Difficulties In Numerical Simulation Research Articles

A Proper Discretization of Hydrodynamic Equations in Cylindrical Coordinates for Astrophysical Simulations

Abstract Cylindrical coordinates are often used in computational fluid dynamics, particularly when one is considering gas flow accreting onto a central object. Although cylindrical coordinates have several advantages in describing rotation, they have apparent singularity along the axis at the coordinate origin (z-axis). This singularity introduces difficulties in numerical simulations. First, it is difficult to reproduce the flow across the z-axis. Second, the time step is extremely shortened by the CFL condition near the z-axis because the numerical cell thereof is narrow in the azimuthal direction for a given angular resolution. Here, we propose a new discretization scheme to overcome these difficulties. In our new scheme, we consider changes in the direction of the unit vector within a cell when evaluating the flux across each cell surface. Besides, we evaluate the source term in the radial component of the momentum equation from the thermal and dynamic pressures acting on the azimuthal cell surface. The new scheme is designed to be free-stream-preserving so that flow with uniform density, pressure, and velocity is an exact solution of the discretized equation. These improvements are essential to using a lower angular resolution in the innermost area and thus to elongating each time step. Our examples demonstrate that the innermost circular region around the axis can be resolved by only six numerical cells. We present an application to an accreting compact star surrounded by a disk, in addition to Sod shock tube and rotating outflow tests.

Експериментальна система для дослідження впливу компонування заповнювачів негерметичних відсіків на густину потоків газовиділення

In recent years, components and materials of industrial and commercial quality have been in frequent use in the assembly of small spacecraft to reduce the cost of space projects. This may affect spacecraft and onboard equipment reliability and performance quality. In particular, a high risk may arise from unpredictable performance characteristics of components and materials of this type caused by outgassing flows of unregulated density. The situation is aggravated by difficulties in numerical simulation of the complex internal geometry of actual nonhermetic small spacecraft. The most efficient way to resolve this problem is to conduct laboratory tests. This paper presents an experimental system for studying the dependence of degassing processes on the geometry of arrangement of electronics boards and various structural materials in a nonhermetic module. The theoretic basis for experiments is a differential realization of the accumulation method, which allows one to differentiate the density of degassing flows from particularly arranged specimens and elements of the module’s inner surface. To reproduce the design features of nonhermetic nano- and picosatellites, models of nonhermetic modules were refined, and a special laboratory module was made on their basis. For this purpose, a unit of synchronous module evacuation control was updated, equipment was developed for positioning plate-type fillers of nonhermetic modules, the recording of gas-dynamic parameters of a nonhermetic module’s own internal atmosphere was provided, and an automated experiment control system was prepared. The laboratory module was integrated into the automated measuring system of the VAU-2M vacuum aerodynamic installation of the Institute of Technical Mechanics of the National Academy of Sciences of Ukraine and the State Space Agency of Ukraine. The experimental system developed was tested to show a satisfactory operability of both its components and the system as a whole. The system developed allows one to study outgassing from the surface of various components and materials, to measure the density of outgassing flows for various equipment arrangements, and to study the dynamics of outflow of the products of the own internal atmosphere through calibrated orifices, which simulate the outflow of the own internal atmosphere of a nonhermetic module through its unregulated clearances.

Relaxing Numerical Difficulties Arising from Discretization in the Cylindrical Coordinates

The cylindrical coordinates, (r, ϕ, z), are often used in our numerical simulations, in particular, when gas accretes onto a central object. Although the cylindrical coordinates have several advantages in describing rotation, they have apparent singularity at r = 0. The singularity induces some difficulties in numerical simulations. First, it is difficult to solve the flow across the axis numerically. Second, the time interval between two successive time steps is restricted by the innermost cell size, resulting in large numbers of time steps to follow dynamics in the outer regions. Here, we present a new discretization scheme to overcome these difficulties. First, we evaluate the centrifugal force from the dynamical pressure working on the azimuthal cell surface. Our method of the evaluation is based on the vector analysis. Second, we reduce the angular resolution appreciably. Our examples demonstrate that the innermost circular region around the axis can be resolved by only six numerical cells. We increase the angular resolution with the increase in the radius so that each numerical cell has an aspect ratio close to unity. Then all the numerical cells have nearly the same size, and the CFL condition is relaxed. We show our application to an accreting protostar after showing a Sod shock tube problem and a uniform flow. The test problems demonstrate that the apparent singularity around the axis have no critical effects on the results.

