Moving grids for magnetic reconnection via Newton–Krylov methods

This paper presents a set of computationally efficient, adaptive grids for magnetic reconnection phenomenon where the current density can develop large gradients in the reconnection region. Four-field extended MagnetoHydroDynamics (MHD) equations with hyperviscosity terms are transformed so that the curvilinear coordinates replace the Cartesian coordinates as the independent variables, and moving grids' velocities are also considered in this transformed system as a part of interpolating the physical solutions from the old grid to the new grid as time advances. The curvilinear coordinates derived from the current density through the Monge–Kantorovich (MK) optimization approach help to reduce the resolution requirements during the computation.     

Large displacement analysis of plates by a stress-based finite element approach

A mixed-hybrid incremental variational formulation, involving orthogonal rigid rotations and a symmetric stretch tensor, is proposed for finite deformation analysis of thin plates and shells. Isoparametric eight-noded elements are based upon the Kirchhoff-Love hypotheses, the assumption of plane stress, and the moderately large rotations of Von Karman plate theory. Semilinear elastic isotropic material properties are assumed, and the right polar decomposition of the deformation gradient is used. The symmetrized Biot-Luré (Jaumann) stress measure gives a unique complementary energy density and a set of variational principles with a priori satisfaction of linear momentum balance, a posteriori angular momentum balance, and interelement traction reciprocity by means of Lagrange multipliers. The incremental modified Newton-Raphson solution procedure is generated by a truncated Taylor series expansion of the functional in a total Lagrangian formulation. The theory is applied to laterally loaded and buckled thin plates, and numerical results are compared with truncated series solutions.     

Optimal Mass Transport for Registration and Warping

Image registration is the process of establishing a common geometric reference frame between two or more image data sets possibly taken at different times. In this paper we present a method for computing elastic registration and warping maps based on the Monge–Kantorovich theory of optimal mass transport. This mass transport method has a number of important characteristics. First, it is parameter free. Moreover, it utilizes all of the grayscale data in both images, places the two images on equal footing and is symmetrical: the optimal mapping from image A to image B being the inverse of the optimal mapping from B to A. The method does not require that landmarks be specified, and the minimizer of the distance functional involved is unique; there are no other local minimizers. Finally, optimal transport naturally takes into account changes in density that result from changes in area or volume. Although the optimal transport method is certainly not appropriate for all registration and warping problems, this mass preservation property makes the Monge–Kantorovich approach quite useful for an interesting class of warping problems, as we show in this paper. Our method for finding the registration mapping is based on a partial differential equation approach to the minimization of the L ² Kantorovich–Wasserstein or Earth Mover's Distance under a mass preservation constraint. We show how this approach leads to practical algorithms, and demonstrate our method with a number of examples, including those from the medical field. We also extend this method to take into account changes in intensity, and show that it is well suited for applications such as image morphing.     

Continuous finite element schemes for a phase field model in two-layer fluid Bénard–Marangoni convection computations

In this article, we study a phase field model for a two-layer fluid where the temperature dependence of both the density (buoyancy forces) and the surface tension (Marangoni effects) is considered. The phase field model consisting of a modified Navier–Stokes equation, a Cahn–Hilliard phase field equation and an energy transport equation is derived through an energetic variational procedure. An appropriate variational form and a continuous finite element method are adopted to maintain the underlying energy law to its greatest extent. A few examples for Bénard–Marangoni convection in an Acetonitrile and n-Hexane two-layer fluid system heated from above will be computed to justify our phase field model and further show the good performance of our methods. In addition, an interesting experiment will be performed to show the competition between the Marangoni effects and the buoyancy forces.     

A spectral collocation method for a rotating Bose–Einstein condensation in optical lattices

We extend the study of spectral collocation methods (SCM) in Li et al. (2009) [1] for semilinear elliptic eigenvalue problems to that for a rotating Bose–Einstein condensation (BEC) and a rotating BEC in optical lattices. We apply the Lagrange interpolants using the Legendre–Gauss–Lobatto points to derive error bounds for the SCM. The optimal error bounds are derived for both H¹-norm and L²-norm. Extensive numerical experiments on a rotating Bose–Einstein condensation and a rotating BEC in optical lattices are reported. Our numerical results show that the convergence rate of the SCM is exponential, and is independent of the collocation points we choose.     

