Hybrid fundamental-solution-based FEM for piezoelectric materials

In this paper, a new type of hybrid finite element method (FEM), hybrid fundamental-solution-based FEM (HFS-FEM), is developed for analyzing plane piezoelectric problems by employing fundamental solutions (Green’s functions) as internal interpolation functions. A modified variational functional used in the proposed model is first constructed, and then the assumed intra-element displacement fields satisfying a priori the governing equations of the problem are constructed by using a linear combination of fundamental solutions at a number of source points located outside the element domain. To ensure continuity of fields over inter-element boundaries, conventional shape functions are employed to construct the independent element frame displacement fields defined over the element boundary. The proposed methodology is assessed by several examples with different boundary conditions and is also used to investigate the phenomenon of stress concentration in infinite piezoelectric medium containing a hole under remote loading. The numerical results show that the proposed algorithm has good performance in numerical accuracy and mesh distortion insensitivity compared with analytical solutions and those from ABAQUS. In addition, some new insights on the stress concentration have been clarified and presented in the paper.     

A new special element for stress concentration analysis of a plate with elliptical holes

Special hole elements are presented for analyzing the stress behavior of an isotropic elastic solidcontaining an elliptical hole. The special hole elements are constructed using the special fundamental solutions for an infinite domain containing a single elliptical hole, which are derived based on complex conformal mapping and Cauchy integrals. During the construction of the special elements, the interior displacement and stress fields are assumed to be the combination of fundamental solutions at a number of source points, and the frame displacement field defined over the element boundary is independently approximated with conventional shape functions. The hybrid finite element model is formulated based on a hybrid functional that provides a link between the two assumed independent fields. Because the fundamental solutions used exactly satisfy both the traction-free boundary conditions of the elliptical hole under consideration and the governing equations of the problems of interest, all integrals can be converted into integrals along the element boundary and there is no need to model the elliptical hole boundary. Thus, the mesh effort near the elliptical hole is significantly reduced. Finally, the numerical model is verified through three examples, and the numerical results obtained for the prediction of stress concentration factors caused by elliptical holes are extremely accurate.     

Special finite elements with holes and internal cracks

For the numerical treatment of stress concentration problems in plane elasticity, special finite elements with circular and elliptic holes and internal cracks have been developed. Two different variational formulations have been used to construct elements, which may be combined with conventional displacement elements. Using complex functions and conformal mapping techniques the systematic construction of trial functions is shown which not only satisfy a priori the governing differential equations but also the boundary conditions on such influential boundary portions as hole or crack surfaces. For the evaluation of the stiffness matrices of the special elements, only boundary integral computations arc necessary. The numerical results of various examples are very accurate for both functionals.     

Numerical implementation of local effects due to two-dimensional discontinuous loads using special elements based on boundary integrals

In this paper, special-purpose elements are developed for solving local effects caused by discontinuous loads such as concentrated forces, line loads and patch loads applied in plane elastic structures. During the derivation of the special-purpose elements, the interior displacement and stress fields are composed of two parts: (1) the homogeneous solution part, which is represented by a linear combination of fundamental solutions at a number of source points outside the element domain; and (2) the particular load-dependent part, which is analytically represented by suitable local solutions. Meanwhile the independent frame displacements defined over the element boundary are approximated by conventional shape functions. The linkage between the two independent fields is established through use of a newly constructed hybrid variational functional, in which discontinuous loads are treated as generalized body forces. Using the property of delta function, the domain integral associated with discontinuous loads in the variational functional can be removed. The advantage of such special-purpose elements is that a large element, independent of the location of discontinuous loads, can be used to avoid the requirement of mesh refinement in the vicinity of the area with local loads. Numerical experiments are carried out to verify the special-purpose elements and to investigate their effectiveness in terms of mesh reduction and accuracy.     

Hybrid displacement function element method: a simple hybrid‐Trefftz stress element method for analysis of Mindlin–Reissner plate

In order to develop robust finite element models for analysis of thin and moderately thick plates, a simple hybrid displacement function element method is presented. First, the variational functional of complementary energy for Mindlin–Reissner plates is modified to be expressed by a displacement function F, which can be used to derive displacement components satisfying all governing equations. Second, the assumed element resultant force fields, which can satisfy all related governing equations, are derived from the fundamental analytical solutions of F. Third, the displacements and shear strains along each element boundary are determined by the locking‐free formulae based on the Timoshenko's beam theory. Finally, by applying the principle of minimum complementary energy, the element stiffness matrix related to the conventional nodal displacement DOFs is obtained. Because the trial functions of the domain stress approximations a priori satisfy governing equations, this method is consistent with the hybrid‐Trefftz stress element method. As an example, a 4‐node, 12‐DOF quadrilateral plate bending element, HDF‐P4‐11 β, is formulated. Numerical benchmark examples have proved that the new model possesses excellent precision. It is also a shape‐free element that performs very well even when a severely distorted mesh containing concave quadrilateral and degenerated triangular elements is employed. Copyright © 2014 John Wiley & Sons, Ltd.     

