Implicit Iterative Scheme Research Articles

Dynamic characteristics of pulsating turbulent flow of compressible gas in a channel

A model of turbulent transfer that allows for the effect of flow-rate oscillations on the turbulent stress is used to investigate pulsating turbulent flow of a compressible gas in a narrow channel. An algorithm for solving numerically the system of equations that describes this process by the finite-difference method with the use of an implicit iteration scheme is proposed. The effect of operating parameters on the amplitude-frequency characteristic is considered.

Solute distribution during rapid solidification into an undercooled melt

Rapid solidification experiments show that a solute-rich core generally exists in a solid rapidly solidified from an undercooled melt. Several simplified models have been proposed to explain and predict this phenomenon. This paper presents a generalized model that includes mass as well as heat diffusion in both solid and liquid phases and considers nonequilibrium solidification kinetics including solute-trapping to treat the local recalescence. For given local temperature gradients and cooling rates, the model leads to a one-dimensional moving boundary problem with a strongly coupled boundary condition at the solid/liquid interface. The solution is obtained by employing an implicit iterative scheme that uses a coordinate transformation. The model predicts successfully a solute-rich core as observed in the experiments. The results show that both melt undercooling and cooling rates strongly affect the solute distribution in the solidified solid. Selected results for dilute aluminum-copper alloys are presented to illustrate the unique features of solute distribution in rapidly solidified alloys.

Unsteady cascade flow calculations of using dual time stepping and the k-$\omega$ turbulence model

A numerical study on two-dimensional unsteady transonic cascade flow has been performed by adopting dual time stepping and the k-.omega. turbulence model. An explicit 4 stage Runge-Kutta scheme for the compressible Navier-Stokes equations and an implicit Gauss-Seidel iteration scheme for the k-.omega. turbulence model are proposed for fictitious time stepping. This mixed time stepping scheme ensures the stability of numerical computation and exhibits a good convergence property with less computation time. Typical steady-state convergence accelerating schemes such as local time stepping, residual smoothing and multigrid combined with dual time stepping shows good convergence properties. Numerical results are presented for unsteady laminar flow past a cylinder and turbulent shock buffeting problem for bicircular arc cascade flow is discussed.

Numerical Error Estimation of Time Integration Schemes on Inviscid Flows around an Oscillating Airfoil.

The actual time-accuracy of several time integration schemes is investigated for the unsteady compressible Euler equations. Runge-Kutta, implicit-approximate factorization, and implicit iterative schemes have been applied to several unsteady problems of transonic flows over a pitching airfoil. The flux difference splitting scheme of Roe with the MUSCL algorithm is used for determining numerical fluxes. The Runge-Kutta scheme produces little numerical error in its stable limit. However, no advantage of its third-order accurate form over its second-order accurate form can be found. The Beam-Warming scheme with approximate flux Jacobians is essentially first-order accurate in time and is not suitable for unsteady flow simulations. In order to preserve the accuracy of the numerical solutions, higher-order time-accurate forms as well as Newton-type iterations are necessary for the implicit time integration scheme.

Flow separation structure in a nonstationary viscous gas flowing over a body

This paper studies the structure of the nonstationary pulsating flow of a viscous compressible heat-conducting gas flowing over a cylinder with a cylindrical step projection on its front end. In a stationary incident stream, a projecting step usually produces a separated vortex flow. In a pulsating incident stream, the overall flow structure, and the structure of the separated flow in particular, changes depending on the frequency and the amplitude of pulsations, and this change substantially affects the aerodynamic characteristics of the immersed body. Such nonstationary processes have been studied by numerical solution of two-dimensional nonstationary Navier-Stokes equations using an implicit iterative difference scheme [1], which approximates the original systems of equations with second-order accuracy in space and in time.

Contribution to Inverse Kinematics of Flexible Robot Arms

An inverse kinematics problem of flexible manipulators is considered. The problem is of importance for compensation of tip deflections when the manipulators perform quasi-static operations. Basic equations based on both kinematic and static relationships are derived and explicit expressions of the mathematical model are presented. Based upon this model, a definition of the inverse kinematic task of flexible manipulators is formulated. Specific features of the task are considered, and a conventional approach applicable for resolving the task is analyzed. To avoid the disadvantages of the conventional approach, an implicit iterative scheme based upon step-by-step compensation for the relative elastic deflections is proposed. Corresponding convergence conditions of the scheme are derived, and applicability of the scheme is verified through simulation.

