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.     

Stokes flow around cylinders in a bounded two‐dimensional domain using multipole‐accelerated boundary element methods

The multipole technique has recently received attention in the field of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from O(N²) (iterative solvers) or O(N³) (gaussian elimination) to O(N log N) or O(N), where N is the number of nodes in the discretized system. Such a reduction in the growth of the calculation time and data storage is crucial in applications where N is large, such as when modelling the macroscopic behaviour of suspensions of particles. In such cases, a minimum of 1000 particles is needed to obtain statistically meaningful results, leading to systems with N of the order of 10 000 for the smallest problems. When only boundary velocities are known, the indirect boundary element formulation for Stokes flow results in Fredholm equations of the second kind, which generally produce a well‐posed set of equations when discretized, a necessary requirement for iterative solution methods. The direct boundary element formulation, on the other hand, results in Fredholm equations of the first kind, which, upon discretization, produce ill‐conditioned systems of equations. The model system here is a two‐dimensional wide‐gap couette viscometer, where particles are suspended in the fluid between the cylinders. This is a typical system that is efficiently modelled using boundary element method simulations. The multipolar technique is applied to both direct and indirect formulations. It is found that the indirect approach is sufficiently well‐conditioned to allow the use of fast multipole methods. The direct approach results in severe ill‐conditioning, to a point where application of the multipole method leads to non‐convergence of the solution iteration. Copyright © 1999 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.

     

Adaptive modification of objective function for lagrange multiplier method in constrained optimization problems

Abstract

This paper proposes an adaptive modification method to transform the objective function with a stationary point to an objective function with a minima point, such that search methods can be used to find the stationary point. The stationary point can be a saddle point in addition to a minima or a maxima. Therefore, this method can be used to transform a constrained optimization by applying Lagrange multipliers to an unconstrained optimization problem. A quadratic term, ½(X — X_N ) ^{T D} (X—X_N ), is added to the original function such that the modified function is a minima at the Newton point X_N of the original function, where D is a diagonal matrix to make the modified Hessian matrix H_O + D positive definite, and H_O is the original Hessian matrix at the initial point X_O .     

Fast multipole method as an efficient solver for 2D elastic wave surface integral equations

The fast multipole method (FMM) is very efficient in solving integral equations. This paper applies the method to solve large solid-solid boundary integral equations for elastic waves in two dimensions. The scattering problem is first formulated with the boundary element method. FMM is then introduced to expedite the solution process. By using the FMM technique, the number of floating-point operations of the matrix-vector multiplication in a standard conjugate gradient algorithm is reduced from O(N ²) to O(N ^1.5), where N is the number of unknowns. The matrix-filling time and the memory requirement are also of the order N ^1.5. The computational complexity of the algorithm is further reduced to O(N ^4/3) by using a ray propagation technique. Numerical results are given to show the accuracy and efficiency of FMM compared to the boundary element method with dense matrix.     

Parallel multipole implementation of the generalized Helmholtz decomposition for solving viscous flow problems

The evaluation of a domain integral is the dominant bottleneck in the numerical solution of viscous flow problems by vorticity methods, which otherwise demonstrate distinct advantages over primitive variable methods. By applying a Barnes–Hut multipole acceleration technique, the operation count for the integration is reduced from O(N²) to O(NlogN), while the memory requirements are reduced from O(N²) to O(N). The algorithmic parameters that are necessary to achieve such scaling are described. The parallelization of the algorithm is crucial if the method is to be applied to realistic problems. A parallelization procedure which achieves almost perfect scaling is shown. Finally, numerical experiments on a driven cavity benchmark problem are performed. The actual increase in performance and reduction in storage requirements match theoretical predictions well, and the scalability of the procedure is very good. Copyright © 2003 John Wiley Sons, Ltd.     

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.     

New splitting algorithms for geometric transformations of digital images and their error analysis

For digital images and patterns under the nonlinear geometric transformation, T: (ξ, η) → (x, y), this study develops the splitting algorithms (i.e., the pixel‐division algorithms) that divide a 2D pixel into N × N subpixels, where N is a positive integer chosen as N = 2 ^k(k ≥ 0) in practical computations. When the true intensity values of pixels are known, this method makes it easy to compute the true intensity errors. As true intensity values are often unknown, the proposed approaches can compute the sequential intensity errors based on the differences between the two approximate intensity values at N and N/2. This article proposes the new splitting–shooting method, new splitting integrating method, and their combination. These methods approximate results show that the true errors of pixel intensity are O(H), where H is the pixel size. Note that the algorithms in this article do not produce any sequential errors as N ≥ N₀, where N₀ (≥2) is an integer independent of N and H. This is a distinctive feature compared to our previous papers on this subject. The other distinct feature of this article is that the true error bound O(H) is well suited to images with all kinds of discontinuous intensity, including scattered pixels. © 2011 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 21, 323–335, 2011     

