**2. Theory**

240 Acoustic Waves – From Microdevices to Helioseismology

slippage (Zhuang et al., 2008), coverage area (Johanssmann et al., 2008), sensitivity profile (Edvardsson et al., 2005) and penetration depth of the shear acoustic wave (Kunze et al.,

Various theoretical models have been developed for quantitative characterization of the TSM sensor response to interfacial interactions. Nunalee et al (2006) developed model to predict of the TSM sensor response to a generalized viscoelastic material spreading at the sensor surface in a liquid medium. Cho et al (2007) created a model system to study the viscoelastic properties of two distinct layers, a layer of soft vesicles and a rigid bilayer. Urbakh and Daikhin (2007) developed a model to characterize the effect of surface morphology of non-uniform surface films on TSM sensor response in contact with liquid. Hovgaard et al (2007) have modeled TSM sensor data using an extension to Kevin-Voigt viscoelastic model for studying glucagon fibrillation at the solid-liquid interface. Kanazawa and Cho (2009) discussed the measurement methodologies and analytical models for

The physical description based on a wave propagation concept in a one-dimensional approximation has been proven as the best model of thickness shear mode (TSM) sensors. The fundamentals have been published in several books (Rosenbaum, 1998). Martin et al. have (1994) applied this background to sensors by using Mason's equivalent circuit to describe the thickness shear mode sensor itself and transmission lines as well as lumped elements for viscoelastic coatings, semi-infinite liquids etc.. Follow-up papers have introduced a more straightforward definition of the elements of the BVD-model (Behling et al, 1998) as well as several additional approximations, e.g. based on perturbation theory, to derive less complex equations, have suggested a simplified notation to separate the mass from so-called nongravimetric effects, or have applied the transmission line model to several subsystems (Voinova et al, 2002) for demonstration of specific situations just to call some examples. More recent papers deal with deviations from the one-dimensional approximations, e.g. by introducing generalized parameters by deriving specific solutions

TSM sensors combined with the theoretical models mentioned above were used to determine the properties of liquids (Lin et al., 1993), high protein concentration solutions

For viscoelastic layers, their mechanical impedance depends upon the density, thickness, and the complex shear modulus of the loading. Identification of the all the system parameters from the impedance measurements has been very challenging and uncertain without a priori knowledge of the thicknesses and/or some of the material properties

Furthermore, Kwoun (2006) showed the beneficial features of the multi-resonance operation of the TSM (called as "multi-resonance thickness shear mode) sensor to study the formation of biological samples, specifically collagen and albumin, on the sensor surface. In this work, it was demonstrated that the different harmonic frequency clearly showed the different characteristics of mechanical properties, especially shear modulus, of the biological sample. Although this work was one of the pioneer studies to demonstrate the strengths of the MTSM measurement technique, it is limited as it is a semi-quantitative method. Exact values of mechanical properties of anisotropic collagen and albumin samples were not able to be defined due to complexity of the non-linear simultaneous equations of the model. An improved MTSM technique combined with an advanced data analysis technique was proposed by Ergezen et al (2010). A new approach merging the multi-harmonic thickness

characterizing macromolecular assembly dynamics.

e.g. for surface roughness or with discontinuity at boundaries.

(Saluja et al., 2005), and thin polymer films (Katz et al., 1996).

(Lucklum et al. 1997).

2006).

### **2.1 Multi-Harmonic Thickness Shear Mode (MTSM) sensor**

Piezoelectric MTSM sensors transmit acoustic shear waves into a medium under test, and the waves interact with the medium. Shear waves monitor local properties of a medium in the vicinity of the sensor and of the medium/sensor interface (on the order of nm - μm); thus, they provide a very attractive technique to study interfacial processes. Measured parameters of acoustic waves are correlated with medium properties such as interfacial mass/density, viscosity, or elasticity changes taking place during chemical or biological processes.

The shear acoustic wave penetrates the medium over a very short distance. The square of the depth of penetration of an acoustic shear wave in MTSM sensor is related to medium viscosity, elasticity, density and the frequency of the wave (please see Appendix IA.) (Kwoun et al. 2006). Figure 1a shows the acoustic wave penetrating the adjacent medium and Figure 1b shows that the depth of penetration decreases at higher harmonic frequencies in a semi-infinite medium.

