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Abstract: This is the second paper out of a series of papers devoted to model- and hpq-adaptive finite element methods assigned for the modeling and analysis of elastic structures of complex mechanical description. In our previous publication we investigated the issue of hierarchical models and approximations of such structures. We applied 3D or 3D-based mechanical models, hierarchical modeling, and hierarchical approximations within the proposed finite element formulation. Furthermore, we assumed that the mechanical model and discretization parameters (such as: the size h of the element, and the longitudinal and transverse approximation orders, p and q) could vary locally, i.e. they could be different in each finite element. The a posteriori error estimation discussed in the present paper is based on the generalization of the residual equilibration method on the models with internal constraints. The generalized method is applied to the assessment of the total and approximation errors, while the modeling error is calculated as the difference between the former two errors. The corresponding error-controlled adaptive procedures are based on a three-step strategy, with possible iterations of h- and p-steps. [Copyright &y& Elsevier]

Abstract: This is a paper out of a series of three papers devoted to model- and hpq-adaptive finite element methods assigned for modeling and analysis of complex structures. In this paper we focus our attention on hierarchical models and hierarchical approximations, while the issues of error estimation and adaptivity control will be presented in the next papers of the series. In all the papers we consider the elastic structures of complex mechanical description. We apply 3D or 3D-based mechanical models, hierarchical modeling, and hierarchical approximations within our finite element formulation. In the proposed approach, the mechanical model and discretization parameters, such as the size h of the element, and the longitudinal and transverse approximation orders, p and q, can vary locally, i.e. they can be different in each finite element. The a posteriori error estimation is based on the residual equilibration methods applied to the assessment of the modeling and approximation errors. The error-controlled adaptive procedures are derived from a four-step strategy, with possible iterations on h- and p-steps. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study from the numerical point of view a strain-adaptive bone remodelling that couples the displacements and the apparent density (the porosity) of the bone. The rate of this density at a particular location is described as an objective function, which depends on a particular stimulus at that location. The variational problem is written as a coupled system of a nonlinear variational equation for the displacement field and a nonlinear parabolic variational inequality for the apparent density. Then, fully discrete approximations are provided by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are proved, from which, under adequate regularity conditions, the linear convergence of the algorithm is deduced. Finally, some numerical simulations involving one- and two-dimensional test examples are presented to demonstrate the accuracy of the approximation and the behaviour of the solution. [Copyright &y& Elsevier]

Abstract: In this paper the grad–div stabilization for the incompressible Navier–Stokes finite element approximations is considered from two different viewpoints: (i) as a variational multiscale approach for the pressure subgrid modeling and (ii) as a stabilization procedure of least-square type. Some new error estimates for the linearized problem with the grad–div stabilization are proved with the help of norms induced by the pressure Schur complement operator. We discuss the stabilization parameter choice arising in the frameworks of least-square and multiscale methods and consider assumptions which allow to relate both approaches. [Copyright &y& Elsevier]

Abstract: In this paper, we study numerical locking effects in the finite element approximation of shell layers. The focus is on four-node degenerated elements that are perhaps the most widely used in practical computations. Our approach is based on simplified reformulations of the original 3D elements in the context of a classical shallow shell model, where the deformation of the shell is described in terms of the three displacements of the middle surface and two dimensionless rotations related to transverse shear deformations. Within that model we compute finite element approximations of layers generated by smooth line segments such as the boundary line. Our numerical results show that approximation failure occurs when bilinear degenerated elements are used on anisotropically refined meshes and that the accuracy depends substantially on the specific geometry of the shell. By analyzing the layers as functions of the thickness of the shell, we obtain theoretical error estimates in the energy norm framework that predict error amplification for small values of the thickness. The error estimates are then verified by conducting further numerical experiments using overrefined meshes. [Copyright &y& Elsevier]

Abstract: A series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper. These methods are often referred to in the literature as projection methods, and can be classified into three classes, namely the pressure-correction methods, the velocity-correction methods, and the consistent splitting methods. For each class of schemes, theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed. The essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners. [Copyright &y& Elsevier]

Abstract: In this paper, we develop a nonconforming finite element methodology using a three-field formulation to analyze a fluid–structure interaction problem. The methodology is used to couple a Lagrangian model describing the structure with the arbitrary Lagrangian–Eulerian strategy used to describe the fluid in order to simulate a full unsteady physical phenomenon. Consistency error estimates are obtained which show that the numerical scheme employed yields a first order approximation for the solution to the fluid–structure interaction problem. Finally, we present a discrete energy estimate to demonstrate the stablity of the proposed method. [Copyright &y& Elsevier]