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Numerical prototyping methods in microsystem accelerometers design
Microelectronics Reliability 51 (2011) 1276–1282
Contents lists available at ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
Numerical prototyping methods in microsystem accelerometers design Łukasz Dowhan´ a,⇑, Artur Wymysłowski a, Stanisław Kalicin´ski b, Paweł Janus b a b
Wrocław University of Technology, Faculty of Microsystem Electronics and Photonics, Janiszewskiego 11/17, 50-372 Wrocław, Poland Institute of Electron Technology, Al. Lotników 32/46, 02-668 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 6 December 2010 Received in revised form 9 February 2011 Accepted 8 March 2011 Available online 2 April 2011
a b s t r a c t The authors of this research would like to present the numerical prototyping methods in reference to design of microsystem silicon accelerometer. As an example device the capacitive accelerometer was taken into account. The accelerometer was prepared as a FEM parametric model. The model was then processed by the numerical optimization algorithms in order to find the optimal design parameters. For this purpose, the so-called numerical multi-objective optimization process was carried out. The idea was to find the optimal solution in reference to more than one optimizing criterion. As a result the set of optimal solutions was obtained and this method seems to be promising for similar purposes. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Numerical prototyping methods are used in many fields of engineering [4]. Apart from the finite element method that is widely used for many years, the numerical optimization algorithms in reference to FE-models seem to be very promising and powerful tools. Additionally, in complex tasks to solve, there is a need for applying so-called multi-objective optimization, where more than one criterion is taken into account as the objective function. This optimization process seems to be neglected in the modern engineer’s practice. Therefore, the goal of this research is the application of such numerical prototyping process including the finite element modeling as well as the numerical multi-objective optimization to the design of example microsystem accelerometer. Firstly, the designed device is presented and then the applied numerical methods in reference to the FE-model are carried out.
Pressure and displacement have frequently been evaluated by this means, and a number of manufacturers now supply capacitive accelerometers. These accelerometers are claimed to give better performance than conventional types for measuring low-frequency, low-level acceleration. Capacitive accelerometers are micro-machined from single-crystal silicon (Fig. 1). Usually, a conducting layer is deposited onto one surface of a silicon block. A second conducting layer is laid down on one slide of a second block, which acts as the seismic mass. The seismic mass is supported by beams and is separated from the base by an air gap. The two halves of the sensor are electrostatically bonded together. Signal processing electronics can be incorporated within the sensor package [1].
2.1. Structure design of accelerometer 2. Description of selected accelerometer There are many different types of accelerometers. In each design the transducer relies on detecting the motion of a seismic mass to measure acceleration. However, there are differences in the way in which this motion is sensed. These differences give rise to variations in parameters such as the bandwidth, sensitivity or susceptibility to interference of the sensor [2]. Taking the application and the measuring parameters into account the capacitive accelerometer was chosen for further design and investigations. Additionally, the use of a capacitance change is well established in sensor design as a measuring technique. ⇑ Corresponding author. Tel.: +48 71 355 9866x64. E-mail address: [email protected] (Ł. Dowhan´). 0026-2714/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2011.03.017
The designed example accelerometer consists of a seismic mass bonded to the silicon frame by 4 flexible silicon springs. The mass is designed as a comb-mass and the combs together with the stationary fingers are the plates of the capacitors. Therefore, the mass moves horizontally (in-plane). In order to release the mass from the substrate, in the etching process the layer with holes was applied and the holes were etched in the mass. The filling factor between the mass with and without holes is 0.49. The accelerometer is supposed to measure the capacitance change in order to extract the acceleration in the range 0–20g. The accelerometer layout is shown in Fig. 2. The outer dimensions of the designed accelerometer are 3.8 mm 4.5 mm 0.36 mm and the inner dimensions are 1.4 mm 1.2 mm 0.36 mm. The seismic mass thickness is set to 15 lm and the gap between the mass and the substrate is set
´ et al. / Microelectronics Reliability 51 (2011) 1276–1282 Ł. Dowhan
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finite element method. The model type was defined as a 3D solid. To create the mesh the 20-node elements were used. They were referred as the stress–strain quadratic elements with reduced integration (C3D20R). These elements were distinguished by three active degrees of freedom (ux, uy, uz). The model was elaborated in Abaqus software. The meshed model is presented in Fig. 4. In order to speed-up the calculations only the symmetric 1=4 of the accelerometer was taken into consideration. The boundary conditions were defined so, that the model behaved as the complete silicon structure. 3.1. Preliminary calculations The simulation was carried out by applying the force as a load. Knowing the mass and the force, the applied acceleration can be calculated as follows
F ¼ma)a¼
F m
ð3Þ
The force F was applied to the seismic mass in order to produce its in-plane movement. As the response the displacement d of the capacitor plates was obtained. With this value it is possible to calculate the capacitance after loading as
Fig. 1. Typical comb drive sensor (ITE Poland).
