Crack detection in a pipe by adaptive subspace iteration algorithm and least square support vector regression

A new combination method of beam-type finite element multiwavelet-based algorithm and least square support vector regression (LSSVR) algorithm is proposed for detecting the location and size of a crack in a pipe. According to operators of engineering problems, Rayleigh-Euler and Rayleigh-Timoshenko beam-type multiwavelets are constructed using the stable completion in the multiresolution finite element space. A rotational spring model is used for cracked pipe modeling and the local flexibility due to the crack is calculated by discrete approximation method. An adaptive subspace iteration algorithm (ASIA) is applied to efficiently approximate the exact solution of pipe model by adding new beam-type multiwavelets in each scale. To avoid the difficulty of constructing well-defined mathematical models, the normalized crack location and depth is detected by using LSSVR algorithm. The numerical and experimental results verify that the presented method can accurately identify the location and depth of crack in a pipe

Spatially adaptive multiwavelet representations on unstructured grids with applications to multidimensional computational modeling

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Includes bibliographical references (p. 130-134).In this thesis, we develop wavelet surface wavelet representations for complex surfaces, with the goal of demonstrating their potential for 3D scientific and engineering computing applications. Surface wavelets were originally developed for representing geometric objects in a multiresolution format in computer graphics. However, we further extend the construction of surface wavelets and prove the existence of a large class of multiwavelets in Rn with vanishing moments around corners that are well suited for complex geometries. These wavelets share all of the major advantages of conventional wavelets, in that they provide an analysis tool for studying data, functions and operators at different scales. However, unlike conventional wavelets, which are restricted to uniform grids, surface wavelets have the power to perform signal processing operations on complex meshes, such as those encountered in finite element modeling. This motivates the study of surface wavelets as an efficient representation for the modeling and simulation of physical processes. We show how surface wavelets can be applied to partial differential equations, cast in the integral form. We analyze and implement the wavelet approach for a model 3D potential problem using a surface wavelet basis with linear interpolating properties.(cont.) We show both theoretically and experimentally that an O(h2/n) convergence rate, hn being the mesh size, can be obtained by retaining only O((logN)7/2 N) entries in the discrete operator matrix, where N is the number of unknowns. Moreover our theoretical proof of accuracy vs compression is applicable to a large class of Calderón-Zygmund integral operators. In principle, this convergence analysis may be extended to higher order wavelets with greater vanishing moment. This results in higher convergence and greater compression.by Julio E. Castrillón Candás.Ph.D

Operator-adapted finite element wavelets : theory and applications to a posteriori error estimation and adaptive computational modeling

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2005.Includes bibliographical references (leaves 166-171).We propose a simple and unified approach for a posteriori error estimation and adaptive mesh refinement in finite element analysis using multiresolution signal processing principles. Given a sequence of nested discretizations of a domain we begin by constructing approximation spaces at each level of discretization spanned by conforming finite element interpolation functions. The solution to the virtual work equation can then be expressed as a telescopic sum consisting of the solution on the coarsest mesh along with a sequence of error terms denoted as two-level errors. These error terms are the projections of the solution onto complementary spaces that are scale-orthogonal with respect to the inner product induced by the weak-form of the governing differential operator. The problem of generating a compact, yet accurate representation of the solution then reduces to that of generating a compact, yet accurate representation of each of these error components. This problem is solved in three steps: (a) we first efficiently construct a set of scale-orthogonal wavelets that form a Riesz stable basis (in the energy-norm) for the complementary spaces; (b) we then efficiently estimate the contribution of each wavelet to the two-level error and finally (c) we select a subset of the wavelets at each level to preserve and solve exactly for the corresponding coefficients. Our approach has several advantages over a posteriori error estimation and adaptive refinement techniques in vogue in finite element analysis. First, in contrast to the true error, the two-level errors can be estimated very accurately even on coarse meshes. Second, mesh refinement is carried out by the addition of wavelets rather than element subdivision.(cont.) This implies that the technique does not have to directly deal with the handling of irregular vertices. Third, the error estimation and adaptive refinement steps use the same basis. Therefore, the estimates accurately predict how much the error will reduce upon mesh refinement. Finally, the proposed approach naturally and easily accommodates error estimation and adaptive refinement based on both the energy norm as well any bounded linear functional of interest (i.e., goal-oriented error estimation and adaptivity). We demonstrate the application of our approach to the adaptive solution of second and fourth- order problems such as heat transfer, linear elasticity and deformation of thin plates.by Raghunathan Sudarshan.Ph.D

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

An operator-customized wavelet-finite element approach for the adaptive solution of second-order partial differential equations on unstructured meshes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Civil and Environmental Engineering, 2005.Includes bibliographical references (p. 139-142).Unlike first-generation wavelets, second-generation wavelets can be constructed on any multi-dimensional unstructured mesh. Instead of limiting ourselves to the choice of primitive wavelets, effectively HB detail functions, we can tailor the wavelets to gain additional qualities. In particular, we propose to customize our wavelets to the problem's operator. For any given linear elliptic second-order PDE, and within a Lagrangian FE space of any given order, we can construct a basis of compactly supported wavelets that are orthogonal to the coarser basis functions with respect to the weak form of the PDE. We expose the connection between the wavelet's vanishing moment properties and the requirements for operator-orthogonality in multiple dimensions. We give examples in which we successfully eliminate all scale-coupling in the problem's multi-resolution stiffness matrix. Consequently, details can be added locally to a coarser solution without having to re-compute the coarser solution.The Finite Element Method (FEM) is a widely popular method for the numerical solution of Partial Differential Equations (PDE), on multi-dimensional unstructured meshes. Lagrangian finite elements, which preserve C⁰ continuity with interpolating piecewise-polynomial shape functions, are a common choice for second-order PDEs. Conventional single-scale methods often have difficulty in efficiently capturing fine-scale behavior (e.g. singularities or transients), without resorting to a prohibitively large number of variables. This can be done more effectively with a multi-scale method, such as the Hierarchical Basis (HB) method. However, the HB FEM generally yields a multi-resolution stiffness matrix that is coupled across scales. We propose a powerful generalization of the Hierarchical Basis: a second-generation wavelet basis, spanning a Lagrangian finite element space of any given polynomial order.by Stefan F. D'Heedene.Ph.D

Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems

Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward neural networks, recurrent neural networks, or convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications