Solving Inverse Problem By Genetic Algorithm

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6th World Congresses of Structural and Multidisciplinary Optimization Rio de Janeiro, 30 May - 03 June 2005, Brazil

Solving Inverse Problems in Structural Damage Identification Using Advanced Genetic Algorithm Representations Tamas R. Liszkai1, Anne M. Raich2 (1) Material and Structural Analysis, Framatome ANP, 155 Mill Ridge Road, Lynchburg, VA, USA 24502 (2) Department of Civil Engineering, Texas A&M University, College Station, TX, USA 77843 Abstract Accurately identifying damage is difficult after an extreme event since damage often cannot be directly measured. Sensors, however, can collect data that reflects damage by measuring changes in structural response. Therefore, damage identification becomes an inverse problem in which the goal is to find system properties using limited response information. The inverse problem is formulated as an optimization problem and is solved using an implicit redundant representation genetic algorithm (IRR GA). Element damage is introduced into the finite element model using a damage indicator to represent a loss of stiffness. The set of damage indicators are optimized by minimizing the difference between frequency response functions obtained from the damaged structure and the updated structural model. Although an accurate structural model may require thousands of finite elements, the number of damaged elements is typically much smaller. The problem, however, is that the number of damaged elements and their locations are not known beforehand, which leads to an unstructured optimization problem. To solve this problem the IRR GA uses redundant segments to allow the number of design variables (damage indicators) to change during optimization. A damage scenario can then be represented by only a small subset of damage indicators, instead of all possible damaged elements. The performance of the method on two-span beam and multi-story frame structures was evaluated using several imposed damage scenarios. High damage identification accuracy was obtained with limited sensor data even when measurement noise was present in the sensor data. The method also was able to identify damage in larger systems that are difficult to solve using existing methods due to the large number of design variables. Keywords: damage identification, inverse problems, optimization, genetic algorithm 1. Introduction Overloading, corrosion, material aging, or other unexpected events are inevitable over the lifetime of many structures. Unfortunately these acts result in structural damage. The goal of structural damage identification methods (SDIM) performing in this environment is to accurately assess the presence of damage in order define rehabilitation and maintenance needs. Over the past twenty years, significant research advances have been made in being able to detect, locate, and quantify damage in structural systems. Although many of these studies found success and even application in the field, there remain two problems that restrict the use of many proposed SDIMs: sensitivity to noise and difficulty in scaling well with problem size. Therefore, this research is focused on developing a robust SDIM that uses minimal measurement information to efficiently identify damage in larger structural systems. The damage identification problem used in this research is posed as an inverse problem in which the goal is to find system properties using limited response information obtained from sensors. An IRR GA optimization method is applied to solve the inverse problem. The SDIM presented provides a Level 3 Method [1] since it is capable of assessment by precisely locating and quantifying the severity of damage. The same methodology is beneficial in other unstructured problem domains such as sensor optimization [2], structural control, system identification, and the design of frame structures [3]. 2. Background The underlying principle of many SDIMs is that vibration signatures, e.g. frequency response functions (FRF) or modal data, are sensitive indicators of structural integrity. This reasoning is based on early research efforts that found that damage changes the dynamic stiffness of structures, and therefore, natural frequency measurements can be used to detect damage using a finite element model [4]. In the last decade, the majority of research has focused on SDIMs that use modal information [5,6,7]. Several researchers, however, have investigated using FRF in system and damage identification. Wang et al. [8] used FRF data obtained before and after damage to develop an algorithm based on nonlinear perturbation equations, which were weighted at selected locations and frequencies to minimize the influence of errors. An iterative procedure was proposed to overcome difficulties associated with incomplete measurement data. An FRF-based SDIM was introduced by Lee and Shin [9] using the dynamic equation motion for the damaged and undamaged beams. An iterative reduced domain strategy was developed to eliminate regions of undamaged areas from the solution domain. Thyagarajan et al. [10] used gradient-based optimization with FRF measurements to localize damage in a structural finite element model. Their research stated that a more robust optimization algorithm was desired for noisy measurements and methods that reduce the optimization domain by eliminating undamaged elements would be beneficial for larger problems. More recently, genetic algorithms (GA) have been used in SDIMs. A common feature of all GA-based methods is that they employ an error function between the measured data and the discrete analytical model. Mares and Surace [11] used GAs to maximize an objective function based on the residual force method, which is computed from measured natural frequencies and mode shapes. Dunn [12] studied the performance of GAs and stochastic hillclimbing on a finite element model identification problem in which the error function between the measures and analytical FRFs was minimized. Moslem and Nafaspour [13] considered the unstructured nature inherent in damage identification problems by applying a two-stage identification procedure in which areas with possible damage are first identified and then only elements of that area are included in the optimization domain. In the research presented, the damage identification problem is formulated as an optimization problem, which is solved using both a SGA [14] and an implicit redundant representation (IRR) GA [3], which takes advantage of the unstructured nature of damage

