Shape Functions For Approximation Research Articles

An accurate solution method for the dynamic characteristics of three-layered partial-interaction composite beams

In the dynamic analysis of three-layered partial interaction composite beams (PICBs), it is not clear whether the effects of shear deformation and rotary inertia all need to be considered, and the shear-locking problems often occurs in existing finite element methods. To improve computational efficiency, accuracy and stability, an analytical method is proposed by considering the shear slips at the interface and the independent rotary angles among the sub-elements. In the proposed method, a matrix containing the natural frequencies of the three-layered PICB is appropriately established by using the solutions to differential equations of motion rather than the shape function in finite element methods. The main improvement of the proposed method is the analyticity of solutions, which can effectively avoid shear-locking problems often mentioned in finite element methods and obtain accurate solutions due to no introduction of approximate shape functions. The present results agree well with the analytical and numerical results in the literature, which validates the accuracy of the proposed method. The effects of shear deformation and rotary inertia on the dynamic behaviors of three-layered PICBs are discussed. It reveals that the shear deformation can not be ignored but rotary inertia can be ignored in the dynamic analysis of three-layered PICBs, which can effectively reduce the computation complexity. In general, the proposed method has practical value in natural frequency analysis of the three-layered PICB with high computational efficiency, accuracy and stability.

The HCVEFG method for 3D steady convection–diffusion–reaction problems

In this paper, a hybrid complex variable element-free Galerkin (HCVEFG) method is proposed for solving three-dimensional (3D) steady convection–diffusion–reaction equations. By introducing the dimension splitting method (DSM), a 3D convection–diffusion–reaction problem is changed into a series of two-dimensional (2D) ones. For each 2D form, the improved complex variable moving least-squares (ICVMLS) method is applied to obtain the required approximation shape functions, and the penalty method is used to apply the essential boundary conditions, thus we can obtain the 2D discretized equations by using the improved complex variable element-free Galerkin (ICVEFG) method. The finite difference method (FDM) is used in one-dimensional splitting direction. Thus, the final solution formula about the numerical solution for 3D convection–diffusion–reaction problem can be obtained. Three numerical examples show that the HCVEFG algorithm can cut down the CPU cost of the improved element-free Galerkin (IEFG) method greatly for solving 3D steady convection–diffusion–reaction problems.

A novel adaptive mesh free approach for even- parity neutron transport equation

In this paper, a novel spatially adaptive Mesh free approach using local nodes and radial basis functions, based on weak-form formulation is presented to solve the even-parity neutron transport equation. A residual based error indicator is proposed and used in an adaptive scheme for error estimation. The local balance of neutrons is used as an indicator to determine the appropriate regions of problem for node refinement. In this regard our previous computer program, MFreent, is upgraded for solving neutron transport equation using K+ variational principle in an adaptive scheme for two-dimensional X-Y geometry. The multi-quadrics radial basis function is used to construct the radial point interpolation shape functions for spatial approximation of even-parity angular flux. Also, the angular part of flux is implemented via spherical harmonics expansion. Some numerical test cases are investigated to evaluate the performance of the locally adaptive node-refinement approach for both fixed source and eigenvalue problems solution. The obtained results confirm the solution accuracy and efficiency of the proposed method.

MESHLESS SIMULATION OF HEAT CONDUCTION IN FUNCTIONALLY GRADED MATERIALS SUBJECTED TO TEMPERATURE-DEPENDENT HEAT SOURCES

In this paper, an efficient meshless local B-spline based finite difference (FD) method for analysis of functionally graded materials (FGMs) subjected to temperature-dependent heat sources is presented. The favourable properties of B-spline basis functions in having arbitrary degree for resolution of solution, partition of unity and the Kronecker delta properties are combined with high accuracy and low computational effort of differential quadrature method in approximating shape functions and their derivatives. In this study, the FGMs are assumed to have temperature-dependent materials properties that vary as a function of radial distance. The homogenized properties are evaluated with power-law mixture rule. The nonlinearities from material properties and heat source terms are handled by the predictor-corrector method along with the Crank-Nicolson scheme for time integration. Case of nonlinear 2D heat conduction in FGM due to temperature dependentheat sources is examined. The method is shown to be accurate and efficient for complex thermal analysis of FGMs taking into account temperature dependency of material properties and heat sources.

