. The maxima of the transmission loss all show a nearly constant magnitude, with their position dependent on the length of the fractal and, consequently, on a higher Hausdorff dimension [START_REF] He | Hilbert cube model for fractal spacetime[END_REF] . Despite the complexity of the acoustic behaviour of these fractal-shaped metamaterials, it is possible to identify some general trends in their TL response. Increased fractal orders feature larger numbers of TL peaks with a very similar magnitude (see Table 2 and Fig. 3c)). This behaviour can be explained by observing that the performance of the TL magnitude of the metamaterial is mainly dependent upon the geometry of the opening slot, rather than on the fractal order itself.

Gap Width η

Fractal Table 1 shows the dependence of the magnitude of the transmission loss peak values from the FEM simulations versus the normalised gap width, the fractal orders and the porosity of the metamaterials. One can notice that the most significant factor affecting the values of the transmission losses is the gap width, regardless of the fractal order or porosity considered (see also Fig. 3c). Of particular interest is also the effect of the fractal orders on the frequencies corresponding to the peak of the transmission losses, which are also known to affect the acoustic absorption properties of Hilbert fractal metamaterials [START_REF] Comandini | Sound absorption in hilbert fractal and coiled acoustic metamaterials[END_REF] . The results suggest that while the order of the fractal pattern can impact both peak TL magnitudes and related frequencies, it is not the dominant factor to control those frequencies in configurations with varying normalised gap widths. Furthermore, the results indicate that variations of the porosity have a limited effect on the transmission losses (Fig. 3c and Table 1).

The results from the experimental and numerical transmission losses show that the frequencies corresponding to the peak of the TL can be approximated using equation (Eq.1), which is also adopted for the design of musical (wind) instruments [START_REF] Ruiz | Hearing the transformation of conical to closed-pipe resonances[END_REF][START_REF] Ayers | The conical bore in musical acoustics[END_REF] . Formula (Eq. 1) is related to open-closed one-dimensional resonating cavities and considers even harmonics only (i.e., r = m + 1 with m ∈ N) [START_REF] Ruiz | Hearing the transformation of conical to closed-pipe resonances[END_REF] . The resonant frequency for an open-closed cavity representing the fractal path is equal to:

f n = rc 4(L + ∆L) (1)
In Eq. 1, c is the speed of sound in air, 343 m/s and L = n(2 ξ ) is the length of the fractal cavity inside the metamaterial sample, and ∆L is the extra length of the fractal channel that needs to be considered due to thermoviscous losses [START_REF] Ward | Boundary-layer effects on acoustic transmission through narrow slit cavities[END_REF][START_REF] Jaouen | Length correction of 2d discontinuities or perforations at large wavelengths and for linear acoustics[END_REF] . The term ∆L is equal to 8w 3π [START_REF] Ward | Boundary-layer effects on acoustic transmission through narrow slit cavities[END_REF] . A comparison of the normalised standard deviations of the of the frequencies related to peaks of the TL and calculated via FEM and the analytical formulas is shown in Fig. S6 of the Supplementary Information. The analytical formulas include the terms ∆L = 0 (Fabry-Pérot model) and ∆L = 8

√

nw [START_REF] Jaouen | Length correction of 2d discontinuities or perforations at large wavelengths and for linear acoustics[END_REF] .

At the frequencies corresponding to the various cavity modes of the fractal patterns, the peak of the TL is always close to a value of 37 dB; this is true also within the low-frequency range within 250Hz -400Hz if the percentage opening on both sides of the metamaterial is 2%, no matter the fractal order considered (Table 1). This simplification is valid, especially for the case of the first peaks related to the first acoustic modes in the lower frequency range (Table 1). Table 2 summarises the numerical (narrow-acoustic approximation [START_REF] User's | version 5.6, comsol[END_REF] ) and analytical results related to the TL for different gap widths and fractal orders. The vibrational modes corresponding to the TL peaks are also indicated. The standard deviation of the distribution of frequencies associated with transmission loss peaks for the different gap widths is normalised against the value of 1688 Hz, which corresponds to the average frequency of the TL peak related to the 0 th order when the gap width (η) is varied from 2% to 20% (Fig. 3d-g). Notably, the frequency corresponding to the TL peaks predicted by the analytical open-closed resonator formula is also close to the numerical FE one (Table 2, fifth column). The number of vibrational modes within the frequency range investigated here increases with the increasing fractal order, and the frequencies related to the TL peaks shift to higher 5/10 values. The magnitudes of TL peaks are again almost constant, with only slight variations attributed to the change in the length of the fractal pattern inside the metamaterial. These findings imply that by increasing the fractal order, it is possible to achieve more resonant peaks, resulting in more effective broadband transmission loss, and also sound absorption [START_REF] Comandini | Sound absorption in hilbert fractal and coiled acoustic metamaterials[END_REF] . Moreover, both numerical and analytical modelling approaches provide a good approximation of the actual behaviour of these fractal acoustic metamaterials (Fig. 2, and Fig. 3ab).

