Convergence study of DGFEM SN based 2D/1D coupling method for solving neutron transport k-eigenvalue problems with Fourier analysis

A domain decomposition method with Lagrange multipliers for the Stokes problem is developed and analysed. A common approach to solve the Stokes problem, termed the Uzawa algorithm, is to decouple the velocity and the pressure. This approach yields the Schur complement system for the pressure Lagrange multiplier which is solved with an iterative solver. Each outer iteration of the Uzawa procedure involves the inversion of a Laplacian in each spatial direction. The objective of this paper is to effectively solve this inner system (the vector Laplacian system) by applying the finite‐element tearing and interconnecting (FETI) method. Previously calculated search directions for the FETI solver are reused in subsequent outer Uzawa iterations. The advantage of the approach proposed in this paper is that pressure is continuous across the entire computational domain. Numerical tests are performed by solving the driven cavity problem. An analysis of the number of outer Uzawa iterations and inner FETI iterations is reported. Results show that the total number of inner iterations is almost numerically scalable since it grows asymptotically with the mesh size and the number of subdomains. Copyright © 2003 John Wiley & Sons, Ltd.

The two-stage iterative methods for symmetric positive definite matrices

Fourier analysis of the continuous infinite homogenous multigroup (MG) formulation is investigated in this paper for the time-dependent Boltzmann transport equation using discrete ordinates formulation. In addition, a continuous two-group (2G) and one-group (1G) Fourier analysis is performed to generate an analytical spectral radius and provide the basis for a theoretical analysis of the convergence. The discrete 1G formulation is then presented, and the theoretical analysis is found to predict the same spectral radius as the continuous 1G formulation. A typical pressurized water reactor pin cell problem with 47-group library is then homogenized with reflective boundary conditions, and the numerical spectral radius is calculated using the MPACT code. The theoretical predictions and the numerical results from the pin cell case agree very well and are found to have the following behavior: (1) The spectral radius is usually very close to unity for standard parameters for realistic transient application, (2) the spectral radius generally decreases as a function of inners per outer M, (3) the spectral radius generally decreases as a function of time-step size and then increases beyond unity for extremely small time steps, and (4) the spectral radius is almost constant as a function of the inserted reactivity. Good agreement is observed with the MG Fourier analysis. Finally, it is shown that the group sweeping coarse mesh finite difference method is theoretically and numerically very slow to converge the time-dependent neutron transport equation and that it is necessary to move the right-hand-side fission and transient source to the left-hand side and to solve the entire matrix form of the system.

We consider the computation of a few eigenvectors and corresponding eigenvalues of a large sparse nonsymmetric matrix using shift-invert Arnoldi's method with and without implicit restarts. For the inner iterations we use preconditioned GMRES as the inexact iterative solver. The costs of the solves are measured by the number of inner iterations needed by the iterative solver at each outer step of the algorithm. We first extend the relaxation strategy developed by Simoncini [SIAM J. Numer. Anal., 43 (2005), pp. 1155–1174] to implicitly restarted Arnoldi's method, which yields an improvement in the overall costs of the method. Secondly, we apply a new preconditioning strategy to the inner solver. We show that small rank changes to the preconditioner can produce significant savings in the total number of iterations. The combination of the new preconditioner with the relaxation strategy in implicitly restarted Arnoldi produces enhancement in the overall costs of around 50 percent in the examples considered here. Numerical experiments illustrate the theory throughout the paper.

This work presents an efficient technique for simultaneously reconstructing an image of a moving object from X-ray projections along with estimating the motion parameters. Current state-of-the-art iterative methods rely on objective functions that contain the solution of an iterative procedure as one of its terms. This complicates the reconstruction process as it leads to nested iterations and makes analytic differentiation impractical. The presented technique relies on an objective function that simultaneously depends on the image and the motion parameters. The derivatives of this objective function towards the image and the motion parameters are known exactly and implemented matrix free and in parallel. Moreover, the stepsizes of both the iterative reconstruction and motion parameter estimation schemes are chosen following a mathematical formulation that guarantees a fast convergence speed. The result is a method that not only needs less iterations, but also removes the need to tune the finite differences stepsize and the number of inner iterations, which allows for efficient, scalable reconstruction using modern optimizers, without nested iterations.

Effect of time-step size on flow around generic car models

In this paper, we apply the projective integration method to moment models of the Boltzmann-BGK equation and investigate the numerical properties of the resulting scheme. Projective integration is an explicit, asymptotic-preserving scheme that is tailored to problems with a large spectral gap between fast and slow eigenvalues of the model. A spectral analysis of the moment model shows a clear spectral gap and reveals the multi-scale nature of the model. The new scheme overcomes the severe time step constraint of standard explicit schemes like the forward Euler scheme by performing a number of inner iterations and then extrapolating the solution forward in time. The projective integration scheme is non-intrusive and yields fast and accurate solutions, as demonstrated using a 1D shock tube test case. These observations open up many possibilities for further use of the scheme for high-resolution discretizations and different collision models.

The monotonicity of two-stage iterative methods

A performance of high-order Cellwise Relaxation Implicit Discontinuous Galerkin (DGCRI) scheme on unsteady flow computation is investigated. The inviscid Taylor-Green vortex is first computed to demonstrate the capability of the high-order DG-CRI scheme on the homogeneous turbulence computation. The higher-order DG-CRI scheme gives better resolved small vortex structures, which is well indicated in the later decay of the kinetic energy and the higher enstrophy. In addition, the effects of the CFL number and the number of inner iterations are discussed by comparing with the results by explicit time integration. Then, the high-order DG method is applied for the LES computation of the turbulent channel flow, and the advantages of the high-order DG method in predicting accurate turbulent statistics are shown.