Computing the velocity of a rotating flow

To compute the time-dependent flow of a rotating incompressible fluid we consider the velocity–vorticity formulation of the Navier–Stokes equations in cylindrical coordinates. In the numerical method employed the velocity field at each time-step is found as the least squares solution of an overdetermined system of linear equations, Ax=b. We consider how to compute x using the preconditioned conjugate gradient algorithm for least squares (PCGLS) on a distributed parallel computer. The various aspects of using a parallel computer are discussed, and results for a wide range of parallel computers are presented. The parallel speed-up depends on the architecture but is typically about 80% of the number of processors used.     

Least-squares finite element formulation for shear-deformable shells

We present a least-squares based finite element formulation for the numerical analysis of shear-deformable shell structures. The variational problem is obtained by minimizing the least-squares functional, defined as the sum of the squares of the shell equilibrium equations residuals measured in suitable norms of Hilbert spaces. The use of least-squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise, i.e. stability requirements such as inf–sup conditions never arise. The proposed formulation retains the generalized displacements and stress resultants as independent variables and, in view of the nature of the variational setting upon which the finite element model is built, allows for equal-order interpolation. A p-type hierarchical basis is used to construct the discrete finite element model based on the least-squares formulation. Exponentially fast decay of the least-squares functional is verified for increasing order of the modal expansions. Several well established benchmark problems are solved to demonstrate the predictive capability of the least-squares based shell elements. Shell elements based on this formulation are shown to be effective in both membrane- and bending-dominated states.     

Electron momentum density distribution in Cd3P2

A study of electron momentum density distribution in Cd₃P₂ is reported in this work. The measurement of Compton profile is carried out on a polycrystalline sample using 59.54 keV gamma-rays emanating from an ²⁴¹Am radioisotope. The theoretical calculations are performed using linear combination of atomic orbitals method following the Hartree–Fock and a posteriori density functional theories. The spherically averaged theoretical Compton profiles are in good agreement with the measurement. The best agreement is, however, shown by the Hartree–Fock scheme. Simple ionic model calculations for a number of configurations ₃(Cd+x)₂(P−3x/2) (0.0?x?2.0 in step of 0.2) are also performed utilizing free atom profiles. The ionic model supports transfer of 2.0 electrons per Cd atom from 5s state to 3p state of P.     

Multiscale temporal integration

We study the application of the variational multiscale method to the problem of temporal integration, with the final goal of designing integration schemes that go beyond the classical notion of upwinding.We develop a formulation based on a mixed hybrid finite element method, where the fine scale mode problems automatically decouple at the element level without the need to resort to a localization assumption. We give general orthogonality conditions for the trial and test spaces that allow to construct hierarchical p methods.We test some simple ideas for the modeling of the unresolved scales. The resulting algorithms are analyzed using classical analytical measures.     

Variational density matrix optimization using semidefinite programming

We discuss how semidefinite programming can be used to determine the second-order density matrix directly through a variational optimization. We show how the problem of characterizing a physical or N-representable density matrix leads to matrix-positivity constraints on the density matrix. We then formulate this in a standard semidefinite programming form, after which two interior point methods are discussed to solve the SDP. As an example we show the results of an application of the method on the isoelectronic series of Beryllium.     

Parametric optimal control problems with weighted <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-norm in the cost function

In the paper, an optimal control problem with weighted L ₁-norm in the cost function is studied. The problem is considered as a parametric problem where L ₁-norm weight ratio is treated as a parameter. We analyze the dependence of solution to the mentioned optimization problem on values of the parameter. A theorem that describes properties of the solution under small parameter perturbations is proved. Differential properties of the solution are investigated. Under assumption that a solution to unperturbed problem is known, rules for construction of solutions to perturbed optimization problems are given.     

