Implicit Iterative Algorithm Research Articles

Simulation of fish swimming and manoeuvring by an SVD-GFD method on a hybrid meshfree-Cartesian grid

In this paper we present the development of the Singular-Value Decomposition (SVD) based Generalized Finite Difference (GFD) method for the simulation of fluid–structure interaction (FSI) problems in a viscous fluid. The class of FSI problems is exemplified by the self-propulsion (swimming) and dynamic manoeuvring of deforming (undulating and flexing) bodies in a fluid medium. Computation is carried out on a hybrid grid comprising meshfree nodes around the undulating swimming body and Cartesian nodes in the background. The meshfree nodes are convected in tandem with the changing shape and motion of the swimming body. The resultant locomotion of the swimmer is governed by fully-coupled dynamic interaction between the deforming body and the fluid in accordance with Newton’s laws. Time integration of motion is carried out by a Crank–Nicolson based implicit iterative algorithm, which fully couples the changing position of the swimming body with the evolving flow field, for numerical stability. The numerical scheme is applied to the steady swimming/cruising and sharp turning manoeuvres of a two-dimensional carangiform fish. The Strouhal number approaches values for efficient steady swimming reported in Fish and Lauder (2006) and Triantafyllou and Triantafyllou (1993) [3,6] at high Reynolds number. An illustrative example shows the numerical carangiform swimmer executing a sharp turn through an angle of 70° from straight coasting within a space of about one body length. The results obtained are consistent with available literature. In steady swimming, the momentumless wake theoretically anticipated by Wu (2001) [57] is successfully reproduced here, as opposed to the inverse von Karman vortex street generally predicted by inviscid flow models. The momentumless wake, characterized by an aligned series of alternately-signed shed vortices, is symptomatic of a state of average equilibrium between drag acting on the body of the fish and thrust produced by its undulating tail fin. Guided swimming towards targets based on a simple feedback control scheme is also demonstrated.

The ALE Method for Oil/Water Two-Phase Flow in Deforming Porous Media

Abstract The main objective of this paper is to establish a consistent geometrical frame focusing on the coupling between hydro-mechanical aspects. An arbitrary Lagrangian and Eulerian method (ALE) has been used to deal with the numerical simulation for oil/water flow in deforming porous media. When the mesh motion is equal to the velocity of the deformed porous media, the convection term referred to as the relative velocity between fluid flow and the skeleton deformation is removed from the Finite Element Method (FEM) formula. Therefore, some difficulty in FEM numerical simulation has been solved. A classical oil/water flow coupled with solid deformation is simulated by ALE code with an iterative scheme. An ALE with a fully implicit and sequential iterative algorithm [Implicit Pressure Explicit Saturation (IMPES) method], which calculates the oil phase pressure and water phase saturation with least squares FEM, is used to simulate the oil/water two-phase fluid flow. Compared with general reservoir simulation, the ALE method could reach a consistent frame description or a more rigorous definition of the parameters for the solid phase and the flow phase. This may be very important for larger deformations in stratum induced by petroleum production and tracking the moving interface in multi-phase fluid flow. Introduction Such independent variables such as stress, strain and deformation in a continuum are described by Lagrangian or Eulerian coordinates. The Lagrangian description, generally, is employed extensively for solid mechanics, and the Eulerian description is preferred for situations which involve large flows and large distortions such as fluid mechanics. The coupling models or mathematical equations for two-phase flow in deforming porous media belong to the inner coupling. It is recognized that the independent variables for a solid phase and a fluid phase are difficult to distinguish and they should be regarded as overlapping contiuums. So a consistent coordinate frame description for this kind of coupled model is necessary, and many results have been presented. The mass equation, introduced by Bear(1), employed an Eulerian description through continuum theory for multi-phase flow. Cooper(2) derived the one-dimensional governing equations of groundwater flow in Eulerian and deforming coordinates. In the deforming coordinates, the form of governing equations does not involve the convective term which is introduced by the motion of porous media. The two-phase flow in deformed porous media was simulated under deforming coordinates by Xue(3). When the assumption with a steady and a small displacement is given, some results have been obtained with Eulerian mesh. The governing equations for oil/water flow in deformed porous media are derived using the assumption that presented by a number of authors (4–9). Although the deforming ordinates concept is proposed, a consistent coordinate frame description for three-dimensional flow in deformed porous and finite deformations has not been satisfactorily resolved. The main object of this paper is to build a consistent coordinate frame description through the ALE theory for the coupling of two-phase flow and deforming porous media. Furthermore, an injection water example is given with the finite element method.

Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings

In this paper, we consider the weak and strong convergence of an implicit iterative process with errors for two finite families of asymptotically nonexpansive mappings in the framework of Banach spaces. Our results presented in this paper improve and extend the recent ones announced by many others.

Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings

Let X be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. First purpose of this paper is to introduce a modified viscosity iterative process with perturbation for a continuous pseudocontractive self-mapping T and prove that this iterative process converges strongly to x ∗ ∈ F( T) ≔ { x ∈ X∣ x = T( x)}, where x ∗ is the unique solution in F( T) to the following variational inequality: 〈 f ( x ∗ ) - x ∗ , j ( v - x ∗ ) 〉 ⩽ 0 for all v ∈ F ( T ) . Second aim of the paper is to propose two modified implicit iterative schemes with perturbation for a continuous pseudocontractive self-mapping T and prove that these iterative schemes strongly converge to the same point x ∗ ∈ F( T). Basically, we show that if the perturbation mapping is nonexpansive, then the convergence property of the iterative process holds. In this respect, the results presented here extend, improve and unify some very recent theorems in the literature, see [L.C. Zeng, J.C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear Anal. 64 (2006) 2507–2515; H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291; Y.S. Song, R.D. Chen, Convergence theorems of iterative algorithms for continuous pseudocontractive mappings, Nonlinear Anal. (2006)].

A new composite implicit iterative algorithm for nonself asymptotically nonexpansive mappings

Let E be a real uniformly convex and smooth Banach space with P as a sunny nonexpansive retraction, K be a nonempty closed convex subset of E. Let {Si}i=1N, {Ti}i=1N:K→K be two finite families of weakly inward and asymptotically nonexpansive mappings with respect to P. It is proved that the composite implicit iteration process converges weakly and strongly to a common fixed point of {Si}i=1N, {Ti}i=1N under certain conditions. The results of this paper improve and extend some well known corresponding results.

Strong Convergence of an Implicit Iteration Algorithm for a Finite Family of Pseudocontractive Mappings

Strong convergence theorems for approximation of common fixed points of a finite family of pseudocontractive mappings are proven in Banach spaces using an implicit iteration scheme. The results presented in this paper improve and extend the corresponding results of Osilike, Xu and Ori, Chidume and Shahzad, and others.

Robust Analog/RF Circuit Design With Projection-Based Performance Modeling

In this paper, a robust analog design (ROAD) tool for post-tuning (i.e., locally optimizing) analog/RF circuits is proposed. Starting from an initial design derived from hand analysis or analog circuit optimization based on simplified models, ROAD extracts accurate performance models via transistor-level simulation and iteratively improves the circuit performance by a sequence of geometric programming steps. Importantly, ROAD sets up all design constraints to include large-scale process and environmental variations, thereby facilitating the tradeoff between yield and performance. A crucial component of ROAD is a novel projection-based scheme for quadratic (both polynomial and posynomial) performance modeling, which allows our approach to scale well to large problem sizes. A key feature of this projection-based scheme is a new implicit power iteration algorithm to find the optimal projection space and extract the unknown model coefficients with robust convergence. The efficacy of ROAD is demonstrated on several circuit examples

Explicit incremental-update algorithm for modeling crystal elasto-viscoplastic response in finite element simulation

