hybrid-Trefftz Formulation Research Articles

Alternative Low-Order Triangular Elements for Thick and Thin Shell Analysis

In this paper, the authors discuss the development of six low-order triangular elements for the linear static analysis of thick and thin shell structures. The elements combine two triangular plane stress elements with three triangular thick plate bending elements to produce three-node flat shell elements with 18 degrees of freedom. The two plane stress components are based on the free formulation and the template formulation, while the three plate bending components are based on the free formulation, the deviatoric strain formulation and the hybrid-Trefftz formulation. The theoretical backgrounds and formulations of all elements are briefly discussed to provide a concise summary, while further details are given in the quoted references. The elements are used to analyze five standard shell benchmark examples, and the results obtained are compared to previous work in this area. The elements provide accurate and robust results for the benchmarks, with only small differences for the results as the meshes are refined. While all the proposed elements can be recommended for general shell analysis, the combination of the template plane stress element and the hybrid-Trefftz plate bending element generally provides the best results.

Splitting method and hybrid-Trefftz formulation for multisite damage analysis in two-dimensional domains

The aim of this paper is to present a new strategy for multiple-site damage analysis by combining the splitting method and hybrid-Trefftz stress formulation in a two-dimensional domain. The first is a decomposition method originally conceived to obtain the solution of a cracked solid. Accordingly, the given problem is split into three subproblems: an uncracked global problem (PG(0)), a local problem (PL(k)) considering a single crack located in a reduced domain and a global problem (PG(k)) aiming to account for the relative interaction effects among all the cracks. Most subproblems are independent and the solution results from a non-iterative strategy, which is one of the main benefits of this method. The hybrid-Trefftz stress formulation is a non-conventional finite element method where stresses and displacements are approximated, independently, in the domain and boundary of the element, respectively. The stress fields are hereby approximated by a basis function that solves the Navier equation and includes analytical fracture mechanics functions, so that the values of stress intensity factors are computed from the solution of the linear system. The hybrid-Trefftz formulation is used in the local problem PL(k) of the splitting method, therefore improving the performance of the method and providing accurate solutions with very coarse meshes. In conclusion, it is shown through the numerical examples that the splitting method and hybrid-Trefftz formulation framework hereby proposed allows conducing accurate analysis. Once exploring a parallel processing, taking advantage of the independence of each subproblem, the framework can be extended to consider different scenarios of multisite cracked solids with a low computation cost.

Hybrid-Trefftz formulation for analysis of anisotropic and symmetric laminated plates

In this paper, two novel and efficient elements are proposed for analysis of anisotropic and symmetric laminated thick plates based on hybrid-Trefftz formulation. In this approach, two independent displacement fields are defined for internal and boundary of the elements. The internal field is selected in such a manner that the governing equation of thick plates could be satisfied. At the first step, the governing differential equations of anisotropic and symmetric laminated thick plates are derived. Then, the analytical solution of these equations is considered. By using these analytical functions, the internal displacement field of proposed elements is achieved. Also, the boundary field is related to the nodal degrees of freedom by the boundary interpolation functions. For this aim, the interpolation functions of tow-node laminated Timoshenko beam element are calculated for boundary field. By utilizing these displacement fields and hybrid-Trefftz functional, a triangular element THT-7 and quadrilateral element QHT-11 are proposed. These elements have 9 and 12 degrees of freedom, respectively. The results of the numerical testes are proven the absence of the shear locking and high accuracy and efficiency of the proposed elements for analysis of anisotropic and symmetric laminated plates.

Vibration and stability analysis of thick orthotropic plates using hybrid-Trefftz elements

This paper discusses the vibration and stability analysis of thick orthotropic plate structures using finite elements based on the hybrid-Trefftz formulation. While the formulation can be used for elements of arbitrary geometry, the paper concentrates on the use of a simple and robust triangular element. The key feature of the formulation is to use element interpolations that are consistent for all values of the plate thickness, including the limit when it goes to zero. This eliminates the locking problem automatically and ensures a robust approximation for thick and thin plates. Results for various problems are included to demonstrate the accuracy and efficiency of the element.

