Non-iterative solution of finite-element equations in incompressible solids

Summary. Finite-element equations for incompressible and near-incompressible media give rise to a matrix with a diagonal block of zeroes or very small numbers. The matrices are not amenable to conventional techniques involving pivoting on diagonal entries. Uzawa methods have been applied to the associated linear systems. They are iterative and converge when the matrix is nonsingular. In the current study an alternate form of the matrix is used which is amenable to a solution without iteration. It likewise is applicable whenever the matrix is nonsingular. The solution process consists of a block LU factorization, followed by Cholesky decomposition of a positive definite diagonal block together with several forward and backward substitution operations. Two illustrative examples are developed.     

Cluster first-sequence last heuristics for generating block diagonal forms for a machine-part matrix

Before detailed cell design analyses, rearranging the binary (or 0-1) machine-part matrix into a compact block diagonal form (BDF) is useful for controlling combinatorial explosion during subsequent decision-making involving machine duplication, subcontracting, intercell layout design, etc. Several authors have shown that a compact BDF corresponds to the implicit clusters in both dimensions being expressed as row and column permutations. A traditional approach for solving this problem has been to obtain the two permutations independently by solving the permutation problem in each dimension as a (unidimensional) travelling salesman problem (TSP). This paper describes cluster first-sequence last heuristics which combine the properties of the minimal spanning tree (MST) (clusters only) and TSP (sequence only) for improved permutation generation. The BDFs obtained with these heuristics were compared with those obtained using the TSP, linear placement problem (LPP), single linkage cluster analysis (SLCA), rank order clustering (ROC) and occupancy value (OV) algorithms. The ratio of variances (β), which was the variance along the minor axis (V_Y) divided by the variance along the major axis (V_x) of the BDF, was used to evaluate the compactness of all the BDFs without assuming any explicit knowledge of the clusters in either dimension.     

Iterative solution of large linear systems with non‐smooth submatrices using partial wavelet transforms and split‐matrix matrix–vector multiplication

The iterative solution of large linear systems with highly irregular matrices cannot be accelerated by wavelet transformation and subsequent sparsification if the transformed matrix is still highly irregular. In this paper we show that if the irregularity of the original matrix is limited to a relatively small known set of rows or columns (as is the case in significant applications), then acceleration can be achieved by a mixed approach in which only the ‘smooth’ submatrix is transformed and iterative solution is implemented using a novel ‘split‐matrix’ form of matrix–vector multiplication. Copyright © 2003 John Wiley & Sons, Ltd.     

Parallel solution of closely coupled systems

The odd—even permutation and associated unitary transformations for reordering the matrix coefficient A is employed as a means of breaking the strong seriality which is characteristic of closely coupled systems. The nested dissection technique is also reviewed, and the equivalence between reordering A and dissecting its network is established. The effect of transforming A with odd—even permutation on its topology and the topology of its Cholesky factors is discussed. This leads to the construction of directed graphs showing the computational steps required for factoring A , their precedence relationships and their sequential and concurrent assignment to the available processors. Expressions for the speed-up and efficiency of using N processors in parallel relative to the sequential use of a single processor are derived from the directed graph. Similar expressions are also derived when the number of available processors is fewer than required.     

Non-iterative load-flow method as a tool for voltage stability studies

Here the use of a non-iterative (NI) method for load-flow solutions is investigated. The method, previously proposed in the literature, presents some advantages in comparison with the iterative approaches usually employed. There, because the Taylor expansion is used, the power flow Jacobian matrix is not updated along the process. Here, some improvements in the implementation are executed, and few control actions are incorporated into the formulation. The method is then applied for voltage stability studies, aiming to reduce the computational time associated with     

An efficient diagonal preconditioner for finite element solution of Biot's consolidation equations

Finite element simulations of very large‐scale soil–structure interaction problems (e.g. excavations, tunnelling, pile‐rafts, etc.) typically involve the solution of a very large, ill‐conditioned, and indefinite Biot system of equations. The traditional preconditioned conjugate gradient solver coupled with the standard Jacobi (SJ) preconditioner can be very inefficient for this class of problems. This paper presents a robust generalized Jacobi (GJ) preconditioner that is extremely effective for solving very large‐scale Biot's finite element equations using the symmetric quasi‐minimal residual method. The GJ preconditioner can be formed, inverted, and implemented within an ‘element‐by‐element’ framework as readily as the SJ preconditioner. It was derived as a diagonal approximation to a theoretical form, which can be proven mathematically to possess an attractive eigenvalue clustering property. The effectiveness of the GJ preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three‐dimensional footing problem. This paper casts a new perspective on the potentialities of the simple diagonal preconditioner, which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix. Copyright © 2002 John Wiley & Sons, Ltd.     

