Trefftz Method Research Articles

Development of a two-dimensional mathematical model of flow boiling heat transfer in micro- and minichannels

The paper concerns flow boiling heat transfer in micro- and minichannels. In the mathematical model, the steady state heat transfer process in a single asymmetrically heated minichannel was considered. Calculations with the use of Trefftz functions were based on the data from own experiments. The temperature of the heater and the refrigerant were assumed to satisfy the Laplace equations and the energy equation respectively. The problem was solved by the Trefftz method using two sets of Trefftz functions. The known heater and refrigerant temperature distributions were used to determine the heat transfer coefficient at the heater – refrigerant contact. To verify the proposed mathematical model, data from experiments were applied to calculations. The essential part of the experimental stand was the test section which comprises a minichannel heat sink. The heated element for Fluorinert FC-72 flowing along minichannels was a thin foil. The temperature of its outer side was measured using infrared thermography. Thermocouples and pressure transducers installed at the inlet and outlet of the test section monitored fluid temperature and pressure. Mass flow rate, the current supplied to the heater and the voltage drop were also recorded. The resulting graphs presented thermograms of measured temperature on outer surface of the heater, temperature distributions of fluid temperature and local values of the heat transfer coefficient.

Численные эксперименты двойственным методом нулевого поля в задаче Дирихле для уравнения Лапласа в эллиптических областях с эллиптическими отверстиями

Dual techniques have been used in many engineering papers to deal with singularity and ill-conditioning of the boundary element method (BEM). In the first part of the two-part article, our efforts were focused on studying the theoretical aspects of this problem, including the analysis of errors and the study of stability. We provided the theoretical analysis for Laplace equation in elliptic domains with elliptic holes. To bypass the degenerate scales of Dirichlet problems, the second and the first kinds of the null field methods (NFM) are used for the exterior and the interior boundaries, simultaneously. This approach is called the dual null field method (DNFM). This paper is the second part of the study. Numerical results for degenerate models of an elliptic domain with one elliptic hole at 𝑎 + 𝑏 = 2 are carried out to verify the theoretical analysis obtained. The collocation Trefftz method (CTM) is also designed for comparisons. Both the DNFM and the CTM can provide excellent numerical performances. The convergence rates are the same but the stability of CTM is excellent and can achieve the constant condition numbers, Cond = 𝑂(1).

Buckling and wrinkling of thin membranes by using a numerical solver based on multivariate Taylor series

Buckling and wrinkling of thin structures often lead to very complex response curves that are hard to follow by standard path-following techniques, especially for very thin membranes in a slack or nearly slack state. Many recent papers mention numerical difficulties encountered in the treatment of wrinkling problems, especially with path-following procedures and often these authors switch to pseudo-dynamic algorithms. Moreover, the numerical modeling of many wrinkles leads to very large size problems. In this paper, a new numerical procedure based on a double Taylor series is presented, that combines path-following techniques and discretization by a Trefftz method: Taylor series with respect to a load parameter (Asymptotic Numerical Method) and with respect to space variables (Taylor Meshless Method). The procedure is assessed on buckling benchmarks and on the difficult problem of a sheared rectangular membrane.

An efficient localized Trefftz-based collocation scheme for heat conduction analysis in two kinds of heterogeneous materials under temperature loading

This paper presents a novel localized collocation Trefftz method (LCTM) for heat conduction analysis in two kinds of heterogeneous materials (functionally graded materials and multi-medium materials) under temperature loading. In contrast to the conventional collocation Trefftz method (CTM), the proposed LCTM divides the whole domain into many stencil support domains consisting of several discretization nodes. An efficient technique, the generalized reciprocity method (GRM), is proposed to derive the problem-dependent T-complete functions for approximating the particular solution of the nonhomogeneous equations in the stencil support domains. Based on the moving least square (MLS) technique and T-complete functions, the LCTM numerical differentiation formulation at a certain node can be derived by using a linear combination of the T-complete functions at its adjacent discretization nodes in the related stencil support domain. It inherits the semi-analytical property from the conventional CTM and avoids the ill-conditioned dense matrix problem. Besides, the quadrant criterion is applied to guarantee the stability of the proposed LCTM, and the domain decomposition method (DDM) is introduced to divide the multi-medium domains into several single-medium sub-domains. Numerical results demonstrate the accuracy and efficiency of the proposed LCTM in comparison with the known analytical solutions and the finite element method (FEM) results.

