Optimizing sensor placement for structural health monitoring based on information entropy or total modal energy

In an example embodiment, an analysis application is used to optimize sensor placement by minimizing information entropy or maximizing total modal energy. These objectives are achieved by implementing a two-part optimization procedure, involving generating an evaluation database that stores an information matrix, and using the evaluation in conjunction with a genetic algorithm to produce an optimized sensor location set.

BACKGROUND

Technical Field

The present disclosure relates generally to structural health monitoring, and more specifically to techniques for optimizing sensor placement for structural health monitoring based on information entropy or total modal energy.

Background Information

Structural deterioration is inevitable for structures (e.g., bridges, dams, buildings, etc.) that are subjected to adverse operational and environmental conditions over long service lives. For example, in the year 2006, over 26% of the 600,905 bridges in the U.S. were rated as either structurally deficient or functionally obsolete. As a result of economic considerations, most of these aging structures are still in service. If existing deficiencies are not improved, for example, damage and cracks detected and repaired at an early stage, minor deficiencies may grow and lead to expensive repairs or, if unaddressed for too long, to catastrophic failures.

To try to address these issues, many structures are periodically inspected for structural deterioration. For example, in the case of bridges in the United States, biennial bridge inspection is mandated by the Federal Highway Administration (FHWA). Typically, such inspection is a manual process, performed primarily visually by skilled engineers. The visual inspections are often quite time-consuming and labor-intensive, and even if diligently performed, generally cannot detect small-size cracks or cracks hidden under paint. Visual inspections may miss many types of hidden deterioration, and seldom reveals the underlying causes of structural damage. Accordingly, they provide an inadequate and unreliable solution to the problem of detecting structural deterioration.

A number of automated structural health monitoring (SHM) systems have been developed, that have the potential to improve upon visual inspection. A typical SHM system includes a collection of sensors (e.g., accelerometers, strain gauges, corrosion sensors, etc.) placed on a structure, which are connected via cabling to one or more data acquisition units. The SHM system may constantly monitor the structure, and alert engineers if sensor readings indicate possible structural damage. Use of a SHM system may potentially allow engineers to move from current time-based maintenance programs to condition-based maintenance programs, which, in theory, could be more cost-effective.

Unfortunately, initial deployment of SHM systems may be quite expensive, reducing any potential overall cost savings. Such expense may be directly related to the number of sensors deployed. In addition to the cost of each sensor itself, additional costs are generally incurred for cabling back to data acquisition units, and for installation labor. Some deployed systems have used large numbers of sensors, in attempts to observe all potentially relevant behavior. Although SHM systems may provide valuable measurements of structural health, the expense involved in their initial deployment has prevented them from achieving widespread use.

It would be desirable to utilize only a limited number of sensors with a SHM system. In order to utilize a limited number of sensors, each sensor should be optimally placed on the structure to maximize the value of the information they are able to collect. However, determining such optimal placements may prove challenging for engineers in the field. Accordingly, there is a need for more accessible techniques for optimizing sensor placement.

SUMMARY

In an example embodiment, an analysis application is used to optimize sensor placement by minimizing information entropy or maximizing total modal energy. These objectives are achieved by implementing a two-part optimization procedure, involving generating an evaluation database that stores an information matrix, and using the evaluation database in conjunction with a genetic algorithm to produce an optimized sensor location set.

More specifically, in an example embodiment, an information matrix generation module of an analysis application generates an evaluation database that stores an information matrix usable to determine information entropy or mode shape for sensor placement. For a user selected number of sensors, an optimized sensor location set is determined. To determine the optimized sensor location set, an optimization module of the analysis application utilizes a genetic algorithm to determine successive candidate sensor location sets. A sensor placement evaluation module of the analysis application computes performance indicators for the candidate sensor location sets by applying the candidate sensor location sets to the evaluation database. The performance indicators are used as fitness values to evolve the sensor location sets, and search for an optimized sensor location set. Once an optimized sensor location set is found that minimizes information entropy or maximizes total modal energy, a user interface (UI) module of the analysis application displays such optimized sensor location set to a user. Based on the display, actual sensors may be applied to the structure at the locations to configure a SHM system.