Conclusion

This work has shown the relationship between the fractal order and the transmission loss in acoustic metamaterials with Hilbert fractal patterns. The equivalent porosity and gap widths play a crucial role in determining the values of the transmission loss (Fig. 3c), with the gap width being the most significant factor impacting the magnitudes of the TL (Fig. 3ab and Table 1) that create the impedance mismatch responsible for the generation of the TL maxima. Our experimental and numerical results show that a decrease in the transmission loss is generated by the increase of the gap width, with a consequent reduction of the performance of the acoustic metamaterial. We have also investigated the influence of fractal order on transmission loss. Our simulations show that the fractal order plays a role in determining the number of TL peaks, but not in their magnitude. Finally, the frequencies corresponding to the peaks of the transmission losses can be well approximated and predicted by considering the fractal patterns as open-closed one-dimensional cavities and calculating the corresponding resonance frequencies. The formula 1 makes it possible at the design stage to relate those frequencies to the fractal order and nondimensional gap width of the fractal acoustic metamaterial.

Methods

All the samples were printed using the Raise3D Pro3 Plus 3D Printer. The machine was set up with a 0.1 mm layer height, 100% infill density, and an extrusion nozzle of 0.4 mm. 3DJake recycled PLA was used as the filament. The transmission loss have been measured following the ASTM E2611-09 standard [START_REF] Astm | Standard test method for measurement of normal incidence sound transmission of acoustical materials based on the transfer matrix method[END_REF] , using a Brüel & Kjaer impedance tube (Fig. 1i) with a four microphones configuration and double load measurements. The gap widths of the metamaterials samples are 1 mm, 2 mm, 4 mm and 8 mm, corresponding to 2% (Fig. 3d), 4% (Fig. 3e), 8% (Fig. 3 f ) and 16% (Fig. 3g). All the tests were performed by mounting the fractal metamaterial sample in a cylindrical holder (Fig. 1 fh). The FEM simulations were performed using the Comsol Multiphysics 5.6 software 59 with a 3D model of the impedance tube and the metamaterials specimens [START_REF] Comandini | Sound absorption in hilbert fractal and coiled acoustic metamaterials[END_REF] . The FEM model does not consider the presence of the PLA material, only the fluid cavity is modelled, with loss-less, thermoviscous and narrow-viscous domains. The part of the impedance tube model outside the test room was represented with a lossless air fluid, with probes simulating the position of the four microphones. The frequency range investigated in both FEM and experiments was from 200 Hz to 1600 Hz.

S.4 How the fractal acoustic metamaterial generates sound transmission loss

Hilbert fractal acoustic metamaterials operate as pass filters at specific frequencies. As illustrated in Fig. S4, from (a) to (d), when a minimum of the reflection coefficient occurs, a reduction in transmission loss happens. A single fractal geometry cavity can have multiple maxima and minima of the impedance mismatch, generating transmission loss peaks according to the overall cavity length, fractal order, side dimensions n of the cube, and gap width w.

200 400 600 800 1000 1200 1400 1600

S.5 Particle velocity field

To better understand how thermoviscous air dissipation impacts the propagation of acoustic waves within narrow channels, we show the particle velocity field in the lower section of the fractal pattern where the inlet and outlet are positioned (Fig. S5) This plots demonstrate a distinct contrast between the finite element model accounting for the thermoviscous dissipations and the lossless model. This difference becomes particularly apparent as the fractal order increases, and hence the channel length becomes larger. The data clearly show significant discrepancy between predicted and experimented transmission loss occurs when ignoring the effect of the thermoviscous dissipation, as underlined by Ward, et al.

Figure 1 .

