Model Problem Analysis Research Articles

Neutronic Analysis of the Conceptual Molten Uranium Breeder Reactor Using MCNP and SCALE Tools

The Molten Uranium Breeder Reactor (MUBR) is a radically new reactor concept with a mixed-energy spectrum. MUBR is fueled with molten uranium metal in large-diameter fuel tubes and is cooled by circulating molten uranium fuel through a heat exchanger. The reactor has heavy water as moderator, and the reactivity of the reactor is primarily controlled by the voiding effect of the moderator through an innovative control cavity structure design. Because the MUBR design is vastly different from most existing fission reactors, neutronics analysis must be performed for many different combinations of design parameters to identify viable and optimum design configurations. To facilitate the neutronics analysis, a proprietary program called MUBR6gen is being developed to provide a pipeline tool to expedite the process. MUBR6gen employs two well-established neutronics codes, i.e., MCNP and SCALE, to perform standard neutronics calculations for MUBR by automating input preparation and output processing. In addition, MUBR6gen ensures consistency of the MCNP and SCALE inputs and compares the outputs of the two codes to warrant the simulation results. Augmented with MUBR6gen, standard neutronics analysis was carried out on a small-scale MUBR design, which serves as a model problem in the paper. The neutronics performance characteristics of the model reactor were obtained and discussed in a code-to-code pattern. An overall very good agreement between the results of the two neutronics codes was established. Based on the success of the model problem analysis, further neutronics analysis using MUBR6gen was extended for a set of MUBR variant designs. Meaningful and promising fuel cycle analysis results for the 10 different designs were achieved and discussed. These results are used to identify the best MUBR candidates in terms of fuel lifetime and utilization efficiency for future applications.

New iterative and staggered solution schemes for incompressible fluid‐structure interaction based on Dirichlet‐Neumann coupling

SummaryIn the presence of strong added mass effects, partitioned solution strategies for incompressible fluid‐structure interaction are known to lack robustness and computational efficiency. A number of strategies have been proposed to address this challenge. However, these strategies are often complicated or restricted to certain problem classes and generally require intrusive modifications of existing software. In this work, the well‐known Dirichlet‐Neumann coupling is revisited and a new combined two‐field relaxation strategy is proposed. A family of efficient staggered schemes based crucially on a force predictor is formulated alongside the classical iterative approach. Both methodologies are rigorously analyzed on the basis of a linear model problem derived from a simplified fluid‐conveying elastic tube. The investigation suggests that both the robustness and the efficiency of a partitioned Dirichlet‐Neumann coupling scheme can be improved by a relatively small nonintrusive modification of a standard implementation. The relevance of the model problem analysis for finite element‐based computational fluid‐structure interaction is demonstrated in detail for a submerged cylinder subject to an external force.

Stability analysis of stationary variational and hemivariational inequalities with applications

In this paper, we provide a comprehensive stability analysis for stationary variational inequalities, hemivariational inequalities, and variational-hemivariational inequalities. With contact mechanics as application background, stability is analyzed for solutions with respect to combined or separate perturbations in constitutive relations, external forces, constraints, and non-smooth contact boundary conditions of the inequality problems. The stability result is first proved for a general variational-hemivariational inequality. Then, stability results are obtained for various variational inequalities and hemivariational inequalities as special cases. Finally, we illustrate applications of the theoretical results for the stability analysis of model problems in contact mechanics.

A Stable Added-Mass Partitioned (AMP) Algorithm for Elastic Solids and Incompressible Flow: Model Problem Analysis

An analysis is made of a new partitioned scheme for solving fluid-structure interaction problems involving viscous incompressible flow and compressible elastic-solids. The new scheme is stable, wit...