Influence matrix diagonal block elements for two‐dimensional elastodynamic problems

A method is described in this article to calculate the diagonal block elements of the influence matrices used in the boundary element method for two‐dimensional elastodynamic problems. Currently, a method that is used in the calculation of diagonal block elements is a combination of rigid‐body translation and Gaussian quadrature for the adjoining elements to the singularity point. This paper proposes to show an alternative method for the calculation of diagonal block elements of the traction influence matrix in elastodynamic problems where linear elements are used. In addition, the method is applied for use in the displacement influence matrix. The accuracy of the method is demonstrated in an example problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical solution of a parabolic system with coupled nonlinear boundary conditions

A numerical technique has been developed to solve a system that consists of m linear parabolic differential equations with coupled nonlinear boundary conditions. Such a system may represent chemical reactions, chemical lasers and diffusion problems. An implicit finite difference scheme is adopted to discretize the problem, and the resulting system of equations is solved by a novel technique that is a modification of the cyclic odd–even reduction and factorization (CORF) algorithm. At each time level, the system of equations is first reduced to m nonlinear algebraic equations that involve only the m unknown grid points on the nonlinear boundary. Newton's method is used to determine these m unknowns, and the corresponding Jacobian matrix can be computed and updated easily. After convergence is achieved, the remaining unknowns are solved directly. The efficiency of this technique is illustrated by the numerical computations of two examples previously solved by the cubic spline Galerkin method.     

On a bounded solution to a pair of coupled nonlinear ordinary differential equations

We present a bounded, decaying solution to a pair of coupled, nonlinear second-order ordinary differential equations arising in the theory of natural convection. The solution is found by transforming the problem into a non-autonomous system in the phase-plane. A uniqueness proof is given for the bounded solution.     

On a bounded solution to a pair of coupled nonlinear ordinary differential equations

We present a bounded, decaying solution to a pair of coupled, nonlinear second-order ordinary differential equations arising in the theory of natural convection. The solution is found by transforming the problem into a non-autonomous system in the phase-plane. A uniqueness proof is given for the bounded solution.     

Simulation of coupled heat and mass transfer during freezing of a porous humid matrix

A numerical simulation using Lees's three-level scheme and a fixed grid was carried out to predict heat and mass transfer during the freezing of a porous humid matrix. The simulations predict local moisture content, temperature and weight loss of the foods throughout the freezing process. The model accommodates the effects of temperature dependent variables such as apparent specific heat, enthalpy, thermal conductivity, and water activity. The model was validated by experiments on a cellulose sponge. The results of the model showed good agreement with experimental data.     

Direct determination of tellurium in geological samples by inductively coupled plasma mass spectrometry using ethanol as a matrix modifier

Direct determination of tellurium in geological samples by inductively coupled plasma mass spectrometry (ICP-MS) is often complicated by its low abundance, poor analytical sensitivity, and the presence of xenon interferences. Therefore, a simplified and rapid method for direct determination of nanogram levels of tellurium in geological samples using ICP-MS by reduction of interferences and improvement of sensitivity was developed. It is impossible to resolve 126Te and 128Te from isotope interferences of Xe even by currently available high-resolution magnetic mass spectrometry due to the extremely small mass difference (0.001-0.002 amu). However, the addition of 4% ethanol was found to suppress the interferences of Xe by a factor of 6 and increases the sensitivity of Te determination in ICP-MS by a factor of 3 relative to the values obtained in conventional 3% (v/v) HNO3 solution at the corresponding optimum operating conditions, respectively. The detection limits of 126Te and 128Te were reduced by a factor of 7.2 and 8.8, respectively, and the limit of quantitation (LOQ) for 126Te in the presence of 4% ethanol was 1.5 ng g(-1) (the LOQ is expressed as the concentration in the solid samples, thereby taking into account the dilution factor of 1000). The agreement between the determined Te concentration values (r = 0.998) in various geological samples (n = 140) by using isotopes of 126Te and 128Te indicates negligible contributions of polyatomic interferences produced by the addition of ethanol at these m/z. The proposed method was successfully applied to the direct determination of nanogram levels of Te in a series of international geological reference materials.     

Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium

Based on nonlocal theory, this article discusses vibration of CDVGS¹ systems. The properties of each single layer graphene sheet (SLGS) are assumed to be orthotropic and viscoelastic. The two SLGSs are simply supported and coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak layer. This model is aimed at representing dynamic interactions in nanocomposite materials with dissipation effect. By considering the Kirchhoff plate theory and Kelvin–Voigt model, the governing equation is derived using Hamilton's principle. The equation is solved analytically to obtain the complex natural frequency. The parametric study is thoroughly performed, concentrating on the series effects of viscoelastic damping structure, aspect ratio, visco-Pasternak medium, and mode number. In this system, in-phase (IPV) and out-of-phase (OPV) vibrations are investigated. The numerical results of this article show a perfect correspondence with those of the previous researches.     

