An efficient numerical algorithm with extrapolation for three-dimensional axisymmetric elasticity problems

A three-dimensional axisymmetric elasticity problem pertaining to the contact stresses between a smooth rigid sphere and an infinite elastic solid with a smooth spherical cavity of the same diameter has been considered. Uniaxial loading is applied to the solid at infinity, resulting in a separation along a portion of the boundary between the sphere and the solid. The problem has been considered as a mixed boundary-value problem of elasticity. The angle of contact and the stress distributions along the contact surface are determined by solving a set of dual-series equations associated with Legendre polynomials. Numerical results are presented.

In this paper the applications of equilibrium equation to geometric modeling is exploited and efficient numerical algorithms are proposed for solving the equilibrium equation. First we show that from diverse geometric modeling applications the equilibrium system can be extracted as the central framework. Second, by exploiting in-depth the special structures inherent in the geometric applications, we present simplified analytic solutions to the resulting geometric equilibrium equations via system decomposition. Finally, given the observation that the geometric equilibrium systems are extremely sensitive to both perturbations in input data and round off errors, efficient, stable and accurate numerical algorithms are proposed.

The three-dimensional axisymmetric elastic problems of two materially dissimillar wedges of arbitrary angles that are bonded together along a common edge are treated. The dependence of the order of the singularity in the stress field at the apex on the wedge angles and material constants is considered. The order of the stress singularities is determined by use of local coordinates at the top of the wedge. The expression to determine the order of stress singularities in these problems is obtained.

ABSTRACTA new method based on topology optimization is proposed for solving a three-dimensional axisymmetric inverse problem regarding the configuration of electric currents used in the electromagnetic casting technique of the metallurgical industry. These electric currents must generate an electromagnetic field such that when certain mass of liquid metal is placed in the field it levitates acquiring a predefined axisymmetric shape. The method proposed is able to obtain a suitable configuration of electric currents, and is very attractive from the computational point of view, since it is based on a sparse convex quadratic programming formulation for which there are very efficient interior-point solution algorithms.

This paper presents a novel variant of the boundary element method, here called the boundary contour method, applied to three-dimensional problems of linear elasticity. In this work, the surface integrals on boundary elements of the usual boundary element method are transformed, through an application of Stokes’ theorem, into line integrals on the bounding contours of these elements. Thus, in this formulation, only line integrals have to be numerically evaluated for three-dimensional elasticity problems—even for curved surface elements of arbitrary shape. Numerical results are presented for some three-dimensional problems, and these are compared against analytical solutions.

The three-dimensional elastic problems in finite deformations are not known to have been analyzed by the usual stress function and displacement function. By applying Hasegawa's presentation and Adkins perturbation method, we propose a new analytical method of three-dimensional elastic problems for compressible materials and incompressible materials, respectively, using the displacement function for axisymmetrical elastic problems in finite deformations with surface force or body force. Further, this analytical method is examined by a simple example.

In this thesis a variety of integral equations and partial differential equations which describe two-dimensional and three-dimensional axisymmetric forced convection problems are studied. The correspondence between the integral equation formulation and partial differential equation formulation of such boundary value problems is investigated and this correspondence is used to develop a new technique for the solution of hitherto intractable Fredholm integral equations of the first kind on a finite interval. Basic properties of the physical models are also established by studying fundamental solutions of the partial differential equations. The problems are developed by considering the transport of heat in an inviscid, incompressible, heat conducting fluid whose thermal constants are assumed invariant in temperature, space and time. The transport mechanisms are those of convection in a prescribed velocity field and diffusion, and the temperature field is given by the solution of the partial differential equation (and, less directly, by the integral equation) which is a heat conservation equation and characterises the balance between the diffusion and convection processes. While such problems are highly idealised they have more practical applications in several special cases in which the partial differential equation takes the form of a linear approximation of the Navier - Stokes equations. Circulation is transported by diffusion and convection in a viscous fluid and in certain circumstances the temperature model can be regarded as a model of a viscous fluid. The integral equations also arise in an elastic half-space problem which is described in the first chapter.The two-dimensional problem has been the subject of-many previous investigations. New contributions are made in the domain of the integral equation whose solution has been given explicitly for the first time, and in the dependence of the solutions on the Peclet number, a non-dimensional parameter which characterises the ratio of the diffusive flux to the convective flux, where certain physically motivated results have been confirmed by a more strict mathematical approach. Similar, but considerably more extensive, investigations are made of a family of three-dimensional axisymmetric problems. In these problems, the first order radial derivative term in the partial differential equation has a singular coefficient whose role in the equation is interpreted as that due to a radial component in the forced convection field (‘radial' means 'radial direction in a cylindrical polar coordinate system') The crucial problem here is the realisation of a full understanding of the relationship between the partial differential equation and the integral equation. This is achieved by rewriting the partial differential equation in what is essentially its adjoint form and interpreting the new partial differential equation as a conservation equation which also expresses the balance of diffusion and convection of a quantity called <<-heat in a prescribed convection field. Physically motivated arguments are used to suggest the relationship between the original partial differential equation and the integral equation formulation which is then proved using rigorous mathematical arguments. A Peclet number is defined in this problem and the dependence of the solutions on this parameter is analysed. Representation and uniqueness theorems are also given in classical cases and a discussion of the extension of these results into generalised function spaces is included.

