In this chapter, we consider theoretical and implementation difficulties in application of the hierarchical modelling and hp-adaptive finite element approach to elasticity, dielectricity and piezoelectricity. The main feature of the applied methodology is its generalizing character which is reflected by application of the same or analogous algorithms to three mentioned physical problems, including multi-physics problem of piezoelectricity, simple and complex physical description as well as simple and complex geometries. In contrast to the most common approaches dealing with a single physical phenomenon, described by a single physical model, within a single geometrical part, this chapter presents the ideas which brake and overcome such a simplicity. This presented chapter generalizes author’s hitherto accomplishments, in hierarchical models and hp-approximations of linear elasticity, onto dielectricity and piezoelectricity. The same refers to error estimation and adaptivity control. In this context, the main similarities and differences of three physical problems are of interest in this work.
Part of the book: Perusal of the Finite Element Method
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.
Part of the book: Finite Element Method