Trefftz Finite Element Method Research Articles

Hybrid Trefftz finite element method for heat conduction in cylindrical composite laminates

Laminated composites are made of continuous fibers and matrix which can provide required engineering properties. This paper proposed a hybrid Trefftz finite element method for the analysis of conductive heat transfer in cylindrical composite laminates by assuming two independent temperature fields. The non-conforming intra-element field which is approximated by a linear combination of T-complete solutions satisfying the governing equation and undetermined coefficients is defined within the element domain, while the auxiliary conforming frame field which is interpolated by standard one-dimensional shape functions is defined on the element boundary. Due to the fabrication characteristics of laminates, each layer can be treated as an orthotropic medium for which the hybrid functional is employed to enforce the continuity of temperature and heat flux across the interface of adjacent elements. The proposed method is tested on three numerical examples and compared with exact or ABAQUS solutions. It was found that the proposed method exhibits low sensitivity to mesh distortion and achieves more acceptable results compared with ABAQUS.

Boundary geometry reconstruction for orthotropic heat conduction problems based on HT-FEM

It is usually inconvenient to directly measure the inner boundary geometry shape of thermal pipeline and furnace wall in engineering. A non-iterative algorithm is proposed to reconstruct the geometry shape of these structures for orthotropic heat conduction problems in nondestructive evaluation. First, the temperature of measurement points in the real domain is determined by utilizing the hybrid Trefftz finite element method (HT-FEM). Then, a virtual inner boundary is introduced into forming a virtual domain. The deviation between the measured temperature and the estimated temperature is defined as an objective function. The temperature on the virtual boundary is obtained by calculating the minimum of the objective function. Finally, the virtual boundary temperature is substituted into the direct problem to acquire the temperature distribution in the global domain. And the inner boundary geometry shape is identified by searching the isothermal curve. Several numerical examples are provided to verify the stability and effectiveness of the proposed method. The merit of this algorithm is that the unknown geometry shape can be directly and accurately reconstructed without the complex iterative process.

Hybrid Trefftz finite element method for axisymmetric elasticity problems under torsion

An eight-node axisymmetric element AX88T is proposed for predicting the torsional response of axisymmetric elasticity based on the hybrid fundamental solution based finite element method (HFS-FEM). In this approach, two displacement fields are independently assumed in the domain and on the boundary of the element, respectively. The intra-element displacement field is expressed in terms of fundamental solutions satisfying the governing equation of the problem while a conforming frame displacement field is related to the nodal degrees of freedom by standard one-dimensional interpolation functions. By applying the Gauss divergence theorem to the hybrid functional, the resultant finite element formulation completely avoids the need for domain integration, which renders the AX88T element insensitive to distortion. The accuracy and efficiency of HFS-FEM are demonstrated through three numerical examples which includes thick-walled cylinder, flange-shaped cylinder and stepped cylindrical shaft.

Special hole elements for simulating the heat conduction in two-dimensional cellular materials

The hybrid Trefftz finite element method (HT-FEM) employs the T-complete solutions as domain approximation bases, providing a cheap boundary-type approach to simulate the heat conduction in two-dimensional cellular materials. The proposed method assumes two independent temperature fields inside the element and on its boundary. A conformal mapping function is introduced so that the T-complete solutions are established for the domain with a triangle- or maize-like hole. With the aid of the modified variational functional, the two assumed temperature fields are linked together at the element level and the conformity is enforced across inter-element boundaries. The resultant finite element formulation involves boundary integrals only, which makes a great flexibility in mesh generation with polygons. Numerical examples demonstrate that the proposed method can capture the local effects around the triangle- or maize-like holes precisely with extremely coarse meshes. Compared to ABAQUS, the proposed method exhibits higher convergence rate and computational efficiency.

Thermal analysis of elliptical fiber-reinforced composites by the hybrid Trefftz finite element method

A hybrid Trefftz finite element method (HT-FEM) is developed for microstructural modeling of fiber-reinforced composites and porous materials. In this method, each polygon contains an elliptical inclusion (fiber) and serves as an element. A frame temperature field is assumed along the element outer-boundary, while intra-element temperature fields in the matrix as well as in the inclusion are independently assumed as a linear combination of T-complete functions. Using the complex functions and conformal mapping technology, systematic construction of T-complete functions which satisfy not only the governing equations but also the continuity of temperature on the inclusion-matrix interface is conducted. Unlike the studies in the literature, only one hybrid functional is required to link the two assumed fields in the inclusion and surrounding matrix. The resultant element stiffness equation involves the integrals along the element outer-boundary only. The proposed method is verified against ABAQUS solutions. Numerical results show that HT-FEM can capture the severe perturbation of otherwise regular distribution of temperature and heat flux due to the presence of inclusions or voids. The size and inclination of the elliptical inclusion have a significant influence on the effective thermal conductivity of the composite. It is also pointed out that the effective thermal conductivity is a cosine function in terms of two times the inclination angle.

