Two-equation (k, ϵ) turbulence computations by the use of a finite element model

A finite element technique is presented and applied to some one- and two-dimensional turbulent flow problems. The basic equations are the Reynolds averaged momentum equations in conjunction with a two-equation (k, ?) turbulence model. The equations are written in time-dependent form and stationary problems are solved by a time iteration procedure. The advection parts of the equations are treated by the use of a method of characteristics, while the continuity requirement is satisfied by a penalty function approach. The general numerical formulation is based on Galerkin's method. Computational results are presented for one-dimensional steady-state and oscillatory channel flow problems and for steady-state flow over a two-dimensional backward-facing step.     

Iterative-method performance evaluation for multiple vectors associated with a large-scale sparse matrix

ABSTRACT

Ensemble computing, which is an instance of capacity computing, is an effective computing scenario for exascale parallel supercomputers. In ensemble computing, there are multiple linear systems associated with a common coefficient matrix. We improve the performance of iterative solvers for multiple vectors by solving them at the same time, that is, by solving for the product of the matrices. We implemented several iterative methods and compared their performance. The maximum performance on Sparc VIIIfx was 7.6 times higher than that of a naïve implementation. Finally, to deal with the different convergence processes of linear systems, we introduced a control method to eliminate the calculation of already converged vectors.     

A SPECTRAL METHOD FOR UNSTEADY FLOW AND HEAT TRANSFER PAST A SPHERICAL DROPLET: PRIMITIVE VARIABLE FORMULATION

Abstract

A spectral method is developed based on the primitive variables for the time-dependent solution of the flow and the temperature past a spherical droplet. Both Chebyshev and Legendre polynomials are used to expand the velocity, pressure, and temperature in the radial and angular directions, respectively. The fractional time-stepping method suggested by Orszag (Orszag et al., 1980) is used for solving the flow and the pressure fields. Euler backward differencing is used for the integration of the energy equation. The computed steady-state drag coefficients are compared to those found in the literature for Reynolds numbers in the range from 0.5 to 50 for both the continuous and the dispersed phase. The transient drag coefficients and Nusselt numbers are compared with our previous study using a stream function-vorticity formulation (Nguyen et al, 1993). The comparison indicates that the present model is capable of predicting the correct nature of the flow and heat transfer associated with a droplet.     

NUMERICAL SOLUTIONS OF OPTIMAL CONTROL FOR THERMALLY CONVECTIVE FLOWS

We study the numerical solution of optimal control problems associated with two-dimensional viscous incompressible thermally convective flows. Although the techniques apply to more general settings, the presentation is confined to the objectives of minimizing the vorticity in the steady state case and tracking the velocity field in the non-stationary case with boundary temperature controls. In the steady state case we develop a systematic way to use the Lagrange multiplier rules to derive an optimality system of equations from which an optimal solution can be computed; finite element methods are used to find approximate solutions for the optimality system of equations. In the time-dependent case a piecewise-in-time optimal control approach is proposed and the fully discrete approximation algorithm for solving the piecewise optimal control problem is defined. Numerical results are presented for both the steady state and time-dependent optimal control problems. © 1997 John Wiley & Sons, Ltd.     

Relaxation and Recovery in Hydrogel Friction on Smooth Surfaces

Background

Hydrogels are crosslinked polymer networks that can absorb and retain a large fraction of liquid. Near a critical sliding velocity, hydrogels pressed against smooth surfaces exhibit time-dependent frictional behavior occurring over multiple timescales. The origin of these dynamics is unresolved

Objective

Here, we characterize this time-dependent regime and show that it is consistent with two distinct molecular processes: sliding-induced relaxation and quiescent recovery.

Methods

Our experiments use a custom pin-on-disk tribometer to examine poly(acrylic acid) hydrogels on smooth poly(methyl methacrylate) surfaces over a variety of sliding conditions, from minutes to hours.

