Finite-difference time-domain modeling of curved material interfaces by using boundary condition equations method

To deal with the staircase approximation problem in the standard finite-difference time-domain（FDTD） simulation,the two-dimensional boundary condition equations（BCE） method is proposed in this paper.In the BCE method,the standard FDTD algorithm can be used as usual,and the curved surface is treated by adding the boundary condition equations.Thus,while maintaining the simplicity and computational efficiency of the standard FDTD algorithm,the BCE method can solve the staircase approximation problem.The BCE method is validated by analyzing near field and far field scattering properties of the PEC and dielectric cylinders.The results show that the BCE method can maintain a second-order accuracy by eliminating the staircase approximation errors.Moreover,the results of the BCE method show good accuracy for cylinder scattering cases with different permittivities.

A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries

In Li and Ren(Int.J.Numer.Methods Fluids 70:742–763,2012),a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain,in which the high-order numerical accuracy and the oscillations-free property can be achieved.In this paper,the method is extended to solve steady state problems imposed in a curved physical domain.The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations,and a geometrical multigrid method to solve the derived linear system.To achieve high-order non-oscillatory numerical solutions,the classical k-exact reconstruction with k=3 and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables.The non-uniform rational B-splines(NURBS)curve is used to provide an exact or a high-order representation of the curved wall boundary.Furthermore,an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state.A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.

Constrained multi-degree reduction of triangular Bézier surfaces

This paper proposes and applies a method to sort two-dimensional control points of triangular Bezier surfaces in a row vector. Using the property of bivariate Jacobi basis functions, it further presents two algorithms for multi-degree reduction of triangular Bezier surfaces with constraints, providing explicit degree-reduced surfaces. The first algorithm can obtain the explicit representation of the optimal degree-reduced surfaces and the approximating error in both boundary curve constraints and corner constraints. But it has to solve the inversion of a matrix whose degree is related with the original surface. The second algorithm entails no matrix inversion to bring about computational instability, gives stable degree-reduced surfaces quickly, and presents the error bound. In the end, the paper proves the efficiency of the two algorithms through examples and error analysis.

A second order isoparametric finite element method for elliptic interface problems

A second order isoparametric finite element method （IPFEM） is proposed for elliptic interface problems. It yields better accuracy than some existing second-order methods, when the coefficients or the flux across the immersed curved interface is discontinuous. Based on an initial Cartesian mesh, a mesh optimization strategy is presented by employing curved boundary elements at the interface, and an incomplete quadratic finite element space is constructed on the optimized mesh. It turns out that the number of curved boundary elements is far less than that of the straight one, and the total degree of freedom is almost the same as the uniform Cartesian mesh. Numerical examples with simple and complicated geometrical interfaces demonstrate the efficiency of the proposed method.

A Hybrid WENO Scheme for Steady Euler Equations in Curved Geometries on Cartesian Grids

For steady Euler equations in complex boundary domains,high-order shockcapturing schemes usually suffer not only from the difficulty of steady-state convergence but also from the problem of dealing with physical boundaries on Cartesian grids to achieve uniform high-order accuracy.In this paper,we utilize a fifth-order finite difference hybrid WENO scheme to simulate steady Euler equations,and the same fifth-order WENO extrapolation methods are developed to handle the curved boundary.The values of the ghost points outside the physical boundary can be obtained by applying WENO extrapolation near the boundary,involving normal derivatives acquired by the simplified inverse Lax-Wendroff procedure.Both equivalent expressions involving curvature and numerical differentiation are utilized to transform the tangential derivatives along the curved solid wall boundary.This hybrid WENO scheme is robust for steady-state convergence and maintains high-order accuracy in the smooth region even with the solid wall boundary condition.Besides,the essentially non-oscillation property is achieved.The numerical spectral analysis also shows that this hybrid WENO scheme has low dispersion and dissipation errors.Numerical examples are presented to validate the high-order accuracy and robust performance of the hybrid scheme for steady Euler equations in curved domains with Cartesian grids.

Influence of equiatomic Zr/(Ti,Nb)substitution on microstructure and ultra-high strength of(Ti,Zr,Nb)C medium-entropy ceramics at 1900℃

High-temperature mechanical properties of medium-entropy carbide ceramics have attracted significant attention.Tailoring the microstructure is an effective way to improve these high-temperature mechanical properties,which can be affected by the evolution of the enthalpy and entropy,as well as by lattice distortion and sluggish diffusion.In this study,the effects of equiatomic Zr/(Ti,Nb)substitution(Zr content of 10-40 at%)on the microstructure and high-temperature strength of(Ti,Zr,Nb)C medium-entropy ceramics were investigated.The grain size of the(Ti,Zr,Nb)C medium-entropy ceramics was refined from 9.4±3.7 to 1.1±0.4μm with an increase in the Zr content from 10.0 to 33.3 at%.A further increase in the Zr content to 40 at%resulted in a slight increase in the grain size.At 1900℃,the(Ti,Zr,Nb)C medium-entropy ceramics with the Zr contents of 33.3 and 40 at%exhibited ultra-high flexural strengths of 875±43 and 843±71 MPa,respectively,which were higher than those of the transition metal carbides previously reported under similar conditions.Furthermore,relatively smooth grain boundaries,which were detected at a test temperature of 1000℃,transformed into curved and serrated boundaries as the temperature increased to 1900℃,which may be considered the primary reason for the improved high-temperature flexural strength.The associated mechanism was analyzed and discussed in detail.