Superfluous matrices in linear complementarity

Superfluous matrices were introduced by Howe (1983) in linear complementarity. In general, producing examples of this class is tedious (a few examples can be found in Chapter 6 of Cottle, Pang and Stone (1992)). To overcome this problem, we define a new class of matrices and establish that in superfluous matrices of any ordern 4 can easily be constructed. For every integerk, an example of a superfluous matrix of degreek is exhibited in the end.     

Multigrid for the Galerkin least squares method in linear elasticity

In SIAM J. Numer. Anal. 28 (1991) 1680-1697, Franca and Stenberg developed several Galerkin least squares methods for the solution of the problem of linear elasticity. That work concerned itself only with the error estimates of the method. It did not address the related problem of finding effective methods for the solution of the associated linear systems. In this work, we prove the convergence of a multigrid method. This multigrid is robust in that the convergence is uniform as the parameter ν goes to 1/2. Computational experiments are included.     

Algebraic Dirichlet-to-Neumann mapping for linear elasticity problems with extreme contrasts in the coefficients

The convergence of iterative based domain decomposition methods is linked with the absorbing boundary conditions defined on the interface between the sub-domains. For linear elasticity problems, the optimal absorbing boundary conditions are associated with non-local Dirichlet-to-Neumann maps. Most of the methods to approximate these non-local maps are based on a continuous analysis. In this paper, an original algebraic technique based on the computation of local Dirichlet-to-Neumann maps is investigated. Numerical experiments are presented for linear elasticity problems with extreme contrasts in the coefficients.     

An auxiliary space preconditioner for linear elasticity based on the generalized finite element method

We construct and analyze a preconditioner of the linear elasticity system discretized by conforming linear finite elements in the framework of the auxiliary space method. The auxiliary space preconditioner is based on two auxiliary spaces corresponding to discretizations of the scalar Poisson equation by linear finite elements and the generalized finite element method. Copyright © 2011 John Wiley & Sons, Ltd.     

Stabilized dual-mixed method for the problem of linear elasticity with mixed boundary conditions

We extend the applicability of the augmented dual-mixed method introduced recently in Gatica (2007), Gatica et al. (2009) to the problem of linear elasticity with mixed boundary conditions. The method is based on the Hellinger–Reissner principle and the symmetry of the stress tensor is imposed in a weak sense. The Neumann boundary condition is prescribed in the finite element space. Then, suitable Galerkin least-squares type terms are added in order to obtain an augmented variational formulation which is coercive in the whole space. This allows to use any finite element subspaces to approximate the displacement, the Cauchy stress tensor and the rotation.     

Displacement decomposition—incomplete factorization preconditioning techniques for linear elasticity problems

Two preconditioning techniques for the numerical solution of linear elasticity problems are described and studied. Both techniques are based on spectral equivalence approach. The first technique consists in an incomplete factorization of the separate displacement component part of the stiffness matrix. The second technique uses an incomplete factorization of the isotropic approximation to the stiffness matrix. Results concerning existence, stability and efficiency of these preconditioning techniques are presented. The efficiency and robustness of the described techniques are illustrated by numerical experiments.     

Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

We consider an augmented mixed finite element method applied to the linear elasticity problem and derive a posteriori error estimators that are simpler and easier to implement than the ones available in the literature. In the case of homogeneous Dirichlet boundary conditions, the new a posteriori error estimator is reliable and locally efficient, whereas for non-homogeneous Dirichlet boundary conditions, we derive an a posteriori error estimator that is reliable and satisfies a quasi-efficiency bound. Numerical experiments illustrate the performance of the corresponding adaptive algorithms and support the theoretical results.     

On the frame invariance of linear elasticity theory

It is widely believed that classical linear elasticity theory does not conform to the Galilean frame invariance of general (non-relativistic) field theories. This view is traced here to an interpretation of the relationship between the deformation gradient and the displacement gradient which does not reflect the tensor character of the variables involved. Frame invariance is shown to follow if tensor character is imposed on this relationship at the outset.     

Hybridized weak Galerkin finite element method for linear elasticity problem in mixed form

A hybridization technique is applied to the weak Galerkin finite element method (WGFEM) for solving the linear elasticity problem in mixed form. An auxiliary function, the Lagrange multiplier defined on the boundary of elements, is introduced in this method. Consequently, the computational costs are much lower than the standard WGFEM. Optimal order error estimates are presented for the approximation scheme. Numerical results are provided to verify the theoretical results.     

Explicit expressions for three-dimensional boundary integrals in linear elasticity

On employing isoparametric, piecewise linear shape functions over a flat triangle, exact formulae are derived for all surface potentials involved in the numerical treatment of three-dimensional singular and hyper-singular boundary integral equations in linear elasticity. These formulae are valid for an arbitrary source point in space and are represented as analytical expressions along the edges of the integration triangle. They can be employed to solve integral equations defined on triangulated surfaces via a collocation method or may be utilized as analytical expressions for the inner integrals in a Galerkin technique. A numerical example involving a unit triangle and a source point located at various distances above it, as well as sample problems solved by a collocation boundary element method for the Lamé equation are included to validate the proposed formulae.     

Approximate boundary controllability for the system of linear elasticity

The approximate controllability for variable coefficients, isotropic, evolution elasticity system is considered. The appropriate unique continuation theorem for solutions of the system is stated. Copyright © 2004 John Wiley & Sons, Ltd.     

