Difference Method For Solution Research Articles

Viscoelastic Simulation of Flow of Rubber Compounds

Abstract A hybrid scheme has been developed for numerical simulation of a nonisothermal viscoelastic flow in an arbitrary planar geometry of uniform thickness during extrusion and injection molding. This formulation is based on the control-volume finite-element method for solution of the continuity and momentum equations and finite difference method for solution of the energy equation. Application of this numerical technique for simulation of an extrusion and injection molding process was performed in a slit die and a quarter of a circular disk cavity, respectively. Development of pressures, shear and normal stresses, velocities, and temperature fields were calculated during nonisothermal flow of the rubber compound. Furthermore, the relaxation of stresses was calculated after cessation of the flow. The location of the meltfront was predicted during the cavity filling process. The contribution of normal stresses was studied by comparing results following from the viscoelastic and inelastic simulations.

Relations between maximum-entropy/ H∞ control and combined H∞/LQG control

The maximum-entropy/ H ∞-control problem and a combined H ∞/LQG-control problem are presented in a unified way and shown to be equivalent, in the case of a full-order controller and equalized H ∞/LQG weights. The result links two separate approaches which have distinctly different methods of solution. Only two decoupled algebraic Riccati equations are needed to state the controller, maximum value of the entropy and minimum value of the auxiliary cost.

Damage in composite laminates: Effects of transverse cracks

Two different methods of solution are used to study the effects of transverse cracks in cross-ply composite laminates. The results of an approximate analytical solution are compared with those obtained using a finite element analysis in order to study the effects of transverse cracks on the degradation of elastic and thermal coefficients as well as stress distributions. In particular, it is shown that transverse cracks cause significant degradation of the Poisson's ratio and shear modulus of the laminates, and also affect some stress distributions in a peculiar manner. Theoretical results are compared with existing experimental results where appropriate.

Simulation of foaming in reaction injection molding

AbstractBlowing agents are often used in the reaction injection molding process to compensate for the shrinkage that occurs upon polymerization, thereby creating a structural foam part. Controlling the thickness of the solid skin and foamed core is essential to achieve the intended mechanical properties of the molded part. A numerical model was developed to predict the foaming behavior in reaction injection molding. This algorithm employs a novel primitive cell construction to enable it to analyze complex rectangular geometries, including inserts, with a two‐dimensional, finite difference solution method. The analysis was applied to foaming of polyurethane in a rectangular cavity. The predicted skin thickness was found to be in good agreement with actual structural foam parts. Foaming as a function of cavity thickness was also treated. The algorithm Is useful for understanding and interpreting nonlinear phenomena of rapid, exothermic polymerizations such as foam formation adjacent to the mold wall or around a metal insert. The results can be used to formulate design guidelines for achieving desired skin/core thicknesses as a function of design, material, and process parameters.

Detecting Voronoi (area-of-influence) polygons

Voronoi, or area-of-influence, polygons are convex, space-filling polygons constructed around a set of points (Voronoi centers) such that each polygon contains all points closer to its Voronoi center than to the center of any other polygon. The relationship of Voronoi centers to edges of Voronoi polygons is used to test whether any convex tessellation consists of Voronoi polygons. This test amounts to finding Voronoi centers that best fit the given tessellation. Voronoi centers are found by solving two systems of linear equations. These equations represent (1) conditions on the slope of polygon edges relative to the slope of lines through Voronoi centers, and (2) conditions on the distance from edges to Voronoi centers. Least squares and constrained least-squares solutions are used to solve the two systems. Different methods of solution can provide insight as to how a tessellation varies from Voronoi polygons. A goodness-of-fit statistic is derived and examined by testing randomly generated convex tessellations. Some polygonal ice cracks provide an example of naturally occurring polygons that are approximated closely by Voronoi polygons.

A Full Potential Rotor Analysis With Wake Influence Using an Inner‐Outer Domain Technique

A three-dimensional, quasi-steady, full potential flow solver was adapted to include realistic rotor wake influence for the aerodynamic analysis of helicopter rotors. The method uses an inner/outer domain technique to accommodate wake effects. Nonlinear flow is computed in the inner domain using a finite difference solution method. The wake is modeled using prescribed wake techniques to allow for the inclusion of realistic wake geometries. Portions of the wake passing inside the inner domain are treated using an embedded vortex technique. The procedure couples the wake influence with the inner domain solution in a consistent and efficient solution process. Correlation with measured lifting transonic data in hover and forward flight is shown which demonstrates the merits of the approach.

