Optimal sensor placement (OSP) technique plays a key role in the structural health monitoring (SHM) of large-scale structures. Based on the criterion of the OSP for the modal test, an improved genetic algorithm, called “generalized genetic algorithm (GGA)”, is adopted to find the optimal placement of sensors. The dual-structure coding method instead of binary coding method is proposed to code the solution. Accordingly, the dual-structure coding-based selection scheme, crossover strategy and mutation mechanism are given in detail. The tallest building in the north of China is implemented to demonstrate the feasibility and effectiveness of the GGA. The sensor placements obtained by the GGA are compared with those by exiting genetic algorithm, which shows that the GGA can improve the convergence of the algorithm and get the better placement scheme.
Civil engineering structures, particularly high-rise structures, are especially susceptible to random vibrations, whether they are due to large ground accelerations, strong wind forces, or abnormal loads such as explosions [
As known, finding the rational sensor locations is a complicated nonlinear optimization problem, and the global optimal solution is often difficult to obtain [
This paper is aimed at adopting a kind of the improved GA called the generalized genetic algorithm (GGA) that could be used practically by engineers to solve the OSP problem. The remaining part of the paper is organized as follows. Section
The sensor placement optimization can be generalized as follows: “given a set of
Inspired by Darwin’s theory of evolution, the GAs try to imitate natural evolution by assigning a fitness value to each candidate solution of the problem and by applying the principle of survival of the fittest [
Flowchart of the GA.
GA has been proved to be a powerful tool for OSP, but it also has some faults that need to be improved. For example, two or more sensors may be placed in one sensor location or sensor number is not equal to a certain number[
The difference between the GGA and simple genetic algorithm (SGA) mainly exists in the evolutionary process. In short, the evolutionary process of the SGA is as follows.
Two-parent selection · crossover · mutation · survival selection · next generation
While the process of the GGA is as follows.
Two-parent selection · crossover · a family of four · two-quarter selection · mutation · a family of four · two-quarter selection · next generation
It can be found that the two-quarter selection is introduced in the GGA. The parents are allowed to compete with the children during the process of crossover and mutation, and only the best one could enter next competition, which can ensure the stability of iterative procedure and complete the function of realizing global optimum. In addition, the crossover and mutation of GGA have little difference with the SGA. In the SGA, the crossover may operate according to a certain probability, while, for the GGA, since the parents are certain to join in the competition, the crossover probability remains 1. The evolutionary process of the GGA has two stages: the gradual change and sudden change. During the gradual change, the local optimum could be achieved mainly by crossover and selection in which the single-point crossover and swap mutation are generally used and the sequence of operations is first crossover and then mutation. In the sudden change, the global optimum may be reached mainly by the mutation and selection which realize the escape from one local optimum to better local optimum, in which the uniform crossover and inversion mutation are mainly used and the sequence of operations is first mutation and then crossover.
From the view of mathematics, the OSP is a kind of particular knapsack problem, which places specified sensors at optimal locations to acquire more structural information. Its mathematic model is a 0-1 programming problem, if the value of the
The dual-structure coding method is shown in Table
Dual-structure coding method.
Append code | |||||||
---|---|---|---|---|---|---|---|
Variable code |
For example, an OSP problem with 10 sensors and the randomly generated order of append code is (4, 3, 5, 8, 6, 10, 2, 7, 9, 1); therefore, the dual-structure code is presented as shown in Table
Example of a dual-structure coding method.
4 | 3 | 5 | 8 | 6 | 10 | 2 | 7 | 9 | 1 |
---|---|---|---|---|---|---|---|---|---|
0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
In the case under investigation, the fitness function is a weighting function that measures the quality and the performance of a specific sensor location design. This function is maximized by the GA system in the process of evolutionary optimization. As known, the measured mode shape vectors in the SHM have to be as much as possible linearly independent, which is a basic requirement to distinguish measured or identified modes. Moreover, the linear independency is particularly important when the test results are to validate or to update the FE model. The simple way to check this linear dependence of mode shapes is to calculate the modal assurance criterion (MAC) [
The MAC can be defined as in (
Then the MAC fitness function is given as
The first step in the GGA is to create an initial population of randomly generated individuals. The “group in group” scheme is used here; that is, for the initially generated population of
The crossover is the process whereby new chromosomes are generated from existing individuals by cutting each old chromosome at a random location (crossover point) and replacing the tail of one string with that of the other. In this paper, the order crossover (OX) method is applied. The OX involves two parents creating two children at the same time. This operator allows the order in which the parts are placed into the creation vat to be changed.
The crossover enables the method to extract the best genes from different individuals and recombine them into potentially superior children. The mutation is a random process whereby values of element within a genetic string is changed. Considering that there are gradual change and sudden change in the GGA, the swap mutation and inversion mutation are used herein. The mutation process adds to the diversity of a population and thus reduces the chance that the optimization process will become trapped in local optimal regions.
