Special finite elements for advanced modelling of engineering problems

Since 1968, the development of new finite elements has not stopped. This has led to a plethora of finite elements that, although they are now available in most commercial codes, none of them seems to be the best. During the last decade, new added values have been introduced, at variational level and/or inside the element’s domain and its boundaries, making these finite elements very simple to implement and efficient in terms of accuracy. Recently, a new generation of special finite elements appeared and continues to be developed. They may be associated to the introduction or use of:

•	Local and particular kinematics allowing for instance to identify one or more new useful degrees of freedom (DOF), which are related to the considered application (material’s forming processes, fluid–structure interactions, etc.).
•	Non-linear multi-physical constitutive models needing other DOFs than classical displacements and rotations (temperature, electric potential, magnetic potential, etc.).
•	Non-local approaches needing additional DOFs corresponding to displacement and stress gradients or global values of internal variables associated to non-local constitutive models.
•	Bubble functions to improve stability and accuracy, without introducing additional DOFs.
•	Wave functions for the enrichment of the classical displacement shape functions to tackle medium- and high-frequency ranges.
•	New variational approaches (hybrid, full mixed, partial-mixed, etc.) for single or multi-physical finite element modeling.

The present focused issue proposes to establish a state of the art in terms of advanced and innovative finite element approaches which put in value the contribution of the “special” aspect of the element formulation on the improvement of the simulation results. Hereafter, the contents of the double (sometimes triple) reviewed, revised and accepted contributions are briefly described:

Cherouat, Moreau, Ayad, and Ben Zineb (2012) adapted an appropriate bi-component finite element (Fibre rotation quadrilateral element connected to truss elements) to model pre-impregnated woven fabrics. Their approach allows capturing the deformation of the meso-structure of the fabric without explicitly modeling every fibre. It has been validated across some achieved experiments.

Wane, Urquiza, Fortin, and Pelletier (2012) showed in their work how quadratic hierarchical elements could be used to develop efficient iterative methods for an accurate simulation of turbulent flows on strongly anisotropic meshes.

Bognet, Leygue, and Chinesta (2012) proposed a particular approach where the specific element aspects are interpreted within the manner to explore new discretisation strategies able to alleviate the drawbacks related to mesh-based discretisations of fully 3D models defined in plate or shell domains.

Abed-Meraim, Trinh, and Combescure (2012) proposed a 6-node prismatic solid–shell element for the analysis of geometric and material non-linearities in solids and structural problems. Their approach is motivated by the possibility of using a natural mesh connection in problems where both structural and continuum elements need to be used. Another major interest is to complement meshes that use hexahedral finite elements, especially when free mesh generation tools are employed.

Bois, Fortin, Fortin, and Couët (2012) proposed via an intelligent use of hierarchical elements, a whole general approach leading to optimal meshes. They showed how to estimate the error on a finite element solution of k degree using hierarchical basis for Lagrange finite element polynomials. This information is then used to produce optimal anisotropic meshes.

Jayabal and Menzel (2012) elaborated an application of a polygonal finite element formulation to 3D elastic mechanical problems with a special emphasis on the numerical implementation of the method and the construction of the element stiffness matrix. A specific property of Voronoi-based discretisations in combination with a hybrid finite element approach was also investigated by the authors.

Andrianarison and Benjeddou (2012) proposed a partial-mixed special finite element for the static analysis of multilayer composite and functionally graded material plates. The corresponding partial-mixed variational formulation, retaining as primary variables the translational displacements augmented with the transverse stresses, is performed by reformulating the 3D elasticity equations. Their multilayer solution, applied to some representative problem tests, seemed to be satisfactory regarding convergence and accuracy.

Sedira, Ayad, Sabhi, Hecini, and Sakami (2012) proposed an enhanced discrete four-node finite element model for Reissner/Mindlin composite plates, introducing a zigzag form in order to improve plane and shear stress accuracy. Their model is based on a piecewise linear variation of displacement which allows to fulfil the stress continuity requirements. Performances of the proposed special element are shown for some representative problems, highlighting an improvement of thickness stress distributions, by comparison with the initial model without zigzag function.

Ghomari, Meftah, Ayad, and Talbi (2012) developed a special axisymmetric finite element formulation for the non-linear analysis of hyperelastic solids and structures. It is based on the kinematic of a space fiber, for which the authors attribute a concept named SFR (space fiber rotation). The derived four-node element improves in a significant way the accuracy of the classical bi-linear four-node element. It can also be compared, for both accuracy and CPU time, with the higher order quadratic eight-node element.

It is worthy to mention that the idea of editing this focused issue emanated from discussions between guest editors prior to the mini-symposium dedicated to special finite elements within the 4th International Congress: Design and Modeling of Mechanical Systems (CMSM’2011) held in Sousse, Tunisia, from 30 May to 1 June 2011. The latter was conceived as an opportunity to hear some potential contributions before their submissions. Therefore, the guest editors are thankful to Professor Abdelmajid BEN AMARA, the Chairman of CMSM2011, for giving them the possibility to organise, in an autonomous way, such a mini-symposium.

Finally, the guest editors would like to thank the authors and the reviewers for their valuable contributions to this focused issue of the European Journal of Computational Mechanics (EJCM). They address also special thanks to Professor Marc Bonnet, the Editor-in-Chief of EJCM, for giving them the total freedom to manage all publication process, from submission of selected contributions to the finalisation of accepted papers, passing by the nomination of the reviewers.