Results

We show that at a fixed sliding velocity, the friction coefficient decays exponentially and reaches a steady-state value. The time constant associated with this decay varies exponentially with the sliding velocity, and is sensitive to any precedent frictional shearing of the interface. This process is reversible; upon cessation of sliding, the friction coefficient recovers to its original state. We also show that the initial direction of shear can be imprinted as an observable “memory”, and is visible after 24 hrs of repeated frictional shearing.

Conclusions

We attribute this behavior to nanoscale extension and relaxation dynamics of the near-surface polymer network, leading to a model of frictional relaxation and recovery with two parallel timescales.

     

An efficient ψ-v scheme for two-dimensional laminar flow past bluff bodies on compact nonuniform grids

We recently proposed a second-order accurate ψ-v formulation of the steady-state Navier-Stokes (N-S) equations on compact Cartesian nonuniform grids. In the current work, we extend the ideas of the aforesaid formulation and propose a second-order spatially compact, implicit, stable ψ-v formulation for the unsteady incompressible N-S equations. Contrary to the existing ψ-v finite difference formulations which use grid transformation, the proposed scheme is developed for nonuniform Cartesian grids without transformation specifically designed for two-dimensional laminar flow past bluff bodies. It has been implemented on problems of internal flows inside curved regions as well as those involving fluid-embedded body interaction. However, the robustness of the scheme is highlighted by the accurate resolution of a host of complex flows past bluff bodies with different physical set-ups and boundary conditions. It was seen to handle problems involving both uniform and accelerated flows across a wide range of structures of varied shape, namely, a flat plate, a circular cylinder, inclined square cylinder, and a wedge in channel hinged to the wall. Apart from elegantly capturing all the details of the shedded vortex structures under different circumstances, the scheme was also able to handle both Dirichlet and Neumann boundary with equal ease. In all the cases, our results are found to be extremely close to the available numerical and experimental results.     

Computations of Optimal Controls for Incompressible Flows

Abstract

This paper is concerned with numerical solutions of optimal control problems for unsteady, viscous, incompressible flows. In general, controls can be of the distributed type (external body force) or Dirichlet type 7lpar;e.g., boundary velocity). Here, wc only consider the former case, although most of what we present is also applicable to the latter. Two different optimization objectives and associated solution methodologies are described. One involves a global-in-time functional, the other a local-in-time functional. Which method is preferred depends on the specific application. Some test computational results are presented.     

时变振动系统数值积分多步方法

本文利用加权残数法,首次获得了时变系统数值积分多步方法的一般形式,并进而推导出了可实际应用的二步方法.然后通过两例说明该方法在求解时变系统初值问题及周期时变系统稳态解方面的应用.     

Numerical design of multimodal axisymmetric hypersonic nozzles for wind tunnels

A method for design of hypersonic nozzles for wind tunnels is developed and implemented on the basis of solving direct problems with various models of the medium and numerical methods of integration of gas-flow equations. Multimodal nozzles for operation in Mach number ranges M _out =8–14 and M _out =14–20 satisfying specified requirements are designed.     

Superconvergence of nonconforming finite element penalty scheme for Stokes problem using L 2 projection method

A modified penalty scheme is discussed for solving the Stokes problem with the Crouzeix-Raviart type nonconforming linear triangular finite element. By the L ² projection method, the superconvergence results for the velocity and pressure are obtained with a penalty parameter larger than that of the classical penalty scheme. The numerical experiments are carried out to confirm the theoretical results.     

Phragmén-lindelöf type results for harmonic functions with nonlinear boundary conditions

This paper is concerned with investigating the asymptotic behavior of harmonic functions defined on a three-dimensional semi-infinite cylinder, where homogeneous nonlinear boundary conditions are imposed on the lateral surface of the cylinder. Such problems arise in the theory of steady-state heat conduction. The classical Phragmén-Lindelöf theorem states that harmonic functions which vanish on the lateral surface of the cylinder must either grow exponentially or decay exponentially with distance from the finite end of the cylinder. Here we show that the results are significantly different when the homogeneous Dirichlet boundary condition is replaced by the nonlinear heatloss or heat-gain type boundary condition. We show that polynomial growth (or decay) or exponential growth (or decay) may occur, depending on the form of the nonlinearity. Explicit estimates for the growth or decay rates are obtained.     

