Treatment Of Material Discontinuities Research Articles

Thin plate bending analysis and treatment of material discontinuities using the generalised RKP-FSM

A finite strip method (FSM) utilising the generalised reproducing kernel particle method (RKPM) [Behzadan, Shodja, and Khezri (2011)] is developed for the bending analysis of thin plates. In this innovative approach, the spline functions in the conventional spline finite strip method (SFSM) are replaced with generalised RKPM 1-D shape functions in the longitudinal direction, while the transverse cubic functions which are used in the conventional formulations are retained. Since the generalised RKPM is one of the class of meshfree methods which deal efficiently with derivative-type essential boundary conditions, its introduction in the FSM is beneficial for solving boundary value problems such as the bending of thin plates in which a number of essential boundary conditions can include first derivatives of the displacement function. In this paper, the formulation for the generalised RKP-FSM is derived for the analysis of thin plates, and its accuracy and convergence are examined through a series of numerical studies. Moreover, by modifying the concept of the augmented corrected collocation method [Shodja, Khezri, Hashemian, and Behzadan (2010)] a new feature is added to the conventional SFSM and the generalised RKP-FSM which allows for the exact treatment of material discontinuities in bending analysis. Oscillatory behaviour of solutions near the interface of material discontinuities due to Gibb's phenomenon is successfully eliminated in both methods.

RKPM with Augmented Corrected Collocation Method for Treatment of Material Discontinuities

RKPM with Augmented Corrected Collocation Method for Treatment of Material Discontinuities

ACE—A FORTRAN subroutine for analytical computation of effective grid parameters for finite-difference seismic waveform modeling with standard Earth models

Despite the broad use of the heterogeneous finite-difference (FD) method for seismic waveform modeling, accurate treatment of material discontinuities inside the grid cells has been a serious problem for many years. One possible way to solve this problem is to introduce effective grid elastic moduli and densities (effective parameters) calculated by the volume harmonic averaging of elastic moduli and volume arithmetic averaging of densities in grid cells. This scheme enables us to place a material discontinuity in an arbitrary position in the spatial grids. Standard Earth models have made a significant contribution to synthetic seismogram calculations with a variety of numerical procedures such as the FD method. For the FD computation of seismic waveform with these models, we must first ensure accurate treatment of material discontinuities in the radius (or depth). The present paper introduces a FORTRAN subroutine ACE which calculates effective parameters analytically for an arbitrary spatial region in either the radius or depth direction for four major standard Earth models, namely, the PREM, IASP91, SP6, and AK135. This program is intended for all FD users who are concerned with seismic wave simulation for these models.

Meshless Local Petrov-Galerkin Formulation for Problems in Composite Micromechanics

In this paper we present the meshless local Petrov-Galerkin formulation for the generalized plane strain problem with specific emphasis on micromechanics of composite materials containing material discontinuities. The problem requires the introduction of an extra discrete degree of freedom, the out-of-plane uniform normal strain. The treatment of material discontinuity at the interface between the two phases of the composite is presented by means of direct imposition of interface boundary conditions. The meshless local Petrov-Galerkin method is used in the micromechanical model for predicting the elastic constants of the composite. To our knowledge, this is the first study in which the meshless local Petrov-Galerkin method is formulated for the so-called meshless local Petrov-Galerkin method based micromechanical analysis. Examples are presented to illustrate the effectiveness of the current method, and it is validated by comparing the results with available analytical and numerical solutions. The current method has the potential for use in micromechanics, especially for textile composites, where the meshing of the unit cell has been quite difficult.

A local search algorithm for natural neighbours in the natural element method

A local basis algorithm for searching natural neighbours in Natural Element Method (NEM) is presented for solving the elasticity problems in this paper. Comparison with the global sweep algorithm used in natural element method or Natural Neighbour Method (NNM) for searching natural neighbours, the proposed algorithm is more expedient and convenient in the constructions and computation of natural neighbour interpolations. In the proposed NEM based on local search, the Laplace (non-sibson) interpolations are constructed with respect to the natural neighbour nodes of the given point which have been locally defined. The shape functions from the Laplace approximations have the delta function property and the Laplace interpolants are strictly linear between adjacent nodes, which facilitate imposition of essential boundary conditions and treatment of material discontinuity with ease as it is in the conventional finite element method. The Laplace interpolants derived from the local algorithm and the global algorithm in NEM are identical because of the uniqueness of the Voronoi diagram. Numerical results and convergence studies also show that the present NEM based on local search algorithm possesses the same accuracy and rate of convergence as they are in previous NEM.

