Showing papers by "Vladimir Sladek published in 2006"

Heat Conduction Analysis of 3-D Axisymmetric and Anisotropic FGM Bodies by Meshless Local Petrov–Galerkin Method

Abstract: The meshless local Petrov–Galerkin method is used to analyze transient heat conduction in 3-D axisymmetric solids with continuously inhomogeneous and anisotropic material properties. A 3-D axisymmetric body is created by rotation of a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduces the original 3-D boundary value problem into a 2-D problem. The cross section is covered by small circular subdomains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function, in order to derive local integral equations on the boundaries of the chosen subdomains, called local boundary integral equations. These integral formulations are either based on the Laplace transform technique or the time difference approach. The local integral equations are nonsingular and take a very simple form, despite of inhomogeneous and anisotropic material behavior across the analyzed structure. 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 method. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.

Inverse heat conduction problems by meshless local Petrov–Galerkin method

Abstract: 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.

Meshless Local Petrov-Galerkin (MLPG) Method for Shear Deformable Shells Analysis

Abstract: A meshless local Petrov-Galerkin (MLPG) method is applied to solve bending problems of shear deformable shallow shells described by the Reissner theory. Both static and dynamic loads are considered. For transient elastodynamic case the Laplace-transform is used to eliminate the time dependence of the field variables. A weak formulation with a unit test function transforms the set of governing equations into local integral equations on local subdomains in the mean surface of the shell. Nodal points are randomly spread on that surface and each node is surrounded by a circular subdomain to which local integral equations are applied. The meshless approximation based on the Moving LeastSquares (MLS) method is employed for the implementation. Unknown Laplace-transformed quantities are computed from the local boundary integral equations. The time-dependent values are obtained by the Stehfest’s inversion technique. keyword: Reissner theory, local boundary integral equations, Laplace-transform, Stehfest’s inversion, MLS approximation, static and impact loads

Meshless local Petrov-Galerkin method for continuously nonhomogeneous linear viscoelastic solids

Abstract: A meshless method based on the local Petrov-Galerkin approach is proposed for the solution of quasi-static and transient dynamic problems in two-dimensional (2-D) nonhomogeneous linear viscoelastic media. A unit step function is used as the test functions in the local weak form. It is leading to local boundary integral equations (LBIEs) involving only a domain-integral in the case of transient dynamic problems. The correspondence principle is applied to such nonhomogeneous linear viscoelastic solids where relaxation moduli are separable in space and time variables. Then, the LBIEs are formulated for the Laplace-transformed viscoelastic problem. The analyzed domain is covered by small subdomains with a simple geometry such as circles in 2-D problems. The moving least squares (MLS) method is used for approximation of physical quantities in LBIEs.

Analysis of thick functionally graded plates by local integral equation method

Abstract: Analysis of functionally graded plates under static and dynamic loads is presented by the meshless local Petrov–Galerkin (MLPG) method. Plate bending problem is described by Reissner–Mindlin theory. Both isotropic and orthotropic material properties are considered in the analysis. 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 considered on local subdomains in the mean surface of the plate. Nodal points are randomly spread on this surface and each node is surrounded by a circular subdomain, rendering integrals which can be simply evaluated. The meshless approximation based on the moving least-squares (MLS) method is employed in the numerical implementation. Numerical results for simply supported and clamped plates are presented. Copyright © 2006 John Wiley & Sons, Ltd.

Transient heat conduction analysis by triple-reciprocity boundary element method

Abstract: The paper deals with 2D initial-boundary value problems in the linear theory of transient heat conduction. A pure boundary element formulation is developed systematically. The time-dependent fundamental solution of the diffusion operator is employed together with higher-order polyharmonic fundamental solutions. The pseudo-initial temperature and/or heat sources density are approximated by using the triple-reciprocity formulation. All the time integrations are performed analytically in the time-marching scheme with integration within one time step and constant interpolation. The spatial discretization is reduced to boundary elements and free scattering of interior nodal points without any connectivity.

Analysis of orthotropic thick plates by meshless local Petrov–Galerkin (MLPG) method

Abstract: A meshless local Petrov–Galerkin (MLPG) method is applied to solve static and dynamic problems of orthotropic plates described by the Reissner–Mindlin theory. Analysis of a thick orthotropic plate resting on the Winkler elastic foundation is given too. 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 surface of the plate and each node is surrounded by a circular subdomain to which local integral equations are applied. The meshless approximation based on the Moving Least–Squares (MLS) method is employed in the numerical implementation. The present computational method is applicable also to plates with varying thickness. Numerical results for simply supported and clamped plates are presented. Copyright © 2006 John Wiley & Sons, Ltd.

