A point interpolation method for two‐dimensional solids

A point interpolation method (PIM) is presented for stress analysis for two‐dimensional solids. In the PIM, the problem domain is represented by properly scattered points. A technique is proposed to construct polynomial interpolants with delta function property based only on a group of arbitrarily distributed points. The PIM equations are then derived using variational principles. In the PIM, the essential boundary conditions can be implemented with ease as in the conventional finite element methods. The present PIM has been coded in FORTRAN. The validity and efficiency of the present PIM formulation are demonstrated through example problems. It is found that the present PIM is very easy to implement, and very flexible for obtained displacements and stresses of desired accuracy in solids. As the elements are not used for meshing the problem domain, the present PIM opens new avenues to develop adaptive analysis codes for stress analysis in solids and structures. Copyright © 2001 John Wiley & Sons, Ltd.     

A meshless local natural neighbour interpolation method for analysis of two-dimensional piezoelectric structures

A novel meshless method applied to solve two-dimensional piezoelectric structures is presented and discussed in this paper. It is called meshless local natural neighbour interpolation (MLNNI) method, which is derived from the generalized meshless local Petrov–Galerkin (MLPG) method as a special case. In the present method, nodal points are spread on the analysed domain and each node is surrounded by a polygonal sub-domain, which can be conveniently constructed with Delaunay tessellations. The spatial variation of the displacements and the electric potential are interpolated by the natural neighbour interpolation. As the shape functions so constructed possess the delta function property, the essential boundary conditions can be imposed by directly substituting the corresponding terms in the system of equations. Furthermore, the usage of three-node triangular FEM shape functions as test functions reduces the order of integrands involved in domain integrals. Numerical examples are presented at the end to demonstrate the applicability and accuracy of the present approach in analysing two-dimensional piezoelectric structures.     

Point interpolation collocation method for the solution of partial differential equations

This paper presents a truly meshless method for solving partial differential equations based on point interpolation collocation method (PICM). This method is different from the previous Galerkin-based point interpolation method (PIM) investigated in the papers [G.R. Liu, (2002), mesh free methods, Moving beyond the Finite Element Method, CRC Press. G.R. Liu, Y.T. Gu, A point interpolation method for two-dimension solids, Int J Numer Methods Eng, 50, 937–951, 2001. G.R. Liu, Y.T. Gu, A matrix triangularization algorithm for point interpolation method, in Proceedings Asia-Pacific Vibration Conference, Bangchun Weng Ed., November, Hangzhou, People's Republic of China, 2001a, 1151–1154. 1–3.], because it is based on collocation scheme. In the paper, polynomial basis functions have been used. In addition, Hermite-type interpolations called as inconsistent PIM has been adopted to solve PDEs with Neumann boundary conditions so that the accuracy of the solution can be improved. Several examples were numerically analysed. These examples were applied to solve 1D and 2D partial differential equations including linear and non-linear in order to test the accuracy and efficiency of the presented method based on polynomial basis functions. The h-convergence rates were computed for the PICM based on different model of regular and irregular nodes. The results obtained by polynomial PICM show the presented schemes possess a considerable perfect stability and good numerical accuracy even for scattered models while matrix triangularization algorithm (MTA) adopted in the computed procedure.     

A sub-domain RPIM with condensation of degree of freedom

A sub-domain radial point interpolation method is proposed to simulate problem of linear elasticity. In present method, the problem domain is firstly divided into sub-domains with arbitrary shape, and then, nodes without connectivity are imbedded in every sub-domain. The local variational weak formulation is established over sub-domains, in which nodes within the sub-domain are used for approximation. Local discrete equations of weak form are simplified by condensation of degree of freedom, which transfers equations of inner nodes to equations of boundary nodes based on sub-domains. Compatibility of displacement in adjacent sub-domains and convergence of present method are discussed. And displacements and its gradient are continuous in the entire problem domain. In contrast to an early formulation of RPIM based on Galerkin weak form, which is proposed by Liu and coworkers, certain modifications are presented to increase its computational efficiency in this paper. Numerical examples show that computational efficiency of present method is higher than that of standard RPIM based on Galerkin weak form, and good accuracy, high convergence can also be obtained.     