Error estimates for projection-based dynamic augmented Lagrangian boundary condition enforcement, with application to fluid–structure interaction

In this work, we analyze the convergence of the recent numerical method for enforcing fluid–structure interaction (FSI) kinematic constraints in the immersogeometric framework for cardiovascular FSI. In the immersogeometric framework, the structure is modeled as a thin shell, and its influence on the fluid subproblem is imposed as a forcing term. This force has the interpretation of a Lagrange multiplier field supplemented by penalty forces, in an augmented Lagrangian formulation of the FSI kinematic constraints. Because of the non-matching fluid and structure discretizations used, no discrete inf-sup condition can be assumed. To avoid solving (potentially unstable) discrete saddle point problems, the penalty forces are treated implicitly and the multiplier field is updated explicitly. In the present contribution, we introduce the term dynamic augmented Lagrangian (DAL) to describe this time integration scheme. Moreover, to improve the stability and conservation of the DAL method, in a recently-proposed extension we projected the multiplier onto a coarser space and introduced the projection-based DAL method. In this paper, we formulate this projection-based DAL algorithm for a linearized parabolic model problem in a domain with an immersed Lipschitz surface, analyze the regularity of solutions to this problem, and provide error estimates for the projection-based DAL method in both the [Formula: see text] and [Formula: see text] norms. Numerical experiments indicate that the derived estimates are sharp and that the results of the model problem analysis can be extrapolated to the setting of nonlinear FSI, for which the numerical method was originally proposed.

A stable partitioned FSI algorithm for rigid bodies and incompressible flow. Part I: Model problem analysis

A stable partitioned algorithm is developed for fluid–structure interaction (FSI) problems involving viscous incompressible flow and rigid bodies. This added-mass partitioned (AMP) algorithm remains stable, without sub-iterations, for light and even zero mass rigid bodies when added-mass and viscous added-damping effects are large. The scheme is based on a generalized Robin interface condition for the fluid pressure that includes terms involving the linear acceleration and angular acceleration of the rigid body. Added-mass effects are handled in the Robin condition by inclusion of a boundary integral term that depends on the pressure. Added-damping effects due to the viscous shear forces on the body are treated by inclusion of added-damping tensors that are derived through a linearization of the integrals defining the force and torque. Added-damping effects may be important at low Reynolds number, or, for example, in the case of a rotating cylinder or rotating sphere when the rotational moments of inertia are small. In this first part of a two-part series, the properties of the AMP scheme are motivated and evaluated through the development and analysis of some model problems. The analysis shows when and why the traditional partitioned scheme becomes unstable due to either added-mass or added-damping effects. The analysis also identifies the proper form of the added-damping which depends on the discrete time-step and the grid-spacing normal to the rigid body. The results of the analysis are confirmed with numerical simulations that also demonstrate a second-order accurate implementation of the AMP scheme.

Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation

Many important physical phenomena can be described by the Helmholtz equation. We investigate to what extent the convergence of the shifted Laplacian preconditioner for the Helmholtz equation can be accelerated using deflation with multigrid vectors. We therefore present a unified framework for two published algorithms. The first deflates the preconditioned operator and requires no further preconditioning. The second deflates the original operator and combines deflation and preconditioning in a multiplicative fashion. We pursue two scientific contributions. First we show, using a model problem analysis, that both algorithms cluster the eigenvalues. The new and key insight here is that the near-kernel of the coarse grid operator causes a limited set of eigenvalues to shift away from the center of the cluster with a distance proportional to the wave number. This effect is less pronounced in the first algorithmic variant at the expense of a higher computational cost. In the second contribution we quantify for the first time the large amount of reduction in CPU-time that results from the clustering of eigenvalues and the reduction in iteration count. We report to this end on the findings of an implementation in PETSc on two and three-dimensional problems with constant and variable wave number.

An algebraic multifrontal preconditioner that exploits the low‐rank property

SummaryWe present an algebraic structured preconditioner for the iterative solution of large sparse linear systems. The preconditioner is based on a multifrontal variant of sparse LU factorization used with nested dissection ordering. Multifrontal factorization amounts to a partial factorization of a sequence of logically dense frontal matrices, and the preconditioner is obtained if structured factorization is used instead. This latter exploits the presence of low numerical rank in some off‐diagonal blocks of the frontal matrices. An algebraic procedure is presented that allows to identify the hierarchy of the off‐diagonal blocks with low numerical rank based on the sparsity of the system matrix. This procedure is motivated by a model problem analysis, yet numerical experiments show that it is successful beyond the model problem scope. Further aspects relevant for the algebraic structured preconditioner are discussed and illustrated with numerical experiments. The preconditioner is also compared with other solvers, including the corresponding direct solver. Copyright © 2015 John Wiley & Sons, Ltd.

