Trefftz Method Research Articles

The polynomial Trefftz method for solving backward and inverse source wave problems

The Trefftz method is a truly meshless boundary-type method, because the trial solutions automatically satisfy the governing equation. In order to stably solve the high-dimensional backward wave problems and the one-dimensional inverse source problems, we develop a multiple-scale polynomial Trefftz method (MSPTM), of which the scales are determined a priori by the collocation points. The MSPTM can retrieve the missing initial data and unknown time varying wave source. The present method can also be extended to solve the higher-dimensional wave equations long-term through the introduction of a director in the two-dimensional polynomial Trefftz bases. Several numerical examples reveal that the MSPTM is efficient and stable for solving severely ill-posed inverse problems of wave equations under large noises.

Trefftz method in solving Fourier-Kirchhoff equation for two-phase flow boiling in a vertical rectangular minichannel

This paper presents the results of investigations into flow boiling heat transfer in an asymmetrically heated vertical minichannel of 1.7 mm depth. The heated element for FC-72 flowing in the minichannel was an alloy plate 0.45 mm thick, microstructured on one side, in direct contact with the flowing fluid. The computational part of the study contains approximate steady state solutions of the heat transfer problems described by Poisson.s equation and the energy equation for the heated plate and the fluid, respectively. For both equations, the boundary conditions were specified on the basis of experimental data. Temperature of the outer plate surface, measured by infrared thermography, and heat losses to ambient air were included in the calculations. For the energy equation we assumed parabolic profile of fluid velocity and the equality of temperatures and heat fluxes at the interface between the heated surface and the fluid. The void fraction was taken from a single-phase flow model. Two-dimensional temperature distributions were obtained by the Trefftz method and, due to the Robin condition at the interface between them, it was possible to calculate the heat transfer coefficient. Its values were compared to those obtained by other correlations known from literature.

Computer simulation of the effective viscosity in Brinkman filtration equation using the Trefftz method

Computer simulation of the effective viscosity in Brinkman filtration equation using the Trefftz method

Radial basis functions in mathematical modelling of flow boiling in minichannels

The paper addresses heat transfer processes in flow boiling in a vertical minichannel of 1.7 mm depth with a smooth heated surface contacting fluid. The heated element for FC-72 flowing in a minichannel was a 0.45 mm thick plate made of Haynes-230 alloy. An infrared camera positioned opposite the central, axially symmetric part of the channel measured the plate temperature. K-type thermocouples and pressure converters were installed at the inlet and outlet of the minichannel. In the study radial basis functions were used to solve a problem concerning heat transfer in a heated plate supplied with the controlled direct current. According to the model assumptions, the problem is treated as twodimensional and governed by the Poisson equation. The aim of the study lies in determining the temperature field and the heat transfer coefficient. The results were verified by comparing them with those obtained by the Trefftz method.

A Trefftz Polynomial Space-Time Discontinuous Galerkin Method for the Second Order Wave Equation

A new space-time discontinuous Galerkin (dG) method utilizing special Trefftz polynomial basis functions is proposed and fully analyzed for the scalar wave equation in a second order formulation. The dG method considered is motivated by the class of interior penalty dG methods, as well as by the classical work of Hughes and Hulbert [Comput. Methods Appl. Mech. Engrg., 66 (1988), pp. 339--363; Comput. Methods Appl. Mech. Engrg., 84 (1990), pp. 327--348]. The choice of the penalty terms included in the bilinear form is essential for both the theoretical analysis and for the practical behavior of the method for the case of lowest order basis functions. A best approximation result is proven for this new space-time dG method with Trefftz-type basis functions. Rates of convergence are proved in any dimension and verified numerically in spatial dimensions d = 1 and d = 2. Numerical experiments highlight the effectivness of the Trefftz method in problems with energy at high frequencies.

