黎曼流形上光滑函数的Hessian在共形度量下的关系式

根据黎曼流形上光滑函数的Hessian以及共形黎曼度量的定义,通过计算,直接推导出黎曼流形上光滑函数的Hessian的分量在共形黎曼度量下的关系式.

基于各向异性非结构网格的超声速流动自适应计算

基于各向异性非结构网格,实现了自适应求解技术,将其应用于超声速楔形体绕流及超声速横向喷流问题的计算研究,对各向异性网格自适应计算过程的收敛性和求解精度进行了对比分析,展示了各向异性网格自适应算法在降低问题求解规模。提高各向异性流场分辨率等方面的优势及存在的问题。超声速楔形体无粘绕流场的自适应计算,清晰地捕捉到了激波附近的流场信息,激波前后的马赫数、压力、密度。温度等物理量与理论分析解吻合很好。超声速横向喷流流场存在激波、分离涡、边界层等流场结构的相互干扰。计算研究表明,单纯基于Hes-sian度量张量的各向异性网格生成及自适应算法不能有效模拟边界层内的流动情况,是将来需要进一步开展的研究挑战。

The Minkowski norm and Hessian isometry induced by an isoparametric foliation on the unit sphere

Let M_(t) be an isoparametric foliation on the unit sphere(S^(n−1)(1),g^(st))with d principal curvatures.Using the spherical coordinatesinduced by M_(t),we construct a Minkowski norm with the representation F=r√2f(t),which generalizes the notions of(α,β)-norm and(α1,α2)-norm.Using the technique of the spherical local frame,we givean exact and explicit answer to the question when F=r√2 f(t)really defines a Minkowski norm.Using the similar technique,we study the Hessian isometry Φ between two Minkowski norms induced by M_(t),which preservesthe orientation and fixes the spherical ξ-coordinates.There aretwo ways to describe this Φ,either by a system of ODEs,or by its restriction toany normal plane for M_(t),which is then reduced to a Hessian isometry between Minkowski norms on R^(2) satisfying certain symmetry and(d)-properties.When d>2,we prove that this Φ can be obtained by gluing positive scalar multiplications and compositions of the Legendre transformation and positive scalar multiplications,so it must satisfy the(d)-property for any orthogonal decomposition R^(n)=V'+V'',i.e.,for any nonzero x=x'+x'' and Φ(x)=x=x'+x''with x',x'∈V'and x'',x''∈V'',we have g_(x)^(F1)(x'',x)=g_(x)^(F2)x(x'',x).As byproducts,we prove the following results.On the indicatrix(S_(F,g)),where F is a Minkowski norm induced by M_(t) and g is the Hessian metric,the foliation N_(t)=S_(F)∩R>_(0)M_(0) is isoparametric.Laugwitz Conjecture is valid for a Minkowski norm F induced by M_(t),i.e.,if its Hessian metric g is flat on R^(n)\{0}with n>2,then F is Euclidean.

仿射空间R^（n＋1）中的非退化二次函数的水平曲面

本文利用Ｈｅｓｓｉａｎ度量和横截向量场，给出了非退化二次函数的水平曲面的一个充要条件．

Some Connections in Almost Hermitian Manifold

The idea of this research is to study different types of connections in an almost Hermite manifold. The connection has been established between linear connection and Riemannian connection. Three new linear connections ∇1, ∇2, ∇3 are introduced. The necessary and sufficient condition for ∇1, ∇2, ∇3 to be metric is discussed. A new metric s* (X,Y) has been defined for (Mn,F,g*) and additional properties are discussed. It is also proved that for the quarter symmetric connection ∇ is unique in given manifold. The hessian operator with respect to all connections defined above has also been discussed.

各向异性网格自适应计算在超燃模拟中的应用

采用各向异性网格自适应求解技术,将其应用于DLR超燃冲压发动机燃烧室中的超燃模拟.开展了3个算例,包括采用滑移壁面条件的燃烧室冷流场模拟、采用无滑移壁面条件的冷流场模拟及采用无滑移壁面条件的反应流模拟.模拟中,各向异性网格自适应计算捕捉到了如激波、射流、边界层、火焰面等具有各向异性特征的大梯度区域,并利用各向异性网格进行了很好的加密.对比利用各向同性网格的初始流场计算,各向异性网格自适应计算使基于滑移条件、无滑移条件的冷流计算及反应流计算的网格单元数量分别下降了36.2%,36.4%和36.8%,有效降低了计算规模,而且流场大梯度区域的计算结果更准确,辨析度更好.结果表明:对于像超燃这类具有各向异性特征的问题,各向异性网格系统比各向同性网格系统有更好的计算效率及准确性,同时也表明基于Mach数场构造的各向异性网格系统可以有效应用于超燃计算.