Exact finite element solutions to the Helmholtz equation

For one-, two- and three-dimensional co-ordinate systems finite element matrices for the wave or Helmholtz equation are used to produce a single difference equation holding at any point of a regular mesh. Under homogeneous Dirichlet or Neumann boundary conditions, these equations are solved exactly. The eigenfunctions are the discrete form of sine or cosine functions and the eigenvalues are shown to be in error by a term of + O(h²ⁿ) where n is the order of the polynomial approximation of the wave function. The solutions provide the means of testing computer programs against the exact solutions and allow comparison with other difference schemes.     

Analysis of two‐grid methods for reaction‐diffusion equations by expanded mixed finite element methods

We present two efficient methods of two‐grid scheme for the approximation of two‐dimensional semi‐linear reaction‐diffusion equations using an expanded mixed finite element method. To linearize the discretized equations, we use two Newton iterations on the fine grid in our methods. Firstly, we solve an original non‐linear problem on the coarse grid. Then we use twice Newton iterations on the fine grid in our first method, and while in second method we make a correction on the coarse grid between two Newton iterations on the fine grid. These two‐grid ideas are from Xu's work (SIAM J. Sci. Comput. 1994; 15 :231–237; SIAM J. Numer. Anal. 1996; 33 :1759–1777) on standard finite element method. We extend the ideas to the mixed finite element method. Moreover, we obtain the error estimates for two algorithms of two‐grid method. It is showed that coarse space can be extremely coarse and we achieve asymptotically optimal approximation as long as the mesh sizes satisfy H =??(h^¼) in the first algorithm and H =??(h^?) in second algorithm. Copyright © 2006 John Wiley & Sons, Ltd.     

Cost‐effective algorithms for processor arrays with reconfigurable bus systems

Abstract

A processor array with a reconfigurable bus system (abbreviated to PARBS) is a computation model which consists of a processor array and a reconfigurable bus system. It is a very powerful computation model in that it possesses the ability to solve many problems efficiently. However, most existing efficient algorithms on PARBS's use a large number of processors to solve problems. For example, to determine the maximum (minimum) of n data items in O(l) time, O(n ²) processors are required [12]. To solve the all‐pairs shortest paths and the minimum spanning tree problems in O(log n) time, O(n ⁴) processors are required [20]. These networks will therefore become very expensive for large n. In this paper, we introduce the concept of iterative‐PARBS, which is similar to the FOR‐loop construct in sequential programming languages. The iterative‐PARBS is a building block through which the processing data can be routed several times. We can think of it as a “hardware subroutine.’’ Based on this scheme, it is possible to explore more cost‐effective, time‐efficient parallel algorithms for use in a PARBS. The following new results are derived in this study: 1. The minimum (maximum) of n data items can be determined in O(l) time on a PARBS with O(n ^1+? ) processors for any fixed 8 > 0.

2. The all‐pairs shortest paths and the minimum spanning tree problems can be solved in O (log n) time on a PARBS with O(n ^3+? ) processors for any fixed 8 > 0.

     

Centroidal Voronoi tessellation‐based finite element superconvergence

In this article, a finding on finite element superconvergence is reported. The Laplacian operator with Dirichlet boundary condition is considered. The linear finite element solutions have an O(h^2+α)(α≈0.5)‐superconvergence in l₂ norm at nodes on an almost equilateral triangular mesh generated based on centroidal Voronoi tessellation, for an arbitrary 2D bounded domain. Extensive numerical examples are presented to demonstrate the superconvergence property. Copyright © 2008 John Wiley & Sons, Ltd.     

Role of Redox‐Inactive Transition‐Metals in the Behavior of Cation‐Disordered Rocksalt Cathodes

Owing to the capacity boost from oxygen redox activities, Li‐rich cation‐disordered rocksalts (LRCDRS) represent a new class of promising high‐energy Li‐ion battery cathode materials. Redox‐inactive transition‐metal (TM) cations, typically d⁰ TM, are essential in the formation of rocksalt phases, however, their role in electrochemical performance and cathode stability is largely unknown. In the present study, the effect of two d⁰ TM (Nb⁵⁺ and Ti⁴⁺) is systematically compared on the redox chemistry of Mn‐based model LRCDRS cathodes, namely Li_1.3Nb_0.3Mn_0.4O₂ (LNMO), Li_1.25Nb_0.15Ti_0.2Mn_0.4O₂ (LNTMO), and Li_1.2Ti_0.4Mn_0.4O₂ (LTMO). Although electrochemically inactive, d⁰ TM serves as a modulator for oxygen redox, with Nb⁵⁺ significantly enhancing initial charge storage contribution from oxygen redox. Further studies using differential electrochemical mass spectroscopy and resonant inelastic X‐ray scattering reveal that Ti⁴⁺ is better in stabilizing the oxidized oxygen anions (O^n?, 0 < n < 2), leading to a more reversible O redox process with less oxygen gas release. As a result, much improved chemical, structural and cycling stabilities are achieved on LTMO. Detailed evaluation on the effect of d⁰ TM on degradation mechanism further suggests that proper design of redox‐inactive TM cations provides an important avenue to balanced capacity and stability in this newer class of cathode materials.     

