The finite difference technique for solving crack problems

A detailed examination of the Finite Difference method for solving crack problems is presented and discussed. The three classical mode I configurations (i.e. Centered Crack Plate, Double Edge Notch and Single Edge Notch) as well as an uncommon case (A Penny Shape Crack embedded in a circular plate in bending) are solved and discussed. The Stress Intensity Factors are computed by taking more than one (first) term in William's Series, using two or three points near the tip. This technique improves the accuracy and frees one from relying on the very first point near the tip as a measure base. In most cases, the accuracy was found to be between 1–3% for uniform mesh size in the order of 5% from the half crack length. No special imposed functions were used near the tip, which makes the technique competitive to the Finite Element method, especially for 3-D problems or cases where the degree of singularity is not known. The solution is found iteratively (a two step SOR method) and some techniques for quick convergence are discussed.     

A higher order compact finite difference algorithm for solving the incompressible Navier–Stokes equations

On the basis of the projection method, a higher order compact finite difference algorithm, which possesses a good spatial behavior, is developed for solving the 2D unsteady incompressible Navier–Stokes equations in primitive variable. The present method is established on a staggered grid system and is at least third‐order accurate in space. A third‐order accurate upwind compact difference approximation is used to discretize the non‐linear convective terms, a fourth‐order symmetrical compact difference approximation is used to discretize the viscous terms, and a fourth‐order compact difference approximation on a cell‐centered mesh is used to discretize the first derivatives in the continuity equation. The pressure Poisson equation is approximated using a fourth‐order compact difference scheme constructed currently on the nine‐point 2D stencil. New fourth‐order compact difference schemes for explicit computing of the pressure gradient are also developed on the nine‐point 2D stencil. For the assessment of the effectiveness and accuracy of the method, particularly its spatial behavior, a problem with analytical solution and another one with a steep gradient are numerically solved. Finally, steady and unsteady solutions for the lid‐driven cavity flow are also used to assess the efficiency of this algorithm. Copyright © 2011 John Wiley & Sons, Ltd.     

An algorithm for adaptive refinement of triangular element meshes

A simple algorithm is developed for adaptive and automatic h refinement of two-dimensional triangular finite element meshes. The algorithm is based on an element refinement ratio that can be calculated from an a posteriori error indicator. The element subdivision algorithm is robust and recursive. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. Several example problems are presented to illustrate the utility of the approach.     

Difference schemes for second order hyperbolic equations

Implicit difference methods for the wave equation in two space variables have been discussed with the help of a stability diagram. The difference methods of intermediate accuracy 0(h⁴+k²) have been determined. A method of order of accuracy 0(h²+k²) with minimum truncation error has also been found.     

Equivalence groups for second order balance equations

The groups of equivalence transformations for a family of second order balance equations involving arbitrary number of independent and dependent variables are investigated. Equivalence groups are much more general than symmetry groups in the sense that they map equations containing arbitrary functions or parameters onto equations of the same structure but with different functions or parameters. Our approach to attack this problem is based on exterior calculus. The analysis is reduced to determine isovector fields of an ideal of the exterior algebra over an appropriate differentiable manifold dictated by the structure of the differential equations. The isovector fields induce point transformations, which are none other than the desired equivalence transformations, via their orbits which leave that particular ideal invariant. The general scheme is applied to a one-dimensional nonlinear wave equation and hyperelasticity. It is shown that symmetry transformations can be deduced directly from equivalence transformations.     

AGTHOM—automatic generation of triangular and higher order meshes

The development and implementation of a comprehensive interactive computer package for the generation of finite element meshes for two-dimensional problems is described. The package, AGTHOM, minimizes the user defined input and attempts to maximize the flexibility for the user with regard to modifying the mesh. An important feature of AGTHOM is its independence of expensive graphics hardware by using approximate terminal plots. Versions are available in both extended BASIC and FORTRAN.     

Generalized finite difference method for solving two-dimensional non-linear obstacle problems

In this study, the obstacle problems, also known as the non-linear free boundary problems, are analyzed by the generalized finite difference method (GFDM) and the fictitious time integration method (FTIM). The GFDM, one of the newly-developed domain-type meshless methods, is adopted in this study for spatial discretization. Using GFDM can avoid the tasks of mesh generation and numerical integration and also retain the high accuracy of numerical results. The obstacle problem is extremely difficult to be solved by any numerical scheme, since two different types of governing equations are imposed on the computational domain and the interfaces between these two regions are unknown. The obstacle problem will be mathematically formulated as the non-linear complementarity problems (NCPs) and then a system of non-linear algebraic equations (NAEs) will be formed by using the GFDM and the Fischer–Burmeister NCP-function. Then, the FTIM, a simple and powerful solver for NAEs, is used solve the system of NAEs. The FTIM is free from calculating the inverse of Jacobian matrix. Three numerical examples are provided to validate the simplicity and accuracy of the proposed meshless numerical scheme for dealing with two-dimensional obstacle problems.     

Comparative study of six explicit and two implicit finite difference schemes for solving one-dimensional parabolic partial differential equations

Eight finite difference schemes used in solving parabolic partial differential equations are compared with respect to accuracy, execution time and programming effort. The analysis presented is useful in selecting the appropriate numerical scheme depending on the emphasis placed upon accuracy, execution time or programming effort.     

On consistent finite difference formulae for ordinary differential equations

This paper presents a finite difference method for two-point boundary value problems described by fourth-order ordinary differential equations which results in consistency of truncation errors. It is demonstrated that the order of the formulae used to approximate the boundary conditions must be higher than those used for similar derivative terms in the differential equation. A generalization of the method to differential equations of order n is discussed. The procedure is illustrated with a numerical example.     

