Higham (2002, IMA J. Numer. Anal., 22, 329–343) considered two types of nearest correlation matrix problems, namely the W-weighted case and the H-weighted case. While the W-weighted case has since been well studied to make several Lagrangian dual-based efficient numerical methods available, the H-weighted case remains numerically challenging. The difficulty of extending those methods from the W-weighted case to the H-weighted case lies in the fact that an analytic formula for the metric projection onto the positive semidefinite cone under the H-weight, unlike the case under the W-weight, is not available. In this paper we introduce an augmented Lagrangian dual-based approach that avoids the explicit computation of the metric projection under the H-weight. This method solves a sequence of unconstrained convex optimization problems, each of which can be efficiently solved by an inexact semismooth Newton method combined with the conjugate gradient method. Numerical experiments demonstrate that the augmented Lagrangian dual approach is not only fast but also robust.
lunes, 18 de enero de 2010
Dual-grid-based tree/cotree decomposition of higher-order interpolatory H([nabla][and], [Omega]) basis
This work extends the zeroth-order tree/cotree (TC) decomposition method into higher order (HO) interpolatory elements and develops the constraints operator required for the elimination of spurious solutions for general HO spectral basis. Earlier methods explicitly enforce the divergence condition that requires a mixed finite element (FE) formulation with both H1 and H([nabla][and]) expansions and involves repeated solutions of the Poisson equation. A recent approach, which avoids the mixed formulation and the Poisson problem, uses TC decomposition of edge DoF over the primal graph and construction of integration and gradient matrices. The approach is easily applied to HO hierarchical elements but becomes quite complex for HO spectral elements. In the presence of internal DoF, it is difficult to utilize the primal graph for an explicit decomposition of the spectral DoF. In contrast, this work utilizes the dual grid, resulting in an explicit decomposition of DoF and construction of constraint equations from a fixed element matrix. Thus, mixed formulation and the Poisson problems are avoided while eliminating the need for evaluation of integration and gradient matrices. The proposed constraints matrix is element-geometry independent and possesses an explicit sparsity formulation reducing the need for dynamic memory allocation. Numerical examples are included for verification. Copyright © 2010 John Wiley & Sons, Ltd.
FLEXMG: A new library of multigrid preconditioners for a spectral/finite element incompressible flow solver
A new library called FLEXMG has been developed for a spectral/finite element incompressible flow solver called SFELES. FLEXMG allows the use of various types of iterative solvers preconditioned by algebraic multigrid methods. Two families of algebraic multigrid preconditioners have been implemented, namely smooth aggregation-type and non-nested finite element-type. Unlike pure gridless multigrid, both of these families use the information contained in the initial fine mesh. A hierarchy of coarse meshes is also needed for the non-nested finite element-type multigrid so that our approaches can be considered as hybrid. Our aggregation-type multigrid is smoothed with either a constant or a linear least-square fitting function, whereas the non-nested finite element-type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand-alone solvers or coupled with a GMRES method. After analyzing the accuracy of the solutions obtained with our solvers on a typical test case in fluid mechanics, their performance in terms of convergence rate, computational speed and memory consumption is compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in the study. Copyright © 2010 John Wiley & Sons, Ltd.
Three-dimensional sloshing: A consistent finite element approach
This paper presents a new computational methodology based on Legendre's polynomials to predict the slosh and acoustic motion in nearly incompressible fluids in both rigid and flexible structures with free surface. Here, we have used a finite element formulation based on Lagrangian frame of reference to model the fluid motion derived using Hamiltonian equation of the fluid system. We formulated three hexahedral finite elements based on strain fields expressed in terms of extended Legendre's polynomials. Sloshing and acoustic motion of liquid is investigated using these newly formulated elements and inf-sup test is performed on these new elements to check the performance of these elements in modeling sloshing under two severe constraints, namely incompressibility and irrotationality. Comparisons of slosh and acoustic frequencies, and mode shapes with exact solutions are given. Dynamic analysis with earthquake and harmonic kind of forcing function is carried out to validate the formulated hexahedral elements to analyze the sloshing response. Numerical results obtained with these new finite elements, and with the present finite element formulation of the mathematical model agree well with the exact solution and as well as with published experimental literature. Copyright © 2010 John Wiley & Sons, Ltd.
A two-phase adaptive finite element method for solid-fluid coupling in complex geometries
In this paper we present a method to solve the Navier-Stokes equations in complex geometries, such as porous sands, using a finite-element solver but without the complexity of meshing the porous space. The method is based on treating the solid boundaries as a second fluid and solving a set of equations similar to those used for multi-fluid flow. When combined with anisotropic mesh adaptivity, it is possible to resolve complex geometries starting with an arbitrary coarse mesh. The approach is validated by comparing simulation results with available data in three test cases. In the first we simulate the flow past a cylinder. The second test case compares the pressure drop in flow through random packs of spheres with the Ergun equation. In the last case simulation results are compared with experimental data on the flow past a simplified vehicle model (Ahmed body) at high Reynolds number using large-eddy simulation (LES). Results are in good agreement with all three reference models. Copyright © 2010 John Wiley & Sons, Ltd.
4D variational data assimilation for locally nested models: Complementary theoretical aspects and application to a 2D shallow water model
We consider the application of a four-dimensional variational data assimilation method to a numerical model, which employs local mesh refinement to improve its solution. We focus on structured meshes where a high-resolution grid is embedded in a coarser resolution one, which covers the entire domain. The formulation of the nested variational data assimilation algorithm was derived in a preliminary work (Int. J. Numer. Meth. Fluids 2008; under review). We are interested here in complementary theoretical aspects. We present first a model for the multi-grid background error covariance matrix. Then, we propose a variant of our algorithms based on the addition of control variables in the inter-grid transfers in order to allow for a reduction of the errors linked to the interactions between the grids. These formulations are illustrated and discussed in the test case experiment of a 2D shallow water model. Copyright © 2010 John Wiley & Sons, Ltd.
