Filters

2006 - 20084

More filters

20194
Reset filters

Settings

Type: Text × Subject: Poisson integral formula ×

Search Results

Now showing 1 - 4 of 4
  • Thumbnail Image
    Item
    Stochastic simulation method for a 2D elasticity problem with random loads
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Sabelfeld, Karl; Shalimova, Irina; Levykin, Alexander
    We develop a stochastic simulation method for a numerical solution of the Lamé equation with random loads. To treat the general case of large intensity of random loads, we use the Random Walk on Fixed Spheres (RWFS) method described in our paper citesab-lev-shal-2006. The vector random field of loads which stands in the right-hand-side of the system of elasticity equations is simulated by the Randomization Spectral method presented in citesab-1991 and recently revised and generalized in citekurb-sab-2006. Comparative analysis of RWFS method and an alternative direct evaluation of the correlation tensor of the solution is made. We derive also a closed boundary value problem for the correlation tensor of the solution which is applicable in the case of inhomogeneous random loads. Calculations of the longitudinal and transverse correlations are presented for a domain which is a union of two arbitrarily overlapped discs. We also discuss a possibility to solve an inverse problem of determination of the elastic constants from the known longitudinal and transverse correlations of the loads.
  • Thumbnail Image
    Item
    Random walk on fixed spheres for Laplace and Lamé equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Sabelfeld, Karl; Shalimova, Irina; Levykin, Alexander
    The Random Walk on Fixed Spheres (RWFS) introduced in our previous paper is presented in details for Laplace and Lamé equations governing static elasticity problems. The approach is based on the Poisson type integral formulae written for each disc of a domain consisting of a family of overlapping discs. The original differential boundary value problem is equivalently reformulated in the form of a system of integral equations defined on the intersection surfaces (arches, in 2D, and caps, if generalized to 3D spheres). To solve the obtained system of integral equations, a Random Walk procedure is constructed where the random walks are living on the intersecting surfaces. Since the spheres are fixed, it is convenient to construct also discrete random walk methods for solving the system of linear equations approximating the system of integral equations. We develop here two classes of special Monte Carlo iterative methods for solving these systems of linear algebraic equations which are constructed as a kind of randomized versions of the Chebyshev iteration method and Successive Over Relaxation (SOR) method. It is found that in this class of randomized SOR methods, the Gauss-Seidel method has a minimal variance ...
  • Thumbnail Image
    Item
    Elastic half plane under random boundary excitations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Shalimova, Irina; Sabel'fel'd, Karl K.
    We study in this paper a respond of an elastic half-plane to random boundary excitations. We treat both the white noise excitations and more generally, homogeneous random fluctuations of displacements prescribed on the boundary. Solutions to these problems are inhomogeneous random fields which are however homogeneous with respect to the longitudinal coordinate. This is used to represent the displacements as series expansions involving a complete set of deterministic functions with corresponding random coefficients. We construct the Karhunen-Loève (K-L) series expansion which is based on the eigen-decomposition of the correlation operator. The K-L expansion can be used to calculate the statistical characteristics of other functionals of interest, in particular, the strain and stress tensors and the elastic energy tensor.
  • Thumbnail Image
    Item
    Stochastic analysis of an elastic 3D half-space respond to random boundary displacements : exact results and Karhunen-Loéve expansion
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Shalimova, Irina A.; Sabelfeld, Karl K.
    A stochastic response of an elastic 3D half-space to random displacement excitations on the boundary plane is studied. We derive exact results for the case of white noise excitations which are then used to give convolution representations for the case of general finite correlation length fluctuations of displacements prescribed on the boundary. Solutions to this elasticity problem are random fields which appear to be horizontally homogeneous but inhomogeneous in the vertical direction. This enables us to construct explicitly the Karhunen-Loève (K-L) series expansion by solving the eigen-value problem for the correlation operator. Simulation results are presented and compared with the exact representations derived for the displacement correlation tensor. This paper is a complete 3D generalization of the 2D case study we presented in J. Stat. Physics, v.132 (2008), N6, 1071-1095.