Novel uncertainty quantification methods for stochastic isogeometric analysis

The main objective of this study is to develop novel computational methods for general high-dimensional uncertainty quantification (UQ) with a focus on stochastic isogeometric analysis. The objective is pursued by: (1) development of an isogeometric collocation method for random field discretization, (2) generalization of isogeometric methods for random field discretization on arbitrary multipatch domains, (3) establishment of a spline dimensional decomposition for high-dimensional UQ, (4) stochastic isogeometric analysis in linear elasticity, (5) stochastic isogeometric analysis on arbitrary multipatch domains, and (6) UQ in linear dynamical systems. This thesis reports research undertaken and associated findings in six stages. First, an isogeometric collocation method for a computationally expedient random field discretization on single-patch domains by means of the K-L expansion. The method involves a collocation projection onto a finite-dimensional subspace of continuous functions over a bounded domain, basis splines (B-splines) and non-uniform rational B-splines (NURBS) spanning the subspace, and standard methods of eigensolutions. Similar to the existing Galerkin isogeometric method, the isogeometric collocation method preserves an exact geometrical representation of the physical or computational domain and exploits the regularity of isogeometric basis functions delivering globally smooth eigensolutions. However, in the collocation method, the construction of the system matrices for a d-dimensional eigenvalue problem asks for at most $d$-dimensional domain integration, as compared with 2d-dimensional integrations required in the Galerkin method. Therefore, the introduction of the collocation method for random field discretization offers a huge computational advantage over the existing Galerkin method. Numerical examples illustrate the accuracy and convergence properties of the collocation method for obtaining the eigensolutions. Second, isogeometric Galerkin and collocation methods are introduced for solving the Fredholm integral eigenvalue problem on arbitrary multipatch domains, delivering the K-L expansion for random field discretization. In both methods, the unknown eigenfunctions are projected onto concomitant finite-dimensional approximation spaces, where NURBS and analysis-suitable T-splines reside. Numerical analyses indicate that the collocation method, by eliminating one d-dimensional domain integration in forming the system matrices, produces eigensolutions markedly more economically than the Galerkin method. Highly effective in large-scale applications, the isogeometric collocation method imparts a tremendous boost to computational expediency. Third, a spline dimensional decomposition (SDD) of a square-integrable random variable is debuted comprising hierarchically ordered measure-consistent orthonormal B-splines in independent random variables. Using the modulus of smoothness, the SDD approximation is shown to converge in mean-square to the correct limit. Analytical formulae are proposed to calculate the second-moment properties of a truncated SDD approximation for a general output random variable in terms of the expansion coefficients involved. Numerical results indicate that a low-order SDD approximation produces probabilistic characteristics of an output variable with an accuracy matching or surpassing that obtained from a high-order approximation from polynomial dimensional decomposition. Fourth, a new stochastic method is proposed by integrating SDD and IGA for solving stochastic boundary-value problems from linear elasticity. The method, referred to as SDD-IGA, involves Galerkin IGA as a deterministic solver for governing partial differential equations and SDD for UQ purposes. For the stochastic part of the SDD-IGA method, an innovative dimension-reduction integration technique is presented for efficiently estimating the expansion coefficients. Numerical examples demonstrate the capability of a low-order SDD-IGA in efficiently delivering probabilistic solutions with an approximation quality as good as, if not better than, that obtained from a high-order polynomial dimensional decomposition. The proposed SDD-IGA method is most suitable in the presence of locally nonlinear or nonsmooth behavior commonly found in applications. Fifth, for solving more complicated engineering problems, a new stochastic method is developed by integrating SDD of a high-dimensional random function and IGA on arbitrary multipatch geometries. The method, referred to as SDD-mIGA, involves (1) analysis-suitable T-splines with significant approximating power for geometrical modeling, random field discretization, and stress analysis; (2) Bezier extraction operator for isogeometric mesh refinement; and (3) the novel SDD method. SDD-mIGA can handle arbitrary multipatch domains in IGA and uses standard least-squares regression method to efficiently estimate the SDD expansion coefficients for UQ applications. Numerical results demonstrate that a low-order SDD–mIGA is capable of efficiently delivering accurate probabilistic solutions when compared with the benchmark results from crude Monte Carlo simulation. Finally, by leveraging orthonormal splines, expansion methods are implemented for solving UQ problems from linear structural dynamics. The resulting methods, premised on spline chaos expansion and SDD, are capable of capturing high nonlinearity and non-smoothness, if they exist, in a stochastic dynamic response markedly better than the polynomial chaos expansion method. However, due to the tensor-product structure, SCE, like PCE, also suffers from the curse of dimensionality. In contrast, SDD, equipped with a desirable dimensional hierarchy of input variables, deflates the curse of dimensionality to a great extent. Numerical results from frequency response analysis indicate that a low-order SCE with fewer basis functions removes or markedly reduces the spurious oscillations generated by high-order PCE in estimating the response statistics. Finally, a high-dimensional modal analysis of a fighter jet comprising 110 random variables demonstrates the ability of SDD in solving large-scale UQ problems.