Inverse heat conduction problems in three-dimensional anisotropic functionally graded solids

A meshless method based on the local Petrov–Galerkin approach is applied to inverse transient heat conduction problems in three-dimensional solids with continuously inhomogeneous and anisotropic material properties. The Heaviside step function is used as a test function in the local weak form, leading to the derivation of local integral equations. Nodal points are randomly distributed in the domain analyzed, and each node is surrounded by a spherical subdomain in which a local integral equation is applied. A meshless approximation based on the moving least-squares method is employed in the implementation. After performing spatial integrations, we obtain a system of ordinary differential equations for certain nodal unknowns. A backward finite-difference method is used for the approximation of the diffusive term in the heat conduction equation. A truncated singular-value decomposition is used to solve the ill-conditioned linear system of algebraic equations at each time step. The effectiveness of the meshless local Petrov–Galerkin (MLPG) method for this inverse problem is demonstrated by numerical examples.     

New implementation of MLBIE method for heat conduction analysis in functionally graded materials

The meshless local boundary integral equation (MLBIE) method with an efficient technique to deal with the time variable are presented in this article to analyze the transient heat conduction in continuously nonhomogeneous functionally graded materials (FGMs). In space, the method is based on the local boundary integral equations and the moving least squares (MLS) approximation of the temperature and heat flux. In time, again the MLS approximates the equivalent Volterra integral equation derived from the heat conduction problem. It means that, the MLS is used for approximation in both time and space domains, and we avoid using the finite difference discretization or Laplace transform methods to overcome the time variable. Finally the method leads to a single generalized Sylvester equation rather than some (many) linear systems of equations. The method is computationally attractive, which is shown in couple of numerical examples for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.     

Local BIEM for transient heat conduction analysis in 3-D axisymmetric functionally graded solids

An advanced computational method for transient heat conduction analysis in 3-D axisymmetric continuously nonhomogeneous functionally graded materials (FGM) is proposed. The analysed domain is covered by small circular subdomains. On each subdomain local boundary integral equations for the transient heat conduction problem are derived in the Laplace transform domain. The meshless approximation based on the moving least-squares method is employed for the numerical implementation. The Stehfest algorithm is applied for the numerical Laplace inversion to obtain the temporal variation. Numerical results are presented for finite full and hollow cylinders with an exponential variation of material parameters with spatial coordinates. The authors acknowledge the support by the Slovak Science and Technology Assistance Agency registered under number APVT-51-003702, and the Project for Bilateral Cooperation in Science and Technology supported jointly by the International Bureau of the German BMBF and the Ministry of Education of Slovak Republic under the project number SVK 01/020.     

Interface integral BEM for solving multi-medium heat conduction problems

In this paper, a new and simple boundary element method, called interface integral boundary element method (IIBEM), is presented for solving heat conduction problems consisting of multiple media. In the method, the boundary integral equation is derived by a degeneration technique from domain integrals involved in varying heat conductivity problems into interface integrals in multi-medium problems. The main feature of the presented technique is that only a single boundary integral equation is used to solve heat conduction problems with different material properties. The effect of nonhomogeneity between adjacent materials is embodied in the interface integrals including the material property difference between the two adjacent materials. Comparing with conventional multi-domain boundary integral equation techniques, the presented method is more efficient in computational time, data preparing, and program coding. Numerical examples are given to verify the correctness of the presented technique.     

Elastodynamic problems by meshless local integral method: Analytical formulation

In this paper, analytical forms of integrals in the meshless local integral equation method in the Laplace space are derived and implemented for elastodynamic problems. The meshless approximation based on the radial basis function (RBF) is employed for implementation of displacements. A weak form of governing equations with a unit test function is transformed into local integral equations. A completed set of the local boundary integrals are obtained in closed form. As the closed form of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically. Several examples including dynamic fracture mechanics problems are presented to demonstrate the accuracy of the proposed method in comparison with analytical solutions and the boundary element method.     

层合板分析的无网格局部Petrov-Galerkin方法

基于Kirchhoff均匀各向异性板控制方程的等效积分弱形式和对挠度函数采用移动最小二乘近似函数进行插值, 进一步研究无网格局部Petrov-Galerkin方法在纤维增强对称层合板弯曲问题中的应用。该方法不需要任何形式的网格划分, 所有的积分都在规则形状的子域及其边界上进行,其问题的本质边界条件采用罚因子法来施加。通过数值算例和与其他方法的结果比较, 表明无网格局部Petrov-Galerkin法求解层合薄板弯曲问题具有解的精度高、收敛性好等一系列优点。      

Meshless boundary integral equations with equilibrium satisfaction

A known feature of any mixed interpolation boundary integral equations (BIE)-based methods is that equilibrium is not generally guaranteed in the numerical solution. Here, a complete meshless technique, based on the boundary element-free method (BEFM) with complete equilibrium satisfaction for 2D elastostatic analysis is proposed. The BEFM adopted is a meshless method based on boundary integral equations such as local boundary integral equation (LBIE) method and boundary node method (BNM), differing from them with respect to the integration domain and the approximation scheme.     