Therefore, by changing the frequency, one can control the distance at which the wave probes the medium. Multi-harmonic operation of MTSM sensor will enable to control the interrogating depth into the biological processes. Therefore it will provide a more in depth

Modeling of Biological Interfacial Processes Using Thickness–Shear Mode Sensors 243

An example of the MTSM's magnitude response in the vicinity of the fundamental resonant frequency is given below (figure 3a). When the TSM sensor is loaded with a biological media, there will be a shift in resonant frequency and a decrease in the magnitude. These changes can be correlated with changes in the mechanical and geometrical properties of the medium such as thickness, viscosity, density and stiffness. Depending on the changes at the interface of the sensor surface-medium interface, a positive and/or negative shift can be

Fig. 3. (a)Demonstration of a typical qualitative frequency-dependent response curve for the MTSM sensor in the vicinity of the resonant frequency; n = harmonic number, αRn=Initial maximum magnitude, fRn=Initial resonant frequency, (b) In the case of both positive and negative frequency shifts throughout the experiment, αRnI, αRnII =Instantaneous maximum magnitudes of loaded MTSM sensor at time t1 and t2 respectively, fRnI, fRnII =Instantaneous resonant frequencies of the loaded MTSM sensor at time t1 and t2 respectively (Inlet)

This section will be structured in the following manner; first, the general structure of a genetic algorithm will be explained. Second, advantages of genetic algorithm over other techniques will be discussed. Finally, implementation of MTSM-GA technique for

Basic definitions of GA terms are defined in Appendix IB. Genetic algorithm (GA) is based on the genetic processes of biological organisms (figure 4). GA works with a population of individuals, each representing a possible solution to a given problem. Each individual is assigned a fitness score according to how good a solution to the problem it is. The highly-fit individuals are given opportunities to reproduce, by cross breeding with other individuals in the population. This produces new individuals as offspring, which share some features

Complex models are ubiquitous in many applications in the fields of engineering and science. Their solution often requires a global search approach. Therefore the objective of optimization techniques is to find the globally best solution of models, in the possible

resonant frequency and magnitude are monitored as a function of time

seen in the frequency response (Figure 3b).

**2.3 MTSM/GA data processing technique** 

taken from each parent.

determination of material parameters will be explained. **Principles of operation of a genetic algorithm (GA)** 

**Comparison of GA to other data processing techniques** 

characterization of the biological interfacial processes. For example, it was suggested that cell adhesion on extra cellular matrix should be modeled as a multi-layered structure (Wegener et al. 2000). Therefore MTSM sensors can provide information about mechanical and structural properties of the biological processes from different depths (slicing the medium).

Fig. 1. a) Acoustic wave penetrating into the medium b) depth of penetration decreases at higher harmonic frequencies

It should be noted that it was assumed that the medium is semi-infinite and the mechanical properties are not frequency dependent in fig. 1.

### **2.2 Electrical response of MTSM sensor**

The MTSM sensor is a piezoelectric-based sensor which has the property that an applied alternating voltage (AC) induces mechanical shear strain and vice versa. By exciting the sensor with AC voltage, standing acoustic waves are produced within the sensor, and the sensor behaves as a resonator. The electrical response of the MTSM sensor in air over a wide frequency range is shown in figure 2, where S21 is the magnitude response of the MTSM sensor (|S21|=20log(100/(100+Zt)), Zt=total electromechanical impedance of the MTSM sensor (Rosenbaum 1998). As an example, the magnitude and phase responses of MTSM sensor are presented at the first (5 MHz), third (15 MHz), fifth (25 MHz) and seventh (35 MHz) harmonics in air.

