Three-dimensional Benchmark Research Articles

A Vector Form Exchange-Area-Based Method for Computation of Anisotropic Radiative Transfer

Exchange-area-based methods (EAMs), such as the zonal method, the combined Monte-Carlo and zonal method, etc., are widely used in the numerical computation of thermal radiative transfer. In view of their inefficiency in treating with the anisotropy of radiation, researchers have been devoted to developing generalized models (GEAMs). In this article a vector form GEAM is proposed, which has concise formulations, and is found to be very efficient for handling the anisotropic scattering and reflection. In the model some vectorlike weighted-summations take account of the directional distribution of radiant energy. The summations give rise to correction coefficients to the radiosities. The heat transfer problem is solved by a coupled computation of the radiosities and the correction coefficients. The model was validated by both a one-dimensional and a three-dimensional benchmark radiation problem. The computational time is decreased to an order of that for computing isotropic radiation by conventional EAM. Such high efficency has never been achieved by the GEAMs proposed to date.

Free-edge stresses in general composite laminates

In the present study, by starting from the reduced form of elasticity displacement field for a long flat laminate, an analytical method is developed in order to accurately calculate the interlaminar stresses near the free edges of generally laminated composite plates under extension. The constant parameter appearing in the reduced displacement field, which describes the global rotational deformation of a laminate, is appropriately obtained by employing an improved first-order shear deformation theory. The accuracy and effectiveness of the proposed first-order theory are verified by means of comparison with the results of Reddy's layerwise theory as a three-dimensional benchmark. Reddy's layerwise theory is then utilized for analytical and numerical investigations of the boundary-layer stresses within arbitrarily laminated composite plates. Various numerical examples are presented for the interlaminar normal and shear stresses along the interfaces and through the thickness of laminates in the vicinity of the free edges. The effects of end conditions of laminates, fibers orientation angles as well as the stacking sequences of the layers within laminates, and geometric parameters on the boundary-layer stresses are presented and discussed.

An immersed boundary method for smoothed particle hydrodynamics of self-propelled swimmers

We present a novel particle method, combining remeshed Smoothed Particle Hydrodynamics with Immersed Boundary and Level Set techniques for the simulation of flows past complex deforming geometries. The present method retains the Lagrangian adaptivity of particle methods and relies on the remeshing of particle locations in order to ensure the accuracy of the method. In fact this remeshing step enables the introduction of Immersed Boundary Techniques used in grid based methods. The method is applied to simulations of flows of isothermal and compressible fluids past steady and unsteady solid boundaries that are described using a particle Level Set formulation. The method is validated with two and three-dimensional benchmark problems of flows past cylinders and spheres and it is shown to be well suited to simulations of large scale simulations using tens of millions of particles, on flow-structure interaction problems as they pertain to self-propelled anguilliform swimmers.

Generalized Runge–Kutta method for two- and three-dimensional space–time diffusion equations with a variable time step

An extensive knowledge of the spatial power distribution is required for the design and analysis of different types of current-generation reactors, and that requires the development of more sophisticated theoretical methods. Therefore, the need to develop new methods for multidimensional transient reactor analysis still exists. The objective of this paper is to develop a computationally efficient numerical method for solving the multigroup, multidimensional, static and transient neutron diffusion kinetics equations. A generalized Runge–Kutta method has been developed for the numerical integration of the stiff space–time diffusion equations. The method is fourth-order accurate, using an embedded third-order solution to arrive at an estimate of the truncation error for automatic time step control. In addition, the A( α)-stability properties of the method are investigated. The analyses of two- and three-dimensional benchmark problems as well as static and transient problems, demonstrate that very accurate solutions can be obtained with assembly-sized spatial meshes. Preliminary numerical evaluations using two- and three-dimensional finite difference codes showed that the presented generalized Runge–Kutta method is highly accurate and efficient when compared with other optimized iterative numerical and conventional finite difference methods.

