(2019). Analysis of a fractal ultrasonic transducer with a range of piezoelectric length scales.

The transmission and reception sensitivities of most piezoelectric ultrasonic transducers are enhanced by their geometrical structures. This structure is normally a regular, periodic one with one principal length scale, which, due to the resonant nature of the devices, determines the central operating frequency. There is engineering interest in building wide-bandwidth devices, and so it follows that, in their design, resonators that have a range of length scales should be used. This paper describes a mathematical model of a fractal ultrasound transducer whose piezoelectric components span a range of length scales. There have been many previous studies of wave propagation in the Sierpinski gasket but this paper is the first to study its complement. This is a critically important mathematical development as the complement is formed from a broad distribution of triangle sizes, whereas the Sierpinski gasket is formed from triangles of equal size. Within this structure, the electrical and mechanical fields fluctuate in tune with the time-dependent displacement of these substructures. A new set of basis functions is developed that allow us to express this displacement as part of a finite element methodology. A renormalization approach is then used to develop a recursion scheme that analytically describes the key components from the discrete matrices that arise. Expressions for the transducer’s operational characteristics are then derived and analysed as a function of the driving frequency. It transpires that the fractal device has a significantly higher reception sensitivity (18 dB) and a significantly wider bandwidth (3 MHz) than an equivalent Euclidean (standard) device.


Introduction
Ultrasonic transducers are devices, which convert mechanical vibrations into electrical energy and conversely convert electrical energy into mechanical vibrations (Beyer & Letcher, 1969;Yang, 2006a,b). One of the main uses of ultrasonic transducers is the interrogation of optically opaque media, by emission of a mechanical wave (converted from electrical to mechanical energy) and the reception of the same wave (after navigating the medium of interest, the wave is converted back from mechanical to electrical energy). Piezoelectric ultrasonic transducers are the most common devices used to assess the structural integrity of safety critical components in, e.g. nuclear plants and are used extensively in medical imaging and therapies; there is therefore considerable interest in improving the design of these sensors and the global market for such devices is set to reach $6 billion by 2020 (iRAP, 2016). Piezoelectric ultrasonic transducers ordinarily use regular, periodic geometrical structures (Algehyne & Mulholland, 2015b;Alippi et al., 1993;Hayward, 1984;Mulholland, 2008;Orr et al., 2007). 606 E. A. ALGEHYNE AND A. J. MULHOLLAND Many biological species produce and receive ultrasound; however, generally speaking, their auditory physiology is highly complex in structure and often consists of resonators with a range of length scales (Algehyne & Mulholland, 2015a;Barlow et al., 2016;Chiselev et al., 2009;Eberl et al., 2000;de Espinosa et al., 2005;Miles & Hoy, 2006;Müller, 2004;Müller et al., 2006;Nadrowski et al., 2008;Robert & Göpfert, 2002). Due to this range of length scales, these natural transducers are able to operate over a wide range of frequencies (so-called broadband transducers). There is a strong motivation therefore to design a device whose structure spans a range of length scales. Fractal structures are a natural choice for such a device as they consist of a variety of length scales Orr et al., 2008) and are therefore a prime candidate for investigation. Various authors have analysed wave propagation in fractal media (Abdulbake et al., 2003(Abdulbake et al., , 2004Barlow, 1998;Derfel et al., 2012;Falconer & Hu, 2001;Giona, 1996;Giona et al., 1996;Kigami, 2001;Strichartz, 1999) and in particular the Sierpinski gasket and its graph counterpart (Algehyne & Mulholland, 2015a, 2017. However, the Sierpinski gasket consists of a series of triangles that are all of the same size. To circumvent this disadvantage, this paper studies wave propagation in the complement of the Sierpinski gasket as this consists of triangles that span a broad range of length scales. This appears to be the first time wave propagation has been studied in this structure and opens up new possibilities for mathematicians who wish to study other types of field equations in a structure with multiple length scales. So this paper marks a significant departure from the previous works of Algehyne & Mulholland (2015a, 2017 as the introduction of piezoelectric resonators covering a broad range of length scales will produce a set of resonances across a range of frequencies and so provide the physical basis for a broadband device. This switch to the piezoelectric material being located in the complement of the Sierpinski gasket is the critical step in producing a marked improvement in the device performance. A new graph is derived here to describe this fractal and it transpires that it has the same adjacency matrix as its dual but now the edges are weighted. The physical model consists of the elastodynamic equations, Gauss's law for the electrical activity and a coupling of the electrical and mechanical waves through the piezoelectric constitutive equations. To describe this model on the fractal graph, a new set of finite element basis functions are derived whose support consists of the edges of the graph. The discrete equations that arise require the inversion of a large (in the case of a pre-fractal) or infinite-dimensional (for the fractal case) matrix in order to describe the dynamics of the device. Fortunately, the self-similar structure of the graph enables a renormalization approach to be utilized, which leads to a new set of nonlinear recursion relations. These recursion relations can be used to study the pre-fractal structures (for a finite number of iterations) or the fractal structure (by studying their steady states). This paper then compares the key operational characteristics of the pre-fractal ultrasound transducer with that of a standard transducer design. It transpires that the fractal device has a significantly higher reception sensitivity (18 dB) and a significantly wider bandwidth (3 MHz) than an equivalent Euclidean (standard) device (Hayward, 1984).