A hybrid-Trefftz formulation for plane elasticity with selective enrichment of the approximations

This work is related to the so-called non-conventional finite element formulations. Essentially, a methodology for the enrichment of the initial approximation which is typical of the meshless methods and based on the clouds concept is introduced in the hybrid-Trefftz formulation for plane elasticity. The formulation presented allows for the approximation and direct enrichment of two independent fields: stresses in the domains and displacements on the boundaries of the elements. Defined by a set of elements and interior boundaries sharing a common node, the cloud notion is employed to select the enrichment support for the approximation fields. The numerical analysis performed reveals an excellent performance of the resulting formulation, characterized by the good approximation ability and a reduced computational effort. Copyright © 2009 John Wiley & Sons, Ltd.

Nonlocal damage theory in hybrid-displacement formulations

This paper discusses three hybrid-displacement finite element formulations for the simulation of strain localization based on nonlocal damage theory. An isotropic integral nonlocal damage model is chosen. The hybrid finite element formulations adopted in this work are developed from first principles of Mechanics. The first one defines the domain approximations using the Trefftz functions derived for the linear elastic regime. When damage appears the hybrid-Trefftz displacement formulation degenerates into an hybrid-displacement formulation. The second formulation uses an enriched Trefftz basis with the consideration of local Heaviside functions. The third formulation uses orthonormal Legendre polynomials for the domain approximations. A set of benchmark tests is presented and discussed in order to compare the performance and accuracy of the different models. It is shown that the proposed hybrid-Trefftz formulation allows the reproduction of the general behavior of the structure but does not lead to a correct simulation of the strain tensor evolution. The hybrid-displacement formulation that uses orthonormal Legendre polynomials gives coherent results, so it appears to be a promising field of investigation.

Hybrid-Trefftz stress and displacement elements for axisymmetric incompressible biphasic media

The stress and displacement models of the hybrid-Trefftz finite element formulation are applied to the solution of incompressible axisymmetric saturated porous media. The use of a Trefftz approximation basis ensures that all domain conditions of the problem are satisfied in a strong form, namely the equilibrium, the constitutive and the strain–displacement relations and the mixture incompressibility condition. The alternative stress and displacement models are fully complementary in terms of approximation criteria. The stress (displacement) model is derived from the direct approximation of the stress and pressure fields (the displacements in the solid and fluid phases) in the domain of the element. The displacements of the solid phase and the normal displacement of the fluid phase are approximated independently on the boundary of the stress element and used to enforce in a weak form the inter-element and boundary equilibrium conditions on the forces in the solid phase and on the fluid pressure. The boundary approximation used in the displacement element is on the solid surface forces and the fluid pressure, and is used to enforce on average the inter-element and boundary displacement continuity conditions. The resulting finite element governing systems are sparse, well-suited to adaptive refinement and parallel processing, and their coefficients are defined by boundary integral expressions. The energy statements associated with the formulation are recovered and sufficient conditions for the uniqueness of the finite element solutions are stated. Benchmark tests on hydrated soft tissue modelling are used to assess the performance of the alternative stress and displacement models of the hybrid-Trefftz formulation.

A corotational hybrid-Trefftz stress formulation for modelling cohesive cracks

This paper presents a corotational hybrid-Trefftz formulation for modelling propagating cohesive cracks and captures moderate rotations but small strains. The formulation is characterised by the fact that stresses are approximated within the domain of the element and the stiffness can be expressed via a boundary rather than a domain integral. Thus, compared to their FEM counterpart, hybrid-Trefftz stress elements exhibit faster convergence of the stress fields. Furthermore, the displacements are approximated on element boundaries and the displacement basis is defined independently on each element interface. Thus, the overall bandwidth of the stiffness matrix is very small and computationally efficient to solve. A corotational formulation for hybrid-Trefftz elements is also introduced in order to capture the effect of geometric nonlinearities in the form of moderate rotations. The model’s performance is demonstrated on three examples, illustrating crack propagation and the influence of geometric nonlinearities.

Toward a Hybrid-Trefftz element with a hole for elasto-plasticity?

