Selected Benchmark Problems Research Articles

Analysis of Stabilization Techniques for CMFD Acceleration of Neutron Transport Problems

The coarse-mesh finite difference (CMFD) method is one of the most widely used methods for accelerating the convergence of numerical transport solutions. However, in some situations, iterative methods using CMFD can become unstable and fail to converge. We present and evaluate three different modifications of the CMFD scheme that provide enhanced stability: multiple transport sweeps, artificial diffusion, and relaxing the flux update. We present the Fourier analysis on each of these schemes for an idealized problem to characterize the stability and rate of convergence for both fixed-source and fission-source problems. Comparisons of the effectiveness of these methods are also performed numerically for a variety of benchmark boiling water reactor and pressurized water reactor problems using the Consortium for Advanced Simulation of Light Water Reactors neutronics code MPACT. We demonstrate a means of stabilizing CMFD by modifying the diffusion coefficient to make the iteration behave more like the partial-current CMFD (pCMFD) method, which is unconditionally stable, and show through a sequence of numerical experiments that the CMFD method performs similarly to the pCMFD method for the selected benchmark problems. We also show, both theoretically and experimentally, that modifying the diffusion coefficient in the CMFD equations is similar to underrelaxing the scalar flux update. The theoretical and experimental results show that many of the known techniques for stabilizing CMFD are fundamentally very closely related.

An improved and efficient finite volume scheme for bivariate aggregation population balance equation

In this work, a finite volume scheme for the numerical solution of bivariate pure aggregation population balance equations on non-uniform meshes is derived. The new method has a simple mathematical structure and it provides high accuracy with respect to the number density distribution as well as different moments. The method relies on weights to conserve the total mass of the system. The new method is compared to a recently developed finite volume scheme by Forestier-Coste and Mancini (2012) for some selected benchmark problems. It is shown that the proposed method is not only computationally more efficient but also more accurate than the method by Forestier-Coste and Mancini (2012).

Benchmark solutions and assessment of variable kinematics models for global and local buckling of sandwich struts

This paper investigates the periodic global buckling and wrinkling of sandwich struts subjected to uniaxial loads. An exact elasticity solution is obtained in the plane strain setting for selected benchmark problems and serves as reference for a systematic assessment of several displacement-based sandwich plate models. The Sublaminate Generalized Unified Formulation is used to generate the algebraic two-dimensional governing equations of the variable kinematics models, for which a Navier-type solution is adopted.

A volume‐consistent discrete formulation of aggregation population balance equations

In this paper, a new finite volume scheme for the numerical solution of the pure aggregation population balance equation, or Smoluchowski equation, on non‐uniform meshes is derived. The main feature of the new method is its simple mathematical structure and high accuracy with respect to the number density distribution as well as its moments. The new method is compared with the existing schemes given by Filbet and Laurençot (SIAM J. Sci. Comput., 25 (2004), pp. 2004–2028) and Forestier and Mancini (SIAM J. Sci. Comput., 34 (2012), pp. B840–B860) for selected benchmark problems. It is shown that the new scheme preserves all the advantages of a conventional finite volume scheme and predicts higher‐order moments as well as number density distribution with high accuracy. Copyright © 2015 John Wiley & Sons, Ltd.

Novel performance metrics for robust multi-objective optimization algorithms

Performance metrics are essential for quantifying the performance of optimization algorithms in the field of evolutionary multi-objective optimization. Such metrics allow researchers to compare different algorithms quantitatively. In the field of robust multi-objective optimization, however, there is currently no performance metric despite its significant importance. This motivates our proposal of three novel specific metrics for measuring the convergence, coverage, and success rate of robust Pareto optimal solutions obtained by robust multi-objective algorithms. The proposed metrics are employed to quantitatively evaluate and compare Robust Multi-objective Particle Swarm Optimization (RMOPSO) and Robust Non-dominated Sorting Genetic Algorithm (RNSGA-II) on seven selected benchmark problems. The results show that the proposed metrics are effective in quantifying the performance of robust multi-objective algorithms in terms of convergence, coverage, and the ratio of the robust/non-robust Pareto optimal solutions obtained.

