Análise de técnicas de recuperação aplicadas em soluções aproximadas em problemas termomecânicos unidimensionais

• Main Author: Biagi, Fernanda Domingues
• Publication Date: 2025
• Format: Bachelor thesis
• Language: por
• Source: Repositório Institucional da UTFPR (da Universidade Tecnológica Federal do Paraná (RIUT))
• Download full: http://repositorio.utfpr.edu.br/jspui/handle/1/37916
• topic: Método dos elementos finitos; Erros - Estimativas; Tensões residuais; Finite element method; Errors - Estimates; Residual stresses; CNPQ::ENGENHARIAS::ENGENHARIA MECANICA

The analysis of thermally induced stresses in engineering applications is essential for evaluating the structural integrity and reliability of mechanical components, particularly under conditions involving significant temperature variations and displacement constraints. These stresses manifest in a wide range of applications, including bridges, railway tracks, shafts, and structural components, characterizing thermomechanical problems. The stress analysis can be conducted using analytical methods, which are appropriate for simpler problems, or via numerical approximation techniques, which are susceptible to errors. In this context, analyzing error estimators is crucial to improve the accuracy of the approximate solutions. This study examines the application of a displacement recovery technique to one-dimensional thermomechanical problems, aiming to assess the quality of the error estimator by analyzing the convergence rate of stress solutions and evaluating parameters such as the energy norm of the error, effectiveness index, mean value, and deviation from unity. For verification, a nodal comparison is conducted between the analytical solution and the solution obtained using the Finite Element Method (FEM), considering different cases with variations in the heat source and prescribed boundary conditions, with the goal of evaluating the behavior of the displacement and stress solutions. The engineering problem studied is composed of two distinct cases: in the first, the heat source is distributed in quadratic form while the external load follows a sinusoidal distribution; in the second, the heat source is sinusoidally distributed and the external load is distributed quadratically. The solutions are performed in the Matlab® program using the FEM, where five types of stress field solutions are evaluated: analytical stress; analytical* stress; stress computed via linear FEM; stress computed via quadratic FEM; and stress obtained through displacement recovery using the Superconvergent Patch Recovery Displacement (SPRD) technique. The results demonstrate that the SPRD technique provides effective performance in stress recovery, exhibiting a higher convergence rate, reduced energy error norm compared to the solution obtained by the linear FEM, in addition to a good behavior of the effectiveness index. It was observed that the application of SPRD on solutions with linear functions can be more beneficial than directly using quadratic approximations, particularly in problems that require highly refined meshes, since the SPRD solution presents convergence rates of the order of 2.2 ± 0.1.