Diagonalization Scheme Research Articles

OPTIMIZATION ISSUES IN FINITE ELEMENT CODES FOR SOLVING OPEN DOMAIN 3D ELECTROMAGNETIC SCATTERING AND CONFORMAL ANTENNA PROBLEMS

SUMMARY The first part of the paper presents the implementation and performance of a new absorbing boundary condition (ABC) for truncating finite element meshes. This ABC can be applied conformally to the surface of the structure for scattering and antenna radiation calculations. Consequently, the computational domain is reduced dramatically, thus allowing the simulation of much larger structures, and results are presented for three-dimensional bodies. The latter part of the paper discusses optimization issues relating to the solver's CPU speed on parallel and vector processors. It is shown that a jagged diagonal storage scheme leads to a four-fold increase in the FLOP rate of the code, and a standard matrix profile reduction algorithm substantially reduces the inter-processor communication.

Molecular dynamics with quantum forces: Vibrational spectra of localized systems.

We present molecular-dynamics simulations with quantum forces for localized systems using a real-space method. We illustrate calculations for the vibrational modes of small molecules and clusters. Unlike other real-space methods using adaptive grids, this procedure does not require any Pulay corrections for the forces, nor does it suffer from the complication of redefining a grid after each time step of the molecular-dynamics simulation. Our method is based on combining higher order finite difference methods with ab initio pseudopotentials. We also introduce an iterative diagonalization scheme based on preconditioned Krylov techniques. Compared to plane-wave--supercell methods, this method is more efficient and simpler to implement. Examples are presented for the ${\mathrm{CO}}_{2}$ molecule and a silicon cluster, ${\mathrm{Si}}_{4}$. \textcopyright{} 1996 The American Physical Society.

Iterative diagonalization within the multi-configurational time-dependent Hartree approach: calculation of vibrationally excited states and reaction rates

An iterative diagonalization scheme for operators involving wavepacket propagations employing the multi-configurational time-dependent Hartree (MCTDH) approach is presented. The scheme is based on a Lanczos-type recursion relation, which is modified to account for the different sets of single-particle functions employed in different MCTDH wavefunctions. Two possible applications are given: the calculation of energies and eigenfunctions of vibrationally excited states, and the direct calculation of thermal rate constants. The electronic ground state of CH3I including four vibrational coordinates and the collinear H + H2 reaction have been used as testing examples.

Iterative diagonalization within the multi-configurational time-dependent Hartree approach: calculation of vibrationally excited states and reaction rates

An iterative diagonalization scheme for operators involving wavepacket propagations employing the multi-configurational time-dependent Hartree (MCTDH) approach is presented. The scheme is based on a Lanczos-type recursion relation, which is modified to account for the different sets of single-particle functions employed in different MCTDH wavefunctions. Two possible applications are given: the calculation of energies and eigenfunctions of vibrationally excited states, and the direct calculation of thermal rate constants. The electronic ground state of CH 3I including four vibrational coordinates and the collinear H + H 2 reaction have been used as testing examples.

Theoretical study of the X‐ray photoionization spectra of polycycloalkanes

The dependence of X-ray photoionization spectra on structural characteristics is assessed by one-particle Green's function calculations on the nearly isomeric adamantane, 1,4-dimethyl bicyclo [2,2,2] octane and 1,1,4,4-tetramethyl cyclohexane compounds. These calculations are carried out using the quasi-particle approximation and the diagonal 2ph-TDA renormalization scheme for the self-energy. The simulated spectra fingerprint remarkably the atomic connectivity from their inner valence part, through the interplay of energy degeneracies and the influence of torsional strains on electron binding energies. The boundary region between the inner and outer valence bands provides also specific signatures for molecular characteristics pertaining to conformational questions, namely the enforced conversion from the chair to the boat form of the cyclohexane units generating these compounds. © 1996 John Wiley & Sons, Inc.