Trefftz function-based thermal solution of inverse problem in unsteady-state flow boiling heat transfer in a minichannel

The paper shows the results for flow boiling heat transfer in a 1.7mm deep minichannel vertically oriented with FC-72 Fluorinert as a working fluid. The heated wall of the minichannel was formed with a thin foil. Thermocouples mounted at 18 points monitored the outer foil surface temperature. All experimental parameters were controlled using data acquisition stations. The measurements were performed at 0.01s intervals. The observations of the flow structures were carried out concurrently on the internal surface of the foil contacting the fluid. The aim of the numerical calculations was to determine the heat transfer coefficient on the contact surface between the heated foil and FC-72. The heat transfer coefficient was calculated with the use of the Robin boundary condition. To do that, the foil and fluid temperatures and foil temperature gradient had to be known. The foil temperature was found by solving an unsteady-state two-dimensional inverse boundary problem with the use of the Trefftz method in time-space subdomains. A linear combination of Trefftz functions was used to approximate the foil temperature. The unknown coefficients of the Trefftz function linear combination were determined by minimizing the functional. Error propagation in time and mean relative differences determined between the measured and computed heated foil temperatures were presented.

A novel Trefftz method of the inverse Cauchy problem for 3D modified Helmholtz equation

We solve an overspecified as well as underspecified inverse Cauchy problems of the modified Helmholtz equation in an arbitrary 3D-bounded domain with a smooth boundary using a multiple/scale/direction Trefftz method (MSDTM), of which the directions are uniformly distributed on , and the scales are determined a priori by the collocation points on boundary to satisfy the specified boundary conditions. The three-dimensional numerical examples confirm the efficiency of the MSDTM. Although under a large noise up to 30%, the solutions of inverse Cauchy problems are quite accurate. More importantly, the present method converges very fast and saves much CPU time.

Comparison of different approaches in the Trefftz method for analysis of fluid flow between regular bundles of cylindrical fibres

In the paper three different approaches for the Trefftz method are compared in analysis of the fluid flow between regular bundles of cylindrical fibres. The approximate solution is a linear combination of such trial functions which fulfil exactly the governing equations. The trial functions can be defined in the Cartesian coordinate system (the first approach), in the cylindrical coordinate system (and can fulfil also some boundary conditions - the second approach) or be defined as a fundamental solutions (the third approach - the method of fundamental solutions).The average velocity and the product of the friction factor and the Reynolds number ƒ ⋅ Re are compared for selected parameters of a considered region.

An application of the non-continuous Trefftz method to the determination of heat transfer coefficient for flow boiling in a minichannel

The paper presents an application of the semi-analytical method, called the non-continuous Trefftz method, to the calculation of the heat transfer coefficients. It is very effective method for solving direct and inverse problems. The results obtained by this method are consistent with the results obtained by using complicated methods: the FEM and Beck method. Sought local heat transfer coefficients between the heating surface and the boiling liquid flowing through 1 mm deep minichannel were calculated from the Robin boundary condition. The temperature of the heating surface and the derivative of the temperature were was found from solving the inverse problem. The study is limited to the identification of the heat transfer coefficient in the subcooled and the saturated nucleate boiling regions. The article presents also the measurement stand and methodology of conducting the experiment. Presented issues allows verification of state-of-the-art methods of solving the inverse problem by using the authors’ empirical data from the experiment.

Multipole Trefftz method implementation in free vibration of a graphene sheet with multiple vacancy defects

The single and multiple vacancy defects in a single layer graphene sheet can affect its mechanical properties. Also, these defects can be inverted to greater holes as amorphous construction when the graphene sheet is working in room temperature. In some cases, these defects can be modelled as circular holes. In this article, multipole Trefftz method as well as translational addition theorem are employed to consider effects of circular defects on behavior of freely vibrating circular graphene sheet. Nonlocal thin plate theory is used to solve the problem. According to the direct-searching approach, the natural frequencies are determined as the minimum singular value of the truncated finite system using the singular value decomposition (SVD) technique. The computational efficiency and convergence of the present analytical method are examined. Accuracy and stability of results are validated by literature. Effects of boundary conditions, geometrical properties and nonlocal parameter on vibrational modes are investigated. Results reveal that the eccentricity, nonlocality and number of holes have significant effects on the natural frequencies.

