In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and fo...In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.展开更多
A one-dimensional non-intrusive Polynomial Chaos （PC） method is applied in Uncertainty Quantification （UQ） studies for CFD-based ship performances simulations. The uncertainty properties of Expected Value （EV） a...A one-dimensional non-intrusive Polynomial Chaos （PC） method is applied in Uncertainty Quantification （UQ） studies for CFD-based ship performances simulations. The uncertainty properties of Expected Value （EV） and Standard Deviation （SD） are evaluated by solving the PC coefficients from a linear system of algebraic equations. The one-dimensional PC with the Legendre polynomials is applied to： （1） stochastic input domain and （2） Cumulative Distribution Function （CDF） image domain, allowing for more flexibility. The PC method is validated with the Monte-Carlo benchmark results in several high-fidelity, CFD-based, ship UQ problems, evaluating the geometrical, operational and environmental uncertainties for the Delft Catamaran 372. Convergence is studied versus PC order P for both EV and SD, showing that high order PC is not necessary for present applications. Comparison is carried out for PC with/without the least square minimization when solving the PC coefficients. The least square minimization, using larger number of CFD samples, is recommended for current test cases. The study shows the potentials of PC method in Robust Design Optimization （RDO） and Reliability-Based Design Optimization （RBDO） of ship hydrodynamic performances.展开更多
Based on the observation data of CHAMP satellite from 2006 to 2009, a 2D crustal magnetic anomaly model in China is established to study the distribution characteristics of crustal magnetic anomaly. In this paper, the...Based on the observation data of CHAMP satellite from 2006 to 2009, a 2D crustal magnetic anomaly model in China is established to study the distribution characteristics of crustal magnetic anomaly. In this paper, the 2D anomaly model is derived from the Legendre polynomial expansion of harmonic term N =6-50. The result shows that many elaborate structures reflected in magnetic anomaly map well correspond to the geologic structures in China and its adjacent area. The magnetic anomaly at low satellite height behaves complexly, which is mainly caused by the magnetic disturbance of shallow rocks.In contrast, the magnetic field isolines at high satellite height are relatively sparse and only magnetic anomalies of deep crust are reflected. This fact implies that the 2D model of crustal magnetic anomaly provides an important method of the space prolongation of geomagnetic field, and is of theoretical and practice importance in geologic structure analysis and geophysical prospecting.展开更多
文摘In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.
基金Project supported by the National Natural Science Foundation of China(Grant No.50979060)
文摘A one-dimensional non-intrusive Polynomial Chaos （PC） method is applied in Uncertainty Quantification （UQ） studies for CFD-based ship performances simulations. The uncertainty properties of Expected Value （EV） and Standard Deviation （SD） are evaluated by solving the PC coefficients from a linear system of algebraic equations. The one-dimensional PC with the Legendre polynomials is applied to： （1） stochastic input domain and （2） Cumulative Distribution Function （CDF） image domain, allowing for more flexibility. The PC method is validated with the Monte-Carlo benchmark results in several high-fidelity, CFD-based, ship UQ problems, evaluating the geometrical, operational and environmental uncertainties for the Delft Catamaran 372. Convergence is studied versus PC order P for both EV and SD, showing that high order PC is not necessary for present applications. Comparison is carried out for PC with/without the least square minimization when solving the PC coefficients. The least square minimization, using larger number of CFD samples, is recommended for current test cases. The study shows the potentials of PC method in Robust Design Optimization （RDO） and Reliability-Based Design Optimization （RBDO） of ship hydrodynamic performances.
基金supported by the National Natural Science Foundation of China (No.41274079,41074048,41374076)National High Technology Research and Development Program of China (863 Program) (2012AA061403,2012AA09A201)
文摘Based on the observation data of CHAMP satellite from 2006 to 2009, a 2D crustal magnetic anomaly model in China is established to study the distribution characteristics of crustal magnetic anomaly. In this paper, the 2D anomaly model is derived from the Legendre polynomial expansion of harmonic term N =6-50. The result shows that many elaborate structures reflected in magnetic anomaly map well correspond to the geologic structures in China and its adjacent area. The magnetic anomaly at low satellite height behaves complexly, which is mainly caused by the magnetic disturbance of shallow rocks.In contrast, the magnetic field isolines at high satellite height are relatively sparse and only magnetic anomalies of deep crust are reflected. This fact implies that the 2D model of crustal magnetic anomaly provides an important method of the space prolongation of geomagnetic field, and is of theoretical and practice importance in geologic structure analysis and geophysical prospecting.