1

identification. The IRR representation, which allows the dynamic change of optimization variables, was applied in [15]. In their approach, however, the authors utilized static equilibrium equations instead of vibration data to set up the optimization problem. 3. Structural Damage Identification Methodology A FRF-based SDIM is presented that can accurately identify both the location and severity of damage in structural systems using minimal measurement information. The inverse problem of damage identification is formulated as an optimization problem. In general, the method has simularites to model parameter updating techniques that try to adjust the physical parameters of an analytical model, (such as stiffness, mass and damping properties), using an iterative process. The difficulty presented with these methods, however, is that as the problem size increases, the number of design variables that must be optimized (possible damaged members) also increases. This research looks into the benefits of using an IRR GA to represent an unstructured problem domain in order to reduce the effect of problem size on the solution obtained. The proposed SDIM works with a linear finite element model of the undamaged structure. Element damage is introduced into the finite element model using a damage indicator to represent a loss of stiffness. The set of damage indicators are optimized by minimizing the difference between FRFs obtained from the damaged structure and from the updated structural model. Although an accurate structural model may require thousands of finite elements, the number of damaged elements is typically much smaller. The problem, however, is that the number of damaged elements and their locations are not known beforehand, which leads to an unstructured optimization problem. The use of the IRR GA is beneficial due to its use of redundant segments to allow the number of design variables (damage indicators) to change during optimization. The equations of motion of an n-degree of freedom, viscously damped system excited by a set of sinusoidal forces with ω circular frequency, but with different amplitudes and phases, can be derived using virtual work. To reduce the computational effort involved in the calculation of the receptance matrix, proportional damping and a linear structure are assumed. In this case, a generalized eigenvalue problem can be formulated

Kφ j = ω 2j Mφ j

(1)

where M and K are the mass and stiffness matrices of the undamaged structure, ωj is the jth circular natural frequency of the structure, and ϕj is the corresponding n × 1 eigenvector. The expression for the receptance matrix, R, defined as a function of ω can be determined as.

⎛ ⎞ T 1 R = Φ diag ⎜ 2 Φ ⎜ ω − 2iωζ ω − ω 2 ⎟⎟ j j ⎝ j ⎠

(2)

where ζj is the jth modal damping ratio, diag() represents a diagonal matrix, and Φ indicates the eigenvector matrix representing the eigenvectors. The analytical computation of this signature uses modal decomposition to reduce the computational expenses incurred by using the elementary definition of frequency response functions. Using the receptance matrix, the corresponding mobility and accelerance matrices can be obtained. 3.1 Damage Indicators Although the source of damage often is unknown, it is often accepted (with certain limitations) that damage affects the vibration characteristics of structures. In developing the FRF-based SDIM, it was assumed that damage only affects the stiffness of structural members. Damage that may change the stiffness properties of structural members includes the loss of cross-sectional area, material softening due to cyclic loading, loss of members, and loosening of bolted connections. Undetectable damages may include cracks that remain closed while vibration measurements are taken. To introduce damage, a damage indicator, x sj , was assigned to each finite element, j, in the model. The stiffness of a damaged element is computed by means of the damage index.

(

)

(3)

k j = 1 − xj k j , 0 ≤ xj ≤ 1 dam

where

kj

dam

s

s

is the stiffness matrix of the jth damaged finite element, kj is the stiffness matrix of the jth finite element without damage,

and x sj is the damage indicator of the jth finite element. The positive damage indicators may be interpreted as percentile damage values from zero to 100 percent. The equation could be modified to accommodate stiffness increases due to material hardening. 3.2 Use of FRF Measurement Data in Damage Identification The goal of the SDIM is to obtain the vector of damage indicators, xs, for all finite elements in the structure. To detect damage, the stiffness parameters of the intact model are updated until the analytically computed FRFs match the corresponding measured FRFs. In this case, the solution of the minimization problem will result in identifying a damage indicator vector that correctly identifies the location and severity of damage(s) in the structure. From the definition of FRF matrices it is obvious that the dimensionality of such matrices may become very large as the number of DOFs in a finite element model increases. As a result, only certain entries of an FRF matrix can be measured in practice. In general the jkth member of an FRF matrix represents the response (displacement, velocity or acceleration) of the jth DOF when the excitation is applied at the kth DOF. In order to formulate the damage identification problem as an optimization problem, the objective is defined as minimizing the error between the measured and analytical FRF data. kn ⎛ ϖ1 ⎞ min f = ∑ ⎜ ∫ H jk (ω ) − H jk (ω ) dω ⎟ ⎜ ⎟ k = k1 ⎝ ϖ 0 ⎠