Fast parallel algorithms for the Maximum Empty Rectangle problem

We present efficient parallel algorithms for the maximum empty rectangle problem in this paper. On crew pram, we solve the area version of this problem inO(log ² n) time usingO(nlogn) processors. The perimeter version of this problem is solved inO(logn) time usingO(nlog ² n) processors. On erew pram, we solve both the problems inO(logn) time usingO(n ²/logn) processors. We also present anO(logn) time algorithm on a mesh-of-trees architecture.     

Mapping stereo and image matching algorithms onto parallel architectures

In this paper we present parallel implementations of two vision tasks; stereo matching and image matching. Linear features are used as matching primitives. These implementations are performed on a fixed size mesh array and achieve processor-time optimal performance. For stereo matching, we proposeO(Nn ³/P ²) time algorithm on aP ×P processor mesh array, whereN is the number of line segments in one image,n is the number of line segments in a window determined by the object size, andP ⩽n. The sequential algorithm takesO(Nn ³) time. For image matching, a partitioned parallel implementation is developed.O[((nm/P ²) +P)nm] time performance is achieved on aP ×P processor mesh array, whereP ² ⩽nm. This leads to a processor-time optimal solution forP ⩽ (nm)^1/3. This research was supported in part bynsf under grantiri-9145810 and in part bydarpa andafosr contracts F-49260-89-C-0126 and F-49620-90-C-0078.     

A wideband fast multipole algorithm for two‐dimensional volume integral equations

This paper describes a wideband fast multipole algorithm (FMA) for the computation of two‐dimensional volume integral equations. Our previous paper presented the wideband FMA by switching between the diagonal and non‐diagonal forms according to cell size and required accuracy. In order to improve the efficiency of the algorithm, we use interpolation and filtering techniques. Moreover, we introduce a simple and efficient way to store sequences of the special functions and their discrete Fourier transforms. Numerical examples show that the computational and memory complexities are reduced from O(N²) to O(N), where N is the number of square elements followed by the discretization of the volume integral equations. The computation results show very good agreement with the analytical solutions. We present some numerical results for the computation of scattering from a cylindrical object with sharp edges and a Gaussian‐like inhomogeneous cylinder. Copyright © 2008 John Wiley & Sons, Ltd.     

Surface wavelets: a multiresolution signal processing tool for 3D computational modelling

In this paper, we provide an introduction to wavelet representations for complex surfaces (surface wavelets), with the goal of demonstrating their potential for 3D scientific and engineering computing applications. Surface wavelets were originally developed for representing geometric objects in a multiresolution format in computer graphics. These wavelets share all of the major advantages of conventional wavelets, in that they provide an analysis tool for studying data, functions and operators at different scales. However, unlike conventional wavelets, which are restricted to uniform grids, surface wavelets have the power to perform signal processing operations on complex meshes, such as those encountered in finite element modelling. This motivates the study of surface wavelets as an efficient representation for the modelling and simulation of physical processes. We show how surface wavelets can be applied to partial differential equations, stated either in integral form or in differential form. We analyse and implement the wavelet approach for a model 3D potential problem using a surface wavelet basis with linear interpolating properties. We show both theoretically and experimentally that an O(h) convergence rate, h_n being the mesh size, can be obtained by retaining only O((logN) ^7/2N) entries in the discrete operator matrix, where N is the number of unknowns. The principles described here may also be extended to volumetric discretizations. Copyright © 2001 John Wiley & Sons, Ltd.     

Use of Thermal Ionization Mass Spectrometry for Measuring the Oxygen Isotope Ratio in Uranium Oxides1,2

A method was developed for precise determination of oxygen isotope ratios in uranium oxides. Thermal ionisation mass spectrometry (TIMS) was used for direct measurements of n(U^{1 8}O⁺)/n(U^{1 6}O⁺) using the molecular species UO⁺. Suitable sample handling and filament preparation techniques were developed in order to obtain reproducible results and avoid oxygen contamination. The actual measurements were performed using the total evaporation method. By this technique the achieved precision for n(U^{1 8}O⁺)/n(U^{1 6}O⁺) measurements was in the range of 0.04%. The peak jump techniques was also used for measurements, and the results of both techniques are discussed. The TIMS measurement are verified by comparative measurements using secondary ion mass spectrometry (SIMS).     