CðdÞ ¼
A e d0 þ d
ð4Þ
where A is the area of the capacitor plates, d0 – the distance between plates before loading, e – permittivity, which equals
e ¼ er e0
ð5Þ
where er is the relative permittivity of the material (er = 1 for air), e0 is the permittivity for the vacuum (e0 ¼ 8:85 1012 F=m). The dependence between the distance d and acceleration a is as follows
d¼
m a k
ð6Þ
where m is the mass and k is the spring constant dependent on the material and dimensions. With the Eq. (3) the dependence between capacitance C and the acceleration a can be defined:
CðaÞ ¼ Fig. 2. The accelerometer layout (ITE Poland).
ð1Þ
The perforated mass weight (with holes) can be therefore calculated knowing its volume and the silicon density (qSi = 2329.6 kg/ m3) and equals
m ¼ 1:03 105 g
ð7Þ
Finally, the capacitance change of the sensor during acceleration can be calculated from the following equation:
to 32 lm. The perforated mass volume can be obtained from the dimensions and multiplication by the filling factor and equals
V m ¼ 4:43 103 mm3
A d0 þ mk a
DCðaÞ ¼ C 0 CðaÞ )
A A e e d0 d0 þ mk a
ð8Þ
Similar calculations referring the capacitive accelerometers are also presented in [5]. Using the finite element simulations of the designed sensor and the above equations the capacitance changes were calculated and taken to the optimization process.
ð2Þ
The outer dimensions are presented in Fig. 3. The yellow1 area represents the perforated mass. 3. Finite element modeling In order to simulate and predict the acceleration of the designed sensor the numerical model was elaborated. It was made using the 1 For interpretation of color in Fig. 3 and Table 2, the reader is referred to the web version of this article.
4. Design for optimization Optimization is in fact a separate branch of knowledge which focuses on development an application of different algorithms in order to find out an optimal solution due to a selected criteria and defined constraints. Advanced optimization problems arise most often in case of a large number of input variables, non-linear output behavior and multi-extreme responses with local and global aspects. The latter is the key selection criteria of an optimization algorithms as the global extreme is most often a final expectation.
´ et al. / Microelectronics Reliability 51 (2011) 1276–1282 Ł. Dowhan
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Fig. 3. Dimensions of the designed accelerometer.
Moreover, in complex tasks there is a need for finding a solution that satisfies more than one objective function (the goal of optimization). In such cases the optimization process is called the multiobjective optimization. In contrast to the single-objective optimization (with one objective function only), the multi-objective optimization can be defined as follows:
min FðXÞ ¼ ff1 ðXÞ; f2 ðXÞ; . . . ; fm ðXÞg for
X ¼ ðx1 ; x2 ; . . . ; xn ÞT
ci ðXÞ ¼ 0; i ¼ 1; 2; . . . ; m ci ðXÞ P 0; i ¼ m þ 1; . . . ; n
ð9Þ
where fi(X) are the objective function and the F(X) is the optimization criterion based on the set of m objective functions; X – vector of optimization variables; ci – equality and inequality constraints functions. The main difference between these two types of optimization is that multi-objective optimization gives as a result a set of equivalent solutions (called the Pareto set) and single-objective methods give in response only one solution. In engineering applications the multi-objective optimal solution can formulated in a form of a set, which is optimal in the Pareto sense. The set means that there is not only one optimal solution but a whole set of solutions, which are equally acceptable from an engineering point of view and equivalent from mathematical point of view. The solution is optimal in the Pareto sense, if there is not a better solution in reference to at least one criterion without worsening the solution in reference to all other ones. The idea of Pareto set is shown in Fig. 5. The values x and y on the right hand-side represent the input space and the f(x,y), g(x,y) are the objective functions dependent on these input (x,y) values. The optimization process in this research was designed and carried out in Optimus software. The goal was to find the optimal geometrical parameters in reference to three selected criteria:
Fig. 4. FE-model of the designed silicon accelerometer.