It should be understood that a variety of additional features and alternative embodiments may be implemented other than those discussed in this Summary. This Summary is intended simply as a brief introduction to the reader for the further description which follows, and does not indicate or imply that the examples mentioned herein cover all aspects of the disclosure, or are necessary or essential aspects of the disclosure.

DETAILED DESCRIPTION

FIG. 1is a block diagram of an example electronic device100(e.g., a computer) that may be used with the present techniques. The electronic device100includes at least one processor110coupled to a host bus120. The processor110may be any of a variety of commercially available processors, such as an Intel x86 processor, or another type of processor. A volatile memory130, such as a Random Access Memory (RAM) is also coupled to the host bus via a memory controller125. When in operation, the memory130stores software (i.e. processor-executable instructions) and data that are provided to the processor110. An input/output (I/O) bus150is coupled to the host bust120via a bus controller145. A variety of additional components are coupled to the I/O bus150. For example, a video display subsystem155is coupled to the I/O bus150. The video display subsystem may include a display screen170and hardware to drive the display screen. At least one input device160, such as a keyboard, a touch sensor, a touchpad, a mouse, etc., is also coupled to the I/O bus. A persistent storage device165, such as a hard disk drive, a solid-state drive, or another type of persistent data store, is further attached, and persistently stores processor-executable instructions and data, that are loaded into the memory130when needed. Still further, a network interface180is coupled to the I/O bus150. The network interface enables communication over a computer network, such as the Internet, between the electronic device100and other devices, using any of a number of well-known networking protocols. Such communication may enable collaborative, distributed, or remote computing with functionality (including the functionality discussed below) spread across multiple electronic devices.

Working together, the components of the electronic device100(and other electronic devices in the case of collaborative, distributed, or remote computing) may execute a number of different software applications. For example, the memory130may store at least a portion of software for an analysis application140used to determine, for a user-provided number of sensors, optimized locations on a structure that minimize information entropy or maximize total modal energy. The analysis application140utilizes data stored in the memory130such as a structural model132of the structure and an evaluation database134, in conjunction with software modules or other sub-parts of the analysis application140, such as an information matrix generation module142, a structural analysis and design library143, an optimization module144, a sensor placement evaluation module146, and a UI module148, etc.

The structural model132may be a parameterized structural model, such as finite element (FE) model or a modal model, which describes the input-output behavior of the structure. The evaluation database134stores an information matrix, (e.g., a Fisher information matrix (FIM)) defined according to an entropy-based approach or an energy-based approach. The information matrix generation module142calculates an information matrix using the structural model132that is stored in the evaluation database134. The structural analysis and design library143includes functions for performing simulation runs for the scenarios, to produce results. The optimization module144may employ a genetic algorithm to determine candidate sensor location sets, and evolve those candidate sensor location sets based on fitness values, until an optimized sensor location set is determined. In one implementation, the optimization module144is implemented as a generic optimization framework, such as the Darwin Optimization Framework available from Bentley Systems Inc. of Exton Pa. The sensor placement evaluation module146computes performance indicators for sensor location sets using information from the evaluation database132. For example, according to an entropy-based approach the sensor placement evaluation module146may generate performance indicators that measure information entropy. Likewise, according to an energy-based approach the sensor placement evaluation module146may generate performance indicators that measure total modal energy. The sensor placement evaluation module146provides the performance indicators back to the optimization module144for use as fitness values. The sensor placement evaluation module146may be an independent software module, or may be implemented as a portion of the optimization module144. The UI module148displays graphical UIs on the display screen170, in which a user may select parameters and view optimized sensor location results, among other tasks.

The analysis application140and its modules142-148operate to solve a mathematically defined sensor placement optimization problem. The sensor placement optimization problem may be formulated to minimizes information entropy or maximizes total modal energy. A spatial error covariance model may be applied in either approach to avoid redundant information provided by neighboring sensors.