Extension of the variational theory of complex rays to orthotropic shallow shell structures

Nowadays, the interest of aerospace and automotive industries on virtual testing of medium-frequency vibrational behavior of shallow shell structures is growing. The development of software capable of predicting the vibrational response in such frequency range is still an open question because classical methods (i.e., FEM, SEA) are not fully suitable for the medium-frequency bandwidth. In this context the Variational Theory of Complex Rays (VTCR) is taking place as an ad-hoc technique to address medium-frequency problems. It is a Trefftz method based on a weak variational formulation. It allows great flexibility because any shape function that satisfies the governing equations can be used. This work further develops such theory. In particular, orthotropic materials are introduced in the VTCR formulation for shallow shell structures. A significant numerical example is proposed to show the strategy.

Trefftz method for a polynomial-based boundary identification in two-dimensional Laplacian problems

The paper addresses a two-dimensional boundary identification (reconstruction) problem in steady-state heat conduction. Having found the solution to the Laplace equation by superpositioning T-complete functions, the unknown boundary of a plane region is approximated by polynomials of an increasing degree. The provided examples indicate that sufficient accuracy can be obtained with a use of polynomials of a relatively low degree, which allows avoidance of large systems of nonlinear equations. Numerical simulations for assessing the performance of the proposed algorithm show better than 1% accuracy after a few iterations and very low sensitivity to small data errors.

Numerical solutions of elliptic partial differential equations using Chebyshev polynomials

We present a simple and effective Chebyshev polynomial scheme (CPS) combined with the method of fundamental solutions (MFS) and the equilibrated collocation Trefftz method for the numerical solutions of inhomogeneous elliptic partial differential equations (PDEs). In this paper, CPS is applied in a two-step approach. First, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then the resulting homogeneous equation is solved by boundary type methods including the MFS and the equilibrated collocation Trefftz method. Numerical results for problems on various irregular domains show that our proposed scheme is highly accurate and efficient.

Convection-adapted BEM-based FEM

We present a new discretization method for homogeneous convection-diffusion-reaction boundary value problems in 3D that is a non-standard finite element method with PDE-harmonic shape functions on polyhedral elements. The element stiffness matrices are constructed by means of local boundary element techniques. Our method, which we refer to as a BEM-based FEM, can therefore be considered a local Trefftz method with element-wise (locally) PDE-harmonic shape functions. The Dirichlet boundary data for these shape functions is chosen according to a convection-adapted procedure which solves projections of the PDE onto the edges and faces of the elements. This improves the stability of the discretization method for convection-dominated problems both when compared to a standard FEM and to previous BEM-based FEM approaches, as we demonstrate in several numerical experiments.

An analytic adjoint Trefftz method for solving the singular parabolic convection–diffusion equation and initial pollution profile problem

In this paper we develop a novel method for the solution of a singularly perturbed convection–diffusion equation by inserting the adjoint Trefftz functions as test functions into the derived domain/boundary integral equation. The trial solution is expressed in terms of the Trefftz series (i.e., the spatial part of the Trefftz test functions), endowing with time-dependent expansion coefficients. With the aid of the weighting orthogonality of the Trefftz series we can find these coefficients in closed-form, which are integrals of initial condition, boundary conditions and source function. Hence, we have an analytical method to find the singular solution, very close to the exact one if more series terms are used. On the other hand, when the recovery of initial pollution profile by using the measured data at a final time is concerned, the number of the Trefftz series must be suitably chosen to overcome the ill-posed behavior of the initial pollution profile problem. The present results are better than that obtained by the Lie-group shooting method.

Trefftz method for an inverse geometry problem in steady-state heat conduction

Trefftz method for an inverse geometry problem in steady-state heat conduction

Bending Solution for Simply Supported Annular Plates Using the Indirect Trefftz Boundary Method

This paper presents the bending analysis of annular plates by the indirect Trefftz boundary approach. The formulation for thin and thick plates is based on the Kirchhoff plate theory and the Reissner plate theory. The governing equations are therefore a fourth-order boundary value problem and a sixth-order boundary value problem, respectively. The Trefftz method employs the complete set of solutions satisfying the governing equation. The main benefit of the Trefftz boundary method is that it does not involve singular integrals because of the properties of its solution basis functions. It can therefore be classified into the regular boundary element method. The present method is simple and efficient in comparison with the other methods. In addition, the boundary conditions can be embedded in this method. Finally, several numerical examples are shown to illustrate the efficiency and simplicity of the current approach.