Higher-order accurate difference solutions of vortex generation from a circular cylinder in an oscillatory flow

A higher-order difference solution method is presented for viscous flow around an oscillating body. The Navier-Stokes equation in a rotational form, which conserves both momentum and kinetic energy, is approximated into a difference form by a fourth-order compact differencing in a general boundary-fitted curvilinear coordinate system. A time-marching procedure is derived and the high resolution property is demonstrated with the computations of some simple problems. The simulation of a viscous flow around a circular cylinder sinusoidally oscillating at low Keulegan-Carpenter number elucidates fairly well the sequential viscous flow mechanism of boundary layer development, flow separation, and vortex shedding with pairing.

Numerical Simulation of a Viscous Flow with a Free Surface around a Ship Model

A new, finite difference solution method for the Navier-Stokes equations is developed to simulate viscous flow with a free surface around a ship. A body-fitted coordinates system that does not fit to the free surface is used for easy application to ships of arbitrary form. The nonlinear free surface condition is implemented by a special method. Simulated results for free surface flows around a Wigley hull at low Reynolds number are presented and compared with experimental data.

Mathematical optimization in the determination of the dissociation constants of a tribasic organic acid by 13C NMR spectroscopy

The calculation of dissociation constants from the chemical shifts of (13)C NMR spectra leads to a complicated non-linear equation. Two different mathematical methods for solution of this equation have been chosen--an iterative step method and a matrix pseudo-inversion method. When the iteration method is used the initial guesses for the parameters, the initial value of the step size and the escalation of the iteration must be optimized. For comparison the matrix pseudo-inversion method was used because it gives a unique result. With the optimized step method the results were as accurate or even better than those obtained with the matrix method. Although it takes time to optimize the system, the step method is the more suitable method of solving the problem. The matrix inversion can be done only with a computer with 13 significant digits and exponent capacity larger than +/- 38.

Comparisons between different methods of solution of the Boltzmann equation adapted to the calculation of swarm parameters in a weakly ionised medium

Two numerical solutions of the Boltzmann equation are developed and applied to the calculations of swarm time-of-flight parameters in molecular model gases. It is shown that the two methods not only give near similar results but are in good agreement with calculations carried out by some other authors for the same situations. Errors already pointed out in some other works are confirmed and the need of using high-order methods of solution of the Boltzmann equation is stressed.

Diffusion in a bistable potential: A comparative study of different methods of solution

The problem of diffusion in a bistable potential is studied by considering the associated nonlinear Langevin equation and its equivalent Fokker-Planck equation. Two numerically exact methods of solution, namely, the Monte Carlo solution of the nonlinear Langevin equation and the solution of the Fokker-Planck equation via the finite difference technique, are considered. The latter method has the advantage that it directly gives the evolution of the probability distribution function. Approximate analyses of the fluctuations using the system size expansion, the Gaussian decoupling procedure, and the scaling approach are also carried out. These investigations are performed on a representative problem for two specific cases: (1) evolution from intrinsically unstable states and (2) evolution from extensive regime. The fluctuations obtained using these approximate methods are compared with those obtained via the numerically exact methods. The study brings out the advantages and limitations of each of the methods considered.

Residual stresses in a CrMoV-2CrMo pipe weld: Part 1—The as-welded condition

There is a complete lack of reliable data concerning the through thickness distribution of residual stress in welds, both for the as-welded and the stress relieved conditions. As a result, the assessment of defects in welds and current stress relieving practice are largely based on surface measurements of residual stress. To improve this situation a technique has been established which permits the complete distribution of residual stress to be determined. This technique has been applied to a ferritic main steam pipe weld, in the as-welded condition. In establishing this technique, which uses experimental measurements combined with finite element analyses, three different methods of solution were investigated involving line loads, band loads and triangular loads. Only the last was found to give acceptable accuracy. The results of the analysis show that, in general, the residual hoop stresses over the outer two-thirds of the weld are tensile, being compressive over the inner third. Axial stresses are tensile over the outer third and compressive over the inner two-thirds. The approximations and assumptions used in the analysis are discussed, together with the implications that the results have on defect assessment and stress relieving.

