**Sponsor:** U.S. National Science Foundation

**Project No.:** CMMI-1607398

**Duration: **July 15, 2016 – June 30, 2021

**Principal Investigator:** Professor Sharif Rahman

**Graduate Student:**Messrs. Ramin Jahanbin and Steven Dixler

**SUMMARY**

The objective of this project is to advance the theory of isogeometric analysis, accompanied by robust numerical algorithms, for uncertainty quantification of a high-dimensional response from complex materials and structures. The proposed effort will involve: (1) new randomized non-uniform rational B-splines (NURBS) for the stochastic matrix equation and NURBS-based random field discretization for a material body; (2) new stochastic isogeometric methods entailing the hierarchical B-spline sparse grids for high-dimensional function interpolation; and (3) new formulae and scalable algorithms for predicting the statistical moments and probability density functions of a complex structural response. The research will bridge geometric modeling, stress analysis, and stochastic simulation by interacting natively upon the same mathematical building blocks, forming a seamless uncertainty quantification pipeline of the future. Due to innovative formulation of the sparse grid interpolation, the resulting stochastic method will be efficiently implemented regardless of the size of an uncertainty quantification problem. New computational algorithms will be generated for efficiently estimating the statistical moments and probability density function of a structural response, including error estimates that will result in a rigorous assessment of the sparse grid approximation. The overall effort will effectively integrate research, education, training, and outreach.

**Task 1:**

**A Galerkin Isogeometric Method for Random Field Discretization:**A Galerkin isogeometric method was developed for solving an integral eigenvalue problem, resulting in an effective discretization of random fields by the well-known Karhunen-Loeve expansion. The method employs a Galerkin discretization, which projects the eigensolutions onto a finite-dimensional subspace of a Hilbert space. Using B-splines and NURBS functions as the basis of the subspace, a concomitant matrix eigenvalue problem is formulated, where the system matrices are constructed by domain integrations. Finally, the eigensolutions are obtained using standard methods. Although there exist similar Galerkin methods, such as the finite-element and mesh-free methods, the NURBS-based isogeometric method offers a few computational advantages. First, many physical or computational domains, such as freeform and sculptured surfaces and conic sections, are exactly represented by NURBS. In consequence, potential numerical errors originating from imprecise geometry, accepted in the finite-element and mesh-free methods, are avoided. Second, as NURBS functions have higher-order continuity, the eigensolutions derived from isogeometric analysis is usually globally smoother than those derived from finite-element analysis. The smoothness can be controlled by judiciously selecting or adjusting the polynomial order of the underlying B-splines as well as the multiplicity of knots. Therefore, the introduction of the isogeometric method for random field discretization is not only novel, but it also presents an attractive alternative to existing methods. More importantly, using NURBS for random field discretization enhances the isogeometric paradigm. In consequence, one can envision developing a seamless uncertainty quantification pipeline, where geometric modeling, stress analysis, and stochastic simulation are all consolidated using the same building blocks of NURBS. Numerical results, obtained for three random field discretization problems in all three dimensions, indicate that the isogeometric method developed provides accurate and convergent eigensolutions.

