Low-order Problem Research Articles

Positive Solutions for a Fractional Boundary Value Problem with Lidstone Like Boundary Conditions

We consider a higher order fractional boundary value problem with Lidstone like boundary conditions, where the nonlinearity is an L1-Carathèodory function. We first consider the lower order problem. Then, by using a convolution to construct the Green’s function for the higher order problem, we are able to apply a recent fixed point theorem to show the existence of positive solutions of the boundary value problem.

Stability and Performance of the X-CMFD Method for Multiphysics Reactor Calculations

This paper presents an innovative approach to efficiently perform deterministic direct whole-core transport calculations with multiphysics feedback for steady-state problems. Traditionally, Picard iteration combined with coarse mesh finite difference (CMFD) acceleration has been used, but it can suffer from instability and inefficiency in certain scenarios. In this work, we introduce the X-CMFD method, supported by Fourier analysis, to enhance the stability of the multiphysics iteration scheme. A new and efficient variation of the X-CMFD method for practical simulations is also present. Additionally, we explore the theoretical convergence rates of nonlinear fully coupled diffusion acceleration (NFCDA), a class of diffusion acceleration methods that formalizes similar ideas of previous research. NFCDA uses a low-order diffusion problem that is fully coupled with equivalent nonlinear multiphysics feedback to accelerate the high-order transport problem with feedback. The theoretical analysis shows that NFCDA offers similar convergence rates to nonlinear diffusion acceleration (NDA) in problems without feedback. This provides theoretical support for numerical experiments conducted by other researchers. X-CMFD, which is a discretized form of NFCDA, leverages typical coarse mesh concepts and operators from CMFD while applying feedback to cross sections in the low-order diffusion problem at each power iteration of the low-order problem. To reduce computational costs, we optimize the implementation of X-CMFD in MPACT by introducing an equivalent low-order approximation to the cross-section updates in the nonlinear low-order problem. Numerical results from pressurized water reactor problems demonstrate that X-CMFD, along with its practical implementation, outperforms current relaxed Picard iteration methods in terms of robustness and efficiency, irrespective of the presence of multiphysics feedback.

Low-Order Preconditioning for the High-Order Finite Element de Rham Complex

In this paper we present a unified framework for constructing spectrally equivalent low-order-refined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in $H^1$, $H({\rm curl})$, and $H({\rm div})$, and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for N\'ed\'elec and Raviart-Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowest-order (piecewise constant) preconditioner is developed for high-order interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any "black-box" preconditioner for low-order problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, based on an implmentation in the finite element library MFEM. The theoretical properties of these preconditioners are corroborated, and the flexibility and scalability of the method are demonstrated on a range of challenging three-dimensional problems.

A hybrid neutronics method with novel fission diffusion synthetic acceleration for criticality calculations

A novel Fission Diffusion Synthetic Acceleration (FDSA) method is developed and implemented as a part of a hybrid neutronics method for source convergence acceleration and variance reduction in Monte Carlo (MC) criticality calculations. The acceleration of the MC calculation stems from constructing a synthetic operator and solving a low-order problem using information obtained from previous MC calculations. By applying the P1 approximation, two correction terms, one for the scalar flux and the other for the current, can be solved in the low-order problem and applied to the transport solution. A variety of one-dimensional (1-D) and two-dimensional (2-D) numerical tests are constructed to demonstrate the performance of FDSA in comparison with the standalone MC method and the coupled MC and Coarse Mesh Finite Difference (MC-CMFD) method on both intended purposes. The comparison results show that the acceleration by a factor of 3–10 can be expected for source convergence and the reduction in MC variance is comparable to CMFD in both slab and full core geometries, although the effectiveness of such hybrid methods is limited to systems with small dominance ratios.