A linearly conforming point interpolation method (LC‐PIM) for three‐dimensional elasticity problems

Linearly conforming point interpolation method (LC‐PIM) is formulated for three‐dimensional elasticity problems. In this method, shape functions are generated using point interpolation method by adopting polynomial basis functions and local supporting nodes are selected based on the background cells. The shape functions so constructed have the Kronecker delta functions property and it allows straightforward imposition of point essential boundary conditions. Galerkin weak form is used for creating discretized system equations, and a nodal integration scheme with strain‐smoothing operation is used to perform the numerical integration. The present LC‐PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method using linear elements, the LC‐PIM can achieve better efficiency, and higher accuracy especially for stresses. Copyright © 2007 John Wiley & Sons, Ltd.     

Analytical solution for the axisymmetric plane strain electroelastic dynamics of a special non-homogeneous piezoelectric hollow cylinder

By virtue of the introduction of a dependent variable and the separation of variables technique, the axisymmetric plane strain electroelastic dynamic problem of a special non-homogeneous piezoelectric hollow cylinder is transformed to a Volterra integral equation of the second kind about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solutions of displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a piezoelectric hollow cylinder with an arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are finally presented.     

平面应变轴对称压电圆柱壳的随机响应

提出求解随机激励轴对称压电圆柱壳响应的一种方法，并导出相应的解析表达式。首先给出压电圆柱壳在边界随机激励下的基本方程；然后通过位移与电势的变换，将随机激励变换到运动方程中；再利用Legendre多项式展开位移，应用Galerkin法化偏微分的运动方程为常微分方程组；最后根据随机振动理论，得到压电圆柱壳位移与加速度响应的均方值，由此可计算随机响应、分析有关因素的影响与机电耦合关系等。分析说明了存在的机电耦合项，及由此产生广义刚度的非对称性。     

A Trefftz function approximation in local boundary integral equations

 In the present paper the Trefftz function as a test function is used to derive the local boundary integral equations (LBIE) for linear elasticity. Since Trefftz functions are regular, much less requirements are put on numerical integration than in the conventional boundary integral method. The moving least square (MLS) approximation is applied to the displacement field. Then, the traction vectors on the local boundaries are obtained from the gradients of the approximated displacements by using Hooke's law. Nodal points are randomly spread on the domain of the analysed body. The present method is a truly meshless method, as it does not need a finite element mesh, either for purposes of interpolation of the solution variables, or for the integration of the energy. Two ways are presented to formulate the solution of boundary value problems. In the first one the local boundary integral equations are written in all nodes (interior and boundary nodes). In the second way the LBIE are written only at the interior nodes and at the nodes on the global boundary the prescribed values of displacements and/or tractions are identified with their MLS approximations. Numerical examples for a square patch test and a cantilever beam are presented to illustrate the implementation and performance of the present method. Received 6 November 2000     

Boundary element formulation for the coupled stretching-bending analysis of thin laminated plates

The boundary integral equations for the coupled stretching-bending analysis of thin laminated plates involve an integral which will be singular when the field point approaches the source point. To avoid the singular problem occurring in the numerical programming, the boundary integral equations are modified in which the integrals of singular part are integrated analytically. The analytical solutions for the free term coefficients and singular integrals are obtained in explicit closed-form. By dividing the boundary into elements and using suitable interpolation polynomials for basic functions, the set of equations necessary for boundary element programming are written explicitly for regular nodes and corner nodes. The equations for the determination of displacements and stresses at internal points are also presented in this paper.     

Meshless Local Petrov-Galerkin Method for Plane Piezoelectricity

Piezoelectric materials have wide range engineering applications in smart structures and devices. They have usually anisotropic properties. Except this complication electric and mechanical fields are coupled each other and the governing equations are much more complex than that in the classical elasticity. Thus, efficient computational methods to solve the boundary or the initial-boundary value problems for piezoelectric solids are required. In this paper, the Meshless local Petrov-Galerkin (MLPG) method with a Heaviside step function as the test functions is applied to solve two-dimensional (2-D) piezoelectric problems. The mechanical fields are described by the equations of motion with an inertial term. To eliminate the time-dependence in the governing partial differential equations the Laplace-transform technique is applied to the governing equations, which are satisfied in the Laplace-transformed domain in a weak-form on small subdomains. Nodal points are spread on the analyzed domain and each node is surrounded by a small circle for simplicity. The spatial variation of the displacements and the electric potential are approximated by the Moving Least-Squares (MLS) scheme. After performing the spatial integrations, one obtains a system of linear algebraic equations for unknown nodal values. The boundary conditions on the global boundary are satisfied by the collocation of the MLS-approximation expressions for the displacements and the electric potential at the boundary nodal points. The Stehfest's inversion method is applied to obtain the final time-dependent solutions.     