The Sierpinski gasket dual
This transducer starts off as a piezoelectric crystal in a triangular structure, connected to three half-sized copies of itself (see Fig. 1). The next generation (n = 2) connects three half-sized copies of the smaller triangles to each of these triangles. Continuing in this way, the complement (or dual) of the standard Sierpinski gasket is produced. Using the complement (the black triangles in Fig. 1) is vital as it has a range of triangle sizes, whereas the Sierpinski gasket is composed of triangles of the same size (the white triangles in Fig. 1) for a given fractal generation level.

Problem statement
Consider the vibrations of a Sierpinski gasket dual (denote this domain by Ω) made from a piezoelectric material (we will focus on PZT-5H) (Yang, 2006a) whose constitutive equations can be used to describe the interplay between the electrical and mechanical behaviour in Ω (Yang, 2006a,b) where the stress tensor is T ij , the stiffness tensor is c ijkl , the strain tensor is S kl , the piezoelectric tensor is e kij , the electric field vector is E k , the electrical displacement tensor is D i and the permittivity tensor is ε ik (where the Einstein summation convention is adopted). The strain tensor describes the displacement gradients u i, j by S ij = (u i, j + u j,i )/2 and the electric field vector is related to the electric potential φ via E i = −φ ,i . Governing the piezoelectric material's dynamics is the following equation: where the density is ρ T and the i th component of displacement is u i . At this point, we will consider the special case where there is only an out-of-plane displacement u = (0, 0, u 3 (x 1 , x 2 , t)) (this is the dominant direction of displacement in engineering applications). The piezoelectric material is polarized in the x 3 direction and the electrodes are arranged so that E = (E 1 (x 1 , x 2 ), E 2 (x 1 , x 2 ), 0). For the piezoelectric material, we choose PZT-5H (Auld, 1973). For this material (in Voigt notation), the various tensors above satisfy c 45 = c 54 = 0, c 44 = c 55 , ε ik = 0 apart from ε 11 = ε 22 and ε 33 , and e ikl = 0 apart from a small number of elements, which include e 113 = e 131 = e 223 = e 232 (which in Voigt notation is e 24 ). Taking all of these restrictions on board then combining (1. where ∇ 2 = ∂ 2 /∂x 2 1 + ∂ 2 /∂x 2 2 , c T = c T 44 /ρ T is the shear wave velocity and c T 44 = c 44 + e 2 24 /ε 11 is the piezoelectrically stiffened shear modulus. We apply the boundary conditions of continuity of force and displacement on ∂Ω as well as the initial conditions u 3 (x, 0) =u 3 (x, 0) = 0. The challenge is to solve the above equations to calculate the dynamics of this device and to then use that to assess its performance as an ultrasonic transducer.