Fig. 5. The idea of Pareto set.
´ et al. / Microelectronics Reliability 51 (2011) 1276–1282 Ł. Dowhan
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Fig. 6. Optimization diagram.
Table 1 Input parameters for the optimization process. Parameter
Minimal
Nominal
Maximal
Spring_thickness (lm) Mass_thickness (lm) capacitor_plate_len (lm)
1.0 5.0 80.0
3.0 15.0 120.0
6.0 25.0 160.0
linearity, sensitivity and resonant frequency. In order to carry out such optimization process the so-called multi-objective optimization was applied. The optimization’s diagram is shown in Fig. 6. 4.1. Input parameters As the input parameters for the optimization the three geometrical parameters were selected: spring thickness, seismic mass thickness and the capacitor’s plate length. The values are shown in Table 1. Input parameters are also presented in Fig. 7. The capacitor’s plate length is referenced to all combs in the sensor. 4.2. Output parameters As it was mentioned before three output parameters (optimization’s objectives) were selected: linearity, sensitivity, resonant frequency.
Fig. 7. Input parameters.
Linearity and sensitivity supposed to be as high as possible (the maximization problem) and the resonant frequency should be as close as possible to the value of 5 kHz. It is worth to mention that for the input parameters the linearity (given by the R coefficient
Fig. 8. Dependence of linearity and sensitivity on accelerometer’s spring thickness.
´ et al. / Microelectronics Reliability 51 (2011) 1276–1282 Ł. Dowhan
1280
from the least-squares method) and the sensitivity have the opposing sense (input values for the maximal linearity are simultaneously the values for the minimal sensitivity). Fig. 8 shows such example dependence obtained in FEM calculations by changing the spring thickness from 1 to 6 lm (with nominal input values of mass thickness and plates length). This opposite relation between the goals is also the reason of using the multi-objective optimization.
4.3. Multi-objective optimization To perform the multi-objective optimization, the normalboundary intersection method was applied. This algorithm is dedicated to find a Pareto front (set) in non-linear multi-objective optimization problems [3]. This method gives as a response the so-called Pareto set of the given problem. The points on the Pareto set are even spread due to the fact that this method uses a scalarization scheme. This scheme is performed in this way, that a uni-
form spread in parameters will give rise to a near uniform spread in points on the efficient boundary [3]. The idea of this algorithm is presented in Fig. 9. In this algorithm the points that belong to the Pareto set are found by carrying out the optimizations that are restricted to lines that intersect perpendicularly the section between the ideal objective values. It is proved that this algorithm is independent of the relative scales of the functions [3]. The described algorithm belongs to the group of multi-objective methods that use the weighting of the objective functions in order to find every Pareto point. It means that the algorithm optimizes a function, which is a weighted combination of the defined objective functions [6]. The sum of the weights is always 1 and for every set of weights values, the algorithm gives 1 Pareto point. In some cases one objective can be favoured and the other can be neglected (for example, the following set of weights: w1 = 0.7, w2 = 0.2, w3 = 0.1), but in the ‘‘multi-objective’’ meaning, the combined optimized function gets as optimal solution as for the other set of weights. For every weight the values are changed from 0 to 1 (in the way that the sum of weights always has to achieve 1) and therefore every objective function has a chance to be favoured. As a result, the set of Pareto points is obtained and in order to choose one solution, some additional conditions have to be applied (or even socalled ‘‘decision support system’’ in complex tasks). This research only demonstrates the usage of such optimization approach and therefore the external condition is defined as relatively simple: the significant solution is the one that gives the resonant frequency value as close as possible to the defined one (5 kHz). 5. Results
Fig. 9. Idea of normal-boundary intersection algorithm.
Using the finite element simulations of the designed sensor and the Eqs. (1)–(8) the capacitance changes were measured for the acceleration from 1 to 20g. The spring area of the FE-model (with nominal input parameters) for the acceleration 20g is shown in Fig. 10. The scale presents the displacement of the seismic mass.
Fig. 10. Displacement of the seismic mass (deformation scale factor = 50).
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Fig. 11. Pareto points in reference to three selected criteria.
Fig. 12. Example dependence diagram: mass and spring thickness dependent on resonant frequency.
For the acceleration value 20g the displacement of the mass was observed as 112 nm. For this value, the capacitance change was calculated as
DC a¼20g ¼ 1:14 104 F
ð10Þ
Such capacitance value is relatively small, but it can be measured by the MS3110 Universal Capacitive ReadoutTM IC provided by Irvine Sensors.