Meshless Local Petrov-Galerkin Method for Linear Coupled Thermoelastic Analysis

Abstract: The Meshless Local Petrov-Galerkin (MLPG) method for linear transient coupled thermoelastic analysis is presented. Orthotropic material properties are considered here. A Heaviside step function as the test functions is applied in the weak-form to derive local integral equations for solving two-dimensional (2-D) problems. In transient coupled thermoelasticity an inertial term appears in the equations of motion. The second governing equation derived from the energy balance in coupled thermoelasticity has a diffusive character. To eliminate the time-dependence in these equations, the Laplace-transform technique is applied to both of them. Local integral equations are written on small sub-domains with a circular shape. They surround nodal points which are distributed over the analyzed domain. The spatial variation of the displacements and temperature are approximated by the Moving LeastSquares (MLS) scheme. After performing the spatial integrations, a system of linear algebraic equations for unknown nodal values is obtained. The boundary conditions on the global boundary are satisfied by the collocation of the MLS-approximation expressions for the displacements and temperature at the boundary nodal points. The Stehfest’s inversion method is then applied to obtain the final time-dependent solutions. keyword: Transient coupled thermoelasticity, Orthotropic materials, Moving least-squares interpolation, 2-D problems, Laplace-transform, Stehfest’s inversion 1 Indroduction Dynamic thermoelasticity is relevant for many engineering problems since thermal stresses play an important role in the integrity of structures. In the case of tra1 Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia 2 Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany 3 Department of Mechanical & Aerospace Engineering, Carleton University, Ottawa, Canada ditional materials thermal effects on a body are limited to strains due to the temperature gradient. For sophisticated materials such as high performance composites thermal effects can include heat production due to the strain rate, i.e. the thermoelastic dissipation. Several computational methods have been proposed over the past years to analyze thermoelasticity problems. Many of them have been directed to uncoupled problems in steady or transient heat conduction states. Few investigations have been done successfully for coupled thermoelasticity. Domain-based approaches, particularly those involving the finite element method (FEM), have been developed and applied to thermoelasticity [Keramidas and Ting (1976); Prevost and Tao (1983); Cannarozzi and Ubertini (2001)]. The boundary element method (BEM), recognized since many years as a powerful tool in numerical analysis, was applied for the first time to transient uncoupled thermoelasticity by Rizzo and Shippy (1977). Shiah and Tan (1999) applied the BEM for 2-D uncoupled thermoelasticity in anisotropic solids. Thermomechanical crack growth has been investigated by Prasad (1998) using a dual BEM. Particular integral formulations for 2-D and 3-D transient uncoupled thermoelastic analyses have been presented by Park and Banerjee (2002). The BEM has been successfully applied also to coupled thermoelastic problems [Sladek and Sladek (1984); Dargush and Banerjee (1991); Chen and Dargush (1995); Suh and Tosaka (1989); Hosseini-Tehrani and Eslami (2000)]. Dual reciprocity BEM has been presented by Gaul et al. (2003), and Kögl and Gaul (2000, 2003). Recently developed sophisticated materials with thermoelastic dissipation are composites with anisotropic properties. Governing equations for coupled thermal and mechanical fields with anisotropic material properties are much more complex than those in uncoupled thermoelasticity for isotropic materials. Thus, efficient computational methods are required to solve the boundary or the initial-boundary value thermoelastic problems for 58 Copyright c © 2006 Tech Science Press CMES, vol.16, no.1, pp.57-68, 2006 anisotropic solids. In recent years, an increasing attention has been paid to the numerical analysis of coupled thermoelasticity problems. In spite of the great success of the FEM and BEM as effective numerical tools for the solution of boundary or initial-boundary value problems in elasticity, there is still a growing interest in the development of new advanced methods. In particular, meshless formulations are becoming popular due to their high adaptivity and low costs to prepare input and output data for numerical analyses. A variety of meshless methods has been proposed so far and some of them also applied to transient heat conduction problems [Batra et al. (2003); Sladek et al. (2003a,b, 2004a, 2006); Qian and Batra (2004); Wang et al. (2006)] or to thermoelastic problems [Sladek et al. (2001); Bobaru and Mukherjee (2003); Qian and Batra (2004)]. The meshless method can be obtained from a weak-form formulation on either