Trefftz Method Research Articles

Some remarks on Trefftz type approximations

For the Trefftz method sequences of linearly independent functions satisfying the governing differential equations are needed. For two- and three-dimensional elasticity problems some useful options for obtaining Trefftz trial functions are discussed. The discussion also includes some very useful particular solutions. For three-dimensional elasticity problems four methods are considered: the displacement representation of Papkovich/Neuber, Piltner's complex representation, a hypercomplex displacement representation of Bock, Gürlebeck, Weisz-Patrault, Legatiuk, and H.M. Nguyen, as well as Slobodyanskii’s representation written in real and complex form. For two-dimensional problems the option of using discretized Cauchy integrals is illustrated. Very briefly Trefftz type Radial Basis Functions are mentioned satisfying a homogeneous or inhomogeneous differential equation. This paper is not meant as a complete survey or review on Trefftz methods. The paper presents a collection of personal choices to help future developers of numerical methods based on Trefftz trial functions.

Thermal analysis of the car windscreen

The task of the ventilation and heating/cooling system in cars is to maintain the thermal comfort conditions in the passenger compartment to ensure safe driving. Designing such systems requires knowledge of many physical parameters, which often have to be specified individually. Such factors include heat transfer coefficients. The paper presents the results concerning the determination of the local heat transfer coefficients at the interface between the car windscreen and the cooling air from the inner side of the passenger compartment. The temperature of the outer side of the vehicle windscreen was measured using infrared thermography. The 2D mathematical approach describing the steady state heat transfer on a car windscreen was proposed. The temperature distribution was determined by the Trefftz method, and the heat transfer coefficient at the air–vehicle windscreen interface was calculated using the third type boundary condition. Graphs were used to represent thermographic images of the vehicle windshield, its temperature distribution, and corresponding values of local heat transfer coefficients as a function the length of the windshield. Results are presented and discussed.

A Novel Boundary-Type Meshless Method for Solving the Modified Helmholtz Equation

This paper presents a novel boundary-type meshless method for solving the two-dimensional modified Helmholtz equation in multiply connected regions. Numerical approximation is obtained by the superposition principle of the non-singular basis functions satisfied the governing equation. The advantage of the proposed method is that the locations of the source points are not sensitive to the results. The novel concept may resolve the major issue for the method of fundamental solutions (MFS). In contrast to the collocation Trefftz method (CTM), the Trefftz order of the non-singular basis functions can be reduced since the multiple source points are adopted. To solve the physical problems in a doubly and multiply connected regions, the boundary-type meshless method with the incorporation of the domain decomposition method (DDM) is proposed. The numerical examples for the validation are conducted including simply, doubly, and multiply connected regions. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient for modeling modified Helmholtz equation. The results also reveal that it resolves the restrictions of the MFS and the CTM for the application in the engineering problems.

Modifications of the Trefftz Method for Estimating the Contribution of the Hall Effect on Magnetotelluric Sounding

Abstract —Possible manifestations of the Hall effect in the Earth’s magnetic field during magnetotelluric sounding are considered. Numerical calculations of the magnitude the effect for a three-dimensional heterogeneous earth, using modifications of the Trefftz method suitable for accounting for anisotropy are made for. Versions of the measurement that allow easy detection of manifestations of the Hall effect are analyzed.

Determining 2D temperature field in flow boiling with the use of Trefftz functions

The paper proposes the use of Trefftz method to solve the triple coupled heat conduction problem in flow boiling of refrigerant in an asymmetrically heated minichannel. A mathematical model of heat transfer in a rectangular minichannel is suggested. Two sets of Trefftz functions were used to determine 2D temperature fields at a fluid flow in the minichannel tilted at a known angle. The procedure for the calculation of the liquid temperature was coupled with the process of determining temperature fields in two adjacent elements of the experimental stand with the minichannel, i.e. in the glass pane and the heating foil. Heat transfer in the glass, foil and liquid is described using various 2D differential equations with an adequate set of boundary conditions. Solving those equations led to the solving of the triple coupled heat conduction problem made up of one direct and two subsequent inverse problems. The results are presented as: (1) 2D temperature of the glass pane, the heating foil, the flowing liquid, (2) mean square errors between temperature approximations and selected boundary conditions, (3) the heat transfer coefficient versus the distance from the minichannel inlet.

