The boundary contour method for two-dimensional Stokes flow and incompressible elastic materials

 A variant of the boundary element method, called the boundary contour method (BCM), offers a further reduction in dimensionality. Consequently, boundary contour analysis of two-dimensional (2-D) problems does not require any numerical integration at all. While the method has enjoyed many successful applications in linear elasticity, the above advantage has not been exploited for Stokes flow problems and incompressible media. In order to extend the BCM to these materials, this paper presents a development of the method based on the equations of Stokes flow and its 2-D kernel tensors. Potential functions are derived for quadratic boundary elements. Numerical solutions for some well-known examples are compared with the analytical ones to validate the development. Received 28 August 2001 / Accepted 15 January 2002     

Boundary-integral equation analysis of twisted internally cracked axisymmetric bimaterial elastic solids

 Green's function is obtained for the infinite bimaterial elastic solid, containing an internal circular interface crack, loaded by a unit tangential co-axial circular source. An axisymmetric direct boundary integral equation (BIE) is used for the analysis of a finite bimaterial axisymmetric body containing an internal circular interface crack and a finite homogeneous cracked cylinder, both under torsional loading. Using the proposed technique, no discretization of the crack surface is necessary. Numerical results for both examples as obtained by the proposed method are presented and discussed. Received: 29 October 2001 / Accepted: 29 May 2002     

A three-dimensional FE analysis of large deformations for impact loadings using tetrahedral elements

 A three-dimensional dynamic program for the anaysis of large deformations in contact-penetration problems is developed using the finite element Lagrangian method with explicit time integration. By incorporating a tetrahedral element, which allows a single-point integration without a special hourglass control scheme, this program can be more effective to the present problem. The position code algorithm is used to search contact surface. Eroding surfaces are also considered. The defense node algorithm was slightly modified for the calculation of contact forces. A study of obliquity effects on metallic plate perforation and ricochet processes in thin plates impacted by a sphere was conducted. It is well simulated that on separation of two parts of the sphere, the portion still within the crater tends to perforate, while the portion in contact with the plate surface ricochets. This deformation pattern is observed in experiments, especially at high obliquities. A long rod that impacts an oblique steel plate at high impact velocity was also simulated in order to study the dynamics of the rod caused by the three dimensional asymmetric contact. The agreement between simulated and experimental results is quite good. Fracture phenomena occuring at high obliquity deserves further investigations. Received: 20 February 2002 / Accepted: 20 September 2002     

Analytical integrations in 3D BEM: preliminaries

 This work provides a preliminary contribution in the context of analytical integrations of strongly and hyper singular kernels in boundary element methods (BEMs) in 3D. It concerns the integral of 1/r ³ over a triangle in R ³, that plays a fundamental role in BEMs in 3D, especially for the Galerkin implementation. Because the existence of the aforementioned integral depends on the position of the source point, all significant instances of the position of the source point will be considered and detailed. For its interest in the context of BEM, the integral is also considered in the more general sense of finite part of Hadamard. Received 6 August 2001     

Modeling of 3D transversely piezoelectric and elastic bimaterials using the boundary element method

 This paper presents a numerical model for three-dimensional transversely isotropic bimaterials based on the boundary element formulation. The point force solutions expressed in a united-form for distinct eigenvalues are studied for transversely isotropic piezoelectricity and pure elasticity. A boundary integral formulation is implemented for the modeling of two-phase materials. In this study, the stress distributions are computed for a near interface flaw. The influences of the shape and location of the flaw on the the stress concentration are examined. The accuracy of the numerical procedures is validated through selected example problems and comparison studies. Received 3 October 2001 / Accepted 9 April 2002     

Cubic Bezier splines for BEM heat transfer analysis of the 2-D continuous casting problems