Mechanical model of lung parenchyma as a two-phase porous medium

The anatomy and geometry of the lung at the micro- and macroscopic level have been described briefly. A notion of lung parenchyma — a macroscopically continuous medium whose mechanical properties result from those of microstructural components — has been adapted. Simplifying assumptions propounded in the constitutive model have been discussed. Two phases have been distinguished in the medium: the solid phase — a highly deformable, nonlinearly elastic skeleton in the form of a thin-walled tissue structure on the micro-scale — and the fluid phase — perfect gas (air) filterating through the structure. General constitutive relations for both phases and their mechanical interactions have ben formulated. Further, the fundamental set of differential equations of the quasi-static coupled problem has been developed. Large deformations, material nonlinearities, and dependence of permeability on skeleton deformation have been included. Matrix formulation of the problem has been presented from the point of view of the finite element method. An implicit iterative time integration scheme has been proposed. The algorithm has been illustrated with results of simple numerical tests.

Prediction of three‐dimensional general shape extrudates by an implicit iterative scheme

AbstractThis paper presents a numerical technique for solving three‐dimensional free surface problems in extrusion applications. The method is fully implicit in the sense that a Newton‐Raphson scheme is applied on all variables, and geometrically general. In particular, the die section shape may be complex and contains multiple corners: very few restrictions apply on the mesh generation because the method does not require the nodes to be located on straight lines (spines). A clear distinction is introduced between the directions associated with the kinematic condition and the remeshing rules. As a difference with respect to earlier publications, these concepts are handled separately. Only Stokes problems are solved in this paper and we have not introduced surface tension. Therefore corners in the die section propagate discontinuities in the extrudate shape, an a method for relocating corners without losing the quadratic convergence of the scheme is presented. Data structures used for the implementation are briefly discussed.We present results on the extrusion of various profiles, including a rectangular die (a benchmark problem) and various complex sections containing multiple corners.

Strong convergence theorems for accretive operators in Banach spaces

Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm and let A ⊂ E × E be m-accretive. Assume that S: cl( D( A)) → E is bounded, strongly accretive, and continuous. For x ϵ E and t > 0, let x t be the unique solution of the inclusion x ϵ Sx t + tAx t . It is proved that if there exists a nonexpansive retraction P of E onto cl( D( A)), then the strong lim t → 0 x t exists. Moreover, it is shown that if every weakly compact convex subset of E has the fixed point property for nonexpansive mappings and 0 ϵ R( A), then the strong lim t → ∞ x t exists and belongs to A −10. An interesting application to a convergence result for an implicit iterative scheme is also included.

EPISO — An implicit non-iterative solution procedure for the calculation of flows in reciprocating engine chambers

This paper deals with the extensions that are required to be made to the implicit non-iterative PISO solution procedure in order to calculate flows in motored reciprocating engines. Firstly the threedimensional conservation equations governing the flow processes in asymmetric engine chambers equipped with piston bowls are given in full. These equations are then cast into finite volume forms, using orthogonal curvilinear computational grids. The operator splitting technique required to handle the velocity-pressure coupling is then given in detail, along with a similar technique for the turbulence model, in this case the turbulence kinetic energy and its dissipation rate. The EPISO (Engine PISO) algorithm is then tested on a number of flow cases in order to (i) assess its performance in comparison with earlier iterative methods, (ii) examine the temporal accuracy of the operator splitting technique, and (iii) illustrate the capability of the method to handle complex geometries and turbulent flow situations. The method is found to produce time step independent solutions at the same time step as fully implicit iterative schemes, but it is nearly an order of magnitude faster than procedures based on the SIMPLE algorithm. The algorithm is also shown to be capable of calculating three-dimensional turbulent flows in engines equipped with curvilinear piston bowls.

Efficient numerical methods for infiltration using Richards' equation

Two efficient finite difference methods for solving Richards' equation in one dimension are presented, and their use in a range of soils and conditions is investigated. Large time steps are possible when the mass‐conserving mixed form of Richards' equation is combined with an implicit iterative scheme, while a hyperbolic sine transform for the matric potential allows large spatial increments even in dry, inhomogeneous soil. Infiltration in a range of soils can be simulated in a few seconds on a personal computer with errors of only a few percent in the amount and distribution of soil water. One of the methods adds points to the space grid as an infiltration or redistribution front advances, thus gaining considerably in efficiency over the other fixed grid method for infiltration problems. In 17‐s computing, this advancing front method simulated infiltration, redistribution, and drainage for 50 days in an inhomogeneous soil with nonuniform initial conditions. Only 16 space and 21 time steps were needed for the simulation, which included early ponding with the development and dissipation of a perched water table.

Application of an implicit iterative difference scheme for the solution of nonstationary Navier-Stokes equations

An implicit iterative difference scheme of second-order accuracy in space and in time is proposed for numerical solution of nonstationary Navier—Stokes equations. The scheme has been applied to model a nonstationary axisymmetric flow in the near wake produced by injection through an annular nozzle from the tail of a supersonic body.