A mesh-free treatment for even parity neutron transport equation

Common family of numerical methods such as finite element and finite difference involve the discretization of domain into regular grids or meshes. Mesh generation and re-meshing process, however, for obtaining required accuracy are complicated and time-consuming. Recently, Mesh Free (MFree) methods have been developed to overcome the drawbacks of such conventional numerical methods. In the present work, a novel approach is developed for solving multi-group neutron transport equation in two-dimensional X-Y geometry based on a mesh free scheme using the Radial Point Interpolation Method (RPIM). The K+ principle which is an extremum variational principle for solving even parity neutron transport equation is our choice to implement RPIM. The directional dependence of even-parity angular flux is expanded by the spherical harmonic polynomials which leads to PN method. The multi-quadrics radial basis function is used to construct the RPIM shape functions for spatial approximation of angular flux to obtain the discretized MFree weak-form of neutron transport equation. As the RPIM shape functions possess the Kronecker delta function property, essential boundary conditions is enforced as efficiently as in the finite element method (FEM). The obtained results are compared with some mesh-based methods such as conventional finite element method to evaluate the performance of the presented approach. It is demonstrated that the proposed method is robust, stable, reliable and efficient for treatment of neutron transport.

Numerical solutions of strongly non-linear generalized Burgers–Fisher equation via meshfree spectral technique

In this article, a meshfree spectral interpolation technique combined with Crank–Nicolson difference scheme is proposed to solve a class of strongly non-linear Burgers–Fisher type equation numerically. The proposed technique utilizes meshless shape functions for approximation of unknown spatial function and its derivatives. These shape functions are obtained by combining radial basis functions and point interpolation method in the spectral framework. The Crank–Nicolson finite difference scheme is employed for time integration. Stability of the proposed method is analysed theoretically and supported by numerical evidences for RBFs shape parameter , which is an equally important task. Measure of fitness quality is assessed via , and error norms. Efficiency and accuracy of the proposed technique is further examined via variation of time-step size and number of nodal points N. Comparison made with existing techniques in the literature confirms excellent performance of the proposed scheme.

Structural shape optimization by IGABEM and particle swarm optimization algorithm

In this paper, a new approach is developed for structural shape optimization, which consists in coupling the particle swarm optimization (PSO) algorithm and the isogeometric boundary element method (IGA-BEM). The IGA-BEM is based on the combination of the isogeometric analysis (IGA) and the boundary element method (BEM), where Non-Uniform Rational B-Splines (NURBS) are employed as shape functions for geometry parameterization and approximation of the field variables. The method inherits the main advantage of the IGA-based shape optimization, i.e., the control points are used as design variables, thus the design model, analysis model and optimization model are uniformly described with the NURBS, providing easy communication between the three models and resulting in a smooth optimized boundary. However, the main feature of the proposed method is the use of PSO, which provides an attractive gradient-free alternative to complicated sensitivity analysis. The efficiency and accuracy of the proposed approach are demonstrated through four 2D shape optimization examples.

Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions

Solving partial differential equations with discontinuous solutions is an important challenging problem in numerical analysis. To this end, there are some methods such as finite volume method, discontinuous Galerkin approach and particle technique that are able to solve these problems. In the current paper, the moving Kriging element-free Galerkin method has been combined with the variational multiscale algorithm to obtain acceptable and high-resolution solutions. For testing this technique, we select some PDEs with discontinuous solution such as Burgers’, Sod’s shock tube, advection–reaction–diffusion, Kuramoto–Sivashinsky, Boussinesq and shallow water equations. First, we obtain a time-discrete scheme by approximating time derivative via finite difference technique. Then we introduce the moving Kriging interpolation and also obtain their shape functions. We use the element-free Galerkin method for approximating the spatial derivatives. This method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the classical element-free Galerkin method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, we employ the shape functions of moving Kriging interpolation and radial point interpolation technique which have the mentioned property. Also, in the element-free Galekin method, we do not use any triangular, quadrangular or other type of meshes. The element-free Galerkin method is a global method while finite element method is a local one. This technique employs a background mesh for integration which makes it different from the truly mesh procedures. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed schemes.

