**Adaptive Modeling and Simulation of Elastic, Dielectric and Piezoelectric Problems** Adaptive Modeling and Simulation of Elastic, Dielectric and Piezoelectric Problems

Grzegorz Zboiński

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156 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

Additional information is available at the end of the chapter Grzegorz Zboiński Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72265

### Abstract

In this chapter, theoretical and implementation details of the algorithms of hierarchical modeling and hierarchical hp-approximations, residual error estimation methods and four-step adaptive procedures are considered in the context of their application to modeling and simulation of the problems of elasticity, dielectricity and piezoelectricity. In the hierarchical modeling, 3D-based hierarchical elastic and dielectric models are applied. The adaptive discretization process is based on the hierarchical shape functions and the constrained approximations. In the error estimation, the equilibrated residual method is applied, which serves the total and approximation error assessment. These errors control the model and hp-adaptivity. In the case of adaptive algorithms, four-step procedure is utilized. It includes global solutions on the initial mesh, mesh modified in order to remove some undesired numerical phenomena, the intermediate h-refined mesh and the final (or target) p-enriched mesh. Examples demonstrating the effectivity of the mentioned modeling and approximation, error estimation and adaptivity control parts of the overall simulation algorithm in the three classes of problems are presented.

Keywords: adaptivity, modeling, simulation, finite elements, hierarchical models, hierarchical approximations, error estimation, adaptivity control, algorithms, effectivity, elasticity, dielectricity, piezoelectricity