Comparison between genetic algorithm and prey-predator algorithm

Metaheuristic algorithms are useful in solving complex optimization problems. Genetic algorithm (GA) is one of the well known and oldest metaheuristic algorithms. It was introduced in 1975 and has been used in many applications varying from engineering to management and many other fields as well. However, Prey-Predator algorithm (PPA) is one of recently introduced algorithm, in 2012, inspired by the interaction between preys and their predator. The motivation and the search mechanism for these two algorithms are different. In this paper the comparison of these two algorithms both from theoretical aspects and using simulation on selected benchmark problems is presented. According to the results, PPA performs better than GA in the selected test problems.

A binary differential evolution algorithm learning from explored solutions

Although real-coded differential evolution (DE) algorithms can perform well on continuous optimization problems (CoOPs), designing an efficient binary-coded DE algorithm is still a challenging task. Inspired by the learning mechanism in particle swarm optimization (PSO) algorithms, we propose a binary learning differential evolution (BLDE) algorithm that can efficiently locate the global optimal solutions by learning from the last population. Then, we theoretically prove the global convergence of BLDE, and compare it with some existing binary-coded evolutionary algorithms (EAs) via numerical experiments. Numerical results show that BLDE is competitive with the compared EAs. Further study is performed via the change curves of a renewal metric and a refinement metric to investigate why BLDE cannot outperform some compared EAs for several selected benchmark problems. Finally, we employ BLDE in solving the unit commitment problem (UCP) in power systems to show its applicability to practical problems.

A Fully Implicit, Lower Bound, Multi-Axial Solution Strategy for Direct Ratchet Boundary Evaluation: Implementation and Comparison

Ensuring sufficient safety against ratcheting is a fundamental requirement in pressure vessel design. However, determining the ratchet boundary using a full elastic-plastic finite element analysis can be problematic and a number of direct methods have been proposed to overcome difficulties associated with ratchet boundary evaluation. This paper proposes a new lower bound ratchet analysis approach, similar to the previously proposed hybrid method but based on fully implicit elastic-plastic solution strategies. The method utilizes superimposed elastic stresses and modified radial return integration to converge on the residual state throughout, resulting in one finite element model suitable for solving the cyclic stresses (stage 1) and performing the augmented limit analysis to determine the ratchet boundary (stage 2). The modified radial return methods for both stages of the analysis are presented, with the corresponding stress update algorithm and resulting consistent tangent moduli. Comparisons with other direct methods for selected benchmark problems are presented. It is shown that the proposed method evaluates a consistent lower bound estimate of the ratchet boundary, which has not previously been clearly demonstrated for other lower bound approaches. Limitations in the description of plastic strains and compatibility during the ratchet analysis are identified as being a cause for the differences between the proposed methods and current upper bound methods.

Forchheimer’s correction in modelling flow and transport in fractured porous media

The scope of this manuscript is to investigate the role of the Forchheimer correction in the description of variable-density flow in fractured porous media. A fractured porous medium, which shall be also referred to as “the embedding medium”, represents a flow region that is made macroscopically heterogeneous by the presence of fractures. Fractures are assumed to be filled with a porous medium characterized by flow properties that differ appreciably from those of the embedding medium. The fluid, which is free to move in the pore space of the entire flow region, is a mixture of water and brine. Flow is assumed to be a consequence of the variability of the fluid mass density in response to the generally nonuniform distribution of brine, which is subject to diffusion and convection. The fractures are assumed to be thin in comparison with the characteristic sizes of the embedding medium. Within this framework, some benchmark problems are solved by adopting two approaches: (i) the fractures are treated as thin but $$\mathrm{d}$$ -dimensional flow subregions, with $$d$$ being the geometric dimension of the embedding medium; (ii) the fractures are regarded as $$(\mathrm{d}-1)$$ -dimensional manifolds. In the first approach, the equations of variable-density flow are written in the same, $$d$$ -dimensional form both in the fractures and in the embedding medium. In the second approach, instead, new equations are obtained by averaging the $$\mathrm{d}$$ -dimensional ones over the fracture width. The reliability of the second approach is discussed by comparing the results of the $$(\mathrm{d}-1)$$ -dimensional numerical simulations of the selected benchmark problems with those obtained by using the $$d$$ -dimensional approach. Moreover, the deviations of the results determined by accounting for the Forchheimer correction to flow velocity are compared with those predicted by Darcy’s law.