Galerkin discretization
Non-dimensionalizing via θ = c T t/h (n) and using the Laplace transform L : θ → q then gives (dropping the subscript on u) (1.8) We will seek a weak solutionū ∈ H 1 (Ω) where on the boundaryū =ū ∂Ω ∈ H 1 (∂Ω). We multiply by a test function integrate over the region Ω and use Green's first identity Ω ψ ∇ 2 φ dv = ∂Ω ψ(∇φ . n) dr − Ω ∇φ . ∇ψ dv, where n is the outward pointing unit normal of surface element dr. So, we seekū ∈ H 1 (Ω) such that . Following a standard Galerkin approach, we employ the finite-dimensional subspaces S S and S B = S S ∩ H 1 B (Ω) rather than H 1 (Ω) and H 1 B (Ω) and U B ∈ S S to approximateū ∂Ω on the boundary ∂Ω. The resulting discrete problem then requiresŪ ∈ S S to be found where where {φ 1 , φ 2 , . . . , φ N n } forms a basis of S B . The dynamics of each triangular component in Ω will be described here by the dynamics of a single point at the centre of each of these components. Each of the basis functions will be centred on each of these points. Each triangular component in Ω is connected to some adjacent components and this connectivity will be captured here by joining these adjacent single points (or vertices) via edges to form a graph. The Sierpinski gasket graph of degree 3, SG, is the graph equivalent of the Sierpinski gasket (Schwalm & Schwalm, 1988). The dual graph SG is introduced in this paper and is constructed by a process that starts from the order n = 1 design (which consists of four piezoelectric triangles) and assigns a vertex to the centre of each of the smaller triangles and, by connecting these vertices together with edges whose length is the side length of the larger triangle, the SG weighted graph at generation level n = 1 is constructed (see Fig. 2).  The Sierpinski gasket has side length L units, which remain constant as the generation level n increases. The length of the smallest edge in the weighted graph SG (n) is h (n) = L/2 n and the longest edge length that connects the three SG (n−1) graphs is h (1) = L/2. Then the overall diameter of the graph L (n) SG = nL/2 and the total number of vertices is N n = 3 n . The vertex degree is 3 apart from the boundary vertices (input/output vertices), which have degree 2 and M n = 3(3 n − 1)/2 denotes the total number of edges at generation level n. The boundary vertices are used to interact with external loads (both electrical and mechanical) and fictitious vertices are introduced to cope with these interfacial boundary conditions. Denote by Ω n the set of points lying on the edges or vertices of the weighted graph SG (n) and denote the region's boundary by ∂Ω n .
Definition 1.1 Denote the set of vertices in Ω n as V Ω n , the set of fictitious vertices as V ∂Ω n (these are vertices N n + 1, N n + 2 and N n + 3), the set of interior vertices as V Ω o n (so V Ω o n = V Ω n \ V ∂Ω n ) and the set of input/output vertices as V ∂Ω o n (these are vertices 1, m n = (N n + 1)/2 and N n ) (see Figs 3 and 4). E Ω n is the set of edges in Ω n , the set of edges joining vertices in V ∂Ω o n to vertices in V ∂Ω n is   Ω n are defined similarly using the complete set of edges in Ω n . Also,Ē ∂Ω n is three copies of the boundary edges E ∂Ω n .
for p n (Fig. 5 graphically illustrates this identity) and Let φ i , i ∈ V Ω n form a set of basis functions for the vertex displacements and let (1.10) Hence, (1.9) becomes (1.15) Lemma 1.1 The basis function φ j for vertex j in the element (the edge) joining vertices j and k at fractal generation level n is given by where, for edges of length greater than h (n) and n 2, the two other adjacent vertices to vertex j are l and m. For interior edges of length h (n) and n 1, then vertex l is equal to vertex j and vertex m is the vertex that is connected to vertex j by the other edge of length h (n) . For exterior edges-these will have length h (n) -then vertices l and m are also the two interior vertices adjacent to vertex j. Hence, (1.17) Proof. The basis function is chosen to be linear having a value of one at vertex j and a value of zero at vertex k. So φ (n) j is a straight line lying in a plane S P containing the points P 2 = (x k , y k , 0) and P 1 = (x j , y j , 1). To make this plane unique, a third point is required. When edge jk is longer than h (n) , then, to retain the symmetry inherent to the SG graph, this third point is chosen as the centroid of the triangle formed by the two other vertices (x m and x l ) connected to vertex j and vertex j itself. When an interior edge jk has length h (n) , then this third point is chosen as the centroid of the triangle formed by the two interior vertices that are connected to vertex j by edges of length h (n) . So let P 3 be the point This is a scalar equation with the unknowns being the components of the vector x = (x, y, z). This equation is manipulated to make z the subject of the equation and the basis function in (1.16) is then given by the right-hand side of this equation. The gradient then follows.
For a particular element of length h (p) lying between vertex j and vertex k, the isoparametric representation, given by