2

2

(4)

where j is the excitation DOF, k is the DOF where the response is measured, Hjk is the jkth FRF function in the finite element model, th H jk is jk measured FRF function on the damaged structure, k1, k2,…kn are the DOF’s where measurements are taken, ϖ0 and ϖ1 are the lower and upper frequencies of the measured frequency range, and the symbol | | indicates complex magnitude. In this research it is assumed that there is only one excitation location, but measurements may be taken simultaneously at multiple locations. If there are multiple excitation locations, then another summation on j can be taken to obtain a valid objective function. The Hjk FRF is a function of the element stiffness matrices, and therefore, it also is a function of the damage indicators. Since the goal of damage identification is to find the damage indicators for each finite element; consequently the damage indicators become the unknown variables to be optimized. 4. Optimization Using Genetic Algorithms The damage identification problem defined can be classified as an unconstrained NLP optimization problem with simple bounds on the variables. Recovery of characteristic properties of a continuous system from noisy, discrete ,and often limited, information results in an ill-posed problem for which a unique solution often does not exist. In optimization terms, the objective has multiple local optima suggesting that traditional local search algorithms may result in solutions that are not representations of the actual damage scenario. To overcome the difficulties associated with uniqueness of solutions, GAs have been chosen as the primary tool to solve the optimization problem. A hybrid genetic algorithm (GA) and hillclimbing optimization method is used to facilitate exploration of the search space early on in the optimization process, while performing a more localized search in later stages. Two distinct GA representations are investigated. The simple genetic algorithm (SGA) optimizes a structured problem formulation domain. The implicit redundant representation (IRR) GA takes advantage of the true unstructured nature of damage identification and optimizes an unstructured problem formulation. Reducing the size of the search space using the IRR GA without having to use subjective assumptions provides significant benefits during optimization and results in an effective SDIM method for relatively large structures. 4.1 Introduction to Genetic Algorithms GAs have been applied successfully to a variety of optimization problems. Pioneering research in the topic was performed by Holland [16] and Goldberg [14]. GAs mimic nature’s evolutionary mechanism and Darwin’s “survival of the fittest” theory. The most common operations in GAs are selection, crossover and mutation. Design variables are encoded into individuals and each individual in the population represents a complete solution. Tournament selection is used to pick individuals for reproduction. Crossover combines the features of two individuals to create two similar, but new, individuals by swapping segments of the encoded string. Mutation introduces new genetic information to facilitate exploration of the search space. Finally, convergence of the population is checked at the end of each generation. A detailed description of the advanced genetic operators implemented can be found in Michalewicz [17]. The operators implemented include adaptive and equal probability multi-point crossover, uniform and non-uniform mutation, tournament selection, elitism, and binary and Gray coding of variables. Non-uniform mutation is only applied for the fixed GA representation trials. 4.2 Comparison of Fixed and Implicit Redundant Representation Genetic Algorithms Each finite element in the structural model is assigned a unique damage indicator. As a result, the number of independent variables the SDIM optimizes is equal to the number of finite elements. The fixed GA representation for damage identification encodes each of the damage indicators as design variables. The gene length using the fixed GA representation is defined by the number of finite elements in the model and the number of binary digits required by the precision of the damage indicators. The length of each individual using the fixed GA representation, therefore, can become very large since hundreds of finite elements may be required to model a structure accurately. Figure 1 shows an example of the fixed GA representation in which the binary-encoded damage indicators are concatenated to form an individual. For complex problems, the SGA has to tackle a search space defined by a large number of variables, which results in poor convergence or failure of the search. Typically even when the number of finite elements in the model becomes very large, the number of actual damaged elements is only a fraction of the total number of finite elements. The main problem, however, is that neither the number of damaged elements, nor their locations or severity, are known. This unique situation defines an unstructured optimization problem in which the number of independent variables is unknown. The IRR GA [3] introduces redundant segments into each individual that allow the number of variables to dynamically change during optimization. The redundant and useful segments in an IRR string interact dynamically by using a string length that is longer than the length required to encode the parameter values. The self-organizing capability of the IRR provides a beneficial representation for unstructured problem domain investigated in this research. This property is beneficial for damage identification since neither the number of damaged elements nor their locations are known beforehand. The dynamic redundancy of the IRR provides additional benefits such as less disruption of crossover and mutation due to the presence of redundant segments, dynamic enlargement and reduction of the search space, and exploration of fit members through the activation of redundant material into gene instances. Encoded damage indicator for Element 2

1 0 0 1 0 1 1 0 0 0 1 0 1 1 0 1 1 0 1 1 ... 1 0 0 1 0 1 1 0 0 0

Encoded damage indicator for Element 1

Encoded damage indicator for Element ne

3

Figure 1. Fixed representation for damage identification A typical IRR GA representation is depicted in Figure 2. In this representation, three parts of the string can be identified: a predefined gene locator (GL) pattern; a gene instance; and redundant segments. Each gene instance consists of two parts, a segment encoding the finite element number (location of damage) and a segment encoding the damage indicator for that finite element (magnitude of damage). Redundant segments occur between any two consecutive gene instances. The redundant segment may become a useful gene instance in later generations through the actions of genetic operations. Gene instances encode the parameter values similar to other GAs. To decode a damage indicator, the IRR string is parsed until a GL pattern is found that identifies a gene instance. The selected GL pattern used in this research is a [1 1 1]. Then the finite element number identifying the damage location is decoded along with the corresponding damage indicator value. If more than one gene instance identifies the same finite element then the average of the damage indicators is taken. For the IRR GA, each individual in the population still represents one complete solution, but the solution is defined only by the damage indicators for a small subset of the finite elements in the model. For the IRR GA, the string length is obtained from the number of significant digits for the damage indicator and the user defined expected number of gene instances in the string. Further details concerning the IRR GA are provided in [3]. Gene instances