First, looking to an entropy-based approach, Let θϵNθbe the vector of freeparameters (physical or modal parameters) that need to be estimated using measured data D collected from a collection of sensors (e.g., accelerometers, strain gauges, corrosion sensors, etc.). Let D={yk, k=1, . . . , N} be measured sampled response time histories data, where ykϵN0refers to output data and N0is a number observed degrees of freedom (DOFs). N is total number of sampled data. Let xkϵNd, k=1, . . . , N be sampled response time histories computed at all Ndmodel DOFs from the structural model132that corresponds to a specific value θ of the model parameters. The measured response satisfies the equation:
yk=Lxk(θ)+Lek(θ)  (1)
where ek (θ) is the prediction error due to modeling error and measurement noise. The matrix LϵNθ×Ndis an observation matrix comprised of zeros and ones and maps model DOFs to measured DOFs. The matrix L therefore defines the location of the sensors in the structure. The information entropy measures uncertainty in the model parameter estimates. Asymptotically for very large number of data (NN0→∞), the information entropy depends on the determinant of the Fisher information matrix (FIM). As such, the determinant (as opposed to the trace or other scalar measures of FIM) may be used, providing:

h⁡(L;Σ,D)∼H⁡(L;θ0,Σ)=12⁢Nθ⁢ln⁡(2⁢π)-12⁢Nθ⁢ln⁡(det⁢Q⁡(L;θ0,Σ))(2)
where θ0={circumflex over (θ)}(L; Σ, D) is the optimal value of the parameter set θ that minimizes the measure of fit J(θ, Σ, D), which is given as:

Q⁡(L,θ0,Σ),=∑k=1N⁢⁢{L⁢∇θ⁢xk}T⁢{L⁢⁢Σ⁢⁢LT}-1⁢{L⁢∇θ⁢xk}(4)
in which ∇θis the gradient vector with respect to the parameter set θ. Σ is the error covariance matrix of ek(θ) composed of measurement errors and model errors such that:
Σ=Σm+Σs(5)
where Σmand Σsare the covariance matrices of the measurement and model errors, respectively. It may be assumed that the measurement error is independent of the location of sensors, so that the covariance matrix Σm=s2I, where I is the identity matrix. However, a certain degree of correlation should be expected for the model errors between two neighborhood locations arising from the underlining model dynamics. This correlation can be taken into account by selecting a non-diagonal covariance matrix Σs. The correlation Σi,jsbetween DOFs i and j can be expressed as:

∑i,js⁢⁢=∑i,is⁢⁢∑i,js⁢⁢R⁡(di,j)(6)
where di,jis the spatial distance between DOFs i and j. The correlation function R(di,j) can be written in a general form

R⁡(di,j)=exp⁡(-di,jλ)(7)
where λ is a measure of the correlation lengths. The spatial correlation of the prediction error tends to shift the sensors away from each other.

Sensor locations are desired that provides the most information in order to estimate the modal coordinate vector θ=ξϵm. Thus, Nθ=m. Applying modal analysis, the response vector xϵNdis given with respect to the parameter set θ in the form x=Φθ, where ΦϵNd×mis the mode shape matrix for m contributing modes. Noting that ∇θx=Φ and substituting into Eq. (4), the information matrix takes the form:
Q(L,θ0,Σ)=Q(L,Σ)={LΦ}T{LΦLT}−1{LΦ}(8)
which is independent of the nominal parameter values θ0. Based on the form of equation (8), a non-singular fisher information matrix Q(L,Σ) is obtained only if the number of sensors, N0, is at least equal to the number of contributing modes, m. Otherwise, for N0<m, the matrix Q(L,Σ) in equation (8) is singular and the determinant of the FIM will be zero for any sensor configuration. One way to optimally place sensors in the structure for N0<m is to maximize the product of the N0non-zero eigenvalues in the FIM, instead of maximizing the product of all eigenvalues.