Fig. 2. A typical a) frequency vs. magnitude response and b) frequency vs. phase response characteristic and the associated resonance harmonics for the MTSM sensor, spanning a wide frequency range (5 MHz to 35 MHz). (Insets) Magnified view of magnitude and phase response at 5 MHz

characterization of the biological interfacial processes. For example, it was suggested that cell adhesion on extra cellular matrix should be modeled as a multi-layered structure (Wegener et al. 2000). Therefore MTSM sensors can provide information about mechanical and structural properties of the biological processes from different depths (slicing the

Fig. 1. a) Acoustic wave penetrating into the medium b) depth of penetration decreases at

It should be noted that it was assumed that the medium is semi-infinite and the mechanical

The MTSM sensor is a piezoelectric-based sensor which has the property that an applied alternating voltage (AC) induces mechanical shear strain and vice versa. By exciting the sensor with AC voltage, standing acoustic waves are produced within the sensor, and the sensor behaves as a resonator. The electrical response of the MTSM sensor in air over a wide frequency range is shown in figure 2, where S21 is the magnitude response of the MTSM sensor (|S21|=20log(100/(100+Zt)), Zt=total electromechanical impedance of the MTSM sensor (Rosenbaum 1998). As an example, the magnitude and phase responses of MTSM sensor are presented at the first (5 MHz), third (15 MHz), fifth (25 MHz) and seventh (35

Fig. 2. A typical a) frequency vs. magnitude response and b) frequency vs. phase response characteristic and the associated resonance harmonics for the MTSM sensor, spanning a wide frequency range (5 MHz to 35 MHz). (Insets) Magnified view of magnitude and phase

medium).

higher harmonic frequencies

MHz) harmonics in air.

response at 5 MHz

properties are not frequency dependent in fig. 1.

**2.2 Electrical response of MTSM sensor** 

An example of the MTSM's magnitude response in the vicinity of the fundamental resonant frequency is given below (figure 3a). When the TSM sensor is loaded with a biological media, there will be a shift in resonant frequency and a decrease in the magnitude. These changes can be correlated with changes in the mechanical and geometrical properties of the medium such as thickness, viscosity, density and stiffness. Depending on the changes at the interface of the sensor surface-medium interface, a positive and/or negative shift can be seen in the frequency response (Figure 3b).

Fig. 3. (a)Demonstration of a typical qualitative frequency-dependent response curve for the MTSM sensor in the vicinity of the resonant frequency; n = harmonic number, αRn=Initial maximum magnitude, fRn=Initial resonant frequency, (b) In the case of both positive and negative frequency shifts throughout the experiment, αRnI, αRnII =Instantaneous maximum magnitudes of loaded MTSM sensor at time t1 and t2 respectively, fRnI, fRnII =Instantaneous resonant frequencies of the loaded MTSM sensor at time t1 and t2 respectively (Inlet) resonant frequency and magnitude are monitored as a function of time

### **2.3 MTSM/GA data processing technique**

This section will be structured in the following manner; first, the general structure of a genetic algorithm will be explained. Second, advantages of genetic algorithm over other techniques will be discussed. Finally, implementation of MTSM-GA technique for determination of material parameters will be explained.

### **Principles of operation of a genetic algorithm (GA)**

Basic definitions of GA terms are defined in Appendix IB. Genetic algorithm (GA) is based on the genetic processes of biological organisms (figure 4). GA works with a population of individuals, each representing a possible solution to a given problem. Each individual is assigned a fitness score according to how good a solution to the problem it is. The highly-fit individuals are given opportunities to reproduce, by cross breeding with other individuals in the population. This produces new individuals as offspring, which share some features taken from each parent.

### **Comparison of GA to other data processing techniques**

Complex models are ubiquitous in many applications in the fields of engineering and science. Their solution often requires a global search approach. Therefore the objective of optimization techniques is to find the globally best solution of models, in the possible

Modeling of Biological Interfacial Processes Using Thickness–Shear Mode Sensors 245

ranges should be reasonable for each parameter in order to determine accurate solutions. For example, for a Newtonian liquid the stiffness is 0, therefore one should not set the range to be between 1e5 N/m2 and 1e7 N/m2. If this were done the algorithm will not converge to

As shown by Kwoun (2006), the viscoelastic materials can be divided in to four regimes, namely; liquid like, soft rubber, hard rubber and solid like. As seen from table 1, the viscosity values might change between 0.001 and 0.1 kg/m.s and stiffness value changes between 0 – 1e9 N/m2 . Typical range of density values for a polymer was determined to be

> **Phase η (kg/m.s) C (N/m^2)**  Liquid like 0.001 – 0.01 0-1e5 Soft Rubber 0.01 – 0.1 0-1e5 Hard Rubber 0.01 – 0.1 1e5 – 1e7 Solid Like - 0.1 1e7 – 1e9

This section will be divided into three sections. First, the GA's main parameters such as number of populations, crossovers, mutation rates and genes per chromosome will be analyzed. Then the fitness function of the GA will be explained. Finally, the technique combination of sub-spacing and zooming to determine the values for four variables will be

Different combinations of the GA parameters were evaluated. Here, the combination that gives the best result is presented. Each variable was represented by a binary chromosome that contains 16 genes. A random population of 100 individuals was generated. Tournament

a solution because of the inappropriate choice of ranges.