This study presents a fast meshless method called the hybrid reproducing kernel particle method (HRKPM) for the solution of three-dimensional (3D) elasticity problems. The equilibrium equations of 3D elasticity are divided into three groups of equations, and two equilibrium equations are contained in each group. By coupling the discrete equations for solving two arbitrary groups of equations, the complete solution of 3D elasticity can be obtained. For an arbitrary group of equations, the 3D elasticity problem is transformed into a series of associated two-dimensional (2D) ones, which is solved by the RKPM to derive the discrete formulae. The discrete equations of 2D problems are combined using the difference method in dimension splitting direction. Then, arbitrarily choosing another group of equilibrium equations, the discrete equation of another group of 2D problems can be obtained similarly. By combining the discrete equations for these two groups of 2D problems, the solution to an original 3D problem will be reached. The numerical results show that the HRKPM performs better than RKPM in solution efficiency.

This paper constructs an e‐cient high-fldelity model for plates made of functionally graded material. By taking advantage of an inherent small parameter, the ratio of the thickness to the characteristic wavelength of the deformation of the reference surface, we apply the variational asymptotic method to rigorously decouple the original threedimensional anisotropic elasticity problem into a one-dimensional through-the-thickness analysis and a two-dimensional plate analysis. The through-the-thickness analysis provides constitutive relations for the plate analysis as well as the recovery information for the three-dimensional flelds, linking the original, complex three-dimensional anisotropic heterogeneous elasticity problem to a simple two-dimensional plate model which achieves the best compromise between e‐ciency and accuracy. Furthermore, the derived models are geometrically exact and valid for large deformations and global rotations with the restriction that strains are small. Excellent accuracy of present model has been validated by comparing the displacement and stress distributions with exact solutions both for the cylindrical bending of an isotropic plate and the behavior of a thick, simply-supported, two-constituent metal-ceramic functionally grated rectangular plate.

The method of fundamental solutions for three-dimensional inverse geometric elasticity problems

Investigation on near-boundary solutions for three-dimensional elasticity problems by an advanced BEM

This paper shows the development of a displacement potential function based on the Galerkin potential using complex variables. The displacement potential function results in a more suitable method for numerical calculations since it avoids the strenuous integration process associated with stress potential methods. Completeness of the displacement potential function is demonstrated. The displacement potential function was applied to the solution of the first fundamental problem of elasticity over a three-dimensional domain with known boundary conditions. It’s application for numerical calculations is demonstrated by solving the pure shear problem over a three-dimensional unit hexahedral cell. Finally, the obtained numerical results are compared against finite element results, proving the validity of the displacement potential function in solving three-dimensional linear elasticity problems.

On a three-dimensional axisymmetric boundary-value problem of nonlinear elastic deformation: Asymptotic solution and exponentially small error

The three-dimensional axisymmetric problem of the indentation of a thin compressible linear elastic layer bonded to a rigid foundation is considered. Approximate analytical solutions of the problem that incorporate a large portion of the singular deformation gradients near the edge of the indenter are presented. An accurate closed-form expression for the deformation as well as the deformation gradient throughout the layer is provided and its effectiveness in solving the problem numerically is demonstrated. By incorporating the approximate solution into the numerical scheme the accuracy and convergence rate increase dramatically.

In the previous chapter we derived boundary integral equations relating the known boundary conditions to the unknowns. For practical problems, these integral equations can only be solved numerically. The simplest numerical implementation is using line elements, where the knowns and unknowns are assumed to be constant inside the element. In this case, the integral equation can be written as the sum of integrals over elements. The integrals over the elements can then be evaluated analytically. In the previous chapter we have presented constant elements for the solution of two-dimensional potential problems only. The analytical evaluation over elements would become quite cumbersome for two- and three-dimensional elasticity problems. Constant elements were used in the early days of the development, where the method was known under the name Boundary Integral Equation (BIE) Method1. This is similar to the development of the FEM, where triangular and tetrahedral elements, with exact integration, were used in the early days. In 1968, Ergatoudis and Irons2 suggested that isoparametric finite elements and numerical integration could be used to obtain better results, with fewer elements. The concept of higher order elements and numerical integration is very appealing to engineers because it alleviates the need for tedious analytical integration and, more importantly, it allows the writing of general purpose software with a choice of element types. Indeed, this concept will allow us to develop one single program to solve two- and three-dimensional problems in elasticity and potential flow, or any other problem for which we can supply a fundamental solution (see Chapter 18).KeywordsBoundary Element MethodNumerical ImplementationBoundary Integral EquationIntegration RegionCollocation PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.