This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).
This paper presents a numerical study on optimal voltages and optimal placement of piezoelectric actuators for shape control of beam structures. A finite element model, based on Timoshenko beam theory, is developed to characterize the behavior of the structure and the actuators. This model accounted for the electromechanical coupling in the entire beam structure, due to the fact that the piezoelectric layers are treated as constituent parts of the entire structural system. A hybrid scheme is presented based on great deluge and genetic algorithm. The hybrid algorithm is implemented to calculate the optimal locations and optimal values of voltages, applied to the piezoelectric actuators glued in the structure, which minimize the error between the achieved and the desired shape. Results from numerical simulations demonstrate the capabilities and efficiency of the developed optimization algorithm in both clamped−free and clamped−clamped beam problems are presented.
Smart or adaptive structures with integrated self−monitoring and control capabilities are of great technological interest due to the increasing requirements on structural performance. The self−monitoring capability of smart structures has numerous applications in shape and vibration control of structures, noise reduction, damage identification, and structural “health” monitoring. Notable references among others are [
Piezoelectrics are the most popular smart materials, which can be used both as sensors and actuators. The coupled electromechanical properties of piezoelectric materials, along with their possibility to be integrated in various structures, make them suitable for use in advanced smart structures.
The recent advances in smart structures have prompted interest in modification and correction of the shape of mechanical structures, e.g., for the correction of the shape and curvature of mirrors/antennas for high pointing accuracy or for maintaining desired shapes of aerospace flexible structures,
One main objective of piezoelectric shape control is to optimize some control parameters (e.g., the number, location and size of the piezoelectric patches, the amount of electric potential to be applied,
In this work, the use of piezoelectric actuators for shape control and correction of static deformations is considered. The models widely used for this kind of problems are based on the Euler beam theory and the Kirchhoff−Love theory of plates with or without electromechanical coupling (e.g., [
Besides establishing an accurate mathematical model for shape control applications, a critical factor for the success and performance of the smart structure is the determination of the optimal location of the piezoelectric actuators together with the optimal actuation voltages. Next, the finite element model developed is used in static shape control. Shape control (SC) is defined here as the determination of the applied voltages of actuators and their layout, such that the structure that is activated using these parameters will conform as closely as possible to the desired shape. The problem is formulated as mixed discrete−continuous programming with a quadratic cost function as objective. A genetic algorithm is used as the optimization technique.
Genetic algorithms (GAs) is a well known optimization method [
Consider a laminate formed from two or more layers bonded together to act as a single layer material and sandwiched between two piezoelectric layers. The bond between two layers is assumed to be perfect, so that the displacements remain continuous across the bond. The classical formulation of laminated materials is followed [
For a laminated beam with midplane symmetry, the displacement field, using the first−order deformation theory, is expressed as functions of two independent nodal degree of freedom of the middle axis,
A constant transverse electrical field is assumed for the piezoelectric layers and the remaining in−plane components are supposed to vanish. Consequently, the electric field inside the
The linear constitutive equations coupling the elastic and the electric fields in a piezoelectric medium are expressed by the direct and the converse piezoelectric equations, which are given as follows:
For a one−dimensional beam where the width in the
To derive the equations of motion for the laminated composite beam with surface bonded sensor and actuator layers, Hamilton’s principle is used:
The kinetic energy, the potential energy and the total work done due to virtual displacements are given as follows:
A two−node finite element is considered with two mechanical degrees of freedom,
Using the variational principle, given by Equation (6), governing equations of an element can be written as:
The global equations can be obtained by assembling the elemental Equation (12). Equation (12) can be used in smart structures applications such as vibration control, static or dynamic shape control,
A computer code is developed, based on the aforementioned finite element model. A special numbering scheme is used to denote the elements with piezoelectric layers. Elements with piezoelectric layers are denoted by 1, while the remaining layers are denoted by 0 for the identification during the assembly process.
The most general problem of SC of smart structures considers as design variables the applied voltages of actuators, their layout and their number. The aim is to find the optimal design values so that the difference between achieved and desired shape is minimized. In essence, SC is an inverse problem where the output, which is the desired shape, is known and the input actuation parameters are to be determined. Therefore, iterative heuristic methods are quite suitable to this task. In this work, this problem is solved by a hybrid genetic algorithm.