BEM for crack-hole problems in thermopiezoelectric materials

The boundary element formulation for analysing interaction between a hole and multiple cracks in piezoelectric materials is presented. Using Green's function for hole problems and variational principle, a boundary element model (BEM) for a 2-D thermopiezoelectric solid with cracks and holes has been developed and used to calculate stress intensity factors of the crack-hole problem. In BEM, the boundary condition on the hole circumference is satisfied a priori by Green's function, and is not involved in the boundary element equations. The method is applicable to multiple-crack problems in both finite and infinite solids. Numerical results for stress and electric displacement intensity factors at a particular crack tip in a crack-hole system of piezoelectric materials are presented to illustrate the application of the proposed formulation.     

Numerical modelling of elastic wave scattering in frequency domain by the partition of unity finite element method

In this paper, we investigate a numerical approach based on the partition of unity finite element method, for the time‐harmonic elastic wave equations. The aim of the proposed work is to accurately model two‐dimensional elastic wave problems with fewer elements, capable of containing many wavelengths per nodal spacing, and without refining the mesh at each frequency. The approximation of the displacement field is performed via the standard finite element shape functions, enriched by superimposing pressure and shear plane wave basis, which incorporate knowledge of the wave propagation. A variational framework able to handle mixed boundary conditions is described. Numerical examples dealing with the radiation and the scattering of elastic waves by a circular body are presented. The results show the performance of the proposed method in both accuracy and efficiency. Copyright © 2008 John Wiley & Sons, Ltd.     

Hybrid Finite Element Method Based on Novel General Solutions for Helmholtz-Type Problems

This paper presents a hybrid finite element model (FEM) with a new type of general solution as interior trial functions, named as HGS-FEM. A variational functional corresponding to the proposed general solution is then constructed for deriving the element stiffness matrix of the proposed element model and the corresponding existence of extremum is verified. Then the assumed intra-element potential field is constructed by a linear combination of novel general solutions at the points on the element boundary under consideration. Furthermore, the independent frame field is introduced to guarantee the intra-element continuity. The present scheme inherits the advantages of hybrid Trefftz FEM (HT-FEM) over the conventional FEM and BEM, and avoids the difficulty in choosing appropriate terms of Trefftz functions in HT-FEM and also removing the troublesome for determining fictitious boundary in hybrid fundamental solution-based FEM (HFS-FEM). The efficiency and accuracy of the proposed model are assessed through several numerical examples.     

Thermoelastic state of layered thermosensitive bodies of revolution for the quadratic dependence of the heat-conduction coefficients

We propose an analytic-numerical method for the solution of one-dimensional static problems of thermoelasticity for layered cylinders and balls subjected to the action of the surface loads for various modes of heating with regard for the quadratic dependence of the heat-conduction coefficients and arbitrary dependences of the other physicomechanical characteristics on temperature. Independently of the number of layers, the problems of heat conduction are reduced, by using the constructed exact solutions of special problems, to the solution of a single nonlinear algebraic equation or a system of two equations of this sort. The solutions of the problems of thermoelasticity are obtained by approximating the coefficients of equations continuous inside each layer by piecewise constant functions with subsequent application of Green’s functions of the problems of statics for many-layer cylinders and balls. We perform the numerical analysis of the temperature fields and the thermoelastic state in two-layer bodies whose outer surface is heated by convective-radiation heat exchange and the inner surface is kept at a constant temperature.     

