A boundary method of Trefftz type with approximate trial functions

This paper presents a new numerical technique which belongs to the group of the Trefftz type methods. It differs from the other boundary techniques by the use of approximate solutions of the corresponding partial differential equations (PDE) as trial functions. This permits to extend the field of application of this boundary method to the problem where one cannot find an appropriate set of exact solutions. In particular, the application of the method presented to a general PDE of the elliptic type is considered in the paper. The method has also been found to work well for certain boundary value and initial value problems including PDEs of the fourth order and equations of the non-stationary heat transfer with moving boundaries.     

Trefftz boundary element method for domains with slits

This paper is concerned with the implementation of the Trefftz boundary element method, for the analysis of two-dimensional potential problems in domains with thin internal or edge cavities. Each cavity is treated as an infinitely thin slit defined in a single region, for which a particular solution of the problem is known. This solution procedure which does not generate new unknowns in the problem, satisfies exactly the specified boundary conditions along the slit boundaries. Furthermore, the strength of the singularity, defined at the tip of each slit, is directly computed by the solver. Several examples are analysed with this procedure for both the collocation and Galerkin techniques of the Trefftz boundary element method. The accuracy and efficiency of the implementations described herein make the Trefftz boundary element method ideal for the study of potential problems with degenerated geometries.     

Trefftz method: Fitting boundary conditions

The paper presents various ways of fitting the boundary conditions in the T-complete functions method. The authors point out the distinct advantages of the orthogonal collocation in comparison to the equidistant collocation and the integral fit. The convergence of the Collatz error measures and the conditioning of the solution matrices are investigated in detail.     

Trefftz interpolation based multi-domain boundary point method

The paper describes meshfree boundary point solution of problems based on Trefftz interpolation functions. The solution is a collocation method, which can use any type of Trefftz functions smooth enough for approximation of domain fields. Polynomial, Kelvin and Boussinesq functions are used in our formulations. A multi-domain boundary formulation with continuity of tractions and displacements over discrete points of the inter-domain surface is shown. The method is applied to linear elastostatics. 2D and 3D examples illustrate the formulation. Discussion about advantages and drawbacks is included. As the conservation equations (force and moment equilibrium) are not satisfied in integral sense, some benchmark tests are proposed for meshless formulations to obtain a view about the accuracy.     

Indirect Trefftz method for boundary value problem of Poisson equation

Trefftz method is the boundary-type solution procedure using regular T-complete functions satisfying the governing equation. Until now, it has been mainly applied to numerical analyses of the problems governed with the homogeneous differential equations such as the two- and three-dimensional Laplace problems and the two-dimensional elastic problem without body forces. On the other hand, this paper describes the application of the indirect Trefftz method to the solution of the boundary value problems of the two-dimensional Poisson equation. Since the Poisson equation has an inhomogeneous term, it is generally difficult to determine the T-complete function satisfying the governing equation. In this paper, the inhomogeneous term containing an unknown function is approximated by a polynomial in the Cartesian coordinates to determine the particular solutions related to the inhomogeneous term. Then, the boundary value problem of the Poisson equation is transformed to that of the Laplace equation by using the particular solution. Once the boundary value problem of the Poisson equation is solved according to the ordinary Trefftz formulation, the solution of the boundary value problem of the Poisson equation is estimated from the solution of the Laplace equation and the particular solution. The unknown parameters included in the particular solution are determined by the iterative process. The present scheme is applied to some examples in order to examine the numerical properties.     

A boundary integral Trefftz formulation with symmetric collocation

The Galerkin and collocation methods are combined in the implementation of a boundary integral formulation based on the Trefftz method for linear elastostatics. A finite element approach is used in the derivation of the formulation. The domain is subdivided in regions or elements, which need not be bounded, simply connected or convex. The stress field is directly approximated in each element using a complete solution set of the governing Beltrami condition. This stress basis is used to enforce on average, in the Galerkin sense, the compatibility and elasticity conditions. The boundary of each element is, in turn, subdivided into boundary elements whereon the displacements are independently approximated using Dirac functions. This basis is used to enforce by collocation the static admissibility conditions, which reduce to the Neumann conditions as the stress approximation satisfies locally the domain equilibrium condition. The resulting solving system is symmetric and sparse. The coefficients of the structural matrices and vectors are defined either by regular boundary integral expressions or determined by direct collocation of the trial functions.     

