APPLICATION OF ADAPTIVE METHODS IN NUMERICAL MODELING OF MECHANICS PROBLEMS
Abstract
The article investigates the application of adaptive hp-methods within the finite element method for modelling the spa- tial contact interaction between a perfectly rigid punch and a homogeneous isotropic elastic half-space. The study considers the problem of indentation of a rigid punch into an elastic half-space with a doubly connected contact region in the form of a symmetrically deformed elliptical ring. For the numerical implementation, several variants of adaptive meshes with differ- ent approximation orders were constructed. Adaptation is performed through local h-refinement in zones with intensive stress gradients and p-enrichment to improve approximation accuracy. An integrated software package has been developed. Its architecture is implemented as a multimodule system combining the capabilities of Ansys for solving the mechanical problem, Gmsh and APDL for adaptive mesh generation, MATLAB for post-processing, and Python/batch scripts for automating the computational cycle. The control module ensures the sequential execution of iterations, monitoring of intermediate data, and logging. The weight-matrix module constructs an importance field that identifies local regions requiring mesh refinement. The adaptive meshing module generates meshes with variable local ele- ment size, while the conversion module ensures correct transfer of meshes into a format compatible with Ansys. The package supports result processing, generation of plots and tables, and formation of data structures for subsequent adaptation iterations. Despite limitations on the available approximation orders (p = 1 and p = 2), the implemented concept enables effective control of both h- and p-adaptation. The analysis shows that increasing the approximation order significantly reduces the error and improves the efficiency index of the a posteriori error estimate, while maintaining rational use of compu- tational resources. The results obtained confirm the effectiveness of the hp-adaptive approach for contact mechanics problems with complex geometry and demonstrate its potential for further use in engineering computations.
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