40 years FORM: Some new aspects?
Section snippets
FORM and the HL–RF algorithm
Given a continuous LSF (limit state function) in the standard normal space, i.e. the n-dimensional Euclidean space with PDF (probability density function)The idea of FORM was to approximate the failure domain by a halfspace. This halfspace was obtained by linearizing the LSF at the point where the limit state surface has minimal distance to the origin, which means that the PDF is maximal there,
Numerical optimization methods
Most deterministic minimization methods for differentiable functions are line search or trust region methods [20, Chapters 3 and 4]. Here only the former will be considered. Line search means that for finding the minimum of a function a sequence of points, which should converge towards a minimum, is calculated iteratively in the following way:Here is the present iteration point, the gradient, a symmetric and positive definite matrix and αk the step
Modified HL–RF algorithm
Due to convergence problems with the original HL–RF algorithm modifications were developed to overcome these deficiencies. In fact, as will be shown now, these modified approaches are variants of an older method, described in the following.
In 1970 Pshenichnyj [21] (see also [22]) proposed a method for constrained minimization. He considered the linearizations of the target function f and the constraint function gSince in general this optimization
The SORM factor
The FORM and SORM approximations for the failure probability areThe SORM factor is given in [5] in the following form as above:Later, some authors replaced the minus sign before the term by a plus sign and wrote instead . This leads to confusions and also to wrong results if the meaning of the factor is not clearly understood. To explain the problem, there are two different definitions for
Dimension reduction and FORM/SORM
A well-known problem in the analysis of the structure of data is to separate the important from the unimportant, to try to find a simple structure. With the increasing complexity of models it becomes more and more substantial to carve out a skeleton which describes the essentials of a system in a form as simple as possible without too much loss of information.
Given many variables describing a system by , can a simpler structure having much less variables be found without too
Two examples for dimension reduction
Let be given a linear LSF of three variablesFrom a three-dimensional random vector with a standard normal distribution a sample of 1000 realizations is taken. Using now the SAVE method and slicing the values of the LSF into 10 slices given by , the eigenvectors areand the eigenvaluesSo the vector is the eigenvector with the largest
Conclusions
There are still some aspects of FORM/SORM as beta point search methods and dimension reduction problems which have not been researched thoroughly. Concerning the search algorithms for the beta point, it might be useful to investigate methods which can achieve superlinear convergence rates and study if the increased computational effort pays off.
The computing of the SORM factor should be done with the result in Eq. (21) to avoid problems with the definition of curvature. In publications using
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