Numerical Benchmarks for Very High-Temperature Reactors Based on the CNPS Critical Experiments

An evaluation of the Compact Nuclear Power Source (CNPS) experiments conducted at Los Alamos National Laboratory in the 1980s has been done using information available in the open literature. The MCNP4C Monte Carlo results for critical test configurations are in good agreement with the experimental values; the keff values are generally within 0.5% of the experimental values. The calculated total and differential rod worths and material worths were also found generally close to experimental values. These good results motivated the utilization of the experimental test data for the specification of two- and three-dimensional numerical benchmark cases that could be used for the verification and validation of core physics codes developed for Very High Temperature Reactor (VHTR) analysis, particularly the deterministic lattice and whole-core physics codes. To define the benchmark cases, the irregular arrangement of channels in the actual CNPS core was simplified to a regular Cartesian geometry arrangement in the benchmark cases, while preserving the important neutronics characteristics of the CNPS. The results of deterministic calculations using the HELIOS/DIF3D code package were compared to MCNP4C results to show the usefulness of the numerical benchmark cases.

A solution-adaptive central-constraint transport scheme for magnetohydrodynamics

The central-constraint transport scheme for magnetohydrodynamics in Ziegler [J. Comput. Phys. 196 (2004) 393] is made adaptive employing a block-structured mesh refinement method. Based on the guidelines in Berger and Collela [J. Comput. Phys. 82 (1989) 64] a mesh refinement variant has been developed which combines a flexible grid adaptation by using small blocks as refinement elements with integration speed by reducing the inherent overhead via block clustering techniques. The algorithms are discussed in detail and the efficiency of the implementation is benchmarked in terms of an efficiency parameter which takes into account both the obtained speedup factor and an error estimate. The three-dimensional benchmark problems are a spherical implosion problem and a shock-cloud collision problem in a magnetic medium. Further examples of astrophysical interest are presented which demonstrate the robustness and versatility of the new adaptive grid code.

Micro–macro simulations of three-dimensional plane contraction flow

Micro–macro calculations of viscoelastic flows offer some advantages in specific cases when compared against classical finite elements calculations based on the constitutive equation formulation. On the other hand, they are much more demanding than these in terms of memory and CPU time for the same calculation. The steep advance in computer technology makes it possible to tackle realistic problems of interest for the industry based on the micro–macro concept. In this work, micro–macro calculations are applied for the first time to the calculation of a three-dimensional benchmark problem: the plane 4 : 1 : 4 three-dimensional contraction flow. The size of the problem is very large and some technical issues need to be carefully handled, but it can be easily adapted to a parallel computing environment.

A HIGH-EFFICIENCY PARALLEL SOLUTION OF THE RADIATIVE TRANSFER EQUATION

Solution of the radiative transfer equation for nontrivial geometry and optical properties is a massive numerical problem, and can benefit from parallel processing to reduce the time to solution. Spatial parallelization is presented, applied to the DOTS algorithm with finite-volume directional discretization, and an explicit pseudo-time-stepping iterative solution with multigrid acceleration. Two- and three-dimensional benchmark problems were investigated with up to 16 processors on a shared-memory vector machine. In contrast to previous studies, high parallel efficiency was demonstrated up to the highest number of processors. The reasons for the good performance can be traced to the explicit iterative algorithm, which permits easy parallelization without affecting the rate of convergence; and to the use of high-bandwidth shared memory, which significantly reduces communication overhead.

Development of Three-Dimensional PML Boundary Conditions for Aeroacoustics Applications

Perfectly Matched Layer (PML) boundary conditions are derived in generalized curvilinear coordinates for three-dimensional aeroacoustic applications. The resulting governing equations are solved numerically by a four-stage Runge-Kutta scheme, with 4th/6th order compact finite difference formulation. The PML equations are programmed in a subroutine, which is easily incorporated to the main program LINEULER (Linearized Euler's equation solver). Two and three-dimensional benchmarks problems are solved to investigate the efficiency and accuracy of the PML boundary conditions. Investigations on the PML parameters have been conducted to determine the optimum combination of parameters used in the computations.