Derivation of the matrix recursions
Using the basis function derived in Lemma 1.1, it can be shown that the matrix H at fractal generation level n can be related to its counterpart at level n − 1.
is the indicator function, which equals 1 if a ∈ A and 0 otherwise) and P (n) Proof. By using (1.13) and (1.18) for edge jk of length h (p) , then From (1.16), the basis function at vertex j (which has coordinates (a, b)) along a typical edge jk ∈ E and at vertex k (which has coordinates (a A similar calculation can be conducted for the case p = n albeit the basis function calculation is slightly different and follows the prescription in the proof of Lemma 2. So (2.2) becomes, for jk ∈ E (p) Ω n , n 1 and p n jk H (n,p) cd We now consider the basis functions at input/output vertices V ∂Ω o n and the integration of (2.2) in the boundary edges jk ∈ E ∂Ω n . Equation (2.2) becomes The matrix H (n) cd is assembled element by element (edge by edge) as follows: It can be shown that jk H is a block diagonal matrix of dimension N n × N n consisting of three blocks given by H (n−1) cd of dimension N n−1 × N n−1 . Now the second term on the right-hand side of (2.7) can be written as Combining this with (2.8), then (2.7) becomeŝ ji , (n) = 2 n−1 (12 + 2 4−2n + 3(2 3−n )) − 28 and χ (n) = 2 n−1 (12 + 2 3−2n + 3(2 3−n )).
Proof. By using (1.15) and (1.18) for edge ji, then where j ∈ V ∂Ω o n . From (1.16), the basis function at vertex j ∈ V ∂Ω o n (without loss of generality, we will examine the vertex with coordinates (a, 0)) along a typical edge ji ∈ E ∂Ω n is and at vertex i ∈ V ∂Ω n (which has coordinates (a + h (n) , 0)) is (2.20) Substituting (

Application of the mechanical boundary conditions
The transducer is electrically coupled to a power supply and is immersed in a mechanical load and appropriate electrical and mechanical boundary conditions can be applied (Algehyne & Mulholland, 2015a).
represents the Green's transfer matrix,B

28)
Z B is the mechanical impedance of the backing material, Z L = μ L A r /c L is the mechanical impedance of the load, Z T = c T 44 A r /c T , ζ = e 24 /ε 11 , Q is the electrical charge, ξ = A r /h (n) , A r is the cross-sectional area of the electrode, A L is the amplitude of the incoming wave that is received by the transducer (in transmission mode A L is zero), μ L is the shear modulus of the load material and c L is its wave speed. Hence,

Renormalization and the transducer operating characteristics
Using Theorem 2.2, a recursion relationship can be derived that relates the pivotal elements of the matrix whereĜ (n) = (Â (n) ) −1 andḠ (n) is a block diagonal matrix consisting of three blocks of matrixĜ (n) .

