Practical engineering issues such as design optimization, design space exploration, sensitivity analyses, and reliability analyses need many simulations. If a single simulation is very time-consuming, engineers cannot perform the thousands or even millions of simulations needed for such analyses. The polynomial chaos expansion (PCE) method is an effective method that allows analyses of complex problems. This paper introduces the mathematical theory of the PCE method and presents a structural reliability analysis example. The performance response function for the structural reliability analysis is expressed as a PCE using Hermite polynomials. A general form of the Hermite polynomial, which is suitable for use in a computer program, is used to generalize the PCE analysis program and the adaptive selection of the polynomial order. Then, the accuracy and applicability of the surrogate model are verified using structural reliability analysis examples with explicit performance functions. The results show that the model has an excellent convergence rate with higher order PCE giving higher accuracy. The examples also show that the direct use of explicit performance functions is the easiest way to investigate PCE surrogate models.

Neither the finite element method nor the discontinuous deformation analysis method can solve problems very well in rock mechanics and engineering due to their extreme complexities. A coupling method combining both of them should have wider applicability. Such a model coupling the discontinuous deformation analysis method and the finite element method is proposed in this paper. In the model, so-called line blocks are introduced to deal with the interaction via the common interfacial boundary of the discontinuous deformation analysis domain with the finite element domain. The interfacial conditions during the incremental iteration process are satisfied by means of the line blocks. The requirement of gradual small displacements in each incremental step of this coupling method is met through a displacement control procedure. The model is simple in concept and is easy in numerical implementation. A numerical example is given. The displacement obtained by the coupling method agrees well with those obtained by the finite element method, which shows the rationality of this model and the validity of the implementation scheme.