Comprehensive two-dimensional river ice model based on boundary-fitted coordinate transformation method

River ice is a natural phenomenon in cold regions, influenced by meteorology, geomorphology, and hydraulic conditions. River ice processes involve complex interactions between hydrodynamic, mechanical, and thermal processes, and they are also influenced by weather and hydrologic conditions. Because natural rivers are serpentine, with bends, narrows, and straight reaches, the commonly-used one-dimensional river ice models and two-dimensional models based on the rectangular Cartesian coordinates are incapable of simulating the physical phenomena accurately. In order to accurately simulate the complicated river geometry and overcome the difficulties of numerical simulation resulting from both complex boundaries and differences between length and width scales, a two-dimensional river ice numerical model based on a boundary-fitted coordinate transformation method was developed. The presented model considers the influence of the frazil ice accumulation under ice cover and the shape of the leading edge of ice cover during the freezing process. The model is capable of determining the velocity field, the distribution of water temperature, the concentration distribution of frazil ice, the transport of floating ice, the progression, stability, and thawing of ice cover, and the transport, accumulation, and erosion of ice under ice cover. A MacCormack scheme was used to solve the equations numerically. The model was validated with field observations from the Hequ Reach of the Yellow River. Comparison of simulation results with field data indicates that the model is capable of simulating the river ice process with high accuracy.

Self-Piercing Riveting Process Simulation of Joining Aluminium Alloy Sheets

In this study, a numerical model was built to simulate the self-piercing riveting (SPR) process using commercial LS-DYNA finite element code. The difficulties in numerical simulation of the SPR process, such as large deformation and fracture, were resolved by the use of an explicit solution process combined with the r-adaptive meshing method. The model was applied to joining two sheets of 6061 aluminium alloy. The effects of die geometry, rivet material properties and the adaptive step size in numerical calculation were studied, using 6061 aluminum sheets as a model system. The simulation agrees well with the experimental results in terms of geometric characterisitcs of the cross-sections of the joints formed.

Hyperelastic Muscle Simulation

Biological soft tissues like muscles and cartilages are anisotropic, inhomogeneous, and nearly incompressible. The incompressible material behavior may lead to some difficulties in numerical simulation, such as volumetric locking and solution divergence. Mixed u-P formulations can be used to overcome incompressible material problems. The hyperelastic materials can be used to describe the biological skeletal muscle behavior. In this study, experiments are conducted to obtain the stress-strain behavior of a solid silicone rubber tube. It is used to emulate the skeletal muscle tensile behavior. The stress-strain behavior of silicone is compared with that of muscles. A commercial finite element analysis package ABAQUS is used to simulate the stress-strain behavior of silicone rubber. Results show that mixed u-P formulations with hyperelastic material model can be used to successfully simulate the muscle material behavior. Such an analysis can be used to simulate and analyze other soft tissues that show similar behavior.

Numerical Simulations for Transport Aircraft High-Lift Configurations Using Adaptive Cartesian Grid Methods

Computations of complex flow fields around three-dimensional high-lift configurations using an adaptive Cartesian grid method and a cell-center finite-volume method are carried out and compared to wind-tunnel experiments. Because of complexities in both configuration and flowfield, there are great difficulties in numerical simulation of flowfields around transport aircraft high-lift configurations with leading slats and trailing flaps. In this paper, a modified Cartesian grid method is introduced to simulate the complex flow using the adaptive grid technique. The main emphasis is given to a multizone technique and a face-to-face algorithm and its applications. The multizone technique is employed to simplify the great difficulties in grid generation. The face-to-face algorithm is constructed to precisely establish the information exchange of flowfields on interfaces between zones. We solve Euler equations with a conventional algorithm, which includes a cell-center finite-volume method and an explicit four-stage Runge-Kutta time-stepping scheme in combination with a dual-time stepping approach. For three three-dimensional high-lift configurations, the numerical results are presented and analyzed. The computational results are in good agreement with the available experimental data show that the present method is feasible and effective for solving the flow around complex three-dimensional high-lift systems.

A mortar finite element approximation for the linear Poisson–Boltzmann equation

In this paper, a new numerical method is presented for solving the linear Poisson–Boltzmann equation. A mortar finite element method with fundamental solution and artificial boundary condition are used to deal with the difficulties in numerical simulation and optimal error estimates are obtained.