An extended Galerkin weak form and a point interpolation method with continuous strain field and superconvergence using triangular mesh

A point interpolation method (PIM) with continuous strain field (PIM-CS) is developed for mechanics problems using triangular background mesh, in which PIM shape functions are used to construct both displacement and strain fields. The strain field constructed is continuous in the entire problem domain, which is achieved by simple linear interpolations using locally smoothed strains around the nodes and points required for the interpolation. A general parameterized functional with a real adjustable parameter α are then proposed for establishing PIM-CS models of special property. We prove theoretically that the PIM-CS has an excellent bound property: strain energy obtained using PIM-CS lies in between those of the compatible FEM and NS-PIM models of the same mesh. Techniques and procedures are then presented to compute the upper and lower bound solutions using the PIM-CS. It is discovered that an extended Galerkin (x-Galerkin) model, as special case resulted from the extended parameterized functional with α = 1, is outstanding in terms of both performance and efficiency. Intensive numerical studies show that upper and lower bound solutions can always be obtained, there exist α values at which the solutions of PIM-CS are of superconvergence, and the x-Galerkin model is capable of producing superconvergent solutions of ultra accuracy that is about 10 times that of the FEM using the same mesh.     

A new hybrid finite element approach for plane piezoelectricity with defects

In this paper, a new type of hybrid fundamental solution-based finite element method (HFS-FEM) is developed for analyzing plane piezoelectric problems with defects by employing fundamental solutions (or Green’s functions) as internal interpolation functions. The hybrid method is formulated based on two independent assumptions: an intra-element field covering the element domain and an inter-element frame field along the element boundary. Both general elements and a special element with a central elliptical hole or crack are developed in this work. The fundamental solutions of piezoelectricity derived from the elegant Stroh formalism are employed to approximate the intra-element displacement field of the elements, while the polynomial shape functions used in traditional FEM are utilized to interpolate the frame field. By using Stroh formalism, the computation and implementation of the method are considerably simplified in comparison with methods using Lekhnitskii’s formalism. The special-purpose hole element developed in this work can be used efficiently to model defects such as voids or cracks embedded in piezoelectric materials. Numerical examples are presented to assess the performance of the new method by comparing it with analytical or numerical results from the literature.     

An h-p- multigrid method for finite element analysis

The finite element method generates solutions to partial differential equations by minimizing a strain energy based functional. Strain energy based techniques for adaptive mesh refinements are not always effective, however. The adaptive refinement technique proposed in this paper uses strain energy but also incorporates advantages from the h- and p- finite element methods, the multigrid method and a Delaunay based mesh generation method. The refinement technique converged rapidly and was numerically efficient when applied to determining stress concentrations around the circular hole of a thick plate under tension.     

基于解析试函数有限单元法的研究进展

基于解析试函数的有限单元法是一种将有限单元的离散法与解析法成果有机融合的方法,在有限单元理论的几个传统问题中取得了一些进展。该文介绍近几年该类方法在克服剪切闭锁以及消除网格畸变对单元性能影响等方面的研究进展;通过运用含应力函数变分原理,得到了一类不受网格畸变影响的高次精度精确单元;利用特征微分方程解法,给出了一个在弹性力学问题中构造独立完备解析试函数的通用方法。     

Null-field integral equation approach for free vibration analysis of circular plates with multiple circular holes

In this paper, a semi-analytical approach for the eigenproblem of circular plates with multiple circular holes is presented. Natural frequencies and modes are determined by employing the null-field integral formulation in conjunction with degenerate kernels, tensor rotation and Fourier series. In the proposed approach, all kernel functions are expanded into degenerate (separable) forms and all boundary densities are represented by using Fourier series. By uniformly collocating points on the real boundary and taking finite terms of Fourier series, a linear algebraic system can be constructed. The direct searching approach is adopted to determine the natural frequency through the singular value decomposition (SVD). After determining the unknown Fourier coefficients, the corresponding mode shape is obtained by using the boundary integral equations for domain points. The result of the annular plate, as a special case, is compared with the analytical solution to verify the validity of the present method. For the cases of circular plates with an eccentric hole or multiple circular holes, eigensolutions obtained by the present method are compared well with those of the existing approximate analytical method or finite element method (ABAQUS). Besides, the effect of eccentricity of the hole on the natural frequency and mode is also considered. Moreover, the inherent problem of spurious eigenvalue using the integral formulation is investigated and the SVD updating technique is adopted to suppress the occurrence of spurious eigenvalues. Excellent accuracy, fast rate of convergence and high computational efficiency are the main features of the present method thanks to the semi-analytical procedure.     

Computation of shear mode weight functions for circular and elliptical cracks

Three-dimensional shear mode fundamental fields in finite bodies with mixed boundary conditions are analyzed by a special finite element method for circular and elliptical cracks. A procedure for determining the Fourier coefficients of the stress intensity factor for circular cracks is presented. A special series is proposed to represent the computed crack face weight functions for elliptical cracks.     

