Weizhuo Wang

Adaptive moment descriptors for full-field strain and displacement measurements

Recent advances in measurement techniques such as digital image correlation, automated photoelasticity, electronic speckle pattern interferometry and thermoelastic stress analysis allow full-field maps (images) of displacement or strain to be obtained easily. This generally results in the acquisition of large volumes of highly redundant data. Fortunately, image decomposition offers feasible techniques for data condensation while retaining essential information. This permits data processing such as the validation of computational models, modal testing or structural damage assessment efficiently and in a straightforward way. The selection, or construction, of decomposition bases (kernel) functions is essential to data reduction and has been shown to produce features, or attributes, of the full-field image that are effective in reproducing the measured information, succinct in condensation and robust to measurement noise. Among the most popular kernel functions are the orthogonal Fourier series, wavelets and Legendre polynomials, which are defined on continuous rectangular domains, and Zernike polynomials and Fourier–Mellin functions, which are defined on continuous circular domains. The discrete orthogonal polynomials include Tchebichef, Krawtchouk and Hahn functions that are directly applicable to digital images and avoid the approximate numerical integration that becomes necessary with the sampling of continuous kernel functions. In practice, full-field measurements of the engineering components are usually non-planar within irregular domains – neither rectangular nor circular, so that the classical kernel functions are not immediately applicable. To address this problem, a complete methodology is described, consisting of (1) surface parameterisation for the mapping of three-dimensional surfaces to two-dimensional planar domains, (2) Gram–Schmidt orthogonalisation for the construction of orthogonal kernel functions on arbitrary domains and (3) reconstruction of localised image features, such as regions of high strain gradient, by a windowing technique. Application of this methodology is demonstrated in a series of illustrative examples

Clustering of Parameter Sensitivities: Examples from a Helicopter Airframe Model Updating Exercise

The need for high fidelity models in the aerospace industry has become ever more important as increasingly stringent requirements on noise and vibration levels, reliability, maintenance costs etc. come into effect. In this paper, the results of a finite element model updating exercise on a Westland Lynx XZ649 helicopter are presented. For large and complex structures, such as a helicopter airframe, the finite element model represents the main tool for obtaining accurate models which could predict the sensitivities of responses to structural changes and optimisation of the vibration levels. In this study, the eigenvalue sensitivities with respect to Youngs modulus and mass density are used in a detailed parameterisation of the structure. A new methodology is developed using an unsupervised learning technique based on similarity clustering of the columns of the sensitivity matrix. An assessment of model updating strategies is given and comparative results for the correction of vibration modes are discussed in detail. The role of the clustering technique in updating large-scale models is emphasised.

Analysis of displacement fields from a high-speed impact using shape descriptors

A composite bonnet liner subject to a high-velocity (70 m/s), low-energy (<300 J) impact by a 50-mm-diameter projectile has been investigated using computational simulation and by experiment. High-speed digital image correlation was employed to generate maps of displacement fields over the 1-m2 bonnet at 0.2 ms increments for 0.1 s, that is, 500 datasets, and the results have been compared to those predicted by finite element analysis. Image decomposition was utilised to reduce the dimensionality of both datasets by representing them using adaptive geometric moment descriptors; these descriptors were used to perform quantitative comparisons of the datasets and to test the validity of the model based on all the available data. The model was found to be a good representation of the physical experiment during the first half of the impact event but a less good representation in the remainder of the test, probably because damping effects were not adequately incorporated into the simulation. The methodologies for data comparison and evaluation of model validity proposed and demonstrated in this study represent a significant advance in procedures for ensuring model fidelity and for creating model credibility in the simulation of dynamic engineering events.

Construction of shape features for the representation of full-field displacement/strain data

The achievement of high levels of confidence in finite element models involves their validation using measured responses such as static strains or vibration mode shapes. A huge amount of data with a high level of information redundancy is usually obtained in both the detailed finite element prediction and the full-field measurements so that achieving a meaningful validation becomes a challenging problem. In order to extract useful shape features from such data, image processing and pattern recognition techniques may be used. One of the most commonly adopted shape feature extraction procedures is the Fourier transform in which the original data may be expressed as a set of coefficients (coordinates) of the decomposition kernels (bases) in the feature space. Localised effects can be detected by the wavelet transform. The acquired shape features are succinct and therefore simplify the model validation, based on the full-field data, allowing it to be achieved in a more effective and efficient way. In this paper, full-field finite element strain patterns of a plate with a centred circular hole are considered. A special set of orthonormal shape decomposition kernels based on the circular Zernike polynomials are constructed by the Gram-Schmidt orthonormalization process. It is found that the strain patterns can suitably be represented by only a very small number of shape features from the derived kernels.

Bias reduction in sub-pixel image registration based on the anti-symmetric feature

A simple but effective bias reduction technique is developed based on the anti-symmetric feature of the sub-pixel image registration bias. Depending on the error propagation theory, the anti-symmetric feature is mathematically derived through a classical subset-based digital image correlation algorithm considering the most common error sources i.e. the grey-intensity interpolation scheme and random noise. This leads to the sub-pixel registration bias formulated in the form of an analytic expression that consists of the interpolation-induced phase error and the random noise induced bias, which is also further illustrated by numerical simulations. Bias reduction is achieved by compensating the bias at a certain sub-pixel displacement with the bias at the corresponding anti-symmetric sub-pixel displacement where the Fourier shift theorem is employed to alter the displacement without introducing extra bias. The performance of proposed method is validated using numerical case studies with different interpolation schemes and noise levels, by which the sub-pixel registration bias is shown to be significantly reduced.

