The paper shows the essential unity between the known approximation procedures to differential equations. The boundary type approximations associated with Trefftz are shown to be a particular form of the weighted residual approximation and can provide a very useful basis for generating hybrid `finite elements'. These can be used in combination with other finite element approaches.

Trefftz approximations are known to require very few degrees of freedom to give an order of magnitude of the solution. In this paper, we show that it is possible to take advantage of this situation in two ways: (i) we show that it is also possible to get accurate solutions (especially for the pressure) at a reasonable cost and (ii) we show that the low number of degrees of freedom needed for this accuracy allows an easy domain optimisation, which is illustrated here for the free boundary problem in extrusion. A construction of Trefftz polynomials associated with Stokes problem for plane strains is also given with some recurrence properties which is usefull for computing them at a low cost. Moreover a domain decomposition method which has shown to be efficient for compressible elastic material has been extended here to the case of incompressible linear viscous fluids.

In both boundary element methods and Trefftz-type finite element methods a discrete problem on the boundary of the domain and possibly the boundaries between subdomains. We consider a Trefftz element formulation which is based on the complementary energy functional, and we compare different regularizations of the interelement continuity conditions. Also starting from the complementary energy functional, mixed finite elements can be constructed such that the stresses satisfy equilibrium a priori. We describe a coupling of these elements with the by now classical symmetric Galerkin-BEM.

Two alternative models of the hybrid-Trefftz finite element formulation to solve linear elastodynamic problems are presented. In the displacement model, the displacements are approximated in the domain of the element and the tractions are approximated on its boundary. The fields selected for approximation in the complementary stress model are the stresses in the domain and the displacements on the boundary of the element. In both models the domain approximation functions are constrained to satisfy locally the governing wave equation. A Fourier time approximation is used to uncouple the solving system in the frequency domain. The formulations are derived from the fundamental relations of elastodynamics and the associated energy statements are obtained a posteriori. Sufficient conditions for the existence and uniqueness of statically and kinematically admissible solutions are presented. Numerical implementation is briefly discussed.

The author's algebraic theory of boundary value problems has permitted systematizing Trefftz method and expanding its scope. The concept of TH-completeness has played a key role for such developments. This paper is devoted to revise the present state of these matters. Starting from the basic concepts of the algebraic theory, Green-Herrera formulas are presented and Localized Adjoint Method (LAM) derived. Then the classical Trefftz method is shown to be a particular case of LAM. This leads to a natural generalization of Trefftz method and a special class of domain decomposition methods: Trefftz-Herrera domain decomposition.

This paper presents a method for a quick evaluation of stresses and displacements for elastostatic problems. A set of polynomial Trefftz functions and a variational formulation are introduced for solving elastostatic problems for simple star-shaped domains. It is shown, through examples, that this approximation allows the computation of the interior large wavelength effects. By a procedure for coupling separate domains, this method is extended to more complex structures, which is a natural extension of the above variational formulation. A discretization of the structure into large substructures, an easy to use and quick computation of the interior solution justify that this method can be termed `simplified'. Comparisons with other similar methods are also shown.

The numerical solution of Helmholtz' equation at large wavenumber is very expensive if attempted by "traditional" discretisation methods (FDM, standard Galerkin FEM). For reliable results, the mesh has to be very fine. The bad performance of the traditional FEM for Helmholtz problems can be related to the deterioration of stability of the Helmholtz differential operator at high wavenumber. As an alternative, several non-standard FEM have been proposed in the literature. In these methods, stabilisation is either attempted directly by modification of the differential operator or indirectly, via improvement of approximability by the incorporation of particular solutions into the trial space of the FEM. It can be shown that the increase in approximability can make up for the stability loss, thus improving significantly the convergence behavior of the knowledge based FEM compared to the standard approach. In our paper, we refer recent results on stability and convergence of h- and h-p-Galerkin (`standard') FEM for Helmholtz problems. We then review, under the label of `knowledge-based' FEM, several approaches of stabilised FEM as well as high-approximation methods like the Partition of Unity and the Trefftz method. The performance of the methods is compared on a two-dimensional model problem.