Considering beam element, the shape of a structure is primarily described by the shape of its middle axis, which itself is described by the transverse displacement of the finite element mesh nodes. Therefore, a reasonable cost function is
In the above equation,
However, the static shape control criteria in [
It should be noted that the simultaneous usage of displacements and rotations in the cost function
In general the displacement field is a function of the electric potential, the layout, the geometry of actuators and the number of actuators. In this framework, two kinds of shape control (SC) problems of a beam with various boundary conditions are studied. The Voltage Problem and the Location and Voltage Problem.
The first SC problem (the Voltage Problem) consists in finding a set of actuation voltages
The second SC problem (the Location and Voltage Problem) is more general. It consists in finding the optimal position and electric potential simultaneously for a given number of actuators, which minimize the cost function
Genetic Algorithms (GAs) are a category of heuristic optimization algorithms that mimics the way traits pass from parents to offspring resulting in the development of characteristics that give an evolutionary advantage to certain members of the population. GAs are an established method of non exact optimization, meaning that a GA is usually able to find very good solutions to hard combinatorial optimization problems when it is difficult or even impossible for an exact optimization method like Linear or Integer Programming to address the same problem within reasonable solving time. A detailed treatment of GAs can be consulted in [
Great Deluge Algorithm (GDA) is a local search optimization method, which was initially proposed by Dueck [
The pseudo−code for generating part of the initial population of the GA using GDA is shown in
Function
As voltage assumes values between a lower and an upper limit, e.g., between 0 Volts and 400 Volts, when
Great Deluge Algorithm for initial population generation.
A decision directly related to the success of a GA in a specific application is the chromosome encoding, which is the encoded form of each individual belonging to the population. Chromosomes are combined in order to breed new individuals or mutated so as to incorporate direct changes. In any kind of problem examined in this work, the beam is divided in 30 equally spaced positions where the actuators can be positioned.
In the first kind of the problem (the Voltage Problem) every six consecutive positions become a group and the same voltage applies to all actuators of the same group. So, the chromosome encoding is just a sequence of decimal numbers equal to the number of groups. Each value of the chromosome is associated with the voltage that will be applied to all actuators of the group in the same place. Given that four different settings are tested with two, three, four, or five groups of actuators active respectively the chromosome length becomes two, three, four, or five.
Chromosome encoding for the location and voltage problem.
In the second case of the problem (the Location and Voltage Problem) the chromosome consists of two parts (
There are numerous GA implementations available in the form of callable libraries, frameworks or integrated environments. In this paper, MatLab’s Global Optimization Toolbox R2012a, which includes the Genetic Algorithm Toolbox was used. MatLab’s Global Optimization Toolbox starting from version R2011b has the capability of defining integer constraints out of the box. This was very convenient in our case given that the Location and Voltage Problem is a Mixed Integer optimization problem.
This section presents numerical results from several representative problems. First, a benchmark problem is considered in order to validate the present optimization algorithm. Next, several illustrative optimization problems are investigated using the developed algorithm. All application examples focus on beams with surface bonded piezoelectric patches as actuators. The host beam is made of T300/976 graphite/epoxy and the piezoelectric layers are PZT G1195N. The length of the beam is equal to 300 mm, the depth is equal to 9.6 mm and width is equal to 40 mm. The thickness of the actuators is equal to 0.2 mm. The elastic constants of T300/976 graphite/epoxy are:
The smart beam structure.
The problem studied by Hadjigeorgiou
The optimization problem for both problem cases is solved using the present developed genetic algorithm. The genetic algorithms were run for 100 generations and 40 individuals. Of several test cases run, the one exhibiting the best fitness is presented. The optimal values of voltages for the most efficient combinations of actuator groups to shape control of the beam are shown in
A beam with similar material and geometric properties as described in
In order to assess the behavior of the algorithmic approach, several runs were performed. For example, for the case of the Clamped Free problem, where the number of actuators is 18, 55 runs were performed using initial values generated with different random seed for each run. Results showed that for this particular case the maximum value of fitness function was
Inclusion of the Great Deluge phase before the Genetic Algorithm added value to the approach. This is demonstrated by the following experiment scenario: For the case of the Clamped Free problem, 100 runs were performed using the Genetic Algorithm including the injection of solutions generated by the Great Deluge and another 100 runs were performed by deactivating the Great Deluge phase and letting the Genetic Algorithm form the initial population randomly. All runs where executed on a Windows7/64 bit machine equipped with Intel i7 860 processor and 16 GB of RAM. Each run took about eight minutes to complete. Results showed that the best 24 generated fitness values during the experiment were all achieved using the configuration of the solver that included the Great Deluge stage. Thus, a small number of good solutions (10% or less of the population size) injected to the initial population seem to drive the Genetic Algorithm to better solutions.
