Solution Accuracy Improvement And Adaptive Mesh Refinement For Natural Frequency Analysis Using Exact Finite Elements

The finite element method is the main method for the analysis of the natural vibration of structures,but the process of establishing the dynamic equations of structural finite elements will generate discretization errors.Therefore,it is of great significance to explore how to efficiently improve the accuracy of the traditional finite element method for the analysis of structural free vibration.Based on the exact finite element stiffness matrix and mass matrix,this paper studies the high efficiency and accuracy improvement method and adaptive mesh refinement for the natural frequency analysis of frame structures.The main contents of the study are as follows:(1)The development of natural frequency error estimation and adaptive analysis is summarized from two aspects of linear static problem and structural dynamic problem.The error estimation analysis methods for structural dynamic problems,such as the advanced spectral finite element method,the smoothing element stress field and the modified mass matrix method,the block recovery technology and the new finite element superconvergent post-processing algorithm based on the element energy projection,are introduced.The basic concept of exact finite element is introduced.(2)The exact solution of axial vibration problem of elastic rod and vibration problem of frame structure are studied.Based on the results of the conventional finite element analysis,a new algorithm——fast iterative correction algorithm,which uses the exact finite element mass matrix and stiffness matrix to calculate Rayleigh quotient,is proposed,which cansignificantly improve the accuracy of frequency calculation.(3)The validity and accuracy of the method are studied and verified by using the MATLAB program.Including non-uniform grid cantilever elastic bar,end with concentrated mass cantilever elastic bar and spring support cantilever elastic bar axial vibration natural frequency analysis;Natural frequency analysis of vibration of simply supported beam and cross frame.(4)The method of adaptive mesh refinement is studied.Based on the exact finite element theory,the accurate error estimation of the element energy can be obtained,and the adaptive mesh refinement of the structure is carried out based on the principle that the element with the largest energy variation is the one that needs to be refined most.Better natural frequency values can be improved by less element refinement.(5)Numerical examples are presented for the verification of the effectiveness of the method,using MATLAB programs.