A point interpolation method for two‐dimensional solids

A point interpolation method (PIM) is presented for stress analysis for two‐dimensional solids. In the PIM, the problem domain is represented by properly scattered points. A technique is proposed to construct polynomial interpolants with delta function property based only on a group of arbitrarily distributed points. The PIM equations are then derived using variational principles. In the PIM, the essential boundary conditions can be implemented with ease as in the conventional finite element methods. The present PIM has been coded in FORTRAN. The validity and efficiency of the present PIM formulation are demonstrated through example problems. It is found that the present PIM is very easy to implement, and very flexible for obtained displacements and stresses of desired accuracy in solids. As the elements are not used for meshing the problem domain, the present PIM opens new avenues to develop adaptive analysis codes for stress analysis in solids and structures. Copyright © 2001 John Wiley & Sons, Ltd.     

A three-dimensional shape optimization for transient acoustic scattering problems using the time-domain boundary element method

We develop a three-dimensional shape optimization (SO) framework for the wave equation with taking the unsteadiness into account. Resorting to the adjoint variable method, we derive the shape derivative (SD) with respect to a deformation (perturbation) of an arbitrary point on the target surface of acoustic scatterers. Successively, we represent the target surface with non-uniform rational B-spline patches and then discretize the SD in term of the associated control points (CPs), which are useful for manipulating a surface. To solve both the primary and adjoint problems, we apply the time-domain boundary element method (TDBEM) because it is the most appropriate when the analysis domain is the ambient air and thus infinitely large. The issues of the severe computational cost and instability of the TDBEM are resolved by exploiting the fast and stable TDBEM proposed by the present authors. Instead, since the TDBEM is mesh-based and employs the piecewise-constant element for space, we introduce some approximations in evaluating the discretized SD from the two solutions of TDBEM. By regarding the evaluation scheme as the computation of the gradient of the objective functional, given as the summation of the absolute value of the sound pressure over the predefined observation points, we can solve SO problems with a gradient-based non-linear optimization solver. To assess the developed SO system, we performed several numerical experiments from the perspective of verification and application with satisfactory results.     

On the choice of interpolation functions used in the dual-reciprocity boundary-element method

The dual-reciprocity boundary-element method is a very powerful technique for solving general elliptic equations of the type ²u=b. In this method, a series of interpolation functions is used to approximate b in order to convert the associated domain integral, which it is necessary to evaluate in a traditional boundary-element analysis, into boundary integrals only. Hence the choice of interpolation functions has direct effects on the numerical results. According to Partridge and Brebbia, the adoption of a comparatively simple form of interpolation function gives the best results. Unfortunately, when b contains partial derivatives of the unknown function u(x, y), the adoption of such a type of interpolation function inevitably leads to the creation of singularities on all boundary and internal nodes used in a dual-reciprocity boundary-element analysis, as was pointed out by Zhu and Zhang in 1992. To avoid this problem, a functional transformation, which applies only to linear governing equations, can be employed to eliminate these derivative terms and thus to obtain better numerical results. In this paper, two new interpolation functions are proposed and examined; they are proven to be generally applicable and satisfactory.     

Combination of Newmark method and natural element method for elastodynamics

The natural element method (NEM) is a meshless method. The trial and test functions of the NEM are constructed using natural neighbor interpolations which are based on the Voronoi tessellation of a set of nodes. The NEM interpolation is linear between adjacent nodes on the boundary of the convex hull, which makes imposition of essential boundary conditions easy to implement. We investigate the performance of the NEM combined with the Newmark method for problems of elastodynamics in this article. Applications are considered for a cantilever beam with different initial load conditions. The NEM numerical results are compared with the finite element method. NEM shows promise for these applications.     

A finite element semi‐Lagrangian method with L2 interpolation

High‐order accurate methods for convection‐dominated problems have the potential to reduce the computational effort required for a given order of solution accuracy. The state of the art in this field is more advanced for Eulerian methods than for semi‐Lagrangian (SLAG) methods. In this paper, we introduce a new SLAG method that is based on combining the modified method of characteristics with a high‐order interpolating procedure. The method employs the finite element method on triangular meshes for the spatial discretization. An L² interpolation procedure is developed by tracking the feet of the characteristic lines from the integration nodes. Numerical results are illustrated for a linear advection–diffusion equation with known analytical solution and for the viscous Burgers’ equation. The computed results support our expectations for a robust and highly accurate finite element SLAG method. Copyright © 2012 John Wiley & Sons, Ltd.     