Applications of the Trefftz method to the anti-plane fracture of 1D hexagonal piezoelectric quasicrystals

In terms of solving fracture mechanics problems of quasicrystals, the boundary element method known as Trefftz Method is used for the first time. In this paper, the collocation Trefftz method as an indirect Trefftz method has been demonstrated to be an excellent numerical method for the anti-plane fracture problems in quasicrystals plate, which can exactly satisfy the governing equations and approximately satisfy the boundary conditions. The algorithms of the collocation Trefftz method and the trial functions for the anti-plane fracture problems of piezoelectric quasicrystals are presented. The effects of the geometry size, the loadings and the materials properties on the fracture results are studied. For the materials with three dimensional constitutive relations, such as piezoelectric quasicrystals, the Trefftz method is an effective numerical method to solve the fracture problems.

Singularity problems from source functions as source nodes located near boundaries; numerical methods and removal techniques

Consider the Dirichlet problem for Laplace’s/Poisson’s equation in a bounded simply-connected domain S. The source function qln|PQ*¯| is a fundamental solution (FS), and it can be found in many physical problems. The singularity occurs when the boundary value data affected by qln|PQ*¯| as the source node Q* is located near the boundary Γ(=∂S). So far, there is no comprehensive study on this kind of singularity. In this paper, the solution singularity is explored and the reduced convergence rates are derived for the method of particular solutions (MPS) and the method of fundamental solutions (MFS). Classic domains, such as disks, ellipses and polygons, are discussed for analysis and computation. For this new kind of solution singularity, the convergence rates of the MFS and the MPS are very low. The errors caused by numerical integration are critical to the solution accuracy. A new analytic framework for the collocation Trefftz method (CTM) involving numerical integration is established in this paper; this is an advanced development of our previous study [19]. Since the numerical solutions are poor in accuracy, removal techniques are essential in applications. New removal techniques are proposed for a node Q* located near Γ. In this paper, an additional FS as, d0ln|PQ0¯|, is added to the original source nodes in the traditional MFS, and the point charge d0(=q) and the source node Q0 are unknowns to be sought by nonlinear solvers (such as the secant method). When the source node Q* is located inside S but near Γ, both simple domains (such as disks, ellipses and squares) and complicated domains (such as amoeba-like domains) are studied. The validity of the new removal techniques is supported by numerical experiments. The removal techniques in this paper may also be applied to solve source identification problems. A comprehensive study has been completed in this paper for the solitary source function qln|PQ*¯| as the source node Q* is located near Γ.

Trefftz Method of Solving a 1D Coupled Thermoelasticity Problem for One- and Two-Layered Media

This paper discusses a 1D one-dimensional mathematical model for the thermoelasticity problem in a two-layer plate. Basic equations in dimensionless form contain both temperature and displacement. General solutions of homogeneous equations (displacement and temperature equations) are assumed to be a linear combination of Trefftz functions. Particular solutions of these equations are then expressed with appropriately constructed sums of derivatives of general solutions. Next, the inverse operators to those appearing in homogeneous equations are defined and applied to the right-hand sides of inhomogeneous equations. Thus, two systems of functions are obtained, satisfying strictly a fully coupled system of equations. To determine the unknown coefficients of these linear combinations, a functional is constructed that describes the error of meeting the initial and boundary conditions by approximate solutions. The minimization of the functional leads to an approximate solution to the problem under consideration. The solutions for one layer and for a two-layer plate are graphically presented and analyzed, illustrating the possible application of the method. Our results show that increasing the number of Trefftz functions leads to the reduction of differences between successive approximations.

