A Kenmotsu Metric as a *-conformal Yamabe Soliton with Torse Forming Potential Vector Field

The purpose of the present paper is to deliberate *-conformal Yamabe soliton,whose potential vector field is torse-forming on Kenmotsu manifold.Here,we have shown the nature of the soliton and find the scalar curvature when the manifold admitting *-conformal Yamabe soliton on Kenmotsu manifold.Next,we have evolved the characterization of the vector field when the manifold satisfies *-conformal Yamabe soliton.Also we have embellished some applications of vector field as torse-forming in terms of *-conformal Yamabe soliton on Kenmotsu manifold.We have developed an example of *-conformal Yamabe soliton on 3-dimensional Kenmotsu manifold to prove our findings.

Energy <i>β</i>-Conformal Change and Special Finsler Spaces

The main aim of the present paper is to establish an intrinsic investigation of the energy β-conformal change of the most important special Finsler spaces, namely, Ch-recurrent, Cv-recurrent, C0-recurrent, Sv-recurrent, quasi-C-reducible, semi-C-reducible, C-reducible, P-reducible, C2-like, S3-like, P2-like and h-isotropic, ···, etc. Necessary and sufficient conditions for such special Finsler manifolds to be invariant under an energy β-conformal change are obtained. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

C1-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

This paper is devoted to Professor Benyu Guo's open question on the C1-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the C1-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories-interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global C1-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the C1-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov-Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a C1-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.

On Uniform Domains

A characteristic parameter α_D of a domain D in R^n is introduced. D is unifonm if and only if α_D<∞. And α_D=1 if D is a ball or a half space. It is proved that the decomposability and the uniformity of a domain in R^n are still equivalent. Some problems on uniform domains and decomposition domains are discussed.

Formatted PVDF in lamellar composite solid electrolyte for solidstate lithium metal battery

Solid polymer electrolytes(SPEs)hold great application potential for solid-state lithium metal battery because of the excellent interfacial contact and processibility,but being hampered by the poor room-temperature conductivity(~10^(−7)S·cm^(−1))and low lithium-ion transference number(tLi+).Here,a lamellar composite solid electrolyte(Vr-NH_(2)@polyvinylidene fluoride(PVDF)LCSE)withβ-conformation PVDF is fabricated by confining PVDF in the interlayer channel of-NH_(2)modified vermiculite lamellar framework.We demonstrate that the conformation of PVDF can be manipulated by the nanoconfinement effect and the interaction from channel wall.The presence of-NH_(2)groups could induce the formation ofβ-conformation PVDF through electrostatic interaction,which serves as continuous and rapid lithium-ion transfer pathway.As a result,a high room-temperature ionic conductivity of 1.77×10^(−4)S·cm^(−1)is achieved,1-2 orders of magnitude higher than most SPEs.Furthermore,Vr-NH_(2)@PVDF LCSE shows a high tLi+of 0.68 because of the high dielectric constant,~3 times of that of PVDF SPE,and surpassing most of reported SPEs.The LiNi_(0.8)Co_(0.1)Mn_(0.1)O_(2)||Li cell assembled by Vr-NH_(2)@PVDF LCSE obtains a discharge specific capacity of 137.1 mAh·g^(−1)after 150 cycles with a capacity retention rate of 93%at 1 C and 25℃.This study may pave a new avenue for high-performance SPEs.

Uniformly convergent H(div)-conforming rectangular elements for Darcy-Stokes problem

We consider a singular perturbation problem which describes 2D Darcy-Stokes flow. An H（div）- conforming rectangular element, DS-R14, is proposed and analyzed first. This element has 14 degrees of freedom for velocity and is proved to be uniformly convergent with respect to perturbation constant. We then simplify this element to get another H（div）-conforming rectangular element, DS-R12, which has 12 degrees of freedom for velocity. The uniform convergence is also obtained for this element. Finally, we construct a de Rham complex corresponding to DS-R12 element.

Divergence-Free H(div)-Conforming Hierarchical Bases for Magnetohydrodynamics (MHD)

In order to solve the magnetohydrodynamics(MHD)equations with a H(div)-conforming element,a novel approach is proposed to ensure the exact divergence-free condition on the magnetic field.The idea is to add on each element an extra interior bubble function from a higher order hierarchicalH(div)-conforming basis.Four such hierarchical bases for theH(div)-conforming quadrilateral,triangular,hexahedral,and tetrahedral elements are either proposed(in the case of tetrahedral)or reviewed.Numerical results have been presented to show the linear independence of the basis functions for the two simplicial elements.Good matrix conditioning has been confirmed numerically up to the fourth order for the triangular element and up to the third order for the tetrahedral element.

ON THE CONSTRUCTION OF WELL-CONDITIONED HIERARCHICAL BASES FOR TETRAHEDRAL H（curl）-CONFORMING NEDELEC ELEMENT

A partially orthonormal basis is constructed with better conditioning properties for tetrahedral H（curl）-conforming Nedelec elements. The shape functions are classified into several categories with respect to their topological entities on the reference 3-simplex. The basis functions in each category are constructed to achieve maximum orthogonaiity. The numerical study on the matrix conditioning shows that for the mass and quasi-stiffness matrices, and in a logarithmic scale the condition number grows linearly vs. order of approximation up to order three. For each order of approximation, the condition number of the quasi-stiffness matrix is about one order less than the corresponding one for the mass matrix. Also, up to order six of approximation the conditioning of the mass and quasi- stiffness matrices with the proposed basis is better than the corresponding one with the Ainsworth-Coyle basis Internat. J. Numer. Methods. Engrg., 58：2103-2130, 2003. except for order four with the quasi-stiffness matrix. Moreover, with the new basis the composite matrix μM ＋ S has better conditioning than the Ainsworth-Coyle basis for a wide range of the parameter μ.

A Well-Conditioned Hierarchical Basis for Triangular H(curl)-Conforming Elements

We construct a well-conditioned hierarchical basis for triangular H(curl)-conforming elements with selected orthogonality.The basis functions are grouped into edge and interior functions,and the later is further grouped into normal and bubble functions.In our construction,the trace of the edge shape functions are orthonormal on the associated edge.The interior normal functions,which are perpendicular to an edge,and the bubble functions are both orthonormal among themselves over the reference element.The construction is made possible with classic orthogonal polynomials,viz.,Legendre and Jacobi polynomials.For both the mass matrix and the quasi-stiffness matrix,better conditioning of the new basis is shown by a comparison with the basis previously proposed by Ainsworth and Coyle[Comput.Methods.Appl.Mech.Engrg.,190(2001),6709-6733].

On the Construction of Well-Conditioned Hierarchical Bases for H(div)-Conforming Rn Simplicial Elements

Hierarchical bases of arbitrary order for H(div)-conforming triangular and tetrahedral elements are constructedwith the goal of improving the conditioning of the mass and stiffness matrices.For the basis with the triangular element,it is found numerically that the conditioning is acceptable up to the approximation of order four,and is better than a corresponding basis in the dissertation by Sabine Zaglmayr[High Order Finite Element Methods for Electromagnetic Field Computation,Johannes Kepler Universit¨at,Linz,2006].The sparsity of the mass matrices from the newly constructed basis and from the one by Zaglmayr is similar for approximations up to order four.The stiffness matrix with the new basis is much sparser than that with the basis by Zaglmayr for approximations up to order four.For the tetrahedral element,it is identified numerically that the conditioning is acceptable only up to the approximation of order three.Compared with the newly constructed basis for the triangular element,the sparsity of the massmatrices fromthe basis for the tetrahedral element is relatively sparser.