As the optimization results the so-called Pareto set was obtained (Fig. 12). As it was mentioned before, all the points are Pareto-optimal, which means that they are equal from the mathematical point of view. In order to choose the best solution from this set, the additional condition defined above was used (the point with the closest value of resonant frequency to 5 kHz). The chosen Pareto point is distinguished by the red color and its values (input and output) are presented in Table 2.
´ et al. / Microelectronics Reliability 51 (2011) 1276–1282 Ł. Dowhan
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Table 2 Optimal design parameters in reference to 3 selected criteria. Input parameter
Nominal
Optimal
Spring_thickness (lm) Mass_thickness (lm) capacitor_plate_len (lm)
3.0 15.0 120.0
1.63 12.23 83.80
Objectives Linearity [1] Sensitivity (F/g) Resonant frequency (Hz)
Nominal 0.99976 5.69E16 13278
Optimal 0.9921 1.64E15 5101.6
Fig. 13. Contribution of input parameters in optimization process.
The example diagram with dependence of resonant frequency on the mass and spring thickness is shown in Fig. 11. This diagram was made by interpolation of the optimization’s FEM results. As it can be seen on the diagram, the highest resonant frequency is obtained by higher values of spring thickness and maximal value of mass thickness. The optimal resonant frequency is obtained in the middle of the mass thickness range. Table 2 shows the optimal output parameters which correspond to the input parameters taken into account during the optimization process. It can be noticed that the response from the nominal input parameters is better in reference to linearity and sensitivity. However, the value of resonant frequency is higher than 13 kHz. The goal was to find the input parameters, which give the best linearity and sensitivity as possible but also the resonant frequency as close as possible to 5 kHz. Thus, the solution presented in Table 2 satisfies all selected criteria (with the emphasis on resonant frequency). Taking into account the contribution of input parameters in the optimization process, it turned out that the spring thickness influenced the most on the results. Second parameter was the mass thickness. Capacitor’s plate length influenced the least (Fig. 13). The disadvantage of this optimization method is the time needed to find the optimal solutions. It is mainly due to the fact, that the applied normal-boundary intersection method needs many iterations (about 800) to achieve the goal. In this approach
one iteration means one FEM calculation, which is time-consuming (about 1 min. 30 s). The solution would be the application of some indirect optimization method (Design of Experiments – DOE) or different direct method (simulated annealing, for instance). 6. Summary In the frame of this research the numerical prototyping was applied to design the example acceleration sensor. The device was prepared as a FEM model and then optimized in order to find the best design parameters in reference to linearity, sensitivity and resonant frequency. The goal was to carry out the optimization process with more than one criterion and therefore the multiobjective optimization algorithm was used. The algorithm called normal-boundary intersection gave the Pareto set from which one solution was selected. The spring thickness – one of the input parameters taken to the optimization process influenced the most on the results. The potential results of this research can be dedicated mainly to the automotive industry. One of the goals is the fuel reduction by the real time wheel balancing measurement. Such measurement can be done by the acceleration measurements. It seems that such numerical approach of designing the microsystem accelerometers is promising. Therefore in the future, the research in the field of applying numerical prototyping in microelectronics will be continued. Other numerical optimization algorithms will be tested in reference to obtain better efficiency and optimization time. Acknowledgements This work was performed in a frame of the ‘‘Nanoelectronics for Safe, Fuel Efficient and Environment Friendly Automotive Solutions (SE2A)’’ project; ENIAC proposal no. 12009. Authors acknowledge Wroclaw Centre for Networking and Supercomputing (WCSS) for the possibility of using modeling software and hardware. References [1] Galayko D, Kaiser A, Buchaillot L. Design, realization and testing of micromechanical resonators in thick-film silicon technology with postprocess electrode-to-resonator gap reduction. J Micromech Microeng 2003;13:134–40. [2] Galayko D, Kaiser A, Buchaillot L. High-frequency high-Q micro-mechanical resonators in thick epipoly technology with post-process gap adjustment. IEEE 2002. [3] Shukla PK. On the normal boundary intersection method for generation of efficient front. Springer; 2007. p.310–7. [4] Erickson D. Towards numerical prototyping of labs-on-chip: modeling for integrated microfluidic devices. Microfluid Nanofluid 2005;1(4):301–18. [5] Xi-hong M, Jun L. Design of hybrid integrated capacitive acceleration sensor. ICECE Proc 2010:744–7. [6] Ehrgott M. Multicriteria optimization. Springer-Verlag; 2005.