Proof. From (2.25) and (2.27), it is clear that we only need to know the elements of G (n)
ji in columns 1, m n and N n . In addition, we will only require elements U j , j ∈ V ∂Ω o n and so we only need to be able to calculate the pivotal Green's functions G (n) ij , i, j ∈ V ∂Ω o n . If we temporarily ignore matrixB in (2.26) (associated with the application of the boundary conditions), then, due to the symmetries of the weighted SG graph, we have Hence, using the N n×1 × N n×1 identity matrix denoted by I n+1 , To calculate G (n) ji , the boundary conditions must be reintroduced and it can be shown that (Algehyne & Mulholland, 2017) (3. 2) The renormalization recursion relationships for the pivotal Green's functions arise from the system of linear equations inĜ (n+1) ji . The three subgraphs of Fig. 2 have a single connection point to one another at the corners and, due to the symmetries of the SG graph, the recursions in (3.1) give rise to only two pivotal Green's functions, known as, corner-to-same corner Ĝ (n) ii =x, say, where i ∈ V ∂Ω o n and cornerto-other-corner Ĝ (n) jk =ŷ, say, where j, k ∈ V ∂Ω o n , j = k ; the so-called input/output vertices. For ease of notation, let,X =Ĝ The matrix is symmetrical and, consequently,Ĝ Lemma 3.2 The renormalization recursion relations for the pivotal Green's functions (ignoring temporarily the boundary conditions) are given bŷ Proof. The (i, j) th element of the matrix (3.1) can be written aŝ (3.5) From the definitions of W where be, dr ∈ E (1) Ω n and in particular b = (N n + 1)/2 = m n , d = N n , e = N n + 1 and r = 2N n + 1.

