Evolutionary optimization in thermoelastic problems using the boundary element method

 The paper is devoted to application of evolutionary algorithms and the boundary element method to shape optimization of structures for various thermomechanical criteria, inverse problems of finding an optimal distribution of temperature on the boundary and identification of unknown boundary. Design variables are specified by Bezier curves. Several numerical examples of evolutionary computation are presented. Received 6 November 2000     

A boundary elements pseudo heat source method formulation for inverse analysis of solidification problems

 An efficient methodology is presented to solve inverse solidification problems. In the procedure, the latent heat effects are implemented by introducing pseudo heat sources near the moving interface. The material properties can be temperature dependent. To account for the nonlinear part of the governing differential equations, a finite-boundary element formulation is employed. To reduce the oscillations in the solution, a sequential regularization scheme is used. A procedure for proper selection of regularization parameters is presented. To smooth the solutions further, a secondary regularization scheme is introduced and employed. Two complete examples are presented to demonstrate the applicability and the accuracy of the methods. Received: 1 March 2002 / Accepted: 10 February 2003     

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     

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     

Boundary element methods for boundary condition inverse problems in elasticity using PCGM and CGM regularization

For an isotropic linear elastic body, only displacement or traction boundary conditions are given on a part of its boundary, whilst all of displacement and traction vectors are unknown on the rest of the boundary. The inverse problem is different from the Cauchy problems. All the unknown boundary conditions on the whole boundary must be determined with some interior points' information. The preconditioned conjugate gradient method (PCGM) in combination with the boundary element method (BEM) is developed for reconstructing the boundary conditions, and the PCGM is compared with the conjugate gradient method (CGM). Morozov's discrepancy principle is employed to select the iteration step. The analytical integral algorithm is proposed to treat the nearly singular integrals when the interior points are very close to the boundary. The numerical solutions of the boundary conditions are not sensitive to the locations of the interior points if these points are distributed along the entire boundary of the considered domain. The numerical results confirm that the PCGM and CGM produce convergent and stable numerical solutions with respect to increasing the number of interior points and decreasing the amount of noise added into the input data.     

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     

An inverse estimation of multi-dimensional load distributions in thermoelasticity problems via dual reciprocity BEM

Abstact The paper deals with inverse thermoelasticity problems. A general, robust, and numerically efficient technique, for retrieving the multi-dimensional, highly varying distributions of boundary conditions is presented. In the class of considered inverse problems, both input data and sought-for quantities are usually specified at the domain boundary, only. As a sequence of forward sub-solutions underlies the inverse analysis, the numerical method of choice for solving the field problem is the boundary element method (BEM). The derived inverse technique is capable of retrieving boundary condition distributions in steady state and transient problems. The accuracy and stability of the algorithm are verified by considering problems involving constant, functionally graded, and temperature dependent material properties. Strain components and temperatures, subject to uncertainties, are used as input data. Presented numerical examples show that the method is capable of reconstructing mechanical and thermal loads with reasonable accuracy. while on leave from Cracow University of Technology, 31-155 Cracow, Poland     

Recursive solution of inverse natural convection problems by means of mode reduction

 A recursive method based on the Kalman filtering is developed to solve inverse natural convection problems of estimating the unsteady nonuniform wall heat flux from temperature measurements in the flow. By employing the Karhunen–Loève Galerkin procedure that reduces the Boussinesq equation to a small set of ordinary differential equations, the computational difficulties associated with the Kalman filtering for the partial differential equations are overcome. The present method is assessed through several numerical experiments, and is found to yield satisfactory results. Received 20 January 2001 / Accepted 31 May 2001     