Analytical solution of radiative transfer in the coupled atmosphere-ocean system with a rough surface

Using the computationally efficient discrete-ordinate method, we present an analytical solution for radiative transfer in the coupled atmosphere-ocean system with a rough air-water interface. The theoretical formulations of the radiative transfer equation and solution are described. The effects of surface roughness on the radiation field in the atmosphere and ocean are studied and compared with satellite and surface measurements. The results show that ocean surface roughness has significant effects on the upwelling radiation in the atmosphere and the downwelling radiation in the ocean. As wind speed increases, the angular domain of sunglint broadens, the surface albedo decreases, and the transmission to the ocean increases. The downward radiance field in the upper ocean is highly anisotropic, but this anisotropy decreases rapidly as surface wind increases and as ocean depth increases. The effects of surface roughness on radiation also depend greatly on both wavelength and angle of incidence (i.e., solar elevation); these effects are significantly smaller throughout the spectrum at high Sun. The model-observation discrepancies may indicate that the Cox-Munk surface roughness model is not sufficient for high wind conditions.     

Transmission-line matrix solution of waveguides with dielectric losses

The transmission-line matrix method is a time-domain numerical method for solving wave problems. This paper describes how a minor change in the computer program for loss-free dielectrics extends the method to include dielectric losses. Results are given in terms of the wave impedance of a waveguide with losses, and also in terms of power decay due to losses in the dielectric of a resonant cavity.     

Effective parameters of static coupled physicomechanical field in matrix composites

Conclusions It is shown that the problem of evaluating the effective parameters of composites for a wide range of stationary coupled fields is reduced to examining uncoupled fields.The multiparticle method of the effective field was generalized with special reference to the problems of examining the static coupled physicomechanical fields in composites.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 27, No. 4, pp. 105–111, July–August, 1991.     

Single-element solution comparisons with a high-performance inductively coupled plasma optical emission spectrometric method.

A solution-based inductively coupled plasma optical emission spectrometric (ICP-OES) method is described for elemental analysis with relative expanded uncertainties on the order of 0.1% relative. The single-element determinations of 64 different elements are presented, with aggregate performance results for the method and parameters for the determination of each element. The performance observed is superior to that previously reported for ICP-OES, resulting from a suite of technical strategies that exploit the strengths of contemporary spectrometers, address measurement and sample handling noise sources, and permit rugged operation with small uncertainty. Taken together, these strategies constitute high-performance ICP-OES.     

Fully coupled solution of the equations for incompressible recirculating flows using a penalty-function finite-difference formulation

This paper describes two new solution algorithms for steady recirculating flows that use a penalty formulation to eliminate the pressure from the finite difference form of the governing equations. One algorithm uses successive substitution to linearize the equations, while the other employs the Newton-Raphson linearization. In both cases, the equations are solved in a fully coupled manner using a sparse matrix form of LU decomposition. The D'Yakonov iteration is used to avoid unnecessary factorizations of the coefficient matrix, significantly improving the computational efficiency. The Newton-Raphson linearization leads to faster convergence, but the execution times of the two methods are comparable. The algorithms converge rapidly and are robust to changes in grid size and Reynolds number. In a number of laminar two-dimensional flows, the new methods proved to be two to ten times faster than some conventional iterative methods.List of symbols A coefficient matrix - A _e area of east control-volume face - a coefficient in the discretization equations - â coefficients in the modified momentum equations in the penalty formulation, Eqs. (13) - b constant term in the discretization equations - F symbolic form of nonlinear system of equations, F ₍₎=A –b=0 - H characteristic length - J Jacobian matrix - n iteration number - P dimensionless pressure - p pressure - Re Reynolds number - U dimensionless u velocity - u x-direction velocity - V dimensionless v velocity - v y-direction velocity - X, Y dimensionless coordinates, X=x/H, Y=y/H - x, y physical coordinates - x x-direction width of the control volume - normalized error in the unconverged solution, Eqs. (19) - dimensionless penalty parameter, Eq. (7) - dimensional penalty parameter, Eq. (10) - viscosity - density     

The importance of diagonal dominance in the iterative solution of equations generated from the boundary element method

The unsymmetric matrix equations generated from the boundary element method (BEM) can be solved iteratively, with convergence to the correct solution guaranteed, if the boundary element system of equations can be first transformed into an equivalent, diagonally dominant system. A transformation is presented which selectively annihilates terms in the coefficient matrix of the system Ax = b until an equivalent, row diagonally dominant system, if available, is obtained. The new, row diagonally dominant system is well suited for use with Jacobi and Gauss-Seidel point iterative equation solvers. The diagonal dominizing transformation presented in this work is not suitable for large systems of equations but is useful as a research tool for studying the importance of diagonal dominance in the iterative solution of equations generated from the BEM. A simple Laplacian problem is used to examine the structure of the BEM equations and to introduce the diagonal dominizing transformation. The importance of diagonal dominance is shown by comparing iterative convergence of positive-definite, symmetric positive-definite and diagonally dominant systems of BEM equations obtained from a plane strain elasticity problem.