Boundary Value Problems Of Elasticity Research Articles

Regularized boundary element solution for an inverse boundary value problem in linear elasticity

AbstractIn this paper the Tikhonov regularization method is analysed for the ill‐conditioned system of linear equations which arises from the discretization of an ill‐posed boundary value problem in linear elasticity. The choice of the regularization parameter is achieved according to the L‐curve method. Then the regularization method is numerically implemented using the boundary element method (BEM). Copyright © 2002 John Wiley & Sons, Ltd.

On the inverse boundary value problem for linear isotropic elasticity

We derive three results on the inverse problem of determining the Lamé parametersλ(x)and μ(x)for an isotropic elastic body from its Dirichlet-to-Neumann map.

Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition

In this paper the singular value decomposition (SVD), truncated at an optimal number, is analysed for obtaining approximate solutions to ill-conditioned linear algebraic systems of equations which arise from the boundary element method (BEM) discretisation of an ill-posed boundary value problem in linear elasticity. The regularisation parameter, namely the optimal truncation number, is chosen according to the discrepancy principle. The numerical results obtained confirm that the SVD+BEM produces a convergent and stable numerical solution with respect to decreasing the mesh size discretisation and the amount of noise added into the input data.

Exact solution of the displacement boundary-value problem of elasticity for a torus

This paper presents an exact analytical solution to the displacement boundary-value problem of elasticity for a torus. The introduced form of the general solution of elastostatics equations allows to solve exactly a broad class of boundary-value problems in coordinate systems with incomplete separation of variables in the harmonic equation. The original boundary-value problem for a torus is reduced to infinite systems of linear algebraic equations with tridiagonal matrices. An analytical technique for solving systems of diagonal form is developed. Uniqueness of the solutions of vector boundary-value problems involving the generalized Cauchy-Riemann equations is investigated, and it is shown that the obtained solution for the displacement boundary-value problem for a torus is unique due to the specific properties of the suggested general solution. The analogy between problems of elastostatics and steady Stokes flows is demonstrated, and the developed elastic solution is used to solve the Stokes problem for a torus.

Nonlocal elasticity and related variational principles

Abstract The Eringen model of nonlocal elasticity is considered and its implications in solid mechanics studied. The model is refined by assuming an attenuation function depending on the `geodetical distance' between material particles, such that in the diffusion processes of the nonlocality effects certain obstacles as holes or cracks existing in the domain can be circumvented. A suitable thermodynamic framework with nonlocality is also envisaged as a firm basis of the model. The nonlocal elasticity boundary-value problem for infinitesimal displacements and quasi-static loads is addressed and the conditions for the solution uniqueness are established. Three variational principles, nonlocal counterparts of classical ones, i.e. the total potential energy, the complementary energy, and the mixed Hu–Washizu principles, are provided. The former is used to derive a nonlocal-type FEM (NL-FEM) in which the (symmetric) global stiffness matrix reflects the nonlocality features of the problem. An alternative standard-FEM-based solution method is also provided, which consists in an iterative procedure of the type local prediction/nonlocal correction, in which the nonlocality is simulated by an imposed-like correction strain. The potentialities of these analysis methods are pointed out, their numerical implementations being the object of an ongoing research work.

Boundary element method for the Cauchy problem in linear elasticity

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput Maths Math Phys 32 (1991) 45] for obtaining approximate solutions to ill-posed boundary value problems in linear elasticity is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results obtained confirm that the iterative 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. An efficient stopping regularizing criterion is given and in addition, the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. Analytical formulae for the integration constants resulting from the direct application of the BEM for an isotropic linear elastic medium are also presented.

Elastic fields of quantum dots in subsurface layers

In this work, models based on conventional small-strain elasticity theory are developed to evaluate the stress fields in the vicinity of a quantum dot or an ordered array of quantum dots. The models are based on three different approaches for solving the elastic boundary value problem of a misfitting inclusion embedded in a semi-infinite space. The first method treats the quantum dot as a point source of dilatation. In the second approach we approximate the dot as a misfitting oblate spheroid, for which exact analytic solutions are available. Finally, the finite element method is used to study complex, but realistic, quantum dot configurations such as cuboids and truncated pyramids. We evaluate these three levels of approximation by comparing the hydrostatic stress component near a single dot and an ordered array of dots in the presence of a free surface, and find very good agreement except in the immediate vicinity of an individual quantum dot.

A Numerical Method for the Solution of the Second Boundary Value Problem of Elasticity for Solids with Cracks

A numerical method for solving the second boundary value problem of elasticity for solids with cracks is developed. The method is based on a class of Gaussian approximating functions that drastically simplify the construction of the matrix of the linear algebraic system of the discretized problem. The results of the application of the method to 2D-problems of elasticity are compared with exact and numerical solutions existing in the literature.