Numerical estimation of a Mode III fracture Mechanics parameter for three-dimensional elliptic boundary value problems with a crack singularity

In this study two spatial problems of solid bodies with a surface crack singularity (V-notch) are examined. These are problems in the domain of linear elastic fracture Mechanics which are reduced to Laplace equation problems by considering a Lamé potential. The boundary singularity is numerically treated by first determining the general solution of the governing equation, in the vicinity of the surface crack and by expressing it as an asymptotic expansion, the coefficients of which are approximated by polynomials. The remaining numerical steps are followed according to the singular function boundary integral method (SFBIM), which in the literature is known as one of the so-called Trefftz methods. Very fast convergence and very high accuracy are observed in implementing the method to solve a general 3-D model problem of a solid body having a cross section in the form of an ellipse and a specific spatial model problem of a steel rivet, which connects metal members of a structure. In fact, the CPU time and the numerical error recorded with this numerical technique are significantly smaller than those achieved with the finite element method (FEM) which was used to solve the same problems. The calculated value of Mode III fracture Mechanics parameter (FMP) indicates that there is no danger of crack propagation. Thus, the extension of the method to this category of problems is quite interesting since it involves a novel application of this algorithm in fracture Mechanics, a domain in which the method demonstrates its capability to treat problems with crack discontinuities and provide results more efficiently (less computing effort and greater accuracy) than the classical FEM with grid refinement. This advantage of the SFBIM can be exploited for research or design purposes, in several fields of applied Mechanics, by implementing its algorithm with subroutines embedded in engineering packages and aiming to make them more efficient in solving specific problems.

Time-Dependent Heat Transfer Calculations with Trefftz and Picard Methods for Flow Boiling in a Mini-Channel Heat Sink

The intensification of heat transfer using two-phase boiling flow in mini-channels is widely used to dissipate the high heat fluxes in miniaturized electronic devices. However, the process itself is not fully recognized and still requires experimental studies and developing computation methods appropriate for them. The main aim of this work was the mathematical modeling of time-dependent heat transfer process in FC-72 flow boiling in a mini-channel heat sink with five parallel mini-channels of 1 mm depth. Channels have an asymmetrically heated wall while its outer temperature was measured by infrared thermography. The opposite wall of the mini-channels was transparent, helping to record flow patterns due to a high-speed digital camera. The objective of the numerical calculations was to determine the heat transfer coefficient on the wall-fluid contact surface from the Robin boundary condition. The problem was solved using methods based on the Trefftz-type functions. Three mathematical methods were applied in calculations: the FEM with Trefftz type basis functions, the Classical Trefftz Method, and the Hybrid Picard-Trefftz Method. The results were compared with the values of the heat transfer coefficient obtained from theoretical correlations from the literature.

Application of the Trefftz Method for Pool Boiling Heat Transfer on Open Microchannel Surfaces

In the paper, the results of tests on pool boiling heat transfer on open microchannel surfaces were discussed. Nucleate pool boiling from copper microchannel surfaces was examined using ethanol as a working fluid. Parallel microchannels fabricated by machining were about 0.2 to 0.4 mm wide and 0.2 to 0.5 mm deep. In the mathematical model the heat transfer process was in steady state. The heat transfer in copper block was assumed axisymmetric while the specimen temperature distribution was not. The temperature of the copper block and the specimen were assumed to satisfy Laplace’s equations with adequate boundary conditions. The problem was solved by the Trefftz method with two sets of Trefftz functions. Graphs were used to represent the copper block and the specimen temperature distributions, local values of heat transfer coefficients as a function of the heat flux and fin temperature distribution. Uncertainty analysis was presented.

Analysis free surface of nonlinear seepage using the MQRBF method

In order to solve the characteristics of low accuracy and slow efficiency in traditional numerical solution the free surface problem, the multiquardatic radial base function collocation method(MQ RBF) is used to analyze the constant seepage and unsteady seepage of the homogeneous earth dam. Computation of transient problem of free surface of earth dam by the linear derivation of Richards equation. The results show that the calculation accuracy of the MQRBF is higher than that of the traditional numerical method. The solution process does not involve numerical integral calculation and grid reorganization, which greatly reduces the calculation amount. Compared with the Trefftz method, it has the advantage of solving boundary values and internal values at the same time. It is not limited by the solution of the Laplace equation, and its application is wider and simpler.

Amplitude-based Generalized Plane Waves: New Quasi-Trefftz Functions for Scalar Equations in two dimensions

Generalized plane waves (GPWs) were introduced to take advantage of Trefftz methods for problems modeled by variable coefficient equations. Despite the fact that GPWs do not satisfy the Trefftz pro...