 1 Figure 1. The Hilbert fractal acoustic metamaterials and the impedance tube used in this work. (a) The 0 th order with underlined n, the side dimension of the cube -50.8 mm, in this case. (b) 1 st , (c) 2 nd , (d) 3 rd , and (e) 4 th orders of the Hilbert fractal. (f) Sample holder used for the samples inside the test room of the impedance tube. The holder has an external diameter m of 100 mm and thickness n of 50.8 mm. (g) Example of the cubic specimen (d), having been partially inserted into the holder (f), showing the gap width w. (h) Holder and sample completely assembled. (i) The impedance tube.

Figure 2 .

 2 Figure 2. Comparison of the transmission loss between experimental results (black) and FEM (red) for the various Hilbert fractal orders. All the measurements and simulations were made taking into account the cylindrical holder of Fig.1 f . (a) Only the holder (Fig.1 f ). (b) Zeroth (Fig.1a). (c) First (Fig.1b). (d) Second (Fig.1c). (e) Third (Fig.1d). (f) Fourth (Fig.1e) orders.

Figure 3 .

 3 Figure 3. Experimental (a) and FE (b) contour plots of the TL as a function of the frequencies and normalised gap widths. (c) Variation of the transmission loss peaks calculated via FEM for the fractal orders and their porosity when the normalised gap changes. Example of the 2 nd order Hilbert fractal with a gap width of (d) 1 mm, (e) 2 mm, (f) 4 mm, and (g) 8 mm.

Figure S4 .

 S4 Figure S4. FEM results related to the transmission loss and reflection coefficients for w = 1 mm gap width. Subfigure (a) presents data related to the first-order Hilbert fractal, (b) for the second, (c) for the third and (d) for the fourth Hilbert fractal orders. Across all subplots, the minima of the reflection coefficient (R) correspond with drops of transmission loss (TL).

Table 1 .

 1 Normalized

		Order ξ	Porosity ϕ	TL [dB]
	2%	0 th , 1 st , 2 nd , 3 rd , 4 th	2%, 4%, 8%, 16%, 31%	37
	4%	0 th , 1 st , 2 nd , 3 rd , 4 th	4%, 8%, 16%, 31%, 63%	31
	6%	0 th , 1 st , 2 nd , 3 rd , 4 th 6%, 12%, 24%, 47%, 94%	27
	8%	0 th , 1 st , 2 nd , 3 rd	8%, 16%, 31%, 63%	25
	10%	0 th , 1 st , 2 nd , 3 rd	10%, 20%, 39%, 79%	23
	12%	0 th , 1 st , 2 nd , 3 rd	12%, 24%, 47%, 94%	21
	14%	0 th , 1 st , 2 nd	14%, 28%, 55%	19
	16%	0 th , 1 st , 2 nd	16%, 31%, 63%	18
	18%	0 th , 1 st , 2 nd	18%, 35%, 71%	17
	20%	0 th , 1 st , 2 nd	20%, 39%, 79%	16

gap, fractal orders, porosity, and TL values for the classes of fractal metamaterials evaluated in this work. The leftmost column shows the normalized gap values, while the second column displays the corresponding fractal orders, which are proportional to the gap width. The third column contains the equivalent porosity of each fractal order in terms of relative gap width. The fourth column shows the values of the peaks of the transmission loss calculated via FEM and related to the first cavity mode of the fractals for a given gap width.

Table 2 .

 2 Cavity resonances predicted via Finite Element (narrow viscous model) and the analytical formulation for the various fractal orders. The Finite Element results related to each mode are calculated by averaging the values of those modes over the different gap widths considered in this work. The standard deviations of those frequencies are normalised against the corresponding average value. The last column refers to the analytical resonance frequency related to an open-closed cavity.

Acknowledgements G.C. acknowledges the support of UK EPSRC through the EPSRC Centre for Doctoral Training in Advanced Composites for Innovation and Science [EP/L016028/1], and Rafael Heeb, Emanuele De Bono and Martin Gillet for technical assistance. M.O.

acknowledges the support of EIPHI (ANR-17-EURE-0002). F.S. acknowledges the support of ERC-2020-AdG-NEUROMETA (No. 101020715). V.T. acknowledges funding from the EPSRC (EP/R01650X/1). All the Authors would like to thank the anonymous referees for the constructive suggestions that have helped to improve the quality of the manuscript.