State-Space Generalized Predictive Control for Redundant Parallel Robots

Abstract

The article deals with the design and properties of generalized predictive control (GPC) for path control of redundant parallel robots. Redundant parallel classification means redundant number of actuators, i.e., more actuators than degrees of freedom of the robot. Control of such structures suffers from several new control problems like potential inconsistency of steady state positions or nonuniqueness of control actions. The article explains classical direct derivation of GPC and its modification based on square root two-step design of control actions for solving the control problems. As an example for verification of algorithms, a prototype of a planar redundant parallel robot is used. Both design approaches are compared and several possibilities of extensions are presented for taking into consideration additional requirements, like smooth course of actuators or fulfillment of the anti-backlash condition.     

Anisotropic slope limiting for discontinuous Galerkin methods

In this paper, we present an anisotropic version of a vertex‐based slope limiter for discontinuous Galerkin methods. The limiting procedure is carried out locally on each mesh element utilizing the bounds defined at each vertex by the largest and smallest mean value from all elements containing the vertex. The application of this slope limiter guarantees the preservation of monotonicity. Unnecessary limiting of smooth directional derivatives is prevented by constraining the x and y components of the gradient separately. As an inexpensive alternative to optimization‐based methods based on solving small linear programming problems, we propose a simple operator splitting technique for calculating the correction factors for the x and y derivatives. We also provide the necessary generalizations for using the anisotropic limiting strategy in an arbitrary rotated frame of reference and in the vicinity of exterior boundaries with no Dirichlet information. The limiting procedure can be extended to elements of arbitrary polygonal shape and three dimensions in a straightforward fashion. The performance of the new anisotropic slope limiter is illustrated by two‐dimensional numerical examples that employ piecewise linear discontinuous Galerkin approximations. Copyright © 2017 John Wiley & Sons, Ltd.     

An optimal method for seismic drift design of concrete buildings using gradient and Hessian matrix calculations

This paper describes a novel seismic optimal design method for the reinforced concrete frame. First, an optimal mathematical model with time-dependent constraints, i.e., inter-story drift constraints, is established for achieving minimum weight design. Second, the inequality constraint problem with time-dependent constraints is converted into a sequence of appropriately formed unconstrained problems using the integral interior point penalty function method. Third, an efficient algorithm of the first and second derivatives of the inter-story drift with respect to design variables is formulated based on Newmark-β method. Gradient and Hessian matrix of the integral interior penalty function are also computed. Fourth, Marquardt’s method is employed to solve a sequence of unconstrained problems. Finally, the minimum weight design of a three-story, two-bay planar frame is demonstrated using the new optimization method and the augmented Lagrange multiplier method. The comparative results show the seismic optimal design method presented in this paper is more efficient than the augmented Lagrange multiplier method in terms of computational time. The proposed new method is an effective and efficient approach for minimum weight design of the reinforced concrete frames subjected to earthquake excitation.     

Inverse problems of 3D ultrasonic tomography with complete and incomplete range data

The paper focuses on the development of efficient methods for solving inverse problems of 3D ultrasound tomography as coefficient inverse problems for the wave equation. The idea of standard tomographic approaches to solving tomography problems is to analyze the 3D objects by their two-dimensional cross sections. This scheme is perfectly implemented in the case of X-ray tomography. Unlike X-ray tomography, ultrasonic tomography has to deal with diffraction and refraction effects, which limit the possibility of solving 3D problems by analyzing 2D cross sections. We propose efficient methods for solving inverse problems of ultrasound tomography directly in the 3D formulation. The proposed algorithms are based on the direct computation of the gradient of the residual functional. The algorithms are primarily oriented toward the development of ultrasound tomographs for differential diagnosis of breast cancer. Computer simulations demonstrated the high efficiency of the developed algorithms. The algorithms are implemented on GPU-based supercomputers. We analyze various schemes of 3D ultrasonic tomographs including those without rotating elements and with fixed positions of the sources and receivers. The algorithms developed can be used for solving inverse problems of seismology, acoustics, and electromagnetic sounding.     