The Equal-Norm Multiple-Scale Trefftz Method for Solving the Nonlinear Sloshing Problem with Baffles

In this paper, the equal-norm multiple-scale Trefftz method combined with the implicit Lie-group scheme is applied to solve the two-dimensional nonlinear sloshing problem with baffles. When considering solving sloshing problems with baffles by using boundary integral methods, degenerate geometry and problems of numerical instability are inevitable. To avoid numerical instability, the multiple-scale characteristic lengths are introduced into T-complete basis functions to efficiently govern the high-order oscillation disturbance. Again, the numerical noise propagation at each time step is eliminated by the vector regularization method and the group-preserving scheme. A weighting factor of the group-preserving scheme is introduced into a linear system and then used in the initial and boundary value problems (IBVPs) at each time step. More importantly, the parameters of the algorithm, namely, the T-complete function, dissipation factor, and time step, can obtain a linear relationship. The boundary noise interference and energy conservation are successfully overcome, and the accuracy of the boundary value problem is also improved. Finally, benchmark cases are used to verify the correctness of the numerical algorithm. The numerical results show that this algorithm is efficient and stable for nonlinear two-dimensional sloshing problems with baffles.

A variational domain decomposition scheme for the natural sloshing modes in the baffled tanks

A variational domain-decomposition scheme is presented to obtain the analytically approximate natural sloshing modes in the rigid baffled tanks. The scheme consists of three steps: (i) dividing the unperturbed complicated liquid domain into several simple sub-domains by introducing artificial interfaces, and auxiliary artificial non-dimensional densities of fluids in these sub-domains are also introduced; (ii) obtaining spectral matrix problem by combing the approximate variational formulation of the approximate natural sloshing problem with the Trefftz method, eigenvalues of the spectral matrix problem are computed and plotted; (iii) extracting the pursued analytically approximate natural sloshing modes from the eigenfunctions given by the spectral matrix problem with artificial non-dimensional densities sufficiently close to 1. The fluid velocity potential in each sub-domain is expressed as a linear combination of a set of harmonic functions with unknown weight coefficients. A variational principle for natural sloshing of multi-layer immiscible fluid with multi free surface is developed and modified to be employed in step (ii). The extraction of the pursued analytically approximate natural sloshing modes is realized by classifying the obtained spectral parameters into two classes, one of which is dominated by the free-surface wave effect, whereas the other one is dominated by the effect of the artificial interfaces between adjacent sub-domains. To verify the present scheme, the results obtained are compared with results published or given by finite element analysis.

Numerical solutions of two-dimensional Laplace and biharmonic equations by the localized Trefftz method

In this paper, the localized Trefftz method (LTM) is proposed to accurately and efficiently solve two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complex domains. The LTM is formed by combining the classical indirect Trefftz method and the localization approach, so the LTM, free from mesh and numerical quadrature, has great potential for solving large-scale problems. For problems in multiply-connected domains, the solutions expressions in the proposed LTM is much simpler and more compact than that in the conventional indirect Trefftz method due to the localization concept and the overlapping subdomains. In the proposed LTM, both of the interior nodes and boundary nodes are required and the algebraic equation at each node, represents the satisfaction of governing equation or boundary condition, can be derived by implementing the Trefftz method at every subdomain. By enforcing the satisfaction of governing equations at every interior node and of boundary conditions at every boundary node, a sparse system of linear algebraic equations can be yielded. Then, the numerical solution of the proposed LTM can be efficiently obtained by solving the sparse system. Several numerical examples in simply-connected and multiply-connected domains are provided to demonstrate the accuracy and simplicity of the proposed LTM. Besides, the extremely-accurate solutions of the LTM are simultaneously demonstrated.