Boundary Point Method in the Dynamic Problems of Elasticity for Plane Areas with Cracks

Boundary point method is applied to the solution of the second boundary value problem of dynamic elasticity for solids with cracks. The method is based on a class of Gaussian approximating functions that drastically simplify the construction of the matrix of the linear algebraic system of the discretized problem. The results of the application of the method to 2D-problems of dynamic elasticity are compared with the results of other methods existing in the literature.

Plane Elastic Boundary Value Problem Posed by Displacement and Force Orientations on a Closed Contour

A boundary value problem (BVP) of the plane elasticity posed in terms of the orientations of forces and displacements is considered. The main aim of the present paper is to investigate the solvability of BVPs of this kind. Firstly, analysis of two cases is performed: the case of a circle with special orientations of force and displacement vectors on the circumference and the case of an arbitrary contour with coaxial orientations of these vectors. The solutions obtained indicate that the problem can have a certain number of solutions or be unsolvable. Then the BVP is reduced to a boundary integral equation and its solvability is investigated for the general case of a smooth simple-connected closed contour. As a result, the number of linearly independent solutions is determined. This number only depends upon the angle between the force and displacement vectors.

A Monte Carlo solution method for linear elasticity

A probabilistic method is proposed and implemented to solve linear elasticity problems. The method, called walk on the boundary method (WBM), uses the same governing equations as the boundary element method. Unlike in finite element and boundary element methods, WBM does not require any meshing. Also, error estimates for WBM are easier to obtain than in finite element and boundary element methods. Furthermore, WBM obtains a point solution at a specific point of interest instead of a full field solution as in finite element and boundary element methods. WBM is developed for general traction boundary value problems in antiplane shear, plane strain, and 3D elasticity. Numerical implementations are performed for three example problems: (1) antiplane shear problem with a centrally located circular hole being loaded by uniformly applied traction, (2) plane strain problem with centrally located circular hole being loaded by uniform tension, and (3) 3D elasticity problem with a centrally located spherical cavity being loaded by uniform tension. Results from the three example problems compared favorably with results from the analytical and finite element solution. Three critical issues associated with the WBM for linear elasticity are pointed out for its further improvement.

Three-dimensional variational theory of laminated composite plates and its implementation with Bernstein basis functions

A new three-dimensional (3-D) variational theory aimed at the stress analysis of thick laminated rectangular plates with anisotropic layers is presented. The developed theory is versatile, it allows one to use various types of basis functions and apply them independently to each of the three coordinate directions. Also, the theory is material-adaptive, enabling us to impose the conditions of displacement continuity between all of the discretization elements (3-D bricks) within the body and, in addition, the conditions of stress continuity between adjacent bricks made from the same material. The theory can be applied, in principle, to any boundary value problem of linear elasticity, considering arbitrarily distributed static surface forces and external kinematic boundary conditions. One specific realization of the theory, elaborated in this work, applies Bernstein basis functions of an arbitrary degree for the displacement approximation in the three coordinate directions. This type of basis function, which are a rarity in the structural analysis, deserve more attention. They provide computational efficiency and unique analytical elegance to the mathematical algorithms. The developed theory can be also used for generating new types of higher-order hexahedral finite element, as illustrated here on the example of a novel 8-node “Bernstein finite element”. Numerical examples given in this work show that, following the concept of material-adaptive inter-element continuity conditions, a higher smoothness and accuracy of the computed stresses can be achieved by incorporating additional inter-element constraints between the same material bricks. However, it is also easy to spoil the solution by adding the same type constraints between adjacent bricks having distinct material properties. Finally, numerical comparison for the benchmark 3-D problem of transverse bending of simply supported 3-layer cross-ply laminated plate validates that the present analysis is significantly more accurate and computationally efficient than ANSYS SOLID 46 element.

A note on a boundary element method for the numerical solution of boundary value problems in isotropic inhomogeneous elasticity

A boundary element method is derived for the solution of boundary value problems for inhomogeneous isotropic elastic materials. Some particular problems are considered to illustrate the application of the method.

Some recent improvements in meshfree methods for incompressible finite elasticity boundary value problems with contact

Two major difficulties are encountered in the meshfree solution of incompressible boundary value problems. The first is due to the employment of higher-order quadrature rules that leads to an over-constrained discrete system in incompressible problems. The second is associated with the treatment of essential boundary conditions and contact conditions owing to the loss of Kronecker delta properties in the meshfree shape functions. This paper discusses some recent enhancements in meshfree methods for incompressible boundary value problems, carries out numerical convergence analysis, and compares accuracy and efficiency improvement of these methods. Presented methods are a pressure projection method to remedy the over-constrained discrete system, and a mixed transformation method and a boundary singular kernel method for imposition of essential boundary conditions and contact constraints. Several linear and nonlinear problems were analyzed to demonstrate the effectiveness of the new approaches.