Optimal values of voltages (clamped−free beam) for
Number of used Actuator Groups  Method  Voltage of the Actuator Groups (V)  Fitness 


1  2  3  4  5  
432.61  0.00  389.20  0.00  0.00  17.86  
Present (case 1)  432.60  389.21  17.86  
Present (case 2)  438.87  400.55  17.76  
(1,2,3)  [ 
324.83  217.58  272.11  0.00  0.00  18.99  
Present (case 1)  324.91  217.41  272.20  18.99  
Present (case 2)  325.35  216.01  277.59  18.89  
(1,2,3,4)  [ 
320.15  243.17  169.88  122.44  0.00  22.14  
Present (case 1)  320.20  243.30  169.51  122.57  22.14  
Present (case 2)  320.33  243.39  169.21  123.79  22.03  
(1,2,3,4,5)  [ 
320.54  240.47  178.09  98.57  41.94  23.41  
Present (case 1)  321.06  241.18  173.58  107.45  30.37  23.61  
Present (case 2)  320.18  243.09  171.98  107.91  31.20  23.56 
First the case of a beam, which is clamped at the left−hand side and is subjected to a concentrated load equal to 4 N at the free right end, is considered. In this case, the lower limit of the voltage is set to be 0 V and the upper limit is set to be 240 V (limit imposed due to depoling of actuators). The desired shape is given by
Optimal Location and voltages of actuators within the 30 finite element mesh for Clamped−Free Beam.
Number of Actuators in Use  

8  12  18  24  30  
1  240.00  240.00  72.88  170.6295  211.79  178.82  150.63  172.21  196.49  171.86 
2  0  240.00  205.33  166.2474  137.64  240.00  181.99  164.26  135.51  165.25 
3  240.00  240.00  57.59  239.5262  224.25  0  233.47  160.70  200.80  159.43 
4  240.00  0  227.76  0  0  237.75  142.78  152.82  106.08  154.16 
5  240.00  0  240.00  235.2183  236.74  150.84  62.97  148.71  172.72  147.67 
6  240.00  240.00  240.00  240.00  157.88  141.92  153.98  141.55  134.09  142.17 
7  240.00  240.00  240.00  0  144.00  136.04  133.83  137.12  175.95  136.62 
8  240.00  0  0  240.00  101.20  131.17  202.06  129.91  77.37  130.25 
9  240.00  0  240.00  0  111.33  124.13  82.88  125.06  133.95  125.01 
10  0  240.00  0  240.00  154.75  179.14  155.75  118.98  147.23  118.82 
11  0  240.00  0  240.00  124.40  0  20.06  113.21  89.92  112.97 
12  0  0  240.00  0  138.96  220.14  142.41  107.06  88.79  107.72 
13  0  240.00  240.00  0  0  0  215.45  101.73  163.93  101.21 
14  0  0  0  240.00  232.81  149.81  0  95.90  54.65  95.94 
15  0  0  0  0  0  89.29  0  89.52  66.10  89.88 
16  0  0  240.00  0  0  84.59  223.23  84.19  137.71  84.19 
17  0  0  0  240.00  180.02  116.37  54.80  78.72  16.61  78.05 
18  0  0  0  0  126.31  0  50.74  71.97  126.12  72.91 
19  0  0  0  240.00  0  167.23  135.11  99.19  72.24  66.33 
20  0  0  0  0  118.10  0  0  0  17.67  61.24 
21  0  0  0  0  0  0  0  87.00  79.27  54.98 
22  0  0  240.00  0  0  132.40  133.63  72.41  27.59  49.26 
23  0  0  0  0  194.29  0  28.23  0  60.20  43.54 
24  0  0  0  0  0  124.34  36.36  77.41  55.35  37.67 
25  0  0  0  0  0  0  0.12  0  21.19  31.62 
26  0  0  0  0  0  0  79.99  68.89  3.47  26.56 
27  0  0  0  0  0  0  0  0  31.47  20.06 
28  0  0  0  0  0  0  0  0  27.63  14.41 
29  0  0  0  0  122.02  0  17.03  0  0.53  8.90 
30  0  0  0  186.51  1.87  46.15  2.03  22.71  0.38  2.59 











As can be shown in
The centerline of the cantilever smart beam under the action of various numbers of actuators for
Second, the case where the beam clamps on both sides and is subjected to a concentrated load equal to 40 N at the center is considered. The lower limit of the voltage is set to be −240 V and the upper limit is set to be 240 V. The pre−defined displacement field (desired shape) is given by
A mathematical model of a laminated composite beam with bonded piezoelectric patches used as actuators is considered in this study. The model is built using finite element method and is applied as a platform for the investigation of shape control of the beam. Shape control was applied to a beam structure with different boundary conditions. The optimal values for the locations of the piezo−actuators and optimal voltages for shape control are determined for clamped−free and clamped−clamped beams by using a genetic optimization procedure. A two−step process including Great Deluge and then a Genetic Algorithm has been performed in order to improve search efficiency. The results presented above demonstrate the capability of the proposed hybrid GA approach in determining optimal voltages and locations of control actuators within a large number of possible positions.