On two-dimensional unsteady incompressible fluid flow in an infinite strip

In this paper, we study two-dimensional incompressible fluid flow in an infinite strip. The stream function form of Navier–Stokes equation is considered, which keeps the physical boundary condition and avoids some difficulties in numerical simulations. The existence and uniqueness of global solution are proved. Some results on the regularity of solution are obtained. Copyright © 2000 John Wiley & Sons, Ltd.

岩盤挙動の過去へのシミユレーション

Simulation of rock behaviour going back to the past is proposed in this paper. Usually computer simulation is carried out solving partial differential equations as time goes by, that is, starting at tA and ending at tB with tA < tB. In the proposed method, computer simulation is performed from tB to tA. Analytical consideration may suggests that solution ascending time is similarly possible as that descending time. Numerical solution ascending time, however, has some difficulties because information of initial state is gradually dispersing as time goes by, therefore, very small difference of initial values at tB results in very large difference of solutions at tA. The difficulties in numerical simulation are discussed in this paper. For demonstrating and evaluating the feature of the proposed method, relaxation of rock stress is selected as an example case. A selected site is peripheral of Beijing, China where 100 million years ago a large rock mass was intruded and considered to cause large rock stress around the site. Supposing that rock stress continuously relaxed up to the present, a simulation back to the past is carried out obtaining material constants, then, simulation on to the future starting at the present is also carried out to estimate the creep strain after 10 thousand years. In the proposed method, only a model at the present is necessary. In the conventional simulation, however, requires modeling at the past which is considered to be more difficult and time consuming than modeling at the present. As stated in this paper, the proposed method has some difficulty in numerical solution, however, we considered this method is one of the promising methods to examine the long-term rock behaviour.

Indirect analysis of boundary stresses

One of the crucial difficulties in numerical simulation of metal forming is to describe precisely the boundary conditions, especially friction. Adequate friction laws have to be engineered and implemented but need to be ascertained experimentally. This contribution puts forward for use in massive forming a new model test based on the bulging test. To determine the local boundary stresses in the contact area at any given time, we make use of the fact that the tool's elastic deformation is readily measurable. It will be demonstrated that this permits both a qualitative evaluation of friction and a quantification of unknown factors in a friction law - the latter by analysing numerically the whole system.

Hyperbolic heat-conduction problems - Numerical simulations via explicit Lax-Wendroff-based finite element formulations

The paper describes numerical simulations for hyperbolic heat-conduction problems involving non-Fourier effects via explicit self-starting Lax-Weiidroff-based finite element formulations. For cases involving extremely short transient durations or for very low temperatures near absolute zero, the classical Fourier diffusion model for heat conduction breaks down since the wave nature of thermal energy transport becomes dominant. Major difficulties in numerical simulations include severe oscillatory solution behavior in the vicinity of the propagating shocks. The present paper describes an alternate methodology and different computational perspectives for effective modeling/analysis of hyperbolic heat-conduction models involving non-Fourier effects. In conjunction with the proposed formulations, smoothing techniques are incorporated to stabilize the oscillatory solution behavior and to accurately predict the propagating thermal disturbances. The capability of exactly capturing the propagating thermal disturbances at characteristic time-step values is noteworthy. Numerical test cases are presented to validate the proposed concepts for hyperbolic heat-conduction problems.

Stress intensity factors for cracks emanating from a curvilinear hole in an anisotropic plane under arbitrary loadings

Based on a specific fundamental solution, which is due to a point force or a concentrated moment acting in an infinite anisotropic plane with a crack, stress intensity factors for problems of a single or two collinear edge cracks emanating from a curvilinear hole can be obtained by satisfying the traction boundary condition on the hole. The fundamental solution used in this study considers inherently the crack geometry and the resultant boundary solutions apply only to the smooth hole. Thus the difficulty in numerical simulation arisen from crack tip singularity can be circumvented. Numerical examples evaluating stress intensity factors for the cracks originating from a circular or an elliptical hole subjected to various loadings are illustrated. The results are in excellent agreement with known solutions. Depending on geometry and the type of loading, the analysis shows that the effect of anisotropy on stress intensity factors may be strong or insignificant. Even under the uniaxial tension or uniform shear loading condition, mixed-mode deformation can still be observed due to the in-plane normal-shear coupling in anisotropy.