... : Redundant segments

1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 Gene locator (GL) pattern Encoded finite element number

Encoded damage indicator value

Figure 2. Implicit redundant representation for damage identification

4.3 Local Hillclimbing Algorithm GAs provide a global, stochastic search method. In certain situations, however, GAs will have a slower convergence rate near the global optimum. To overcome this problem, a local hillclimbing search technique is implemented to allow fine-tuning of the results obtained by the GA. A reduced version of hillclimbing was implemented, in which hillclimbing is only carried out on variables with damage indices greater than zero.

5. Multi-objective Optimization of Sensor and Excitation Layouts Sensor and excitation layout design, as well as many other complex design problems, can be stated as a multi-objective optimization problem. It is essential to design the sensor layout in such a way that the information contained in the collected vibration data is maximized with respect to the SDIM used. Maximizing the diagnostic information provided will enhance the efficiency and reliability of the SDIM. This objective, however, contradicts the desire to minimize the number of sensors used in order to reduce equipment, preparation, and labor costs. The tradeoff between the number of sensors and the quality of information provided can be investigated using a multi-objective (MO) optimization approach. The optimization of sensor and excitation layouts can be performed with two objectives: Minimize the number of sensors used, min f1(x) = nmeas, and maximize the total diagnostic information provided by the set of measurements, max f2(x) = I, subject to 0 < nmeas ≤ nu,meas where nu,meas is a constraint on the maximum number of sensors allowed. The sensor decision variables are logical variables that act as an “on/off” switches for each unrestrained DOF in a structure where a sensor can be placed and nmeas is number of sensors in the current layout. In addition, there is a discrete decision variable, x0, that defines the excitation DOF location. Typically if there are multiple objectives, there is not a single optimal solution that minimizes all objectives simultaneously. When multiple solutions exist, the optimum set of solutions is a collection of “trade-off” or Pareto-optimal solutions. A vector of decision variables x*, contained in the set of feasible solutions, is Pareto optimal if there does not exist any feasible solution x such that fi(x) ≤ fi(x*) for all i = 1, 2, …, s and fi(x) < fi(x*) for at least one j assuming that all objective are to be minimized, where f(x) is the vector function consisting of s objective functions. A vector of decision variables x* satisfying these conditions is considered to be nondominated and belongs to the set of nondominated solutions defining the Pareto front [18]. In this research, a multi-objective genetic algorithm is used to solve the MO optimization problem by evolving discrete sets of Pareto-optimal sensor and excitation layout designs. A significant benefit of using GAs is that they work with a population of solutions and therefore a set of Pareto optimal solutions can be evolved in a single run. Having a set of optimal solutions allows the designer to select the “best” alternative with respect to a specific environment by examining the set of Pareto optimal solutions. 5.1 Diagnostic Information Measure Satisfying the first objective of the optimum sensor layout problem requires a diagnostic information measure for FRF data. The information measure is based on the sensitivities of FRFs with respect to individual damage indicators (finite elements). The sensitivity of the FRF matrix with respect to the jth finite element is stated.

4

∂H (ω ) ∂x sj

= − H (ω )

∂Z (ω ) ∂x sj

H (ω )

(5)

where H(ω) is an FRF matrix and Z(ω) is the inverse of the FRF matrix under investigation. Equation 5, however, is computationally prohibitive for large structures. In these cases, a numerical differentiation approach for the calculation of the FRF function sensitivities can be used. ∂H km (ω ) H km (ω ) x sj = 0 − H km (ω ) x sj =∆x sj (6) ≈ ∂x sj ∆x sj where Hkm(ω) is an FRF function (kmth element of H(ω)) for which the excitation is located at the kth degree of freedom (DOF) and s the measurement is taken at the mth DOF, and ∆x j is a small perturbation at element j. The diagnostic information measure was defined for a pair of excitation, kth, and measurement, mth, DOFs using the FRF function sensitivities with respect to the damage indicator vector xs.

ιkm

⎛ ϖ1 ∂H km (ω ) ⎞ dω ⎟ = ∑⎜ ∫ s ⎜ ⎟ ∂x j j =1 ϖ 0 ⎝ ⎠ ne

2

(7)