Corneal topography matching by iterative registration

Videokeratography is used for the measurement of corneal topography in overlapping portions (or maps) which must later be joined together to form the overall topography of the cornea. The separate portions are measured from different viewpoints and therefore must be brought together by registration of measurement points in the regions of overlap. The central map is generally the most accurate, but all maps are measured with uncertainty that increases towards the periphery. It becomes the reference (or static) map, and the peripheral (or dynamic) maps must then be transformed by rotation and translation so that the overlapping portions are matched. The process known as registration, of determining the necessary transformation, is a well-understood procedure in image analysis and has been applied in several areas of science and engineering. In this article, direct search optimisation using the Nelder–Mead algorithm and several variants of the iterative closest/corresponding point routine are explained and applied to simulated and real clinical data. The measurement points on the static and dynamic maps are generally different so that it becomes necessary to interpolate, which is done using a truncated series of Zernike polynomials. The point-to-plane iterative closest/corresponding point variant has the advantage of releasing certain optimisation constraints that lead to persistent registration and alignment errors when other approaches are used. The point-to-plane iterative closest/corresponding point routine is found to be robust to measurement noise, insensitive to starting values of the transformation parameters and produces high-quality results when using real clinical data.

Image Analysis for Full-Field Displacement/Strain Data: Method and Applications

Recent advances in measurement techniques, including digital image correlation, automated photoelasticity, electronic speckle pattern interferometry and thermoelastic stress analysis, permit full-field maps of displacement or strain to be obtained easily. They provide large volumes of mostly redundant data, which should be condensed to the essential information to permit straightforward processes such as validations of computational models or damage assessments. A way to do this is by image processing, an important aspect of which is the definition of an orthogonal basis (orthogonal kernel functions). Generally, this is problem dependent and requires some skill from the analyst if the number of image features (the coefficients of the orthogonal basis) is to be restricted to a suitably small number. Advantage may be taken of patterns of symmetry, for example cyclically symmetric patterns are well-suited to treatment by Zernike polynomials and rectangular patterns are well-suited to treatment by Fourier series. The Zernike and Fourier kernels are continuous polynomials with orthogonality properties that require integration and must be discretised. The discrete Tchebichef polynomials are ideal for the treatment of full-field information at multiple discrete data points. In many cases the data field is localised around a particular feature, such as local strain around a hole in a tension-test specimen. In this case, the polynomial basis should similarly be localised by various forms of scaling – this requires the application of the Gram-Schmidt procedure to maintain orthogonality. The image features (sometimes called shape features) are meaningful and may be used to identify particular patterns in the data – e.g. for detecting cracks or other forms of damage. When assembled in a feature vector, the distance between feature vectors from measured and numerical results are useful for refining numerical models. In this paper the principles of image analysis, as applied to full-field displacement/strain data are explained and experimental examples are used to illustrate the practical usefulness of the method. The applications include (i) vibration mode shapes of laminated honeycomb structures and, (ii) strain in an aluminium plate with a central hole in tension.

Shape descriptors for mode-shape recognition and model updating

The most widely used method for comparing mode shapes from finite elements and experimental measurements is the Modal Assurance Criterion (MAC), which returns a single numerical value and carries no explicit information on shape features. New techniques, based on image processing (IP) and pattern recognition (PR) are described in this paper. The Zernike moment descriptor (ZMD), Fourier descriptor (FD), and wavelet descriptor (WD), presented in this article, are the most popular shape descriptors having properties that include efficiency of expression, robustness to noise, invariance to geometric transformation and rotation, separation of local and global shape features and computational efficiency. The comparison of mode shapes is readily achieved by assembling the shape features of each mode shape into multi-dimensional shape feature vectors (SFVs) and determining the distances separating them.

Model Updating Using Shape Descriptors from Full-Field Images

The comparison of structural responses (natural frequencies, mode shapes or strain maps) between predictions and measurements is an important step in finite element (FE) model updating. Full-field measurement techniques such as digital image correlation (DIC) provide detailed, global displacement data. It is necessary to compress huge amounts of full-field data before implementing the comparison procedures. Image decomposition using orthogonal kernel functions is one of the most common approaches. Appropriate selection or construction of the kernels generates shape feature terms capable of accurate image reproduction. Thus, the discrepancies between data and FE predictions may be assessed by using distance measures between the shape feature vectors. Two examples of model updating using shape features are described. In the first example vibration mode shapes of a composite panel from a structure to be deployed in outer space are measured by a DIC system. FE model updating is carried out using natural frequencies and Tchebichef moment descriptors. In the second example a square plate with a circular hole subject to a uniaxial tensile load is considered. Model updating of nonlinear elasto-plastic material properties is carried out using modified Zernike moment descriptors.

Principles of Image Processing and Feature Recognition Applied to Full-Field Measurements

Recent advances in measurement techniques such as digital image correlation (DIC) allow full-field maps (images) of vibration shapes or strain to be obtained easily. This generally results in the acquisition of large volumes of highly redundant data. Fortunately, image decomposition offers feasible techniques for data condensation while retaining essential information. The selection, or construction, of decomposition basis (kernel) functions is essential to data reduction and has been shown to produce descriptions of the full-field image capable of accurate reproduction of the original data, very efficiently. Image descriptors are robust to measurement noise. Classical orthogonal kernel functions include Fourier series, wavelets and Legendre, Zernike and Tchebichef polynomials, defined on either rectangular or circular domains. In practice full-field measurements of the engineering components are usually non-planar within irregular domains, so that the classical kernel functions are not immediately applicable. This problem may be addressed using a methodology based on adaptive geometric moment descriptors (AGMD) as will be demonstrated in a series of illustrative examples. Model updating from full-field measurements and modal testing in the shape-feature domain are enabled with attendant advantages of the full-field data over measurements taken with a limited number of sensors at discrete locations.