**Task 2:**

**An Isogeometric Collocation Method for Random Field Discretization:**An isogeometric collocation method was developed for solving an integral eigenvalue problem, resulting in a computationally efficient discretization of random fields by means of the well-known Karhunen-Loeve expansion. The method employs a collocation projection onto a finite-dimensional subspace of continuous functions over a bounded domain. Using basis splines (B-splines) and non-uniform rational B-splines (NURBS) functions as the basis of the subspace, a finite-dimensional matrix eigenvalue problem is formulated, where the system matrices are constructed by basis functions and domain integration. Finally, the eigensolutions are obtained using standard methods of linear algebra. Similar to the existing Galerkin isogeometric method, the isogeometric collocation method preserves exact geometrical representation of the physical or computational domain and exploits regularity of isogeometric basis functions delivering globally smooth eigensolutions. However, the chief advantage of the collocation method is the computational efficiency in constructing the system matrices and the resulting matrix equations, where the cost of numerical integration in the Galerkin method is greatly diminished. By avoiding the need for a weak form of the governing integral equations, the collocation method shaves off one dimension-order of domain integration in establishing the finite-dimensional matrix equations. Consequently, given a d-dimensional integral eigenvalue problem, the collocation method needs only a d-dimensional domain integration, whereas a 2d-dimensional domain integration is required in the Galerkin method. Therefore, the introduction of the collocation method for random field discretization offers a hefty computational benefit over the existing Galerkin method. The results from three numerical examples, illustrating random field problems in all three dimensions, indicate that the isogeometric collocation method developed provides accurate, convergent, and computationally efficient eigensolutions.**A Spline Dimensional Decomposition Method for Uncertainty Quantification Analysis:**A new spline dimensional decomposition (SDD) method was developed for solving high-dimensional UQ problems subject to random input following an arbitrary probability distribution. The method is based on (1) the classical analysis-of-variance (ANOVA) decomposition of a high-dimensional function; (2) a Fourier-spline expansion of the ANOVA component functions entailing bases from orthonormal B-splines and NURBS; and (3) an innovative dimension-reduction integration for approximating the associated expansion coefficients. Since SDD stems from the ANOVA decomposition, SDD inherits all desirable properties the latter, including its orthogonal structure. Due to the dimensional hierarchy of SDD, subsequent stochastic analysis of a high-dimensional, large-scale complex systems can be conducted efficiently. Compared with the previous variants, such as polynomial dimensional decomposition (PDD), SDD does not suffer the handicap of polynomials, since it is defined piecewise. Yet, for a moderately large degree, SDD is equipped to capture locally abrupt changes in stochastic responses better than globally smooth PDD. Therefore, a long-standing UQ problem associated with the curse of dimensionality and non-smooth stochastic responses will be alleviated to an appreciable magnitude with positive ramifications in engineering and applied sciences.