Results
As a result of the equations presented above, a comparison between the key operating characteristics of the transmission and reception spectra of the fractal and standard (Euclidean) designs (Hayward, 1984) can now be evaluated and compared. The key parameters of interest to be compared between both models include the amplitude (or gain) and the bandwidth (the range of frequencies over which a certain gain, expressed in decibels, is exceeded). To evaluate the transmission and reception sensitivities of the fractal device, the fractal generation levels were gradually increased and the differences in these parameters observed. However, with the aim of physically producing the device, we will focus on the fractal generation level n = 3. From a practical perspective, these fractal transducers will only be able to be manufactured at low fractal generation levels. To perform a fair comparison, the volume of piezoelectric material in the standard design (vol E ) and fractal design (vol F ) is kept consistent. The volume of the piezoelectric material in the standard design is vol E = L 2 d E = LA E , where L is the length of the front face, d E is the thickness and A E is the area occupied by the electrode. The volume of the piezoelectric material in the fractal design is vol F = S n d F , where S n is the area of the front face of the fractal piezoelectric design at generation level n (the black area in Fig. 6) and d F is the thickness; the area S n is then given by S n = √ 3L 2 1 − (3/4) n+1 /4 and equating vol E and vol F gives d F = L 2 d E /S n . The fractal transducer has one electrode of area A F = d F h (n) /2 at one face and two electrodes of area A F each on the opposite face (see Fig. 6). As the device operates essentially as a capacitor in this circuit, and since the total capacitance of two capacitors in parallel is just the sum of those two capacitances, then we take the total area to be A F = 2A F = d F h (n) = d F L/2 n . By choosing a particular value for the fractal generation level n, the thickness d E and the length L, this sets d F , which in turn sets A E , A F and the volume of piezoelectric material. The behaviour of the device is dictated by its electrical Fig. 6. A three-dimensional schematic of the Sierpinski gasket ultrasonic transducer at fractal generation level n = 2. The black triangles are the front faces of the piezoelectric material. The device is connected to an electrical circuit as shown where each electrode has surface area A F . Fig. 7. Non-dimensionalized electrical impedance (equation (3.14)) versus frequency for the SG transducer at fractal generation level n = 3 (dashed line). The non-dimensionalized electrical impedance of the standard (Euclidean) transducer (Hayward, 1984) is plotted for comparison (full line). Parameter values are given in Table 1. resonances, which in turn depend on its capacitance. The capacitance depends on the distance between the electrodes, which remains fixed in the conventional design. So, to assist in the comparison between the standard design and the fractal design, the distance between the electrodes was held fixed in the fractal case too. A different scenario could be studied where the surface area remains fixed and the edge length L is varied. To enable a frequency domain analysis, we set q → σ + jω where σ , ω ∈ R, σ is a damping term (set to zero in this paper), ω is the angular frequency and j is the square root of −1. The characteristic profile of the electrical impedance spectrum (magnitude) for both models where n = 3 is illustrated in Fig. 7. The fractal design, given by (3.14), is represented by the dashed line while the equivalent standard design is represented by the full line (Hayward, 1984). It can be seen that the behaviour is predominantly that of a capacitor it decreases according to a 1/(C 0 f ) profile where C 0 is the capacitance and f is the frequency with a few resonant peaks superimposed on this. From the above analysis, A E = √ 3 (1 − (3/4) n+1 ) 2 (n−3) A F and the coefficient of A F is monotonically increasing as a function of the fractal generation level n and is greater than one for all n 3. Hence, Table 1 Parameter values for the front and back mechanical loads and the electrical load (Mulholland et al., 2007; (Hayward, 1984) is plotted for comparison (full line). Parameter values are given in Table 1. for reasonable values of n, Since the standard design has a larger capacitance, then this explains why its electrical impedance is in general lower. The locations and magnitudes of the first maximum ( f a ) and first minimum ( f r ) turning points are the critical features for a design engineer. In transmission mode, the device should be driven at the mechanical resonance frequency (the frequency at the first minimum) where it will deliver the maximum force on the load. In reception mode however, the device should be driven at the frequency of the first maximum (the electrical resonance frequency). As shown in Fig. 7, for the standard design (full line) f r = 0.9 MHz, |Ẑ E ( f r )| = 34 dB, f a = 1.2 MHz and |Ẑ E ( f a )| = 51 dB. As discussed above, these frequencies correspond precisely to the first maximum in the transmission sensitivity plot ( Fig. 8; full line) and the reception sensitivity plot ( Fig. 9; full line). Fig. 9. Non-dimensionalized reception sensitivity (equation (3.16)) versus frequency for the SG transducer at fractal generation level n = 3 (dashed line). The non-dimensionalized reception sensitivity of the standard (Euclidean) transducer (Hayward (1984)) is plotted for comparison (full line). Parameter values are given in Table 1.

Table 2 A comparison between the operating characteristics of a fractal transducer and an equivalent standard design at fractal generation level n = 3. The mechanical resonance frequency is denoted by f r (MHz), the electrical resonance frequency is denoted by f a (MHz), the transmission sensitivity gain is denoted by ψ max (dB), the reception sensitivity gain is denoted by φ max (dB), the transmission sensitivity bandwidth is denoted by BW T (MHz) and the reception sensitivity bandwidth is denoted by BW R (MHz)
Design The piezoelectrically stiffened velocity (c T ) in (1.7), calculated from the parameter values for PZT-5H (see Auld, 1973), is 2370 min/s giving an electrical resonance frequency f a = c T /(2L) = 1.2 MHz for a device of length L = 1 mm. This agrees reasonably well with the reception sensitivity maximum for the homogenized estimate for f a (the full line in Fig. 9). The fractal device produces an electrical impedance spectrum that has a richer set of resonances and this is due to the many length scales inherent to its structure. To show all of the resonances, the zero damping case is shown (σ = 0); when even a small level of damping is added, the resonances that have a very narrow support (such as those at frequencies 2.2 MHz and around 6 MHz) do not appear in these plots. Note however that the electrical impedance gain of the fractal design is lower than the standard design by around 2 dB at the optimal reception frequency ( f (3) r = 4.2 MHz, |Ẑ E ( f r ; 3)| = 33 dB, f (3) a = 4.3 MHz and |Ẑ E ( f a ; 3)| = 49 dB). Figure 8 shows that the transmission sensitivity of the fractal design has a maximum amplitude (gain) that is higher than the Euclidean case (standard design) at its lower operating frequency (37 dB at 4.2 MHz compared to 32 dB at 3.5 MHz for the Euclidean case). However, the bandwidth is reduced; at 3 dB below the maximum amplitude of the standard design, i.e. 29 dB, the bandwidth of the fractal design is 0.1 MHz and the standard design is 0.2 MHz. Figure 9 shows that the reception sensitivity spectrum has a much larger gain; there is an 18 dB improvement in the reception sensitivity gain from the standard design to the fractal design. This peak in the reception sensitivity also results in an enhanced bandwidth; if we take the noise floor to be 3 dB below the peak gain of the standard design (i.e. the 14 dB level Downloaded from https://academic.oup.com/imamat/article-abstract/84/3/605/5381041 by guest on 16 October 2019 in this plot), then the operational bandwidth of the standard design is 0.2 MHz, whereas the fractal design has an operational bandwidth of around 3.2 MHz (a summary of these comparisons is provided in Table 2).