Exact three-dimensional free vibration analysis of thick homogeneous plates coated by a functionally graded layer

Exact closed-form solutions are carried out for both in-plane and out-of-plane free vibration of thick homogeneous simply supported rectangular plates coated by a functionally graded (FG) layer, based on three-dimensional elasticity theory. The elasticity modulus and mass density of the FG coating are assumed to vary exponentially through the thickness of the coating layer, whereas Poisson’s ratio is remaining constant. The equations of motion are solved using two proposed displacement fields for the in-plane and out-of-plane vibration modes. By inserting the displacement fields in the 3-D elasto-dynamic equations, some independent ordinary equations are obtained and solved analytically. Natural frequencies are extracted by satisfying boundary conditions of interface and surfaces of the structure. The solution procedure is validated by comparing the obtained results with corresponding results of a 3-D finite element analysis. Finally, the influence of the FG coating layer on the natural frequencies of the structure is investigated and discussed. Clearly, the present closed-form solutions can exactly predict both in-plane and out-of-plane vibration modes of thick FG coated plates.     

A mixed finite element for the nonlinear analysis of in-plane loaded masonry walls

A mixed membrane eight-node quadrilateral finite element for the analysis of masonry walls is presented. Assuming that a nonlinear and history-dependent 2D stress-strain constitutive law is used to model masonry material, the element derivation is based on a Hu-Washizu variational statement, involving displacement, strain, and stress fields as primary variables. As the behavior of masonry structures is often characterized by strain localization phenomena, due to strain softening at material level, a discontinuous, piecewise constant interpolation of the strain field is considered at element level, to capture highly nonlinear strain spatial distributions also within finite elements. Newton's method of solution is adopted for the element state determination problem. For avoiding pathological sensitivity to the finite element mesh, a novel algorithm is proposed to perform an integral-type nonlocal regularization of the constitutive equations in the present mixed formulation. By the comparison with competing serendipity displacement-based formulation, numerical simulations prove high performances of the proposed finite element, especially when coarse meshes are adopted.     

A shape‐free 8‐node plane element unsymmetric analytical trial function method

The unsymmetric FEM is one of the effective techniques for developing finite element models immune to various mesh distortions. However, because of the inherent limitation of the metric shape functions, the resulting element models exhibit rotational frame dependence and interpolation failure under certain conditions. In this paper, by introducing the analytical trial function method used in the hybrid stress‐function element method, an effort was made to naturally eliminate these defects and improve accuracy. The key point of the new strategy is that the monomial terms (the trial functions) in the assumed metric displacement fields are replaced by the fundamental analytical solutions of plane problems. Furthermore, some rational conditions are imposed on the trial functions so that the assumed displacement fields possess fourth‐order completeness in Cartesian coordinates. The resulting element model, denoted by US‐ATFQ8, can still work well when interpolation failure modes for original unsymmetric element occur, and provide the invariance for the coordinate rotation. Numerical results show that the exact solutions for constant strain/stress, pure bending and linear bending problems can be obtained by the new element US‐ATFQ8 using arbitrary severely distorted meshes, and produce more accurate results for other more complicated problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Study on the element with the hole and crack

A new element is proposed for describing a discontinuous medium, such as holes and cracks, inside the region of the element. The underlying idea is to construct numerically the base functions of the discontinuous region by capturing the results calculated by fine finite elements in small-scale and then to construct the element in macro-scale with the crack and hole based on the theories of the multi-scale finite element method and the extended finite element method. Some numerical analysis is performed. The results show that the proposed element can well describe the field of displacement, strain, and stress intensity of the discontinuous region inside the element and can significantly decrease the number of elements and nodes of the calculated porous structure. The precision of the proposed element is also acceptable.     

A time-marching scheme based on implicit Green’s functions for elastodynamic analysis with the domain boundary element method

The present work presents an alternative time-marching technique for boundary element formulations based on static fundamental solutions. The domain boundary element method (D-BEM) is adopted and the time-domain Green’s matrices of the elastodynamic problem are considered in order to generate a recursive relationship to evaluate displacements and velocities at each time-step. Taking into account the Newmark method, the Green’s matrices of the problem are numerically and implicitly evaluated, establishing the Green–Newmark method. At the end of the work, numerical examples are presented, verifying the accuracy and potentialities of the new methodology.