Stress analysis in anisotropic functionally graded materials by the MLPG method

A meshless method based on the local Petrov–Galerkin approach is proposed for stress analysis in two-dimensional (2D), anisotropic and linear elastic/viscoelastic solids with continuously varying material properties. The correspondence principle is applied for non-homogeneous, anisotropic and linear viscoelastic solids where the relaxation moduli are separable in space and time. The inertial dynamic term in the governing equations is considered too. A unit step function is used as the test functions in the local weak-form. It leads to local boundary integral equations (LBIEs). The analyzed domain is divided into small subdomains with a circular shape. The moving least squares (MLS) method is adopted for approximating the physical quantities in the LBIEs. For time-dependent problems, the Laplace-transform technique is utilized. Several numerical examples are given to verify the accuracy and the efficiency of the proposed method.     

Evaluation of fracture parameters in continuously nonhomogeneous piezoelectric solids

A contour integral method is developed for computation of stress intensity and electric intensity factors for cracks in continuously nonhomogeneous piezoelectric body under a transient dynamic load. It is shown that the asymptotic fields in the crack-tip vicinity in a continuously nonhomogeneos medium is the same as in a homogeneous one. A meshless method based on the local Petrov-Galerkin approach is applied for computation of physical fields occurring in the contour integral expressions of intensity factors. A unit step function is used as the test functions in the local weak-form. This leads to local integral equations (LBIEs) involving only contour-integrals on the surfaces of subdomains. The moving least-squares (MLS) method is adopted for approximating the physical quantities in the LBIEs. The accuracy of the present method for computing the stress intensity factors (SIF) and electrical displacement intensity factors (EDIF) are discussed by comparison with available analytical or numerical solutions.     

Meshless local boundary integral equation method for 2D elastodynamic problems

A new meshless method for solving transient elastodynamic boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation (MLS), is proposed in this paper. The LBIE with the MLS is applied to both transient and steady‐state (Laplace transformed) elastodynamics. Applying the MLS approximation for spatially dependent terms in the first approach, the LBIEs are transformed into a system of ordinary differential equations for nodal unknowns. This system of ordinary differential equations is solved by the Houbolt finite difference scheme. In the second formulation, the time variable is eliminated by using the Laplace transformation. Unknown Laplace transforms of displacements and traction vectors are computed from the LBIEs with the MLS approximation. The time‐dependent values are obtained by the Durbin inversion technique. Copyright © 2003 John Wiley & Sons, Ltd.     

Fracture analysis of cracks in magneto-electro-elastic solids by the MLPG

A meshless method based on the local Petrov–Galerkin approach is proposed for crack analysis in two-dimensional (2-D) and three-dimensional (3-D) axisymmetric magneto-electric-elastic solids with continuously varying material properties. Axial symmetry of geometry and boundary conditions reduces the original 3-D boundary value problem into a 2-D problem in axial cross section. Stationary and transient dynamic problems are considered in this paper. The local weak formulation is employed on circular subdomains where surrounding nodes randomly spread over the analyzed domain. The test functions are taken as unit step functions in derivation of the local integral equations (LIEs). The moving least-squares (MLS) method is adopted for the approximation of the physical quantities in the LIEs. The accuracy of the present method for computing the stress intensity factors (SIF), electrical displacement intensity factors (EDIF) and magnetic induction intensity factors (MIIF) are discussed by comparison with numerical solutions for homogeneous materials.     

Inverse heat conduction problems by meshless local Petrov–Galerkin method

The meshless local Petrov–Galerkin (MLPG) method is used to solve stationary and transient heat conduction inverse problems in 2-D and 3-D axisymmetric bodies. A 3-D axisymmetric body is generated by rotating a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduce the original 3-D boundary value problem to a 2-D problem. The analyzed domain is covered by small circular subdomains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function in deriving the local integral equations (LIEs) on the boundaries of the chosen subdomains. The time integration schemes are formulated based on the Laplace transform technique and the time difference approach, respectively. The local integral equations are non-singular and take a very simple form. Spatial variation of the temperature and heat flux (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. Singular value decomposition (SVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after MLS approximation. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.     

