Polynomial Chaos Expansion(PCE)has gained significant popularity among engineers across various engineering disciplines for uncertainty analysis.However,traditional PCE suffers from two major drawbacks.First,the ortho...Polynomial Chaos Expansion(PCE)has gained significant popularity among engineers across various engineering disciplines for uncertainty analysis.However,traditional PCE suffers from two major drawbacks.First,the orthogonality of polynomial basis functions holds only for independent input variables,limiting the model’s ability to propagate uncertainty in dependent variables.Second,PCE encounters the"curse of dimensionality"due to the high computational cost of training the model with numerous polynomial coefficients.In practical manufacturing,compressor blades are subject to machining precision limitations,leading to deviations from their ideal geometric shapes.These deviations require a large number of geometric parameters to describe,and exhibit significant correlations.To efficiently quantify the impact of high-dimensional dependent geometric deviations on the aerodynamic performance of compressor blades,this paper firstly introduces a novel approach called Data-driven Sparse PCE(DSPCE).The proposed method addresses the aforementioned challenges by employing a decorrelation algorithm to directly create multivariate basis functions,accommodating both independent and dependent random variables.Furthermore,the method utilizes an iterative Diffeomorphic Modulation under Observable Response Preserving Homotopy regression algorithm to solve the unknown coefficients,achieving model sparsity while maintaining fitting accuracy.Then,the study investigates the simultaneous effects of seven dependent geometric deviations on the aerodynamics of a high subsonic compressor cascade by using the DSPCE method proposed and sensitivity analysis of covariance.The joint distribution of the dependent geometric deviations is determined using Quantile-Quantile plots and normal copula functions based on finite measurement data.The results demonstrate that the correlations between geometric deviations significantly impact the variance of aerodynamic performance and the flow field.Therefore,it is crucial to consider these correlations for accurately assessing the aerodynamic uncertainty.展开更多
This paper proposes a novel recursive partitioning method based on constrained learning neural networks to find an arbitrary number (less than the order of the polynomial) of (real or complex) roots of arbitrary polyn...This paper proposes a novel recursive partitioning method based on constrained learning neural networks to find an arbitrary number (less than the order of the polynomial) of (real or complex) roots of arbitrary polynomials. Moreover, this paper also gives a BP network constrained learning algorithm (CLA) used in root-finders based on the constrained relations between the roots and the coefficients of polynomials. At the same time, an adaptive selection method for the parameter d P with the CLA is also given. The experimental results demonstrate that this method can more rapidly and effectively obtain the roots of arbitrary high order polynomials with higher precision than traditional root-finding approaches.展开更多
Groebner basis theory for parametric polynomial ideals is explored with the main objec- tive of nfinicking the Groebner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive GrSbne...Groebner basis theory for parametric polynomial ideals is explored with the main objec- tive of nfinicking the Groebner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive GrSbner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a GrSbnerbasis of the associated specialized polynomial ideal. For various specializations of parameters, structure of specialized ideals becomes qualitatively different even though there are significant relationships as well because of finiteness properties. Key concepts foundational to GrSbner basis theory are reexamined and/or further developed for the parametric case： （i） Definition of a comprehensive Groebner basis, （ii） test for a comprehensive GrSbner basis, （iii） parameterized rewriting, （iv） S-polynomials among parametric polynomials, （v） completion algorithm for directly computing a comprehensive Groebner basis from a given basis of a parametric ideal. Elegant properties of Groebner bases in the classical ideal theory, such as for a fixed admissible term ordering, a unique GrSbner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Groebner basis of an ideal, turn out to be a major challenge to generalize for parametric ideals; issues related to these investigations are explored. A prototype implementation of the algorithm has been successfully tried on many examples from the literature.展开更多
基金the National Science and Technology Major Project of China(No.J2019-I-0011)the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University,China(No.CX2023057)for supporting the research work.
文摘Polynomial Chaos Expansion(PCE)has gained significant popularity among engineers across various engineering disciplines for uncertainty analysis.However,traditional PCE suffers from two major drawbacks.First,the orthogonality of polynomial basis functions holds only for independent input variables,limiting the model’s ability to propagate uncertainty in dependent variables.Second,PCE encounters the"curse of dimensionality"due to the high computational cost of training the model with numerous polynomial coefficients.In practical manufacturing,compressor blades are subject to machining precision limitations,leading to deviations from their ideal geometric shapes.These deviations require a large number of geometric parameters to describe,and exhibit significant correlations.To efficiently quantify the impact of high-dimensional dependent geometric deviations on the aerodynamic performance of compressor blades,this paper firstly introduces a novel approach called Data-driven Sparse PCE(DSPCE).The proposed method addresses the aforementioned challenges by employing a decorrelation algorithm to directly create multivariate basis functions,accommodating both independent and dependent random variables.Furthermore,the method utilizes an iterative Diffeomorphic Modulation under Observable Response Preserving Homotopy regression algorithm to solve the unknown coefficients,achieving model sparsity while maintaining fitting accuracy.Then,the study investigates the simultaneous effects of seven dependent geometric deviations on the aerodynamics of a high subsonic compressor cascade by using the DSPCE method proposed and sensitivity analysis of covariance.The joint distribution of the dependent geometric deviations is determined using Quantile-Quantile plots and normal copula functions based on finite measurement data.The results demonstrate that the correlations between geometric deviations significantly impact the variance of aerodynamic performance and the flow field.Therefore,it is crucial to consider these correlations for accurately assessing the aerodynamic uncertainty.
文摘This paper proposes a novel recursive partitioning method based on constrained learning neural networks to find an arbitrary number (less than the order of the polynomial) of (real or complex) roots of arbitrary polynomials. Moreover, this paper also gives a BP network constrained learning algorithm (CLA) used in root-finders based on the constrained relations between the roots and the coefficients of polynomials. At the same time, an adaptive selection method for the parameter d P with the CLA is also given. The experimental results demonstrate that this method can more rapidly and effectively obtain the roots of arbitrary high order polynomials with higher precision than traditional root-finding approaches.
基金supported by the National Science Foundation under Grant No.DMS-1217054
文摘Groebner basis theory for parametric polynomial ideals is explored with the main objec- tive of nfinicking the Groebner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive GrSbner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a GrSbnerbasis of the associated specialized polynomial ideal. For various specializations of parameters, structure of specialized ideals becomes qualitatively different even though there are significant relationships as well because of finiteness properties. Key concepts foundational to GrSbner basis theory are reexamined and/or further developed for the parametric case： （i） Definition of a comprehensive Groebner basis, （ii） test for a comprehensive GrSbner basis, （iii） parameterized rewriting, （iv） S-polynomials among parametric polynomials, （v） completion algorithm for directly computing a comprehensive Groebner basis from a given basis of a parametric ideal. Elegant properties of Groebner bases in the classical ideal theory, such as for a fixed admissible term ordering, a unique GrSbner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Groebner basis of an ideal, turn out to be a major challenge to generalize for parametric ideals; issues related to these investigations are explored. A prototype implementation of the algorithm has been successfully tried on many examples from the literature.