Data availability

The datasets used and analysed during the current study are available from the corresponding author on reasonable request.

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Author contributions statement

Competing interests

The authors declare no conflicts of interest exist for the development of this work.

Supplementary Information

S.1 Comparison between experiments and FE models

Fig. S1 shows a comparison between experimental results and Finite Element Method (FEM) simulations for three different metamaterials configurations. The first FEM model (lossless) considers only the Helmholtz equations into account. The second FEM model (thermoviscous) accounts for the acoustic thermoviscous dissipation caused by the linearized Navier-Stokes equations. The third FEM model (narrow acoustics) takes into account the overall effects of the linearized Navier-Stokes equations, but at lower fidelity in narrow geometries. The Lossless model provides a good approximation of the experimental results in the case of the sample's holder (Fig. S1a), order zero (Fig. S1b) and the first Hilbert fractal order (Fig. S1c). However, the lossless modelling approach becomes less reliable in prediction of the TL values from the second fractal order onwards (Fig. S1d-f). Hilbert fractal acoustic metamaterials considered here range from 0 th order to the 4 th order. The PLA bulk cube has a nominal near-zero porosity (i.e., maximum infill ratio). Notably, the transmission losses for Hilbert fractal metamaterials from the 2 nd to the 4 th order are larger than those of the PLA cube in specific frequency ranges.

S.3 Benchmark of the experimental results shown in this work against those related to other materials and FEM data

We benchmark here our experimental results against those from melamine foam [START_REF] Droz | Improving sound transmission loss at ring frequency of a curved panel using tunable 3d-printed small-scale resonators[END_REF] and polyurethane foam 56 panels with the same thickness (50.8 mm), plus three different types of Helmholtz resonators. The first resonator uses an origami approach to modulate the air volume of the resonator chamber [START_REF] Abbad | Adaptive helmholtz resonator based on electroactive polymers: modeling, characterization, and control[END_REF] . The second one uses a 3 mm internal porous material [START_REF] Fahy | Foundations of engineering acoustics[END_REF] , while the third is a classic resonator [START_REF] Fahy | Foundations of engineering acoustics[END_REF] . Moreover, we also show a comparison of the FEM transmission losses between the third and fourth Hilbert fractal order, and the transmission loss generated by a multi-slit array with a gap width of 0.2 mm, 0.5 mm, 1 mm, and 2 mm. The design of the slits has been taken from the work of Ward, et al. [START_REF] Ward | Boundary-layer effects on acoustic transmission through narrow slit cavities[END_REF] .

S.6 Normalised standard deviation

Normalised standard deviation for the three different ∆L considered in Eq.1 for every acoustic mode within the frequency range investigated. In the case of ∆L = 0, we observe a fast degeneration of the frequency predicted for acoustic modes after the first. ∆L = 8w 3π is the best predictor for the frequencies corresponding to the TL maxima within the gap width range considered in this work. A possible correction for the channel length is ∆L = 8 3π 3 2

√

nw 60 ; when using this expression, one can observe a slightly worse prediction that the one provided by ∆L = 8w 3π . This is true for all the fractal orders of the metamaterials, apart from the 0 th one, for which we have a simple slit without any ninety-degree angles inside. The calculation to normalise the standard deviation is the following:

Where NSD is the normalised standard deviation along the several gap widths analysed. A is the average value of the acoustic mode with Eq.1. B and C are the average FEM values for the acoustic modes and for the standard deviation.

Figure S6

. Comparative analysis of the three analytical formulas predicting the frequency position of transmission loss peak between 200 Hz and 1600 Hz. The normalised standard deviation is calculated as the difference between the average value of the acoustic modes from the analytical formulas and the FEM simulations, divided by the standard deviation of the frequencies from the FEM simulations. The three formulas differ in considering the correction of the channel length within the fractal. The first formula does not account for any artificial length correction and maintains a reasonably low normalised standard deviation only for the first acoustic mode of the fractals, with an increasing deviation for subsequent modes. The second formula, which considers an artificial elongation as a function of the gap width (w), maintains a low normalised standard deviation across all investigated acoustic modes. The third formula, which considers a length correction as a function of the fractal slit area, also maintains a low normalised standard deviation across all modes but performs worse than the second formula, except for the first acoustic mode of a simple slit without fractal geometry.

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Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐ The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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