Implementation of isogeometric boundary element method for 2-D steady heat transfer analysis

In this paper, the recently developed hybrid numerical method based on isogeometric analysis (IGA) and boundary element method (BEM), named as IGABEM, is further extended to study steady heat transfer problems in two dimensions (2D). The IGA employs spline basis functions (e.g., NURBS) as shape functions for geometric description and field approximation. The NURBS only describe the boundary of objects, and the BEM also directly deals with the boundaries. Therefore, the combined IGABEM is highly attractive as it takes advantages of the common boundary representation of both the IGA and the BEM. The formulation of IGABEM for steady heat transfer is derived, and its performance and accuracy are verified through numerical test cases by comparing the obtained results with analytical and finite element method solutions. The convergence of the developed approach for steady heat transfer problems is also analyzed. The computer codes of the IGABEM developed for steady heat transfer analysis can be accessed at: http://www.idmes.cn/codes.html.

A new method for essential boundary conditions imposition in explicit meshless methods

In this paper we present a new technique of enforcing Essential Boundary Conditions (EBC) in Meshless Methods (MMs) based on the Element Free Galerkin (EFG) principles. Imposing EBC is a fundamental issue in MMs. The imposition of prescribed displacement values on the boundary in MMs based on approximating shape functions is not as straightforward as in the Finite Element Method (FEM) because the meshless shape functions are generally not interpolating at nodes. Furthermore, many techniques of enforcing EBC are not compatible with explicit time integration schemes. This paper describes a new method of imposing EBC in EFG based MMs suitable for explicit time integration, named Essential Boundary Conditions Imposition in Explicit Meshless (EBCIEM). The effectiveness of the proposed method is demonstrated using both 2D and 3D numerical examples.

Effect of radial loads on the natural frequencies of thin-walled circular cylindrical shells

In this paper, we study the variation of natural frequencies for thin walled circular cylindrical shells subject to external radial load (uniform and hydrostatic pressure). The study has been performed for four different types of boundary conditions viz. pinned-pinned, pinned-free, clamped-free and one end is pinned and other end axially constrained. Towards this, we have used approximate shape functions and Galerkin projections of the PDEs governing the shell motion, linearized about a steady state solution due to the applied mean radial pressure. Our results have very good agreement with the experimental and theoretical results available in literature. They have also been verified against finite element analysis (FEA) using ABAQUS. We have found that with increasing magnitude of inward radial load, the natural frequency decreases, however, the circumferential wavenumber corresponding to the lowest natural frequency increases. For outward radial load the opposite happens. Furthermore, the circumferential wavenumbers corresponding to lowest natural frequency and first buckling mode are different for short cylinders with the difference decreasing with increasing length of the cylinder. We have also found an interesting result for the beam mode vibration of cylindrical shells. For short cylinders, the natural frequency of the beam mode increases (decreases) with internal (external) pressure, which is consistent with other circumferential modes. Subsequently, we obtain a critical length at which the applied radial pressure has no effect on the natural frequency. For cylinders longer than the critical length, the effect of the radial load is opposite of that observed for the short cylinders, and this is contrary to the behavior of the other circumferential modes. As a result, buckling of the beam mode happens for an outward radial (internal) pressure.