Multi-objective shape optimization using ant colony coupled computational fluid dynamics solver

An adaptation of a parametric ant colony optimization (ACO) to multi-objective optimization (MOO) is presented in this paper. In this algorithm (here onwards called MACO) the concept of MOO is achieved using the reference point (or goal vector) optimization strategy by applying scalarization. This method translates the multi-objective optimization problem to a single objective optimization problem. The ranking is done using ?-dominance with modified L_p metric strategy. The minimization of the maximum distance from the goal vector drives the solution close to the goal vector. A few validation test cases with multi-objectives have been demonstrated. MACO was found to out perform R-NSGA-II for the test cases considered. This algorithm was then integrated with a meshless computational fluid dynamics (CFD) solver to perform aerodynamic shape optimization of an airfoil. The algorithm was successful in reaching the optimum solutions near to the goal vector on one hand. On the other hand the algorithm converged to an optimum outside the boundary specified by the user for the control variables. These make MACO a good contender for multi-objective shape optimization problems.     

Study of variational inequality and equality formulations for elastostatic frictional contact problems

Summary An overview ofvariational inequality andvariational equality formulations for frictionless contact and frictional contact problems is provided. The aim is to discuss the state-of-the-art in these two formulations and clearly point out their advantages and disadvantages in terms of mathematical completeness and practicality. Various terms required to describe the contact configuration are defined.Unilateral contact law and classical Coulomb’s friction law are given.Elastostatic frictional contact boundary value problem is defined. General two-dimensional frictionless and frictional contact formulations for elastostatic problems are investigated. An example problem of a two bar truss-rigid wall frictionless contact system is formulated as an optimization problem based on the variational inequality approach. The problem is solved in a closed form using the Karush-Kuhn-Tucker (KKT) optimality conditions. The example problem is also formulated as a frictional contact system. It is solved in the closed form using a new two-phase analytical procedure. The procedure avoids use of the incremental/iterative techniques and user defined parameters required in a typical implementation based on the variational equality formulation. Numerical solutions for the frictionless and frictional contact problems are compared with the results obtained by using a general-purpose finite element program ANSYS (that uses variational equality formulation). ANSYS results match reasonably well with the solutions of KKT optimality conditions for the frictionless contact problem and the two-phase procedure for the frictional contact problem. The validity of the analytical formulation for frictional contact problems (with one contacting node) is verified. Thevariational equality formulation for frictionless and frictional, contact problems is also studied in detail. The incremental/iterative Newton-Raphson scheme incorporating the penalty approach is utilized. Studies are conducted to provide insights for the numerical solution techniques. Based on the present study it is concluded that alternate formulations and computational procedures need to be developed for analysis of frictional contact problems.     

Central Limit Theorems for Stochastic Optimization Algorithms Using Infinitesimal Perturbation Analysis

Central limit theorems are obtained for the perturbation analysis Robbins-Monro single run (PARMSR) optimization algorithm, with updates either after every L customers or after every busy period, in the context of the optimization of a GI/GI/1 queue. The PARMSR algorithm is a stochastic approximation (SA) method for the optimization of infinite-horizon models. It is shown that the convergence rate and the asymptotic variance constant of the optimization algorithm, as a function of the total computing budget (i.e., total number of customers), are the same for both updating methods, and independent of L, provided that the step sizes of SA are chosen in the (asymptotically) optimal way for each method.     

Non-linear minimax optimization as a sequence of least pth optimization with finite values of p

Following developments in non-linear least pth optimization by the authors it is possible to derive two new methods of non-linear minimax optimization. Unlike the Polya algorithm in which a sequence of least pth optimizations as p→∞ is taken our methods do not require the value of p to tend to infinity. Instead we construct a sequence of least pth optimization problems with a finite value of p. It is shown that this sequence will converge to a minimax solution. Two interesting minimax problems were constructed which illustrate some of the theoretical ideas. Further numerical evidence is presented on the modelling of a fourth-order system by a second-order model with values of p varying between 2 and 10 000.     