The paper deals with the modelling of riveted assemblies for full-scale complete aircraft crashworthiness. Many comparisons between experiments and FE computations of bird impacts onto aluminium riveted panels have shown that macroscopic plastic strains were not sufficiently developed (and localised) in the riveted shell FE in the impact area. Consequently, FE models never succeed in initialising and propagating the rupture in the sheet metal plates and along rivet rows as shown by experiments, without calibrating the input data (especially the damage and failure properties of the riveted shell FE). To model the assembly correctly, it appears necessary to investigate on FE techniques such as Hybrid-Trefftz finite element method (H-T FEM). Indeed, perforated FE plates developed for elastic problems, based on a Hybrid-Trefftz formulation, have been found in the open literature. Our purpose is to find a way to extend this formulation so that the super-element can be used for crashworthiness. To reach this aim, the main features of an elastic Hybrid-Trefftz plate are presented and are then followed by a discussion on the possible extensions. Finally, the interpolation functions of the element are evaluated numerically.

Quadrilateral and triangular plane elements with rotational degrees of freedom based on the hybrid Trefftz method

In this paper, we suggest flexible, accurate, and robust linear elastic plane elements. We introduce drilling degrees of freedom to the hybrid Trefftz plane element by using Allman's quadratic displacement instead of conventional finite element approximation for the boundary displacement. Added drilling degrees increase the accuracy and robustness to mesh quality though they keep the advantages of the hybrid Trefftz method such as efficiency and freedom of the shape of the element. In addition, due to the hybrid Trefftz formulation, Poisson's locking, which may come from Allman's quadratic displacement, disappears.

Dual hybrid‐Trefftz element formulation based on independent boundary traction frame

AbstractThe paper presents the dual hybrid‐Trefftz (HT) element approach devised so as to enforce more strongly the reciprocity of boundary tractions at the element interfaces than the interelement conformity of displacements. As in the standard HT approach,1,2 a complete set of Trefftz functions is used to represent the internal displacement field of the element (and, as a consequence, the governing Navier‐Lamé differential equations are a priori verified over the element). But rather than employing an auxiliary interelement displacement frame field to enforce the conformity on the internal displacements, an auxiliary boundary traction field is adopted to enforce at the element interfaces the reciprocity of tractions derived from the internal displacement field. Provided that the number of Trefftz functions is sufficiently large, this approach yields an opposite bound on the energy in comparison to the standard HT approach based on the displacement frame. As in the standard HT formulation,3 the dual HT formulation is particularly well suited for generation of highly accurate p‐version elements. Although presented for solid mechanics, the formulation also holds for other problems. Numerical studies which involve comparisons of the two alternative displacement and traction frame HT element models have been performed by considering the Laplace's equation problem in two dimensions.

Hybrid trefftz plane elasticity elements with p ‐method capabilities

AbstractA family of p‐method plane elasticity elements is derived based on the hybrid Trefftz formulation.1 Exact solutions of the Lamé‐Navier equations are used for the intra‐element displacement field together with an independent displacement frame function field along the element boundary. The final unknowns are the parameters of the frame function field consisting of the usual degrees of freedom at corner nodes and an optional number of hierarchic degrees of freedom associated with the mid‐side nodes. Since the element matrices do not involve integration over the element area, the elements have a polygonal contour with an optional number of curved sides. The quadrilateral element has the same external appearance as the conventional p‐method plane elasticity element.2,3 But unlike in the conventional p‐method approach, suitable special‐purpose Trefftz functions are generally used to handle the singularity and/or stress concentration problems rather than a local mesh refinement. The practical efficiency of the new elements is assessed through a series of examples.

Hybrid-trefftz quadrilateral elements for thick plate analysis

A family of thick plate quadrilateral elements is derived based on the hybrid-Trefftz formulation. Exact solutions of the governing equations are used inside the element together with independent displacements and rotations on the boundary of the element. The element stiffness equations are derived using a modified hybrid-stress variational principle which enables all element matrices to be calculated using only boundary integrals. The resulting elements enable the accurate analysis of both thick and thin plate to be performed.