 This paper presents a two-dimensional model for identification of the phase change front in a continuous casting process. The transport phenomena encountered in the considered process are solved by Boundary Element Method (BEM). For the known set of external boundary conditions, the whole problem is solved in two subdomains separated by a phase change front whose position is updated during the iteration process. The solution scheme involves the application of a front tracking procedure based on using sensitivity coefficients to find the correct position of the phase change front modelled by Bezier splines. The main features of the developed algorithms were investigated by several numerical tests, the most important results of which are presented in this article. Received 6 November 2000     

A Trefftz function approximation in local boundary integral equations

 In the present paper the Trefftz function as a test function is used to derive the local boundary integral equations (LBIE) for linear elasticity. Since Trefftz functions are regular, much less requirements are put on numerical integration than in the conventional boundary integral method. The moving least square (MLS) approximation is applied to the displacement field. Then, the traction vectors on the local boundaries are obtained from the gradients of the approximated displacements by using Hooke's law. Nodal points are randomly spread on the domain of the analysed body. The present method is a truly meshless method, as it does not need a finite element mesh, either for purposes of interpolation of the solution variables, or for the integration of the energy. Two ways are presented to formulate the solution of boundary value problems. In the first one the local boundary integral equations are written in all nodes (interior and boundary nodes). In the second way the LBIE are written only at the interior nodes and at the nodes on the global boundary the prescribed values of displacements and/or tractions are identified with their MLS approximations. Numerical examples for a square patch test and a cantilever beam are presented to illustrate the implementation and performance of the present method. Received 6 November 2000     

3D fracture analysis by the symmetric Galerkin BEM

 The subject of this paper is the formulation and the implementation of the symmetric Galerkin BEM for three-dimensional linear elastic fracture mechanics problems. A regularized version of the displacement and traction equations in weak form is adopted and the integration techniques utilized for the evaluation of the double surface integrals appearing in the discretized equations are detailed. By using quadratic isoparametric quadrilateral and triangular elements, some example crack problems are solved to assess the efficiency and robustness of the method. Received 6 November 2000     

A general algorithm for the numerical evaluation of nearly singular boundary integrals of various orders for two- and three-dimensional elasticity

 A general algorithm of the distance transformation type is presented in this paper for the accurate numerical evaluation of nearly singular boundary integrals encountered in elasticity, which, next to the singular ones, has long been an issue of major concern in computational mechanics with boundary element methods. The distance transformation is realized by making use of the distance functions, defined in the local intrinsic coordinate systems, which plays the role of damping-out the near singularity of integrands resulting from the very small distance between the source and the integration points. By taking advantage of the divergence-free property of the integrals with the nearly hypersingular kernels in the 3D case, a technique of geometric conversion over the auxiliary cone surfaces of the boundary element is designed, which is suitable also for the numerical evaluation of the hypersingular boundary integrals. The effects of the distance transformations are studied and compared numerically for different orders in the 2D case and in the different local systems in the 3D case using quadratic boundary elements. It is shown that the proposed algorithm works very well, by using standard Gaussian quadrature formulae, for both the 2D and 3D elastic problems. Received: 20 November 2001 / Accepted: 4 June 2002 The work was supported by the Science Foundation of Shanghai Municipal Commission of Education.     

A comparative study on various time integration schemes for heat wave simulation

The performance of several explicit and implicit time advancement schemes of first-order ODEs are examined for heat wave simulation with different boundary conditions. It is found that the boundary conditions have a considerable influence on the stiffness property of the hyperbolic heat conduction equation, due to the occurrence of thermal shock waves, and hence, according to the type of the enforced boundary conditions, a specific time integration scheme has to be performed in order to obtain an accurate and efficient solution. The results of the considered time integration schemes are also compared with analytical solution and based on the obtained results, some recommendations regarding the numerical simulation of hyperbolic heat conduction are presented.     