The thin plate p-elements considered in this paper are based on assumed displacement field chosen so as to a priori satisfy the governing Lagrange equation within the element. The required C1 conformity is then enforced in a weak sense trough an auxiliary displacement frame defined in terms of nodal and side mode parameters. While thus far the standard approach consisted in using three parameters (one displacement and two rotations) at corner nodes and an optional number of side mode parameters associated with mid-side nodes, other alternative formulations are also possible wherein the number of corner mode parameters is either inferior or superior to three. As compared to the standard frame, such alternative formulations may exhibit some advantages and some shortcomings with respect to accuracy, convergence rate, error distribution, computational efficiency and/or ease of use. The paper surveys and critically assesses some of such formulations and reports the results of extensive numerical studies involving regular and singular plate bending applications.

The reported research presents a finite element formulation for folded plate analysis based on the p-version of the hybrid-Trefftz finite element model. The internal displacement field of the elements consists of a suitably truncated T-complete set of in-plane (u,v) and out-of-plane (w) components which satisfy the respective governing differential plane elasticity and thin plate (Kirchhoff) equations. Conformity is enforced in a weak, weighted residual sense through an auxiliary displacement frame, independently defined at the boundary of the element and consisting of displacement components and normal rotation. The displacement frame parameters are the global Cartesian displacements at corner nodes and the hierarchical side-mode parameters for normal rotation and the global Cartesian displacement components, an optional number of which is allotted, formally, to mid-side nodes. The investigated approach is assessed on numerical examples.

This paper presents the boundary-type schemes of the first- and the second-order sensitivity analyses by Trefftz method. In the Trefftz method, physical quantities are approximated by the superposition of the T-complete functions satisfying the governing equations. Since the T-complete functions are regular, the approximate expressions of the quantities are also regular. Therefore, direct differentiation of them leads to the expressions of the sensitivities. Firstly, the Trefftz method for the two-dimensional potential problem is formulated by means of the collocation method. Then, the first- and the second-order sensitivity analysis schemes are explained with the simple numerical examples for their verification.

The paper propose two adaptive algorithms based on a Trefftz method for two-dimensional Laplace equation satisfying the maximal principle. First one for given the error tolerance and an initial number of terms in the solution expansion, the algorithm computes expansion coefficient by location of boundary conditions and evaluates the maximum absolute error on the boundary. If error exceeds the error the tolerance, additional expansion terms and boundary collocation points are added and process repeated until the tolerance is satisfied. The second one is based on Galerkin formulation of Trefftz method and utilizes the exact potential error norm for predict a new mesh and new solution expansion until the tolerance is satisfied.

The purpose of the paper is to propose of a way of constructing trial functions for the indirect Trefftz method as applied to harmonic problems possessing circular holes, circular inclusions, corners, slits, and symmetry. In the traditional indirect formulation of the Trefftz method, the solution of the boundary-volume problem is approximated by a linear combination of the T-complete functions and some coefficients. The T-complete Trefftz functions satisfy exactly the governing equations, while the unknown coefficients are determined so as to make the boundary conditions satisfied approximately. The trial functions, proposed and systematically constructed in this paper, fulfil exactly not only the differential equation, but also certain given boundary conditions for holes, inclusions, cracks and the conditions resulting from symmetry. A list of such trial functions, unavailable elsewhere, is presented. The efficiency is illustrated by examples in which three Trefftz-type procedures, namely the boundary collocation, least square, and Galerkin is used.

We introduce the hybrid-Trefftz FE formulations for linear statics of solids as well as for linear (slow velocity) steady fluid dynamics. Moving least square procedure is given to obtain continuous secondary fields (such as stresses for solids), which improves the results. For nonlinear problems the governing equations are satisfied in the discrete least square residual form. Also for such problems the hybrid FE formulation is shown.

A recent geometric presentation of a general and efficient methodology for recovering equilibrating tractions and stress fields from 2-D conforming displacement finite element models is reviewed, and further considered in the context of plate elements in 2-D and 3-D. This methodology requires the resolution of corner nodal forces/moments, and this presents localised problems which are solved in a simple way by exploiting Maxwell force diagrams. For more complex 3-D models including solid elements where higher degrees of element connectivity occur, the geometric procedure is adapted so as to retain the computational efficiency gained from recognising the topological and geometrical properties of a finite element model. Graph-theoretic and algebraic topological concepts are invoked in this context. The equilibrating tractions recovered for each element enable statically admissible stresses to be computed element by element, and local Trefftz fields may be exploited.