On refined nonlinear theories of laminated composite structures with piezoelectric laminae

In this paper geometrically nonlinear theories of laminated composite plates with piezoelectric laminae are developed. The formulations are based on thermopiezoelectricity, and include the coupling between mechanical deformations, temperature changes, and electric displacements. Two different theories are presented: one based on an equivalent-single-layer third-order theory and the other based on the layerwise theory, both of which were developed by the senior author for composite laminates without piezoelectric laminae. In the present study, they are extended to include piezoelectric laminae. In both theories, the electric field is expanded layerwise through the laminate thickness. The dynamic version of the principle of virtual displacements (or Hamilton’s principle) is used to derive the equations of motion and associated boundary conditions of the two theories. These theories may be used to accurately determine the response of laminated plate structures with piezoelectric laminae and subjected to thermomechanical loadings.     

Modeling of porous piezoelectric structures by the meshless local Petrov-Galerkin method

A meshless method based on the local Petrov-Galerkin approach is proposed to solve initial-boundary value problems of porous piezoelectric solids. Constitutive equations for porous piezoelectric materials possess a coupling between mechanical displacements and electric intensity vectors in both solid and fluid phases. Stationary and transient 2-D and 3-D axisymmetric problems are considered in this article. Nodal points are spread on the problem domain, and each node is surrounded by a small circle for simplicity. The spatial variation of displacements and electric potentials for both phases is approximated by the moving least-squares scheme. After performing the spatial integration, one obtains a system of ordinary differential equations for certain nodal unknowns. The resulting system is solved numerically by the Houbolt finite-difference scheme as a time stepping method. The proposed method is applied to bending problems associated with a porous piezoelectric 2-D plate and 3-D axisymmetric cylinder under simply supported and clamped boundary conditions.     

Variable Kinematic Shell Elements for the Analysis of Electro-Mechanical Problems

The present article considers the linear static analysis of both composite plate and shell structures embedding piezoelectric layers by means of a shell finite element with variable through-the-thickness kinematic. The refined models used are grouped in the Unified Formulation by Carrera (CUF) and they permit to accurately describe the distribution of displacements and stresses along the thickness of the multilayered shell. The shell element has nine nodes and the mixed interpolation of tensorial components (MITC) method is employed to contrast the membrane and shear locking phenomenon. The governing equations are derived from the principle of virtual displacement (PVD) and the finite element method (FEM) is employed to solve them. Cross-ply multilayered plates and cylindrical shells embedding piezoelectric layers are analyzed with simply-supported boundary conditions and subjected to sensor and actuator configurations. Various thickness ratios are considered. The results, obtained with different theories contained in the CUF, are compared with both the elasticity solutions given in literature and the analytical solutions obtained using the CUF and the Navier’s method. From the analysis, one can conclude that the shell element based on the CUF is very efficient and its use is mandatory with respect to the classical models in the study of multilayered structures embedding piezo-layers.     

Three-step multi-domain BEM solver for nonhomogeneous material problems

A three-step solution technique is presented for solving two-dimensional (2D) and three-dimensional (3D) nonhomogeneous material problems using the multi-domain boundary element method. The discretized boundary element formulation expressed in terms of normalized displacements and tractions is written for each sub-domain. The first step is to eliminate internal variables at the individual domain level. The second step is to eliminate boundary unknowns defined over nodes used only by the domain itself. And the third step is to establish the system of equations according to the compatibility of displacements and equilibrium of tractions at common interface nodes. Discontinuous elements are utilized to model the traction discontinuity across corner nodes. The distinct feature of the three-step solver is that only interface displacements are unknowns in the final system of equations and the coefficient matrix is blocked sparse. As a result, large-scale 3D problems can be solved efficiently. Three numerical examples for 2D and 3D problems are given to demonstrate the effectiveness of the presented technique.     

A geometrically non‐linear piezoelectric solid shell element based on a mixed multi‐field variational formulation

This paper is concerned with a geometrically non‐linear solid shell element to analyse piezoelectric structures. The finite element formulation is based on a variational principle of the Hu–Washizu type and includes six independent fields: displacements, electric potential, strains, electric field, mechanical stresses and dielectric displacements. The element has eight nodes with four nodal degrees of freedoms, three displacements and the electric potential. A bilinear distribution through the thickness of the independent electric field is assumed to fulfill the electric charge conservation law in bending dominated situations exactly. The presented finite shell element is able to model arbitrary curved shell structures and incorporates a 3D‐material law. A geometrically non‐linear theory allows large deformations and includes stability problems. Linear and non‐linear numerical examples demonstrate the ability of the proposed model to analyse piezoelectric devices. Copyright © 2005 John Wiley & Sons, Ltd.     