Nodal High-Order Methods on Unstructured Grids: I. Time-Domain Solution of Maxwell's Equations

We present a convergent high-order accurate scheme for the solution of linear conservation laws in geometrically complex domains. As our main example we include a detailed development and analysis of a scheme for the time-domain solution of Maxwell's equations in a three-dimensional domain. The fully unstructured spatial discretization is made possible by the use of a high-order nodal basis, employing multivariate Lagrange polynomials defined on the triangles and tetrahedra, while the equations themselves are satisfied in a discontinuous Galerkin form with the boundary conditions being enforced weakly through a penalty term. Accuracy, stability, and convergence of the semidiscrete approximation to Maxwell's equations is established rigorously and bounds on the growth of the global divergence error are provided. Concerns related to efficient implementations are discussed in detail. This sets the stage for the presentation of examples, verifying the theoretical results, and illustrating the versatility, flexibility, and robustness when solving two- and three-dimensional benchmark problems in computational electromagnetics. Pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated.

Second-order distributed plasticity analysis of space steel frames

This paper provides benchmark solutions of space steel frames using second-order distributed plasticity analysis. The majority of available benchmark solutions of steel frames in the past were only of two-dimensional frames. Therefore, three-dimensional benchmark solutions are needed to extend the knowledge of this field. Details of the modeling including element type, mesh discretization, material model, residual stresses, initial geometric imperfections, boundary conditions, and load applications are presented. Case studies of Vogel's portal frame and space steel frames are performed. The ultimate loads obtained from the proposed analysis and Vogel agree well within 1% error. The ultimate loads of the space steel frames obtained from the proposed analysis and experiment compare well within 3∼5% error. The benchmark solutions of the space steel frames are useful for the verification of various simplified second-order inelastic analyses. It is observed that the load carrying capacities calculated by the AISC-LRFD method are 25∼31% conservative when compared with those of the proposed analysis. This difference is attributed to the fact that the AISC-LRFD approach does not consider the inelastic moment redistribution, but the analysis includes the inelastic redistribution effect.

NEA Organised Validation of Multi-Dimensional Transport Codes and Cross-Section Sets Used Presently in Reactor Shielding

A Task Force of the OECD/NEA Nuclear Science Committee performed benchmark calculations to validate two- and three-dimensional transport codes and cross-section sets used for radiation shielding. Based on careful measurements from the VENUS critical facility of SCK/CEN-Mol/Belgium, the Task Force started with two-dimensional benchmark calculations representing the arrangement of VENUS-1, which is a mock-up of the Westinghouse three loop reactor. About 20 different calculations were presented. Several 2D-SN-codes were used including some in-house codes and the Monte Carlo program MCNP-4A. The transport cross-sections were taken from ENDF/B-VI or JEF 2.2 and the dosimetry data mainly from IRDF-90v2. The most challenging task was the validation of the latest versions of three-dimensional transport codes with the complex arrangement of VENUS-3. Approximately 14 independent benchmark calculations were contributed world-wide. Several versions of 3D-SN-codes and the Monte Carlo code MCNP were applied. In conclusion the results of the three-dimensional benchmark (VENUS-3) are in general much closer to the experimental values than for the two-dimensional benchmark (VENUS-1). This shows the inherent deficiencies of two-dimensional transport calculations, due to approximations such as diffusion based axial buckling corrections and improper averaging of source and shielding materials, since real configurations are always three-dimensional.

EXAMINATION OF CONVENTIONAL AND EVEN-PARITY FORMULATIONS OF DISCRETE ORDINATES METHOD IN A BODY-FITTED COORDINATE SYSTEM

The conventional radiative transfer equation (RTE) and the even-parity formulation (EPF) of the RTE in a general body-fitted coordinate system have been developed and they are used to simulate multi-dimensional radiative heat transfer in irregular geometries by the discrete ordinates method (DOM). The discrete ordinates equations for the EPF are second-order differential equations and they are spatially discretized using a second-order central difference scheme. At the boundary, a higher-order upwind scheme is employed to prevent solution instability and minimize errors. The spatially discretized equations are solved by a preconditioned conjugate gradients method. To investigate the accuracy and efficiency of the conventional RTE and the even-parity RTE in a body-fitted coordinate system, five two-dimensional and three-dimensional benchmark problems with absorbing-emitting and scattering media enclosed by irregular walls are considered. Compared to the conventional RTE, the EPF of the DOM is found to be more sensitive to the grid layout and it requires a clustered grid near the wall in order to provide accurate results in radiative wall heat flux. With an appropriate selection of a grid, the even-parity solution generally has an accuracy close to the conventional discrete ordinates solution. However, it usually requires more CPU time and iterations to converge, especially for the case with a small optical thickness. The present study indicates that the conventional RTE of the DOM is a more computationally robust radiation model than the EPF of the DOM in a body-fitted coordinate system.

