Optimal Convergence Analysis of Two-Level Nonconforming Finite Element Iterative Methods for 2D/3D MHD Equations

Several two-level iterative methods based on nonconforming finite element methods are applied for solving numerically the 2D/3D stationary incompressible MHD equations under different uniqueness conditions. These two-level algorithms are motivated by applying the m iterations on a coarse grid and correction once on a fine grid. A one-level Oseen iterative method on a fine mesh is further studied under a weak uniqueness condition. Moreover, the stability and error estimate are rigorously carried out, which prove that the proposed methods are stable and effective. Finally, some numerical examples corroborate the effectiveness of our theoretical analysis and the proposed methods.     

Modeling and Performance Optimization of an Irreversible Two-Stage Combined Thermal Brownian Heat Engine

Based on finite time thermodynamics, an irreversible combined thermal Brownian heat engine model is established in this paper. The model consists of two thermal Brownian heat engines which are operating in tandem with thermal contact with three heat reservoirs. The rates of heat transfer are finite between the heat engine and the reservoir. Considering the heat leakage and the losses caused by kinetic energy change of particles, the formulas of steady current, power output and efficiency are derived. The power output and efficiency of combined heat engine are smaller than that of single heat engine operating between reservoirs with same temperatures. When the potential filed is free from external load, the effects of asymmetry of the potential, barrier height and heat leakage on the performance of the combined heat engine are analyzed. When the potential field is free from external load, the effects of basic design parameters on the performance of the combined heat engine are analyzed. The optimal power and efficiency are obtained by optimizing the barrier heights of two heat engines. The optimal working regions are obtained. There is optimal temperature ratio which maximize the overall power output or efficiency. When the potential filed is subjected to external load, effect of external load is analyzed. The steady current decreases versus external load; the power output and efficiency are monotonically increasing versus external load.     

Ply-by-Ply Failure Analysis of Laminates Under Dynamic Loading

Ply-by-ply failure analysis of symmetric and anti-symmetric laminates under uniform sinusoidal transverse dynamic loading is performed for a specified duration. The study investigates the first ply failure load, followed by the detection of successive ply failures along with their failure modes using various failure theories. Some of the well-established failure theories, mostly used by the researchers, are considered for the failure prediction in laminates. The finite element computational model based on higher order shear deformation displacement field is used for the failure analysis and the complete methodology is computer coded using FORTRAN. The ply-discount stiffness reduction scheme is employed to modify the material properties of the failed lamina. The failure theories used in the analysis are compared according to their ability to predict failure load, failed ply, failure mode and progression of failure. The failure analysis is performed for both the cross-ply and angle-ply laminates with all edges simply supported and clamped. The significance of fibre orientation and stacking sequence in terms of the strength of a laminate and failure progression is also highlighted.     

一种新型高双折射光子晶体光纤特性研究

设计了一种高双折射高非线性光子晶体光纤, 采用全矢量有限元法研究了这种光纤的基模模场、双折射、非线性、有效模面积及色散特性. 数值研究发现, 减小孔间距Λ的大小, 在波长1550 nm 处, 该光纤可获得10^-2 数量级的双折射B, 比普通的椭圆保偏光纤高约两个数量级; 同时, 该光纤可获得42 W^-1·km^-1 的高非线性系数γ. 另外,分别在可见光和近红外波段出现了两个零色散波长, 在波长800–2000 nm 之间具有良好的色散平坦特性. 这种设计为获得高双折射高非线性超平坦色散光子晶体光纤提供了一种新的方法, 该光纤在偏振控制、非线性光学和色散控制方面具有广泛的应用前景. 关键词： 光子晶体光纤 高双折射 高非线性 有限元法     

Application of h-adaptive,high order finite element method to solve radial Schrödinger equation

The h-adaptive, high order finite element method is applied to solve a second order one dimension eigenvalue problem. The finite element formulation for the Lobatto basis is given, for which basis functions of arbitrary order can be constructed. The adaptive algorithm is simple, yet very efficient and straightforward to implement. The algorithm is based on the observation that the expansion coefficients of Lobatto basis functions decay rapidly. It allows evaluating the smallest eigenvalues simultaneously with the comparable accuracy for all eigenvalues. The presented algorithm is applied to solve the radial Schrödinger equation with the Coulomb and the Woods–Saxon potentials. For both potentials the convergence rate is presented. After seven adaptive iterations nine-digit accuracy was obtained.     

Semiconductor diamond heater in the Kawai multianvil apparatus: an innovation to generate the lower mantle geotherm

Semiconductor diamond is considered the best heater material to generate ultra-high temperatures in a Kawai cell. In two pioneering studies, a mixture of graphite and amorphous boron (or boron carbide, B₄C) was converted to semiconductor diamond in the diamond stability field and was confirmed to generate 2000°C and 3500°C, respectively. Following these works, we synthesized a homemade boron-doped graphite block with fine machinability. With this technical breakthrough, we developed a semiconductor diamond heater in a smaller Kawai-type cell assembly. Here, we report the procedure for making machinable boron-doped graphite, and the performance of the material as a heater in a Kawai cell at 15?GPa using tungsten carbide anvils and at ～50?GPa using sintered diamond anvils. Furthermore, we present a finite element simulation of the temperature distribution generated by a semiconductor diamond heater, which is much more homogeneous than that generated by a metal heater.     

