A Characteristic Iteration for Solving Coefficient Inverse Problems of a Wave Equation

In this paper, a characteristic iteration for solving coefficient inverse problems has been presented. This method is stable and fast converging, and may be extended to the 2-D case. Excellent numerical results have been obtained by this method.     

Subsonic Euler flows in a divergent nozzle

We characterize a class of physical boundary conditions that guarantee the existence and uniqueness of the subsonic Euler flow in a general finitely long nozzle.More precisely,by prescribing the incoming flow angle and the Bernoulli’s function at the inlet and the end pressure at the exit of the nozzle,we establish an existence and uniqueness theorem for subsonic Euler flows in a 2-D nozzle,which is also required to be adjacent to some special background solutions.Such a result can also be extended to the 3-D asymmetric case.     

The bidimensional interproximation problem and mixed splines

Interproximation methods for surfaces can be used to construct a smooth surface interpolating some data points and passing through specified regions. In this paper we study the use of mixed splines, that is smoothing splines with additional interpolation constraints, to solve the interproximation problem for surfaces in the case of scattered data. The solution is obtained by solving a linear system whose structure can be improved by using “bell-shaped” thin plate splines.     

Limit cycles in a food-chain with inhibition responses

In this paper, by constructing a Lyapunov function, we show the global asymptotical stability of a three-dimensional food-chain model with inhibition response. We then using a corollary to center manifold theorem to show that the system undergoes a 3-D Hopf bifurcation, and obtain the existence of limit cycles for the three-dimensional model. The methods used here can be extended to many other 3-D differential systems.     

A modified method of approximate particular solutions for solving linear and nonlinear PDEs

The method of approximate particular solutions (MAPS) was first proposed by Chen et al. in Chen, Fan, and Wen, Numer Methods Partial Differential Equations, 28 (2012), 506–522. using multiquadric (MQ) and inverse multiquadric radial basis functions (RBFs). Since then, the closed form particular solutions for many commonly used RBFs and differential operators have been derived. As a result, MAPS was extended to Matérn and Gaussian RBFs. Polyharmonic splines (PS) has rarely been used in MAPS due to its conditional positive definiteness and low accuracy. One advantage of PS is that there is no shape parameter to be taken care of. In this article, MAPS is modified so PS can be used more effectively. In the original MAPS, integrated RBFs, so called particular solutions, are used. An additional integrated polynomial basis is added when PS is used. In the modified MAPS, an additional polynomial basis is directly added to the integrated RBFs without integration. The results from the modified MAPS with PS can be improved by increasing the order of PS to a certain degree or by increasing the number of collocation points. A polynomial of degree 15 or less appeared to be working well in most of our examples. Other RBFs such as MQ can be utilized in the modified MAPS as well. The performance of the proposed method is tested on a number of examples including linear and nonlinear problems in 2D and 3D. We demonstrate that the modified MAPS with PS is, in general, more accurate than other RBFs for solving general elliptic equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1839–1858, 2017     

MULTIVARIATE FOURIER SERIES OVER A CLASS OF NON TENSOR－PRODUCT PARTITION DOMAINS

This paper finds a way to extend the well-known Fourier methods,to so-called n 1 directions partition domains in n-dimension.In particular,in 2-D and 3-D cases,we study Fourier methods over 3-direction parallel hexagon partitions and 4-direction parallel paralleogram dodecahedron partitions,respectively.It has pointed that,the most concepts and results of Fourier methods on tensor-product case,such as periodicity,orthogonality of Fourier basis system,Partial sum of Fourier series and its approximation behavior,can be moved on the new non tensor-product partiton case.     