Comparison of the 1D and 2D calculation models used for determination of the heat transfer coefficient during flow boiling heat transfer in a minichannel

The paper discusses the results of the flow boiling heat transfer in a vertical minichannel with rectangular cross-section. The heating element for FC-72 flowing in the minichannel is a thin plate. Infrared thermography is used to determine changes in the temperature on its outer side. The aim of thecalculation is to determine the heat transfer coefficient using 1D and 2D calculation models. Local values of heat transfer coefficient on the surface between the heated plate and boiling fluid are calculated from the Newton`s and Fourier`s laws. In 2D model the plate temperature distribution is obtained by solving the inverse heat conduction problem. The governing equation is solved by means of two methods: the non-continuous Trefftz method and the Beck method. The results are presented as plate temperature and heat transfer coefficient calculated using 1D and 2D models as a function of the distance fromthe minichannel inlet. The analysis of the results revealed that the values and distributions of the heat transfer coefficient calculated by means of both models were similar. This suggests that all mentioned methods are interchangeable.

Theoretical and numerical analysis of selected hybrid Trefftz method with the use of method of subdomains coupling based on “frame” concept

The aim of this paper is theoretical and numerical analysis of Trefftz method in Galerkin version, which symbol is O-S;T-T.The analysis was conducted on the example of two-dimensional Poisson’s problem. Domain of interest was divided into homogeneous subdomains. In this way hybrid method was obtained, that is a combination of the boundary method with conventional finite element known from the Finite Element Method. The method based on idea of “frames” was used to couple subdomains. Finally, the results of numerical experiments obtained for two boundary value problems were presented. The accuracy of the method was analyzed depending on the number of Trefftz coefficients.

Coupling finite elements and auxiliary sources for Maxwell's equations

AbstractThe Multiple Multipole Program is a Trefftz method approximating the electromagnetic field in a domain filled with a homogeneous linear medium. MMP can easily handle unbounded domains; yet, it cannot accommodate inhomogeneous or nonlinear materials, situations well within the scope of the standard finite element method. We propose to couple FEM and MMP to model Maxwell's equations for materials with spatially varying properties in an unbounded domain. In some bounded parts of the domain, we use Nédélec's first family of curl‐conforming elements; in the unbounded complement, multipole expansions. Several approaches are developed to couple both discretizations across the common interface: Least‐squares–based coupling using techniques from PDE‐constrained optimization. Multifield variational formulation in the spirit of mortar finite element methods. Discontinuous Galerkin coupling between the FEM mesh and the single‐entity MMP subdomain. Coupling by tangential components traces. We study the convergence of these approaches in a series of numerical experiments.

A novel space–time meshless method for solving the backward heat conduction problem

This paper presents a novel space–time meshless method for solving the backward heat conduction problem (BHCP). A numerical approximation is obtained using the Trefftz basis function of the heat equation. The Trefftz method, which differs from conventional collocation methods based on a set of unstructured points in space, is used in this study to collocate boundary points in the space–time coordinate system such that the initial and boundary conditions can both be treated as boundary conditions on the space–time domain boundary. Because the solution in time on the boundary of the domain is unknown, the BHCP can be transformed into an inverse boundary value problem. The numerical solution is obtained by superpositioning the Trefftz base functions that automatically satisfy the governing equation. The validity of the proposed method is established for several test problems, including the one-dimensional BHCP and two-dimensional BHCP. The accuracy of the proposed method is compared with that of a conventional time-marching scheme based on the finite difference method. The results demonstrate that highly accurate numerical solutions can be obtained and errors may not accumulate over the entire time domain. Moreover, the boundary data on the inaccessible boundary can be recovered even when the partial data on the final time boundary are absent.

Trefftz method in an inverse problem of two-phase flow boiling in a minichannel

The paper discusses the application of the Trefftz method for solving a coupled inverse problem concerning heat conduction and flow boiling. The T-complete functions were derived for a two-dimensional stationary energy equation in a flowing fluid, with a known velocity profile. The properties of these functions are described, with a focus on their relationship to harmonic polynomials. In view of the problem domain (two adjacent regions with different physical parameters), two subsequent inverse problems were formulated. The former was solved by approximating the temperature with a combination of harmonic polynomials while for the latter problem a twin-track approach was taken. The first method employed was the Trefftz method with newly generated functions. For comparison, the calculation was repeated using the Picard iterative process combined with the Trefftz method based on harmonic functions. The numerical calculations were conducted using experimental data concerning flow boiling of a refrigerant through a rectangular vertical minichannel. The computation resulted in obtaining two-dimensional temperature distributions (in each region) and a heat transfer coefficient at the solid-fluid interface versus the distance from the minichannel inlet. Both numerical approaches gave comparable results. The experimental heat transfer coefficient was compared with some correlations known from the literature.