A truly generalized plane strain meshless method for combined normal and shear loading of fibrous composites

A truly meshless method based on the integral form of equilibrium equations is developed and formulated to study the micro-stresses in the unidirectional fiber reinforced composite material which is subjected to the normal and shear load. The presented meshless method is formulated for a special form of generalized plane strain assumption and is employed for micromechanical modeling of the composite. A small repeating area of the composite including a fiber surrounded by matrix, which is called representative volume element (RVE) is considered as the solution domain. A direct method is proposed for the enforcement of the appropriate periodic boundary conditions to the representative volume element (RVE) in shear and normal loading conditions. The computational efforts of the method are reduced because the domain integration has been eliminated from the formulation. The fully bonded interface condition is investigated and the continuity of displacements and traction is directly imposed to the fiber–matrix interface. Comparison of the predicted results shows excellent agreement with the results in the available literature. Results of this study also revealed that the presented model can provide highly accurate predictions with relatively small number of nodes and small computational time.     

Three dimensional static and dynamic analysis of thick functionally graded plates by the meshless local Petrov–Galerkin (MLPG) method

In this paper, three dimensional (3D) static and dynamic analysis of thick functionally graded plates based on the Meshless Local Petrov–Galerkin (MLPG) is presented. Using the kinematics of a three-dimensional continuum, the local weak form of the equilibrium equations is derived. A weak formulation for the set of governing equations is transformed into local integral equations on local sub-domains using a Heaviside step function as test function. In this case, governing equations corresponding to the stiffness matrix do not involve any domain integration or singular integrals. Nodal points are distributed in the 3D analyzed domain and each node is surrounded by a cubic sub-domain to which a local integral equation is applied. The meshless approximation based on the three-dimensional Moving Least-Square (MLS) is employed as shape function to approximate the field variable of scattered nodes in the problem domain. The Newmark time integration method is used to solve the system of coupled second-order ODEs. Effective material properties of the plate, made of two isotropic constituents with volume fractions varying only in the thickness direction, are computed using the Mori–Tanaka homogenization technique. Numerical examples for solving the static and dynamic response of elastic thick functionally graded plates are demonstrated. As a result, the distributions of the deflection and stresses through the plate thickness are presented for different material gradients and boundary conditions. The effects of the volume fractions of the constituents on the centroidal deflection are also investigated. The numerical efficiency of the proposed meshless method is illustrated by the comparison of results obtained from previous literatures.     

The local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity

 The meshless method based on the local boundary integral equation (LBIE) is a promising method for solving boundary value problems, using an local unsymmetric weak form and shape functions from the moving least squares approximation. In the present paper, the meshless method based on the LBIE for solving problems in linear elasticity is developed and numerically implemented. 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. All integrals in the formulation can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions in the present formulation can be easily imposed even when the non-interpolative moving least squares approximation is used. Several numerical examples are presented to illustrate the implementation and performance of the present method. The numerical examples show that high rates of convergence with mesh refinement for the displacement and energy norms are achievable. No post-processing procedure is required to compute the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.     

A new method for meshless integration in 2D and 3D Galerkin meshfree methods

A method for the evaluation of regular domain integrals without domain discretization is presented. In this method, a domain integral is transformed into a boundary integral and a 1D integral. The method is then utilized for the evaluation of domain integrals in meshless methods based on the weak form, such as the element-free Galerkin method and the meshless radial point interpolation method. The proposed technique results in truly meshless methods with better accuracy and efficiency in comparison with their original forms. Some examples, including linear and large-deformation problems, are also provided to demonstrate the usefulness of the proposed method.     

A meshless BEM for isotropic heat conduction problems with heat generation and spatially varying conductivity

In this paper, a new and simple boundary‐domain integral equation is presented for heat conduction problems with heat generation and non‐homogeneous thermal conductivity. Since a normalized temperature is introduced to formulate the integral equation, temperature gradients are not involved in the domain integrals. The Green's function for the Laplace equation is used and, therefore, the derived integral equation has a unified form for different heat generations and thermal conductivities. The arising domain integrals are converted into equivalent boundary integrals using the radial integration method (RIM) by expressing the normalized temperature using a series of basis functions and polynomials in global co‐ordinates. Numerical examples are given to demonstrate the robustness of the presented method. Copyright © 2005 John Wiley & Sons, Ltd.     