**Task 3:**

**A Stochastic Isogeometric Method for Solving Stochastic Boundary-Value Problems from Linear Elasticity:**A new stochastic method, coupling isogeometric analysis (IGA) and spline dimensional decomposition (SDD), was developed for solving stochastic boundary-value problems from linear elasticity. The method, referred to as SDD-IGA, has, at least, two important novelties. First, measure-consistent orthonormal basis splines (B-splines) were employed for refining the analysis-of-variance dimensional decomposition of a high-dimensional response function. The result is a Fourier-like orthogonal expansion in spline basis, which is equipped to capture locally abrupt changes in stochastic responses, including discontinuity and nonsmoothness, better than globally supported polynomial basis. Second, an innovative dimension-reduction integration technique, meant for computing high-dimensional integrals, was exploited for estimating the expansion coefficients of SDD-IGA. In consequence, high-dimensional stochastic problems, featuring locally prominent, highly nonlinear response functions, can be effectively solved with a polynomial computational complexity. Therefore, a time-honored stochastic problem associated with the curse of dimensionality has been diminished to an appreciable magnitude. Analytical formulae were derived to calculate the second-moment properties of a general SDD-IGA approximation. The significance of the SDD-IGA method lies in harnessing common ingredients of IGA --- B-splines and non-uniform rational B-splines (NURBS) --- in geometrical modeling, random field discretization, stress analysis, and now probabilistic analysis for any square-integrable output random variable of interest. Because of the connection to IGA, the SDD-IGA method preserves exact geometrical representation of many commonly used shapes found in engineering, thereby removing an additional source of imprecision in the discretization process. The numerical results from three elastostatic problems, including a three-dimensional, fifteen-variable horseshoe problem, demonstrate that a low-order SDD-IGA approximation is capable of efficiently delivering probabilistic solutions with an approximation quality as good as, if not better than, that obtained from a high-order polynomial dimensional decomposition method. The SDD-IGA method developed is most suitable in the presence of locally nonlinear or nonsmooth response behavior, not uncommon in engineering applications.**A Spline Chaos Expansion Method for Uncertainty Quantification Analysis:**A new chaos expansion, namely, spline chaos expansion (SCE) of a square-integrable random variable, comprising measure-consistent multivariate orthonormal basis splines (B-splines) in independent random variables, is unveiled. Under prescribed assumptions, a whitening transformation is proposed to decorrelate univariate B-splines in each coordinate direction into their orthonormal version. The transformed set of B-splines was proved to form a basis of a general spline space comprising splines of specified degree and knot vector. Through a tensor-product structure, multivariate orthonormal B-splines were constructed, spanning the space of multivariate splines of specified degrees and knot vectors in all coordinate directions. The result is an expansion of a general L2-function with respect to measure-consistent multivariate orthonormal B-splines. The approximation quality of the expansion was demonstrated in terms of the modulus of smoothness of the function, leading to the mean-square convergence of SCE to the correct limit. The weaker modes of convergence, such as those in probability and in distribution, follow readily. The optimality of SCE, including deriving polynomial chaos expansion (PCE) as a special case of SCE, was demonstrated. Analytical formulae akin to those found in the PCE literature are proposed to calculate the mean and variance of an SCE approximation for a general output variable in terms of the expansion coefficients. Numerical results obtained for one- and two-dimensional uncertainty quantification problems entailing oscillatory, nonsmooth, and nearly discontinuous functions indicate that a low-degree SCE approximation with an adequate mesh size is capable of producing a substantially more accurate estimate of the output variance than a PCE with an overly large degree of approximation.**Novel Isogeometric Galerkin and Collocation Methods for Karhunen-Loeve Expansion of Random Fields on Multi-Patch Domains:**Two new isogeometric methods, one rooted in the Galerkin projection and the other stemming from the collocation projection, were developed for solving the Fredholm integral eigenvalue problem on arbitrary multi-patch domains, enabling random field discretization by means of the well-known Karhunen-Loeve (K-L) expansion. Compared with similar existing works on isogeometric analysis, the methods proposed can now handle markedly more complex geometries, including multiply-connected bodies, that cannot be represented by tensor-product, single-patch domains alone. In both methods, the unknown eigensolutions are projected onto concomitant finite-dimensional approximation spaces, where non-uniform rational B-splines (NURBS) and analysis-suitable rational T-splines are exploited as basis functions. Using these approximation spaces, finite-dimensional matrix eigenvalue problems are formulated, where the system matrices are constructed by NURBS or rational T-splines as basis and subsequent domain integration. Finally, the eigensolutions are obtained using standard methods of linear algebra. Similar to the existing methods but applicable to single-patch domains only, the Galerkin and collocation methods developed here preserve an exact geometrical representation of complex engineering structures and exploit the regularity of isogeometric basis functions to the extent possible, furnishing patchwise smooth but globally continuous eigensolutions. Numerical results of eigensolutions obtained from two- and three-dimensional engineering problems indicate that the collocation method, when compared with the Galerkin method, is not only accurate and convergent, but also substantially more economical. The collocation method achieves this desirable feat by sidestepping one d-dimensional domain integration in forming the system matrices, whereas a 2d-dimensional domain integration is mandated in the Galerkin method. As a result, the collocation method in the context of IGA offers a tremendous boost to computational expediency when generating the K-L expansion.-
**Stochastic Isogeometric Analysis on Arbitrary Multi-patch Domains by Spline Dimensional Decomposition:**A new stochastic method by integrating spline dimensional decomposition (SDD) of a high-dimensional random function and isogeometric analysis (IGA) on arbitrary multi-patch geometries was developed for solving stochastic boundary-value problems from linear elasticity. 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) a novel Fourier-like expansion of a high-dimensional output function in terms of measure-consistent orthonormalized splines. The dimensionwise expansion of the function leads to a polynomial computational complexity, thereby alleviating the curse of dimensionality to a significant extent.The proposed method can handle arbitrary multi-patch domains in IGA and uses standard least-squares regression to efficiently estimate the SDD expansion coefficients for uncertainty quantification applications. Analytical formulae have been derived to calculate the second-moment properties of an SDD-mIGA approximation for a general output random variable of interest. Numerical results, including those obtained for a 54-dimensional, industrial-scale problem, 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.