Numerical results on a benchmark problem validate both the finite element code being used as well as the optimization algorithm. Examples that demonstrate the capabilities and efficiency of the developed optimization algorithm in both clamped−free and clamped−clamped beam problems were presented.
In the near future, our research team plans to apply the proposed hybrid GA to more realistic engineering problems such as plate structures.
Optimal Location and voltages of actuators within the 30 finite element mesh for Clamped−Clamped Beam.
Number of Elements  Number of Actuators in Use  

8  12  18  24  30  
1  240.00  240.00  239.98  240.00  240.00  240.00  240.00  235.43  239.95  211.92 
2  240.00  240.00  0  0  240.00  240.00  240.00  205.09  196.47  180.26 
3  0  0  239.97  0  240.00  213.31  240.00  174.19  141.02  149.30 
4  0  0  0  0  240.00  157.75  99.60  143.31  33.89  118.34 
5  0  0  0  240.00  0  169.21  70.087  112.43  72.03  87.38 
6  0  0  0  0  0  0  3.87  108.23  72.32  56.42 
7  0  0  0  0  157.35  91.23  93.21  0  −51.29  25.46 
8  0  0  0  −168.77  65.58  0  46.64  46.47  48.25  −5.51 
9  0  0  0  0  −172.58  0  −59.20  −11.14  −162.20  −36.47 
10  0  −240.00  0  0  0  0  −240.00  −42.00  1.25  −67.42 
11  −240.00  0  0  0  −240.00  131.61  0  −72.92  −151.25  −98.38 
12  0  0  −239.96  −240.00  0  121.53  0  −103.80  −144.01  −129.35 
13  0  0  239.97  −240.00  −240.00  156.44  −240.00  −134.69  −230.12  −160.31 
14  −240.00  0  239.98  0  −240.00  191.35  −240.00  −165.58  −239.98  −191.27 
15  0  0  239.99  −240.00  0  197.23  −240.00  −167.45  −239.93  −193.21 
16  −240.00  −240.00  239.98  −240.00  −240.00  174.10  −240.00  −140.27  −222.46  −166.13 
17  0  −240.00  0  0  −240.00  150.97  −191.31  −158.49  −186.16  −139.05 
18  0  0  0  0  −192.47  230.14  0  0  −106.64  −111.97 
19  0  0  239.93  0  −199.99  0  −143.09  −125.64  −115.69  −84.89 
20  0  −240.00  0  −240.00  0  0  −96.15  0  −54.88  −57.80 
21  0  0  0  0  0  −184.05  13.02  0  −67.16  −30.72 
22  0  0  0  0  0  0  −46.17  0  −66.87  −3.64 
23  0  0  0  0  129.42  0  14.11  97.55  61.63  23.44 
24  0  0  0  0  0  0  0  0  5.15  50.52 
25  0  0  0  240.00  0  98.34  139.55  144.69  45.88  77.61 
26  0  0  0  0  0  0  0  131.25  95.75  104.68 
27  0  240.00  239.93  240.00  0  0  240.00  158.41  100.36  131.77 
28  240.00  240.00  239.97  240.00  240.00  240.00  0  185.55  102.98  158.85 
29  240.00  0  239.96  0  240.00  0  240.00  214.49  206.31  185.93 
30  240.00  0  239.98  240.00  185.24  211.60  240.00  240.00  239.91  213.64 











The centerline of the clamped−clamped smart beam under the action of various numbers of actuators for
This research has been co−financed by the European Union (European Social Fund–ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)−Research Funding Program: ARCHIMEDES III−Investing in knowledge society through the European Social Fund. The authors gratefully acknowledge this support.
Detailed expressions of the mass and stiffness matrices as well as loading vectors that appear in the paper.