where lkm is the information contained in the Hkm FRF function, ϖ0 and ϖ1 are the lowest and highest measured frequencies, respectively; and ne is the number of finite elements in the model. The information measure is defined with respect to all finite elements in the model; therefore this function gives an overall quality measure of the information contained in the measurement data. The total diagnostic information, I, contained in nmeas sensors, assuming only one excitation DOF, is defined as the sum of the squared lkm terms obtained considering each measurement location using Equation 7. 5.2 Multi-Objective Genetic Algorithms An overview of MO GAs is provided by Coello [18]. A specific MO GA technique called Nondominated Sorting Genetic Algorithm (NSGA) proposed by Srinivas and Deb [19] is used in this research. NSGA employs a different selection procedure then a simple GA [14]. Before selection is performed, the population is ranked based on each individual’s domination using a Pareto ranking. Nondominated individuals are identified from the population and make up the first nondominated front. Individuals in the nondominated front are assigned the same dummy fitness value. The dummy fitness value is used during selection to assign the same reproductive potential to each nondominated individual in the same front. The nondominated individuals are removed from the population and the ranking procedure is repeated for the remaining individuals in the population until all individuals have been ranked. Discrete fitness sharing was used to maintain diversity and prevent premature convergence of the population to local optimum. In the discrete implementation, individuals in the first rank were assigned a dummy fitness of zero and in each subsequent rank the fitness value assigned was increased by one. Fitness sharing was then performed if two individuals had the same number of sensors and the same level of measurement information. Tournament selection selected individuals for the next generation based on the NSGA dummy fitness. Adaptive multi-point crossover, non-uniform mutation, and elitism were imposed. 5.3 GA Genotype Representation for Sensor Optimization In each sensor optimization problem there is a set of DOFs where the sensors and/or actuators can be located. All of the sensor layout optimization trials considered all unrestrained DOFs in the horizontal, x, and vertical, y, directions as potential sensor locations. Rotational DOFs were not considered because of the difficulties in obtaining these measurements. The fixed GA representation shown in Figure 3 was used to encode information for all DOFs in each individual. The first n-bits of the individual encode the excitation DOF. The rest of the individual encodes nsens number of binary digits, or “on/off” switches, that identify the DOFs at which the sensors are located. In all trials, a maximum of 10 sensors was specified, along with one excitation. An individual, therefore, was repaired if the number of sensors encoded was zero or exceeded ten sensors. In early generations, an individual is more likely to be repaired by randomly turning off/on encoded sensor locations until the number of sensors encoded equals a random feasible number of sensors. In later generations, an individual is more likely to be reinitialized by replacing the infeasible individual with a new, randomly encoded individual. Index identifying the excitation DOF 0 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 ... 0 0 1

ON/OFF switches corresponding to sensor locations at elements

Figure 3. GA representation for sensor optimization 6. Description of Structural Damage Case Studies To evaluate the benefit of using the proposed FRF-based SDIM and the optimal sensor layouts, two sets of case studies were investigated. In each trial, the performance of both the fixed GA and the IRR GA representations were compared. The effect of seeding the initial population of the fixed GA with a zero-damage individual was also investigated. The zero damage individual in

5

the fixed GA representation is simply a string in which all damage indicators are encoded with zero values. The benefit of using a zero damage individual in the initial population is that this solution may contain beneficial genetic information that enables the fixed GA to find a better solution. In each trial, the benefit of the optimal excitation and sensor layouts that were evolved using multiobjective genetic algorithms is also investigated [2]. Solving an inverse problem using limited information leads to uniqueness problems of that solution, i.e. several solutions can be found satisfying the inverse problem formulation. Since noise in sensor data further intensifies the difficulty of inverse problems, the effect of noise on SDIM accuracy was studied for 0%, 5%, and 10% noise. Normally distributed random noise was added to the simulated FRF data with zero mean and a variance of unity. The number of significant digits for all case study trials was 6, which required 20 bits to represent a damage indicator with a precision of about 9.44⋅10-7 or 9.44⋅10-5%. Finally, for all case studies the Young’s modulus of steel, E, is 207 GPa (30,000 ksi), Poisson’s ratio is 0.3, and the density is 7780 kg/m3 (0.000728 lb-s2/in4). Table 1. Case study imposed damages scenarios Case Study TWSP FRM II

Damaged Imposed Non-optimal Layout Optimal Layout Element Damage Excitation Sensor Excitation Sensor 6 5% 18-y 14-y 13-y 13-y 10 10% 28 20% 61-y 59, 60, 61, 61-y 59, 60, 61, 16 10% 62, 63, 64 62, 65, 66, 11 10% (all y) 71, 72 4 10% (all y) 10 10% 18 20% y: measurement taken in vertical direction; x: measurement taken in horizontal direction

6.1 Two-span Beam Case Studies The two-span beam (TWSP) case study is outlined in Table 1. This case simulates multiple damages. The finite element model is shown in Figure 4 defined by 20 steel beam elements (W12x65), in which regular numbers indicate node numbers and element numbers are in italics. To obtain the required FRF functions, the first 10 natural frequencies and mode shapes were used. 15.24 m (50 ft)

7.62 m (25 ft)

1

2 1 2

3 3

4

5

4

5

6

7

6

8

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21

9 10

11 12 13 14 15 16 17 18 19 20

Figure 4. Two-span continuous beam model 6.2 Unbraced Frame Case Studies The unbraced steel frame (FRM) case study imposes multiple damages with varying severity that represent beam or joint deterioration. The frame structure has 76 nodes and 81 elements as shown in Figure 5. Columns (W14x132) are modeled using 3 finite elements and beams (W12x65) with 5 finite elements. Joint damage is modeled with short elements (0.3048 m, 1 ft) at the connections. For calculating the FRFs, the first 38 natural frequencies (many of them are multiple or closely spaced) and mode shapes of the structure are used. 9.14 m (30 ft)

