An adaptive wavelet Galerkin scheme for solving contact problems based on elliptic variational inequalities of the first kind
ISSN: 0264-4401
Article publication date: 10 May 2019
Issue publication date: 5 June 2019
Abstract
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.
Keywords
Citation
Ranjan, K.K., Kumar, S., Tyagi, A. and Sharma, A. (2019), "An adaptive wavelet Galerkin scheme for solving contact problems based on elliptic variational inequalities of the first kind", Engineering Computations, Vol. 36 No. 4, pp. 1258-1281. https://doi.org/10.1108/EC-07-2018-0294
Publisher
:Emerald Publishing Limited
Copyright © 2019, Emerald Publishing Limited