Fig. 5. Basic structure of MTSM/GA technique

Table 1. Four regimes of a viscoelastic system **Genetic Algorithm and its internal functions** 

between 1000 – 1400 kg/m3.

**Selection of GA parameters** 

presented.

presence of multiple local optima. Conventional optimization and search techniques include; (1) gradient-based local optimization method, (2) random search, (3) stochastic hill climbing, (4) simulated annealing, (5) symbolic artificial intelligence and (6) genetic algorithms. The detailed information on each technique and comparisons to Genetic Algorithms (GA) are already explained by Depa and Sivanandam (2008). Here, the aim is not to analyze these techniques in detail but to show the suitability of GA as a parameter estimation algorithm. As discussed by Depa and Sivanandam, some of the advantages of GA over other techniques are: (1) it is good for multi-mode problems, (2) it is resistant to becoming trapped in local optima, (3) it performs well in large-scale optimization problems, (4) it handles large, poorly understood search spaces easily. These advantages match with the requirements for an optimization technique to be applied in this application. Therefore GA was chosen as an optimization technique and successfully combined with the MTSM technique.

Fig. 4. Flow chart of a genetic algorithm

### **Structure of the MTSM/GA technique**

The structure of MTSM-GA technique is presented in figure 5. As seen from the figure, there are two inputs to the GA, namely; range of variables and MTSM sensor response. GA outputs the determined values of the variables by using GA functions such as crossover, mutation and fitness evaluation. In the following sections, initially, the inputs to the GA will be explained. Then the structure of GA and its internal functions will be presented.

### **MTSM sensor response**

The first input to the GA is the MTSM sensor response. Both magnitude and phase responses were continuously monitored during the experiments (see materials and methods section). Then the specific points on these responses such as resonant frequency, maximum magnitude, minimum phase, frequency at minimum phase, and phase at maximum magnitude were input to GA for calculating the fitness score for each individual. The changes in these target points were calibrated with the diwater/glycerin changes.

### **Selection of the ranges for variables**

The next step of the technique is to set the ranges for the variables (chromosomes). These ranges represent the bounded space within which the GA will search for solutions. The ranges should be reasonable for each parameter in order to determine accurate solutions. For example, for a Newtonian liquid the stiffness is 0, therefore one should not set the range to be between 1e5 N/m2 and 1e7 N/m2. If this were done the algorithm will not converge to a solution because of the inappropriate choice of ranges.

Fig. 5. Basic structure of MTSM/GA technique

244 Acoustic Waves – From Microdevices to Helioseismology

presence of multiple local optima. Conventional optimization and search techniques include; (1) gradient-based local optimization method, (2) random search, (3) stochastic hill climbing, (4) simulated annealing, (5) symbolic artificial intelligence and (6) genetic algorithms. The detailed information on each technique and comparisons to Genetic Algorithms (GA) are already explained by Depa and Sivanandam (2008). Here, the aim is not to analyze these techniques in detail but to show the suitability of GA as a parameter estimation algorithm. As discussed by Depa and Sivanandam, some of the advantages of GA over other techniques are: (1) it is good for multi-mode problems, (2) it is resistant to becoming trapped in local optima, (3) it performs well in large-scale optimization problems, (4) it handles large, poorly understood search spaces easily. These advantages match with the requirements for an optimization technique to be applied in this application. Therefore GA was chosen as an optimization technique and successfully combined with the MTSM

The structure of MTSM-GA technique is presented in figure 5. As seen from the figure, there are two inputs to the GA, namely; range of variables and MTSM sensor response. GA outputs the determined values of the variables by using GA functions such as crossover, mutation and fitness evaluation. In the following sections, initially, the inputs to the GA will

The first input to the GA is the MTSM sensor response. Both magnitude and phase responses were continuously monitored during the experiments (see materials and methods section). Then the specific points on these responses such as resonant frequency, maximum magnitude, minimum phase, frequency at minimum phase, and phase at maximum magnitude were input to GA for calculating the fitness score for each individual. The

The next step of the technique is to set the ranges for the variables (chromosomes). These ranges represent the bounded space within which the GA will search for solutions. The

be explained. Then the structure of GA and its internal functions will be presented.

changes in these target points were calibrated with the diwater/glycerin changes.

technique.