9.14 m (30 ft)

7.62 m (25 ft)

69

71

72

65

66

61

62

3.66 m (12 ft)

46 36

67

68

59

60

3.66 m (12 ft)

73

63

64

4.57 m (15 ft)

21 38 28 49

6

7 19 16 11 27

2

4 3

5 10

18

34

Figure 5. Three-story, three-bay unbraced frame model

7. Discussion of Two-span Beam Case Study Results The TWSP case study has two damaged elements. Element 6 has 5% damage and element 10 has 10% damage. Measurements obtained from one non-optimal sensor location and from one optimal sensor location were used to identify damage. Three types of representations were investigated: fixed seeded, IRR seeded, and IRR unseeded. The GA parameters used in the TWSP trials are summarized in Table 2.

6

Table 2. GA Parameter Settings for SDIM Case Study Trials TWSP (Fixed and IRR) FRM (IRR only) Population Size 20 200 Tournament Size 4 (Fixed) 6, 8 (IRR) 8 Maximum Generations 200 300 Stop if no Improvement 20 20 Crossover Type Adaptive Adaptive Crossover Sites 6, 8 6 Crossover Rates Primary: 0.9, Secondary: 1.0 Primary: 0.9, Secondary: 1.0 Mutation Type Non-uniform (Fixed) Uniform Mutation Rate 0.005, 0.0075 0.005 Elitism Yes Yes IRR - Number of gene instances was 5 and the gene locator (GL) length is 3

Percent damage

For 0% noise, the fixed representation was not able to found the global optimum after 200 generations for the TWSP case study. Instead the fixed representation obtained a solution in which the correct damaged elements were identified with higher damage indicator values (4.19 and 8.82%) than any other elements. The largest false prediction of 1.48% occurred at element 11. Since the fixed representation encodes all of the damage indicators, there are many falsely identified damaged elements of low severity. Hillclimbing found the global optimum. In comparison for 0% noise, the IRR GA found a near-global optimum with only two false predictions of low severity. Only a few hillclimbing iterations were required to find the global optimum. The damage identification results obtained considering 5% noise are presented in Figure 7. The predicted damages in element 6 and 10 are slightly underestimated with gradually increasing magnitudes as the noise level increases. The IRR GA without initial seed performed the best; finding the global optimum solution 3 times out of 4 (in the sense that hillclimbing could not improve on it) with the optimal sensor layout. With 10% noise imposed, the IRR GA representation using the non-optimal sensor layout predicted the following damages in the true damaged elements: 5.9% in Element 6 and 9.5% in Element 10 as shown in Figure 7. Typically, two elements were falsely identified with damage severities less than 1%. Therefore the SDIM accuracy degraded with measurement noise. Damage predictions using the non-optimal layout tended to overestimate the damage in element 6 (5%), while underestimating the damage in element 10 (10%). These results suggest that the presence of noise has the effect of equalizing damage levels in the SDIM predictions using the non-optimal layout. In comparison, the IRR GA representation using the optimal sensor location predicted a 4.9% damage in Element 6 and 10.0% damage in Element 10 in the 10% noise trials. In addition, using the optimal layout reduced the number and severity of falsely identified elements, since no false damage predictions were obtained typically. The average fitness of the initial population increased by a factor of 3.9 using the optimal layout, which indicates that the measurements used to detect damage contain more diagnostic information. Simulated 5% Damage in Element 6, and 10% Damage in Element 10 12.00% TWSP IIIA, fixed repr. 10.00% TWSP IIIA, IRR seeded 8.00% TWSP IIIA, IRR unseeded TWSP OPT, Fixed

6.00%

TWSP OPT, IRR seeded

4.00%

TWSP OPT, IRR unseeded

2.00% 0.00% 6

10

11

14

15

16

17

Element number

Percent damage

Figure 6. TWSP SDIM results for 5% comparing type of representation and sensor locations Simulated 5% Damage in Element 6, and 10% Damage in Element 10 10.00% TWSP non-optimal layout 8.00% TWSP optimal layout 6.00% 4.00% 2.00% 0.00% 6

10

11

14

15

16

17

Element number

Figure 7. TWSP SDIM results for 10% noise using IRR GA representation comparing sensor locations