Convergence Properties of a Linear Diffusion-Acceleration Method for k-Eigenvalue Transport Problems

Most methods that use low-order operators to accelerate the iterative solution of transport eigenvalue problems employ nonlinear functionals of the transport solution (such as Eddington tensors) in their low-order equations, which are themselves standard eigenvalue problems. Here, we discuss linear diffusion synthetic acceleration (DSA) for -eigenvalue problems, which belongs to a family of methods that has received less attention than its nonlinear counterparts. We review the history of these linear methods as far as we know it and describe theoretical questions that to our knowledge have remained unanswered. With these methods, a low-order problem is solved after each transport step for an updated eigenvalue and an additive correction to the eigenfunction. These low-order problems are not standard eigenvalue problems, for they contain residuals as fixed sources. The low-order problems admit infinitely many solutions (updated and additive correction to the eigenfunction), and the solution that is obtained depends on the initial guess and iterative method chosen for the low-order problems. Experience has shown that when the low-order problems are solved with a powerlike iteration method and certain initial guesses, they yield solutions that cause rapid convergence to the correct high-order solution. We study the convergence properties of this algorithm applied to two model problems: an infinite homogeneous medium and a one-cell problem. For the infinite homogeneous problem, we present a Fourier analysis of the linear DSA method, which demonstrates that when the low-order problems are solved using a powerlike iteration scheme, the linear DSA scheme provides immediate convergence of the -eigenvalue and rapid convergence of the eigenfunction (much like DSA applied to scattering iterations in fixed-source problems). For the one-cell problems, we find that the linear scheme for -eigenvalue problems performs approximately as well as DSA for fixed-source problems. The latter analysis reveals a quantitative bound on the consistency between low- and high-order operators that is necessary and sufficient for convergence of those problems. With some theoretical foundations for the linear methods now established, we turn to numerical testing. We find, as others have before us using different low-order operators, that the method works well in practice. We provide numerical results from reactor problems in which our linear DSA is approximately as effective as the more widely used nonlinear methods. Our theoretical and numerical results add to the body of evidence that the linear methodology offers a simple path to rapid convergence of -eigenvalue problems, especially for codes that already employ linear low-order operators to converge scattering iterations.

A Robust, Relaxation–Free Multiphysics Iteration Scheme for CMFD–Accelerated Neutron Transport k–Eigenvalue Calculations–I: Theory

This paper presents a new robust scheme for coupled physics nuclear reactor calculations. We focus specifically on high-fidelity whole-core transport calculations with coarse mesh finite difference (CMFD) coupled to thermal hydraulics. These simulations traditionally employ rthe Picard iteration for the coupled solution, where it has been observed that the use of CMFD (or nonlinear diffusion acceleration) is detrimental to the overall convergence of the coupled problem. Moreover, (1) if the acceleration equations are tightly converged every iteration, the overall multiphysics iteration becomes less stable and (2) properly loosening the convergence criteria of the acceleration equations at each iteration can stabilize the overall scheme. In this paper, we develop a Fourier analysis for a simplified CMFD-accelerated neutron transport problem with feedback from flux-dependent cross sections to provide a theoretical explanation for, and gain insight into, the aforementioned observations. Furthermore, we establish the theoretical relationship between relaxation and partial convergence of the low-order problem. Using this result, a relaxation-free iteration scheme is then proposed, with a formula to determine the nearly optimal partial convergence of the low-order diffusion problem. The new CMFD method is called the nearly optimally partially converged coarse mesh finite difference (NOPC-CMFD) method. It is shown theoretically that the NOPC-CMFD method in problems with feedback has stability properties comparable to CMFD in problems without feedback and requires no relaxation factor, i.e., is relaxation free. The results presented in this paper provide a theoretical foundation for the development of a robust multiphysics iteration scheme for nuclear reactor modeling. The implementation of the method and application to various test cases are presented in the companion paper.