Application of conformal mappings and the numerical analysis of conditioning of the matrices in Trefftz method for some boundary value problems

Provided that the boundary value problem can be equivalently solved in conformally transformed domain, one can consider its mapping into unit disk domain. The problem can be solved for the new geometry and the solution can be retransformed back to the original problem domain. The paper presents this approach in order to validate whether it can be applied for improving ill-conditioning of the matrices and solutions obtained in indirect Trefftz methods.

A Novel Hybrid Boundary-Type Meshless Method for Solving Heat Conduction Problems in Layered Materials

In this article, we propose a novel meshless method for solving two-dimensional stationary heat conduction problems in layered materials. The proposed method is a recently developed boundary-type meshless method which combines the collocation scheme from the method of fundamental solutions (MFS) with the collocation Trefftz method (CTM) to improve the applicability of the method for solving boundary value problems. Particular non-singular basis functions from cylindrical harmonics are adopted in which the numerical approximation is based on the superposition principle using the non-singular basis functions expressed in terms of many source points. For the modeling of multi-layer composite materials, we adopted the domain decomposition method (DDM), which splits the domain into smaller subdomains. The continuity of the flux and the temperature has to be satisfied at the interface of subdomains for the problem. The validity of the proposed method is investigated for several test problems. Numerical applications were also carried out. Comparison of the proposed method with other meshless methods showed that it is highly accurate and computationally efficient for modeling heat conduction problems, especially in heterogeneous multi-layer composite materials.

THE TREFFTZ TEST FUNCTIONS METHOD FOR SOLVING THE GENERALIZED INVERSE BOUNDARY VALUE PROBLEMS OF LAPLACE EQUATION

The issue of data completion is important for the elliptic type partial differential equation. In the inverse Cauchy problem, we need to complete the boundary data by over-specifying Dirichlet and Neumann data on a portion of the boundary. In this paper, we numerically solve the generalized inverse boundary value problems of Laplace equation in a rectangle with one boundary function and two boundary functions missing, which are more difficult than the inverse Cauchy problem. By using the technique of a boundary integral equation method together with a specially designed Trefftz test function, we can complete the boundary data by requiring minimal extra data. Then solving the Laplace equation with the given data and recovered data by the multiple-scale Trefftz method, we can find the numerical solution in the interior nodal points.

Many names of the Trefftz method

The Trefftz method is understood as an approximate method for solving boundary value problems in which the approximate solution is a linear combination of trial functions satisfying exactly the governing equation. The method was proposed by Erich Trefftz in Zürich in 1926 as an alternative to the Ritz method in which the boundary conditions are satisfied exactly. However such an approach was applied already in some previous papers. There are also examples in the publications published after 1926 in which the authors used this approach however they did not mention the name “Trefftz method”. There are also different modifications of the Trefftz-type methods in which the authors proposed new names because of these modifications. The aim of this paper is to present many variations of the Trefftz method and to collect the names under which they were published, before the original paper of Erich Trefftz in 1926 and after it.

Space–time Trefftz-DG approximation for elasto-acoustics

ABSTRACTDiscontinuous finite element methods have proven their numerical accuracy and flexibility, but they are criticized for requiring high number of degrees of freedom for computation. There have been some advances with Hybridizable Discontinuous Galerkin approximations, but essentially in the time-harmonic regime, because the time integration requires implicit schemes. Another possibility to explore is Trefftz method, which distinguishes itself by the choice of basis functions: they are local solutions of initial equation. Thus, in case of time-dependent problems, space-time meshes are required. Classical Trefftz-type approximation uses the exact solutions of acoustic and elastodynamic systems taken with different frequencies, in order to obtain better numerical errors. We have computed a polynomial basis using the Taylor expansions of generating exponential functions that are the exact solutions of the initial acoustic and elastodynamic systems. This basis, compared to the classical one based on trigonometric functions, requires less degrees of freedom for the same level of accuracy. In the present work, we develop the theory for coupled elasto-acoustic system. We confirm well-posedness of the problem, based on error estimates in mesh-dependent norms. We consider a space-time polynomial basis to implement numerically the method and to perform some numerical experiments showing promising results.