**Supervision of a Graduate Student: **Two graduate students, Messrs. Ramin Jahanbin and Steven Dixler, are involved in this project. They are making satisfactory progress toward completion of their Ph.D. programs. The PI is the advisor of both graduate students.

**Insertion of Research Results into Undergraduate and Graduate Courses: **The newly developed Galerkin isogeometric and collocation methods for random field discretization, spline chaos and dimensional decomposition methods for uncertainty quantification, and stochastic isogeometric method for solving linear elasticity problems are being planned to be integrated into the syllabus of both intermediate- and graduate-level courses. The PI strongly believes that the insertion of new research results will motivate learning by students in the areas of stochastic mechanics and high-performance computing.

**Invited Seminars and Lectures: **Since the beginning of this project, the PI and his students presented 11 technical papers or posters at: (1) 14th U.S. National Congress on Computational Mechanics in Montreal, Quebec (July 2017); (2) 12th International Conference on Structural Safety & Reliability, Vienna, Austria (August 2017); (3) 13th World Congress on Computational Mechanics, New York, NY (July 2018); (4) 8th Conference on Computational Stochastic Mechanics, Paros, Greece (June 2018); (5) 15th U.S. National Congress on Computational Mechanics, Austin, Texas (July-August 2019); (5) 8th Workshop on High-Dimensional Approximation, Zurich, Switzerland (September 2019); (6) 16^{th} U.S. National Congress on Computational Mechanics, Chicago, IL (July 2021); and (7) 1^{st} International Conference on Mechanical System Dynamics, Nanjing, China (November 2021).

**Journal Articles:**

- Lee, D., Jahanbin, R., and Rahman, S., “Robust Design Optimization by Spline Dimensional Decomposition,” submitted to Probabilistic Engineering Mechanics, 2021.
- Jahanbin, R. and Rahman, S., “Stochastic Isogeometric Analysis on Arbitrary Multipatch Domains by Spline Dimensional Decomposition,” submitted to Computer Methods in Applied Mechanics and Engineering, 2020.
- Rahman, S. and Jahanbin, R., “A Spline Dimensional Decomposition for Uncertainty Quantification,” submitted to SIAM Journal on Uncertainty Quantification, 2020.
- Dixler, S., Jahanbin, R., and Rahman, S., “Uncertainty Quantification by Optimal Spline Dimensional Decomposition,” International Journal for Numerical Methods in Engineering, Vol. 122, pp. 5898-5934, 2021.
- Jahanbin, R. and Rahman, S., “Isogeometric Methods for Karhunen-Loeve Representation of Random Fields on Arbitrary Multi-patch Domains,” International Journal for Uncertainty Quantification, Vol. 11, pp. 27-57, 2021.
- Rahman, S. and Jahanbin, R., “Orthogonal Spline Expansions for Uncertainty Quantification in Linear Dynamical Systems,” Journal of Sound and Vibration, Vol. 512, Article 116366, 2021.
- Jahanbin, R. and Rahman, S., “Stochastic Isogeometric Analysis in Linear Elasticity,” Computer Methods in Applied Mechanics and Engineering, Vol. 364, Article 112928, pp. 1-38, 2020.
- Rahman, S., “A Spline Chaos Expansion,” SIAM Journal on Uncertainty Quantification, Vol. 8, pp. 27-57, 2020.
- Jahanbin, R. and Rahman, S., “An Isogeometric Collocation Method for Efficient Random Field Discretization,” International Journal for Numerical Methods in Engineering, Vol. 117, pp. 344-369, 2019.
- Rahman, S., “A Galerkin Isogeometric Method for Karhunen-Loeve Approximation of Random Fields,” Computer Methods in Applied Mechanics and Engineering, Vol. 338, pp. 533-561, 2018.
- Rahman, S., “A Polynomial Chaos Expansion in Dependent Random Variables,” Journal of Mathematical Analysis and Applications, Vol. 464, pp. 749-775, 2018.
- Rahman, S., “Mathematical Properties of Polynomial Dimensional Decomposition,” SIAM/ASA Journal on Uncertainty Quantification, Vol. 6, pp. 816-844, 2018.