7

8. Discussion of Unbraced Frame Case Study Results

Percent damage

Multiple damages with different severities result in increased uniqueness problems that make it increasingly difficult to find the true damaged elements and severities. The case study investigated for the unbraced frame problem is listed in Table 1. There are six damaged elements all located in the first floor beams at the joints, which simulates a damage scenario that may occur in frames sized according to the strong column, weak beam design concept. The damage severity at the exterior beam joints is 20% and the interior beam joints have 10% damage on each side of the interior joints. The GA input parameters used are listed in Table 2. Case study trials show that the IRR GA continues to perform well, while no valid results could be obtained using the fixed GA representation. Previously in the TWSP case study, the initial population of fixed representation SGA had to be seeded with the zero damage individual in order to perform well. In comparison, the good performance of the IRR GA can be traced to its adaptive characteristic that make it well suited for damage identification problems. For 0% noise trials, the number of correctly identified elements out of the six true damaged elements was 4, 5, and 6 using 3, 4, and 5 optimum sensors, respectively. Three interior beam elements (4, 10 and 11) and one exterior beam element (18) were correctly identified as damaged in all three trials and the accuracy of the severity estimates increased when more sensors were used. By using four sensors instead of three, the additional exterior beam element (28) was identified with a severity somewhat higher than the imposed 20%. The addition of the fifth sensor provided enough information for the SDIM to uniquely identify all six damaged elements without any false identification as shown in Figure 8. After hillclimbing, the global optimum was found using the fivesensor optimal layout. By increasing the number of sensors, the SDIM results become more unique. The benefit of the optimal layout is that only a fraction (5) of the total number of possible sensor locations (144) was required to accurately locate and quantify multiple damages with different severities. Additional trials were performed assuming 5% measurement noise. Using a 9-sensor optimal layout, all six true damage elements were identified with damage indicator values reasonably close to the inflicted damages as depicted in Figure 9. After hillclimbing, however, element 28 was assigned a zero damage indicator reducing the correctly located elements to five. The information contained in the measurements using nine sensors was not sufficient to uniquely locate all damage elements with high noise levels. Based on the trials performed, it is concluded that that increasing the number of sensors may not always facilitate damage identification, although typically both localization and quantification of damages becomes more accurate with increasing number of sensors. The results of trials investigated for the noisy measurements suggest that there is an optimal sensor configuration that is dependent on the damage scenario. In other words, different damage locations may impose a different set of optimum sensor locations. For certain civil engineering structures, such as frames, the designer may have a good understanding about possible damage locations before measurements are taken. For instance, joints on lower floors are more likely to be damaged than joints at higher elevations. Although the information measure defined in this research was formulated with respect to all finite elements in the model, this measure could be prioritized based on heuristics. Simulated 10% Damages in Elements 4, 10, 11 and 16, and 20% Damages in Elements 18 and 28 25.00% Trial 5, GA 20.00% Trial 5, Hillclimb 15.00% 10.00% 5.00% 0.00% 4

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11

16

18

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Element number

Figure 8. FRM SDIM results using IRR GA and a 5-sensor optimal layout (0% noise)

Percent damage

Simulated 10% Damages in Elements 4, 10, 11 and 16, and 20% Damages in Elements 18 and 28 50.00% Trial 9, GA

40.00%

Trial 9, Hillclimb

30.00% 20.00% 10.00% 0.00% 2

4

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Element number

Figure 9. FRM SDIM results using IRR GA and a 9-sensor optimal layout (5% noise) There are at least three distinguishable benefits of using optimum measurement layouts in damage identification. First, the damage-balancing phenomenon (i.e. predicted damage indicators of unequal damages tend to approach to a value in between the inflicted damage severities) that is observed for the non-optimal sensor location cases is not apparent in the optimum sensor location cases. The number and severity of falsely identified elements is also very much reduced. In addition, the consistency of the results is