A spectral multipole method for efficient solution of large-scale boundary element models in elastostatics

In this paper we introduce a method to reduce the solution cost for Boundary Element (BE) models from O(N³)operations to O(N²logN) operations (where N is the number of elements in the model). Previous attempts to achieve such an improvement in efficiency have been restricted in their applicability to problems with regular geometries defined on a uniform mesh. We have developed the Spectral Multipole Method (SMM) which can be used not only for problems with arbitrary geometries but also with a variety of element types. The memory necessary to store the required influence coefficients for the spectral multipole method is O(N) whereas the memory required for the traditional Boundary Element method is O(N²). We demonstrate the savings in computational speed and fast memory requirements in some numerical examples. We have established that the break-even point for the method can be as low as 500 elements, which implies that the method is not only suitable for extremely large-scale problems, but that it also provides a useful bridge between the small-scale and large-scale problems. We also demonstrate the performance of the multipole algorithm on the solution of large-scale granular assembly models. The large-scale BE capacity provided by this algorithm will not only prove to be useful in large macroscopic models but it will also make it possible to model microscopic damage processes that form the fundamental mechanisms in plastic flow and brittle fracture.     

Non‐singular boundary integral formulations for plane interior potential problems

In this article, a non‐singular formulation of the boundary integral equation is developed to solve smooth and non‐smooth interior potential problems in two dimensions. The subtracting and adding‐back technique is used to regularize the singularity of Green's function and to simplify the calculation of the normal derivative of Green's function. After that, a global numerical integration is directly applied at the boundary, and those integration points are also taken as collocation points to simplify the algorithm of computation. The result indicates that this simple method gives the convergence speed of order N ^?3 in the smooth boundary cases for both Dirichlet and mix‐type problems. For the non‐smooth cases, the convergence speed drops at O(N ^?1/2) for the Dirichlet problems. Copyright © 2001 John Wiley & Sons, Ltd.     

Supersinglets

Abstract

Supersinglets |S _N ^(d) 〉 are states of total spin zero of N particles of d levels. Some applications of the |S _N ^(N) 〉 and |S _N ⁽²⁾ 〉 states are described. The |S _N ^(N) 〉 states can be used to solve three problems that have no classical solution: The ‘N strangers,’ ‘secret sharing,’ and ‘liar detection’ problems. The |S _N ⁽²⁾ 〉 (with N even) states can be used to encode qubits in decoherence-free subspaces.     

Sweepline algorithm for unstructured-grid generation on two-dimensional non-convex domains

The sweepline algorithm has been modified to triangulate a set of nodes on a non-convex domain. The new method is based on a fragile sweepline which is split and rejoined in a manner which ensures that boundaries are preserved. Both the original and the modified algorithms are introduced using simple examples. It is shown that a set of N nodes is triangulated in O(N log N) time. The validity of the new method has been confirmed through testing on a variety of problems, and two test cases are presented in this paper.     

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.     

Simulation of the Processes of Formation and Dissociation of Neutral Particles in Air Plasma: Kinetics of Neutral Components

The results are given of calculations of the concentration of basic neutral particles and of the energy balance in an air plasma in the positive column of a d.c. discharge at pressures of 30 to 300 Pa and currents of 20 to 110 mA. The calculations are based on a combined solution of the Boltzmann equation; equations of vibrational kinetics for the ground states of molecules of N₂, O₂, and NO; and of the equations of chemical kinetics for these molecules, their excited states, and dissociation products. The results are compared with the measured concentrations of O(³ P) atoms and NO molecules and with the effective vibrational temperatures of N₂(X ¹⁺ _g ). The measured values of gas temperatures and temperatures of the reactor wall are used to find the fractions of energy transforming to heat in the plasma volume, which are compared with the calculation results.     

