Numerical solution of 2D elastostatic problems formulated by potential functions

2D linear elastostatic problems formulated in Cartesian coordinates by potential functions are numerically solved by network simulation method which allows an easy implementation of the complex boundary conditions inherent to this type of formulation. Four potential solutions are studied as governing equations: the general Papkovich–Neuber formulation, which is defined by a scalar potential plus a vector potential of two components, and the three simplified derived formulations obtained by deleting one of the three original functions (the scalar or one of the vector components). Application of this method to a rectangular plate subjected to mixed boundary conditions is presented. To prove the reliability and accurate of the proposed numerical method, as well as to demonstrate the suitability of the different potential formulations, numerical solutions are compared with those coming from the classical Navier formulation.     

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.     

Probability of unique integer solution to a system of linear equations

We consider a system of m linear equations in n variables Ax = d and give necessary and sufficient conditions for the existence of a unique solution to the system that is integer: x ∈ {−1, 1}ⁿ. We achieve this by reformulating the problem as a linear program and deriving necessary and sufficient conditions for the integer solution to be the unique primal optimal solution. We show that as long as m is larger than n/2, then the linear programming reformulation succeeds for most instances, but if m is less than n/2, the reformulation fails on most instances. We also demonstrate that these predictions match the empirical performance of the linear programming formulation to very high accuracy.     

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.     

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.     

Explicit green matrices for linear elasticity in the half space

This paper studies traveling-wave solutions of the partial differential equations which model waves in a horizontal water channel traveling in both directions. The existence of solitary-wave solutions for a number of systems is proved. Interesting new multi-pulsed traveling-wave solutions which consist of an arbitrary number of troughs are found numerically. The bifurcation diagrams of N-trough solutions with respect to phase speed and system parameters are presented.     

A general projection algorithm for solving systems of linear equations

In this paper, we give a general projection algorithm for implementing some known extrapolation methods such as the MPE, the RRE, the MMPE and others. We apply this algorithm to vectors generated linearly and derive new algorithms for solving systems of linear equations. We will show that these algorithms allow us to obtain known projection methods such as the Orthodir or the GCR.     

A procedure for solving some second-order linear ordinary differential equations

The purpose of this work is to introduce the concept of pseudo-exactness for second-order linear ordinary differential equations (ODEs), and then to try to solve some specific ODEs.     

Error propagation of general linear methods for ordinary differential equations

We discuss error propagation for general linear methods for ordinary differential equations up to terms of order p+2, where p is the order of the method. These results are then applied to the estimation of local discretization errors for methods of order p and for the adjacent order p+1. The results of numerical experiments confirm the reliability of these estimates. This research has applications in the design of robust stepsize and order changing strategies for algorithms based on general linear methods.     

Iterative solution of linear equations with unbounded operators

A convergent iterative process is constructed for solving any solvable linear equation in a Hilbert space, including equations with unbounded, closed, densely defined linear operators. The method is proved to be stable towards small perturbation of the data. Some abstract results are established and used in an analysis of variational regularization method for equations with unbounded linear operators. The dynamical systems method (DSM) is justified for unbounded, closed, densely defined linear operators. The stopping time is chosen by a discrepancy principle. Equations with selfadjoint operators are considered separately. Numerical examples, illustrating the efficiency of the proposed method, are given.     

Maximal smoothness for solutions to equilibrium equations in 2D nonlinear elasticity

For a class of variational integrals from 2D nonlinear elasticity, we prove that any weak solution for the equilibrium equations is smooth. Moreover, we present an example showing that the assumption is optimal.

     

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.     

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.     

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.     

The general solution and reduction to diagonal form of a system of equations of linear isotropic elasticity

A simple representation is obtained of the general solution of the Lame system of equations for an isotropic material in the form of a linear combination of the first derivatives of the three functions satisfying three independent wave or harmonic (in the static case) equations. In the two-dimensional case of plane deformation, the found solution directly implies the Kolosov-Muskhelishvili representation of the displacements by two analytic functions of a complex variable. A formula for generation of new solutions is given.     

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 parallel solution of linear elasticity problems: Part I: theory

The discretized linear elasticity problem is solved by the preconditioned conjugate gradient (pcg) method. Mainly we consider the linear isotropic case but we also comment on the more general linear orthotropic problem. The preconditioner is based on the separate displacement component (sdc) part of the equations of elasticity. The preconditioning system consists of two or three subsystems (in two or three dimensions) also called inner systems, each of which is solved by the incomplete factorization pcg-method, i.e., we perform inner iterations. A finite element discretization and node numbering giving a high degree of partial parallelism with equal processor load for the solution of these systems by the MIC(0) pcg method is presented. In general, the incomplete factorization requires an M-matrix. This property is studied for the elasticity problem. The rate of convergence of the pcg-method is analysed for different preconditionings based on the sdc-part of the elasticity equations. In the following two parts of this trilogy we will focus more on parallelism and implementation aspects. © 1998 John Wiley & Sons, Ltd.     

Preconditioned implicit solution of linear hyperbolic equations with adaptivity

This paper describes a method for solving hyperbolic partial differential equations using an adaptive grid: the spatial derivatives are discretised with a finite volume method on a grid which is structured and partitioned into blocks which may be refined and derefined as the solution evolves. The solution is advanced in time via a backward differentiation formula. The discretisation used is second-order accurate and stable on Cartesian grids. The resulting system of linear equations is solved by GMRES at every time-step with the convergence of the iteration being accelerated by a semi-Toeplitz preconditioner. The efficiency of this preconditioning technique is analysed and numerical experiments are presented which illustrate the behaviour of the method on a parallel computer.