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Purpose: The real challenge in the solution of contact problems is the lack of an optimal adaptive scheme. As the contact zone is a priori unknown, successive refinement and iterative method are necessary to obtain a high-accuracy solution. The purpose of this paper is to provide an optimal adaptive scheme based on second-generation finite element wavelets for the solution of non-linear variational inequality of the contact problem. Design/methodology/approach: To generate an elementary multi-resolution mesh, the authors used hierarchical bases (HB) composed of Lagrange finite element interpolation functions. These HB functions are customized using second-generation wavelet techniques for a fast convergence rate. At each step of the algorithm, the active set method along with mesh adaptation is used for solving the constrained minimization problem of contact case. Wavelet coefficients-based error indicators are used, and computation is focused on mesh zones with a high error indication. The authors take advantage of the wavelet transform to develop a parameter-free adaptive scheme to generate an appropriate and optimal mesh. Findings: Adaptive wavelet Galerkin scheme (AWGS), a newly developed method for multi-scale mesh adaptivity in this work, is a combination of the second-generation wavelet transform and finite element method and significantly improves the accuracy of the results without approximating an additional problem of error estimation equations. A comparative study is performed taking a solution on a highly refined mesh and results are generated using AWGS. Practical implications: The proposed adaptive technique can be utilized in the simulation of mechanical and biomechanical structures where multiple bodies come into contact with each other. The algorithm of the method is easy to implement and found to be successful in producing a sufficiently accurate solution with relatively less number of mesh nodes. Originality/value: Although many error estimation techniques have been developed over the past several years to solve contact problems adaptively, because of boundary non-linearity development, a reliable error estimator needs further investigation. The present study attempts to resolve this problem without having to recompute the entire solution on a new mesh. [ABSTRACT FROM AUTHOR]

Micromechanical analysis of polycrystalline microstructures of metals and alloys, using crystal plasticity finite element (CPFE) models is extensively used for predicting deformation and failure under various conditions of strain-rates, creep and fatigue loading. Many CPFE models involve a large number of degrees of freedom for accurate representation of realistic polycrystalline microstructures. This can lead to prohibitively high computational costs to conduct meaningful analyses of phenomena of interest. To overcome this limitation, the authors have recently developed a wavelet enrichment adapted finite element model in Azdoud and Ghosh (2017) for elastic materials. The method adaptively creates an optimal discretization space conforming to the solution profile by projecting the solution field onto a set of scaling and multi-resolution wavelet basis functions. This paper extends this wavelet adapted FE model to finite deformation, crystal plasticity analysis of polycrystalline microstructures. After presenting the formulations, various validation tests are conducted to examine the convergence rates and computational efficiency of this method. [ABSTRACT FROM AUTHOR]

In this paper we have developed a second generation wavelet based finite element method for solving elliptic PDEs on two dimensional triangulations using customized operator dependent wavelets. The wavelets derived from a Courant element are tailored in the second generation framework to decouple some elliptic PDE operators. Starting from a primitive hierarchical basis the wavelets are lifted (enhanced) to achieve local scale-orthogonality with respect to the operator of the PDE. The lifted wavelets are used in a Galerkin type discretization of the PDE which result in a block diagonal, sparse multiscale stiffness matrix. The blocks corresponding to different resolutions are completely decoupled, which makes the implementation of new wavelet finite element very simple and efficient. The solution is enriched adaptively and incrementally using finer scale wavelets. The new procedure completely eliminates wastage of resources associated with classical finite element refinement. Finally some numerical experiments are conducted to analyze the performance of this method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new method is presented to identify crack location and size, which is based on stress intensity factor suitable for pipe structure and finite element method of second generation wavelets (SGW-FEM). Pipe structure is dispersed into a series of nested thin-walled pipes. By making use of stress intensity factor of the thin-walled pipe, a new calculation method of crack equivalent stiffness is proposed to solve the stress intensity factor of the pipe structure. On this basis, finite element method of second generation wavelets is used to establish the dynamic model of cracked pipe. Then we combine forward problem with inverse problem in order to establish quantitative identification method of the crack based on frequency change, which provides a non-destructive testing technology with vibration for the pipe structure. The efficiency of the proposed method is verified by experiments. [Copyright &y& Elsevier]

Abstract: A distinguishing feature of second generation wavelets is that it can be custom designed depending on applications. Based on second generation wavelets, a multiresolution finite element method is discussed, and its adaptive algorithm is constructed. The hierarchical approximation spaces for finite element analysis are produced. The finite element equation is scale-decoupled via eliminating all coupling in the stiffness matrix of element across scales, then resolved in different spaces independently. The coarse solution can be obtained in the coarse approximation space, and refined by adding details in the detail spaces over several levels till the equation is resolved to the desired accuracy. The scale-decoupling condition of the stiffness matrix of element is proposed by introducing wavelet vanishing moments, and the principle of constructing the scale-decoupling wavelet bases is established. The method establishes an important connection between finite element analysis and multiresolution analysis. The numerical examples have illustrated that the proposed method is powerful to analyze the field problems with changes in gradients and singularities. [Copyright &y& Elsevier]