To sum up, the whole flowchart of the genetic search to find the optimal sensor locations presented in this paper is shown in Figure
Flowchart of the dual-structure coding based GGA.
To demonstrate the possible enhancement of the GGA compared to the SGA by optimally allocating the sensors across the structure, a case study to determine the optimal sensor locations on a high-rise building is given here.
The Dalian International Trade Mansion (DITM), currently being constructed in the centre of Dalian city, when completed in the near future, will be the tallest building in the north of China [
Dalian International Trade Mansion.
Bird view of the Mansion
Standard floor plan
First floor plan
In order to provide input data for the OSP method, a three-dimensional FE model of the mansion is built using the ETABS software. The FE model is built considering the bending and shearing deformation of the beam and column and also the axial deformation of the column. The rigid-floor assumption is used. To the strengthened story, for the axial deformation of the column needs to be considered, the corresponding floors are computed as flexible floors. The overall model has 34,308 node elements, 34,791 frame elements and 29,071 shell elements, considering 36 section types and 11 materials’ properties. The mesh representing the model has been studied and is sufficiently fine in the areas of interest to ensure that the developed forces can be accurately determined. Then, the modal analysis is carried out, the periods of the first 6 modes are listed in Table
Calculated periods of DITM using FE model.
Mode | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
Period(s) | 6.80561 | 4.94373 | 2.3094 | 1.53486 | 1.34306 | 0.88573 |
First six mode shapes of the DITM.
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
For the problem at hand, the size of the searching space is the number of nodes on the FE model excluding the constrained nodes and the vibration nodes of the selected modes. Since the structural stiffness of the DITM in two translational directions is obviously different, it should mainly take into account the structural vibration monitoring in the direction of weaker stiffness. On the other hand, although the structure has a large number of DOFs, only translational DOFs are considered for possible sensor installation in the on-site test, as rotational DOFs are usually difficult to measure. Consequently, a total of 79 DOFs are available for sensor installation. In order to improve the results of optimal sensor layout, the first 10 modes of the DITM are selected for calculation.
As known, the GGA has a number of parameters that are problem specific and needs to be explored and tuned so that the best algorithm performance is achieved. These parameters are the population size, leading population size, and number of sudden changes. To find out the most appropriate population size for minimal computation cost, various simulations with different population size have been first done, and population size of 200 has been found to be adequate. In the simulation of the GGA process, the leading population size is defined as one quarter of the population size, and a relative large number of 10 sudden changes are selected to avoid redundant iteration. In order to demonstrate the possible enhancement of the GGA compared to the SGA, the SGA is also used to determine the optimal sensor locations. In the calculation of the SGA process, the population size of 200, the crossover rate of 95% (i.e., 95% parents produce their child), and the mutation rate of 5% (i.e., a selected chromosome bit flips in the mutation with a probability 10%) are used.
Suppose that there are 20 one-dimensional accelerators needed to be installed. It should be noticed that due to the nature of the GA method, the results are usually dependent on the randomly generated initial conditions, which means the algorithm may converge to a different result in the parameter space. For the problem considered in this paper, the GGA and SGA processes have been run for 10 times with a different stochastic initial population, and the best result is shown in Figure
Comparison of the optimal sensor locations of the GGA and SGA.
Sensor no. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
GGA | 4 | 6 | 7 | 9 | 10 | 12 | 13 | 15 | 16 | 29 | 34 | 43 | 45 | 49 | 51 | 60 | 65 | 66 | 75 | 77 |
SGA | 2 | 3 | 6 | 8 | 10 | 11 | 18 | 21 | 23 | 25 | 26 | 32 | 36 | 55 | 57 | 61 | 67 | 68 | 69 | 71 |
Evolution progress of the best fitness value of the GGA and SGA.
GGA
SGA
From Figure
This paper describes the implementation of GGA as a strategy for the optimal placement of a predefined number of sensors. The dual-structure coding method instead of binary coding method is adopted to code the solution. Accordingly, the selection scheme, crossover strategy, and mutation mechanism used in this paper are given in detail. These methods operate only on upper append code, and the lower variable value of offspring is kept fixed to guarantee the number of sensors can be unchanged. Then, the GGA and SGA are used for selecting optimal sensor locations of the tallest building, DITM, in the north of China. The optimal results obtained by the GGA are compared with those by the SGA, which demonstrated that the GGA was particularly effective in solving the OSP problem, and it can get the better results with lower computational iterations.
This research work was jointly supported by the Program for New Century Excellent Talents in University (Grant no. NCET-10), the National Natural Science Foundation of China (Grant no. 50708013), and the Open Fund of State Key Laboratory of Coastal and Offshore Engineering (Grant no. LP0905).