Comparative studies of iterative methods for solving the “optimally diffusive” coarse mesh finite difference accelerated transport equation

The emergence of advanced nuclear reactor systems with increasing complexity and heterogeneity has necessitated detailed high fidelity neutronic analysis. This requires exhaustive solution of the neutron transport equation which has the disadvantage of long computation times, especially for scattering dominant problems. In this paper, the performance behavior and stability analysis of several iterative methods for solving the optimally diffusive coarse mesh finite difference (odCMFD) accelerated transport equation has been presented. Efficacy of the acceleration iterations with varying coarse mesh size and convergence criteria of the low order problem, and their overall performance has been investigated for the 2-group LRA-BWR problem and the 7-group C5G7 problem. Prescriptions for different parameters and the most robust iterative scheme have been proposed based on these studies. The theoretical background of odCMFD based acceleration schemes and the numerical results of their application to different problems using the code DIAMOND have been presented in this paper.

Protection and Identification of Thermoacoustic Azimuthal Modes

Abstract This paper first characterizes the acoustic field of two annular combustors by means of data from acoustic pressure sensors. In particular, the amplitude, orientation, and nature of the acoustic field of azimuthal order n are characterized. The dependence of the pulsation amplitude on the azimuthal location in the chamber is discussed, and a protection scheme making use of just one sensor is proposed. The governing equations are then introduced, and a low-order model of the instabilities is discussed. The model accounts for the nonlinear response of M distinct flames, for system acoustic losses by means of an acoustic damping coefficient α and for the turbulent combustion noise, modeled by means of the background noise coefficient σ. Keeping the response of the flames arbitrary and in principle different from flame to flame, we show that, together with α and σ, only the sum of their responses and their 2n Fourier component in the azimuthal direction affect the dynamics of the azimuthal instability. The existing result that only this 2n Fourier component affects the stability of standing limit-cycle solutions is recovered. It is found that this result applies also to the case of a nonhomogeneous flame response in the annulus, and to flame responses that respond to the azimuthal acoustic velocity. Finally, a parametric flame model is proposed, depending on a linear driving gain β and a nonlinear saturation constant κ. The model is first mapped from continuous time to discrete time, and then recast as a probabilistic Markovian model. The identification of the parameters {α,β,κ,σ} is then carried out on engine time-series data. The optimal four parameters {α,σ,β,κ} are estimated as the values that maximize the data likelihood. Once the parameters have been estimated, the phase space of the identified low-order problem is discussed on selected invariant manifolds of the dynamical system.

Existence of a positive solution for a singular fractional boundary value problem with fractional boundary conditions using convolution and lower order proble

Existence of a positive solution is shown for two singular two-point fractional boundary value problems with fractional boundary conditions using fixed point theory, lower order problems, and convolution of Green's functions. A nontrivial example is included.

Mixed meshless local Petrov–Galerkin (MLPG) collocation methods for gradient elasticity theories of Helmholtz type

Two mixed meshless collocation methods for solving problems by considering a linear gradient elasticity theory of the Helmholtz type are proposed. The solution process is facilitated by employing operator-split procedures, splitting the original 4th-order problem into two uncoupled second-order sub-problems, which are then solved in a staggered manner by applying the mixed meshless local Petrov–Galerkin collocation strategy. Thereby, identical nodal pattern is used for the discretization of both lower-order problems, while the approximation of all unknown field variables is performed by applying the Moving Least Squares functions with interpolatory conditions. The performance of the derived methods is tested by some suitable numerical examples, dealing with elasticity problems that are often treated by gradient theories. Therein, it is demonstrated that both proposed methods are able to capture the size effect and that the strain-based method is able to produce the non-singular strain field around the crack tip, similar to the nonlocal elasticity approach of Eringen. It has been found out that the obtained results agree well with available analytical and numerical solutions.