Fig. 4. Flow chart of a genetic algorithm **Structure of the MTSM/GA technique** 

**Selection of the ranges for variables** 

**MTSM sensor response** 

As shown by Kwoun (2006), the viscoelastic materials can be divided in to four regimes, namely; liquid like, soft rubber, hard rubber and solid like. As seen from table 1, the viscosity values might change between 0.001 and 0.1 kg/m.s and stiffness value changes between 0 – 1e9 N/m2 . Typical range of density values for a polymer was determined to be between 1000 – 1400 kg/m3.


Table 1. Four regimes of a viscoelastic system

## **Genetic Algorithm and its internal functions**

This section will be divided into three sections. First, the GA's main parameters such as number of populations, crossovers, mutation rates and genes per chromosome will be analyzed. Then the fitness function of the GA will be explained. Finally, the technique combination of sub-spacing and zooming to determine the values for four variables will be presented.

### **Selection of GA parameters**

Different combinations of the GA parameters were evaluated. Here, the combination that gives the best result is presented. Each variable was represented by a binary chromosome that contains 16 genes. A random population of 100 individuals was generated. Tournament

Modeling of Biological Interfacial Processes Using Thickness–Shear Mode Sensors 247

thickness of the film, and complex modulus (= GI + jGII ). Therefore there are four independent variables to define the surface acoustic impedance. The MTSM sensor response contributes two parameters by providing real and imaginary part of mechanical impedance. Hence using single harmonic response results in an under-determined problem. Genetic optimization technique has been applied to under-determined problems to obtain approximate solutions with satisfactory accuracy (Wang and Dhawan, 2008). Here genetic algorithm has been improved by combining sub-space and zooming techniques. It was shown that this combination provides very good approximation with less than 1% error. First, sub-spacing method was applied. This method gives a quick idea of where the solution can be and also it decreases algorithm running time dramatically (Garaia and Chaudhurib, 2007). Therefore the solution space was divided in 10 sub-spaces. Genetic algorithm was run 5 times in each subspace. Each subspace's convergence performance was evaluated. The sub-space with the best fitness score was considered to be the candidate solution space. It was observed that the candidate sub-space had a distinct convergence performance compared to the others. This method dramatically increased the efficiency of

Secondly, GA was run 100 times (this number was chosen to have 95% confidence level and 10% confidence interval statistically). The termination criterion for each run was 500 generations. After 100 runs, it was observed that, for two out of four variables, observed points having a uniform distribution (skewness < 0.5) were accumulating around one number in a narrow range (in ±20% of candidate solution point). The average value of the observed points was also equal or very close (<5%) to solution (theoretically shown). Therefore GA was always able to converge to "the most likely" values for two out of four variables after these two steps (from our observations, mostly stiffness and thickness, and sometimes, viscosity and thickness). It was shown theoretically that one can always put these numbers, and calculate the other two variables with the error of less than <15% at this step. Then zooming method was applied to reduce the search space around the candidate optimum solution point. Several zooming methods have been developed for different applications (Ndiritu and Daniel, 2001, Kwon et al. 2003). In this project, the GA was run 30 times, and then the new range was set to be between maximum and minimum numbers of the 30 points. This zooming continued until the error was less than 1% for all variables. This

These results showed that the MTMS/GA technique combined with sub-spacing and zooming methods can be applied successfully to approximate the solution with good

The MTSM/GA technique first experimentally tested with the polymer SU8-2002 layer spin coated on sensor surface. The determined properties of the layer were compared with the values obtained from literature. The technique was then applied to obtain the mechanical and geometrical properties of a protein layer adsorbed on gold layer. The methods and

The SU 8-2002 (MicroChem) polymer solution was spin coated on MTSM sensor by using the following procedure. First, the gold electrode surface of TSM sensors was cleaned using

GA by eliminating the irrelevant solution spaces.

error was achieved after 6 zooming.