A meshfree radial point interpolation method for analysis of functionally graded material (FGM) plates

A meshfree model is presented for the static and dynamic analyses of functionally graded material (FGM) plates based on the radial point interpolation method (PIM). In the present method, the mid-plane of an FGM plate is represented by a set of distributed nodes while the material properties in its thickness direction are computed analytically to take into account their continuous variations from one surface to another. Several examples are successfully analyzed for static deflections, natural frequencies and dynamic responses of FGM plates with different volume fraction exponents and boundary conditions. The convergence rate and accuracy are studied and compared with the finite element method (FEM). The effects of the constituent fraction exponent on static deflection as well as natural frequency are also investigated in detail using different FGM models. Based on the current material gradient, it is found that as the volume fraction exponent increases, the mechanical characteristics of the FGM plate approach those of the pure metal plate blended in the FGM.     

A meshfree method: meshfree weak–strong (MWS) form method,for 2-D solids

A novel meshfree weak–strong (MWS) form method is proposed based on a combined formulation of both the strong-form and the local weak-form. In the MWS method, the problem domain and its boundary is represented by a set of distributed points or nodes. The strong form or the collocation method is used for all nodes whose local quadrature domains do not intersect with natural (Neumann) boundaries. Therefore, no numerical integration is required for these nodes. The local weak-form, which needs the local numerical integration, is only used for nodes on or near the natural boundaries. The locally supported radial point interpolation method and the moving least squares approximation are used to construct the meshfree shape functions. The final system matrix will be sparse and banded for computational efficiency. Numerical examples of two-dimensional solids are presented to demonstrate the efficiency, stability, accuracy and convergence of the proposed meshfree method.     

Three-dimensional analysis of an antiparallel piezoelectric bimorph

Summary Three-dimensional electromechanical responses of a piezoelectric bimorph are studied. The bimorph is antiparallel in the sense that it consists of two identical, plate-like piezoelectric elements with opposite poling directions. Both the top and bottom surfaces of the bimorph are fully covered with negligibly thin conductive electrodes. By introducing a small parameter and using the transfer matrix method it is shown that a three-dimensional solution of the problem can be readily constructed, provided the solution to a set of two-dimensional equations very similar to those in the classic plate theory is obtainable. The three-dimensional solution satisfies all the field equations as well as the boundary conditions on the major surfaces and at the interface between the two piezoelectric plates. In many special cases, the electric edge condition can be fulfilled point by point, and thus the solution is exact in Saint-Venant's sense. The formulation and new analytical results for a strip-shaped cantilever bimorph under the action of applied voltage and end moment are presented.     

Direct imposition of essential boundary conditions and treatment of material discontinuities in the EFG method

A method for direct imposition of essential boundary condition and treatment of material discontinuity in element free galerkin (EFG) method is presented. By using the actual displacements at the nodes on the essential boundary and the material interface in each material domain, the stiffness matrix and load vector at an integral point have been rewritten and transformed. As a result, the proposed method yields a positive, symmetrical and banded global stiffness matrix like it is in finite element methods and has the advantages of stabilization and easy implementation as compared to the penalty method, the Lagrange method, and other methods. Numerical results indicate that the present method is effective and retains high rates of convergence for both displacements and energy.     

Non-linear coupled multi-field mechanics and finite element for active multi-stable thermal piezoelectric shells

In this paper, a coupled multi-field mechanics framework is presented for analyzing the non-linear response of shallow doubly curved adaptive laminated piezoelectric shells undergoing large displacements and rotations in thermal environments. The mechanics incorporate coupling between mechanical, electric and thermal fields and encompass geometric non-linearity effects due to large displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear coordinates and are combined with the kinematic assumptions of a mixed-field shear-layerwise shell laminate theory. A finite element methodology and an eight-node coupled non-linear shell element are developed. The discrete coupled non-linear equations of motion are linearized and solved, using an extended cylindrical arc-length method together with a Newton–Raphson technique, to enable robust numerical predictions of non-linear active shells transitioning between multiple stable equilibrium paths. Validation and evaluation cases on laminated cylindrical strips and cylindrical panels demonstrate the accuracy of the method and its robust capability to predict non-linear response under thermal and piezoelectric actuator loads. Moreover, the results illustrate the capability of the method to model piezoelectric shells undergoing large shape changes by actively jumping between stable equilibrium states and quantify the strong relationship between shell curvature, applied electric potential, applied temperature differential and induced shape change. Copyright © 2008 John Wiley & Sons, Ltd.