Replica time integrators

This paper is concerned with the classical problem of wave propagation in discrete models of nonuniform spatial resolution. We develop a new class of Replica Time Integrators (RTIs) that permit the two‐way transmission of thermal phonons across mesh interfaces. This two‐way transmissibility is accomplished by representing the state of the coarse regions by means of replica ensembles, consisting of collections of identical copies of the coarse regions. In dimension d, RTIs afford an O(n^d) speed‐up factor in sequential mode, and O(n^{d + 1}) in parallel, over regions that are coarsened n‐fold. In this work, we restrict ourselves to the solution of the 3d continuous wave equation, for both linear and non‐linear materials. By a combination of phase‐error analysis and numerical testing, we show that RTIs are convergent and result in exact two‐way transmissibility at the Courant–Friedrichs–Lewy limit for any angle of incidence. In this limit, RTIs allow step waves and high‐frequency harmonics to cross mesh interfaces in both directions without internal reflections or appreciable loss or addition of energy. The possible connections of RTIs with discrete‐to‐continuum approaches and, in particular, with the transition between molecular dynamics and continuum thermodynamics are also pointed to by way of future outlook. Copyright © 2011 John Wiley & Sons, Ltd.     

Attempts to increase the nuclear relaxation time of a3He gas at low temperatures

In an attempt to increase, at temperatures of a few kelvins, the relaxation time of a dilute³He gas enclosed in spherical Pyrex or pure silica bulbs 3 cm in diameter, we have studied the relaxing properties of various coatings, such as solid O₂ and solid N₂ and multiple coatings of H₂ on N₂, H₂ on O₂, and H₂ on H₂O. Contrary to naive expectations, thin films of solid oxygen proved to be very efficient coatings, although single molecules are paramagnetic. When multiple coatings are compared, they even provide the best undercoating to hydrogen films. The longest relaxation time we measured at 4.2 K was 8000 min, i.e., more than 5 days. We also found that Pyrex and pure silica, either bare or coated, give comparable results below 50 K. These experiments enabled us to measure the adsorption energy of³He on O₂, W(³He/O₂)=130±15 K, and of³He on N₂, W(³He/N₂)=200±50 K. In the absence of any theoretical calculation, these energies still need to be confirmed by other experimental techniques.Laboratory associated with the CNRS. U.A. 18.     

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.     

Cobalt Nanoparticles and Atomic Sites in Nitrogen‐Doped Carbon Frameworks for Highly Sensitive Sensing of Hydrogen Peroxide

In situ monitoring of hydrogen peroxide (H₂O₂) during its production process is needed. Here, an electrochemical H₂O₂ sensor with a wide linear current response range (concentration: 5 × 10^?8 to 5 × 10^?2 m ), a low detection limit (32.4 × 10^?9 m ), and a high sensitivity (568.47 µA mm ^?1 cm^?2) is developed. The electrocatalyst of the sensor consists of cobalt nanoparticles and atomic Co‐N_x moieties anchored on nitrogen doped carbon nanotube arrays (Co‐N/CNT), which is obtained through the pyrolysis of the sandwich‐like urea@ZIF‐67 complex. More cobalt nanoparticles and atomic Co‐N_x as active sites are exposed during pyrolysis, contributing to higher electrocatalytic activity. Moreover, a portable screen‐printed electrode sensor is constructed and demonstrated for rapidly detecting (cost ≈40 s) H₂O₂ produced in microbial fuel cells with only 50 µL solution. Both the synthesis strategy and sensor design can be applied to other energy and environmental fields.     

Printable Skin‐Driven Mechanoluminescence Devices via Nanodoped Matrix Modification

Mechanically driven light generation is an exciting and under‐exploited phenomenon with a variety of possible practical applications. However, the current driving mode of mechanoluminescence (ML) devices needs strong stimuli. Here, a flexible sensitive ML device via nanodopant elasticity modulus modification is introduced. Rigid ZnS:M²⁺(Mn/Cu)@Al₂O₃ microparticles are dispersed into soft poly(dimethylsiloxane) (PDMS) film and printed out to form flexible devices. For various flexible and sensitive scenes, SiO₂ nanoparticles are adopted to adjust the elasticity modulus of the PDMS matrix. The doped nanoparticles can concentrate stress to ZnS:M²⁺(Mn/Cu)@Al₂O₃ microparticles and achieve intense ML under weak stimuli of the moving skin. The printed nano‐/microparticle‐doped matrix film can achieve skin‐driven ML, which can be adopted to present fetching augmented animations expressions. The printable ML film, amenable to large areas, low‐cost manufacturing, and mechanical softness will be versatile on stress visualization, luminescent sensors, and open definitely new functional skin with novel augmented animations expressions, the photonic skin.