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This paper presents advances in recent work of the authors to derive a fractional step scheme based on the stabilized finite element method that allows overcoming the above mentioned problem, resulting in a efficient time accurate scheme.

The starting point is the modified governing differential equations for the incompressible turbulent viscous flow and the free surface condition incorporating the necessary stabilization terms via a finite calculus (FIC) procedure developed by the authors .  This technique is based on writting the different balance equations over a domain of finite size and retaining higher order terms. These terms incorporate the ingredients for the necessary stabilization of any transient and steady state numerical solution already at the differential equations level.

The resulting stabilized equations are integrated in space using the standard finite element method, and in time using an implicit and uncoupled second order fractional step method.

Abstract

The paper extends recent work of the authors to include transverse shear effects on rotation-free triangular element for plates (Oñate and Zárate in Int J Numer Methods Eng 83(2):196–227, 2010). Two new shell triangular elements are presented, the EBST+ and the EBST+1. Transverse shear deformation effects are important for thick shells, as well when the shell is laminated or formed by composite material. The ingredients for the element formulation are: a Hu-Washizu type mixed functional and linear interpolation for the displacement field. In both elements presented a finite volume approach is used for computing the bending moments and the curvatures over a patch of elements. The nodal translational degrees of freedom of the original enhanced basic shell triangle (EBST) are extended with the two shear deformation angles via two different approaches. The first one uses a linear interpolation of the rotation angles inside the element (EBST+) and the second one assumes a constant field for the rotation angles (EBST+1). For the thin shell case the shear angles vanish and the new elements reproduce the good behaviour of the original thin EBST element. As a consequence the elements can reproduce the solutions for thick to thin shells situations without exhibiting shear locking. The numerical solution for the thick shell case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin EBST element. Examples of the good performance of the new rotation-free shell triangles are given.

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In this paper some results of a wide experimental program are presented and compared with some finite element solution of sheet metal forming problems using a viscous shell formulation.

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A set of numerical model experiments has been conducted to simulate the circulation driven by oscillatory forcing over a theoretical continental slope configuration used previously in laboratory experiments. The test case considered was the numerical simulation of the flow over a model of a submarine canyon, and the numerical model used in the analysis was a coastal ocean model version based on an adaptation of the finite-calculus–finite-element method (FIC-FEM) approach implemented in the commercial package Tdyn. Two cases were analyzed involving changes in fluid density. Structured and unstructured finite-element spatial discretizations were generated for the same study domain to compare the resulting velocity field with outputs from the laboratory experiments and to assess which mesh provided a better representation of the complex geometry of the channel model and the water circulation process. The comparison between the laboratory results from the reference article and the output of the numerical model showed good agreement in the structure and magnitude of the phaseaveraged and residual velocity fields.

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This paper summarizes the development for a large displacement formulation of a membrance composed of three-node triangular elements. A formulation in terms of the deformation gradient is first constructed in terms of nodal variables. In particular, the use of the right Cauchy-Green deformation tensor is shown to lead to a particulary simple representation in terms of nodal quantities. This may then be used to construct general models for use in static and transient analyses.

Abstract

This article presents the first application of the Finite Calculus (FIC) in a Ritz-FEM variational framework. FIC provides a steplength parametrization of mesh dimensions, which is used to modify the shape functions. This approach is applied to the FEM discretization of the steady-state, one-dimensional, diffusion–absorption and Helmholtz equations. Parametrized linear shape functions are directly inserted into a FIC functional. The resulting Ritz-FIC equations are symmetric and carry a element-level free parameter coming from the function modification process. Both constant- and variable-coefficient cases are studied. It is shown that the parameter can be used to produce nodally exact solutions for the constant coefficient case. The optimal value is found by matching the finite-order modified differential equation (FOMoDE) of the Ritz-FIC equations with the original field equation. The inclusion of the Ritz-FIC models in the context of templates is examined. This inclusion shows that there is an infinite number of nodally exact models for the constant coefficient case. The ingredients of these methods (FIC, Ritz, MoDE and templates) can be extended to multiple dimensions

Abstract

In this paper an automatic remeshing algorithm of triangular finite elements is presented. It is well known that the element sizes of the mesh play an important role in modeling the continuum, particularly when notable material properties differences exist in contiguous areas of the medium. In such cases, the mesh must be fine enough in such areas in order to obtain reliable results. Therefore, in this paper is advanced an algorithm to carry out automatic remeshing of local areas where the “remeshing” criteria is activated to refine the mesh accordingly. Herein the proposed algorithm integrated into a two dimensional finite element computer program is used to analyze a classical geotechnical problem to show the importance of locally refining the mesh and to demonstrate that regardless of the geometric characteristics of the initial mesh, the algorithm yields practically equal results. Computations with the proposed method are compared with the corresponding close form solutions, whenever available, to show the usefulness and reliability of the remeshing algorithm. Furthermore, to show the algorithm’s versality, the initial loading boundary conditions considered for the cases included in this paper are modified in order to show how the automatic local remeshing is capable of adapting the initial mesh configuration into a new one as a function of the new boundary conditions. As shown in the paper, the final resulting meshes for both load boundary conditions considered are appreciably dissimilar from each other, which leads to somehow different results.

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