An improved meshless method with almost interpolation property for isotropic heat conduction problems

In the paper an improved element free Galerkin method is presented for heat conduction problems with heat generation and spatially varying conductivity. In order to improve computational efficiency of meshless method based on Galerkin weak form, the nodal influence domain of meshless method is extended to have arbitrary polygon shape. When the dimensionless size of the nodal influence domain approaches 1, the Gauss quadrature point only contributes to those nodes in whose background cell the Gauss quadrature point is located. Thus, the bandwidth of global stiff matrix decreases obviously and the node search procedure is also avoided. Moreover, the shape functions almost possess the Kronecker delta function property, and essential boundary conditions can be implemented without any difficulties. Numerical results show that arbitrary polygon shape nodal influence domain not only has high computational accuracy, but also enhances computational efficiency of meshless method greatly.     

Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method

Advanced computational method for transient heat conduction analysis in continuously nonhomogeneous functionally graded materials (FGM) is proposed. The method is based on the local boundary integral equations with moving least square approximation of the temperature and heat flux. The initial-boundary value problem is solved by the Laplace transform technique. Both Papoulis and Stehfest algorithms are applied for the numerical Laplace inversion to obtain the time-dependent solutions. Numerical results are presented for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.     

Transient heat conduction in anisotropic and functionally graded media by local integral equations

Reliable computational techniques are developed for the solution of two-dimensional (2-d) transient heat conduction problems in anisotropic media with continuously variable material coefficients. Two kinds of the domain-type interpolation, namely the standard domain elements and the meshless point interpolation, are adopted for the approximation of the spatial variation of the temperature field or its Laplace-transform. The coupling among the nodal values of the approximated field is given by integral equations considered on local sub-domains. Three kinds of local integral equations are derived from physical principles instead of using a weak-form formulation. The accuracy and the convergence of the proposed techniques are tested by several examples and compared with exact benchmark solutions, which are derived too.     

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.     

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

A novel meshless local boundary integral equation (LBIE) method is proposed for the numerical solution of two-dimensional steady elliptic problems, such as heat conduction, electrostatics or linear elasticity. The domain is discretized by a distribution of boundary and internal nodes. From this nodal points’ cloud a “background” mesh is created by a triangulation algorithm. A local form of the singular boundary integral equation of the conventional boundary elements method is adopted. Its local form is derived by considering a local domain of each node, comprising by the union of neighboring “background” triangles. Therefore, the boundary shape of this local domain is a polygonal closed line. A combination of interpolation schemes is taken into account. Interpolation of boundary unknown field variables is accomplished through boundary elements’ shape functions. On the other hand, the Radial Basis Point Interpolation Functions method is employed for interpolating the unknown interior fields. Essential boundary conditions are imposed directly due to the Kronecker delta-function property of the boundary elements’ interpolation functions. After the numerical evaluation of all boundary integrals, a banded stiffness matrix is constructed, as in the finite elements method. Several potential and elastostatic benchmark problems in two dimensions are solved numerically. The proposed meshless LBIE method is also compared with other numerical methods, in order to demonstrate its efficiency, accuracy and convergence.     

Voronoi cell finite element model based on micropolar theory of thermoelasticity for heterogeneous materials

In this paper, a new ‘Voronoi cell finite element model’ is developed for solving steady-state heat conduction and micropolar thermoelastic stress analysis problems in arbitrary heterogeneous materials. The method is based on the natural discretization of a multiple phase domain into basic structural elements by Dirichlet Tessellation. Tessellation process results in a network of polygons called Voronoi polygons. In this paper, formulations are developed for treating these polygons as elements in a finite element mesh. Furthermore, a composite Voronoi cell finite element model is developed to account for the presence of a second phase inclusion within a polygonal element. Various numerical examples are executed for validating the effectiveness of this model in the analysis of the temperature and stress fields for micropolar elastic materials. Effective material properties are derived for microstructures containing different distributions of second phase.     

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

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.     

Transient heat conduction analysis of solids with small open-ended tubular cavities by boundary face method

This paper applies the boundary face method (BFM) to solve transient heat conduction problems for the first time. Rather than using a transformation scheme, a direct solution of the boundary integral equation (BIE) with time domain fundamental solution is performed in this application. To avoid the domain integrals, the boundary integral equation is solved by the time stepping convolution method. For problems on structures that contain a large number of open-ended tubular shaped cavities in small diameters, a curvilinear tube element is employed to approximate the variables on the cavity surface. Furthermore, to perform integration and boundary variable approximation on the end faces that are intersected by the tubular cavity, a triangular element with negative part is adopted. With the two types of specified elements, the BFM is implemented to solve transient heat conduction problems on structures with open-ended tubular shaped cavities of small size which are usually inconvenience in finite element implementations. Three numerical examples on different structures are presented to illustrate the validity and efficiency of the method.