Theory of acoustic radiation near a hyperbolic ridge

A solution to the acoustic field due to a harmonic point source near a hyperbolic ridge with perfectly reflecting boundaries in an isovelocity ocean is derived and developed. An exact expression for the field is given as a modal sum of integrals. The integrals are in terms of eigenfunctions of the reduced wave equation in the three-dimensional elliptical coordinate system. The eigenfunctions are approximated away from the low-frequency limit by standard WKB techniques. The resulting integrals are estimated by first and second order stationary phase asymptotics, which are matched in the vicinity of the caustics, yielding a complete representation of the field. The field is given in terms of standard elliptic functions for which fast numerical routines are available. The solution includes the locations of the caustics and shadow zones, as well as predicting a complicated intramodal interference pattern resulting from the intersection of up to three rays in a given mode. The explicit representation of these features arising from horizontal refraction makes this theoretical model useful as a new fully three-dimensional benchmark solution.

PREDICTION OF RADIATIVE TRANSFER IN GENERAL BODY-FITTED COORDINATES

The discrete ordinates method (DOM) and the finite-volume method (FVM) are employed to simulate two-dimensional and three-dimensional radiation problems in general body-fitted coordinates. Numerical analyses of these two methods indicate that the discretized equations for the DOM and the FVM are very similar. However, the FVM uses the averaged direction and scattering phase function over a control angle rather than the discrete direction and scattering phase function as in the DOM. To examine the accuracy and computational efficiency of the DOM and FVM in general body-fitted coordinates, five two-dimensional and three-dimensional benchmark problems with transparent, absorbing, emitting, and scattering media are considered. The solutions from both methods are found to be in good agreement with other available solutions. In some cases, the results from the FVM seem slightly better than those from the DOM. The required CPU time and iterations for the DOM and FVM are essentially the same for each problem. This finding is slightly different from other studies that claim that the FVM requires more CPU time to converge than the DOM for scattering-medium computation.

Parallel benchmarks of turbulence in complex geometries

In this paper we present benchmark results from the parallel implementation of the three-dimensional Navier-Stokes solver Prism on different parallel platforms of current interest: IBM SP2 (all three types of processors), SGI Power Challenge XL and Cray C90. The numerical method is based on mixed spectral element-Fourier expansions in ( x − y) and z-directions, respectively. Each one (or a group) of the Fourier modes can be computed on a separate processor as the linear contributions in Navier-Stokes equations (Helmholtz solvers) are completely uncoupled. Coupling is obtained via the non-linear contributions (advection terms) and requires a global transpose of the data and one-dimensional multiple-point FFTs. We first analyze the computational complexity of Prism identifying basic computational kernels and communication bottlenecks. These kernels are benchmarked separately on several computer architectures. More specifically, we obtain timings for BLAS routines on all aforementioned processors but also on the SP1, Cray J90, the Intel Paragon, and the DEC AlphaServer 8400 5/300. A two-dimensional version of Prism is also benchmarked on most processors testing both direct and iterative methods of solution. Next, we define two three-dimensional benchmark flow problems in prototype complex geometries, and measure parallel scalability and performance using different message-passing libraries. Special emphasis is placed on runtime data processing, e.g. turbulence statistics computed during the simulation. Our results provide a measure of MFlop s sustained performance of individual processors, and indicate the limitations of the current communications software for typical problems in computational mechanics based on spectral or finite element discretizations.