Consistent vs. reduced integration penalty methods for incompressible media using several old and new elements

The frequently used reduced integration method for solving incompressible flow problems ‘a la penalty’ is critically examined vis-a-vis the consistent penalty method. For the limited number of quadrilateral and hexahedral elements studied, it is shown that the former method is only equivalent to the latter in certain special cases. In the general case, the consistent penalty method is shown to be more accurate. Finally, we demonstrate significant advantages of a new element, employing biquadratic (2-D) or triquadratic (3-D) velocity and linear pressure over that using the same velocity but employing bilinear (2-D) or trilinear (3-D) pressure approximation.     

DYNAMICAL APPROACH STUDY OF SPURIOUS STEADY-STATE NUMERICAL SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS II. GLOBAL ASYMPTOTIC BEHAVIOR OF TIME DISCRETIZATIONS ∗

SUMMARY

The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 × 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how “numerical” basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DCs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless ofthe stability ofthe spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.     

Numerical simulation of supersonic flow past reentry capsules

The flow fields over ARD (ESA's atmospheric reentry demonstrator), OREX (orbital reentry experiments) and spherically blunted cone-flare reentry configurations are numerically obtained by solving time-dependent, axisymmetric, compressible Navier–Stokes equations for freestream Mach numbers range of 1.2–6.0. The fluid dynamics are discretized in spatial coordinates employing a finite volume approach which reduces the governing equations to semi discretized ordinary differential equations. Temporal integration is performed using the multistage Runge–Kutta time-stepping scheme. A local time step is used to achieve steady-state solution. The numerical simulation is carried out on a structured grid. The flow-field features around the reentry capsule, such as bow shock wave, sonic line, expansion fan and recirculating flow in the base region are obtained. A good agreement is found between the calculated value of aerodynamic drag coefficient of the spherically blunted cone/fare reentry configuration with the experimental data. The effects of geometrical parameters, such as radius of the spherical cap, half cone angle, with sharp shoulder edge and with smooth shoulder edge on the flow-field have been numerically investigated for various reentry configuration which will be useful for optimization of the reentry capsule. PACS 47.11.Df, 47.40.Ki     

Time-dependent numerical simulation of viscoelastic flow and stability

The 4×4 mixed finite-element method is extended in order to calculate time-dependent viscoelastic flow. The basic algorithm uses a fully implicit technique with a predictor-corrector control of the time step for monitoring the accuracy. Excellent agreement is found with analytical test cases. Several decoupled schemes are also developed with a view to reducing the cost of the time-dependent calculations. However, none of them reaches that goal on actual flow problems because of a drastic reduction of the time step.The time-dependent algorithm is used for verifying the stability of steady-state viscoelastic flow solutions. The approach consists of using a steady-state solution as a set of initial conditions for a time-dependent algorithm with the hope that, in case of instability, the system will lead to a stable solution. More precisely, we consider the flow of an Oldroyd-B fluid through a four-to-one contraction. Starting from a steady-state flow, we impose a pressure impulse in the entry section and calculate the approach toward another steady-state. For planar contractions, we do not constrain the flow to be symmetric while, for circular contractions, the flow is endowed with swirling capabilities. It is found that the stable or unstable character of the flow depends upon the mesh as well as upon the method of discretization.     

Domain decomposition methods applied to solve frictionless-contact problems for multilayer elastic bodies

A parallel Dirichlet–Dirichlet domain-decomposition algorithm for solving frictionless-contact problems for elastic bodies made of composite materials is proposed and justified. Numerical results that demonstrate the effectiveness of the approach and its software implementation are presented