Contents lists available at ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
Numerical prototyping methods in microsystem accelerometers design Łukasz Dowhan´ a,⇑, Artur Wymysłowski a, Stanisław Kalicin´ski b, Paweł Janus b a b
Wrocław University of Technology, Faculty of Microsystem Electronics and Photonics, Janiszewskiego 11/17, 50-372 Wrocław, Poland Institute of Electron Technology, Al. Lotników 32/46, 02-668 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 6 December 2010 Received in revised form 9 February 2011 Accepted 8 March 2011 Available online 2 April 2011
a b s t r a c t The authors of this research would like to present the numerical prototyping methods in reference to design of microsystem silicon accelerometer. As an example device the capacitive accelerometer was taken into account. The accelerometer was prepared as a FEM parametric model. The model was then processed by the numerical optimization algorithms in order to find the optimal design parameters. For this purpose, the so-called numerical multi-objective optimization process was carried out. The idea was to find the optimal solution in reference to more than one optimizing criterion. As a result the set of optimal solutions was obtained and this method seems to be promising for similar purposes. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Numerical prototyping methods are used in many fields of engineering [4]. Apart from the finite element method that is widely used for many years, the numerical optimization algorithms in reference to FE-models seem to be very promising and powerful tools. Additionally, in complex tasks to solve, there is a need for applying so-called multi-objective optimization, where more than one criterion is taken into account as the objective function. This optimization process seems to be neglected in the modern engineer’s practice. Therefore, the goal of this research is the application of such numerical prototyping process including the finite element modeling as well as the numerical multi-objective optimization to the design of example microsystem accelerometer. Firstly, the designed device is presented and then the applied numerical methods in reference to the FE-model are carried out.
Pressure and displacement have frequently been evaluated by this means, and a number of manufacturers now supply capacitive accelerometers. These accelerometers are claimed to give better performance than conventional types for measuring low-frequency, low-level acceleration. Capacitive accelerometers are micro-machined from single-crystal silicon (Fig. 1). Usually, a conducting layer is deposited onto one surface of a silicon block. A second conducting layer is laid down on one slide of a second block, which acts as the seismic mass. The seismic mass is supported by beams and is separated from the base by an air gap. The two halves of the sensor are electrostatically bonded together. Signal processing electronics can be incorporated within the sensor package [1].
2.1. Structure design of accelerometer 2. Description of selected accelerometer There are many different types of accelerometers. In each design the transducer relies on detecting the motion of a seismic mass to measure acceleration. However, there are differences in the way in which this motion is sensed. These differences give rise to variations in parameters such as the bandwidth, sensitivity or susceptibility to interference of the sensor [2]. Taking the application and the measuring parameters into account the capacitive accelerometer was chosen for further design and investigations. Additionally, the use of a capacitance change is well established in sensor design as a measuring technique. ⇑ Corresponding author. Tel.: +48 71 355 9866x64. E-mail address: [email protected] (Ł. Dowhan´). 0026-2714/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2011.03.017
The designed example accelerometer consists of a seismic mass bonded to the silicon frame by 4 flexible silicon springs. The mass is designed as a comb-mass and the combs together with the stationary fingers are the plates of the capacitors. Therefore, the mass moves horizontally (in-plane). In order to release the mass from the substrate, in the etching process the layer with holes was applied and the holes were etched in the mass. The filling factor between the mass with and without holes is 0.49. The accelerometer is supposed to measure the capacitance change in order to extract the acceleration in the range 0–20g. The accelerometer layout is shown in Fig. 2. The outer dimensions of the designed accelerometer are 3.8 mm 4.5 mm 0.36 mm and the inner dimensions are 1.4 mm 1.2 mm 0.36 mm. The seismic mass thickness is set to 15 lm and the gap between the mass and the substrate is set
´ et al. / Microelectronics Reliability 51 (2011) 1276–1282 Ł. Dowhan
1277
finite element method. The model type was defined as a 3D solid. To create the mesh the 20-node elements were used. They were referred as the stress–strain quadratic elements with reduced integration (C3D20R). These elements were distinguished by three active degrees of freedom (ux, uy, uz). The model was elaborated in Abaqus software. The meshed model is presented in Fig. 4. In order to speed-up the calculations only the symmetric 1=4 of the accelerometer was taken into consideration. The boundary conditions were defined so, that the model behaved as the complete silicon structure. 3.1. Preliminary calculations The simulation was carried out by applying the force as a load. Knowing the mass and the force, the applied acceleration can be calculated as follows
F ¼ma)a¼
F m
ð3Þ
The force F was applied to the seismic mass in order to produce its in-plane movement. As the response the displacement d of the capacitor plates was obtained. With this value it is possible to calculate the capacitance after loading as
Fig. 1. Typical comb drive sensor (ITE Poland).