An Interaction Integral Method for Computing Fracture Parameters in Functionally Graded Magnetoelectroelastic Composites

A contour integral method is developed for the computation of stress intensity, electric and magnetic intensity factors for cracks in continuously nonhomogeneous magnetoelectroelastic solids under a transient dynamic load. It is shown that the asymptotic fields in the crack-tip vicinity in a continuously nonhomogeneos medium are the same as in a homogeneous one. A meshless method based on the local Petrov-Galerkin approach is applied for the computation of the physical fields occurring in the contour integral expressions of intensity factors. A unit step function is used as the test functions in the local weak-form. This leads to local integral equations (LIEs) involving only contour-integrals on the surfaces of subdomains. The moving least-squares (MLS) method is adopted for approximating the physical quantities in the LIEs. The accuracy of the present method for computing the stress intensity factors (SIF), electrical displacement intensity factors (EDIF) and magnetic induction intensity factors (MIIF) are discussed by comparison with numerical solutions for homogeneous materials.     

A generalized complex eigenvector method for dynamic analysis of heterogeneous viscoelastic structures

A generalized complex eigenvector method which can be used to a linear dynamic analysis of viscoelastic structures is described. Here dynamic analysis is understood as transient analysis and frequency response analysis. The generalized complex eigenvector method is based on finite element discretization of structure, approximation of viscoelastic properties by differential operators and mode superposition technique. Coefficients of differential operator are defined from the condition of best coincidence of complex characteristic of viscoelastic material and complex characteristic of differential operator in preset frequency range. Advantage of this method is that it allows to take into account the real changes of the viscoelastic property in frequency range. Also, the generalized complex eigenvector method permit to describe a viscoelastic properties by two functions (complex Young's modulus, complex Poisson's ratio). The method is verified with the help of comparing with solutions obtained by complex modulus method. An influence of viscoelastic Poisson's ratio on transient and frequency responses of structure is demonstrated. Copyright © 2001 John Wiley & Sons, Ltd.     

Meshless Local Petrov-Galerkin (MLPG) Method for Laminate Plates under Dynamic Loading

A meshless local Petrov-Galerkin (MLPG) method is applied to solve laminate plate problems described by the Reissner-Mindlin theory. Both stationary and transient dynamic loads are analyzed here. The bending moment and the shear force expressions are obtained by integration through the laminated plate for the considered constitutive equations in each lamina. The Reissner-Mindlin theory reduces the original three-dimensional (3-D) thick plate problem to a two-dimensional (2-D) problem. Nodal points are randomly distributed over the mean surface of the considered plate. Each node is the center of a circle surrounding this node. The weak-form on small subdomains with a Heaviside step function as the test functions is applied to derive local integral equations. After performing the spatial MLS approximation, a system of ordinary differential equations of the second order for certain nodal unknowns is obtained. The derived ordinary differential equations are solved by the Houbolt finite-difference scheme as a time-stepping method.     

Matrix dominated time dependent failure predictions in polymer matrix composites

Various types of matrix dominated failures in polymer matrix composites (PMC) are reviewed. Current methods to evaluate the modulus degradation of PMC materials are discussed including viscoelastic/plastic and continuum damage models. It is pointed out that in each case the approach is based upon developing an analytical constitutive relation for the material in order to represent a measured stress–strain response. The suggestion is made that care must be used in the measurement of stress–strain behavior such that the modeling represents the true material behavior. New digital imaging methods are suggested as a means to determine in situ properties at a local scale commensurate with the continuum modeling procedure. A little used method to model viscoelastic/plastic (linear and non-linear) effects is discussed and modified to obtain a simple and easy to use time dependent failure law. Also, a little used energy based time dependent failure criterion is presented which can be combined with a non-linear viscoelastic integral approach to provide a prediction method for the time for creep rupture under simple stress states. Each is validated with experimental data for simple stress states but their generality is such that they could be used for complex (3-D) stress states. Advantages and limitations of both are addressed. Finally, a discussion of possible fruitful research areas are presented with the view of providing engineers in industry with an easy to use accelerated life prediction procedure.     

FINITE ELEMENT ALGORITHMS FOR DYNAMIC SIMULATIONS OF VISCOELASTIC COMPOSITE SHELL STRUCTURES USING CONJUGATED GRADIENT METHOD ON COARSE GRAINED AND MASSIVELY PARALLEL MACHINES