A Multiobjective Particle Swarm Optimizer for Constrained Optimization

Constraint handling techniques are mainly designed for evolutionary algorithms to solve constrained multiobjective optimization problems (CMOPs). Most multiojective particle swarm optimization (MOPSO) designs adopt these existing constraint handling techniques to deal with CMOPs. In the proposed constrained MOPSO, information related to particles’ infeasibility and feasibility status is utilized effectively to guide the particles to search for feasible solutions and improve the quality of the optimal solution. This information is incorporated into the four main procedures of a standard MOPSO algorithm. The involved procedures include the updating of personal best archive based on the particles’ Pareto ranks and their constraint violation values; the adoption of infeasible global best archives to store infeasible nondominated solutions; the adjustment of acceleration constants that depend on the personal bests’ and selected global best’s infeasibility and feasibility status; and the integration of personal bests’ feasibility status to estimate the mutation rate in the mutation procedure. Simulation to investigate the proposed constrained MOPSO in solving the selected benchmark problems is conducted. The simulation results indicate that the proposed constrained MOPSO is highly competitive in solving most of the selected benchmark problems.

A scatter search algorithm for scheduling optimisation of job shop problems

The Job Shop Scheduling (JSS) problem is one of the most difficult NP-hard combinatorial optimisation problems. it is very difficult to solve by conventional optimisation techniques. There has been increasing interest in applying metaheuristic methods to solve such hard optimisation problems. In this work, a novel metaheuristic approach called Scatter Search (SS) is applied for the JSS problem, an NP-hard sequencing problem. The approach is used to find a schedule to minimise the makespan (Cmax), that is, the time required to complete all jobs. SS contrasts with other evolutionary procedures by providing a wide exploration of the search space through intensification and diversification. In addition, it has a unifying principle for joining solutions. It exploits the adaptive memory principle to avoid generating or incorporating duplicate solutions at various stages of the problem. The main aim of this research is to explore the potential of SS for scheduling JSS problems. SS provides unifying principles for joining solutions based on generalised path constructions and by utilising strategic designs where other approaches resort to randomisation. In this paper, 11 benchmark problems are taken from the literature. The results available for the various existing metaheuristic methods for the selected benchmark problems are compared with results obtained by the SS method. The proposed framework achieves better results for all 11 problems.

An elasto‐plastic three phase model for partially saturated soil for the finite element simulation of compressed air support in tunnelling

AbstractThis paper presents a fully coupled finite element formulation for partially saturated soil as a triphasic porous material, which has been developed for the simulation of shield tunnelling with heading face support using compressed air. While for many numerical simulations in geotechnics use of a two‐phase soil model is sufficient, the simulation of compressed air support demands the use of a three‐phase model with the consideration of air as a separate phase. A multiphase model for soft soils is developed, in which the individual constituents of the soil—the soil skeleton, the fluid and the gaseous phase—and their interactions are considered. The triphasic model is formulated within the framework of the theory of porous media, based upon balance equations and constitutive relations for the soil constituents and their mixture. An elasto‐plastic, cam–clay type model is extended to partially saturated soil conditions by incorporating capillary pressure according to the Barcelona basic model. The hydraulic properties of the soil are described via DARCY's law and the soil–water characteristic curve after VAN GENUCHTEN. Water is modelled as an incompressible and air as a compressible phase. The model is validated by means of selected benchmark problems. The applicability of the model to geotechnical problems is demonstrated by results from the simulation of a compressed air intervention in shield tunnelling. Copyright © 2009 John Wiley & Sons, Ltd.