CðdÞ ¼
A e d0 þ d
ð4Þ
where A is the area of the capacitor plates, d0 – the distance between plates before loading, e – permittivity, which equals
e ¼ er e0
ð5Þ
where er is the relative permittivity of the material (er = 1 for air), e0 is the permittivity for the vacuum (e0 ¼ 8:85 1012 F=m). The dependence between the distance d and acceleration a is as follows
d¼
m a k
ð6Þ
where m is the mass and k is the spring constant dependent on the material and dimensions. With the Eq. (3) the dependence between capacitance C and the acceleration a can be defined:
CðaÞ ¼ Fig. 2. The accelerometer layout (ITE Poland).
ð1Þ
The perforated mass weight (with holes) can be therefore calculated knowing its volume and the silicon density (qSi = 2329.6 kg/ m3) and equals
m ¼ 1:03 105 g
ð7Þ
Finally, the capacitance change of the sensor during acceleration can be calculated from the following equation:
to 32 lm. The perforated mass volume can be obtained from the dimensions and multiplication by the filling factor and equals
V m ¼ 4:43 103 mm3
A d0 þ mk a
DCðaÞ ¼ C 0 CðaÞ )
A A e e d0 d0 þ mk a
ð8Þ
Similar calculations referring the capacitive accelerometers are also presented in [5]. Using the finite element simulations of the designed sensor and the above equations the capacitance changes were calculated and taken to the optimization process.
ð2Þ
The outer dimensions are presented in Fig. 3. The yellow1 area represents the perforated mass. 3. Finite element modeling In order to simulate and predict the acceleration of the designed sensor the numerical model was elaborated. It was made using the 1 For interpretation of color in Fig. 3 and Table 2, the reader is referred to the web version of this article.
4. Design for optimization Optimization is in fact a separate branch of knowledge which focuses on development an application of different algorithms in order to find out an optimal solution due to a selected criteria and defined constraints. Advanced optimization problems arise most often in case of a large number of input variables, non-linear output behavior and multi-extreme responses with local and global aspects. The latter is the key selection criteria of an optimization algorithms as the global extreme is most often a final expectation.
´ et al. / Microelectronics Reliability 51 (2011) 1276–1282 Ł. Dowhan
1278
Fig. 3. Dimensions of the designed accelerometer.
Moreover, in complex tasks there is a need for finding a solution that satisfies more than one objective function (the goal of optimization). In such cases the optimization process is called the multiobjective optimization. In contrast to the single-objective optimization (with one objective function only), the multi-objective optimization can be defined as follows:
min FðXÞ ¼ ff1 ðXÞ; f2 ðXÞ; . . . ; fm ðXÞg for
X ¼ ðx1 ; x2 ; . . . ; xn ÞT
ci ðXÞ ¼ 0; i ¼ 1; 2; . . . ; m ci ðXÞ P 0; i ¼ m þ 1; . . . ; n
ð9Þ
where fi(X) are the objective function and the F(X) is the optimization criterion based on the set of m objective functions; X – vector of optimization variables; ci – equality and inequality constraints functions. The main difference between these two types of optimization is that multi-objective optimization gives as a result a set of equivalent solutions (called the Pareto set) and single-objective methods give in response only one solution. In engineering applications the multi-objective optimal solution can formulated in a form of a set, which is optimal in the Pareto sense. The set means that there is not only one optimal solution but a whole set of solutions, which are equally acceptable from an engineering point of view and equivalent from mathematical point of view. The solution is optimal in the Pareto sense, if there is not a better solution in reference to at least one criterion without worsening the solution in reference to all other ones. The idea of Pareto set is shown in Fig. 5. The values x and y on the right hand-side represent the input space and the f(x,y), g(x,y) are the objective functions dependent on these input (x,y) values. The optimization process in this research was designed and carried out in Optimus software. The goal was to find the optimal geometrical parameters in reference to three selected criteria:
Fig. 4. FE-model of the designed silicon accelerometer.
Fig. 5. The idea of Pareto set.