the global domain or a set of local subdomains. In the global formulation background cells are required for the integration of the weak-form. In methods based on local weak-form formulation, no background cells are required and therefore they are often referred to as truly meshless methods. The meshless local Petrov-Galerkin (MLPG) method is a fundamental base for the derivation of many meshless formulations, since trial and test functions can be chosen from different functional spaces. By using the fundamental solution as the test function, accurate numerical results can be obtained, which were reported in previous papers for 2-D transient heat conduction problems in isotropic, homogeneous or continuously nonhomogeneous solids [Sladek et al. (2003a,b)], elasticity under static and dynamic loads [Atluri et al. (2000, 2003); Sellountos and Polyzos (2003); Sellountos et al. (2005)], and for 3-D problems in homogeneous and isotropic solids under a static or a dynamic load [Han and Atluri (2004a,b)]. In this paper, the MLPG method with a Heaviside step function as the test functions [Atluri et al. (2003); Atluri (2004); Sladek et al. (2004a,b)] is applied to solve two-dimensional transient coupled thermoelasticity problems. An inertial term exists in the equations of motion for transient thermoelasticity. The second governing equation derived from the energy balance has a diffusive character. To eliminate the time-dependences in both governing partial differential equations, the Laplacetransform technique is applied such that they are satisfied in the Laplace-transformed domain in a weak-form on small fictitious subdomains. If the shape of subdomains has a simple form, numerical integrations over them can be easily carried out. Nodal points are introduced and distributed over the analyzed domain and each of them is surrounded by a small circle for simplicity, but without loss of generality. The integral equations have a very simple nonsingular form. The spatial variations of the displacements and temperature are approximated by the Moving Least-Squares (MLS) scheme [Belytschko et al. (1996); Zhu et al. (1998)]. After performing the spatial integrations, a system of linear algebraic equations for unknown nodal values is obtained. The boundary conditions on the global boundary are satisfied by the collocation of the MLS-approximation expressions for the displacements and temperature at the boundary nodal points. To obtain the final time-dependent solutions, the Stehfest’s inversion method [Stehfest (1970)] is applied. The accuracy and the efficiency of the proposed MLPG method are verified by numerical examples. 2 The MLPG in transient coupled thermoelasticity A homogeneous, orthotropic and linear elastic solid is considered. The equilibrium and the thermal balance equations in transient coupled thermoelasticity [Nowacki (1986)] can be written as σi j, j(x,τ)−ρüi(x,τ)+Xi(x,τ) = 0, (1) [ki j(x)θ, j(x,τ)],i−ρcθ̇(x,τ)−γi jθ0u̇i, j(x,τ)+Q(x,τ) = 0, (2) where σi j , τ, θ ,θ0 ,ui ,Xi and Q are the stress, time, temperature difference, reference temperature, displacement, density of body force vector and density of heat sources, respectively. Also, ρ, ki j , c and γi j and are the mass density, thermal conductivity tensor, specific heat, stresstemperature modulus, respectively. The dots over a quantity indicate the time derivatives. A static problem can be considered formally as a special case of the dynamic one, by omitting the acceleration üi(x,τ) in the equations of motion (1) and the time derivative terms in equation (2). Therefore, both cases are analyzed in this paper. In the case of orthotropic materials, the relation between the stress σi j and the strain εi j when temperature changes are considered, is governed by the well known DuhamelNeumann constitutive equations for the stress tensor σi j(x,τ) = ci jklεkl(x,τ)− γi jθ(x,τ), (3) Meshless Local Petrov-Galerkin Method for Linear Coupled Thermoelastic Analysis 59 where ci jkl are the material stiffness coefficients. The stress-temperature modulus can be expressed through the stiffness coefficients and the coefficients of linear thermal expansion αkl γi j = ci jklαkl . (4) For plane problems the constitutive equation (3) is frequently written in terms of the second-order tensor of elastic constants [Lekhnitskii (1963)]. The constitutive equation for orthotropic materials and plane strain problems has the following form ⎡ ⎣ σ11 σ22 σ12 ⎤ ⎦ = ⎡ ⎣ c11 c12 0 c12 c22 0 0 0 c66 ⎤ ⎦ ⎡ ⎣ ε11 ε22 2ε12 ⎤ ⎦ − ⎡ ⎣ c11 c12 c13 c12 c22 c23 0 0 0 ⎤ ⎦ ⎡ ⎣ α11 α22 α33 ⎤ ⎦θ = C ⎡ ⎣ ε11 ε22 2ε12 ⎤ ⎦− γθ, (5) with γ = ⎡ ⎣ c11 c12 c13 c12 c22 c23 0 0 0 ⎤ ⎦ ⎡ ⎣ α11 α22 α33 ⎤ ⎦ = ⎡ ⎣ γ11 γ22 0 ⎤ ⎦. Equation (5) can be reduced to a simple form for isotropic materials σi j = 2μεi j +λεkkδi j − (3λ+2μ)αθδi j , (6) with Lame’s constants λand μ . The following essential and natural boundary conditions are assumed for the mechanical quantities ui(x,τ) = ũi(x,τ) on Γu, ti(x,τ) = σi j(x,τ)n j(x) = t̃i(x,

