Novel uncertainty quantification methods for stochastic isogeometric analysis

In many engineering applications, there are uncertainties in the way a system behaves because the materials, geometric dimensions, forces, and/or other system parameters are generally random. These parameters are called random variables. Therefore, for a reliable system design, an engineer must first be able to quantify the randomness in the system output, which depends on the random input. This is called uncertainty quantification (UQ) and generally requires an engineer to deal with a random output function of many input random variables.

Oftentimes, there are random quantities with spatial variability. For example, the weight of snow on buildings' roofs in a residential complex is variable not only from building to building, but also from one location to another on any given building roof. This is simply one example of a random field. When an engineer wants to analyze such a problem and quantify the randomness of, say, a building structure's deflections, there is a need to write the random field in terms of a finite number of input random variables. This is called random field discretization. New methods are needed to efficiently and accurately discretize a random field on both simple and complex geometries. This is one of the objectives of this work. Isogeometric methods are used to precisely model many geometrical shapes common in engineering applications, and numerical investigations illustrate the strengths of the proposed isogeometric random field discretization methods.

In many cases, the output random variable depends on numerous input random variables– typically more than ten– which makes the UQ process challenging, as it demands a hefty amount of computer resources in addition to powerful solvers. Solving such problems requires developing efficient and reliable UQ methods that are capable of modeling complex system behaviors. This is another objective of this work. As a remedy, a novel UQ method, referred to as spline dimensional decomposition (SDD) is presented. For systems simulations, isogeometric methods are employed in conjunction with SDD to surpass the well-known polynomial-based UQ methods, such as the polynomial chaos expansion, in terms of both accuracy and efficiency.

Finally, the proposed UQ methods are implemented for solving problems involving complex random functions and large-scale systems from structural dynamics. The numerical examples show that the methods efficiently capture the complex behavior of stochastic dynamic responses including random vibration frequencies.