Abstract
This chapter estimates spatial and the temporal discretization errors, and proposes an effective algorithm that controls the errors automatically and simultaneously by adaptive modification of the mesh distribution and time step size. There are two basic issues for the adaptive finite element analysis: the error estimation and the adaptive control of the error. For elliptic problems, the discretization error occurs from spatial discretization and the control of the error is effectively achieved by an adaptive mesh generation. Unlike the adaptive methods in elliptic problems, more error sources, such as the truncation in time integration, are required for hyperbolic problems such as dynamic problems. Therefore, for hyperbolic problems a combined posteriori error estimate including both space and time discretization is needed. An adaptive analysis should find a discretization in the least cost, such that the local error is uniformly distributed and within a given tolerance over the entire spatial/time domain. If error estimate in discretization does not satisfy the given error tolerance, the time step size is updated according to the local refinement index until the required accuracy is achieved. A single-degree-of-freedom problem was solved to evaluate the accuracy of the proposed time discretization error estimate. The adaptive procedure can be extended to two-dimensional and three-dimensional problems with minimum modification. For this, however, the use of an effective automatic mesh generator is recommended. The chapter also illustrates graphs and tables to further explain the concept of adaptive analysis and the variation of time step sizes and meshes in the bar.
Published Version
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