A residual‐potential boundary for time‐domain problems in computational acoustics.

For many years, researchers have sought theoretically exact time‐domain computational boundaries that are temporally, spatially, and geometrically local. Unfortunately, spatial locality and geometric locality cannot be simultaneously achieved for exact boundaries. Absorbing boundaries offer spatial locality but not geometric locality, whereas impedance boundaries offer the latter but not the former. Following a brief discussion of these topics, a new, theoretically exact impedance boundary is introduced that is based on modal residual potentials for the spherical geometry. The new boundary produces a set of first‐order, uncoupled ODEs for nodal boundary responses and a set of low‐order, uncoupled stepping formulas for modal boundary responses. The two sets are coupled through nodal‐modal transformation based on the orthogonal surface functions for the spherical boundary. Numerical results generated with the boundary are presented for selected benchmark problems. Finally, extension of the method to other separable geometries for the wave equation is discussed.

Computational modeling of cardiac electrophysiology: A novel finite element approach

AbstractThe key objective of this work is the design of an unconditionally stable, robust, efficient, modular, and easily expandable finite element‐based simulation tool for cardiac electrophysiology. In contrast to existing formulations, we propose a global–local split of the system of equations in which the global variable is the fast action potential that is introduced as a nodal degree of freedom, whereas the local variable is the slow recovery variable introduced as an internal variable on the integration point level. Cell‐specific excitation characteristics are thus strictly local and only affect the constitutive level. We illustrate the modular character of the model in terms of the FitzHugh–Nagumo model for oscillatory pacemaker cells and the Aliev–Panfilov model for non‐oscillatory ventricular muscle cells. We apply an implicit Euler backward finite difference scheme for the temporal discretization and a finite element scheme for the spatial discretization. The resulting non‐linear system of equations is solved with an incremental iterative Newton–Raphson solution procedure. Since this framework only introduces one single scalar‐valued variable on the node level, it is extremely efficient, remarkably stable, and highly robust. The features of the general framework will be demonstrated by selected benchmark problems for cardiac physiology and a two‐dimensional patient‐specific cardiac excitation problem. Copyright © 2009 John Wiley & Sons, Ltd.

A scatter search method to minimise makespan of cell scheduling problem

Scheduling problem in a Cellular Manufacturing System (CMS) environment is treated as group scheduling problem, which assumes that all parts in a part family are processed in the same cell and no inter-cellular transfer is needed. In a typical CMS environment, however, there could be some exceptional parts, which need to visit machines in the other cells. This fact limits the applicability of group scheduling approaches. This paper addresses the scheduling of manufacturing cells in which parts may need to visit different cells. These problems are solved in two stages namely intra-cell scheduling and inter-cell scheduling. Through inter-cell scheduling, the sequence of parts within manufacturing cells is determined. In inter-cell scheduling, the sequence of cells is obtained. Scatter Search (SS) is proposed to solve the intra-cell and SVS algorithm is to solve inter-cell scheduling problem. The performance of proposed SS algorithm is evaluated on benchmark problems selected from literature. The results reveal that the SS algorithm performs better than NEH heuristic method in the selected benchmark problems with respect to average makespan.

