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In this paper we construct optimal preconditioners for the discrete mathematical models arising in modeling the elastic responses of lattice block materials. We present extensive numerical experiments to show that the preconditioned system has a uniformly bounded condition number with respect to the size of problem and with respect to the parameter relating the stretching and bending of the beams in a lattice. Using the limiting system of partial differential equations, we show theoretically that for square lattices the proposed preconditioners are efficient by proving a uniform bound on the condition number of the preconditioned system. [ABSTRACT FROM AUTHOR]

This paper describes the mechanics of materials with periodic skeletal micro-structures in infinite domains. The principal technical results consist of certain Korn-type inequalities that provide upper and lower bounds for the linear elastic strain energy in the material. Using these inequalities, existence and uniqueness results for the equations of linear elastic equilibrium are derived, and some asymptotic properties of the solutions are described. Particular attention is paid to the question of when a lattice structure can accurately be modeled as a pin-jointed truss, and when a rigid-node frame model must be employed. A practical technique for how to distinguish between the two types of material is given, and the distinct differences in their mechanical behavior are described. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuška, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the "probability error" with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. [ABSTRACT FROM AUTHOR]

The paper presents a reliable and efficient method for solving problems on a large set of domains with different geometries. In applications these geometries might be idealizations of a mechanical structure with multiple damage. The method is designed, and used, to analyse large three-dimensional problems. The present paper formulates the method and presents the theory for a simple model problem, i.e., the Laplace equation in two dimensions. In the numerical implementation of the splitting scheme, the p-version of the finite element method and strongly graded meshes are used. This approach provides high accuracy and error control. The exponential rate of convergence of the method is shown and practically exemplified. A numerical example shows the effects of various choices of parameters on the accuracy of the method. A complex problem related to aircraft safety, which was solved using the present method, is used to exemplify the generality and the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the computations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method generates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The second method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h- or p-version, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation. [ABSTRACT FROM AUTHOR]