Inverse problems of ultrasound tomography in models with attenuation

We develop efficient methods for solving inverse problems of ultrasound tomography in models with attenuation. We treat the inverse problem as a coefficient inverse problem for unknown coordinate-dependent functions that characterize both the speed cross section and the coefficients of the wave equation describing attenuation in the diagnosed region. We derive exact formulas for the gradient of the residual functional in models with attenuation, and develop efficient algorithms for minimizing the gradient of the residual by solving the conjugate problem. These algorithms are easy to parallelize when implemented on supercomputers, allowing the computation time to be reduced by a factor of several hundred compared to a PC. The numerical analysis of model problems shows that it is possible to reconstruct not only the speed cross section, but also the properties of the attenuating medium. We investigate the choice of the initial approximation for iterative algorithms used to solve inverse problems. The algorithms considered are primarily meant for the development of ultrasound tomographs for differential diagnosis of breast cancer.

Finite Volume Evolution Galerkin Methods for Hyperbolic Systems

The subject of the paper is the derivation and analysis of new multidimensional, high-resolution, finite volume evolution Galerkin (FVEG) schemes for systems of nonlinear hyperbolic conservation laws. Our approach couples a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of the multidimensional hyperbolic system such that all of the infinitely many directions of wave propagation are taken into account. In particular, we propose a new FVEG scheme, which is designed in such a way that for a linear wave equation system the approximate evolution operator calculates any one-dimensional planar wave exactly. This operator improves the stability of the FVEG scheme considerably, leading to a stability limit closer to 1. Using the results obtained for the wave equation system, a new approximate evolution operator for the linearized Euler equations is also derived. The integrals over the cell interfaces also need to be approximated with care; in this case, our choice of Simpson's rule is guided by stability analysis of model problems. Second order resolution is obtained by means of a piecewisebilinear recovery. Numerical experiments confirm the accuracy and multidimensional behavior of the new scheme.

Element-Free AMGe: General Algorithms for Computing Interpolation Weights in AMG

We propose anew general algorithm for constructing interpolation weights in algebraic multigrid (AMG). It exploits a proper extension mapping outside a neighborhood about a fine degree of freedom (dof) to be interpolated. The extension mapping provides boundary values (based on the coarse dofs used to perform the interpolation) at the boundary of the neighborhood. The interpolation value is then obtained by matrix dependent harmonic extension of the boundary values into the interior of the neighborhood. We describe the method, present examples of useful extension operators, provide a two-grid analysis of model problems, and, by way of numerical experiments, demonstrate the successful application of the method to discretized elliptic problems.

Interacting cracks and inclusions in a solid by multipole expansion method

An accurate series solution is obtained of the elastic problem for a solid containing penny-shaped cracks and spheroidal inclusions of cavities. The method of solution is based on the general solution procedure developed by Kushch [(1996) Elastic equilibrium of a solid containing a finite number of aligned inclusions. International Journal of Solids and Structures33, 1175–1189] and consists in representation of the displacement vector by a series of the vectorial partial solutions of Lame's equation, written in a spheroidal basis. By using the addition theorems for these partial solutions the primary boundary-value problem is reduced to an infinite set of linear algebraic equations. An asymptotic analysis of the problem is performed and the series expansion of the opening-mode stress intensity factor is obtained. Numerical analysis of model problems is performed and some results demonstrating the effect on the stress intensity factor of the pair interactions in crack-crack, crack-cavity and crack-inclusion geometries are presented.

