The extension of this approach to hybrid and mixed finite element analysis of membranes, plates, shells and solids was initiated in the mid 1980's.
Research on the boundary element method started in the same period. It follows a conventional, direct approach and has been mainly oriented to the development of adaptive methods and techniques and to the analysis of elastoplastic fracture mechanics problems.
To support the development of the research work, it became necessary to invest a significant effort in the adaptation to the specific problems under analysis of the existing software for computer graphics and for processing of large sparse systems.
The Computational Mechanics Laboratory of the Department of Civil Engineering of IST is currently equipped with IBM 550 servers, IBM 320H and HP 9000/710 workstations and Xstations. This network is served by input/output processing equipment, such as scanners, digitizing tables, plotters and printers. The laboratory is linked to the computer centre of IST and of the National (Portuguese) Foundation for Scientific Computation, served by vector and parallel processing mainframes, and has access to international scientific networks.
The research group participates in international research projects supported by the European Union. However, for operational funding its autonomy relies mainly on research projects directly supported by Portuguese research funding agencies and bodies.
As the approach adopted is not problem dependent, it is easily adaptable to distinct structural forms and constitutive relations and is open to the incorporation of a wide variety of interpolation functions.
Two complementary sets of finite element descriptions can be established from a Navier-type description of boundary value problems. They yield to stress and displacement finite elements, respectively.
In each case, three alternative formulations can be stated, depending on the constraints enforced on the intervening approximation functions.
These formulations are termed hybrid-mixed, hybrid and hybrid-Trefftz.
In the hybrid-mixed finite element formulations the stress and the displacement fields in the domain are independently and simultaneously approximated. What distinguishes the stress and displacement mixed-hybrid versions is the boundary field which is simultaneously approximated: these are the boundary displacements and the boundary tractions, respectively.
The hybrid finite element formulations can be obtained by specialization of the hybrid-mixed models by requiring either the stress approximation to be self-equilibrated or the displacement approximation to induce compatible strains. It is found now that in the stress version only the stresses and the boundary displacement have to be explicitly approximated. Conversely, in the displacement version the displacements in the domain and the boundary tractions are the fields which are explicitly approximated.
The hybrid-Trefftz finite element formulations can be obtained by specialization of the hybrid models. In the stress version the stress potential approximation functions are required to satisfy locally the Beltrami differential governing system. In the displacement version the displacement potential approximation functions are required to satisfy locally the Navier differential governing system.
As they can incorporate a wide variety of approximation functions, the hybrid-mixed formulations for the finite element method under development are designed to generate highly sparse, hierarchical systems particularly suited to processing in parallel and/or vectorial modes.
The typical pattern of the finite element governing system is illustrated in figure 1. Sparsity indices of orders over 99% are frequently obtained.
To exploit the architecture of digital computers and to enhance the implementation of adaptive techniques, Walsh functions and wavelets, shown in figures 2 and 3, have been tested and used in the solution of a wide variety of problems using hybrid-mixed formulations.
The Walsh functions are applied in the interpolation of stress and displacement fields. The progressive Walsh refinement of the stress and displacement estimates in a plate is illustrated in figure 4.
Shown in figure 5 is the plastic yield zone and the stress distribution (at the incipient collapse point indicated in the load-displacement diagram) obtained for a square thick clamped plate subjected to a uniform load.
A hybrid-mixed elastoplastic formulation is used to analyse a quarter of the specimen discretized using a single element. Wavelets are used to approximate the displacement and stress fields. The plastic multiplier distribution is modeled with polynomial functions. An incremental, asymptotic solution procedure is adopted. In each increment, the step length is maximized and automatically adjusted to the onset of plastic straining or unstressing and also subject to a control on the accummulation of truncation errors.
Figure 6 illustrates the elastic stress distribution in a 3 layer cross-ply fibre composite plate subject to a uniform axial deformation; axial stresses are shown on top and shear stresses below. By enforcing symmetry, only an eighth of the plate is analysed using 2 higher-order plate macroelements.
To obtain the formulation for these special elements, the displacement field is expanded in a Taylor series and subsequently interpolated. The 3-D problem is thus uncoupled into a sequence of higher-order plate problems. Kirchhoff and Mindlin theories are recovered as the lower-order particular cases. Legendre polynomials are used to approximate both stress and displacement fields.
An interesting feature of the hybrid formulations is the possibility to generate solutions that satisfy locally either the equilibrium equations or the compatibility equations, together with the associated boundary conditions.
This is illustrated in figures 7 and 8 for a tapered cantilever beam subject to transverse loading. The statically admissible solution, represented in figure 7, is obtained with stress elements. The kinematically admissible solution shown in figure 8 is obtained with displacement elements. In both cases 5 super-elements designed to eliminate spurious modes are used.
Dual stress and displacement hybrid formulations can be used to bound the finite element estimates. Figure 9 illustrates the application of an adaptive mesh optimizer based on dual, hybrid finite element formulations. The geometry shown optimizes an initial, regular mesh of 4x4 elements. Stress trajectories are now used to represent the stress distribution.
Besides the capacity to generate statically or kinematically admissible solutions from symmetric, sparse governing systems, the hybrid-Trefftz formulations also display the interesting property of yielding boundary integral expressions for all the intervening structural matrices and vectors, a feature typical of the boundary element method.
The regular and (weakly) singular stress and displacement distributions derived as closed form solutions for classical elastic and plastic problems, both in quasi-static and dynamic regimes, are used as interpolation functions.
The elastic stress distribution in the cruciform plate shown in figure 10 is obtained using the Chebyshev polynomials present in Michell's solution for 2-D elastostatic stress functions. The stress hybrid-Trefftz formulation is implemented on a single, non-convex element.
Weakly singular stress functions are used to model the concentration of stresses.
The illustration in figure 11 represents the stress pattern obtained in the simulation of the elastic response of a cracked plate when a relatively coarse finite element mesh is used; less than 200 degrees of freedom are sufficient to generate an accurate solution. The stress intensity factors are directly computed as they represent the weights of the crack functions explicitly enforced in the approximation of the stress field.
As in the previous illustrations, the distributions represented in figure 11 are directly obtained from the computed stress values. Stress smoothing is not applied.
This technique is applied in the analysis of soil-structure interaction problems relevant in the design of dams and of piling foundations.
The finite element mesh used to model the structure is coupled with the boundary element mesh adopted in the representation of the response of the foundation.
A second area of research in the boundary element method is the development of p-, h- and hp- hierarchical adaptive to improve the numerical solution of singular and non-singular problems. Numerical testing is performed on potential problems and 2-D elastostatic problems.
The exact and the initial mesh solutions for a potential problem are compared in figure 13 with the solution obtained with an adaptive procedure. To optimize the convergence rate, refinement is not uniformly distributed along the boundary but according to criteria established through the implementation of error indicators, which select the zones where the local error is larger.
The analysis of linear elastic fracture mechanics is an area for which boundary element formulations are particularly suitable. The extension of these formulations for the analysis of elastoplastic fracture mechanics problems is being investigated.
An example of the work in progress is shown in figure 14. The predicted fatigue crack growth rates are obtained using an elastoplastic boundary element formulation derived to model the response of a standard aluminium alloy aircraft component previously prestressed to 80% of its yield stress.
The software for graphical representation has been illustrated throughout this presentation. Figure 15 shows a typical output of the finite mesh element generator being developed.
Figure 15: Boundary discretization and the interior points in a complex three dimensional domain.