An Incomplete Domain Decomposition Preconditioning Method for Nonlinear Nodal Kinetics Calculations

Methods are proposed for the efficient parallel solution of nonlinear nodal kinetics equations. Because the two-node calculation in the nonlinear nodal method is naturally parallelizable, the majority of the effort is devoted to the development of parallel methods for solving the coarse-mesh finite difference (CMFD) problem. A preconditioned Krylov subspace method (biconjugate gradient stabilized) is chosen as the iterative algorithm for the CMFD problem, and an efficient parallel preconditioning scheme is developed based on domain decomposition techniques. An incomplete lowerupper triangular factorization method is first formulated for the coefficient matrices representing each three-dimensional subdomain, and coupling between subdomains is then approximated by incorporating only the effect of the nonleakage terms of neighboring subdomains. The methods are applied to fixed-source problems created from the International Atomic Energy Agency three-dimensional benchmark problem. The effectiveness of the incomplete domain decomposition preconditioning on a multiprocessor is evidenced by the small increase in the number of iterations as the number of subdomains increases. Through the application to both CMFD-only and nodal calculations, it is demonstrated that speedups as large as 49 with 96 processors are attainable in the nonlinear nodal kinetics calculations.

Three-Dimensional Benchmark Tests of ENDF/B-VI for Thermal Reactor Core.

Three-Dimensional Benchmark Tests of ENDF/B-VI for Thermal Reactor Core.

Qualification of the ARROTTA Code for Light Water Reactor Accident Analysis

Qualification efforts have been performed by the Taiwan Power Company (TPC) and the Institute of Nuclear Energy Research (INER) for the three-dimensional spatial kinetics code ARROTTA for light water reactor (LWR) core transient analysis. Together TPC and INER started a 5-yr project in 1989 to establish independent capabilities to perform reload design and transient analysis utilizing state-of-the-art computer programs. As part of the effort, the ARROTTA code was chosen to perform multidimensional kinetics calculations such as rod ejection for pressurized water reactors and rod drop for boiling water reactors (BWR). To qualify ARROTTA for evaluation of the Final Safety Analysis Report licensing basis core transients, ARROTTA has been benchmarked for the static core analysis against plant measured data and SIMULATE-3 predictions, and for the kinetic analysis against available benchmark problems. The static calculations compared include critical boron concentration, core power distribution, and control rod worth. The results indicate that ARROTTA predictions match very well with plant measured data and SIMULATE-3 predictions. The kinetic benchmark problems validated include the Nuclear Energy Agency Committee on Reactor Physics rod ejection problem, the three-dimensional Langenbuch-Maurer-Werner LWR rod withdrawal/insertion problem, and the three-dimensional linear regression analysis BWR transient benchmark problem. The results indicate that ARROTTA'smore » accuracy and stability are excellent as compared with other space-time kinetics codes. It is therefore concluded that ARROTTA provides accurate predictions for multidimensional core transients for LWRs.« less

Monte Carlo calculation of kerma to a point in the vicinity of media interfaces

Recent work has demonstrated that Monte Carlo photon transport (MCPT) simulation is a practical and accurate dosimetry tool for characterizing brachytherapy dose distributions in the presence of complex three-dimensional (3D) geometries. Because of the steep dose gradients encountered in this application, the choice of estimation algorithm, i.e., the algorithm for extracting dose estimates from individual photon histories, is critical to ensure adequate spatial resolution and computational efficiency. While conventional MCPT dose estimation techniques are sufficiently accurate in many cases, they inevitably fail with the point of interest is located near media interfaces, since they require the point of interest to be contained by a region of homogeneous medium of several millimetres in diameter. This severely limits the domain of application of MCPT in brachytherapy dosimetry and prevents dose estimation at points such as those immediately behind lead shields contained in gynaecological applicators, or simulation of small intricately structured detector response. We have successfully adapted to an MCPT code a neutron-flux estimator, the one-more-collided flux estimator (OMCFE), which overcomes this problem. OMCFE is derived from the familiar next-flight estimation algorithm, and, through the use of a photon trajectory resampling scheme, does not require homogeneous media at the point of interest. it is therefore a true point-dose estimator. OMCFE was tested against a variety of one- and three-dimensional benchmark problems. These benchmarks included mono-energetic point sources of 30 and 600 keV, as well as a 3M model 6702 125I seed, a 137Cs tube, and an 192Ir seed in geometries approximating common brachytherapy dosimetric problems. OMCFE agrees closely with conventional MCPT dose estimation algorithms in solving these benchmark problems, while in many cases being much more efficient.