A multiple-direction Trefftz method for solving the multi-dimensional wave equation in an arbitrary spatial domain

In this paper we first express the wave equation in terms of the Minkowskian polar coordinates and generate a set of complete hyperbolic type Trefftz bases: rkcosh⁡(kθ) and rksinh⁡(kθ), which are further transformed to wave polynomials as the trial solution bases for the one-dimensional wave equation. In order to stably solve the wave propagation problems long-term we develop a multiple-scale Trefftz method (MSTM), of which the scales are determined a priori by the collocation points. Then we derive a very simple method of multi-dimensional wave polynomials, equipped with different spatial directions which being the normalized wavenumber vectors, as the polynomial Trefftz bases for solving the multi-dimensional wave equations, which is named a multiple-direction Trefftz method (MDTM). Several numerical examples of two- and three-dimensional wave equations demonstrate that the present method is efficient and stable.

Temperature dependent thermal conductivity determination and source identification for nonlinear heat conduction by means of the Trefftz and homotopy perturbation methods

The homotopy perturbation method (HPM) combined with Trefftz method is employed to find the solution of two kinds of nonlinear inverse problems for heat conduction. The first one is a coefficient inverse problem. The thermal conductivity coefficient, with the prescribed boundary conditions on a part of the boundary and with some measured or anticipated values of the solution in some inner points, is determined. The thermal conductivity is assumed to have a form of a linear function of temperature with unknown coefficients. The number of T-functions in HPM is chosen to obtain the best fitting of the approximate solution to the input data. Minimization of the difference between the input data and the approximate solution of the problem, leads to the values of coefficients describing in an approximate way the unknown coefficients. The second problem is determination of an unknown source term for nonlinear stationary heat conduction with prescribed boundary conditions and some measured or anticipated values of the solution in some inner points. The source term is assumed to have a form of a polynomial with unknown coefficients. Number of the coefficients determines the number of functions in HPM resulting from expansion of H(v,p) with respect to the parameter p in order to find the components of the approximate solution of the problem. The components consist of Trefftz functions for the linear parts of the resultant equations based on powers of p-terms. Minimization of the difference between the values prescribed or measured inside the considered domain and the approximate solution of the problem, leads to the values of coefficients describing in an approximate way the source term.

Experimental investigations and numerical modeling of 2D temperature fields in flow boiling in minichannels

The experimental purpose of the study was to simultaneously measure and record two-dimensional temperature distributions on the heating surface and the images of the corresponding two-phase flow structures. The study was conducted for FC-72 flow boiling in rectangular, vertical and asymmetrically heated minichannels with a high aspect ratio of 40, 20 and 13.3 (0.5, 1.0, 1.5mm deep, 20mm wide and 360mm long), for heat fluxes ranging from 58 to 132kWm−2, absolute pressure from 1.16 to 1.84bar and the mass flux from 185 to 1139kgm−2s−1. The flat heating surface was made of a thin rolled plate. The inlet temperature of the liquid was kept constant at 288K. The CCD camera, mounted on the heated side of the channel, produced color images of liquid crystals. High-speed camera images, taken through the glass window on the opposite side of the channel, were used to measure void fraction for selected cross-sections as a function of variable thermal and flow parameters. The ranges of characteristic two-phase flow structures were determined relative to the hydraulic diameter dh and local void fraction φ(x). The best-agree correlation was found for the relationship between the local void fraction and the local vapor quality in high aspect ratio minichannels.The theoretical part of the study focused on the development of flow boiling heat transfer model based on the Trefftz method. It was formulated to include the minimum number of experimental constants and to numerically solve the manifold inverse problems. The numerical procedure included solving two sequential inverse problems (in the heating foil and in the flowing liquid) and the accompanying direct problem (in the protecting glass pane). To obtain two-dimensional temperature distributions in the boiling liquid for bubbly and bubbly–slug flows, a triple coupled inverse problem was solved using the Trefftz method. The numerical solution was compared with the simplified one, assuming that the entire heat generated in the foil was transferred to the flowing refrigerant. Both solutions gave similar results.