Fringe element reconstruction for front tracking for three-dimensional incompressible flow analysis

Fringe element reconstruction technique for tracking the free surface in three-dimensional incompressible flow analysis was developed. The flow field was calculated by the mixed formulation based on a four-node tetrahedral element with a bubble function at the centroid (P1+/P1). Since an Eulerian approach was employed in this study, the flow front interface was advected by the flow through a fixed mesh. For accurate modelling of interfacial movement, a fringe element reconstruction method developed can provide not only an accurate treatment of material discontinuity but also surface tension across the interface. The effect of surface tension was modelled by imposing tensile stress directly on the constructed surface elements at the flow front interface. To verify the numerical approach developed, the developed algorithm was applied to two examples whose solutions are available in references. Good agreement was obtained between the simulation results and these solutions. Copyright © 2005 John Wiley & Sons, Ltd.

Treatment of material discontinuity in two meshless local Petrov–Galerkin (MLPG) formulations of axisymmetric transient heat conduction

AbstractWe use two meshless local Petrov–Galerkin (MLPG) formulations to analyse heat conduction in a bimetallic circular disk. The continuity of the normal component of the heat flux at the interface between two materials is satisfied either by the method of Lagrange multipliers or by using a jump function. The convergence of the H0 and H1 error norms for the four numerical solutions with an increase in the number of equally spaced nodes and in the number of quadrature points is scrutinized. With an increase in the number of uniformly spaced nodes, the two error norms decrease monotonically for the MLPG5 formulation but are essentially unchanged for the MLPG1 formulation. To our knowledge, this is the first study comparing the performance of the two methods of modelling a discontinuity in the gradient of a field variable at the interface between two different materials. Copyright © 2004 John Wiley & Sons, Ltd.

Direct imposition of essential boundary conditions and treatment of material discontinuities in the EFG method

A method for direct imposition of essential boundary condition and treatment of material discontinuity in element free galerkin (EFG) method is presented. By using the actual displacements at the nodes on the essential boundary and the material interface in each material domain, the stiffness matrix and load vector at an integral point have been rewritten and transformed. As a result, the proposed method yields a positive, symmetrical and banded global stiffness matrix like it is in finite element methods and has the advantages of stabilization and easy implementation as compared to the penalty method, the Lagrange method, and other methods. Numerical results indicate that the present method is effective and retains high rates of convergence for both displacements and energy.

The meshless finite element method

AbstractA meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n, and the condition that the shape functions depend only on the node positions. Furthermore, the method proposed also shares several of the advantages of the finite element method such as: (a) the simplicity of the shape functions in a large part of the domain; (b) C0 continuity between elements, which allows the treatment of material discontinuities, and (c) ease of introduction of the boundary conditions. Copyright © 2003 John Wiley & Sons, Ltd.

Treatment of material discontinuity in the Element-Free Galerkin method

A method is presented for application of the Element-Free Galerkin method (EFG) to solid mechanics problems containing material discontinuities. By using the EFG method, the trial and test functions for the weak form are constructed with moving least-square interpolants in each material domain. Additional constraints are imposed at the variational level to ensure satisfaction of interface conditions. As a result, only nodal data (no elemental connectivity) is needed. Given C 1 continuity of the interpolant weighting function, the dependent variable and its gradient are continuous in each material domain. Two linear elastic bi-material examples are presented to illustrate the effectiveness of the method.

Treatment of material discontinuities in finite element computations : MacKinnon, R J; Carey, G F Int J Num Meth EngngV24, N2, Feb 1987, P393–417

Treatment of material discontinuities in finite element computations : MacKinnon, R J; Carey, G F Int J Num Meth EngngV24, N2, Feb 1987, P393–417

Treatment of material discontinuities in finite element computations

AbstractWe consider finite element analysis of problems with discontinuous material coefficients. For applications in which the material interface crosses an element, we develop special elements with an embedded flux constraint at the interface. This new procedure is compared with the standard finite element method with interface coincident with the element boundary and with an existing method proposed by Steven.1 Supporting numerical studies are conducted and rates of convergence for the solution and interface flux are examined. Some local superconvergence behaviour is observed.