Interior boundary conditions in the Schwarz alternating method for the Trefftz method

New types of the Schwarz alternating methods (SAMs) with overlapping and non-overlapping are proposed, by using different interior boundary conditions, such as the Dirichlet, the Neumann and the Robin conditions. Those SAMs using different interior boundary conditions are called the mixed SAMs. This paper consists of two parts. In the first part, for a simple continuous model: the Laplacian solutions on a sectorial domain, the convergence rates of the mixed SAMs are derived in detail for different interior boundary conditions. Compared with the classic overlapping SAM with the Dirichlet conditions, some of mixed SAMs with the interior Neumann or Robin conditions will converge faster. Moreover, a number of new and better SAMs than those in the existing literature are explored in this paper. In the second part, for solving Motz's problem, the Trefftz method using particular solutions and the finite difference method (FDM) are combined, and the mixed SAMs are applied to implement the algorithms into parallel. Numerical results by the mixed SAMs are given to support the analysis made from the simple models. Hence, we may employ some mixed SAMs proposed in this paper to speed the SAM convergence rates.     

A surface fitting method for three dimensional scattered data

A method is presented that may be used to empirically establish the type of relationship that is present between a response variable and its influencing factors, by fitting a mathematical model to three dimensional scattered data. The generated response surface is composed of continuous triangular planes that are fitted to the corresponding data in the least squares sense. The method may be easily implemented. It requires some fairly large number of scattered data, two initial boundary conditions and a desired accuracy for the band-wise partitioning of the data. The proposed surface fitting technique has been successfully applied to solar radiation modelling for a number of different data combinations.     

Multi-region Trefftz boundary element method for fracture analysis in plane piezoelectricity

This paper presents a multi-region Trefftz boundary element method for fracture analysis in plane piezoelectricity. To model the sub-region that contains the crack, a special set of Trefftz functions that satisfy the traction-free and charge-free conditions along the crack faces are constructed. To model the remaining sub-regions, the basic set of Trefftz functions co-derived previously by the authors are employed. With the two sets of Trefftz functions, the multi-region Trefftz boundary element method is formulated by point collocation. The special set of Trefftz functions exempts all the boundary treatment of the crack faces and enables the direct determination of the electromechanical intensity factors. Numerical examples are presented to illustrate the efficacy of the formulation.     

A Trefftz function approximation in local boundary integral equations

 In the present paper the Trefftz function as a test function is used to derive the local boundary integral equations (LBIE) for linear elasticity. Since Trefftz functions are regular, much less requirements are put on numerical integration than in the conventional boundary integral method. The moving least square (MLS) approximation is applied to the displacement field. Then, the traction vectors on the local boundaries are obtained from the gradients of the approximated displacements by using Hooke's law. Nodal points are randomly spread on the domain of the analysed body. The present method is a truly meshless method, as it does not need a finite element mesh, either for purposes of interpolation of the solution variables, or for the integration of the energy. Two ways are presented to formulate the solution of boundary value problems. In the first one the local boundary integral equations are written in all nodes (interior and boundary nodes). In the second way the LBIE are written only at the interior nodes and at the nodes on the global boundary the prescribed values of displacements and/or tractions are identified with their MLS approximations. Numerical examples for a square patch test and a cantilever beam are presented to illustrate the implementation and performance of the present method. Received 6 November 2000     

Adaptive error estimation technique of the Trefftz method for solving the over-specified boundary value problem

In this paper, numerical solutions are investigated based on the Trefftz method for an over-specified boundary value problem contaminated with artificial noise. The main difficulty of the inverse problem is that divergent results occur when the boundary condition on over-specified boundary is contaminated by artificial random errors. The mechanism of the unreasonable result stems from its ill-posed influence matrix. The accompanied ill-posed problem is remedied by using the Tikhonov regularization technique and the linear regularization method, respectively. This remedy will regularize the influence matrix. The optimal parameter λ of the Tikhonov technique and the linear regularization method can be determined by adopting the adaptive error estimation technique. From this study, convergent numerical solutions of the Trefftz method adopting the optimal parameter can be obtained. To show the accuracy of the numerical solutions, we take the examples as numerical examination. The numerical examination verifies the validity of the adaptive error estimation technique. The comparison of the Tikhonov regularization technique and the linear regularization method was also discussed in the examples.     

Trefftz solution for boundary value problem of three-dimensional Poisson equation

This paper describes the application of the Trefftz method to the solution of three-dimensional Poisson equation. An inhomogeneous term containing the unknown function is approximated with a polynomial function in the Cartesian coordinates to determine the particular solution for the Poisson equation. The solution of the problem is approximated with the superposition of the Trefftz functions of the Laplace equation and the derived particular solution. Unknown parameters included in the approximate solution are determined so that the solution satisfies the boundary conditions. The present scheme is applied to some examples in order to study the numerical properties.     

A hybrid semismooth quasi-Newton method for nonsmooth optimal control with PDEs

Optimization and Engineering - We propose a semismooth Newton-type method for nonsmooth optimal control problems. Its particular feature is the combination of a quasi-Newton method with a...     