MULTIVARIATE FOURIER TRANSFORM METHODS OVER SIMPLEX AND SUPER-SIMPLEX DOMAINS

In this paper we propose the well-known Fourier method on some non-tensor productdomains in R~d, inclding simplex and so-called super-simplex which consists of (d 1)!simplices. As two examples, in 2-D and 3-D case a super-simplex is shown as a parallelhexagon and a parallel quadrilateral dodecahedron, respectively. We have extended mostof concepts and results of the traditional Fourier methods on multivariate cases, such asFourier basis system, Fourier series, discrete Fourier transform (DFT) and its fast algorithm(FFT) on the super-simplex, as well as generalized sine and cosine transforms (DST, DCT)and related fast algorithms over a simplex. The relationship between the basic orthogonalsystem and eigen-functions of a Laplacian-like operator over these domains is explored.     

Efficient Spatial Modeling Using the SPDE Approach With Bivariate Splines

Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associated computational cost can be a bottleneck, especially in realistic applications. It has been shown that computational efficiency can be gained by doing the computations using Gaussian Markov random fields (GMRFs) as the GFs can be seen as weak solutions to corresponding stochastic partial differential equations (SPDEs) using piecewise linear finite elements. We introduce a new class of representations of GFs with bivariate splines instead of finite elements. This allows an easier implementation of piecewise polynomial representations of various degrees. It leads to GMRFs that can be inferred efficiently and can be easily extended to nonstationary fields. The solutions approximated with higher order bivariate splines converge faster, hence the computational cost can be alleviated. Numerical simulations using both real and simulated data also demonstrate that our framework increases the flexibility and efficiency. Supplementary materials are available online.     

二维等谱问题研究的计算数学框架

等谱问题是数学、物理诸学科关注的一个热点问题,本文总结并诠释了二维等谱问题的内在计算数学性质与规律:利用镜像反演讨论等谱对的几何结构(不等距而谱相等);把一般文献中假定的特殊三角形扩展到一般的三角形或者矩形;研究特征函数的正交结构,把特定的Laplace等谱问题扩展到一般零边值的二阶线性椭圆算子等谱问题.指出合理的粗网格对于研究等谱问题及其计算的重要性:两个连续问题等谱成立的充分必要条件是存在自然粗网格使其离散问题谱相等.文中给出的数值例子与特征值近似逼近验证了相应的结论,所用的方法原则上可用于研究三维乃至高维的PDE等谱问题.     

基于透射边界条件的高阶离散型角点条件

对波动方程的数值模拟中，在有限区域建立吸收边界条件，其中对区域角点的处理是一个很重要的问题．随着吸收边界条件阶数的提高，与之匹配的角点条件也越难建立．MTF是一种离散型吸收边界条件．在此，对于二维问题，基于MTF建立离散型高阶角点条件，对计算区域角点处理时，在区域对角线方向上建立N阶MTF公式．问题也可推广到三维．数值结果证实了我们的猜测．     

Variational Interpolation of Subsets

We consider the problem of the variational interpolation of subsets of Euclidean spaces by curves such that the L² norm of the second derivative is minimized. It is well-known that the resulting curves are cubic spline curves. We study geometric boundary conditions arising for various types of subsets such as subspaces, polyhedra, and submanifolds, and we indicate how solutions can be computed in the case of convex polyhedra.     

Numerical optimisation for induction heat treatment processes

This paper aims at introducing new approaches for designing and optimising induction heat treatment processes. Although the final objectives of induction heating processes may deal with some specific mechanical or metallurgical properties for manufactured parts, we shall primarily focus here on achieving an accurate control of temperature distribution and evolution in the Heat Affected Zone (HAZ). This objective can be formalised as a classical optimisation problem: we seek to minimise a cost function which measures the difference between computed and goal temperatures – along with some constraints on process parameters. We deal here with both zero-order algorithms – using a method based on Efficient Global Optimization algorithm which is an optimisation procedure assisted by a meta model – as well as first-order algorithms. These algorithms have been coupled with 2-D and 3-D finite element models developed in our laboratory; this model is based on a coupling procedure between Maxwell equations and heat transfer models, and has been extended to mechanical and metallurgical computations.     