Computing Group Velocities and Group-Velocity Dispersions of Optical Fibers Through Automatic Differentiation of Explicit Forms of Propagation Constants

We present a method to compute differential coefficients of the first and second orders, for propagation constants of nondegenerate and degenerate modes, propagating in an optical fiber to calculate dispersion characteristics of group velocities (GV) and group-velocity dispersions (GVD). This method automatically differentiates explicit forms of eigenvalues extracted from determinants of a generalized eigenvalue problem. This problem is converted from a nonlinear eigenvalue problem derived from the Trefftz method (e.g., transfer matrix and multipole methods) by the Sakurai-Sugiura method. We compute the differential coefficients of the second order of degenerate modes without computing eigenvectors and their first order differential coefficients in the generalized eigenvalue problems. Therefore, in parametric optimization, we can compute the differential coefficients of all propagating modes without remodeling the waveguide cross-section, as analyzing the whole cross-section avoids the symmetric conditions derived from the numerical model. Computed results with the proposed method for step-index and holey fibers validate the method and its effectiveness. In particular, our results show that our method computes the GV and GVD more accurately than computation using numerical differentiation.

Trefftz collocation method for two‐dimensional strain gradient elasticity

AbstractIndirect Trefftz method is proposed for solving two‐dimensional boundary value problems of the strain gradient elasticity theory (SGET). A system of trial functions satisfying the fourth‐order equilibrium equations of SGET are developed based on the generalized Papkovich‐Neuber potentials. The classical part of the displacement solution is represented through the T‐complete system of functions satisfying the Laplace equation. The gradient part of the solution is represented through the system of heuristic functions satisfying the Helmholtz equation. The least squares collocation method is used to enforce the boundary conditions. Numerical examples are presented for the square domain under non‐uniform tensile and bending loads. It is shown, that the advantage of the presented method is that it allows to directly control the accuracy of the fulfillment of all nonstandard boundary conditions, that are prescribed in SGET on the surfaces and edges of the body.

A novel localized collocation solver based on Trefftz basis for potential-based inverse electromyography

This paper introduces a novel localized collocation Trefftz method (LCTM) for potential-based inverse electromyography (PIE). PIE is a noninvasive technique to calculate the internal electrical potentials from measured body surface electromyographic data, which can be considered as an inverse Cauchy problem with potential equation. In the proposed LCTM, the electrical potential at every node is expressed as a linear combination of 3D Trefftz basis in each stencil support, and the sparse linear system is yield by satisfying governing equation at interior nodes and boundary conditions at boundary nodes. The proposed LCTM inherits the properties of easy-to-use and meshless from the collocation Trefftz method (CTM), and mitigates the ill-conditioning resultant matrix encountered in the CTM. Numerical efficiency of the proposed method is investigated in comparison with the CTM and experimental data.

Application of the 2-D Trefftz Method for Identification of Flow Boiling Heat Transfer Coefficient in a Rectangular MiniChannel

The study presents the experimental and numeric heat transfer investigations in flow boiling of water through an asymmetrically heated, rectangular and horizontal minichannel, with transparent side walls. A dedicated system was designed to record images of two-phase flow structures using a high-speed video camera with a synchronous movement system. The images were analyzed with Matlab 2019a scripts for determination of the void fraction for each pattern of two-phase flow structures observed. The experimental data measured during the experimental runs included inlet and outlet temperature, temperature at three internal points of the heater body, volume flux of the flowing water, inlet pressure, pressure drop, current and the voltage drop in the heater power supply. The flows were investigated at Reynolds number characteristic of laminar flow. The mathematical model assumed the heat transfer process in the measurement module to be steady-state with temperature independent thermal properties of solids and flowing fluid. The defined two inverse heat transfer problems were solved with the Trefftz method with two sets of T- functions. Graphs were used to represent: the boiling curves, the local void fraction values, the boiling heat transfer coefficients and the errors of both of them for selected mass fluxes and heat fluxes.

A discontinuous Galerkin Trefftz type method for solving the two dimensional Maxwell equations

Trefftz methods are known to be very efficient to reduce the numerical pollution when associated to plane wave basis. However, these local basis functions are not adapted to the computation of evanescent modes or corner singularities. In this article, we consider a two dimensional time-harmonic Maxwell system and we propose a formulation which allows to design an electromagnetic Trefftz formulation associated to local Galerkin basis computed thanks to an auxiliary Nedelec finite element method. The results are illustrated with numerous numerical examples. The considered test cases reveal that the short range and long range propagation phenomena are both well taken into account.