A SEMI-CONJUGATE MATRIX BOUNDARY VALUE PROBLEM FOR GENERAL ORTHOGONAL POLYNOMIALS ON AN ARBITRARY SMOOTH JORDAN CURVE

In this article, the author characterizes orthogonal polynomials on an arbitrary smooth Jordan curve by a semi-conjugate matrix boundary value problem, which is different from the Riemann-Hilbert problems that appear in the theory of Riemann -Hilbert approach to asymptotic analysis for orthogonal polynomials on a real interval introduced by Fokas, Its, and Kitaev and on the unit circle introduced by Baik, Deift, and Johansson. The author hopes that their characterization may be applied to asymptotic analysis for general orthogonal polynomials by combining with a new extension of steepest descent method which we are looking for.

ALGEBRAIC-GEOMETRY FOUNDATION FOR CONSTRUCTING SMOOTH INTERPOLANTS ON CURVED SIDES ELEMENT

The aim of this paper is to lay a algebraic geometry foundation for constructing smoothing interpolants on curved side element. Some interpolation theorems in polynomial space are given. The main results effectively for CAGD are presented.

Ahlfors-regular curves and Carleson measures

We study the relation between the boundary of a simply connected domainΩbeing Ahlfors-regular and the invariance of Carleson measures under the push-forward operator induced by a conformal mapping from the unit disk△onto the domainΩ.As an application,we characterize the chord-arc curves with small norms and the asymptotically smooth curves in terms of the complex dilatation of some quasiconformal reflection with respect to the curve.

Simulation of 3D fracture propagation under I-II-III mixed-mode loading

Fracture propagation under mixed-mode loading conditions prevails in many natural geological processes and deep engineering projects,while the corresponding numerical simulation is very challenging in rock mechanics,especially in 3D cases.In most previous studies,the complexity of 3D fracture geometry was over-simplified,and model III loading was often not considered.In this study,we propose to use an efficient stress-based Sch€ollmann criterion combined with Displacement Discontinuity Method(DDM)to model 3D fracture propagation under arbitrary I+II+III mixed-mode loading conditions.A novel curve-smoothing algorithm is developed to smoothen the fracture front during propagation,which significantly enhances the model's ability in dealing with complex 3D fracture geometry.In particular,we adopt two different solution schemes,namely staggered and monolithic,to simulate mode I fracture propagation in the case of hydraulic fracturing.The accuracy,efficiency and convergency of the two solution schemes are compared in detail.Our research findings suggest that the degree of coupling between fracture aperture and fluid pressure in hydraulic fracturing lies somewhere between one-way and two-way,which favors the staggered solution scheme.To further test our new model,we provide three additional numerical examples associated with 3D fracture propagation under various mixed-mode loading conditions.Our model shows excellent performance in efficiently locating the new fracture front and reliably capturing the complex 3D fracture geometry.This study provides a generic algorithm to model high-fidelity 3D fracture propagation without simplifying fracture geometry or loading conditions,making it widely applicable to fracture-propagation-related problems.