A fast and robust fictitious domain method for modelling viscous flows in complex mixers: The example of propellant make‐down

This work deals with the development of a fast three‐dimensional numerical strategy for the simulation of viscous fluid flow in complex mixing systems. The proposed method is based on a distributed Lagrange multiplier fictitious domain method and the use of the low‐cost MINI finite element. Contrary to the previous fictitious domain method developed by our group a few years ago, the underlying partial differential equations are solved here in a coupled manner using a consistent penalty technique. The method is discussed in detail and its precision is assessed by means of experimental data in the case of an agitated vessel. A comparison made with our existing fictitious domain method and its decoupled Uzawa‐based solver clearly shows the advantages of resorting to the MINI finite element and fully coupled solution strategy. The new technique is then applied to the simulation of the flow of a Newtonian viscous fluid in a three‐blade planetary mixer in the context of the production of solid propellants. Copyright © 2008 John Wiley & Sons, Ltd.     

Computing the structural buckling limit load by using dynamic relaxation method

The numerical structural analysis schemes are extensively developed by progress of modern computer processing power. One of these approximate approaches is called "dynamic relaxation (DR) method." This technique explicitly solves the simultaneous system of equations. For analyzing the static structures, the DR strategy transfers the governing equations to the dynamic space. By adding the fictitious damping and mass to the static equilibrium equations, the corresponding artificial dynamic system is achieved. The static equilibrium path is required in order to investigate the structural stability behavior. This path shows the relationship between the loads and the displacements. In this way, the critical points and buckling loads of the non-linear structures can be obtained. The corresponding load to the first limit point is known as buckling limit load. For estimating the buckling load, the variable load factor is used in the DR process. A new procedure for finding the load factor is presented by imposing the work increment of the external forces to zero. The proposed formula only requires the fictitious parameters of the DR scheme. To prove the efficiency and robustness of the suggested algorithm, various geometric non-linear analyses are performed. The obtained results demonstrate that the new method can successfully estimate the buckling limit load of structures.     

Upper Bounds on Deformations of Elastic-Workhardening Structures in the Presence of Dynamic and Second-Order Geometric Effects∗

Abstract

Discrete models of elastoplastic structures are considered, Piecewise linear yield conditions and hardening rules are assumed. On this basis, a deformation bounding method resting on the use of fictitious loads as proposed first by Ponter [6, 7], is developed for situations in which: (a) the geometry changes affect the equilibrium equations but their effects may be expressed by bilinear terms in the pre-existing stresses and additional displacements (“second-order geometric effects”); (b) inertia and viscous damping forces play a significant role. Comparisons are made with different bounding methods previously established by the author [3,4], for the same classes of structures and mechanical situations.     

DLT-Lines Based Camera Calibration with Lens Radial and Tangential Distortion

Background

Camera calibration is an essential step for the optical measurement method used in the experimental mechanics. Most plumb line methods focus on solving lens distortions without considering camera intrinsic and extrinsic parameters.

Objective

In this paper, we propose a full camera calibration method to estimate the camera parameters, including camera intrinsic parameters, extrinsic parameters and lens distortion parameters, from a single image with six or more non-coplanar lines.

Methods

We parameterize the 3D lines with the intersection of two planes that allow the direct linear transformation of the lines(DLT-Lines). Based on the DLT-Lines, the projection matrix is estimated linearly, and then the camera intrinsic and extrinsic parameters are extracted from the matrix. The relationship between the distorted 2D lines and the distortion coefficients is derived, based on which the distortion coefficients can be solved linearly. In the last step, a non-linear optimization algorithm is used to jointly refine all the camera parameters, including the distortion coefficients.

Results

Both synthetic and real data are used to evaluate the performance of our method, which demonstrates that the proposed method can calibrate the cameras with radial and tangential distortions accurately.

Conclusions

We propose a DLT-lines based camera calibration method for experimental mechanics. The proposed method can calibrate all the camera parameters from a single image.

     

A HYBRID SPECTRAL/FINITE VOLUME METHOD FOR VISCOUS FLOWS

Abstract

This paper presents a hybrid spectral/finite volume method for steady-state compressible viscous flows. The method is evaluated for accuracy via test cases for various Mach numbers. The domain is divided into a viscous region and an inviscid region. The viscous region uses the full Navier-Stokes equations, while the inviscid region employs the Euler equations. A high order Chebyshev collocation spectral method is developed for the viscous region to resolve boundary layers. This method avoids the dense grids needed by finite-volume methods to resolve the viscous areas. A low order finite-volume method based on a Lax-Wendroff type scheme is employed for the inviscid region. A special interface formulation is developed for coupling the spectral with the finite-volume method. Comparisons with analytic results as well as convergence histories are presented.     

