J. A. Teixeira de Freitas

Non-conventional formulations for the finite element method

Summary This paper reports on hybrid formulations being developed by the Structural Analysis Research Group of Instituto Superior Técnico. Three alternative sets of hybrid finite element formulations are presented. They are termed hybrid-mixed, hybrid and hybrid-Trefftz and differ essentially on the field conditions that the approximation functions are constrained to satisfy locally. Two models, namely the displacement and the stress models, are obtained for each formulation depending on whether the tractions or the boundary displacements are the field chosen to implement inter-element continuity. Because they are derived from a strict hybrid approach released from the conventional node conformity concepts, these formulations allow different fields to be independently approximated, within certain consistency criteria, and enhance the use of a wide range of approximation functions. For simplicity and objectivity, the description of the approach followed in the derivation of the alternative formulations and models is based on the elementary linear elastostatic problem of structural analysis. Their fundamental properties are identified and their patterns of convergence are analysed and compared.

Alternative approach to the formulation of hybrid equilibrium finite elements

Abstract A formulation appropriate to the development of equilibrated hybrid finite elements is presented. Techniques to overcome the problems traditionally associated with these elements are discussed along with the characteristics of the elements obtained. Numerical tests on the elements obtained using this formulation are presented.

Formulation of elastostatic hybrid-Trefftz stress elements

Abstract The hybrid-Trefftz stress (HTS) element is derived from the fundamental conditions of linear elastostatics. The stress and boundary displacement fields are directly approximated. Consistent finite element variables are so defined as to obtain boundary integral expressions for the finite element equilibrium and compatibility conditions, encoded by dual transformations independent of the constitutive relations. Reciprocity is preserved in the finite element elasticity relations, which are also found to be described by boundary integral arrays when the Trefftz constraint is enforced. Consequent upon duality and reciprocity, the resulting boundary integral solving system is symmetric and sparse. The energy statements associated with the hybrid-Trefftz stress element are established and the conditions for the existence, uniqueness, multiplicity and static and kinematic admissibility of the finite element solutions are presented. The HTS clement is compared with the displacement frame based hybrid-Trefftz (HT-D) element of Jirousek. It is shown that the HT-D element is in fact an hybrid stress element. The numerical tests performed to assess the relative performances of the HTS and HT-D elements show that better accuracy levels are obtained using discontinuous boundary approximation bases.

Hybrid finite element formulations for elastodynamic analysis in the frequency domain

Abstract Three alternative sets of hybrid formulations to solve linear elastodynamic problems by the finite element method are presented. They are termed hybrid–mixed, hybrid and hybrid–Trefftz and differ essentially on the field conditions that the approximation functions are constrained to satisfy locally. Two models, namely the displacement and the stress models, are obtained for each formulation depending on whether the tractions or the boundary displacements are the field chosen to implement interelement continuity. A Fourier time discretization is used to uncouple the solving system in the frequency domain. The basic space discretization criterion is implemented directly on the fundamental relations of elastodynamics and used to derive the stress and displacement models of the hybrid–mixed formulation. The hybrid and hybrid–Trefftz formulations are presented in sequence as the variants of the hybrid–mixed formulation obtained by progressively increasing the constraints on the approximation bases. Numerical implementation aspects are briefly discussed and the performance of the finite element models is illustrated with numerical applications.