´ et al. / Microelectronics Reliability 51 (2011) 1276–1282 Ł. Dowhan
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Fig. 6. Optimization diagram.
Table 1 Input parameters for the optimization process. Parameter
Minimal
Nominal
Maximal
Spring_thickness (lm) Mass_thickness (lm) capacitor_plate_len (lm)
1.0 5.0 80.0
3.0 15.0 120.0
6.0 25.0 160.0
linearity, sensitivity and resonant frequency. In order to carry out such optimization process the so-called multi-objective optimization was applied. The optimization’s diagram is shown in Fig. 6. 4.1. Input parameters As the input parameters for the optimization the three geometrical parameters were selected: spring thickness, seismic mass thickness and the capacitor’s plate length. The values are shown in Table 1. Input parameters are also presented in Fig. 7. The capacitor’s plate length is referenced to all combs in the sensor. 4.2. Output parameters As it was mentioned before three output parameters (optimization’s objectives) were selected: linearity, sensitivity, resonant frequency.
Fig. 7. Input parameters.
Linearity and sensitivity supposed to be as high as possible (the maximization problem) and the resonant frequency should be as close as possible to the value of 5 kHz. It is worth to mention that for the input parameters the linearity (given by the R coefficient
Fig. 8. Dependence of linearity and sensitivity on accelerometer’s spring thickness.
´ et al. / Microelectronics Reliability 51 (2011) 1276–1282 Ł. Dowhan
1280
from the least-squares method) and the sensitivity have the opposing sense (input values for the maximal linearity are simultaneously the values for the minimal sensitivity). Fig. 8 shows such example dependence obtained in FEM calculations by changing the spring thickness from 1 to 6 lm (with nominal input values of mass thickness and plates length). This opposite relation between the goals is also the reason of using the multi-objective optimization.
4.3. Multi-objective optimization To perform the multi-objective optimization, the normalboundary intersection method was applied. This algorithm is dedicated to find a Pareto front (set) in non-linear multi-objective optimization problems [3]. This method gives as a response the so-called Pareto set of the given problem. The points on the Pareto set are even spread due to the fact that this method uses a scalarization scheme. This scheme is performed in this way, that a uni-
form spread in parameters will give rise to a near uniform spread in points on the efficient boundary [3]. The idea of this algorithm is presented in Fig. 9. In this algorithm the points that belong to the Pareto set are found by carrying out the optimizations that are restricted to lines that intersect perpendicularly the section between the ideal objective values. It is proved that this algorithm is independent of the relative scales of the functions [3]. The described algorithm belongs to the group of multi-objective methods that use the weighting of the objective functions in order to find every Pareto point. It means that the algorithm optimizes a function, which is a weighted combination of the defined objective functions [6]. The sum of the weights is always 1 and for every set of weights values, the algorithm gives 1 Pareto point. In some cases one objective can be favoured and the other can be neglected (for example, the following set of weights: w1 = 0.7, w2 = 0.2, w3 = 0.1), but in the ‘‘multi-objective’’ meaning, the combined optimized function gets as optimal solution as for the other set of weights. For every weight the values are changed from 0 to 1 (in the way that the sum of weights always has to achieve 1) and therefore every objective function has a chance to be favoured. As a result, the set of Pareto points is obtained and in order to choose one solution, some additional conditions have to be applied (or even socalled ‘‘decision support system’’ in complex tasks). This research only demonstrates the usage of such optimization approach and therefore the external condition is defined as relatively simple: the significant solution is the one that gives the resonant frequency value as close as possible to the defined one (5 kHz). 5. Results
Fig. 9. Idea of normal-boundary intersection algorithm.
Using the finite element simulations of the designed sensor and the Eqs. (1)–(8) the capacitance changes were measured for the acceleration from 1 to 20g. The spring area of the FE-model (with nominal input parameters) for the acceleration 20g is shown in Fig. 10. The scale presents the displacement of the seismic mass.
Fig. 10. Displacement of the seismic mass (deformation scale factor = 50).
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Fig. 11. Pareto points in reference to three selected criteria.
Fig. 12. Example dependence diagram: mass and spring thickness dependent on resonant frequency.
For the acceleration value 20g the displacement of the mass was observed as 112 nm. For this value, the capacitance change was calculated as
DC a¼20g ¼ 1:14 104 F
ð10Þ
Such capacitance value is relatively small, but it can be measured by the MS3110 Universal Capacitive ReadoutTM IC provided by Irvine Sensors.