The Trefftz method for solving eigenvalue problems

For Laplace's eigenvalue problems, this paper presents new algorithms of the Trefftz method (i.e. the boundary approximation method), which solve the Helmholtz equation and then use an iteration process to yield approximate eigenvalues and eigenfunctions. The new iterative method has superlinear convergence rates and gives a better performance in numerical testing, compared with the other popular methods of rootfinding. Moreover, piecewise particular solutions are used for a basic model of eigenvalue problems on the unit square with the Dirichlet condition. Numerical experiments are also conducted for the eigenvalue problems with singularities. Our new algorithms using piecewise particular solutions are well suited to seek very accurate solutions of eigenvalue problems, in particular those with multiple singularities, interfaces and those on unbounded domains. Using piecewise particular solutions has also the advantage to solve complicated problems because uniform particular solutions may not always exist for the entire solution domain.     

Trefftz method for Kirchhoff plate bending problems

Kirchhoff plate bending problems were studied by both the Trefftz direct and indirect approximations in which non-singular, complete Trefftz functions are used as the weighting and/or trial functions. The Trefftz direct method involved only the quantities of engineering interest. Numerical results are given to show the efficiency and the excellent accuracy of the present method.     

A Trefftz based method for solving Helmholtz problems in semi-infinite domains

The wave based method (WBM), which is based on an indirect Trefftz approach, is a deterministic prediction method posed as an alternative to the element-based methods. It uses wave functions, which are exact solutions of the underlying differential equation, to describe the dynamic field variables. In this way, it can avoid the pollution errors associated with the polynomial element-based approximations. As a consequence, a dense element discretization is no longer required, yielding a smaller numerical system. The resulting enhanced computational efficiency of the WBM as compared to the element-based methods has been proven for the analysis of both bounded and unbounded acoustic problems. This paper extends the applicability of the WBM to semi-infinite domains. An appropriate function set is proposed, together with a calculation procedure for both semi-infinite radiation and scattering problems, and transmission or diffraction problems containing a rigid baffle. The resulting technique is validated on two numerical examples.     

A multiple-scale Trefftz method for an incomplete Cauchy problem of biharmonic equation

In this paper we numerically solve both the direct and the inverse Cauchy problems of biharmonic equation by using a multiple-scale Trefftz method (TM). The approximate solution is expressed to be a linear combination of T-complete bases, and the unknown coefficients are determined to satisfy the boundary conditions, by solving a resultant linear equations system. We introduce a better multiple-scale in the T-complete bases by using the concept of equilibrated norm of the coefficient matrix, such that the explicit formulas of these multiple scales can be derived. The condition number of the coefficient matrix can be significantly reduced upon using these better scales; hence, the present multiple-scale Trefftz method (MSTM) can effectively solve the inverse Cauchy problem without needing of the overspecified data, which is an incomplete Cauchy problem. Numerical examples reveal the efficiency that the new method can provide a highly accurate numerical solution even the problem domain might have a corner singularity, and the given boundary data are subjected to a large random noise.     

Differential quadrature Trefftz method for irregular plate problems

Differential quadrature Trefftz method (DQTM) is developed to deal with plate problems defined in irregular domains. DQTM divides the solution into two parts, a particular solution for inhomogeneous biharmonic equation and the general solution for homogeneous biharmonic equation. For the former, differential quadrature method based on the interpolation of the highest derivative (DQIHD) is involved. For the latter, polynomial basis functions are adopted instead of fundamental solutions. We will show that DQTM not only keeps the advantages of traditional differential quadrature method (DQM), high efficiency and accuracy, but also has no difficulties to deal with geometrically irregular domains.     

Comparison of different methods for choosing the collocation points in the boundary collocation method for 2D-harmonic problems with special purpose Trefftz functions

Based on special purpose Trefftz functions, this paper presents a comparison of different methods for choosing collocation points when a boundary collocation method is used to fulfill the boundary conditions. First, it is shown that for some geometries when applying a boundary collocation method with special purpose Trefftz functions, equidistant collocation points give unacceptable results. Next, for four 2-D harmonic boundary value problems, four cases of different placements of the collocation points are described and compared. Based on the results of the comparison of the test methods for the location of the collocation points it is shown that one of the proposed methods for placing the collocation points is simple to implement and accurate. The essence of this method is an adaptive determination of the consequent collocation points.     

Sensitivity analysis scheme of boundary value problem of 2D Poisson equation by using Trefftz method

This paper describes the sensitivity analysis of the boundary value problem of two-dimensional Poisson equation by using Trefftz method. A non-homogeneous term of two-dimensional Poisson equation is approximated with a polynomial function in the Cartesian coordinates to derive a particular solution. The unknown function of the boundary value problem is approximate with the superposition of the T-complete functions of Laplace equation and the derived particular solution with unknown parameters. The parameters are determined so that the approximate solution satisfies boundary conditions. Since the T-complete functions and the particular solution are regular, direct differentiation of the expression results leads to the sensitivity expressions. The boundary-specified value and the shape parameter are taken as the variables of the sensitivity analysis to formulate the sensitivity analysis methods. The present scheme is applied to some numerical examples in order to confirm the validity of the present algorithm.