Thermal analysis of multilayer coated fiber-reinforced composites by the hybrid Trefftz finite element method

A hybrid Trefftz finite element method (HT-FEM) that incorporates the details of heterogeneity, is proposed for capturing the thermal behavior of multilayer coated fiber-reinforced composites. The main idea is to build-up independent two-temperature fields within the domain and along the boundary of the polygonal element. Each element domain is embedded by a fiber with multiple coatings. The Trefftz complete functions analytically satisfying the heat transfer equations are used as intra-element trial functions for the individual phases (fiber, coating and matrix) in a recursive form. The continuity of temperature between adjacent constituent phases is consequently enforced. After introducing the divergence theorem into the single hybrid functional, the degrees of freedom appear only along the element boundaries. This enables polygonal elements to be easily constructed for modeling the heterogeneous materials. Close attention is paid to evaluate the accuracy and efficiency of the HT-FEM through three numerical examples which involve mono- or multilayer coated fibers in different spatial distributions.

Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties

In this paper, the accuracy and the convergence properties of Trefftz finite element method over arbitrary polygons are studied. Within this approach, the unknown displacement field within the polygon is represented by the homogeneous solution to the governing differential equations, also called as the T-complete set. While on the boundary of the polygon, a conforming displacement field is independently defined to enforce the continuity of the field variables across the element boundary. An optimal number of T-complete functions are chosen based on the number of nodes of the polygon and the degrees of freedom per node. The stiffness matrix is computed by the hybrid formulation with auxiliary displacement frame. Results from the numerical studies presented for a few benchmark problems in the context of linear elasticity show that the proposed method yields highly accurate results with optimal convergence rates.

Transparent boundary conditions for a discontinuous Galerkin Trefftz method

The modeling and simulation of electromagnetic wave propagation is often accompanied by a restriction to bounded domains which requires the introduction of artificial boundaries. The corresponding boundary conditions should be chosen in order to minimize parasitic reflections. In this paper, we investigate a new type of transparent boundary condition for a discontinuous Galerkin Trefftz finite element method. The choice of a particular basis consisting of polynomial plane waves allows us to split the electromagnetic field into components with a well specified direction of propagation. The reflections at the artificial boundaries are then reduced by penalizing components of the field incoming into the space–time domain of interest. We formally introduce this concept, discuss its realization within the discontinuous Galerkin framework, and demonstrate the performance of the resulting approximations by numerical tests. A comparison with first order absorbing boundary conditions, that are frequently used in practice, is made. For a proper choice of basis functions, we observe spectral convergence in our numerical test and an overall dissipative behavior for which we also give some theoretical explanation.

Fundamental solution based graded element model for steady-state heat transfer in FGM

A novel hybrid graded element model is developed in this paper for investigating thermal behavior of functionally graded materials (FGMs). The model can handle a spatially varying material property field of FGMs. In the proposed approach, a new variational functional is first constructed for generating corresponding finite element model. Then, a graded element is formulated based on two sets of independent temperature fields. One is known as intra-element temperature field defined within the element domain; the other is the so-called frame field defined on the element boundary only. The intra-element temperature field is constructed using the linear combination of fundamental solutions, while the independent frame field is separately used as the boundary interpolation functions of the element to ensure the field continuity over the inter-element boundary. Due to the properties of fundamental solutions, the domain integrals appearing in the variational functional can be converted into boundary integrals which can significantly simplify the calculation of generalized element stiffness matrix. The proposed model can simulate the graded material properties naturally due to the use of the graded element in the finite element (FE) model. Moreover, it inherits all the advantages of the hybrid Trefftz finite element method (HT-FEM) over the conventional FEM and boundary element method (BEM). Finally, several examples are presented to assess the performance of the proposed method, and the obtained numerical results show a good numerical accuracy.

Computational simulation methods for fiber reinforced composites

Trefftz-finite element method (Trefftz-FEM), adaptive cross approximation BEM (ACA BEM) and continuous source function method (CSFM) are used for the simulation of composites reinforced by short fibers (CRSF) with the aim of showing the possibilities of reducing the problem of complicated and important interactions in such composite materials.

Hybrid FEM with fundamental solutions as trial functions for heat conduction simulation

A new type of hybrid finite element formulation with fundamental solutions as internal interpolation functions, named as HFS-FEM, is presented in this paper and used for solving two dimensional heat conduction problems in single and multi-layer materials. In the proposed approach, a new variational functional is firstly constructed for the proposed HFS-FE model and the related existence of extremum is presented. Then, the assumed internal potential field constructed by the linear combination of fundamental solutions at points outside the elemental domain under consideration is used as the internal interpolation function, which analytically satisfies the governing equation within each element. As a result, the domain integrals in the variational functional formulation can be converted into the boundary integrals which can significantly simplify the calculation of the element stiffness matrix. The independent frame field is also introduced to guarantee the inter-element continuity and the stationary condition of the new variational functional is used to obtain the final stiffness equations. The proposed method inherits the advantages of the hybrid Trefftz finite element method (HT-FEM) over the conventional finite element method (FEM) and boundary element method (BEM), and avoids the difficulty in selecting appropriate terms of T-complete functions used in HT-FEM, as the fundamental solutions contain usually one term only, rather than a series containing infinitely many terms. Further, the fundamental solutions of a problem are, in general, easier to derive than the T-complete functions of that problem. Finally, several examples are presented to assess the performance of the proposed method, and the obtained numerical results show good numerical accuracy and remarkable insensitivity to mesh distortion.