Domain decomposition in the symmetric boundary element analysis

 Recent developments in the symmetric boundary element method (SBEM) have shown a clear superiority of this formulation over the collocation method. Its competitiveness has been tested in comparison to the finite element method (FEM) and is manifested in several engineering problems in which internal boundaries are present, i.e. those in which the body shows a jump in the physical characteristics of the material and in which an appropriate study of the response must be used. When we work in the ambit of the SBE formulation, the body is subdivided into macroelements characterized by some relations which link the interface boundary unknowns to the external actions. These relations, valid for each macroelement and characterized by symmetric matricial operators, are similar in type to those obtainable for the FEM. The assembly of the macroelements based on the equilibrium conditions, or on the compatibility conditions, or on both of these conditions leads to three analysis methods: displacement, force, and mixed-value methods, respectively. The use of the fundamental solutions involves the punctual satisfaction of the compatibility and of the equilibrium inside each macroelement and it causes a stringent elastic response close to the actual solution. Some examples make it possible to perform numerical checks in comparison with solutions obtainable in closed form. These checks show that the numerical solutions are floating ones when the macroelement geometry obtained by subdividing the body changes. Received 26 January 2001     

An iterative boundary element method for Cauchy inverse problems

 We are interested in this paper in recovering an harmonic function from the knowledge of Cauchy data on some part of the boundary. A new inversion method is introduced. It reduces the Cauchy problem resolution to the determination of the resolution of a sequence of well-posed problems. The sequence of these solutions is proved to converge to the Cauchy problem solution. The algorithm is implemented in the framework of boundary elements. Displayed numerical results highlight its accuracy, as well as its robustness to noisy data. Received 6 November 2000     

Shape design sensitivity analysis and optimization of two-dimensional periodic thermal problems using BEM

 A general procedure to perform shape design sensitivity analysis for two-dimensional periodic thermal diffusion problems is developed using boundary integral equation formulation. The material derivative concept to describe shape variation is used. The temperature is decomposed into a steady state component and a perturbation component. The adjoint variable method is used by utilizing integral identities for each component. The primal and adjoint systems are solved by boundary element method. The sensitivity results compared with those by finite difference show good accuracy. The shape optimal design problem of a plunger model for the panel of a television bulb, which operates periodically, is solved as an example. Different objectives and amounts of heat flux allowed are studied. Corresponding optimum shapes of the cooling boundary of the plunger are obtained and discussed. Received 15 August 2001 / Accepted 28 February 2002     

Boundary element model of microsegregation during volumetric solidification of binary alloy

 In the paper the mathematical model of heat and mass transfer processes proceeding in the domain of casting is discussed. In particular, the volumetric solidification of a binary alloy under the assumption that the temperature is only time-dependent is analyzed. From the practical point of view such situation takes place when the casting is produced in typical moulding sand. The differential equation describing the course of solidification and cooling processes is presented in Sect. 1. In this equation the capacity of an internal heat source results from the Mehl–Johnson–Avrami–Kolmogorov theory [1, 2], at the same time the constant number of nuclei is accepted, while the rate of the solid phase growth is proportional to the second power of undercooling below the liquidus temperature. The macroscopic model is coupled with a microsegregation one (Sect. 2). This process is analyzed at the level of a single grain. The distribution of the alloy component in the control volume corresponding to the final grain radius is found on a basis of the boundary element method using discretization in time [3, 4]. The examples of numerical computations are also presented. Received 6 November 2000     

A symmetric Galerkin BEM implementation for 3D elastostatic problems with an extension to curved elements

 Like the finite element method (FEM), the symmetric Galerkin boundary element method (SGBEM) can produce symmetric system matrices. While widely developed for two dimensional problems, the 3D-applications of the SGBEM are very rare. This paper deals with the regularization of the singular integrals in the case of 3D elastostatic problems. It is shown that the integration formulas can be extended to curved elements. In contrast to other techniques, the Kelvin fundamental solutions are used with no need to introduce the new kernel functions. The accuracy of the developed integration formulas is verified on a problem with known analytical solution. Received 6 November 2000     

Stability analysis of an explicit finite element scheme for plane wave motions in elastic solids