Computation of a controlled store separation from a cavity

Coupling of the Reynolds-averaged Navier-Stokes equations, rigid-body dynamics, and a pitch-attitude control law is demonstrated in two and three dimensions. The application problem was the separation of a canard-controlled store from an open-flow rectangular cavity at a freestream Mach number of 1.2. The transient flowfleld was computed using a diagonal scheme in an overset mesh framework, with the resultant aerodynamic loads used as the forcing functions in the nonlinear dynamics equations. The proportional and rate gyro sensitivities were computed via pole placement techniques for the linearized dynamical equations, in which computed aerodynamic stability derivatives were used. In two dimensions, a comparison between full and linearized flow equations for a perturbed pinned missile was made, and a controlled store was found to possess improved separation characteristics over a canard-fixed store. In three dimensions, trajectory comparisons with quasisteady wind-tunnel data for the canard-fixed case were made. Comparisons of canard-fixed and canardactive simulations showed that controlled store offers only modest improvements in cavity separation characteristics for these high-ejection rate cases.

Comment on the diagonalization of Green functions

Real-time thermal field theory is known in the two flavors “closed-time path formalism” and “thermo field dynamics”. With a certain choice of parameters the full two-point functions of these models are identical, hence a scheme to diagonalize the Green function in one model can be applied to the other as well. This allows us to compare the diagonalization schemes that have been discussed in the recent literature in order to select the simplest one. Such comparison speaks in favor of a diagonalization method which is motivated from thermo field dynamics.

A new time-dependent approach to the direct calculation of reaction rates

A wave packet dynamical approach to the direct calculation of the rate constant of a chemical reaction is presented. Based on the position-flux correlation function of Miller, Schwartz, and Tromp [J. Chem. Phys. 79, 4889 (1983)] a reaction rate operator is introduced, which can be viewed as the thermal analog of the energy-dependent reaction probability operator [J. Chem. Phys. 99, 3411 (1993)]. It is shown that this reaction rate operator has in general only a small number of eigenstates with nonvanishing eigenvalues. These eigenstates can be interpreted as the vibrational ground state and the vibrationally excited states of the activated complex. The eigenstates and eigenvalues can efficiently be computed via an iterative (Lanczos) diagonalization scheme. The number of wave packet propagations required equals approximately the number of relevant states of the activated complex, it is considerably smaller as in previous approaches to the calculation of rate constants based on wave packet dynamics. The new approach is illustrated by three examples: transmission through a one-dimensional (Eckart) potential barrier, the collinear model of the H+H2 reaction, and the H+H2 reaction in its full dimensionality for J=0. For temperatures below 1000 K, in all examples presented, the rate constant can be calculated employing only a single wave packet. This result suggests that the approach can efficiently be applied to problems with a larger number of degrees of freedom.

RENORMALIZATION AND BOLTZMANN EQUATIONS IN THERMAL QUANTUM FIELD THEORY

The renormalization scheme in nonequilibrium thermal quantum field theories is reexamined. Instead of the self-energy diagonalization scheme, we propose to diagonalize Green’s function at equal time. This eliminates the problem of on-shell definition related to time-dependent energies and spatially inhomogeneous situations, and yields a Boltzmann equation that contains memory effect. The new diagonalization scheme and the derivation of the Boltzmann equation from it can be applied to any thermal situation. It allows the treatment of a nonequilibrium problem beyond perturbational calculations in a self-consistent manner. The results are applicable to both thermo field dynamics and the closed time path formalism.

Ab initio pseudopotential plane-wave calculations of the electronic structure of YBa2Cu3O7.

We present an ab initio pseudopotential local-density-functional calulation for the stoichiometric high-${\mathit{T}}_{\mathit{c}}$ cuprate ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}$ using the plane-wave basis set. We have overcome well-known difficulties in applying pseudopotential methods to first-row elements, transition metals, and rare-earth materials by carefully generating norm-conserving pseudopotentials with excellent transferability and employing an extremely efficient iterative diagonalization scheme optimized for our purpose. The self-consistent band structures, the total and site-projected densities of states, the partial charges and their symmetry-decompositions, and some characteristic charge densities near ${\mathit{E}}_{\mathit{F}}$ are presented. We compare our results with various existing full-potential linear augmented-plane-wave and full-potential linear muffin-tin orbital calculations and establish that the ab initio pseudopotential method is competitive with other methods in studying the electronic structure of such complicated materials as high-${\mathit{T}}_{\mathit{c}}$ cuprates.