A fast multi‐level convolution boundary element method for transient diffusion problems

A new algorithm is developed to evaluate the time convolution integrals that are associated with boundary element methods (BEM) for transient diffusion. This approach, which is based upon the multi‐level multi‐integration concepts of Brandt and Lubrecht, provides a fast, accurate and memory efficient time domain method for this entire class of problems. Conventional BEM approaches result in operation counts of order O(N²) for the discrete time convolution over N time steps. Here we focus on the formulation for linear problems of transient heat diffusion and demonstrate reduced computational complexity to order O(N^3/2) for three two‐dimensional model problems using the multi‐level convolution BEM. Memory requirements are also significantly reduced, while maintaining the same level of accuracy as the conventional time domain BEM approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Approximate Bayesian methods

Summary This paper develops asymptotic expansions for the ratios of integrals that occur in Bayesian analysis: for example, the posterior mean. The first term omitted isO(n ⁻²) and it is shown how the termO(n ⁻¹) can be of importance.     

Diluted Oxide Interfaces with Tunable Ground States

The metallic interface between two oxide insulators, such as LaAlO₃/SrTiO₃ (LAO/STO), provides new opportunities for electronics and spintronics. However, due to the presence of multiple orbital populations, tailoring the interfacial properties such as the ground state and metal‐insulator transitions remains challenging. Here, an unforeseen tunability of the phase diagram of LAO/STO is reported by alloying LAO with a ferromagnetic LaMnO₃ insulator without forming lattice disorder and at the same time without changing the polarity of the system. By increasing the Mn‐doping level, x, of LaAl_1?xMn_xO₃/STO (0 ≤ x ≤ 1), the interface undergoes a Lifshitz transition at x = 0.225 across a critical carrier density of n_c = 2.8 × 10¹³ cm^?2, where a peak T_SC ≈255 mK of superconducting transition temperature is observed. Moreover, the LaAl_1?xMn_xO₃ turns ferromagnetic at x ≥ 0.25. Remarkably, at x = 0.3, where the metallic interface is populated by only d_xy electrons and just before it becomes insulating, a same device with both signatures of superconductivity and clear anomalous Hall effect (7.6 × 10¹² cm^?2 < n_s ≤ 1.1 × 10¹³ cm^?2) is achieved reproducibly. This provides a unique and effective way to tailor oxide interfaces for designing on‐demand electronic and spintronic devices.     

A two‐grid method for expanded mixed finite‐element solution of semilinear reaction–diffusion equations

We present a scheme for solving two‐dimensional semilinear reaction–diffusion equations using an expanded mixed finite element method. To linearize the mixed‐method equations, we use a two‐grid algorithm based on the Newton iteration method. The solution of a non‐linear system on the fine space is reduced to the solution of two small (one linear and one non‐linear) systems on the coarse space and a linear system on the fine space. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h^1/3). As a result, solving such a large class of non‐linear equation will not be much more difficult than solving one single linearized equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Parameter‐free,weak imposition of Dirichlet boundary conditions and coupling of trimmed and non‐conforming patches

We present a parameter‐free domain sewing approach for low‐order as well as high‐order finite elements. Its final form contains only primal unknowns; that is, the approach does not introduce additional unknowns at the interface. Additionally, it does not involve problem‐dependent parameters, which require an estimation. The presented approach is symmetry preserving; that is, the resulting discrete form of an elliptic equation will remain symmetric and positive definite. It preserves the order of the underlying discretization, and we demonstrate high‐order accuracy for problems of non‐matching discretizations concerning the mesh size h as well as the polynomial degree of the order of discretization p. We also demonstrate how the method may be used to model material interfaces, which may be curved and for which the interface does not coincide with the underlying mesh. This novel approach is presented in the context of the p‐version and B‐spline version of the finite cell method, an embedded domain method of high order, and compared with more classical methods such as the penalty method or Nitsche's method. Copyright © 2014 John Wiley & Sons, Ltd.     

Model simplification of stable discrete‐time systems by the squared magnitude Padé approximation

Abstract

A squared magnitude Padé approximation technique is presented for the model simplification of stable discrete‐time systems. The simplification is started from the squared magnitude function M(e^jTω ) =G(e^jTω )G(e^–jTω ), where G(z) is the z‐transfer function of a given high order discrete‐time system. The method is fully computer‐oriented and leads to a satisfactory approximation while preserving stability and minimum‐phase characteristics.     