Second, looking to an modal energy-based approach, the modal energy for the i-the mode Eicaptured by certain sensor placement L can be defined as following:
Ei(L,Σ)={Lφ}T{LΣLT}−1{Lφi}  (9)
where φiis the i-th mode shape and Σ is the error covariance matrix defined in equation (5). The modal energy defined above is equivalent to the i-th diagonal entry of FIM in equation (9). Therefore the total modal energy E captured by certain sensor placement L can be expressed as:

E⁡(L,Σ)=∑i=1m⁢⁢Ei⁡(L,Σ)=trace⁢{Q(L,Σ}(10)
Following the definition of captured modal energy, the total modal energy of the system can be expressed as:

Ea⁡(Σ)=∑i=1m⁢⁢φiT⁢Σ-1⁢φi(11)
The captured energy ratio can be defined as:

r⁡(L,Σ)=E⁡(L,Σ)Ea⁡(Σ)(12)
Using either the entropy-based approach or the modal energy-based approach, it is desirable to search for a specified number of sensor locations, noted as K, so that the overall performance of the K sensors is maximized. Therefore, the sensor placement optimization may be formulated as:

The analysis application140may solve this problem in two phases including: (1) generation of the information matrix and (2) optimization of sensor placement with a genetic algorithm.

FIG. 2is a flow diagram of an example sequence of steps200that may be implemented by the analysis application140for optimizing sensor placement. At step210, a user select whether an entropy-based approach or a modal energy-based approach is to be performed. The selection may be made in a UI, or by activating a particular version of the analysis application140. At step220, the UI module148presents an evaluation database creation UI and then a parameter setting UI on the display screen170.FIG. 3Ais a screen shot of an example evaluation database creation UI300.FIGS. 3B and 3Care screenshots of an example parameter setting UIs302and306. The evaluation database creation UI300may include a field312for selecting a structural analysis and design library143, a field314for selecting a structural model, a field316for selecting an output destination, as well as fields318for setting at least some parameters. The parameter setting UI302and306may include additional fields320for selecting sensor types and properties, a field324for selecting a number of sensors to be applied to the structure, as well as other fields for setting a variety of other parameters. One such other parameter may be whether existing sensors applied to the structure should be considered as part of an optimized solution. For example, fields322-324may be provided to select whether existing sensors should be considered, and to receive a source of data describing locations of existing sensors.

At step230, using the selected structural model and parameters, the information matrix generation module142calculates an information matrix using the above discussed formulas, and stores the information matrix in the evaluation database134.

Thereafter, at step240, the UI module148present an run optimization UI.FIG. 3Dis a screen shot of an example run optimization UI330. It may include a fields332for selecting optimization methods, fields334indicating problem size, fields336for selecting run status parameters, and fields338for displaying preliminary results, among other fields. A run may be conducted that includes a number of generations. At step250, the optimization module144determines a candidate sensor location set indicating possible locations for each of the number of sensors. The optimization module144passes the sensor location set to the sensor placement module evaluation module146. At step260, the sensor placement module evaluation module146computes a performance indicator for the sensor location set, using the equations provided above and the evaluation database134(specifically to the information matrix represented therein). The performance indicator produced by the sensor placement module evaluation module146is provided back to the optimization module144as a fitness value. With a goal of minimizing information entropy or maximizes total modal energy, according the approach utilized, the optimization module144may produce (according to a genetic algorithm) a subsequent, potentially-improved candidate sensor location set (a next generation), and the steps250-260repeated until an optimized sensor location set is eventually produced. The optimized sensor location set may be provided to the UI module148, which, at step270, may display an optimization run UI.FIGS. 3E and 3Fare screen shots of example results UIs340,342. Among other information, it may display results352including percent coverage of individual sensor location sets (including the optimized sensor location set) and locations of the sensors354for individual sensor location sets (including the optimized sensor location set). Utilizing the results, an engineer may apply actual sensors at the locations on the structure to configure a SHM system.

In general, the analysis application140may be used with a wide variety of different types of structures, and with different numbers of sensors, to produce optimized sensor location sets. As the number of sensors increases, the accuracy of modal identification results improves. Typical properties may be illustrated by example results of an example implementation.