Flow analysis of a ribbed helix lip seal with consideration of fluid–structure interaction

Ribbed helix lip seals for rotating shafts have been widely used to retain oil and exclude contaminants in many applications throughout the industry. The objective of this study is to better understand the basic flow behavior associated with the pumping process of a ribbed helix lip seal. The theoretical model consists of a flow analysis of the lubricating film of the hydraulic fluid in conjunction with a stress analysis of the lip seal distortion. The complicated mechanical interaction between the oil flow and rubber deformation was simulated using a coupled fluid–structure approach implemented in a commercial computational fluid dynamics (CFD) code ESI-CFD, ACE+®. The flow characteristics and rubber deformation around a ribbed helix lip seal were fully resolved in a pumping-rate test environment, where both air and oil sides were filled with oil initially. The three-dimensional pressure field solved by the model via the coupled flow-stress analysis was compared with the predictions obtained from the model via the nondeformable rubber assumption to elucidate the significant effect of the fluid–structure interaction on accurate simulation of the oil pumping behavior. In the rotating speed ranging from 1000 to 6000 rpm, both measured and calculated pumping rates increase with the shaft speed for a ribbed helix lip seal. As compared to the baseline case, calculations with considering the fluid–structure interaction at higher rotary speeds can result in thicker oil films, and in turn produce greater pumping rates.     

Experimental investigation and thermodynamic modeling of the Cu–Mn–Ni system

The phase equilibria of the ternary Cu–Mn–Ni system in the region above 40 at.% Mn at 600 ^°C were investigated by means of optical microscopy, X-ray diffraction, scanning electron microscopy with energy dispersive X-ray spectroscopy and electron probe microanalysis. The isothermal section of the Cu–Mn–Ni system at 600 ^°C consists of 4 two-phase regions (cbcc_A12 +fcc_A1, cub_A13 +fcc_A1, cbcc_A12 + cub_A13, L₁₀$L{1}_0$ +fcc_A1) and 1 three-phase region (cbcc_A12 +cub_A13 +fcc_A1). The disordered fcc_A1 phase exhibits a large continuous solution between γ$\gamma$(Cu,Ni) and γ$\gamma$(Mn). The L₁₀$L{1}_0$ phase only equilibrates with fcc_A1 phase, and the solubility of Cu in L₁₀$L{1}_0$ phase is up to 16 at.%. A thermodynamic modeling for this system was performed by considering reliable literature data and incorporating the current experimental results. A self-consistent set of thermodynamic parameters was obtained, and the calculated results show a general agreement with the experimental data.     

并行马尔可夫随机场计算变分光流方法

目的 基于马尔可夫随机场（MRF）的变分光流计算是一种较为鲁棒的光流计算方法,但是计算效率很低。置信传播算法（BP） 是一种针对MRF较为高效的全局优化算法。本文提出一种MRF变分光流计算模型并采用并行BP方法实现,极大提高计算效率。方法 提出的MRF变分光流计算模型中的数据项采用了Horn等人根据灰度守恒假设得到的光流基本约束方程,并采用非平方惩罚函数进行调整以平滑边界影响。为在CUDA平台上实现高效并行处理,本文提出了一种优化的基于置信传播的MRF并行光流计算方法。该优化方法在采用置信传播最小化MRF光流能量函数时,采用了一种4层的3维网络结构进行并行计算,每层对应MRF4邻域模型中的一个方向的信息传播,同时在每层中为每个像素分配多个线程采用并行降维法计算所要传递的信息,大大降低单线程计算负荷,大幅度提高计算效率。结果 采用旋转小球图像序列进行实验,计算效率提高314倍;采用旋转小球、Yosemite山谷和RubberWhale 3种不同图像序列,与Horn算法、Weickert算法、Hossen并行Lucas算法、Grauer-Gray并行MRF算法进行对比实验,本文方法得到最低的平均端点误差（AEE）,分别为0.13、0.55和0.34。结论 本文提出了一种新的MRF光流计算模型,并在CUDA平台上实现了并行优化计算。实验结果表明,本文提出的并行计算方法在保持计算精度的同时极大提高了计算效率。本文方法对内存需求巨大,在处理高分辨率图像时,限制了采样点数,难以计算大位移。     

The optimal location of replicas in a network using a READ-ONE-WRITE-ALL policy

We consider the problem of locating replicas in a network to minimize communications costs. Under the assumption that the read-one-write-all policy is used to ensure data consistency, an optimization problem is formulated in which the cost function estimates the total communications costs. The paper concentrates on the study of the optimal communications cost as a function of the ratio between the frequency of the read and write operations. The problem is reformulated as a zero-one linear programming problem, and its connection to the p-median problem is explained. The general problem is proved to be NP-complete. For path graphs a dynamic programming algorithm for the problem is presented. Received: May 1993 / Accepted: June 2001