Three-dimensional hybrid-Trefftz stress elements

Abstract

The stress model of the hybrid-Tre!tz "nite 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 de"ned 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 di!erential 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-re"nement and parallel processing. The numerical implementation of these equations is discussed and numerical tests are presented to illustrate the performance of the "nite element formulation. thus obtained can be recognized in the embracing works of Pian and Tong [3] and Brezzi and Fortin [4].The formulation used here was identi"ed later as the stress model of the hybrid-Tre!tz "nite element formulation and applied to di!erent two-dimensional structural analysis problems [5}8]. It is, therefore, directly related with the displacement frame Tre!tz element developed by Jirousek and Stein and their co-workers [9, 10]. The theoretical basis of the formulation is established in [11], where a comparative analysis with the related T-elements is also presented.In the terminology adopted here, a "nite element derived from the direct approximation of the stress "eld is termed a stress "nite element model. The stress "eld is directly related with two fundamental conditions, namely the equilibrium condition in the domain of the element and the di!usivity (Neumann) condition on the associated boundary tractions (Cauchy stresses). Three classes of formulations can be established depending on the equilibrium constraints enforced a priori on the stress basis, namely the hybrid-mixed, the hybrid and the hybrid-Tre!tz formulations of the "nite element stress model [12,13].The hybrid-mixed formulation is the most general. No constraints (besides linear independence and completeness) are placed on the stress approximation basis. Besides the stresses, the displacements are also approximated, and independently, in the domain and on the boundary of the element. These bases are used to enforce on average the equilibrium and the di!usivity conditions, respectively. The hybrid formulation is obtained by constraining a priori the stress approximation to satisfy locally the equilibrium condition. Consequently, the displacement approximation in the domain of the element becomes redundant an...

Simulation of Stress Concentration Problems in Laminated Plates by Quasi-Trefftz Finite Element Models

“…Because of that this element has been named quasi-Trefftz. Legendre and Chebychev modified polynomials are attributed to ψ and ϕ, as explained by Freitas and Bussamra (2000). They form a complete stress approximation basis for degrees lower than seven.…”

“…In particular, the hybrid-Trefftz formulation approximates stresses within the element and displacements on its boundary. The Trefftz constraint consists of assuming that the stress approximation basis is the local solution of the Navier equation (Freitas and Bussamra, 2000). Freitas and Ji (1996) presented highly accurate finite element solution procedures for simulation of singular stress fields with two-dimensional hybrid-Trefftz element.…”

“…Jin et al (2005) have applied Trefftz collocation and Trefftz Galerkin methods to the analysis of plane piezoelectricity. Freitas and Bussamra (2000) have formulated hybrid-Trefftz element for 3-D elasticity. The element presented low sensitivity to geometric irregularities, incompressibility, and produced good estimates for stresses and displacements.…”

Fundamental-solution-based hybrid FEM for plane elasticity with special elements

“…As a variant of the hybrid stress equilibrium finite element methods, the hybrid Trefftz finite element method (HT-FEM) has been established by further constraining the stress basis to be associated with the satisfaction of all the governing equations of the problem in the element domain and has proved to be an efficient computational tool for solving engineering problems with local effects due to loading and/or geometry without troublesome mesh adjustment [2][3][4]. In the past decades, the HT-FEM has been successfully applied to problems of elasticity [5][6][7], plate bending [8,9], elastodynamic problems [10,11], transient heat conduction analysis [12], geometrically nonlinear plates [13], elastoplasticity [14,15], piezoelectric materials [16], and nonlinear minimal surface problems [17]. Unlike the conventional finite element method (FEM) and boundary element method (BEM), the hybrid Trefftz FEM is based on a hybrid model which includes the use of independent auxiliary inter-element frame fields defined on each element boundary and independent intra-element fields chosen so as to a priori satisfy the homogeneous governing differential equations by means of a suitably truncated T-complete function set of homogeneous solutions.…”

Método da partição e formulação híbrido-Trefftz na análise de sólidos bidimensionais contendo múltiplas fissuras

“…Por fim, a formulação híbrido-Trefftz (JIROUSEK, 1978, JIROUSEK, LEON, 1977, JIROUSEK, TEODORESCU, 1982, JIROUSEK, VENKATESH, 1992, JIROUSEK, WRÓBLEWSKI, 1996, FREITAS, 1998, FREITAS, ALMEIDA, PEREIRA, 1999, FREITAS, BUSSAMRA, 2000, BUSSAMRA, NETO, PONCIANO, 2014, BUSSAMRA, NETO, RODRIGUES, 2016 também é um caso particular da híbrido-mista ao impor que as funções de aproximação das tensões no domínio do elemento atendam a equação governativa: a equação de Navier. Apesar da forte restrição na escolha de funções para a aproximação, essa formulação proporciona soluções mais precisas (mesmo com malhas grosseiras) dentre as três formulações híbridas aqui apresentadas, particularmente nos problemas da elasticidade linear.…”