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maintained at all damage levels and the accuracy of damage identification is less sensitive for noise in the measurements. The performance of fixed representation GA degrades as the number of variables increases due to the larger search space that needs to be explored. For example, the size of the search space considering a 20-bit encoding of the damage indicators is 40 S = ( 220 ) = 6.668 ⋅10240 , which is extremely large. In comparison, the IRR GA is able to effectively reduce the search space by adjusting the number of gene instances present in the string. For example in a trial with a single damaged element, the best individual 32 in the initial population may have 8 gene instances (damaged elements), which is equivalent to a ( 220 ) = 4.562 ⋅10192 times reduction in the search space when compared to the fixed GA. In the final population, there may be only a single gene instance in the best individual, which means that the search space has a size of 2 20 = 1, 048, 576 , which is much smaller then the initial search space or the search space for the fixed representation. 9. Conclusions This research addressed the problem of structural damage identification using the inherent information contained in FRF vibration signatures. A robust and efficient SDIM that only requires minimal measurement information to precisely locate and quantify the severity of damage was developed. The SDIM adjusts the stiffness properties of the undamaged linear discrete analytical model by minimizing the difference between the measured and analytically computed FRFs. The problem of damage identification is formulated as an unconstrained optimization problem using continuous variables representing the stiffness reductions of elements. Two GA representations were evaluated on simulated damage case studies involving a two-span continuous beam and an unbraced frame structure. The performance of the fixed GA representation was severely impacted by the size of the search domain and seeding the initial population was required to obtain a solution. The accuracy of damage identification using the fixed GA also degraded with increased measurement noise. The IRR GA in comparison was able to evolve global or near-global optimum solutions in noise-free trials. In the presence of noise, the IRR GA results were considerably less sensitive than the fixed GA results. As a consequence, on average the IRR GA converged faster than the fixed representation GA and its ability to find the true damaged elements was much more stable and accurate. Even for large frame problems, the IRR GA was capable of identifying the true damaged elements due to encoding only a small subset of all possible finite elements. In addition, seeding the initial population with the zero damage individual was not necessary to find the solution for IRR GA, but was advantageous in certain situations. Trials using non-optimal and optimal sensor layouts were performed to identify the benefits of using the optimal sensor layouts with the FRF-based SDIM. The solution to the multi objective optimization problem was a set of optimal sensor layout designs, which defined tradeoffs between the two objectives. Overall, trial results show that there was an increase in the measurement information collected using the optimal layouts. However, the diagnostic information contained in the measurements did not increase proportionally with the increase in the number of sensors. For all trials, the ability of the SDIM to uniquely identify damaged elements was enhanced using the optimal sensor layout designs, even in noisy measurement environments. Significant improvement in the robustness of the damage identification method was found when the optimal sensor layout designs were used for detecting multiple damages in larger frame structures. The FRF-based SDIM developed in this research is distinct from other SDIMs in several aspects. Damage can be localized and quantified using only a small subset of the possible measurement locations using the IRR GA representation. The SDIM developed is robust and is able to accurately locate and quantify multiple damages in relatively large-scale structures. The SDIM provides reduced sensitivity to noise, which allows greater accuracy in predicting damage under realistic noise levels. This research shows that IRR GAs support the development of highly-sophisticated systems capable of identifying crucial changes in the environment. In the future, the use of advanced GAs will help identify new solution techniques, alter the way in which we define the problems, and will broaden the boundaries of smart design technologies.

10. References 1.

Rytter, A., (1993). “Vibration based inspection of civil engineering structures.” Doctoral dissertation, Department of Building Technology and Structural Engineering, University of Aalborg, Denmark. 2. Liszkai, T.R. (2003). “Modern heuristics in structural damage detection using frequency response functions,” PhD Thesis, Dept. of Civil Engineering, Texas A&M University, College Station, TX. 3. Raich, A.M. and Ghaboussi, J. (1997). “Implicit redundant representation in genetic algorithms.” Evolutionary Computation 5 (3), 277-302. 4. Cawley, P., and Adams, R.D., (1979). “The location of defects in structures from measurements of natural frequencies.” Journal of Strain Analysis, 14(2), 49-57. 5. Kim, H.M., and Bartkowicz, T.J., (2001). “An experimental study for damage detection using a hexagonal truss.” Computers and Structures, 79, 173-182. 6. Kim, J.T., and Stubbs, N., (2002). “Improved damage detection identification method based on modal information.” Journal of Sound and Vibration, 252(2), 223-238. 7. Law, S.S., Chan, T.H.T., and Wu, D., (2001). “Efficient numerical model for the damage detection of large scale structure.” Engineering Structures, 23, 436-451. 8. Wang, Z. Lin, R.M. & Lim, M.K. 1997. Structural damage detection using measured FRF data. Computer Methods in Applied Mechanics and Engineering 147: 187-197. 9. Lee, U. & Shin, J. 2002. A frequency response function-based structural damage identification method. Computers & Structures 80: 117-132. 10. Thyagarajan, S.K. Schulz, M.J. & Pai, P.F. 1998. Detecting structural damage using frequency response functions. Journal of Sound and Vibration 210 (1): 162-170.

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11. Mares, C., and Surace, C., (1996). “An application of genetic algorithms to identify damage in elastic structures.” Journal of Sound and Vibration, 195(2), 195-215. 12. Dunn, S.A. 1998. The use of genetic algorithms and stochastic hill-climbing in dynamic finite element model identification. Computers & Structures 66 (4): 489-497. 13. Moslem, K., and Nafaspour, R., (2002). “Structural damage detection by genetic algorithms.” AIAA Journal, 40(7), 1395-1401. 14. Goldberg, D.E. 1989. Genetic algorithm in search, optimization, and machine learning. Reading, Mass.: Addison-Wesley Pub. Co. 15. Chou, J-H., and Ghaboussi, J., (2001). “Genetic algorithm in structural damage detection.” Computers and Structures, 79(14), 1335-1353. 16. Holland, J.H. 1975. Adaptation in natural and artificial systems. Ann Arbor, MI: The University of Michigan Press. 17. Michalewicz, Z. 1996. Genetic algorithms + data structures = evolution programs. Berlin – New York: Springer-Verlag. 18. Coello, C.A.C. (2001). “A short tutorial on evolutionary multiobjective optimization.” First International Conference on Evolutionary Multi-Criterion Optimization, EMO 2001, LNCS 1993, Springer-Verlag, Berlin Heidelberg, 21-40. 19. Srinivas, N. and Deb, K. (1994). “Multiobjective optimization using nondominated sorting in genetic algorithms,” Evolutionary Computation, 2(3), 221-248.

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