**3. Materials and methods** 

accuracy for this under-determined problem.

**a. Deposition of the thin polymer film** 

chemicals used in the experiments are described below.

selection was implemented for selection of individuals for mutation and crossover. In order to carry out the crossovers the entire population is divided into groups of 5 individuals each, these groups are randomly selected. From each group, the individual with the highest fitness together with another individual of this group are selected for crossover. The two selected individuals are the parents and yield two offspring. Both the parents and the offspring pass to the next generation. This idea was implemented in order to reduce the selection pressure.

The crossover between the parents is a simple one meaning that a random crossover point is selected and two kids' genome are formed with the left and right genes of the crossover point of each parent. A relatively high mutation probability (0.5) is present in order to avoid local minimum, otherwise all the individuals might end up having the same genome and this genome corresponding to a not optimal solution. Also elitism was implemented to assure that the best individual of a generation survives to the next generation. This ensures that the algorithm keeps the best solution until a better one is found.

### **Fitness function**

One of the most important parts of a genetic algorithm is the fitness function. The fitness function must reflect the relevant measures to be optimized. This function evaluates the function being searched for the set of parameters of each member of the population. The output of the fitness function is a vector that contains the fitness for each member of the population. This vector helps in the selection of individual for generating new offspring or individuals that will be included in the new generated population.

The approach used, in this study to model biolayers on a MTSM sensor, is Mason's transmission line model (please see Appendix C). This model is a one-dimensional model that describes the electrical characteristics of an acoustic structure wherein, each layer of load can be represented as a T-network of impedances.

Once the initial population is created the algorithm randomly generates a population (includes 100 individuals) chosen from the ranges of the variables (the section titled "selection of the ranges for variables"). Then each individual was input to fitness function (transmission line model). The error between the model (transmission line model) and the experimental results were compared by using the following equation:

$$\text{fit\\_fonc} = \frac{10}{1 + \left(\sqrt{(\alpha\_{\text{ke}} - \alpha\_{\text{ke}})^2} + \sqrt{(f\_{\text{ke}} - f\_{\text{th}})^2} + \sqrt{(P\_{\text{Ak}} - P\_{\text{At}})^2} + \sqrt{(f\_{\text{Ak}} - f\_{\text{At}})^2} + \sqrt{(\alpha\_{\text{Ak}} - \alpha\_{\text{At}})^2} + \sqrt{(f\_{\text{Ak}} - f\_{\text{At}})^2}\right)} + \frac{1}{2}$$

The denominator of this function represents the difference between the model and the experimental data (we use the plus one in order to avoid the eventual division by zero). In this project, rather than fitting the whole magnitude and phase curve, certain points such as αR = maximum magnitude, fR = resonant frequency, PM = minimum phase, fM = resonant frequency at minimum phase, αAR = minimum magnitude, fAR = anti-resonant frequency has been compared between the model and the experimental results. Subscript "e" indicates experimental results and subscript "t" stands for theoretical model. This function is monotonously increasing with the kindness of the solution provided by the genetic algorithm. The algorithm was terminated at after 500 generations.

### **Set-up of the Genetic Algorithm**

Acoustic impedance seen at the sensor/film interface is derived from transmission line theory (Martin and Frye 1991). Surface mechanical impedance is related to density and

selection was implemented for selection of individuals for mutation and crossover. In order to carry out the crossovers the entire population is divided into groups of 5 individuals each, these groups are randomly selected. From each group, the individual with the highest fitness together with another individual of this group are selected for crossover. The two selected individuals are the parents and yield two offspring. Both the parents and the offspring pass to the next generation. This idea was implemented in order to reduce the

The crossover between the parents is a simple one meaning that a random crossover point is selected and two kids' genome are formed with the left and right genes of the crossover point of each parent. A relatively high mutation probability (0.5) is present in order to avoid local minimum, otherwise all the individuals might end up having the same genome and this genome corresponding to a not optimal solution. Also elitism was implemented to assure that the best individual of a generation survives to the next generation. This ensures