Summary and conclusion
Using a design based on fractal geometry, a theoretical model of a piezoelectric ultrasound transducer was produced and analysed in this paper. This showed that the fractal design has certain advantages over the current technology. The critical innovation was to base the design on the complement of the Sierpinski gasket as this introduces a range of length scales within the resonating structure. Using a finite element methodology, the weighted graph counterpart of this fractal (denoted SG) was used to support the electrical and mechanical fields. New basis functions were derived; however, since this is the first finite element analysis on this graph. These are resonating devices and since the standard design typically has a single length scale, while the fractal design has multiple length scales, it is unsurprising that this results in a much richer set of resonance frequencies; it is well known in fact that musical instruments and naturally occurring auditory systems rely precisely on this very principle. The finite element formulation discretized the problem to produce a matrix equation and, benefitting from its block diagonal structure, the inversion of this matrix was achieved using a renormalization approach. The symmetry of the structure is embedded in these renormalization equations and this enabled us to describe the dynamics of the structure via only two coupled recursion relationships. These pivotal Green's functions were then used in expressions for the transmission and reception sensitivities of the device. From a practical perspective, it will only be possible to manufacture these devices for a small number of fractal generation levels. Hence, the results produced in this paper focus on low-generation levels (pre-fractals) but it is worth emphasizing that the dynamics of the true fractal can also be examined by studying the steady states of the recursion relationships. As anticipated, the model predicts that, in comparison to the standard transducer, this pre-fractal transducer has more resonance frequencies. In addition, the fractal transducer also produced a higher amplitude (or gain) in transmission and reception sensitivities than the standard design, a 5 dB increase in transmission mode and an 18 dB increase in reception mode. The reception sensitivity also resulted in a wider bandwidth than the standard design, a 3 MHz increase. The encouraging theoretical results in this paper have led to a manufacturing programme of work. A piezoelectric ultrasonic transducer has been successfully manufactured whose design is the complement of the Sierpinski gasket; full details can be found in the paper by Fang et al. (2018). The manufacturing process involved a combination of 3D printing and a piezoelectric pillar placement methodology. As was done here in this theoretical paper, two equivalent (same volume of piezoelectric material) 1-3 piezocomposite designs were manufactured and experimentally compared, one with a standard design and one with the fractal geometry. It was found that there was a marked improvement in both bandwidth and sensitivity when using the fractal device. The fractal device was also experimentally compared to a commercial transducer and over a 100% bandwidth improvement was found.

Funding
The authors gratefully acknowledge the support given by the University of Tabuk, Ministry of Education in Saudi Arabia and the Royal Embassy of Saudi Arabia in UK. United Kingdom Engineering and Physical Sciences Research Council (EP/P005268/1).