A Transport Method for Treating Three-Dimensional Lattices of Heterogeneous Cells

A new ray-tracing method for the calculation of collision probabilities within arbitrary three-dimensional geometries has been developed. This method is used to discretize the neutron transport equation for heterogeneous rectangular cells containing zones of mixed cylindrical and rectangular geometry. For multicell applications, the interface current (IC) method provides the coupling between cells. The solution to the IC equations over multicell domains consisting of rectangular three-dimensional cells is improved by using an alternate direction implicit iteration scheme with variational acceleration. Results include comparisons of this technique with SHETAN for simple geometries and the analysis of a three-dimensional extension of a two-dimensional 15 × 15 pressurized water reactor benchmark problem.

Isomorphic iterative methods in solving singularly perturbed elliptic difference equations

A new approach for the efficient numerical solution of Singular Perturbation (SP) second order boundary-value problems based on Gradient-type methods is introduced. Isomorphic implicit iterative schemes in conjuction with the Extended to the Limit sparse factorization procedures [9] are used for solving SP second order elliptic equations in two and three-space dimensions. Theoretical results on the convergence rate of these first-degree iterative methods for three-space variables are presented. The application of the new methods on characteristic SP boundary-value problems is discussed and numerical results are given.

Hypersonic viscous flows past general bodies at angle of attack and yaw

Computational results of hypersonic viscous flows over blunt-nosed complex re-entry bodies are presented. An implicit iterative numerical scheme at each marching step is used to solve the parabolized Navier-Stokes (PNS) equations. An existing PNS code has been extended to treat nonaxisymmetric bodies at angle of attack and yaw, multiconic bodies with spin, surface mass transfer, and adiabatic wall conditions. Boundary conditions take into account the periodic condition around the body due to the yaw or spin, and the mass-transfer and/or spin effects at the body surface. The computational results presented include surface-measurable quantities and aerodynamic force and moment coefficients for complex geometries; available data were used for comparison. Grid-size effects on the accuracy of the Magnus force components due to spin are also discussed.

Dispersion in Two-Layer Stratified Water Bodies

A finite element model is developed for the numerical prediction of the dispersion process in a strongly stratified water body idealized as a two-layer system. The physical processes of mixing through the density interface and horizontal dispersion are discussed. Numerical integration in time is based on an implicit iterative trapezoidal scheme. Results for one-dimensional counterflow conditions are verified with analytical solutions. Finally, the model is applied to a large dispersion experiment in Massachusetts Bay.

Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces

LetA be anm-accretive operator in a Banach spaceE. Suppose thatA −10 is not empty and that bothE andE * are uniformly convex. We study a general condition onA that guarantees the strong convergence of the semigroup generated by—A and of related implicit and explicit iterative schemes to a zero ofA. Rates of convergence are also obtained. In Hilbert space this condition has been recently introduced by A. Pazy. We also establish strong convergence under the assumption that the interior ofA −10 is not empty. In Hilbert space this result is due to H. Brezis.

Constructing zeros of accretive operators

We prove a convergence theorem for an implicit iterative scheme and then apply it to an explicit one.

NUMERICAL MODELING OF DISPERSION IN STRATIFIED WATERS

A numerical model is developed for the quantitative description of the dispersion process in a two-layer system which represents an approximation for a natural coastal water body during the summer season when a distinct thermocline usually exists. The formulation is based on the convection-diffusion equation, vertically integrated between the layer boundaries. Layer velocities and thicknesses are assumed to be obtained from a separate hydrodynamic model. The quantification of the physical processes of entrainment and mixing through the density interface as well as the horizontal dispersion mechanisms is discussed. The finite element method is chosen for numerical implementation because of its flexibility in grid layout and easier handling of spatial and temporal variability. Triangular elements with linear interpolation functions are used for the spatial discretization, while a simple implicit iterative scheme based on the trapezoidal rule is employed for time integration. The method is shown to be unconditionally stable, for an arbitrary grid and both one- and two-layer problems, when there is no iteration and the parameters are constant. General convergence criteria required by the iteration procedure are developed and expressed in terms of the basic parameters of the problem and are subsequently confirmed by numerical experiments. Verification of the model is performed by comparison with analytical solutions derived for counterflow conditions. Finally the model is applied to a particle dispersion experiment carried out recently in the Massachusetts Bay and comparisons with field data are presented.

An iterative scheme for solving a quasilinear heat conduction equation

AN implicit iterative difference scheme for the solution of a quasilinear heat conduction equation is considered. An example of the divergence of the iterations is presented.