Heat Transfer Coefficient Identification in Mini-Channel Flow Boiling with the Hybrid Picard–Trefftz Method

This paper summarizes the results of the flow boiling heat transfer study with ethanol in a 1.8 mm deep and 2.0 mm wide horizontal, asymmetrically heated, rectangular mini-channel. The test section with the mini-channel was the main part of the experimental stand. One side of the mini-channel was closed with a transparent sight window allowing for the observation of two-phase flow structures with the use of a fast film camera. The other side of the channel was the foil insulated heater. The infrared camera recorded the 2D temperature distribution of the foil. The 2D temperature distributions in the elements of the test section with two-phase flow boiling were determined using (1) the Trefftz method and (2) the hybrid Picard–Trefftz method. These methods solved the triple inverse heat conduction problem in three consecutive elements of the test section, each with different physical properties. The values of the local heat transfer coefficients calculated on the basis of the Robin boundary condition were compared with the coefficients determined with the simplified approach, where the arrangement of elements in the test section was treated as a system of planar layers.

Trefftz methods with cracklets and their relation to BEM and MFS

In this paper we consider Trefftz methods which are based on functions defined by single layer or double layer potentials, integrals of the fundamental solution, or their normal derivative, on cracks. These functions are called cracklets, and satisfy the partial differential equation, as long as the crack support is not placed inside the domain. A boundary element method (BEM) interpretation is to consider these cracks as elements of the original boundary, in a direct BEM approach, or elements of an artificial boundary, in an indirect BEM approach. In this paper we consider the cracklets just as basis functions in Trefftz methods, as the method of fundamental solutions (MFS). We focus on the 2D Laplace equation, and establish some comparisons and connections between these methods with cracklets and standard approaches like the BEM, indirect BEM, and the MFS. Namely, we propose the enrichment of the MFS basis with the cracklets. Several numerical simulations are presented to test the performance of the methods, in particular comparing the results with the MFS and the BEM.

Identification of the heat transfer coefficient during cooling process by means of Trefftz method

In the article, the Trefftz method, belonging to the analytical-numerical methods, was used to identify the heat transfer coefficient for Inconel during spray cooling at the part of boundary. A mathematical model of the process has been formulated using cylindrical coordinate. The heat transfer coefficient is identified based of the Robin condition at the cooled boundary. Trefftz functions for heat conduction equation in cylindrical dimensionless coordinates has been used to find an approximate solution of the problem. As the internal temperature responses the real data from measurements has been used. The result of calculation is similar to the obtained by the other method.

A Trefftz/MFS mixed-type method to solve the Cauchy problem of the Laplace equation

By using the multiple-scale Trefftz method (MSTM) to solve the Cauchy problem of the Laplace equation in an arbitrary bounded domain, we may lose the accuracy several orders when the noise being imposed on the specified Cauchy data is quite large. In addition to the linear equations obtained from the MSTM, the fundamental solutions play as the test functions being inserted into a derived boundary integral equation. Therefore, after merely supplementing a few linear equations in the mixed-type method (MTM), which is a well organized combination of the Trefftz method and the method of fundamental solutions (MFS), we can improve the ill-conditioned behavior of the linear equations system and hence increase the accuracy of the solution for the Cauchy problem significantly, as explored by two numerical examples.

A Trefftz collocation method for multiple interacting spherical nano-inclusions considering the interface stress effect

In this study, a Trefftz collocation method (TCM) is proposed for modeling multiple interacting nano-scale spherical inhomogeneities considering the interface stress effect. The Papkovich–Neuber (P–N) general solutions are used as Trefftz trial functions, which are expressed in terms of spherical harmonics. Non-singular harmonic functions, and singular harmonics from multiple source points are included, facilitating the study of multiple inclusions. Characteristic lengths are used to scale the Trefftz trial functions, to avoid ill-conditioning of the derived system of linear equations. The collocation method is used to enforce boundary conditions. The displacement continuity and the stress jump across the matrix/inclusion interface, which is described by the generalized Young–Laplace equation for solids, are also enforced by the collocation method. Numerical results by the proposed Trefftz method agree well with the available analytical solutions in the literature. The stress distributions of solids containing nano-inhomogeneities show significant size-dependency, in contrast to those for composites without considering the interface stress effect. Interactions of multiple nano-inclusions are also studied, which can be used as benchmark solutions in future studies.