Optimisation of the high-order problems in evolutionary algorithms: an application of transfer learning

Evolutionary Algorithms (EAs) have been applied to many optimisation problems, among which those with high order are difficult for EAs. The higher the order, the steeper the curve around the optimum is, therefore the more difficult it is. This paper introduces Transfer Learning (TL) aided EAs to conquer the high-order problems more efficiently and effectively by optimum transfer from the low-order problem (as source domain) to high-order problem (as the target domain). The experiments validated this method by comparison of the average number of the convergence generation and an impressive feature was observed: this method is robust against the difficulties of the problems. This method is not only significant for high-order problems, but also useful for other difficult problems by borrowing optimum from other feature-similar easy problems.

Optimisation of the high-order problems in evolutionary algorithms: an application of transfer learning

Evolutionary Algorithms (EAs) have been applied to many optimisation problems, among which those with high order are difficult for EAs. The higher the order, the steeper the curve around the optimum is, therefore the more difficult it is. This paper introduces Transfer Learning (TL) aided EAs to conquer the high-order problems more efficiently and effectively by optimum transfer from the low-order problem (as source domain) to high-order problem (as the target domain). The experiments validated this method by comparison of the average number of the convergence generation and an impressive feature was observed: this method is robust against the difficulties of the problems. This method is not only significant for high-order problems, but also useful for other difficult problems by borrowing optimum from other feature-similar easy problems.

Rescaling Borders of Investment: The Arctic Council and the Economic Development Policies

ABSTRACTThis paper explores the developments that led to the creation of the Arctic Economic Council (AEC) by the Arctic Council, and its relationship to the larger structures of global investment as represented by the World Economic Forum’s Arctic Investment. It argues that the prioritization of environment, and the construction of a regional “map” based primarily upon environmental cooperation positions development as a lower order problem for regional actors to resolve and leads to a particular positioning of economic activities and networks at the regional scale. What this means for the scale of regional cooperation, especially in light of larger Arctic investment initiatives such as the World Economic Forum’s Arctic Investment Protocol (AIP), will be considered.

Step by Step: Biology Undergraduates' Problem-Solving Procedures during Multiple-Choice Assessment.

This study uses the theoretical framework of domain-specific problem solving to explore the procedures students use to solve multiple-choice problems about biology concepts. We designed several multiple-choice problems and administered them on four exams. We trained students to produce written descriptions of how they solved the problem, and this allowed us to systematically investigate their problem-solving procedures. We identified a range of procedures and organized them as domain general, domain specific, or hybrid. We also identified domain-general and domain-specific errors made by students during problem solving. We found that students use domain-general and hybrid procedures more frequently when solving lower-order problems than higher-order problems, while they use domain-specific procedures more frequently when solving higher-order problems. Additionally, the more domain-specific procedures students used, the higher the likelihood that they would answer the problem correctly, up to five procedures. However, if students used just one domain-general procedure, they were as likely to answer the problem correctly as if they had used two to five domain-general procedures. Our findings provide a categorization scheme and framework for additional research on biology problem solving and suggest several important implications for researchers and instructors.

The FFTRR-based fast direct algorithms for complex inhomogeneous biharmonic problems with applications to incompressible flows

We develop analysis-based fast and accurate direct algorithms for several biharmonic problems in a unit disk derived directly from the Green’s functions of these problems and compare the numerical results with the “decomposition” algorithms (see Ghosh and Daripa, IMA J. Numer. Anal. 36(2), 824–850 [17]) in which the biharmonic problems are first decomposed into lower order problems, most often either into two Poisson problems or into two Poisson problems and a homogeneous biharmonic problem. One of the steps in the “decomposition algorithm” as discussed in Ghosh and Daripa (IMA J. Numer. Anal. 36(2), 824–850 [17]) for solving certain biharmonic problems uses the “direct algorithm” without which the problem can not be solved. Using classical Green’s function approach for these biharmonic problems, solutions of these problems are represented in terms of singular integrals in the complex z−plane (the physical plane) involving explicitly the boundary conditions. Analysis of these singular integrals using FFT and recursive relations (RR) in Fourier space leads to the development of these fast algorithms which are called FFTRR based algorithms. These algorithms do not need to do anything special to overcome coordinate singularity at the origin as often the case when solving these problems using finite difference methods in polar coordinates. These algorithms have some other desirable properties such as the ease of implementation and parallel in nature by construction. Moreover, these algorithms have O(logN) complexity per grid point where N2 is the total number of grid points and have very low constant behind this order estimate of the complexity. Performance of these algorithms is shown on several test problems. These algorithms are applied to solving viscous flow problems at low and moderate Reynolds numbers and numerical results are presented.