Geometrically non-linear transient analysis of laminated composite and sandwich shells with a refined theory and C0 finite elements

A C 0 continuous finite element formulation of a higher order shear deformation theory is presented for predicting the linear and geometrically non-linear, in the sense of von Karman, transient responses of composite and sandwich laminated shells. The displacement model accounts for the non-linear cubic variation of the tangential displacement components through the thickness of the shell and the theory requires no shear correction coefficients. In the time domain, the explicit central difference integrator is used in conjunction with the special mass matrix diagonalization scheme which conserves the total mass of the element and includes effects due to rotary inertia terms. Numerical results for central transverse deflection and stresses are presented for composite and sandwich laminated shells with various boundary conditions subjected to different types of loads and are compared with the results from other sources. Some new results are also included for future reference.

Electronic-structure calculations and molecular-dynamics simulations with linear system-size scaling

We present a method for total energy minimizations and molecular dynamics simulations based either on tight-binding or on Kohn-Sham hamiltonians. The method leads to an algorithm whose computational cost scales linearly with the system size. The key features of our approach are (i) an orbital formulation with single particle wavefunctions constrained to be localized in given regions of space, and (ii) an energy functional which does not require either explicit orthogonalization of the electronic orbitals, or inversion of an overlap matrix. The foundations and accuracy of the approach and the performances of the algorithm are discussed, and illustrated with several numerical examples including Kohn-Sham hamiltonians. In particular we present calculations with tight-binding hamiltonians for diamond, graphite, a carbon linear chain and liquid carbon at low pressure. Even for a complex case such as liquid carbon -- a disordered metallic system with differently coordinated atoms -- the agreement between standard diagonalization schemes and our approach is very good. Our results establish the accuracy and reliability of the method for a wide class of systems and show that tight binding molecular dynamics simulations with a few thousand atoms are feasible on small workstations.

Large amplitude free vibration analysis of cross-ply composite and sandwich laminates with a refined theory and [formula omitted] finite elements

A refined higher-order shear deformation theory has been developed for large amplitude, in the sense of von Kármán, free vibration analysis of fibre reinforced cross-ply composite and sandwich laminates by assuming cubic variations of in-plane displacement components and a constant transverse displacement component through the thickness of the laminate. The theory accounts for warping of the transverse cross-section, which cannot be modelled with the Reissner-Mindlin first-order shear deformation theory. The displacement-based finite element method of analysis using C ° isoparametric nine-node quadrilateral elements of the Lagrangian family is adopted. A special mass matrix diagonalization scheme is employed which conserves the total mass of the element and includes the effects due to rotary inertia terms. The validity and efficiency of the present development is then established by obtaining the solutions to a wide range of problems and comparing them with the available two- and three-dimensional closed-form and finite element solutions. Some new results are also generated in a non-linear context for future comparisons.

Shell dynamics with three-dimensional degenerate finite elements

An explicitly through the thickness integrated two-dimensional version of the three-dimensional degenerated shell element is formulated here to study the dynamics of elastic shells. A nine-noded quadrilateral Lagrangian element is used with five degrees of freedom per node. A specialized mass diagonalization scheme, developed by Hinton, Rock and Zienkiewicz, is used which conserves the total mass of the element and also includes the effects of the rotary inertia terms. Hamilton's principle is used to derive the equations of motion. Mode superposition coupled with Duhamel's integral is first employed to obtain a solution of the equations of motion in time. Mode shapes and frequencies are computed by subspace iteration technique. Direct time integration using the implicit Newmark-β method is also carried out. Several examples are presented and the results obtained by mode superposition and direct time integration methods are compared.