The generation of arbitrary order curved meshes for 3D finite element analysis

A procedure for generating curved meshes, suitable for high-order finite element analysis, is described. The strategy adopted is based upon curving a generated initial mesh with planar edges and faces by using a linear elasticity analogy. The analogy employs boundary loads that ensure that nodes representing curved boundaries lie on the true surface. Several examples, in both two and three dimensions, illustrate the performance of the proposed approach, with the quality of the generated meshes being analysed in terms of a distortion measure. The examples chosen involve geometries of particular interest to the computational fluid dynamics community, including anisotropic meshes for complex three dimensional configurations.     

An efficient method for solving diffusion equations

An algorithm for solving diffusion equations is described which has an arbitrary order of approximation in space variables, i.e., its accuracy is the higher, the smoother the solution in space variables. This provides an advantage over difference methods, which have a fixed order of approximation in space variables irrespective of the solution smoothness. In practice, this allows one (with smooth initial data) to carry out calculations on coarse space grids by an explicit scheme with applicable time steps. The method is competitive with difference methods in speed and the amount of information stored. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 70, No. 2, pp. 314–317, March–April, 1997.     

An automatic adaptive refinement procedure using triangular and quadrilateral meshes

An automatic adaptive refinement procedure for finite element analysis for two-dimensional stress analysis problems is presented. Through the combined use of the new mesh generator developed by the authors (to appear) for adaptive mesh generation and the Zienkiewicz-Zhu [Int. J. numer. Meth. Engng31, 1331–1382 (1992)] error estimator based on the superconvergent patch recovery technique, an adaptive refinement procedure can be formulated which can achieve the aimed accuracy very economically in one or two refinement steps. A simple method is also proposed to locate the existence and the position of singularities in the problem domain. Hence, little or no a priori knowledge about the location and strength of the singularities is required. The entire adaptive refinement procedure has been made fully automatic and no user intervention during successive cycles of mesh refinements is needed. The robustness and reliability of the refinement procedure have been tested by solving difficult practical problems involving complex domain geometry with many singularities. We found that in all the examples studied, regardless of the types of meshes employed, triangular and quadrilateral meshes, nearly optimal overall convergence rate is always achieved.     

An embedded mesh method for treating overlapping finite element meshes

A new technique for treating the mechanical interactions of overlapping finite element meshes is presented. Such methods can be useful for numerous applications, for example, fluid–solid interaction with a superposed meshed solid on an Eulerian background fluid grid. In this work, we consider the interaction of two elastic domains: one mesh is the foreground and defines the surface of interaction, the other is a background mesh and is often a structured grid. Many of the previously proposed methods employ surface defined Lagrange multipliers or penalties to enforce the boundary constraints. It has become apparent that these methods will cause mesh locking under certain conditions. Appropriately applied, the Nitsche method can overcome this locking, but, in its canonical form, is generally not applicable to non‐linear materials such as hyperelastics. The relationship between interior point penalty, discontinuous Galerkin and Nitsche's method is well known. Based on this relationship, a nonlinear theory analogous to the Nitsche method is proposed to treat nonlinear materials in an embedded mesh. Here, a discontinuous Galerkin derivative based on a lifting of the interface surface integrals provides a consistent treatment for non‐linear materials and demonstrates good behavior in example problems. Published 2012. This article is a US Government work and is in the public domain in the USA.     

Adaptivity for finite volume on unstructured triangular meshes: a study of thermal injury in teeth

This paper presents a numerical study of thermal injury in teeth, caused both by convective heating, due to drinking of hot beverage and mastication of foods, and by laser irradiation in dental treatment. The numerical study employs an adaptive finite volume method on unstructured triangular meshes to solve the governing equations. An adaptive time stepping methodology was also used in order to control the solution error. Adaptive methodologies are adequate to solve such problems since steep gradients will develop at specific locations in the domain of study. The convective heating results were compared to experimental data available in the literature. Laser treatment results are in agreement to the temperature increasing observed in literature. The simulation results demonstrate that both the error estimate and adaptive methodology herein proposed are suitable and reliable for the controlled solution of parabolic problems. Copyright © 2004 John Wiley & Sons, Ltd.     

An efficient discontinuous Galerkin method on triangular meshes for a pedestrian flow model

In this paper, we develop a discontinuous Galerkin method on triangular meshes to solve the reactive dynamic user equilibrium model for pedestrian flows. The pedestrian density in this model is governed by the conservation law in which the flow flux is implicitly dependent on the density through the Eikonal equation. To solve the Eikonal equation efficiently at each time level, we use the fast sweeping method. Two numerical examples are then used to demonstrate the effectiveness of the algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

Fundamental solutions for second order linear elliptic partial differential equations

This paper is concerned with outlining some fundamental solutions and Green's functions for a system of second order linear elliptic partial differential equations in two independent variables. The fundamental solution and a number of Green's functions are given in relatively elementary closed form for some cases when the coefficients in the equations are constant. When the coefficients are variable the fundamental solution is obtained for some particular classes of equations.     

An improved stress recovery technique for low‐order 3D finite elements

In this paper, we propose a stress recovery procedure for low‐order finite elements in 3D. For each finite element, the recovered stress field is obtained by satisfying equilibrium in an average sense and by projecting the directly calculated stress field onto a conveniently chosen space. Compared with existing recovery techniques, the current procedure gives more accurate stress fields, is simpler to implement, and can be applied to different types of elements without further modification. We demonstrate, through a set of examples in linear elasticity, that the recovered stresses converge at a higher rate than that of directly calculated stresses and that, in some cases, the rate of convergence is the same as that of the displacement field.