Space-Time Reduced Element for Adaptive Finite Element Analysis of Forced Vibration of Elastic String in Maximum Norm

Abstract

This paper presents, by taking the elastic string vibration problem as the model problem, the space–time reduced element and the corresponding adaptive analysis algorithm. The solution of the space–time reduced element is extracted from the standard two-dimensional (spatial and temporal dimensions) polynomial element of the Galerkin-type by omitting the highest-degree terms, which serve as a built-in pointwise error estimator for the reduced solution. Taking the reduced solution as the final solution, the proposed adaptivity algorithm can produce solutions from the reduced element that satisfy the user-preset error tolerances in the maximum norm. Theoretical analysis and formulation are presented. Representative numerical examples, including forced vibrations with damping on an elastic foundation and moving load problems, validate the feasibility, effectiveness, and reliability of the proposed method.