Application of hybrid Trefftz finite element method to non‐linear problems of minimal surface

AbstractThis investigation provides a hybrid Trefftz finite element approach for analysing minimal surface problems. The approach is based on combining Trefftz finite element formulation with radial basis functions (RBF) and the analogue equation method (AEM). In this method, use of the analogue equation approach avoids the difficulty of treating the non‐linear terms appearing in the soap bubble equation, making it possible to solve non‐linear problems with the Trefftz method. Global RBF is used to approximate the inhomogeneous term induced from non‐linear functions and other loading terms. Finally, some numerical experiments are implemented to verify the efficiency of this method. Copyright © 2006 John Wiley & Sons, Ltd.

Formulation of hybrid Trefftz finite element method for elastoplasticity

The present investigation provides a hybrid Trefftz finite element approach for analysing elastoplastic problems. A dual variational functional is constructed and used to derive hybrid Trefftz finite element formulation for elastoplasticity of bulky solids. The formulation is applicable to either strain hardening or elastic-perfectly plastic materials. A solution algorithm based on initial stress formulation is introduced into the new element model. The performance of the proposed element model is assessed by three examples and comparison is made with results obtained by other approaches. The hybrid Trefftz finite element approach is demonstrated to be particularly suited for nonlinear analysis of two-dimensional elastoplastic problems.

The Temperature Coefficient of Frequency and Quality Factors at Resonance and Antiresonance Frequencies of Infinite Interdigital Transducers for Leaky Surface Acoustic Waves on a SiO2/IDT/36° Y–X LiTaO3 Substrate

The temperature coefficient, electromechanical coupling coefficient and quality factors at resonance frequencies of infinite interdigital transducers (IDTs) for leaky or nonleaky surface acoustic waves on SiO2/IDT/36° Y–X LiTaO3 are presented. The electrode materials of IDTs with aluminum, copper, gold, silver, tantalum, titanium or tungsten alloy are examined. Using the hybrid Trefftz finite element method, the radiation conditions of leaky surface acoutic waves are rigorously taken into account in the presented results. Computed values reveal that silver electrode IDTs with a SiO2 film thickness of 0.36p and an electrode height of 0.2p have an electromechanical coupling coefficient of 0.035 and a temperature coefficient of frequency of -10 ppm/°C, where p is the period of IDT fingers.

Dual variational formulation for Trefftz finite element method of elastic materials

A dual variational principle is presented for Trefftz finite element analysis. The proof of the stationary conditions of the variational functional and the theorem on the existence of extremum are provided in this paper. They are boundary displacement condition, surface traction condition and interelement continuity condition. Based on the assumed intraelement and frame fields, element stiffness matrix equation is obtained which can easily be implemented into computer programs for numerical analysis with Trefftz finite element method. Two numerical examples are considered to illustrate the effectiveness and applicability of the proposed element model.

Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach

Applications of the Trefftz finite element method to anti-plane electroelastic problems are presented in this paper. A dual variational functional is constructed and used to derive Trefftz finite element formulation. Special trial functions which satisfy boundary conditions are also used to develop a special purpose element with local defects. The performance of the proposed element model is assessed by an example and comparison is made with results obtained by other approaches. The Trefftz finite element approach is demonstrated to be ideally suited for the analysis of the anti-plane problem.

Trefftz finite elements for electromagnetics

It is shown that the method of minimum autonomous blocks (MAB) of Nikol'skii and Nikol'skaia can be reformulated as the Trefftz finite-element method. Solutions of Maxwell's equations in form of plane waves are used to represent fields inside a finite element. Their projections on a set of basis functions on the surface of the element are used to obtain a descriptor of the element in form of an admittance matrix. It is shown that a point-matching projection procedure gives the frequency-domain transmission-line-matrix formulation and Galerkin-type projection leads to the MAB formulation. Admittance matrix representation of the descriptors of the elements makes it possible to use a finite-element-type global matrix assembling procedure and a sparse matrix solver.

Postbuckling analysis of thin plates by a hybrid Trefftz finite element method

A hybrid Trefftz finite element (HT FE) model for postbuckling analysis of thin plates is presented, based on a modified variational principle. With the aid of incremental form of equilibrium equations, all non-linear terms may be viewed as pseudo-loads, and then the in-plane and out-of-plane equations can be uncoupled. As a consequence, the solution procedure for the postbuckling plate becomes very simple. Two numerical examples are considered to illustrate the correctness and accuracy of the proposed formulation.