We describe a new Quasi Trefftz-type Spectral Method (QTSM) for solving boundary value and initial value problems. QTSM combines the properties of the Trefftz method with the spectral approach. The special feature of QTSM is that we use trial functions which satisfy the corresponding homogeneous equation only approximately. These trial functions are represented in the terms of a truncated series of eigenfunctions of some eigenvalue problem associated with the problem considered. The method has been found to work well for different elliptic problems with the Laplace, the Helmholtz and the biharmonic operators. We also consider some nonstationary parabolic problems including the problem in the domain with moving boundaries. The possibilities of further development of QTSM are also discussed.

Generally, two approaches have been used to study the nonlinear wave-structure interaction in the context of offshore engineering in recent years. One is based on the Stokes perturbation procedure in the frequency domain and has been applied to weak-nonlinear problems. The other is based on a full nonlinear solution to the resulting wave field by a time-stepping procedure with boundary conditions applied on the moving free and body surfaces. In the present work an alternative solution method for nonlinear wave-structure interaction problems is proposed. The method is based on the evolution equations for the free-surface elevation and the free-surface potential, which are solved by the time-stepping procedure. The field problem is solved at each time step by the perturbation method combined with Trefftz approach. The method is applied to the study of the evolution of three-dimensional waves generated by a vertical circular cylinder oscillating in water of constant depth.

A review is presented of a recent formulation of hybrid-equilibrium elements for modelling planar structural problems. The formulation is based on the use of polynomials of general degree to approximate internal stress fields and bounding side displacements. The existence of hyperstatic stress fields and spurious kinematic modes are considered algebraically for both primitive and macro-elements, the latter providing a means of controlling or removing the spurious modes in the former. Consideration is given to stress fields which are statically admissible, and to Trefftz stress fields which are both statically and kinematically admissible.

The aim of the paper is twofold. In the first part, we present an analysis of the approximation properties of `complete systems', that is, systems of functions which satisfy a given differential equation and are dense in the set of all solutions. We quantify the approximation properties of these complete systems in terms of Sobolev norms. As a first step of the analysis, we consider the approximation of harmonic functions by harmonic polynomials. By means of the theory of Bergman and Vekua, the approximation results for harmonic polynomials are then extended to the case of general elliptic equations with analytic coefficients if the harmonic polynomials are replaced with their analogs, `generalized harmonic polynomials'. In the second part of the paper, we present the Partition of Unity Method (PU). This method has the feature that it allows for the inclusion of a priori knowledge about the local behavior of the solution in the ansatz space. Therefore, the PU can lead to very effective and robust methods. We illustrate the PU with an application to Laplace's equation and the Helmholtz equation.

In this paper the possibility of applying the Trefftz-method to thick and thin shells is discussed. A mixed variational formulation is used in which the assumed strain and stress functions are derived from the three-dimensional solution representation for the displacement field. For the construction of the linearly independent Trefftz trial functions both the Neuber/Papkovich solution representation and a complex variable approach of the author are considered. The difficulty in constructing the solution functions for the displacement field consists of two problems: i) How can we choose the functions in order to have a symmetric structure in the displacement field and not to bias the solution in any direction? ii) How can we avoid to get linearly dependent terms for displacements, strains and stresses when seeking polynomial solution terms?

Modeling of elastic thin-walled beams, plates and shells as 1D and 2D boundary value problems is valid in undisturbed subdomains. Disturbances near supports and free edges, in the vicinity of concentrated loads and at thickness jumps cannot be described by 1D and 2D BVP's. In these disturbed subdomains dimensional (d)-adaptivity and possibly model (m)-adaptivity have to be performed and coupled with mixed h- and/or p-adaptivity by hierarchically expanded test spaces in order to guarantee a reliable and efficient overall solution. Using residual error estimators coupled with anisotropic error estimation and mesh refinement, an efficient adaptive calculation is possible. This residual estimator is based on stress jumps along the internal boundaries and residua of the field equation in L2 norms. In this paper, we introduce an equilibrium method for calculation of the internal tractions on local patches using orthogonality conditions. These tractions are equilibrated with respect to the global equilibrium condition of forces and bending moments. We derive a new error estimation based on jumps between the new tractions and the tractions calculated with the stresses of the current finite element solution solution. This posterior equilibrium method (PEM) is based on the local calculation of improved stress tractions along the internal boundaries of element patches with continuity condition in normal directions. The introduction of new tractions is a method which can be regarded as a stepwise hybrid displacement method or as Trefftz method for a Neumann problem of element patches. An additional and important advantage is the local numerical solution and the model error estimation based on the equilibrated tractions.