High-order methods for the numerical solution of the BiGlobal linear stability eigenvalue problem in complex geometries
A high-order computational tool based on spectral and spectral/hp elements (J. Fluid. Mech. 2009; to appear) discretizations is employed for the analysis of BiGlobal fluid instability problems. Unlike other implementations of this type, which use a time-stepping-based formulation (J. Comput. Phys. 1994; 110(1):82-102; J. Fluid Mech. 1996; 322:215-241), a formulation is considered here in which the discretized matrix is constructed and stored prior to applying an iterative shift-and-invert Arnoldi algorithm for the solution of the generalized eigenvalue problem. In contrast to the time-stepping-based formulations, the matrix-based approach permits searching anywhere in the eigenspace using shifting. Hybrid and fully unstructured meshes are used in conjunction with the spatial discretization. This permits analysis of flow instability on arbitrarily complex 2-D geometries, homogeneous in the third spatial direction and allows both mesh (h)-refinement as well as polynomial (p)-refinement. A series of validation cases has been defined, using well-known stability results in confined geometries. In addition new results are presented for ducts of curvilinear cross-sections with rounded corners. Copyright © 2010 John Wiley & Sons, Ltd.
Canonical Monte Carlo simulation of adsorption of O2 and N2 mixture on single walled carbon nanotube at different temperatures and pressures
Adsorption of pure and mixtures of O2 and N2 on isolated single-walled carbon nanotube (SWCNT) have been investigated at the subcritical (77 K) and different supercritical (273, 293, and 313K) temperatures for the pressure range between 1 and 31 MPa using (N,V,T) Monte Carlo simulation. Both O2 and N2 gravimetric storage capacity exhibit similar behaviors, gas adsorption is higher on outer surface of tube, compared to the inner surface. Results are consistent with the experimental adsorption measurements. All adsorption isotherms for pure and mixture of O2 and N2 are characterized by type I (Langmuir shape), indicating enhanced solid-fluid interactions. Comparative studies reveal that, under identical conditions, O2 adsorption is higher than N2 adsorption, due to the adsorbate structure. Excess amount of O2 and N2 adsorption reach to a maximum at each temperature and specified pressure which can be suggested an optimum pressure for O2 and N2 storage. In addition, adsorptions of O2 and N2 mixtures have been investigated in two different compositions: (i) an equimolar gas mixture and (ii) air composition. Also, selectivity of nanotube to adsorption of O2 and N2 gases has been calculated for air composition at ambient condition. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010
Hydrogen uptake capacity of C2H4Sc and its ions: A density functional study
We report the gravimetric hydrogen uptake capacity of C2H4Sc complex and isoelectronic ions using Density Functional Theory. We predict that C2H4Sc+ can bind maximum seven hydrogen molecules in dihydrogen form giving gravimetric uptake capacity of 16.2 wt %, larger by about 2 and 4 wt % than the neutral and anion, respectively. We also found that the interaction of hydrogen molecules with C2H4Sc+ ion is characteristically different than that with neutral and anion. Vibrational spectroscopic study reveals that C2H4Sc and isoelectronic ions are quantum mechanically stable with their characteristic change in respective identified mode. The large gravimetric H2 uptake capacity of C2H4Sc+ is well above the target specified by Department of Energy (DOE) by 2015. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010
Quantum chemical modeling of enzymatic reactions: The case of histone lysine methyltransferase
Quantum chemical cluster models of enzyme active sites are today an important and powerful tool in the study of various aspects of enzymatic reactivity. This methodology has been applied to a wide spectrum of reactions and many important mechanistic problems have been solved. Herein, we report a systematic study of the reaction mechanism of the histone lysine methyltransferase (HKMT) SET7/9 enzyme, which catalyzes the methylation of the N-terminal histone tail of the chromatin structure. In this study, HKMT SET7/9 serves as a representative case to examine the modeling approach for the important class of methyl transfer enzymes. Active site models of different sizes are used to evaluate the methodology. In particular, the dependence of the calculated energies on the model size, the influence of the dielectric medium, and the particular choice of the dielectric constant are discussed. In addition, we examine the validity of some technical aspects, such as geometry optimization in solvent or with a large basis set, and the use of different density functional methods. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010
A new Lamarckian genetic algorithm for flexible ligand-receptor docking
We present a Lamarckian genetic algorithm (LGA) variant for flexible ligand-receptor docking which allows to handle a large number of degrees of freedom. Our hybrid method combines a multi-deme LGA with a recently published gradient-based method for local optimization of molecular complexes. We compared the performance of our new hybrid method to two non gradient-based search heuristics on the Astex diverse set for flexible ligand-receptor docking. Our results show that the novel approach is clearly superior to other LGAs employing a stochastic optimization method. The new algorithm features a shorter run time and gives substantially better results, especially with increasing complexity of the ligands. Thus, it may be used to dock ligands with many rotatable bonds with high efficiency. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010
Billiards in magnetic fields: A molecular dynamics approach
Author(s): M. Aichinger, S. Janecek, and E. Räsänen
We present a computational scheme based on classical molecular dynamics to study chaotic billiards in static external magnetic fields. The method allows us to treat arbitrary geometries and several interacting particles. We test the scheme for rectangular single-particle billiards in magnetic fields...
[Phys. Rev. E 81, 016703] Published Thu Jan 14, 2010
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