DYNAMICS OF RECTANGULAR ORTHOTROPIC PLATES USING CHARACTERISTIC ORTHOGONAL POLYNOMIAL – GALERKIN’S METHODS

The assumed deflection shapes used in the approximate methods such as in the Galerkin’s method were normally formulated by inspection and sometimes by trial and error, until recently, when a systematic method of constructing such a function in the form of Characteristic Orthogonal Polynomial (COPs) was developed by Bhat in 1985. In the vibrational analyses of orthotropic rectangular plates with different boundary conditions, the study used the characteristic orthogonal polynomial theory to obtain satisfactory approximate shape functions for these plates. These functions were applied to Galerkin indirect varational method to obtain new set of fundamental natural frequencies for these plates. The results were reasonable when compared with those in the previous work. All round simply supported thin rectangular plate (SSSS), rectangular clamped plated (CCCC) and rectangular plate with one edge clamped and all others edges simply supported (CSSS) gave 5.172, 9.429 and 6.202 natural frequencies in rad /sec respectively at 0.05%, 0.0% and 22.93% difference with the previous[3] results5.170rad/sec, 9.429rad/sec and 8.048rad/sec for SSSS, CCCC and CSSS. For others like: rectangular plate with one edge simply supported and all other edges clamped (CCSC), rectangular plate simply supported at two opposite sides and clamped at the others (CSCS) and rectangular plate clamped at two adjacent sides and simply supported at the others (CCSS) with no available results, their natural frequencies obtained are 8.041rad/sec, 6.272rad/sec and 7.106rad/sec respectively. http://dx.doi.org/10.4314/njt.v36i1.8

A model for moisture and heat flow in fractured unsaturated porous media

SummaryThe present study investigates propagation of a cohesive crack in non‐isothermal unsaturated porous medium under mode I conditions. Basic points of skeleton deformation, moisture, and heat transfer for unsaturated porous medium are presented. Boundary conditions on the crack surface that consist of mechanical interaction of the crack and the porous medium, water, and heat flows through the crack are taken into consideration. For spatial discretization, the extended finite element method is used. This method uses enriched shape functions in addition to ordinary shape functions for approximation of displacement, pressure, and temperature fields. The Heaviside step function and the distance function are exploited as enrichment functions for representing the crack surfaces displacement and the discontinuous vertical gradients of the pressure and temperature fields along the crack, respectively. For temporal discretization, backward finite difference scheme is applied. Problems solved from the literature show the validity of the model as well as the dependency of structural response on the material properties and loading. Copyright © 2016 John Wiley & Sons, Ltd.

A meshless point collocation method for 2-D multi-term time fractional diffusion-wave equation

In this paper, a meshless collocation method is considered to solve the multi-term time fractional diffusion-wave equation in two dimensions. The moving least squares reproducing kernel particle approximation is employed to construct the shape functions for spatial approximation. Also, the Caputo's time fractional derivatives are approximated by a scheme of order O(ź3źź), 1<ź < 2. Stability and convergence of the proposed scheme are discussed. Some numerical examples are given to confirm the efficiency and reliability of the proposed method.

Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach

In this paper a numerical procedure is presented for solving a class of three-dimensional Turing system. First, we discrete the spatial direction using element free Galerkin (EFG) method based on the shape functions of moving Kriging interpolation. Then, to achieve a high-order accuracy, we use the fourth-order exponential time differencing Runge–Kutta (ETDRK4) method. Using this discretization for the temporal dimension, we obtain an explicit scheme and do not need to solve nonlinear system of equations. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Also building shape functions of MLS approximation is a time consuming procedure. Because of the mentioned reasons we employ the shape functions of moving Kriging interpolation technique which have the mentioned property and less CPU time is required for building them. For testing this method on three-dimensional PDEs, we select some equations and system of PDEs such as Allen–Cahn, Gray–Scott, Ginzburg–Landau, Brusselator models, predator–prey model with additional food supply to predator. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme.

Application of isogeometric method to free vibration of Reissner–Mindlin plates with non-conforming multi-patch

The isogeometric method is used to study the free vibration of thick plates based on Mindlin theory. The Non-uniform Rational B-Spline (NURBS) basis functions are employed to build the thick plate’s geometry models and serve as the shape functions for solution field approximation in finite element analysis. The Reissner–Mindlin plates built with multiple NURBS patches are investigated, in which several patches of the model have multi-interface and different patches may share a common point. In order to solve the non-conforming interface problems, Nitsche method is employed to glue different NURBS patches and only refers to the coupling conditions in this work. Various plate shapes, different boundary conditions and several kinds of thickness-span ratios are considered to verify the validity of the presented method. The dimensionless frequencies for different cases are obtained by solving the eigenvalue equation problems and compared with the existing reference solutions or the results calculated by ABAQUS software. Several numerical examples exhibit the effectiveness of the isogeometric approach. It shows that the natural frequencies of the Reissner–Mindlin plate can be successfully predicted by the combination of isogeometric analysis and Nitsche method.

