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– Ace Ventura, Ace Ventura: Pet Detective (1994)
Concluding Review
The idea that led to Measurement- and Model-based Structural Analysis was the collaboration of two different engineering sciences, geodesy and continuum physics. It was recognized that there are knowledge gaps on both sides and that the gap could be partially closed by the knowledge of the other field of expertise. This led to a lively cooperation with the realisation that although both disciplines dealt with different tasks with different topics, they solve problems using essentially identical methods. It is theVariational Calculusthat connects both disciplines.
The calculus of variations, founded almost 300 years ago by Lagrange and Euler, influenced both disciplines in many ways. In mechanics, it led to the development of theprinciple of virtual displacements, of theLagrangian mechanics, of thefinite element methodand many more. As can be read in Dunnington et al. (2004, p. 11 ff), Gauss was influenced by the work of Lagrange and Euler. It should not be proven that he has taken up the calculus of variations for the development of the least squares method. Nevertheless, based on the generalization of the adjustment calculation by Helmert (1872, p. 173 ff), it can be seen that hiscorrelatesare nothing more than theLagrange multiplier. Because of this compatibility, it can be seen how inherently different methods are related. There are many other methods that have been developed for different reasons, motivations and out of different perspectives independent of the calculus of variations. Nevertheless, the calculus of variations can help to understand both old and newly developed methods. And it is precisely this understanding that leads to appreciation and ultimately to new possibilities and to maturity.
By means of two experiments, the Measurement- and Model-based Structural Analysis shows its promising capabil-ity for damage detection and localization. This capabilcapabil-ity was achieved thanks to the cooperation of two different fields of expertise. A physical model of the samples is expressed in the form of differential equations and boundary conditions using continuum mechanics. In this way, the material parameters could be considered as a possible pa-rameter for the assessment of damage. Furthermore, displacements and their derivatives, inclinations and strains were identified as possible measurands. And it could also be decided which quantity could be seen as fixed values.
Using the finite element method, the differential equations and boundary conditions could be transformed into a system of linear equations on which the functional model of the least squares method can be based. The adjust-ment calculation offers the possibility to determine unknown parameters from the observations and to evaluate the results stochastically, whereby statistical global and local tests could be used to detect and localize damage.
Scientific and Technological Contribution
By examining the variational calculus in the work, it became clear that the finite element method and the least squares method is one single method. The only difference is that both methods solve different problems. The fi-nite element method solves special types of partial differential equations with corresponding boundary conditions, while the least squares method solves overdetermined systems of equations. In Boljen (1993), this insight was already given, but the presentation was incomplete. In the interest of completion, it was shown in this work that the same problem can be formulated in three variants (strong, weak and extremal formulation) and that they can be reformulated among one another. It was also shown that both methods lead to a weak solution of their corre-sponding problem. The term “weak” refers to the approximate respectively most probable property of the solution.
Weak solutions are unable to solve the strongly formulated problem. The exact or analytical solution of the partial differential is generally unknown, but applying finite element method, an approximate solution can be obtained.
Similarly, there is in general no solution for an overdetermined system of equations, but the most probable solution is obtained by using the least squares method. It should also be noted that the viewpoint of variational calculus is by no means limited to the two methods. The different solution methods emerge when the following questions are determined:
• Is it a continuous or discrete problem?
• What is the formulation of the problem and, if necessary, the conditions?
• Which trial function is used for the solution?
• Which test function is used for the solution?
The study of the variational calculus led to the realisation that many methods are principally the same. Old, new and unknown methods from other disciplines may be more accessible if the variational calculus is used as some sort of “template”. In this work, we have shown how the finite element method and the least squares method fit into the scheme of the variational calculus. This insight is intended to expand the adjustment theory and is thus the scientific contribution to the geodetic community.
For the development of this Measurement- and Model-based Structural Analysis, the finite element method and the method of least squares is combined. Equations of system of linear equations from the finite element method were selected depending on the observations, and they are inserted directly into the least squares method. As a result, the finite element method calculation procedure is inverted. For arbitrarily shaped bodies, the elastic parameters can be determined directly from the measured displacement fields. This also offers the ability to compute directly the stochastic properties of the material constants. In addition, this has the advantage that the linear elasticity can be exploited to calculate the gradient of the target function or the Jacobian directly. While the other approach with the outer loop around the finite element method, the individual columns of the Jacobian must be costly calculated from the objective function. The following aspects could be examined with Measurement- and Model-based Structural Analysis:
• Determination of an optimal measurement set-ups.The test structures “Variationsbrücke” and an aluminium beam are characterised using elastostatic equations or the Euler-Bernoulli beam theory. From this, the elastic parameters are considered suitable for damage evaluation. Also, the displacement fields respectively the displacement, inclination and strain have been found to be possible measurands. The capability to di-rectly calculate the stochastic properties of the material constants allows a numerical preliminary examina-tion of displacement, inclinaexamina-tion and strain sensors. It was shown how the precision of the sensors, the placement of the sensors and the number of sensors impacted the parameters. The optimal measurement set-ups can then be determined from these results.
