Use of the conventional and tangent derivative boundary integral equations for the solution of problems in linear elasticity

Regularized forms of the traction and tangent derivative boundary integral equations of elasticity are derived for the case of closed regions. The hypersingular and strongly singular integrals of the displacement gradient representation are regularized independently, through identities of the fundamental solution and its various derivatives, before the boundary integral equations are formed. Besides the displacements and the tractions, only the tangential derivatives of the displacements evaluated at the singular point appear in the regularized equations making them well suited for numerical treatment. The regularization of the hypersingular integrals demands that the displacement components have Holder continuous first derivatives at the singular point. Consistent with this requirement, the regularization of the strongly singular integrals is effective if the tractions and the unit vectors normal and tangent to the surface are continuous at that location.;Higher order elements for two and three dimensional elastostatic problems are implemented through the coincident collocation of regularised forms of the displacement and the tangent derivative equations. The nodal values of the displacements, the fractions and their tangential derivatives are used as the degrees of freedom associated with the functional representation of the boundary variables. The tangential derivatives of the displacements and the tractions at the functional nodes are directly recovered from the boundary solution with comparable accuracy as the primative variables. Hence, the nodal values of the stress components are directly obtained through Hooke\u27s law and need not be determined in a post processing manner. Several numerical examples demonstrate the advantages of the higher order elements versus the conventional ones. In two dimensions, four degrees of freedom per node Hermitian elements are used for functional interpolation only on those portions of the boundary where the gradients are high and quadratic Lagrangian elements are employed for the remaining parts of the modelled region. In three dimensions, nine degrees of freedom per node, incomplete quartic elements are employed for the approximation of the displacements and the tractions. Finally, the methodology presented here is general and can be extended to other problems amenable to a boundary integral formulation

An O(h6) cubic spline interpolating procedure for harmonic functions

An O(h6) method for the interpolation of harmonic functions in rectangular do- mains is described and analysed. The method is based on an earlier cubic spline tech- nique [7], and makes use of recent results concerning the a posteriori correction of interpolatory cubic splines

Set covering and set partitioning: a collection of test problems

It is now well established that set covering and set partitioning models play a central role in many scheduling applications. There are many algorithms which solve these problems. In order to test and validate such implementations we have collected a range of test problems taken from different contexts. A brief description of these models, their applications and summary model data, are supplied in this paper

On the development of an efficient truly meshless discretization procedure in computational mechanics

Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2001.Includes bibliographical references (leaves 157-163).The objective of this thesis is to present an efficient and reliable meshless computational technique - the method of finite spheres - for the solution of boundary value problems on complex domains. This method is truly meshless in the sense that the approximation spaces are generated and the numerical integration is performed without a mesh. While the theory behind meshless techniques is rather straightforward, the generation of a computationally efficient scheme is quite difficult. Computational efficiency may be achieved by proper choice of the interpolation functions, effective ways of incorporating the essential boundary conditions and efficient and specialized numerical integration rules. The pure displacement formulation is observed to exhibit volumetric "locking" during incompressible (or nearly incompressible) analysis. A displacement/pressure mixed formulation is developed to overcome this problem. The stability and optimality of the mixed formulation are tested using numerical inf-sup tests for a variety of discretization schemes. Solutions to several example problems are presented showing the application of the method of finite spheres to problems in solid and fluid mechanics. A very specialized application of the technique to physically based real time medical simulations in multimodal virtual environments is also presented. In the current form of implementation, the method of finite spheres is about five times slower than the finite element techniques for problems in two-dimensional elastostatics.by Suvranu De.Sc.D

On the development of an efficient truly meshless discretization procedure in computational mechanics

Supervised by Mandayam A. Srinivasan and Klaus-Jurgen Bathe.Also issued as Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2001.Includes bibliographical references (leaves 157-163).by Suvranu De

Evolutionary structural pptimisation based on boundary element representation of b-spline geometry