One of the most important parts of a genetic algorithm is the fitness function. The fitness function must reflect the relevant measures to be optimized. This function evaluates the function being searched for the set of parameters of each member of the population. The output of the fitness function is a vector that contains the fitness for each member of the population. This vector helps in the selection of individual for generating new offspring or

The approach used, in this study to model biolayers on a MTSM sensor, is Mason's transmission line model (please see Appendix C). This model is a one-dimensional model that describes the electrical characteristics of an acoustic structure wherein, each layer of

Once the initial population is created the algorithm randomly generates a population (includes 100 individuals) chosen from the ranges of the variables (the section titled "selection of the ranges for variables"). Then each individual was input to fitness function (transmission line model). The error between the model (transmission line model) and the

22 2 2 2 2

α α

100 1 *Re Rt Re Rt Me Mt Me Mt ARe ARt ARe ARt fit\_ func ( ( ) (f f ) (P P ) (f f ) ( ) (f f ) )*

<sup>=</sup> + −+ −+ − + − + − + −

The denominator of this function represents the difference between the model and the experimental data (we use the plus one in order to avoid the eventual division by zero). In this project, rather than fitting the whole magnitude and phase curve, certain points such as αR = maximum magnitude, fR = resonant frequency, PM = minimum phase, fM = resonant frequency at minimum phase, αAR = minimum magnitude, fAR = anti-resonant frequency has been compared between the model and the experimental results. Subscript "e" indicates experimental results and subscript "t" stands for theoretical model. This function is monotonously increasing with the kindness of the solution provided by the genetic

Acoustic impedance seen at the sensor/film interface is derived from transmission line theory (Martin and Frye 1991). Surface mechanical impedance is related to density and

that the algorithm keeps the best solution until a better one is found.

individuals that will be included in the new generated population.

experimental results were compared by using the following equation:

algorithm. The algorithm was terminated at after 500 generations.

load can be represented as a T-network of impedances.

α α

**Set-up of the Genetic Algorithm** 

selection pressure.

**Fitness function** 

thickness of the film, and complex modulus (= GI + jGII ). Therefore there are four independent variables to define the surface acoustic impedance. The MTSM sensor response contributes two parameters by providing real and imaginary part of mechanical impedance. Hence using single harmonic response results in an under-determined problem. Genetic optimization technique has been applied to under-determined problems to obtain approximate solutions with satisfactory accuracy (Wang and Dhawan, 2008). Here genetic algorithm has been improved by combining sub-space and zooming techniques. It was shown that this combination provides very good approximation with less than 1% error.

First, sub-spacing method was applied. This method gives a quick idea of where the solution can be and also it decreases algorithm running time dramatically (Garaia and Chaudhurib, 2007). Therefore the solution space was divided in 10 sub-spaces. Genetic algorithm was run 5 times in each subspace. Each subspace's convergence performance was evaluated. The sub-space with the best fitness score was considered to be the candidate solution space. It was observed that the candidate sub-space had a distinct convergence performance compared to the others. This method dramatically increased the efficiency of GA by eliminating the irrelevant solution spaces.

Secondly, GA was run 100 times (this number was chosen to have 95% confidence level and 10% confidence interval statistically). The termination criterion for each run was 500 generations. After 100 runs, it was observed that, for two out of four variables, observed points having a uniform distribution (skewness < 0.5) were accumulating around one number in a narrow range (in ±20% of candidate solution point). The average value of the observed points was also equal or very close (<5%) to solution (theoretically shown). Therefore GA was always able to converge to "the most likely" values for two out of four variables after these two steps (from our observations, mostly stiffness and thickness, and sometimes, viscosity and thickness). It was shown theoretically that one can always put these numbers, and calculate the other two variables with the error of less than <15% at this step.

Then zooming method was applied to reduce the search space around the candidate optimum solution point. Several zooming methods have been developed for different applications (Ndiritu and Daniel, 2001, Kwon et al. 2003). In this project, the GA was run 30 times, and then the new range was set to be between maximum and minimum numbers of the 30 points. This zooming continued until the error was less than 1% for all variables. This error was achieved after 6 zooming.

These results showed that the MTMS/GA technique combined with sub-spacing and zooming methods can be applied successfully to approximate the solution with good accuracy for this under-determined problem.