On the essence and the evaluation of the shape functions for the smoothed finite element method (SFEM)

This paper is written in response to the recently published paper (Int. J. Numer. Meth. Engng 2008; 76 :1285–1295) at IJNME entitled ‘On the smoothed finite element method’ (SFEM) by Zhang HH, Liu SJ, Li LX. In this paper we

• (1) repeat briefly the important essence of the original SFEM presented in (Comp. Mech. 2007; 39 : 859–877; Int. J. Numer. Meth. Engng 2007; 71 :902–930; Int. J. Numer. Meth. Engng 2008; 74 :175–208; Finite Elem. Anal. Des. 2007; 43 :847–860; J. Sound Vib. 2007; 301 :803–820), and
• (2) examine further issues in the evaluation of the shape functions used in the SFEM.

It will be shown that the ‘SFEM’ presented in paper (Int. J. Numer. Meth. Engng 2008; 76 :1285–1295) is not at all our original SFEM presented in (Comp. Mech. 2007; 39 :859–877; Int. J. Numer. Meth. Engng 2007; 71 :902–930; Int. J. Numer. Meth. Engng 2008; 74 :175–208; Finite Elem. Anal. Des. 2007; 43 :847–860; J. Sound Vib. 2007; 301 :803–820). Therefore, all these ‘Theorems’, ‘Corollaries’ and ‘Remarks’ presented in paper (Int. J. Numer. Meth. Engng 2008; 76 :1285–1295) have nothing to do with our original SFEM. The properties of the original SFEM stand as they were presented in our original papers (Comp. Mech. 2007; 39 :859–877; Int. J. Numer. Meth. Engng 2007; 71 :902–930; Int. J. Numer. Meth. Engng 2008; 74 :175–208; Finite Elem. Anal. Des. 2007; 43 :847–860; J. Sound Vib. 2007; 301 :803–820). Finally, we brief on our advancements made far beyond our original SFEM and our visions on future numerical methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Boundary element method for natural convection in non-Newtonian fluid saturated square porous cavity

The main purpose of this work is to present the use of the Boundary Element Method (BEM) in the analysis of the natural convection in the square porous cavity saturated by the non-Newtonian fluid. The results of hydrodynamic and heat transfer evaluations are reported for the configuration in which the enclosure is heated from a side wall while the horizontal walls are insulated. The flow in the porous medium is modelled using the modified Brinkman extended Darcy model taking into account the non-Darcy viscous effects. The governing equations are transformed by the velocity–vorticity variables formulation enabling the computation scheme to be partitioned into kinematic and kinetic parts. To analyse the effects of the available non-Newtonian viscosity and to evaluate the presented approach, the power law model for shear thinning fluids (n<1), for shear thickening fluids (n>1) and in the limit for the Newtonian fluids (n=1) is considered. Numerical model is tested also for the Carreau model adequate for many non-Newtonian fluids. Solutions for the flow and temperature fields and Nusselt numbers are obtained in terms of a modified Rayleigh number Ra^*, Darcy number Da, and the non-Newtonian model parameters. The agreement between the results obtained with finite difference method is very good indicating that BEM can be efficiently used for solving transport phenomena in saturated porous medium.     

Modeling of interfacial debonding crack in particle reinforced composites using Voronoi cell finite element method

In this paper, Voronoi cell finite element method (VCFEM), introduced by Ghosh and coworkers (1993), is applied to describe the matrix-inclusion interfacial debonding for particulate reinforced composites. In proposed VCFEM, the damage initiation is simulated by partly debonding of the interface under the assumption of the critical normal stress law, and gradual matrix-inclusion separations are simulated with an interface remeshing method that a critical interfacial node at the crack tip is replaced by a node pairs along the debonded matrix-inclusion interface and a more pair of nodes are needed to be added on the crack interface near the crack tip in order to better facilitate the free-traction boundary condition and the jumps of solution. The comparison of the results of proposed VCFEM and commercial finite element packages MARC and ABAQUS. Examples have been given for a single inclusion of gradually interfacial debonding and for a complex structure with 20 inclusions to describe the interfacial damage under plane stress conditions. Good agreements are obtained between the VCFEM and the general finite element method. It appears that this method is a more efficient way to deal with the interfacial damage of composite materials. The financial support by the Special Funds for the National Major Fundamental Research Projects G19990650 and the National Natural Science Foundation of China No. 59871022 are gratefully acknowledged.     

Die shape optimal design in three-dimensional shape metal extrusion by the finite element method

A new approach to die shape optimal design in shape extrusion is presented. In this approach, the design problem is formulated as an optimization problem incorporating the three-dimensional finite element analysis model, and optimization of the die shape is conducted on the basis of the design sensitivities. The approach is applied to the determination of the die shapes for extrusion of parts with various cross sections including polygons and T sections. © 1998 John Wiley & Sons, Ltd.