Evolutionary Structural Optimisation (ESO) has become a well-established technique for determining the optimum shape and topology of a structure given a set of loads and constraints. The basic ESO concept that the optimum topology design evolves by slow removal and addition of material has matured over the last ten years. Nevertheless, the development of the method has almost exclusively considered finite elements (FE) as the approach for providing stress solutions. This thesis presents an ESO approach based on the boundary element method. Non-uniform rational B-splines (NURBS) are used to define the geometry of the component and, since the shape of these splines is governed by a set of control points, use can be made of the locations of these control points as design variables. The developed algorithm creates internal cavities to accomplish topology changes. Cavities are also described by NURBS and so they have similar behaviour to the outside boundary. Therefore, both outside and inside are optimised at the same time. The optimum topologies evolve allowing cavities to merge between each other and to their closest outer boundary. Two-dimensional structural optimisation is investigated in detail exploring multi-load case and multi-criteria optimisation. The algorithm is also extended to three-dimensional optimisation, in which promising preliminary results are obtained. It is shown that this approach overcomes some of the drawbacks inherent in traditional FE-based approaches, and naturally provides accurate stress solutions on smooth boundary representations at each iteration

Determination of suitable values for parameters governing B-spline based evolutionary structural optimisation using the boundary element method

The basic evolutionary structural optimisation concept (ESO) has been developed for several years. Recently, the first ESO algorithm based on the boundary element method (BEM) has been presented. In this thesis, this algorithm is used for the 2D shape optimisation. The aim is to develop a greater understanding of the role of certain governing parameters that drive the optimisation using this algorithm, and to make recommendations as to appropriate values of these parameters that give rise to good optimal solutions most efficiently. Two problems, a short cantilever beam and a fillet, are selected as test cases in this work. By using a wide range of numerical tests, the performance of the optimisation has been evaluated using a variety of methods including mean performance analysis and multi-objective optimisation approaches using Pareto curves and weighted sums. Recommendations are made as to appropriate values of these parameters that give rise to good optimal solutions most efficiently. Sensitivity analysis is another important method in engineering design. In this work a new algorithm to undertake a sensitivity analysis has been developed and used in a small number of investigations for boundary element structural optimisation process. ESO is selected when computational efficiency is thought the most important consideration, since it can reach the optimum in fewer iterations and lower run-time compared with sensitivity analysis in structural optimisation

Vibrations and instabilities of a disk galaxy with modified gravity.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1974.Vita.Bibliography: leaves 177-179.Ph.D

Boundary element analysis of spherical and radome shells

This study presents the application of the Boundary Element Method (BEM) to spherical and radome geometries. The boundary of the solution domain was discretized by using both linear and quadratic elements and the validity of the results were compared against other analytical and numerical methods.Several improvements to the BEM have been presented. These include the efficient evaluation of the singular integrals where new methods have been implemented and compared with other schemes. Improvement is also shown by the implementation of the semi-continuous elements to solve the well known limitation of the Corner Problems present in the BEM. Exhaustive numerical experimentation is carried out to establish the optimum collocation point for the semi-continuous elements and to link this to the quadrature rule used for the integration of that element. The present study also includes the limitations of the BEM in applications involving geometries of long and thin sections. The study shows in detail the circumstances under which accurate results can be expected in the BEM. In this case, the emphasis is placed on the element size and the section thickness. A relationship linking these two parameters in the control of the accuracy of the BEM results is also established. For the surface stresses and strains of the domain, a detailed implementation of a natural cubic spline is illustrated which greatly improved these surface results

3-D inelastic analysis methods for hot section components. Volume 2: Advanced special functions models

This Annual Status Report presents the results of work performed during the third year of the 3-D Inelastic Analysis Methods for Hot Sections Components program (NASA Contract NAS3-23697). The objective of the program is to produce a series of computer codes that permit more accurate and efficient three-dimensional analyses of selected hot section components, i.e., combustor liners, turbine blades, and turbine vanes. The computer codes embody a progression of mathematical models and are streamlined to take advantage of geometrical features, loading conditions, and forms of material response that distinguish each group of selected components