Effect of finite boundary junction mobility on the growth rate of grains in 3D polycrystals

With decreasing grain size, grain boundary junctions become increasingly important for microstructure evolution. We show that the effect of a limited mobility of triple junctions on the growth rate of polycrystals can be implemented in theories of three-dimensional (3D) grain growth. Respective analytical relations are derived on the basis of the average n-hedra approach introduced by Glicksman to describe the volume rate of change of 3D grains in a polycrystalline aggregate under the impact of a limited triple junction mobility. The theoretical predictions were compared to network-model computer simulations, and good agreement was obtained.     

Fastiterative methods for least squares estimations

Least squares estimations have been used extensively in many applications, e.g. system identification and signal prediction. When the stochastic process is stationary, the least squares estimators can be found by solving a Toeplitz or near-Toeplitz matrix system depending on the knowledge of the data statistics. In this paper, we employ the preconditioned conjugate gradient method with circulant preconditioners to solve such systems. Our proposed circulant preconditioners are derived from the spectral property of the given stationary process. In the case where the spectral density functions() of the process is known, we prove that ifs() is a positive continuous function, then the spectrum of the preconditioned system will be clustered around 1 and the method converges superlinearly. However, if the statistics of the process is unknown, then we prove that with probability 1, the spectrum of the preconditioned system is still clustered around 1 provided that large data samples are taken. For finite impulse response (FIR) system identification problems, our numerical results show that annth order least squares estimator can usually be obtained inO(n logn) operations whenO(n) data samples are used. Finally, we remark that our algorithm can be modified to suit the applications of recursive least squares computations with the proper use of sliding window method arising in signal processing applications.Research supported in part by HKRGC grant no. 221600070, ONR contract no. N00014-90-J-1695 and DOE grant no. DE-FG03-87ER25037.     

Asynchronous and corrected-asynchronous finite difference solutions of PDEs on MIMD multiprocessors

A major problem in achieving significant speed-up on parallel machines is the overhead involved with synchronizing the concurrent processes. Removing the synchronization constraint has the potential of speeding up the computation, while maintaining greater computation flexibility (e.g. differences in processors speed; differences in the data input to processors). We construct asynchronous (AS) finite difference schemes for the solution of PDEs by removing the synchronization constraint. We analyze the numerical properties of these schemes. Based on the analysis, we develop corrected-asynchronous (CA) finite difference schemes which are specifically constructed for an asynchronous processing. We present asynchronous (AS) and corrected-asynchronous (CA) finite difference schemes for the multi-dimensional heat equation. Although our discussion concentrates on the Euler scheme it should serve only as a sample, as it can be extended to other schemes and other PDEs.These schemes are implemented on the shared-memory multi-userSequent Balance machine. Numerical results for one and two dimensional problems are presented. It is shown experimentally that synchronization penalty can be about 50% of run time: in most cases, the asynchronous scheme runs twice as fast as the parallel synchronous scheme. In general, the efficiency of the parallel schemes increases with processor load, with the time-level, and with the problem dimension. The efficiency of the AS may reach 90% and over, but it provides accurate results only for steady-state values. The CA, on the other hand, is less efficient but provides more accurate results for intermediate (non steady-state) values. The results show the potential of developing asynchronous finite deference schemes for steady-state as well as non steadystate problems.This research was partially supported by a grant from The Basic Research Foundation administrated by The Israel Academy of Sciences and Humanities.A reduced version of the paper was presented at the 4th SIAM Conference on Parallel Processing for Scientific Computing, Dec. 11–13, 1989, Chicago, USA.The work by this author was supported by research grant 337 of the Israeli National Council for Research and Development in the years 1990–1991.This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NASI-18107 while the author was in residence at the Institute for Computer Applications in Sciences and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665, USA.     

Mixed methods on quadrilaterals and hexahedra

We describe a new family of discrete spaces suitable for use with mixed methods on certain quadrilateral and hexahedral meshes. The new spaces are natural in the sense of differential geometry, so all the usual mixed method theory, including the hybrid formulation, carries over to these new elements with proofs unchanged. Because transforming general quadrilaterals into squares introduces nonlinearity and because mixed methods involve the divergence operator, the new spaces are more complicated than either the corresponding Raviart-Thomas spaces for rectangles or corresponding finite element spaces for quadrilaterals. The new spaces are also limited to meshes obtained from a rectangular mesh through the application of a single global bilinear transformation. Despite this limitation, the new elements may be useful in certain topologically regular problems, where initially rectangular grids are deformed to match features of the physical region. They also illustrate the difficulties introduced into the theory of mixed methods by nonlinear transformations.