Computational stability and efficiency are the key problems for numerical modeling of crystal plasticity, which will limit its development and application in finite element (FE) simulation evidently. Since implicit iterative algorithms are inefficient and have difficulty to determine initial values, an explicit incremental-update algorithm for the elasto-viscoplastic constitutive relation was developed in the intermediate frame by using the second Piola-Kirchoff (P-K) stress and Green stain. The increment of stress and slip resistance were solved by a calculation loop of linear equations sets. The reorientation of the crystal as well as the elastic strain can be obtained from a polar decomposition of the elastic deformation gradient. User material subroutine VUMAT was developed to combine crystal elasto-viscoplastic constitutive model with ABAQUS/Explicit. Numerical studies were performed on a cubic upset model with OFHC material (FCC crystal). The comparison of the numerical results with those obtained by implicit iterative algorithm and those from experiments demonstrates that the explicit algorithm is reliable. Furthermore, the effect rules of material anisotropy, rate sensitivity coefficient (RSC) and loading speeds on the deformation were studied. The numerical studies indicate that the explicit algorithm is suitable and efficient for large deformation analyses where anisotropy due to texture is important.

Fully conservative monotonic numerical scheme for streamer plasma 2D-modelling

A fully conservative numerical scheme for the streamer dynamies modelling, based on a monotonic second order Godunov-type finite difference scheme for charged particles dynamics simulation and an alternative directions, implicit iterative algorithm for electrostatic potential calculations is proposed.

Adaptive finite element procedures for linear and non-linear dynamics

This paper discusses implementation and adaptivity of the Discontinuous Galerkin (DG) finite element method as applied to linear and non-linear structural dynamic problems. By the DG method, both displacements and velocities are approximated as piecewise bilinear functions in space and time and may be discontinuous at the discrete time levels. Both implicit and explicit iterative algorithms for solving the resulted system of coupled equations are derived. They are third-order accurate and, while the implicit procedure is unconditionally stable, the explicit one is conditionally stable. An h-adaptive procedure based on the Zienkiewicz–Zhu error estimate using the SPR technique is applied. Numerical examples are presented to show the suitability of the DG method for both linear and non-linear structural dynamic analysis. Copyright © 1999 John Wiley & Sons, Ltd.

Automatic differentiation of advanced CFD codes for multidisciplinary design

Automated multidisciplinary design of aircraft and other flight vehicles requires the optimization of complex performance objectives with respect to a number of design parameters and constraints. The effect of these independent design variables on the system performance criteria can be quantified in terms of sensitivity derivatives which must be calculated and propagated by the individual discipline simulation codes. Typical advanced CFD analysis codes do not provide such derivatives as part of a flow solution; these derivatives are very expensive to obtain by divided (finite) differences from perturbed solutions. It is shown here that sensitivity derivatives can be obtained accurately and efficiently by using the ADIFOR source translator for automatic differentiation. In particular, it is demonstrated that the 3D, thin-layer Navier-Stokes, multigrid flow solver called TLNS3D is amenable to automatic differentiation in the forward mode even with its implicit iterative solution algorithm and complex turbulence modeling. It is significant that, using computational differentiation, consistent discrete nongeometric sensitivity derivatives have been obtained from an aerodynamic 3D CFD code in a relatively short time, e.g. O (man-week) not O (man-year).

Distributed minimal residual (DMR) method for acceleration of iterative algorithms

A new method for enhancing the convergence rate of iterative schemes for the numerical integration of systems of partial differential equations has been developed. It is termed the Distributed Minimal Residual (DMR) method, and is based on general Krylov subspace methods. The DMR method differs from the Krylov subspace methods by the fact that the iterative acceleration factors are different from equation to equation in the system. At the same time, the DMR method can be viewed as an incomplete Newton iteration method. The DMR method has been applied to Euler equations of gasdynamics and incompressible Navier-Stokes equations. All numerical test cases were obtained using either explicit four stage Runge-Kutta or Euler implicit time integration. The DMR method was found capable of reducing the computation time by 20–80% depending on the test case. When directly compared with an implicit residual smoothing, the DMR method performed consistently better and more reliably. The formulation for the DMR method is general in nature and can be applied to explicit and implicit iterative algorithms for arbitrary systems of partial differential equations.