Evaluation of fracture parameters for crack problems in fgm by a meshless method

Abstract: A meshless method based on the local Petrov-Galerkin approach is proposed for crack analysis in two-dimensional (2D), anisotropic and linear elastic solids with continuously varying material properties. Both quasi-static thermal and transient elastodynamic problems are considered. For time-dependent problems, the Laplace transform technique is utilized. The analyzed domain is divided into small subdomains of circular shapes. A unit step function is used as the test function in the local weak form. It leads to Local Integral Equations (LIE) involving a domain-integral only in the case of transient dynamic problems. The Moving Least Squares (MLS) method is adopted for approximating the physical quantities in the LIE. Efficient numerical methods are presented to compute the fracture parameters, namely, the stress intensity factors and the $ T$-stress, for a crack in Functionally Graded Materials (FGM). The path-independent integral representations for stress intensity f actors and $ T$-stresses in continuously non-homogeneous FGM are presented.

A frequency-domain BEM for 3D non-synchronous crack interaction analysis in elastic solids

Abstract: A frequency-domain boundary element method (BEM) is presented for non-synchronous crack interaction analysis in three-dimensional (3D), infinite, isotropic and linear elastic solids with multiple coplanar cracks. The cracks are subjected to non-synchronous time-harmonic crack-surface loading with contrast frequencies. Hypersingular frequency-domain traction boundary integral equations (BIEs) are applied to solve the boundary value problem. A collocation method is adopted for solving the BIEs numerically. The local square-root behavior of the crack-opening-displacements at the crack-front is taken into account in the present method. For two coplanar penny-shaped cracks of equal radius subjected to non-synchronous time-harmonic crack-surface loading, numerical results for the dynamic stress intensity factors are presented and discussed.

A Comparative Study of Meshless Approximations in Local Integral Equation Method

Abstract: This paper concerns the stability, convergence of accuracy and cost efficiency of four various formulations for solution of boundary value problems in non-homogeneous elastic solids with functionally graded Young’s modulus. The meshless point interpolation method is employed with using various basis functions. The interaction among the elastic continuum constituents is considered in the discretized formulation either by collocation of the governing equations or by integral satisfaction of the force equilibrium on local sub-domains. The exact benchmark solutions are used in numerical tests. keyword: Integral equation method, Integral force equilibrium, Collocation method, Point interpolation method, Radial basis functions, Elasticity, Functionally Graded Materials (FGMs)

Fracture Mechanics Analysis of 2-D FGMs by a Meshless BEM

Abstract: This paper presents a fracture mechanics analysis in continuously non-homogeneous, isotropic, linear elastic and functionally graded materials (FGMs). A meshless boundary element method (BEM) is developed for this purpose. Young’s modulus of the FGMs is assumed to have an exponential variation, while Poisson’s ratio is taken as constant. Since no simple fundamental solutions are available for general FGMs, fundamental solutions for homogeneous, isotropic and linear elastic solids are used in the present BEM, which contains a domain-integral due to the material non-homogeneity. Normalized displacements are introduced to avoid displacement gradients in the domain-integral. The domain-integral is transformed into a boundary integral along the global boundary by using the radial integration method (RIM). To approximate the normalized displacements arising in the domain-integral, basis functions consisting of radial basis functions and polynomials in terms of global coordinates are applied. Numerical results are presented and discussed to show the accuracy and the efficiency of the present meshless BEM.