Fast time‐domain simulation for large‐order linear time‐invariant state space systems

Time‐domain simulation is essential for both analysis and design of complex systems. Unfortunately, high model fidelity leads to large system size and bandwidths, often causing excessive computation and memory saturation. In response we develop an efficient scheme for large‐order linear time‐invariant systems. First, the A matrix is block diagonalized. Then, subsystems of manageable dimensions and bandwidth are formed, allowing multiple sampling rates. Each subsystem is then discretized using a O(n_s) scheme, where n_s is the number of states. Subsequently, a sparse matrix O(n_s) discrete‐time system solver is employed to compute the history of the state and output. Finally, the response of the original system is obtained by superposition. In practical engineering applications, closing feedback loops and cascading filters can hinder the efficient use of the simulation scheme. Solutions to these problems are addressed in the paper. The simulation scheme, implemented as a MATLAB function fastlsim, is benchmarked against the standard LTI system simulator lsim and is shown to be superior for medium to large systems. The algorithm scales close to O(n) for a set of benchmarked systems. Simulation of a high‐fidelity model of (n_s ≈ 2200) the Space Interferometry Mission spacecraft illustrates real world application of the method. Copyright © 2005 John Wiley & Sons, Ltd.     

Fracture and fragmentation of simplicial finite element meshes using graphs

An approach for the topological representation of simplicial finite element meshes as graphs is presented. It is shown that by using a graph, the topological changes induced by fracture reduce to a few, local kernel operations. The performance of the graph representation is demonstrated and analyzed, using as reference the three‐dimensional fracture algorithm by Pandolfi and Ortiz (Eng. Comput. 1998; 14 (4):287–308). It is shown that the graph representation initializes in O(N) time and fractures in O(N) time, while the reference implementation requires O(N) time to initialize and O(N) time to fracture, where N_E is the number of elements in the mesh and N_I is the number of interfaces to fracture. Copyright © 2007 John Wiley & Sons, Ltd.     

Universal meshes: A method for triangulating planar curved domains immersed in nonconforming meshes

We introduce a new method to triangulate planar, curved domains that transforms a specific collection of triangles in a background mesh to conform to the boundary. In the process, no new vertices are introduced, and connectivities of triangles are left unaltered. The method relies on a novel way of parameterizing an immersed boundary over a collection of nearby edges with its closest point projection. To guarantee its robustness, we require that the domain be C²‐regular, the background mesh be sufficiently refined near the boundary, and that specific angles in triangles near the boundary be strictly acute. The method can render both straight‐edged and curvilinear triangulations for the immersed domain. The latter includes curved triangles that conform exactly to the immersed boundary, and ones constructed with isoparametric mappings to interpolate the boundary at select points. High‐order finite elements constructed over these curved triangles achieve optimal accuracy, which has customarily proven difficult in numerical schemes that adopt nonconforming meshes. Aside from serving as a quick and simple tool for meshing planar curved domains with complex shapes, the method provides significant advantages for simulating problems with moving boundaries and in numerical schemes that require iterating over the geometry of domains. With no conformity requirements, the same background mesh can be adopted to triangulate a large family of domains immersed in it, including ones realized over several updates during the coarse of simulating problems with moving boundaries. We term such a background mesh as a universal mesh for the family of domains it can be used to triangulate. Universal meshes hence facilitate a framework for finite element calculations over evolving domains while using only fixed background meshes. Furthermore, because the evolving geometry can be approximated with any desired order, numerical solutions can be computed with high‐order accuracy. We present demonstrative examples using universal meshes to simulate the interaction of rigid bodies with Stokesian fluids. Copyright © 2014 John Wiley & Sons, Ltd.     

Phosphorus Incorporation into Co9S8 Nanocages for Highly Efficient Oxygen Evolution Catalysis

The improvement of activity of electrocatalysts lies in the increment of the density of active sites or the enhancement of intrinsic activity of each active site. A common strategy to realize dual active sites is the use of bimetal compound catalysts, where each metal atom contributes one active site. In this work, a new concept is presented to realize dual active sites with tunable electron densities in monometal compound catalysts. Dual Co²⁺ tetrahedral (Co²⁺(T_d)) and Co³⁺ octahedral (Co³⁺(O_h)) coordination active sites are developed and adjustable electron densities on the Co²⁺(T_d) and Co³⁺(O_h) are further achieved by phosphorus incorporation (P‐Co₉S₈). The experimental results and density functional theory calculations show that the nonmetal P doping can systematically modulate charge density of Co²⁺(T_d) and Co³⁺(O_h) in P‐Co₉S₈ and simultaneously improve the electrical conductivity of Co₉S₈, which substantially enhances oxygen evolution reaction performance of P‐Co₉S₈.