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     

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     

A fast convergent iterative boundary element method on PVM cluster

This paper reports a fast convergent boundary element method on a Parallel Virtual Machine (PVM) (Geist et al., PVM: Parallel Virtual Machine, A Users' Guide and Tutorial for Networked Parallel Computing. MIT Press, Cambridge, 1994) cluster using the SIMD computing model (Single Instructions Multiple Data). The method uses the strategy of subdividing the domain into a number of smaller subdomains in order to reduce the size of the system matrix and to achieve overall speedup. Unlike traditional subregioning methods, where equations from all subregions are assembled into a single linear algebraic system, the present scheme is iterative and each subdomain is handled by a separate PVM node in parallel. The iterative nature of the overall solution procedure arises due to the introduction of the artificial boundaries. However, the system equations for each subdomain is now smaller and solved by direct Gaussian elimination within each iteration. Furthermore, the boundary conditions at the artificial interfaces are estimated from the result of the previous iteration by a reapplication of the boundary integral equation for internal points. This method provides a consistent mechanism for the specification of boundary conditions on artificial interfaces, both initially and during the iterative process. The method is fast convergent in comparison with other methods in the literature. The achievements of this method are therefore: (a) simplicity and consistency of methodology and implementation; (b) more flexible choice of type of boundary conditions at the artificial interfaces; (c) fast convergence; and (d) the potential to solve large problems on very affordable PVM clusters. The present parallel method is suitable where (a) one has a distributed computing environment; (b) the problem is big enough to benefit from the speedup achieved by coarse-grained parallelisation; and (c) the subregioning is such that communication overhead is only a small percentage of total computation time.     

A boundary element method for solving 3D static gradient elastic problems with surface energy

 A boundary element methodology is developed for the static analysis of three-dimensional bodies exhibiting a linear elastic material behavior coupled with microstructural effects. These microstructural effects are taken into account with the aid of a simple strain gradient elastic theory with surface energy. A variational statement is established to determine all possible classical and non-classical (due to gradient with surface energy terms) boundary conditions of the general boundary value problem. The gradient elastic fundamental solution with surface energy is explicitly derived and used to construct the boundary integral equations of the problem with the aid of the reciprocal theorem valid for the case of gradient elasticity with surface energy. It turns out that for a well posed boundary value problem, in addition to a boundary integral representation for the displacement, a second boundary integral representation for its normal derivative is also necessary. All the kernels in the integral equations are explicitly provided. The numerical implementation and solution procedure are provided. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. Advanced algorithms are presented for the accurate and efficient numerical computation of the singular integrals involved. Two numerical examples are presented to illustrate the method and demonstrate its merits. Received: 9 November 2001 / Accepted: 20 June 2002 The first and third authors gratefully acknowledge the support of the Karatheodory program for basic research offered by the University of Patras.     

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     

Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations

 In this paper, an iterative algorithm based on the conjugate gradient method (CGM) in combination with the boundary element method (BEM) for obtaining stable approximate solutions to the Cauchy problem for Helmholtz-type equations is analysed. An efficient regularising stopping criterion for CGM proposed by Nemirovskii [25] is employed. The numerical results obtained confirm that the CGM + BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. Received: 5 November 2002 / Accepted: 5 March 2003 L. Marin would like to acknowledge the financial support received from the EPSRC. The authors would like to thank Professor Dinh Nho Hào and Dr. Thomas Johansson for some useful discussions and suggestions.     

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     

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     

Dual boundary element formulation for inverse potential problems in crack identification

In this paper a new boundary element formulation is presented for the identification of the location and size of internal cracks in two dimensional structures. The method is presented, as a supplement to the experimental non-destructive testing (NDT) methods, for more accuracy in the identification procedure. The identification method is presented, proposing the dual boundary element method (DBEM) as the basis for design sensitivity computation. Examples are presented to demonstrate the performance of the crack identification method for various cracks.     

Radial integration boundary element method for acoustic eigenvalue problems

In this paper, the radial integration boundary element method is developed to solve acoustic eigenvalue problems for the sake of eliminating the frequency dependency of the coefficient matrices in traditional boundary element method. The radial integration method is presented to transform domain integrals to boundary integrals. In this case, the unknown acoustic variable contained in domain integrals is approximated with the use of compactly supported radial basis functions and the combination of radial basis functions and global functions. As a domain integrals transformation method, the radial integration method is based on pure mathematical treatments and eliminates the dependence on particular solutions of the dual reciprocity method and the particular integral method. Eventually, the acoustic eigenvalue analysis procedure based on the radial integration method resorts to a generalized eigenvalue problem rather than an enhanced determinant search method or a standard eigenvalue analysis with matrices of large size, just like the multiple reciprocity method. Several numerical examples are presented to demonstrate the validity and accuracy of the proposed approach.