ACCELERATED COMPUTATION OF VISCOUS INCOMPRESSIBLE FLOWS WITH HEAT TRANSFER

A new method for enhancing the convergence rates of iterative schemes for the numerical integration of systems of partial differential equations has been developed. It is termed the distributed minimal residual (DMR) method. The DMR method has been applied to incompressible Navier-Stokes equations with heat transfer. Alt numerical lest cases were obtained using explicit four-stage Runge-Kutla or Euler implicit lime integration. The DMR method was found to reduce compulation lime by 207ndash;60%, depending on the test case. The formulation for the DMR method is general in nature and can be applied to explicit and implicit iterative algorithms for arbitrary systems of partial differential equations.

Inception of Cavitation From a Solid Surface

This study of cavitation inception from a solid surface includes the presentation of an empirical model for the nuclei, a completely new design for experimental test equipment, a rigorous analysis of liquid conditions within the test section, and quantitative test data. The results prove that it is possible to reduce the effective size of cavitation nuclei, thereby making the interface more resistant to cavitation. The empirical model for cavitation nuclei provides the mechanism for bubble formation from gas-filled, submicroscopic crevices in a solid surface. The size of these nuclei may be reduced by forcing the liquid into these crevices. If the liquid-solid interface is hydrophilic, the pressure treatment is permanent, but the liquid will pull out of a hydrophobic interface thus providing only a temporary reduction of nuclei dimensions. The newly designed experimental equipment for cavitation inception studies has been demonstrated to produce cavitation under controlled conditions. The test section resembles a 180-deg journal bearing constructed in three slices. The middle slice is slid away from the journal to produce a diverging wedge which induces a tensile stress in the liquid. The inception stage of cavitation is detected by an acoustical probe. The temperature and pressure distributions in the thin film are calculated numerically by successively solving the energy and generalized Reynolds equations. A strongly implicit iteration algorithm is employed to solve the large set of simultaneous algebraic equations resulting from placing the nonlinear elliptic partial differential equations in finite difference form. Experimental data from test runs with petroleum oil, glycerin, and a glycerin-soap mixture showed that stress required for inception of cavitation could be determined with a repeatability of ±1.4 × 104 Pa (±2 psi). A linear relationship was experimentally determined to exist between the magnitude of pressure forcing oil into the crevices of a steel surface and a capacity of the oil-steel interface to withstand a tensile stress.

Computation of stable equilibria in minimum- B mirror devices

We present a three-dimensional code that solves for plasma equilibria in open-field geometries, which allows analysis of most minimum- B mirror systems. Open confinement with p = p( B) only requires the hydromagnetic equilibrium equation to reduce to one elliptic, scalar partial differential equation involving a tensor pressure. The finite difference form of the equation is solved by an implicit iterative algorithm similar to the ADI (alternating direction implicit) method. Cylindrical grid coordinates are employed, and the mesh spacing is variable to allow representation of far boundary conditions. We have time-optimized the program, so the memory requirements are large. It is found that, when the threshold for hydromagnetic instability is passed, the hydromagnetic equilibrium equation becomes hyperbolic, which leads to an ill-posed problem as predicted by Grad. Equilibria are found for an increasing sequence of pressures until the instability threshold is reached. The present code utilized planes of inversion symmetry common to many minimum- B devices, to allow a reduction in the domain to be studied; Ioffe bar devices, Baseball II, 2 X-II, and the proposed mirror fusion experiment are representable. We present sample results for Baseball II and the Ioffe bar device, including cases near the high beta instability threshold.