Computational multiresolution wavelet-based methods for transient structural dynamic simulations with intrinsic localization and error estimation capabilities
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Date
2024-12-12
Authors
Δημητρίου, Δημήτρης
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Abstract
The present thesis focuses on the development of a multiresolution wavelet-based computational method for the efficient and multifunctional simulation of transient dynamic structural responses. The proposed method employes the Daubechies scaling and wavelet functions as basis functions for the approximation of field variables. The remarkable mathematical properties of the Daubechies wavelet family result in hierarchical multiresolution equations of motion with two types of solutions: the first solution type is named as “coarse” solution and utilizes the Daubechies scaling functions as basis, while the second type is called “fine” solution and uses the Daubechies wavelet functions as basis. The multiresolution discretized algebraic system starts with the calculation of the coarse solution, that provides an initial approximation, and then this solution can be further enriched by the summation of fine solutions until the desired level of accuracy is achieved. The orthogonality between the Daubechies scaling and wavelet functions leads to resolution-decoupled total mass matrices, namely, the coarse and fine solutions are totally uncoupled with respect to the mass matrix. So, the absence of mass coupling terms results in the independent calculation of the two resolution components (coarse, fine) in each time step, thus reducing the algebraic system size in half. Also, the orthogonality of the Daubechies scaling and wavelet functions yields diagonal mass matrices for both coarse and fine solutions. This is extremely important when it comes to transient dynamics because the utilization of explicit time integration schemes requires the inversion of the mass matrix, which is a rapid operation if the matrix is diagonal.
Concerning the numerical formulation of the multiresolution method, termed as MR-FWD method, two new types of wavelet-based beam elements are developed; one following the first-order shear deformation theory, and the second encompassing a high-order laminate layerwise theory. These novel wavelet-based beam elements are tested against conventional finite elements, time domain spectral finite elements and other wavelet-based elements in numerical case studies of wave propagation in isotropic and composite beams and sandwich strips. In all cases, the proposed method manifests outstanding results considering the requirement of nodes for convergence, as well as the computing times. Moreover, the advanced capacity of the layerwise wavelet-based strip elements is demonstrated through the accurate prediction of fundamental (A0,S0) and higher (A1,S1) guided wave modes in thick sandwich composites. Furthermore, a new wavelet-based plate element is constructed based on the Reissner-Mindlin plate theory. The 2D wavelet-based plate elements reveal extraordinary convergence rates compared to time domain spectral finite elements and other wavelet-based elements in guided wave propagation analyses of isotropic and laminated plate structures. Besides the computational advantages of the method, it is shown that the fine solutions in all element types have additional inherent localization capabilities that emanate from the Daubechies wavelets. These localization properties make possible for the fine solutions to capture different structural responses based on the involved materials, geometric characteristics and wave direction (in 2D cases), among other characteristics.
This notable behavior can lead to intrinsic error estimation through the direct comparison of multiple resolution components of the proposed method. After extensive study of the behavioral patterns of the various resolution components, it is deduced that the ratios of specific solutions can constitute explicit convergence indicators. In that way, the multiresolution procedure can stop at the intended level of accuracy without the need of convergence checks or any sort of comparisons with other results. Apart from that, the developed convergence indices can estimate the error locally, providing the basis for local solutions enrichment strategies. By this means, these indicators are leveraged for the creation of targeted and adaptive multiresolution refinement techniques. The targeted refinement works like a numerical microscope and increases the initial solution’s fidelity on an ad hoc basis. The automatic adaptive refinement utilizes the convergence indicators in order to enhance the solution’s accuracy only in the timesteps and nodes that it is needed. Both the targeted and adaptive refinement techniques exhibit very good accuracy and drastically decrease the solver durations, making the proposed method even faster.
Finally, the intrinsic localization capacity of the method is further leveraged for the development of inverse methodologies for wave-based parameter estimation. The application of the proposed methodologies requires full-field data of the wave propagation phenomenon. The data are decomposed using wavelet decomposition, resulting in approximation and detail datasets, that are directly comparable to the coarse and the fine solutions of the MR-FWD method, respectively. Thus, the proposed inverse methodologies take advantage of the high sensitivity and localization properties of the fine solutions in order to create a model update procedure that compares the fine solutions of the MR-FWD model to the detail data of the experimental measurements for the estimation of the design variables. Numerical case studies of wave-based damage detection in composite strips and 2D solids are performed, manifesting the proposed methods’ superior accuracy compared to traditional methodologies.
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Keywords
Wavelet-based methods, Computational mechanics, Multiresolution analysis, Wave propagation, Convergence indicators, Adaptive refinement, Guided waves, SHM