**Papers in Conferences, Symposiums, and Congresses:**

- Rahman, S., “Uncertainty Quantification in Linear Dynamical Systems by Orthogonal Spline Expansions,” Proceedings of 1
^{st}International Conference on Mechanical System Dynamics, Nanjing, China, November 2021. - Rahman, S. and Jahanbin, R., “Stochastic Isogeometric Analysis on Arbitrary Multipatch Domains by Spline Dimensional Decomposition,” Proceedings of SIAM Conference on Uncertainty Quantification (UQ22), Atlanta, GA, April 2022.
- Dixler, S. and Rahman, S., “An Adaptive-Sparse Spline Dimensional Decomposition Method for High-Dimensional Uncertainty Quantification,” Proceedings of 16
^{th}U.S. National Congress on Computational Mechanics, Chicago, IL, July 2021. - Jahanbin, R. and Rahman, S., “A Stochastic Isogeometric Analysis Method by Spline Dimensional Decomposition for Arbitrary Domains,” Proceedings of 16
^{th}U.S. National Congress on Computational Mechanics, Chicago, IL, July 2021. - Rahman, S. and Jahanbin, R., “A Spline Dimensional Decomposition for High-Dimensional Uncertainty Quantification,” Proceedings of 8th Workshop on High Dimensional Approximation, Zurich, Switzerland, September 2019.
- Jahanbin, R. and Rahman, S., “Stochastic Isogeometric Analysis in Elastostatics,” Proceedings of 15th U.S. National Congress on Computational Mechanics, Austin, Texas, July-August 2019.
- Rahman, S. and Jahanbin, R., “Uncertainty Quantification by a Spline Dimensional Decomposition,” Proceedings of 15th U.S. National Congress on Computational Mechanics, Austin, Texas, July-August 2019.
- Jahanbin, R. and Rahman, S., “An Isogeometric Collocation Method for Karhunen-Loeve Discretization of Random Fields,” Proceedings of 13th World Congress on Computational Mechanics, New York, NY, July 2018.
- Rahman, S., “A Polynomial Dimensional Decomposition for Dependent Random Variables,” Proceedings of 8
^{th}Conference on Computational Stochastic Mechanics, Paros, Greece, June 2018. - Rahman, S., “A Galerkin Isogeometric Method for Karhunen-Loeve Approximation of Random Fields,” Proceedings of 14
^{th}U. S. National Congress on Computational Mechanics, Montreal, Quebec, July 2017. - Rahman, S., “A Generalized Wiener-Hermite Polynomial Expansion for Multivariate Gaussian Probability Measures,” Proceedings of 12th International Conference on Structural Safety & Reliability, Vienna, Austria, August 2017.

**Others:**

- Rahman, S., “Uncertainty Quantification in Computational Science and Engineering,” seminar presented at the UI Applied and Computational Mathematics Program, Iowa City, IA, April 2017.

A number of computer codes were developed for performing various numerical simulations required in this project. For further information, please contact the PI (S. Rahman) at sharif-rahman@uiowa.edu.