Recently much attention has been paid to high-performance computing and the development of parallel computational strategies and numerical algorithms for large-scale problems. In this present study, a finite element procedure for the dynamic analyses of anisotropic viscoelastic composite shell structures by using degenerated 3-D elements has been studied on vector and coarse grained and massively parallel machines. CRAY hardware performance monitors such as Flowtrace and Perftrace tools are used to obtain performance data for subroutine program modules and specified code segments. The performances of conjugated gradient method, the Cray sparse matrix solver and the Feable solver are evaluated. SIMD and MIMD parallel implementation of the finite element algorithm for dynamic simulation of viscoelastic composite structures on the CM-5 is also presented. The performance studies have been conducted in order to evaluate efficiency of the numerical algorithm on this architecture versus vector processing CRAY systems. Parametric studies on the CM-5 as well as the CRAY system and benchmarks for various problem sizes are shown. The second study is to evaluate how effectively the finite element procedures for viscoelastic composite structures can be solved in the Single Instruction Multiple Data (SIMD) parallel environment. CM-FORTRAN is used. A conjugate gradient method is employed for the solution of systems. In the third study, we propose to implement the finite element algorithm in a scalable distributed parallel environment using a generic message passing library such as PVM. The code is portable to a range of current and future parallel machines. We also introduced the domain decomposition scheme to reduce the communication time. The parallel scalability of the dynamic viscoelastic finite element algorithm in data parallel and scalable distributed parallel environments is also discussed. © 1997 by John Wiley & Sons, Ltd.     

Transient elastodynamic analysis of nonhomogeneous anisotropic plane bodies

A boundary-only BEM procedure is employed to solve the transient dynamic analysis of nonhomogeneous anisotropic plane elastic bodies. The response of such bodies is governed by two coupled linear, second-order hyperbolic PDEs with spatially dependent coefficients. The lack of a reliable 2D time-domain elastodynamic fundamental solution is overcome using the principle of the Analog Equation, a method by which the equations of motion of the problem are substituted by two coupled quasi-static Poisson-type equations having as nonhomogeneous terms the components of a fictitious time-dependent load distribution in the specified domain. The standard BEM is employed for the solution of the substitute equations. To avoid the appearance of the domain integral in the integral representation of the solution, the fictitious load distribution is approximated by multiquadrics with unknown time-dependent expansion coefficients, which are calculated at discrete timepoints by collocating the equations of motion at a predefined set of domain interpolation nodes. The obtained numerical results by the proposed method demonstrate its stability and accuracy over other numerical methods.     

Local integral equation method for viscoelastic Reissner–Mindlin plates

A meshless local Petrov-Galerkin (MLPG) method is applied to solve static and dynamic bending problems of linear viscoelastic plates described by the Reissner–Mindlin theory. To this end, the correspondence principle is applied. A weak formulation for the set of governing equations in the Reissner–Mindlin theory with a unit test function is transformed into local integral equations on local subdomains in the mean surface of the plate. Nodal points are randomly spread on the mean surface of the plate and each node is surrounded by a circular subdomain to which local integral equations are applied. A meshless approximation based on the moving least-squares (MLS) method is employed in the numerical implementation.     

A three-dimensional micromechanical model to predict the viscoelastic behavior of woven composites

Polymer matrix based cloth composites are increasingly used in engineering applications. For such composites, significant viscoelastic behavior can be observed for dynamic load conditions. The viscoelastic effect is primarily due to the polymeric matrix used as most of the fibers used in structural applications are elastic. Matrix does not show a major contribution in the axial properties of composites, thus in the traditional modeling its viscoelastic nature is often ignored. However, the effective out of plane properties are influenced by the matrix material and exhibit significant damping characteristics. Therefore, a complete three-dimensional (3-D) model considering the viscoelastic nature of matrix is needed for better understanding of cloth composites. An analytical 3-D micromechanical model based on classical laminate theory (CLT) is verified, in this paper for the prediction of effective elastic and viscoelastic properties of a cloth composite. The method is shown to be accurate. This model is extended to the viscoelastic regime with the use of Laplace transform and correspondence principle. Prony series coefficients for composite cloth are obtained for different volume fractions of fibers in yarn. It is observed from the hysteresis plots that dissipation in out of plane normal and shear modes is significantly higher than the normal directions.     