Hybrid Deterministic/Monte Carlo Neutronics

In this paper we describe a hybrid deterministic/Monte Carlo algorithm for neutron transport simulation. The algorithm is based on nonlinear accelerators for source iteration, using Monte Carlo methods for the purely absorbing high-order problem and a Jacobian-free Newton--Krylov iteration for the low-order problem. We couple the Monte Carlo solution with the low-order problem using filtering to smooth the flux and current from the Monte Carlo solver and an analytic Jacobian-vector product to avoid numerical differentiation of the Monte Carlo results. We use a continuous energy deposition tally for the Monte Carlo simulation. We conclude the paper with numerical results which illustrate the effectiveness of the new algorithm.

Methods of extending lower order problems to higher order problems in the context of smallest eigenvalue comparisons

Methods of extending lower order problems to higher order problems in the context of smallest eigenvalue comparisons

A Generalized KYP Lemma-Based Approach for H ∞ Control of Singularly Perturbed Systems

In this paper, H ? control synthesis of linear time invariant singularly perturbed systems (SPS) based on a generalized KYP lemma approach is investigated. By employing a generalized KYP lemma on the low- and high-frequency domains of SPS, respectively, a slow (low-frequency) controller and a fast (high-frequency) controller are designed to stabilize the slow and fast subsystems, and also to meet the individual H ? performance specifications. A composite controller for the full-order SPS is constructed via the above two above well-defined lower-order problems. Moreover, the effectiveness of the proposed method is verified as compared with the traditional H ? design method by its application to a tracking problem of SPS and an H ? model matching problem for SPS.

Shakedown analysis of spatial frames with parameterized load domain

This work addresses shakedown analysis of geometrically linear spatial frames with linearized yield criteria and parameterized general variable load domain. If the boundary of the load domain is curved, the respective optimization problem has an infinite number of constraints. Using parameterization, a computational bi-level decomposition methodology is developed, which is capable of treating problems of this kind. Two alternative bi-level procedures are presented: the centered minmax technique and the parameterized Maier’s technique. The lower level, shared by both procedures, comprises a series of local constrained optimization problems for the check points across the frame. These low-order problems, solved either algorithmically or analytically, provide data for the upper level, where a linear programming problem of limit analysis type is solved in each procedure for the whole frame. Two examples with ellipsoidal load domain are presented. Several aspects are discussed.

A Nonlinear Space-Angle Multigrid Acceleration for the Method of Characteristics in Unstructured Meshes

A new space-angle multigrid technique has been developed to accelerate the free inner transport iterations based upon the method of characteristics (MOC). We present a two-level scheme that consists of a fine level on which the MOC transport calculation is performed and a more coarsely discretized phase-space in which a low-order problem is solved as an acceleration step. A flux-volume homogenization technique is employed to define the coarse-level cross sections. This entails the nonlinearity of the scheme. Restriction and prolongation operators are defined between the two levels. After each fine transport iteration, a low-order transport problem is iteratively solved on the homogenized grid. A coarser angular representation is used within an MOC-like framework. Discontinuity factors are employed to reconstruct the scalar incoming and outgoing currents on each region of the coarse discretization. The solution of the aforementioned low-order problem is used to correct the angular moments of the flux resulting from the previous free transport sweep. A complete description of the low-order operator and of the grid-to-grid transfer operators is given. A further application of the method to the acceleration of outer transport iterations is also presented. In order to test the effectiveness of this method, numerical tests for given benchmark geometries have been performed. Results are discussed.