Three‐dimensional hybrid‐Trefftz stress elements

The stress model of the hybrid-Trefftz finite element formulation is applied to the linear elastostatic analysis of solids. The stresses are approximated in the domain of the element and displacements on its boundary. Complete, linearly independent, hierarchical polynomial approximation functions are used in both domain and boundary approximations. The displacement basis is defined independently on each inter-element surface. Continuity at the edges and on the corners of the elements is not enforced a priori. The stress basis is constrained to solve locally the Beltrami governing differential equation. It is derived from the associated Papkovitch–Neuber elastic displacement solution. Generalized variables are used to ensure that the approximations are independent of the geometric description of the elements. The solving system is derived directly from the fundamental relations of elastostatics. The solving system is symmetric, when the same property applies to the local elasticity condition, sparse, described by boundary integral arrays and well suited to p-refinement and parallel processing. The numerical implementation of these equations is discussed and numerical tests are presented to illustrate the performance of the finite element formulation. Copyright

Hybrid-Trefftz displacement element for spectral analysis of bounded and unbounded media

The hybrid-Trefftz displacement element is applied to the elastodynamic analysis of bounded and unbounded media in the frequency domain. The displacements are approximated in the domain of the element using local solutions of the wave equation, the Neumann conditions are enforced directly and the surface forces are approximated on the Dirichlet and inter-element boundaries of the finite element mesh. Two alternative elements are developed to model unbounded media, namely a finite element with absorbing boundaries and an unbounded element that satisfies explicitly the Sommerfeld condition. The finite element equations are derived from the fundamental relations of elastodynamics written in the frequency domain. The numerical implementation of these equations is discussed and numerical tests are presented to assess the performance of the formulation.

Numerical implementation of hybrid-Trefftz displacement elements

Abstract The numerical implementation of the displacement model of the hybrid-Trefftz finite element formulation is presented. The geometry of the supporting element is not constrained a priori. Unbounded, non-convex and multiply connected elements can be used. The approximation basis is naturally hierarchical and very rich. It is constructed on polynomial solutions of the governing differential equation, and extended to include the particular terms known to model accurately important local effects, namely the singular stress patterns due to cracks or point loads. Numerical and semi-analytical methods are used to compute the finite element matrices and vectors, all of which present boundary integral expressions. Appropriate procedures to store, manipulate and solve symmetric highly sparse systems are used. The characteristics of the finite element solving system in terms of sparsity and conditioning are analysed, as well as its sensitivity to the effects of mesh distortion, incompressibility and rotation of the local reference systems. Benchmark tests are used also to illustrate the performance of the element in the estimation of displacements, stresses and stress intensity factors.

A kinematic model for plastic limit analysis of solids by the boundary integral method

Abstract A mixed boundary integral formulation for plastic limit analysis by linear programming is presented. The interpolation criteria are so defined as to preserve duality in the description of equilibrium and compatibility and reciprocity in the plastic constitutive relations. The governing systems appears in the form of a symmetric linear complementarity problem. The associated linear programs are derived and identified with the static kinematic theorems of plastic limit analysis as applied to the discretized structure.

The hybrid stress model for mindlin-reissner plates based on a stress optimization condition

Abstract A stress optimization condition for hybrid stress finite elements is applied to develop a Mindlin-Reissner plate element. A modified Hellinger-Reissner energy functional, derived from a stress optimization condition, is employed to relax the patch test constraint of incompatible displacements. An eight-node serendipity thin and moderately thick plate element, which is stable and free from shear locking, is formulated from the optimized stress distribution and the use of incompatible displacements. Numerical results indicate that the resulting element is accurate in both displacements and stresses, and insensitive to mesh distortion.

Elastodynamic analysis with hybrid stress finite elements

Abstract The stress model of the hybrid finite element formulation is applied to the solution of free and forced vibration problems. The formulation develops from the independent approximation of the stresses in the domain of the element and of the displacements on its boundary. The inertia force field is determined from the stress approximation to solve locally the dynamic equilibrium condition. Implementation of the time discretisation procedure leads to dependent approximations for the displacement, velocity and acceleration fields. Duality is used to establish the finite element description of each fundamental condition of elastodynamics. Nodal interpolation is replaced by the use of orthogonal and naturally hierarchical bases associated with generalised finite element variables. The resulting solving systems are symmetric, sparse, p-adaptive and well suited to parallel processing. The performance of the element is assessed through a comprehensive set of two-dimensional testing problems.