ANALYSIS/DESIGN CALCULATIONS OF TRANSONIC FLOWS

SUMMARY

Analysis/design calculations of transonic flow are discussed and several improvements are made. The nonisentropic potential method is used to calculate the inviscid transonic flow analysis problem instead of the traditional potential method. An inverse integral 3D boundary layer method is used to calculate the boundary layer in the viscous transonic flow analysis problem. The viscous/inviscid interaction calculations are carried out by a semi-inverse coupling scheme. In design problem calculations, an improved residual-correction method is used. Three individual methods are combined in a global algorithm and computing code. The improvements speed up the convergence, increase applicability and computational efficiency. Some numerical results are given to illustrate that the present method provides an effective engineering tool of high accuracy and efficiency in three dimensional transonic analysis and design situations.     

Numerical study of the wall effect on particle sedimentation

ABSTRACT

In this article, we investigate the abnormal settling of two-disk systems and elliptical shaped particles in infinite two-dimensional channels filled with an incompressible viscous fluid. We apply a distributed Lagrange multiplier/fictitious domain method (DLM/FDM) for the direct numerical simulation of these particulate flows. Due to the wall effect, the two-disk systems can form chains which settle stably instead of having the particles moving apart. Also, sedimentation with the long axis moving to vertical positions in the middle of the infinite channel has been observed for the elliptic shaped particles. The critical Reynolds number for having such an abnormal settling behaviour decreases as the width of the channel increases.     

Application of a Special Yield Stress Strain Rate Law to Structural Impulse Problems

ABSTRACT

This paper presents approximate solutions to the dynamic response of three impulsively loaded structures: a wire with an impulsively loaded end mass, an impulsively loaded circular ring, and a cantilever beam with a tip mass subjected to an impulsive load at its tip. The material is assumed to be rigid, perfectly plastic with strain rate sensitivity. A proposed power law form of yield stress strain rate relationship is used to simplify the theoretical development. Numerical solutions are presented for mild steel and are compared with previously published results. Elastic effects and wave propogations are ignored.     

Identification method for backbone curve of cantilever beam using van der Pol-type self-excited oscillation

This study presents an experimental method for identification of the backbone curves of cantilevers using the nonlinear dynamics of a van der Pol oscillator. The backbone curve characterizes the nonlinear stiffness and nonlinear inertia of the resonator, so it is important to identify this curve experimentally to realize high-sensitivity and high-accuracy sensing resonators. Unlike the conventional method based on the frequency response under external excitation, the proposed method based on self-excited oscillation enables direct backbone curve identification, because the effect of the viscous environment is eliminated under the linear velocity feedback condition. In this research, the method proposed for discrete systems is extended to give an identification method for continuum systems such as cantilever beams. The actuation is given with respect to both the linear and nonlinear feedbacks so that the system behaves as a van der Pol oscillator with a stable steady-state amplitude. By varying the nonlinear feedback gain, we can produce the self-excited oscillation experimentally with various steady-state amplitudes. Then, using the relationship between these steady-state amplitudes and the corresponding experimentally measured response frequencies, we can detect the backbone curve while varying the nonlinear feedback gain. The efficiency of the proposed method is determined by identifying the backbone curves of a macrocantilever with a tip mass and a macrocantilever subjected to atomic forces, which are representative sources of hardening and softening cubic nonlinearities, respectively.

     

A distributed Lagrange multiplier/fictitious domain method for flows around moving rigid bodies: Application to particulate flow

This article discusses the application of a Lagrange multiplier‐based fictitious domain method to the numerical simulation of incompressible viscous flow modeled by the Navier–Stokes equations around moving rigid bodies; the rigid body motions are due to hydrodynamical forces and gravity. The solution method combines finite element approximations, time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method. The study concludes with the presentation of numerical results concerning four test problems, namely the simulation of an incompressible viscous flow around a NACA0012 airfoil with a fixed center but free to rotate, then the sedimentation of 200 and 1008 cylinders in a two‐dimensional channel, and finally the sedimentation of two spherical balls in a rectangular cylinder. Copyright © 1999 John Wiley & Sons, Ltd.     

Fictitious domain methods for two‐phase flow energy balance computations in nuclear components

This paper is dedicated to the numerical simulation of nuclear components (cores and steam generators) by fictitious domain methods. The fictitious domain approach consists in immersing the physical domain under study in a Cartesian domain, called the fictitious domain, and in performing the numerical resolution on this fictitious domain. The calculation times are then efficiently reduced by the use of fast solvers. In counterpart, one has to handle with an immersed boundary, generally non‐aligned with the Cartesian mesh, which can be non‐trivial. The two fictitious domain methods compared here on industrial simulations and developed by Ramière et al. deal with an approximate immersed interface directly derived from the uniform Cartesian mesh. All the usual immersed boundary conditions (Dirichlet, Robin, Neumann), possibly mixed, are handled through a unique formulation of the fictitious problem. This kind of approximation leads to first‐order methods in space that exhibit a good ratio of the precision of the approximate solution over the CPU time, which is very important for industrial simulations. After a brief recall of the fictitious domain method with spread interface (Ramière et al., CMAME 2007) and the fictitious domain method with immersed jumps (Ramière et al., JCP 2008), we will focus on the numerical results provided by these methods applied to the energy balance equation in a steam generator. The advantages and drawbacks of each method will be pointed out. Generally speaking, the two methods confirm their very good efficiency in terms of precision, convergence, and calculation time in an industrial context. Copyright © 2011 John Wiley & Sons, Ltd.     