Stability analysis of model problems for elastodynamic boundary element discretizations

In the literature there is growing evidence of instabilities in standard time-stepping schemes to solve boundary integral elastodynamic models [1]–[3]. In this article we use three distinct model problems to investigate the stability properties of various discretizations that are commonly used to solve elastodynamic boundary integral equations. Using the model problems, the stability properties of a large variety of discretization schemes are assessed. The features of the discretization procedures that are likely to cause instabilities can be established by means of the analysis. This new insight makes it possible to design new time-stepping schemes that are shown to be more stable. © 1996 John Wiley & Sons, Inc.

Stability analysis of model problems for elastodynamic boundary element discretizations

In the literature there is growing evidence of instabilities in standard time-stepping schemes to solve boundary integral elastodynamic models [1]–[3]. In this article we use three distinct model problems to investigate the stability properties of various discretizations that are commonly used to solve elastodynamic boundary integral equations. Using the model problems, the stability properties of a large variety of discretization schemes are assessed. The features of the discretization procedures that are likely to cause instabilities can be established by means of the analysis. This new insight makes it possible to design new time-stepping schemes that are shown to be more stable. © 1996 John Wiley & Sons, Inc.

Analysis of a multigrid strokes solver

Model-problem analysis or local-mode analysis is used to investigate the two-level convergence rates for multigrid Strokes solvers. The solvers are based on the discretization on the staggered grid and the distributive relaxation of the continuity equation. The influence of the grid geometry is analyzed by also considering the Poisson equation on the semistaggered grid. Various grid transfer operators are compared. For an isotropic ( h x = h y discretization, simple tranfer operators are sufficient to obtain optimal rates, but not in the anisotropic case. The theoretical results are compared with observed multigrid convergence rates.

Aggregation Methods for Solving Sparse Triangular Systems on Multiprocessors

Efficient methods are presented for solving large sparse triangular systems on multiprocessors. These methods use heuristics for the aggregation, mapping, and scheduling of relatively fine-grained computations whose data dependencies are specified by directed acyclic graphs. Results of experiments run on the Encore Multimax, as well as model problem analysis, measure the performance of the partitioning strategies on shared-memory architectures with varying synchronization costs.

Asymptotic analysis of model problems for a coupled system

On etudie le comportement asymptotique quand e→0 + des solutions du systeme d'equations couplees: u″=ν, eν″+f(u,u')ν'=0 pour f(u,u')=u ou u', qui satisfait les conditions de Dirichlet u(0,e)=u(1,e)=0 et ν(0,e)=ν 0 , ν(1,e)=ν 1

A multilevel method with overcorrection by aggregation for solving discrete elliptic problems

A multilevel method with correction by aggregation, originated from additive aggregation correction techniques well-known in the pre-multigrid age, is analysed within the multigrid framework and compared with standard multigrid methods. From the model problem analysis it follows that the overcorrection can substantially improve the reduction of smooth error components and change favourably the properties of the multilevel aggregation correction method. Numerical experiments illustrate the efficiency of the examined method.

Chebyshev Spectral Methods for Limited-Area Models. Part I: Model Problem Analysis

Abstract This study considers how spectral methods can be applied to limited-area models using Chebyshev polynomials as basis functions. We review the convergence of Sturm–Liouville series to motivate the use of the Chebyshev polynomials, and describe the tau and collocation projections which allow the use of general (nonperiodic) boundary conditions. These methods are illustrated for a simple model problem, the linear advection equation in one dimension, and numerical results confirm their high accuracy. Time differencing and efficiency are considered in detail using both asymptotic analysis and numerical result from the model problem. The stability condition for Chebyshev methods with explicit time differencing, often thought to be severe, is shown to be less severe than that for finite difference methods when high accuracy is desired. Fourth-order Runge-Kutta time differencing is the most efficient of the many schemes considered. When the accuracy desired is high enough, Chebyshev spectral methods are ...

The contraction number of a multigrid method with mesh ratio two for solving model problems

The algorithm proposed in [6] is analyzed with the aid of model problem analysis. The explicit expression for the convergence factor and the exact values of the contraction number are obtained.