Two meshless procedures: moving Kriging interpolation and element-free Galerkin for fractional PDEs

In this paper, the element-free Galerkin (EFG) meshless method and moving Kriging collocation meshless technique are applied for finding the numerical solution of a class of two-dimensional (2D) nonlinear time fractional partial differential equations. The Klein–Gordon, sine-Gordon, diffusion wave and Cattaneo equations with Neumann boundary condition are studied. The time fractional derivative has been described in the Caputo’s sense. Firstly, we use a semi-implicit finite difference scheme of convergence order , and then for obtaining a full discrete scheme, the space derivative is discretized with the EFG and moving Kriging collocation techniques. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method, test and trial functions are moving least squares approximation (MLS) shape functions. Also, in the element-free Galekin method, we do not use any triangular, quadrangular, or other types of meshes. The EFG method is a global method while finite element method is a local one. The EFG method is not a truly meshless method and for integration uses a background mesh. We prove the unconditional stability and obtain an error bound for the EFG method using the energy method. Numerical examples are reported which support the theoretical results and the efficiency of the proposed scheme.

The use of element free Galerkin method based on moving Kriging and radial point interpolation techniques for solving some types of Turing models

In this paper two numerical procedures are presented for solving a class of Turing system. Firstly, we obtain a time discrete scheme by approximating time derivative via finite difference technique. Then we introduce the moving Kriging interpolation and radial point interpolation and also obtain their shape functions. We use the element free Galerkin method for approximating the spatial derivatives. This method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the classical element free Galerkin method test and trial functions are moving least squares (MLS) approximation shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus we employ the shape functions of moving Kriging interpolation and radial point interpolation technique which have the mentioned property. Also, in the element free Galerkin method, we do not use any triangular, quadrangular or other type of meshes. The element free Galerkin method is a global method while finite elements method is a local one. This technique employs a background mesh for integration which makes it different from the truly mesh procedures. The coefficient matrix of the element free Galerkin is symmetric. Also, using numerical algorithms, we can conclude that the eigenvalues of the coefficient matrix are positive. Thus, for solving the obtained linear system of equations from the discretization, we use the conjugant gradient method. To keep away from solving a nonlinear algebraic system of equations and obtaining the acceptable numerical results, we use a predictor–corrector algorithm. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed schemes.

Smoothed Point Interpolation Method for Elastoplastic Analysis

This work formulates the node-based smoothed radial point interpolation method (NS-RPIM), a typical model of smoothed point interpolation method, for the elastoplastic analysis of two-dimensional solids with gradient-dependent plasticity. The NS-RPIM uses radial point interpolation shape functions for field approximation and node-based gradient smoothing for strain field construction. The formulation is based on the parametric variational principle (PVP) in the form of complementarity with the gradient-dependent plasticity being represented by means of the linearization of the yield criterion and the flow rule. Numerical study results have demonstrated the accuracy and stability of the proposed approach for elastoplastic analysis.

Developing an element free method for higher order shear deformation analysis of plates

A procedure is presented to eliminate transverse shear locking in analysis of laminated composite plates using Element Free Galerkin (EFG) method based on higher-order transverse shear deformation theory (HSDT). In the procedure, derivatives of the transverse displacement are introduced as independent variables. Thus, a formulation requiring C0 continuity shape functions for approximation is proposed for higher-order transverse shear deformation analysis of plates. Shear locking is avoided considering reduced integration for shear stiffness matrix; a method which is implemented in FEM. Moving Least Squares (MLS) method is utilized for shape functions and the penalty method is implemented to impose approximate boundary conditions. The present solutions are verified with other higher-order shear deformable studies. Moreover, a comparison between the present solutions with those obtained by EFG procedure based on First-order transverse Shear Deformation Theory (FSDT) is performed.