An algebraic multigrid method with interpolation reproducing rigid body modes for semi-definite problems in two-dimensional linear elasticity

Based on the geometric grid information as geometric coordinates, an algebraic multigrid (AMG) method with the interpolation reproducing the rigid body modes (namely the kernel elements of semi-definite operator arising from linear elasticity) is constructed, and such method is applied to the linear elasticity problems with a traction free boundary condition and crystal problems with free boundary conditions as well. The results of various numerical experiments in two dimensions are presented. It is shown from the numerical results that the constructed AMG method is robust and efficient for such semi-definite problems, and the convergence is uniformly bounded away from one independent of the problem size. Furthermore, the AMG method proposed in this paper has better convergence rate than the commonly used AMG methods. Simultaneously, an AMG method that can preserve the quotient space, which means that if the exact solution of original problem belongs to the quotient space of discrete operator considered, then the numerical solution of AMG method is convergent in the same quotient space, is obtained using the technique of orthogonal decomposition.     

A mixed nonconforming finite element for linear elasticity

This article considers a mixed finite element method for linear elasticity. It is based on a modified mixed formulation that enforces the continuity of the stress weakly by adding a jump term of the approximated stress on interior edges. The symmetric stress are approximated by nonconforming linear elements and the displacement by piecewise constants. We establish ??(h) error bound in the (broken) L² norm for the divergence of the stress and ??(h) error bound in the L² norm for both the displacement and the stress tensor. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

A general solution of equations of equilibrium in linear elasticity

A general solution of equations of equilibrium in linear elasticity is presented in cylindrical coordinates in terms of three harmonic functions describing an arbitrary displacement field. The structure of this solution is similar to the general solution given by Love (Kelvin’s solution) in spherical coordinates. Galerkin vector representation of our solution leads to an integral connecting the harmonic functions. The connections to Papkovich–Neuber and Muki’s general representations are also provided. Suitable choices of the harmonic functions in our new representation yield general solutions for axisymmetric deformations due to Love, Boussinesq and Michell. Some unbounded deformations induced by singular forces are tabulated in terms of the scalar harmonic functions to justify the simple nature of our representation. Exact solution of the half-space boundary value problem is also provided to demonstrate the power of our approach. The stress components computed via our solution are also listed (see the Appendix).     

Plane wave approximation in linear elasticity

We consider the approximation of solutions of the time-harmonic linear elastic wave equation by linear combinations of plane waves. We prove algebraic orders of convergence both with respect to the dimension of the approximating space and to the diameter of the domain. The error is measured in Sobolev norms and the constants in the estimates explicitly depend on the problem wavenumber. The obtained estimates can be used in the h- and p-convergence analysis of wave-based finite element schemes.     

On the singular Neumann problem in linear elasticity

The Neumann problem of linear elasticity is singular with a kernel formed by the rigid motions of the body. There are several tricks that are commonly used to obtain a nonsingular linear system. However, they often cause reduced accuracy or lead to poor convergence of the iterative solvers. In this paper, different well‐posed formulations of the problem are studied through discretization by the finite element method, and preconditioning strategies based on operator preconditioning are discussed. For each formulation, we derive preconditioners that are independent of the discretization parameter. Preconditioners that are robust with respect to the first Lamé constant are constructed for the pure displacement formulations, whereas a preconditioner that is robust in both Lamé constants is constructed for the mixed formulation. It is shown that, for convergence in the first Sobolev norm, it is crucial to respect the orthogonality constraint derived from the continuous problem. On the basis of this observation, a modification to the conjugate gradient method is proposed, which achieves optimal error convergence of the computed solution.     

Boundary element-Landweber method for the Cauchy problem in linear elasticity

In this paper, an iterative algorithm based on the Landwebermethod in combination with the boundary element method is developedfor solving the Cauchy problem in isotropic linear elasticity.An efficient regularizing stopping criterion is also employed.The numerical results obtained confirm that the iterative methodproduces a convergent and stable numerical solution with respectto increasing the number of boundary elements and decreasingthe amount of noise added into the input data, respectively.     

Representing the space of linear programs as the Grassmann manifold

Each linear program (LP) has an optimal basis. The space of linear programs can be partitioned according to these bases, so called the basis partition. Discovering the structures of this partition is our goal. We represent the space of linear programs as the space of projection matrices, i.e., the Grassmann manifold. A dynamical system on the Grassmann manifold, first presented in Sonnevend et al. (Math Program 52:527–553), is used to characterize the basis partition as follows: From each projection matrix associated with an LP, the dynamical system defines a path and the path leads to an equilibrium projection matrix returning the optimal basis of the LP. We will present some basic properties of equilibrium points of the dynamical system and explicitly describe all eigenvalues and eigenvectors of the linearized dynamical system at equilibrium points. These properties will be used to determine the stability of equilibrium points and to investigate the basis partition. This paper is only a beginning of the research towards our goal. Research is supported in part by NUS Academic Research Grant R-146-000-084-112. The author wishes to thank Josef Stoer for his valuable comments on the paper and to thank Wingkeung To, Jie Wu, Xingwang Xu, Deqi Zhang and Chengbo Zhu for providing consultations on Differential Geometry and Grassmann manifolds and pointing out useful literature. The author is certainly responsible to all faults in the paper.     

Mixed finite element methods for linear elasticity with weakly imposed symmetry

In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.