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It is well known that the classical theory of heat conduction is scale invariant. On the other hand, it is experimentally evident that size-dependent effects are observable in small samples of micro/nano-scale dimensions. Incorporation of higher-order gradients of primary field variables into constitutive relationships yields a qualitative explanation of size-effects. Form the mathematical point of view, the governing equations are given by the partial differential equations (PDE) with higher-order derivatives in higher-grade continuum theories. As compared with classical theory of continua, the other complication of governing equations occurs in case of continuous media with functional gradation of material coefficients, when the problem is described by the PDE with variable coefficients. The traditional weak formulations are considered in global sense, hence the whole analysed domain/boundary is to be discretized into finite size elements. On the other hand, the strong formulations bring better computational efficiency because of elimination of integrations, but the price which should be paid is the need to approximate higher order derivatives of field variables. One of the most criticized shortcoming of the finite element (FE) approximation is its limited continuity on element intersections. The C 0 continuity is insufficient for calculation of gradients of field variables on element intersections as well as for numerical solution of problems with governing equations of higher than 2 nd order partial differential equations (PDE). Recently widely spread and elaborated mesh-free approximations utilize the higher order continuous shape functions. Another advantage is elimination of discretization of the analysed domain into finite elements, since only the nodes are scattered on the domain and its boundary. Both the strong and weak formulations are applicable in combination with mesh-free approximations. The Moving Finite Element Approximation (MFEA) and its utilization in mesh-free formulations for heat conduction problems is presented in this paper.

Joule heating is occurred in electrically conductive materials. In cracked structures with high concentration of electric current at the crack tip vicinity the Joule heating can be extremely high and even a melting of metal can be observed. It is interesting to investigate thermal stresses caused by this local heating. In micro/nano-sized structures the overheating of structures can be very fast and structure can be damaged. However, in these small structures it is observed the size effect for both heat transport and also for the mechanical balance equation. The thermal transport cannot be described by the classical Fourier’s equation in materials with an internal structure. For mechanical problem due to the size effect the strain gradients have to be considered in the constitutive equations. Then, governing equations for both problems are given by partial differential equations with order of derivatives being higher than in classical case. The collocation mixed FEM is developed here for this multi-physical problem. The present computational method is applied to some crack problem to illustrate compressive stresses at the crack tip vicinity, which are leading to the crack closure.

This paper presents a dynamic analysis of an anti-plane crack in functionally graded piezoelectric semiconductors. General boundary conditions and sample geometry are allowed in the proposed formulation. The coupled governing partial differential equations (PDEs) for shear stresses, electric displacement field and current are satisfied in a local weak-form on small fictitious subdomains. The derived local integral equations involve one order lower derivatives than the original PDEs. All field quantities are approximated by the moving least-squares (MLS) scheme. After performing spatial integrations, we obtain a system of ordinary differential equations for the involved nodal unknowns. It is noted that the stresses and electric displacement field in functionally graded piezoelectric semiconductors exhibit the same singularities at crack tips as in a homogeneous piezoelectric solid. The influence of the initial electron density on the intensity factors and energy release rate is also investigated.

A review is presented for analysis of problems in engineering & the sciences, with the use of the meshless local Petrov-Galerkin (MLPG) method. The success of the meshless methods lie in the local nature, as well as higher order continuity, of the trial function approximations, high adaptivity and a low cost to prepare input data for numerical analyses, since the creation of a finite element mesh is not required. There is a broad variety of meshless methods available today; however the focus is placed on the MLPG method, in this paper. The MLPG method is a fundamental base for the derivation of many meshless formulations, since the trial and test functions can be chosen from different functional spaces. In the last decade, a broad community of researchers and scientists contributed to the development and implementation of the MLPG method in a wide range of scientific disciplines. This paper first presents the basics and principles of the MLPG method, the meshless local approximation techniques for trial and test functions, applications to elasticity and elastodynamics, plasticity, fracture and crack analysis, heat transfer and fluid flow, coupled problems involving multiphase materials, and techniques for increasing the accuracy and computational effectiveness. Various applications to 2-D planar problems, axisymmetric problems, plates and shells or 3-D problems are included. An increased number of published papers in literature in the recent years can be considered as a measure of the growing research activity in the general scope of the MLPG method, and thus, several trends and ideas for future research interest are also outlined.

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.

Biot’s poroelastic theory has been applied for Mindlin plates to model moderately thick plates. If Mindlin’s kinematical assumptions and a power series expansion for the pore pressure in the thickness direction are considered, the original 3D problem is reduced to 2D. A truncated power series expansion with quadratic terms is considered for the pore pressure. Two functional relationships based on the given boundary conditions and one PDE are derived for the expansion coefficients. A meshless method based on the local Petrov–Galerkin approach is proposed to solve the set of governing PDE in the neutral plane of the poroelastic plate. All in-plane quantities are approximated by the moving least-squares scheme. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns.

Although micro-dilatation theory is very suitable and effective in modeling elastic porous materials, the absence of any guidance to evaluate or characterize its porosity-related parameters in the literature limited its use and applicability. This paper is proposing a methodology to characterize two of such important parameters, namely porosity change stress parameter and void stiffness coefficient, in terms of microstructural details such as average void radius and density of voids. Two numerical experiments were used to characterize these parameters, and results were validated by a comparison with a high-resolution finite element model of the microstructure with voids explicitly considered.

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.

A meshless method based on the local Petrov-Galerkin approach is proposed for the solution of steady-state and transient heat conduction problems in a continuously non-homogeneous anisotropic medium. The Laplace transform is used to treat the time dependence of the variables for transient problems. The analyzed domain is covered by small subdomains with a simple geometry. A weak formulation for the set of governing equations is transformed into local integral equations on local subdomains by using a unit test function. Nodal points are randomly distributed in the 3D analyzed domain and each node is surrounded by a spherical subdomain to which a local integral equation is applied. The meshless approximation based on the Moving Least-Squares (MLS) method is employed for the implementation. Several example problems with Dirichlet, mixed, and/or convection boundary conditions, are presented to demonstrate the veracity and effectiveness of the numerical approach.

A meshless method based on the local Petrov-Galerkin approach is proposed to solve initial-boundary-value crack problems in decagonal quasicrystals. These quasicrystals belong to the class of two-dimensional (2-d) quasicrystals, where the atomic arrangement is quasiperiodic in a plane, and periodic in the perpendicular direction. The ten-fold symmetries occur in these quasicrystals. The 2-d crack problem is described by a coupling of phonon and phason displacements. Both stationary governing equations and dynamic equations represented by the Bak’s model with oscillations for phasons are analyzed here. Nodal points are spread on the analyzed domain, and each node is surrounded by a small circle for simplicity. The spatial variation of phonon and phason displacements is approximated by the moving least-squares scheme. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time-stepping method.