FINITE ELEMENT MODELLING OF THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

SUMMARY

The paper presents a finite element model for the solution of the three-dimensional, time-dependent Navier-Stokes equations. Isoparametric brick elements are used, with tri-linear interpolation for both velocity and pressure. A fractional-step algorithm is applied, and the advection-diffusion part of the system is solved using a two-step Taylor-Galerkin formulation. The pressure is computed from a Poisson equation, which is solved numerically using a preconditioned conjugate gradient method.

Computations are performed for the flow about a sphere at Reynolds numbers 50 and 100, and comparisons are made with available measurements.     

Dynamic Subsurface Deformation and Strain of Soft Hydrogel Interfaces Using an Embedded Speckle Pattern With 2D Digital Image Correlation

Background

Subsurface mechanisms can greatly affect the mechanical behavior of biological materials, but observation of these mechanisms has remained elusive primarily due to unfavorable optical characteristics. Researchers attempt to overcome these limitations by performing experiments in biological mimics like hydrogels, but measurements are generally restricted due to the spatio-temporal limitations of current methods.

Objective

Utilization of contemporary 3D printing techniques into soft, transparent, aqueous yield-stress materials have opened new avenues of approach to overcoming these roadblocks. By incorporating digital image correlation with such 3D printing techniques, a method is shown here that can acquire full-field deformation of a hydrogel subsurface in real-time.

Methods

Briefly, the method replaces the solvent of a transparent and low polymer concentration yield-stress material with an aqueous hydrogel precursor solution, then a DIC speckle plane is 3D printed into it. This complex is then polymerized using photoinitiation thereby locking the speckle plane in place.

Results

Full-field deformation measurements are made in real-time as the embedded speckle plane (ESP) responds with the bulk to the applied load. Example results of deformation and strain fields associated with indentation, relaxation, and sliding contact experiments are shown.

Conclusions

This method has successfully observed the subsurface mechanical response in the bulk of a hydrogel and has the potential to answer fundamental questions regarding biological material mechanical behaviors.

     

Real-Time Multibody System Dynamic Simulation: Part II. A Parallel Algorithm and Numerical Results

ABSTRACT

This paper presents a computational algorithm that exploits inherent parallelism in the modified recursive formulation presented in Part I of the paper. Computational data flows to implement the algorithm are defined. By combining the topological analysis method presented in Part 1 of the paper, an efficient general purpose dynamic simulation algorithm is developed. Examples using the code developed show that real-time simulation can be achieved for moderately complex mechanical systems using a shared memory multiprocessor.     

Semi-analytical finite element method for fictitious crack model in fracture mechanics of concrete

Based on the Hamiltonian governing equations of plane elasticity for sectorial domain, the variable separation and eigenfunction expansion techniques were employed to develop a novel analytical finite element for the fictitious crack model in fracture mechanics of concrete. The new analytical element can be implemented into FEM program systems to solve fictitious crack propagation problems for concrete cracked plates with arbitrary shapes and loads. Numerical results indicate that the method is more efficient and accurate than ordinary finite element method.     

Numerical investigation on vortex-induced vibration of an elastically mounted circular cylinder at low Reynolds number using the fictitious domain method

A direct numerical simulation of two-dimensional (2D) flow past an elastically mounted circular cylinder at low Reynolds number using the fictitious domain method had been undertaken. The cylinder motion was modelled by a two degree-of-freedom mass–spring–damper system. The computing code was verified against a benchmark problem in which flow past a stationary circular cylinder is simulated. Then, analyses of vortex-induced vibration (VIV) responses, drag and lift forces and the phase and vortex structures were carried out. Results show that the cylinder's non-dimensional cross-flow response amplitude reaches its summit of 0.572 in the ‘lock-in’ regime. The ‘2S’, instead of the ‘2P’, vortex shedding mode is dominated in the ‘lower’ branch for this 2D low-Re VIV. A secondary oscillation is observed in the lift force when ‘lock-in’ occurs. It is shown that this secondary component changes the phase, offset the energy input by the primary component and thus reduces the cylinder responses. Effects of the Skop–Griffin parameter on cylinder responses were also investigated.