Aspects of Adaptive Remeshing Techniques in Finite Elasticity

AbstractOne of the major difficulties in applying the finite element method to boundary value problems of finite elasticity is the design of a suitable mesh in order to maximize the accuracy of the solution and to minimize the numerical cost. The basic problem in this context is the error estimation of the nonlinear problem. Our aim is to present possible a posteriori error estimators for finite elasticity which operate in the linear tangential space of the nonlinear problem.

First-Order System Least Squares for the Stokes and Linear Elasticity Equations: Further Results

First-order system least squares (FOSLS) was developed in [ SIAM J. Numer. Anal., 34 (1997), pp. 1727--1741; SIAM J. Numer. Anal., 35 (1998), pp. 320--335] for Stokes and elasticity equations. Several new results for these methods are obtained here. First, the inverse-norm FOSLS scheme that was introduced but not analyzed in [ SIAM J. Numer. Anal., 34 (1997), pp. 1727--1741] is shown to be continuous and coercive in the L2 norm. This result is shown to hold for pure displacement or pure traction boundary conditions in two or three dimensions, and for mixed boundary conditions in two dimensions. Next, the FOSLS schemes developed in [ SIAM J. Numer. Anal., 35 (1998), pp. 320--335] are applied to the pure displacement problem in planar and spatial linear elasticity byeliminating the pressure variable in the FOSLS formulations of [SIAM J. Numer. Anal., 34 (1997), pp. 1727--1741]. The idea of two-dimensional variable rotation is then extended to three dimensions to make the intervariable coupling subdominant (uniformly so in the Poisson ratio for elasticity). This decoupling ensures optimal (uniform) performance of finite element discretization and multigrid solution methods. It also allows special treatment of the new trace variable, which corresponds to the divergence of velocity in the case of Stokes, so that conservation can be easily imposed, for example. Numerical results for various boundary value problems of planar linear elasticity are studied in a companion paper [SIAM J. Sci. Comput., 21 (2000), pp. 1706--1727].

First-Order System Least Squares For Linear Elasticity: Numerical Results

Two first-order system least squares (FOSLS) methods based on L2 norms are applied to various boundary value problems of planar linear elasticity. Both use finite element discretization and multigrid solution methods. They are two-stage algorithms that solve first for the displacement flux variable (the gradient of displacement, which easily yields the deformation and stress variables), then for the displacement variable itself. As a complement to a companion theoretical paper, this paper focuses on numerical results, including finite element accuracy and multigrid convergence estimates that confirm uniform optimal performance---even as the material tends to the incompressible limit.

Micro-Macro Modeling of Nonlinear Mechanical Behaviors based on the Homogenization Theory.

A multiscale modeling method based on the mathematical theory of homogenization is introduced for studying the micro-macro coupled mechanical behaviors. After describing a general nature of two-scale boundary value problem for nonlinear elasticity in terms of the generalized variational principle, the multiscale mathematical modeling for several microscopically defined mechanical behaviors are provided by using the generalized convergence theorems in the homogenization theory. With some numerical examples for each modeling, the mathematical structure of this modeling is interpreted in mechanics' points of view. That is, the macrostructure refers its material response to the corresponding microscopic mechanical behavior which defines microscopic boundary value problem. Due to the generality of this modeling method and the numerical algorithm, the micro-macro coupling effects of heterogeneous media can be analyzed to clarify the highly complex mechanisms in nonlinear or inelastic responses.

The Cosserat spectrum for cylindrical geometries: (Part 2: [formula omitted] subspace and applications)

We construct the orthonormal bases of the Cosserat subspace u ̃ −1 corresponding to the eigenvalue of infinite multiplicity ω ̃ =−1 for the first boundary value problems of elasticity for a solid cylinder and a cylindrical rigid inclusion. These bases involve the Jacobi polynomials with different weight functions. An example of non-harmonic heat flow past a cylindrical rigid inclusion shows that the sequence of u ̃ −1 converges fast, thus, the Cosserat spectrum theory is an efficient method for solving elasticity problems of general body force or boundary loading.

A mathematical study on a unified variational principle with two constrained parameters

In this paper, a parameterized variational principle based on a mixed functional obtained by a linear combination of the total potential energy functional, the modified Hellinger-Reissner functional, and the Hu-Washizu functional with two constrained parameters is proposed, and the mathematical characteristics of the variational equation of the principle are investigated for the analysis of boundary value problems in linear elasticity. It is first proved that the Euler-Lagrange equations of the variational equation is identical to the governing equations for the given problem. Then existence of the unique solution of the variational equation is systematically proved by showing that the energy bilinear form is weakly-coercive. As an application, the stress/strain smoothing can be obtained as a form of mixed FEM based on the variational equation.