Influence of mantle anelasticity on the phase and amplitude of Earth tides

The effect of the anelasticity of the mantle on the phase and amplitude of Earth tides is calculated for recent models of the internal structure of the Earth and its rheological characteristics. The anelastic properties of the mantle are modelled by the Maxwell and Knopoff-Lomnitz rheological bodies. For numerical calculations two different methods of solution are used. Results indicate that the effect of mantle anelasticity on tidal amplitudes is practically zero. For both types of rheological models the phase shifts of the functions characterizing solid tides are small, none of them exceeding values of some minutes of arc. These phase shifts have a very weak dependence on the variation of attenuation and viscosity within the mantle. The present study is closely related to an important problem: what proportion of the observed tidal friction arises not in the ocean but is due to the anelasticity of the mantle? The results suggest that dissipation by solid friction at present is an insignificant, almost negligible component of tidal energy sink.

Curvilinear finite difference method for biharmonic equation

AbstractThe essence of the proposed curvilinear finite difference solution method is the direct solution of the transformed fourth order partial differentiall equation using the finite difference expressions based on the flexible structure so that the geometry generality of the finite element method can be approached. The examples of applications indicte that the curvilinear mesh can be used and the boundary condition on the curved boundary can be easily and accurately prescribed.

Convection in a porous cavity

Convection in a porous cavity driven by heating in the horizontal is analysed by a number of different techniques which yield a fairly complete description of the two-dimensional solutions. The solutions are governed by two dimensionless parameters: the Darcy-Rayleigh number R and the cavity aspect ratio A. We first find solutions valid for shallow cavities, A → 0, by using matched asymptotic expansions. These solutions are given up to O(A6R4). For A fixed, we find regular expansions in R by semi-numerical techniques, up to O(R30) in some cases. Series-improvement techniques then enable us to cover the range 0 ≤ R ≤ ∞. A limited result regarding bifurcations is noted. Finally, for R → ∞ with A fixed, we propose a self-consistent boundary-layer theory which extends previous approximate work. The results obtained by these different methods of solution are in good agreement with each other and with experiments.

The use of angles and angular rates

MIT's Lincoln Laboratory has developed a computer driven, rapidly slewing (≃4° s−1), electro-optical (≃3″ resolution) telescope. This enables the rapid measurement of angles and instantaneous angular rates for artificial satellites. The simultaneous acquisition of angles and angular rates constitutes a new initial orbit problem which has been solved. Three different methods of solution are presented including an exact, analytical one. Numerical tests on six widely different satellite orbits indicate that the topocentric distance can be determined to better than 1% (and usually as well as 0.1%) for most satellites after a 5–10 min observation interval.

A comparative study of elasticity, shell and boundary layer solutions applied to axially compressed cylinders

This paper deals primarily with a comparative study based on different methods of solution for the problem of axially compressed cylinders. A comprehensive discussion on the range of validity of these types of solutions is also included, and an extensive numerical analysis has been carried out.

Benchmark Calculations for a Sodium-Cooled Breeder Reactor by Two- and Three-Dimensional Diffusion Methods

This investigation concentrates on the numerical solution of the multigroup neutron diffusion equations by computer codes. For a realistic model liquid-metal fast breeder reactor, several benchmark problems in two and three space dimensions were derived and calculations were performed by eight different computer programs. The effect on keff and the neutron fluxes of the refinement of the discretization mesh is studied.Very good agreement (∼0.05%) of the results was found in those cases where the computer programs use the same discretization scheme of mesh-edged discretization formulas, although the codes employ different methods of solution. On the other hand, minor discrepancies remain between results obtained by codes using mesh-edged and mesh-centered discretization formulas, even for fine-mesh grids. The reasons are not understood in every detail. Fortunately, these discrepancies are very small and more of theoretical than practical interest.The effect of a simple group condensation scheme on keff was also investigated by considering several different energy group structures. Spatial mesh refinements and resolution of the energy range were found to be well decoupled.As the main result, one may take the fact that spatial and energetic mesh refinements may influence the results rather strongly, unless the mesh step is comparable to the minimum diffusion length and unless enough energy groups are used.

The semi-detached system RT Persei and its light curve

The observedV light curve of the eclipsing binary RT Persei has been analysed by four different methods of solution to get the geometrical and photometric elements of the system. The results show a good agreement within about 6%.

Analysis of the V light curve of the system CW Cas

The V light curve of CW Cas, observed during 1972, has been analysed by different methods of solution. The results show great differences between rectifiable and direct methods.