 Expressions for critical timesteps are provided for an explicit finite element method for plane elastodynamic problems in isotropic, linear elastic solids. Both 4-node and 8-node quadrilateral elements are considered. The method involves solving for the eigenvalues directly from the eigenvalue problem at the element level. The characteristic polynomial is of order 8 for 4-node elements and 16 for 8-node elements. Due to the complexity of these equations, direct solution of these polynomials had not been attempted previously. The commonly used critical time-step estimates in the literature were obtained by reducing the characteristic equation for 4-node elements to a second-order equation involving only the normal strain modes of deformation. Furthermore, the results appear to be valid only for lumped-mass 4-node elements. In this paper, the characteristic equations are solved directly for the eigenvalues using <ty>Mathematica<ty> and critical time-step estimates are provided for both lumped and consistent mass matrix formulations. For lumped-mass method, both full and reduced integration are considered. In each case, the natural modes of deformation are obtained and it is shown that when Poisson's ratio is below a certain transition value, either shear-mode or hourglass mode of deformation dominates depending on the formulation. And when Poisson's ratio is above the transition value, in all the cases, the uniform normal strain mode dominates. Consequently, depending on Poisson's ratio the critical time-step also assumes two different expressions. The approach used in this work also has a definite pedagogical merit as the same approach is used in obtaining time-step estimates for simpler problems such as rod and beam elements. Received: 8 January 2002 / Accepted: 12 July 2002 The support of NSF under grant number DMI-9820880 is gratefully acknowledged.     

A complex variable boundary collocation method for plane elastic problems

 Lagrange interpolation is extended to the complex plane in this paper. It turns out to be composed of two parts: polynomial and rational interpolations of an analytical function. Based on Lagrange interpolation in the complex plane, a complex variable boundary collocation approach is constructed. The method is truly meshless and singularity free. It features high accuracy obtained by use of a small number of nodes as well as dimensionality advantage, that is, a two-dimensional problem is reduced to a one-dimensional one. The method is applied to two-dimensional problems in the theory of plane elasticity. Numerical examples are in very good agreement with analytical ones. The method is easy to be implemented and capable to be able to give the stress states at any point within the solution domain. Received: 20 August 2002 / Accepted: 31 January 2003     

A mixed least squares method for solving problems in linear elasticity: theoretical study

 In a previous paper we proposed a mixed least squares method for solving problems in linear elasticity. The solution to the equations of linear elasticity was obtained via minimization of a least squares functional depending on displacements and stresses. The performance of the method was tested numerically for low order elements for classical examples with well known analytical solutions. In this paper we derive a condition for the existence and uniqueness of the solution of the discrete problem for both compressible and incompressible cases, and verify the uniqueness of the solution analytically for two low order piece-wise polynomial FEM spaces. Received: 20 January 2001 / Accepted: 14 June 2002 The authors gratefully acknowledge the financial support provided by NASA George C. Marshall Space Flight Centre under contract number NAS8-38779.     

Time stepping schemes for coupled displacement and pore pressurē analysis

 The paper investigates the performance of various time stepping schemes for coupled displacement and pore pressure analysis. A number of alternative forms of the automatic time stepping method proposed by Sloan and Abbo (1999a) are also presented. These alternative schemes use different updates for the displacements and pore pressures and also adopt different starting conditions for the iterations. The automatic schemes are compared with an implicit θ-method, as well as an explicit method, through analysis of a variety of problems involving undrained loading, drained loading, and consolidation for Mohr-Coulomb and critical state models. As expected, the numerical results confirm that the explicit scheme is neither accurate nor robust. Although the implicit θ-method is accurate and fast, it fails to give a solution in a number of cases where the time step is large. The automatic schemes are shown to be accurate, fast and generally robust. Two of the automatic schemes proposed never fail to furnish a solution for the cases considered. In addition, all the automatic schemes are able to constrain the time-stepping (temporal integration) error in the displacements and pore pressures to lie near a prescribed tolerance, provided the iteration error tolerance is properly chosen. For complex soil models, it is important that the latter is set sufficiently small in order for the schemes to be able to constrain the time-stepping error to lie within a prescribed tolerance. Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years.