Choice of input fields in stochastic finite elements

The problem of the arbitrary choice of variables for random field modelling in structural mechanics or in soil mechanics is treated. For example, it is relevant to ask the question of whether it is best to choose a stiffness field along a beam element or to choose its reciprocal field, the flexibility field, as the input to the stochastic finite element model.

To answer this question the focus should be on the error of the output of the mechanical model rather than on the input field itself when discretizing the held through replacing it by a field defined in terms of a finite number of random variables. Several reported discretization methods define these random variables as integrals of the product of the held and some suitable weight functions.

In particular, the weight functions can be Dirac delta functions whereby the random variables become the field values at a finite set of given points. The replacement field is often defined as the linear regression of the original field on the considered vector of the weighted integrals of the field.

For example, this holds for discretizations obtained by truncation of the Karhunen-Loeve expansion of the field, but only approximately so for truncations of expansions given in terms of any other complete orthogonal function basis. Solely discretizations based on the linear regression method are considered herein.

The solution to the problem of best choice of the vector of weight functions is not universal but depends on the mechanical problem under consideration as well as on the choice of the input field. Obviously it makes a difference whether it is the stiffness field or its reciprocal field that is chosen to be represented by a vector of weighted averages of the field.

As a test example a lognormal stiffness field along the axis of a linear-elastic Bernoulli-Euler column is considered. Then the exact one-to-one conversion from the stiffness held to the flexibility field is directly obtained. From the form of two functionals that have similarity to the potential energy functional and the complementary energy functional, respectively, both derived from the differential equation of the column displacement and the relevant boundary conditions, it can be expected that the discretization of the flexibility field is preferable over the discretization of the stiffness field.

Direct mechanical considerations support this expectation. (C) 1998 Published by Elsevier Science Ltd. All rights reserved.