Geometrically non-linear transient C° finite element analysis of composite and sandwich plates with a refined theory

A <TEX>$C^{\circ}$</TEX> continuous finite element formulation of a higher order displacement theory is presented for predicting linear and geometrically non-linear in the sense of von Karman transient responses of composite and sandwich plates. The displacement model accounts for non-linear cubic variation of tangential displacement components through the thickness of the laminate and the theory requires no shear correction coefficients. In the time domain, the explicit central difference integrator is used in conjunction with the special mass matrix diagonalization scheme which conserves the total mass of the element and included effects due to rotary inertia terms. The parametric effects of the time step, finite element mesh, lamination scheme and orthotropy on the linear and geometrically non-linear responses are investigated. Numerical results for central transverse deflection, stresses and stress resultants are presented for square/rectangular composite and sandwich plates under various boundary conditions and loadings and these are compared with the results from other sources. Some new results are also tabulated for future reference.

Use of sparse diagonal storage in finite element analysis

The use of the sparse diagonal storage technique in finite element analysis is discussed in this paper. Formulas, which can be used in the development of algorithms with sparse diagonal storage scheme in 2-D and 3-D finite element analysis, are presented. In addition, implementation of the diagonal storage scheme on CRAY-2 is demonstrated with the help of an example. It is shown that the proposed vector-matrix multiplication technique, which employs a temporary array, is an improvement over a previous technique available in the literature.

Transient response of isotropic, orthotropic and anisotropic composite-sandwich shells with the superparametric element

The first-order Reissner-Mindlin shear deformation theory (FOST) is employed to investigate the transient response of isotropic, layered orthotropic and anisotropic composite and sandwich shells. The eight-noded Serendipity and nine-noded Lagrangian quadrilateral superparametric shell elements are used. Numerical convergence and stability of the elements are established using an explicit central difference technique with a special mass matrix diagonalization scheme. The effects of transverse shear modulii of stiff layers, length/thickness and radius/length ratios, time step, finite element mesh, orientation of fibers and degree of orthotropy on the transient response of shells are studied. The variety of results presented here, based on realistic material properties of more commonly used advanced laminated composite shells, should serve as references for future investigations.

Ground state properties of a one dimensional frustrated quantum XY model

We have investigated the ground state properties of a frustrated quantum anisotropic XY chain. The ground state energy, short range dimer order parameter and the spin-spin correlation functions are estimated using an exact diagonalisation scheme and a self-consistent Hartree fock method. A phase diagram is also estimated from the spin correlations.

Diagonal implicit scheme for computing flows with finite rate chemistry

A new algorithm for solving steady, finite-rate chemistry, flow problems is presented. The new scheme eliminates the expense of inverting large block matrices that arise when species conservation equations are introduced. The source Jacobian matrix is replaced by a diagonal matrix which is tailored to account for the fastest reactions in the chemical system. A point-implicit procedure is discussed and then the algorithm is included into the LU-SGS scheme. Solutions are presented for hypervelocity reentry and Hydrogen-Oxygen combustion. For the LU-SGS scheme a CFL number in excess of 10,000 has been achieved.

High-lying rovibrational states of floppy X3 triatomics by a new D3h symmetry adapted method: Application to the H+3 molecule

A new D3h symmetry adapted approach to accurate calculation of high-lying rovibrational (J&amp;gt;0) states of floppy X3 triatomic molecules is presented, extending our recent work on J=0 bound states of D3h triatomics [Z. Bačić and J. Z. H. Zhang, Chem. Phys. Lett. 184, 513 (1991)]. Three sets of Jacobi coordinates are employed, allowing construction of basis sets with full S3 permutation symmetry of the three identical nuclei. Inclusion of S3 symmetry reduces substantially the size of the final matrix eigenvalue problem and the computational effort involved. It also assures unambiguous, correct symmetry assignment of the calculated rovibrational levels. Our method is especially suitable for rotating D3h molecules with strongly coupled large amplitude motions of the three atoms. An efficient quasiadiabatic diagonalization and truncation scheme is incorporated into our methodology. Application to H+3 for J=1 produced the total of 426 well converged, symmetry assigned states up to ∼24 000 cm−1 above the ground state, far more than in any previous calculation for rotating H+3.