As the optimization results the so-called Pareto set was obtained (Fig. 12). As it was mentioned before, all the points are Pareto-optimal, which means that they are equal from the mathematical point of view. In order to choose the best solution from this set, the additional condition defined above was used (the point with the closest value of resonant frequency to 5 kHz). The chosen Pareto point is distinguished by the red color and its values (input and output) are presented in Table 2.
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Table 2 Optimal design parameters in reference to 3 selected criteria. Input parameter
Nominal
Optimal
Spring_thickness (lm) Mass_thickness (lm) capacitor_plate_len (lm)
3.0 15.0 120.0
1.63 12.23 83.80
Objectives Linearity [1] Sensitivity (F/g) Resonant frequency (Hz)
Nominal 0.99976 5.69E16 13278
Optimal 0.9921 1.64E15 5101.6
Fig. 13. Contribution of input parameters in optimization process.
The example diagram with dependence of resonant frequency on the mass and spring thickness is shown in Fig. 11. This diagram was made by interpolation of the optimization’s FEM results. As it can be seen on the diagram, the highest resonant frequency is obtained by higher values of spring thickness and maximal value of mass thickness. The optimal resonant frequency is obtained in the middle of the mass thickness range. Table 2 shows the optimal output parameters which correspond to the input parameters taken into account during the optimization process. It can be noticed that the response from the nominal input parameters is better in reference to linearity and sensitivity. However, the value of resonant frequency is higher than 13 kHz. The goal was to find the input parameters, which give the best linearity and sensitivity as possible but also the resonant frequency as close as possible to 5 kHz. Thus, the solution presented in Table 2 satisfies all selected criteria (with the emphasis on resonant frequency). Taking into account the contribution of input parameters in the optimization process, it turned out that the spring thickness influenced the most on the results. Second parameter was the mass thickness. Capacitor’s plate length influenced the least (Fig. 13). The disadvantage of this optimization method is the time needed to find the optimal solutions. It is mainly due to the fact, that the applied normal-boundary intersection method needs many iterations (about 800) to achieve the goal. In this approach
one iteration means one FEM calculation, which is time-consuming (about 1 min. 30 s). The solution would be the application of some indirect optimization method (Design of Experiments – DOE) or different direct method (simulated annealing, for instance). 6. Summary In the frame of this research the numerical prototyping was applied to design the example acceleration sensor. The device was prepared as a FEM model and then optimized in order to find the best design parameters in reference to linearity, sensitivity and resonant frequency. The goal was to carry out the optimization process with more than one criterion and therefore the multiobjective optimization algorithm was used. The algorithm called normal-boundary intersection gave the Pareto set from which one solution was selected. The spring thickness – one of the input parameters taken to the optimization process influenced the most on the results. The potential results of this research can be dedicated mainly to the automotive industry. One of the goals is the fuel reduction by the real time wheel balancing measurement. Such measurement can be done by the acceleration measurements. It seems that such numerical approach of designing the microsystem accelerometers is promising. Therefore in the future, the research in the field of applying numerical prototyping in microelectronics will be continued. Other numerical optimization algorithms will be tested in reference to obtain better efficiency and optimization time. Acknowledgements This work was performed in a frame of the ‘‘Nanoelectronics for Safe, Fuel Efficient and Environment Friendly Automotive Solutions (SE2A)’’ project; ENIAC proposal no. 12009. Authors acknowledge Wroclaw Centre for Networking and Supercomputing (WCSS) for the possibility of using modeling software and hardware. References [1] Galayko D, Kaiser A, Buchaillot L. Design, realization and testing of micromechanical resonators in thick-film silicon technology with postprocess electrode-to-resonator gap reduction. J Micromech Microeng 2003;13:134–40. [2] Galayko D, Kaiser A, Buchaillot L. High-frequency high-Q micro-mechanical resonators in thick epipoly technology with post-process gap adjustment. IEEE 2002. [3] Shukla PK. On the normal boundary intersection method for generation of efficient front. Springer; 2007. p.310–7. [4] Erickson D. Towards numerical prototyping of labs-on-chip: modeling for integrated microfluidic devices. Microfluid Nanofluid 2005;1(4):301–18. [5] Xi-hong M, Jun L. Design of hybrid integrated capacitive acceleration sensor. ICECE Proc 2010:744–7. [6] Ehrgott M. Multicriteria optimization. Springer-Verlag; 2005.