Two-dimensional stress analysis of functionally graded solids using the MLPG method with radial basis functions

The meshless local Petrov–Galerkin (MLPG) method is used for analysing two-dimensional (2D) static and dynamic deformations of functionally graded materials (FGMs) with material response modelled as either linear elastic or as linear viscoelastic. The multiquadric radial basis function (RBF) is employed to approximate the trial solution. Results are computed with two different choices of test functions, namely a fourth-order spline weight function, and a Heaviside step function, each having a compact support. No background mesh is used to numerically evaluate integrals appearing in the weak formulation of the problem, thus the method is truly meshless. A benefit of using RBFs is that they possess the Kronecker delta property; thus it is easy to satisfy essential boundary conditions. For five problems, the computed results are found to match well with those either from their analytical solutions or numerical solutions of other researchers who employed different algorithms. For a dynamic problem, the Laplace-transform technique is utilised. The numerical examples illustrate that displacements and stress distributions in a structure made of an FGM differ considerably from those at the corresponding points in the same structure made of a homogeneous material. Thus, the inhomogeneity in material properties can be exploited to optimise stress distribution, minimise deflection and reduce the maximum stress.     

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.     

Application of local boundary integral equation method into micropolar elasticity

A new meshless method for solving boundary value problems in micropolar elasticity is presented. The method is based on the local boundary integral equation (LBIE) method with the moving least squares approximation of physical quantities. Randomly scattered nodes are utilized for interpolation of field data. Every node is surrounded by a simple surface centered at the collocation point in the LBIE method. On the surface of subdomains the LBIEs are written. Fundamental solutions corresponding to uncoupled governing equations are derived. To eliminate the traction vector in the LBIE, the modified fundamental solution is introduced.     

A 3-D boundary element formulation of viscoelastic media with gravity

This paper presents a boundary element formulation for 3-D linear and viscoelastic bodies subjected to the body force of gravity. The Laplace transformation is first used to suppress the time variable, and solutions of displacements and stresses are found in the transformed domain. The time domain solutions are then found by an accurate and efficient numerical inversion method which requires only real calculations for all quantities. Input and output data, and solutions in the transformed and time domains are connected through an Interactive Data Language code written by the authors. While particular solutions of stresses and displacements related to the body force of gravity (which is applied at time t = 0 and is kept constant) are derived, the Green's functions in the Laplace domain are obtained through the correspondence principle. The new formulation has been implemented into an existing 3-D BEM program, and several numerical examples involving 3-D viscoelastic bodies are presented. Although the discussion in this paper focuses on Maxwell viscoelastic and isotropic media, other linear isotropic and even anisotropic viscoelastic models can also be incorporated, without difficulty, into the 3-D viscoelastic BEM program.     

Three dimensional automatic refinement method for transient small strain elastoplastic finite element computations

In this paper, the refinement strategy based on the “Non-Linear Localized Full MultiGrid” solver originally published in Int. J. Numer. Meth. Engng 84(8):947–971 (2010) for 2-D structural problems is extended to 3-D simulations. In this context, some extra information concerning the refinement strategy and the behavior of the error indicators are given. The adaptive strategy is dedicated to the accurate modeling of elastoplastic materials with isotropic hardening in transient dynamics. A multigrid solver with local mesh refinement is used to reduce the amount of computational work needed to achieve an accurate calculation at each time step. The locally refined grids are automatically constructed, depending on the user prescribed accuracy. The discretization error is estimated by a dedicated error indicator within the multigrid method. In contrast to other adaptive procedures, where grids are erased when new ones are generated, the previous solutions are used recursively to reduce the computing time on the new mesh. Moreover, the adaptive strategy needs no costly coarsening method as the mesh is reassessed at each time step. The multigrid strategy improves the convergence rate of the non-linear solver while ensuring the information transfer between the different meshes. It accounts for the influence of localized non-linearities on the whole structure. All the steps needed to achieve the adaptive strategy are automatically performed within the solver such that the calculation does not depend on user experience. This paper presents three-dimensional results using the adaptive multigrid strategy on elastoplastic structures in transient dynamics and in a linear geometrical framework. Isoparametric cubic elements with energy and plastic work error indicators are used during the calculation.     

A meshless local boundary integral equation method for dynamic anti-plane shear crack problem in functionally graded materials

This paper presents a meshless local boundary integral equation method (LBIEM) for dynamic analysis of an anti-plane crack in functionally graded materials (FGMs). Local boundary integral equations (LBIEs) are formulated in the Laplace-transform domain. The static fundamental solution for homogeneous elastic solids is used to derive the local boundary–domain integral equations, which are applied to small sub-domains covering the analyzed domain. For the sub-domains a circular shape is chosen, and their centers, the nodal points, correspond to the collocation points. The local boundary–domain integral equations are solved numerically in the Laplace-transform domain by a meshless method based on the moving least–squares (MLS) scheme. Time-domain solutions are obtained by using the Stehfest's inversion algorithm. Numerical examples are given to show the accuracy of the proposed meshless LBIEM.