Evolution of Elastic T-stresses of Growing Mixed-Mode Cracks

Finite element simulations of growing mixed-mode central line-cracks (2-D case) and internal penny-shaped cracks (3-D case) have been performed to address the evolution of the elastic T-stresses acting at the crack-tip (the non-singular stress terms in the stress expansion formula at the crack-front). (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral properties of operators with polynomial invariants in real finite-dimensional spaces

We consider linear operators lying in the orthogonal group of a quadratic form and study those spectral properties of such operators that can be expressed in terms of the signature of this form. We show that in the typical case these transformations are symplectic. Some of the results can be extended to the general case when the operator admits a homogeneous form of degree ≥3.     

To the theory of optimal splines

On June 18, 2008 at the Plenary Meeting of the International Conference “Differential Equations and Topology” dedicated to the 100th anniversary of Pontryagin, the report [1] was submitted by Isaev and Leitmann. This report in a summary form included a section dedicated to the research of scientists of TsAGI in the field of automation of full life-cycle (i.e. engineering-design-manufacturing, or CAE/CAD/CAM, or CALS-technologies) of wind tunnel models [2]. Within this framework, methods of geometric modeling [3] and [4] were intensively developed, new classes of optimal splines have been built, including the Pontryagin splines and the Chebyshev splines [5], [6], [7] and [8]. This paper reviews some results on the Pontryagin splines. We also give some results on the Lurie splines, that arise in the problem of interpolation of a cylindrical type surface given by the family of table coplanar planes.     

Local lagrange interpolation with cubic splines on tetrahedral partitions

We describe an algorithm for constructing a Lagrange interpolation pair based on C¹ cubic splines defined on tetrahedral partitions. In particular, given a set of points , we construct a set P containing and a spline space based on a tetrahedral partition whose set of vertices include such that interpolation at the points of P is well-defined and unique. Earlier results are extended in two ways: (1) here we allow arbitrary sets , and (2) the method provides optimal approximation order of smooth functions.     

Two-point Padé approximants for the expansions of Stieltjes functions in real domain

The fundamental inequalities for the sequences of subdiagonal and diagonal one-point Padé approximants to Stieltjes function has been extended to the case of certain two-point Padé approximants. The results can be applied to the theory of inhomogeneous media for calculating the bounds for the effective transport coefficients of two-components heterogeneous materials.     

Non blow-up criterion for the 3-D Magneto-hydrodynamics equations in the limiting case

In this paper, we prove that suitable weak solution(u, b) of the 3-D MHD equations can be extended beyond T if u∈L~∞(0,T; L~3(R~3)) and the horizontal components b_h of the magnetic field satisfies the well-known Ladyzhenskaya-Prodi-Serrin condition, which improves the corresponding regularity criterion by Mahalov-Nicolaenko-Shilkin.     

Synthesizing monochromatic 3-D images by multiple-exposure rainbow holography with vertical area-partition approach

We report for the first time the theoretical analysis and experimental results of a white-light reconstructed monochromatic 3-D image synthesizing tomograms by multiple rainbow holography with vertical-area partition (VAP) approach. The theoretical and experimental results show that 3-D monochromatic image can be synthesized by recording the master hologram by VAP approach without any distortions either in gray scale or in geometrical position. A 3-D monochromatic image synthesized from a series of medical tomograms is presented in this paper for the first time     

Investigation of the variation of the rubber shearing force during extended stationary contact under constant compressive strain

The effect of extended stationary contact on the shearing force has been investigated in the case of rubbers subjected to constant compressive deformation at high and low temperatures. The dependence of the shearing force on the logarithm of the stationary contact time is linear up to the moment of accumulation of 100% relative residual deformation. It is shown that the time-dependent processes that determine the values of the shearing forces can be intensified by raising the temperature. The results of the investigation can be used in predicting the shearing force in the case of rubbers in extended contact with metal under conditions of constant compressive strain.Laboratory for Problems in Polymer Physics, Lenin Moscow State Pedagogical Institute; Leningrad Branch of the Scientific-Research Institute of the Rubber Industry. Translated from Mekhanika Polimerov, No. 4, pp. 629–633, July–August, 1971.