Based on the transient planar pitching method, the dynamic derivative calculation function is implemented on an unstructured computational fluid (CFD) software NNW-FlowStar. Then the pitch-damping dynamic derivatives of the standard model of the ANF missile at zero angles of attack and different Mach numbers (0.5∼4.5) are calculated. The influence of various numerical and physical parameters on the calculation results of dynamic derivatives is studied in detail. It is found that the dynamic derivatives of pitch damping are independent of the values of amplitude (0.125° ≤ A ≤ 1°) and reduction frequency (0.025 ≤ k ≤ 0.2). When the number of calculation steps of the sine oscillation period is 200 and the number of inner iterations is 40, the prediction accuracy and computing efficiency can meet the engineering requirements simultaneously in this case. The influence of different grid resolutions on the dynamic derivative calculation results is studied. Results show that the grid requirements of dynamic derivative calculation are consistent with those of steady-state calculations. Finally, the calculation results are compared with the free flight test results, which verifies the reliability of the methods and parameters suggested in this paper.

A tightly coupled scheme for neutronics and thermal–hydraulics using open-source software

Conventional finite element analysis (FEA) performed in electromagnetic-vibration coupling calculation of motor suffers from several significant drawbacks, such as large memory space, high computational cost and heavy reliance on mesh quality for accurate solution. With the traditional meshless method, special attention needs to be paid to correctly impose the boundary condition like FEA. Besides, the matrix is easily prone to be ill-conditioned due to introducing large amount of higher-order basis functions. We propose a novel multi-physical coupling method combining FEA and optimized meshless method. The proposed methodology is further evaluated on the vibration analysis of a 12-slot 10-pole permanent magnet synchronous motor (PMSM). Firstly, 2D stator electromagnetic force is simulated and derived based on the local Jacobian derivative method through FEA. The electromagnetic force spectrum is calculated using FFT analysis and further imported into commercial meshless structural simulation software SimSolid for stator harmonic response analysis. Correct force boundary condition and data mapping between meshless and FEA simulation interface are key to the accuracy of the proposed combined multi-physical modeling methodology. This is achieved by introducing new high-dimensional ramp function in the transition region between FEA and meshless domains, which are defined with the shape functions composed of the FEA and meshless method. This function satisfies the continuity and consistency of the displacement function and ensures the convergence of coupled FEA-meshless method. Subsequently, construction of basis function is key to the establishment of convention meshless discrete equation for the elastic problem of rotating machinery. This is designed by using moving least square theory in cylindrical coordinate system. A harmonic response with meshless method is analyzed by using the mode superposition method to obtain detailed mode shape data, acceleration and displacement distribution of stator. Finally, the tangential continuity and robustness are not well considered in the traditional simulation with FEA coupled meshless method. To mitigate this problem, we propose an optimized meshless method based on modified local basis functions to recalculate the harmonic response motion. Then, the coupling electromagnetic-vibration simulation results of traditional coupled FEA-meshless method, optimized coupled FEA-meshless method and complete FEA coupled method are compared. It is worth noting that the optimized method significantly improves accuracy, robustness and computational speed at the same time. In short, the proposed electromagnetic-structure coupling calculation method provides a novel alternative for the multi-physical coupling calculation of rotating machinery combining FEA and meshless simulation methods.

We consider the two‐level methods based on Arrow–Hurwicz (A‐H) iteration for solving the stationary incompressible magnetohydrodynamics problem. The methods carry out some A‐H iterations on the coarse mesh, improve the obtained solutions by using one‐step scheme on the fine mesh. Such methods enjoy some advantages including that the number of A‐H iterations can be flexibly and controllably selected, and they are valid for high hydrodynamic Reynolds number. The convergence analyses are established and various numerical experiments are performed to validate and illustrate the stability, accuracy, and efficiency of the new methods.

In the ddCOSMO solvation model for the numerical simulation ofmolecules (chains of atoms), the unusual observation was made thatthe associated Schwarz domain-decomposition method converges independentlyof the number of subdomains (atoms) and this without coarsecorrection, i.e., the one-level Schwarz method is scalable.We analyzed this unusual property for the simplified caseof a rectangular molecule and square subdomains using Fourier analysis,leading to robust convergence estimates in the $L^2$-norm and lateralso for chains of subdomains represented by disks using maximumprinciple arguments, leading to robust convergence estimates in$L^{\\\\infty}$. A convergence analysis in the more natural$H^1$-setting proving convergence independently of the number ofsubdomains was, however, missing. We close this gap in this paperusing tools from the theory of alternating projection methodsand estimates introduced by P.-L. Lions for the study of domaindecomposition methods. We prove that robust convergenceindependently of the number of subdomains is possible also in $H^1$and show furthermore that even for certain two-dimensional domainswith holes, Schwarz methods can be scalable without coarse-space corrections.As a by-product, we review some of the results of P.-L. Lions[On the Schwarz alternating method. I, in DomainDecomposition Methods for Partial Differential Equations, SIAM,Philadelphia, 1988, pp. 1-42]and in some cases provide simpler proofs.

Finite volume coupling strategies for the solution of a Biot consolidation model