FIG. 4is a representation of an example structural model400, specifically a FE model, of an example structure (here one approach span of the Verranzno Narrows Bridge Span in Brooklyn N.Y. City). The example structural model400was created using the STAAD.Pro v8i structural analysis and design application available from Bentley Systems Inc. of Exton Pa., and includes 177 beam elements and 107 plate elements, with 142 nodes. However, it should be understood that a variety of other structural analysis and design applications alternatively could have been utilized to create structure models with different numbers of element and nodes. For both an entropy-based approach and a modal energy-based approach, two representative spatial correlation lengths are considered: λ=0, which indicates uncorrelated measurements from sensors and λ=2.3 meters (91 inches), which coincides with a length between neighboring nodes in the width direction, as marked inFIG. 4

Considering first the entropy-based approach,FIG. 5Ais a plot510of information entropy for different numbers of sensors (2, 5, 8, 10, 15 and 20) with λ=0. The optimal entropy value inFIG. 5Adecreases monotonically as the number of sensors increases, which indicates that the uncertainty in measured modes decreases, and the accuracy of modal identification results improves.

FIGS. 5B-Gare diagrams520-570showing example optimized sensor location sets for a entropy-based approach with different numbers of sensors (2, 5, 8, 10, 15, and 20) with λ=0. As can be seen, some sensors are placed very closely for the scenarios using 8, 15 and 20 sensors. This indicates that some redundant information is likely being captured by those sensors.

FIG. 6Ais a plot610of information entropy for different numbers of sensors (2, 5, 8, 10, 15 and 20) with λ=2.3. The optimal entropy value inFIG. 6Adecreases monotonically as the number of sensors increases, which indicates that the uncertainty in measured modes decreases, and the accuracy of modal identification results improves.

FIGS. 6B-Gare diagrams620-670showing example optimized sensor location sets for a entropy-based approach with different numbers of sensors (2, 5, 8, 10, 15, and 20) with λ=2.3. Compared with the sets ofFIGS. 5B-G, it can be seen that the sensors are better separated, which indicates that the captured model information is likely improved (i.e. there is less redundant information)

Considering next the modal energy-based approach,FIG. 7Ais a plot710of modal energy and a plot720of energy ratio for different numbers of sensors (2, 5, 8, 10, 15 and 20) with λ=0. They both increase monotonically as the number of sensors increases, which indicates that the accuracy of modal identification results improves.

FIGS. 7B-Gare diagrams730-780showing example optimized sensor location sets for a modal energy-based approach with different numbers of sensors (2, 5, 8, 10, 15, and 20) with λ=0. As can be seen, some sensors are placed very closely for the scenarios using 8, 10, 15 and 20 sensors, again indicating that some redundant information is likely being captured by those sensors.

FIG. 8Ais a plot810of modal energy and a plot820of energy ratio for different numbers of sensors (2, 5, 8, 10, 15 and 20) with λ=2.3. They both increase monotonically as the number of sensors increases, which again indicates that the accuracy of modal identification results improves.

FIGS. 8B-Gare diagrams830-880showing example optimized sensor location sets with different numbers of sensors (2, 5, 8, 10, 15, and 20) with λ=2.3. Compared with the sets ofFIGS. 7B-G, it can be seen that the sensors are better separated, which again indicates that the captured model information is likely improved (i.e. there is less redundant information)

In summary, the above description details techniques for optimizing sensor placement for structural health monitoring of a structure using an entropy-based approach or an energy-based approach. It should be understood that various adaptations and modifications may be readily made to the techniques, to suit various implementations. Further, it should be understood that at least some of the techniques may be implemented in software, in hardware, or a combination thereof. A software implementation may include computer-executable instructions stored in a non-transitory computer-readable medium, such as a volatile or persistent memory, a hard-disk, a compact disk (CD), or other storage medium. A hardware implementation may include specially configured processors, logic circuits, application specific integrated circuits, and/or other types of hardware components. Further, a combined software/hardware implementation may include both computer-executable instructions stored in a non-transitory computer-readable medium, as well as one or more specially configured hardware components, for example, processors. Accordingly, it should be understood that the above descriptions are meant to be taken only by way of example.