• Finding a geometrically simple substitute model for a geometric complex body.If the complexity of the body exceeds a certain limit, a finite element meshing of the body is no longer possible due to the memory limit.
By using Measurement- and Model-based Structural Analysis, a substitute model can be determined. The geometric simpler substitute model deforms like the geometrically more complex original under the same conditions. A representative part of the geometrically complex body made of isotropic linear elastic material is cut out. Then, the deformation behaviour of this part is calculated by the finite element method. The resulting displacement field is then evaluated. Using the Measurement- and Model-based Structural Analysis, the elastic parameters of the substitute part made of anisotropic linear elastic material are determined. This allows a substitute model to be derived.
• Detect and localise damage.The undamaged state of the test structure is defined as reference state. In this state, the test structure consists of finite elements that share the same elastic parameter. This is then used as observed unknowns to allow a comparison between the reference and a current state of the test structure.
Theχ2-test respectively the global test jointly evaluates all residuals of the observations to detect significant changes between the states of the test structure. If this is the case, damage is detected. Only then the indi-vidual finite elements are then checked iteratively by releasing them indiindi-vidually and evaluating the target
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function. The minimum value of all possible target function values indicates the position of the damaged elements. In order to circumvent the iterative calculation, the standardised residuals can be used to estimate the location of the damaged elements.
Two experiments were carried out to validate the ability of Measurement- and Model-based Structural Analysis to detect and locate damage. In both experiments, photogrammetry was used to determine the displacement field.
The following test structures were built:
• “Variationsbrücke”. A truss consisting of special construction profiles made of aluminium. Artificial dam-ages were caused by loosening screws. Damage could be detected, but the localization was unsuccessful. It is suspected that due to production errors, the profiles did not meet the required length. Residual stress was inadvertently generated during assembly. The standardized residuals in the results probably indicate the released residual stresses.
• Aluminium beam.Because the results of the previous experiment were unsatisfactory, a bending test exper-iment was carried out. In spite of subsiding bearings, damage could be detected and localized to a certain extent. Still, it was unable to pinpoint the absolute correct location, but it indicates approximately the dam-aged area in consistent manner. It should be noted that the residuals of the displacement observations shows more precise position of the damage than the standardised residuals.
As noticed, due to the time constraints some big leaps had to be made in the development of Measurement- and Model-based Structural Analysis. Although at that time, the theoretical works were unfinished, it was already in-sisted on planning the experiment. Due to the great pressure from the boss and from a very eager colleague, it was decided to build the aluminium truss as a test structure. The result was anything but exhilarating. Nonetheless, this has led to some new techniques and insights such as the substitute model, the introduction of standardised observations to circumvent the determination of the optimal measurement set-up for photogrammetry, the lin-ear relationship between the standardized residuals and the cofactor matrix of observations, and consequently this has led to the second experiment and to the extension of stiffness matrix to determine the boundary condition as unknowns. The Measurement- and Model-based Structural Analysis is by definition a diagnostic method suit-able for monitoring tasks. But, the experiments show that Measurement- and Model-based Structural Analysis is still premature. Nevertheless, the establishment of the fundamentals of Measurement and Model-based Structural Analysis is the technological contribution for structural health monitoring.
Improvement Suggestions and Further Research
The weak link of Measurement- and Model-based Structural Analysis is the physical modelling of the specimen and its external influences. Either the condition can be adapted so that a simplified physical model is sufficient to describe the specimen, or the physical model must be enhanced. For the latter, the following points might of interest for a geodetic re-evaluation:
• Kinematics. A reassessment of the motion description from a geodesist’s point of view could bring new insights, see Sec. 2.2.
• Material laws.The more complex the physical processes, the more complex the material equations are. To-gether with geodesists, the material equations can be examined for their numerical and stochastic properties.
It is then possible to formulate new equations, whereby their parameters can be determined with numeric stability.
• Vibrations.The vibration behaviour of a test bridge is to be examined. It is helpful to use the elastodynamic equations derived in Sec. 2.5.1 as well as their numerical treatment with finite element method in Sec. 2.6.
• Hybrid measurements. Although different types of observation were also treated theoretically, see Sec. 5.1, only the displacement in the experimental examinations could be measured in this dissertation. In further work, hybrid measurements should be studied.
As already mentioned at the beginning, the procedure of the variational calculus could be a universal entry point for many methods. It could be lucrative for adjustment theory to examine the following points:
• Examination of known methods.Fourier-Analysis and Laplace transform, for example, are well-known.
It can be shown that with a small amount of preliminary information, such as problem description and solution approach, it is possible to reach the known methods from the variational calculus procedure.
• Examination of different solution representations. The solution function is often expressed as an infinite series. The most popular representation is the power series. By omitting some summands, other types of infinite series are formed. The relationship between different series should be examined, for example, the Laurent series and Chebyshev polynomials. The development of new series should also be studied, for example, Wavelets or neuronal networks.
• Examination of popular methods.From completely unfamiliar procedures of popular applications, it should be shown that they follow the same scheme. By means of simple examples, it should show how to encrypt and decrypt a number, how to adjust the parameters of a neuronal networks by Gauss-Markov model, etc.
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