Relating time complexity of protein folding simulation to approximations of folding time

Finkelstein and Badretdinov [A.V. Finkelstein, A.Y. Badretdinov, Rate of protein folding near the point of thermodynamic equilibrium between the coil and the most stable chain fold, Fold. Des. 2 (1997) 115–121] approximated the folding time of protein sequences of length n by exp ( λ ⋅ n 2 / 3 ± χ ⋅ n 1 / 2 / 2 ) n s , where λ and χ are constants close to unity. Recently, Fu and Wang [B. Fu, W. Wang, A 2 O ( n 1 − 1 / d ⋅ log n ) time algorithm for d-dimensional protein folding in the HP-model, in: J. Daz, J. Karhumäki, A. Lepistö, D. Sannella (Eds.), Proceedings of the 31st International Colloquium on Automata, Languages and Programming, in: Lecture Notes in Comput. Sci., vol. 3142, Springer-Verlag, Heidelberg, 2004, pp. 630–644] published an exp ( O ( n 1 − 1 / d ) ⋅ ln n ) algorithm for d-dimensional protein folding simulation in the HP-model, which is close to the folding time approximation by Finkelstein and Badretdinov and can be seen as a justification of the HP-model for investigating general complexity issues of protein folding. We propose a stochastic local search procedure that is based on logarithmic simulated annealing. We obtain that after ( m / δ ) a ⋅ D Markov chain transitions the probability to be in a minimum energy conformation is at least 1 − δ , where m ⩽ b ( d ) ⋅ n is the maximum neighbourhood size ( b ( d ) small integer), a is a small constant, and D is the maximum value of the minimum escape height from local minima of the underlying energy landscape. We note that the time bound is instance-specific, and we conjecture D < n 1 − 1 / d as a worst case upper bound. We analyse D < n experimentally on selected benchmark problems for the d = 2 case.

Enhancement of performance of Genetic Algorithm for job shop scheduling problems through inversion operator

It is observed that the optimization technique Genetic Algorithm is gaining more importance over the past several years. With high computing power we are able to apply soft computing techniques to solve complex problems in less time. An approach through Genetic Algorithm to solve job shop scheduling problems using inversion operator has been tried, with make-span objective. Computational experiments of this attempt have shown better solutions coupled with appreciable reduction in computer processing time. A set of 20 selected benchmark problems were tried with the proposed heuristic for validation and the results are encouraging. The inversion operator is found to perform better.

Improved solutions for job shop scheduling problems through genetic algorithm with a different method of schedule deduction

Job shop scheduling problems are one of the challenging combinatorial problems that have drawn the attention of researchers for the last three decades. It is observed that genetic algorithm (GA) is gaining more importance over the past several years. An attempt has been made through GA to solve job shop scheduling problems with job-based, operation-based, and proposed methods of representation and schedule deduction with the make-span objective. Computational experiments of this attempt have yielded better solutions coupled with appreciable reduction in computer processing time. A set of selected benchmark problems have been used with the proposed heuristic for validation and the results show the better performance of the proposed method of representation of jobs and schedule deduction.

Asymptotic deformation modes of benchmark problems suitable for evaluating shell elements

An analytical approach introduced recently concerning the definition of the limiting cases of the asymptotic behaviour of shell elements is critically reviewed. A simple analytical approach for defining the asymptotic behaviour of shell elements is proposed. This approach is employed to define and analyse the asymptotic deformation modes of selected benchmark problems suitable for the evaluation of the asymptotic behaviour of shell elements. A new limit test problem, the hyperbolic paraboloid shell problem, proposed as a strong test for the pure bending asymptotic behaviour of shell elements, is analysed in the context of the proposed approach. Explicit expressions for characteristic asymptotic displacements are derived for the shell benchmark problems considered. These expressions are proposed for testing the asymptotic behaviour of finite shell elements in membrane-dominated, bending-dominated and mixed membrane-bending mode shell problems.

Evaluation of the FRANCO finite element fuel rod analysis code

Knowledge of the temperature distribution in a nuclear fuel rod is required to predict the thermal and mechanical response of the strongly temperature-dependent fuel elements. In this research, the FRANCO ( F inite Element Fuel R od AN alysis CO de) computer code was developed for use on an IBM-PC to predict thermal and mechanical fuel performance characteristics for reactor operating conditions. A cross-sectional area of a fuel rod is discretized using constant strain triangular elements. To simplify the time-dependent fuel performance analysis, this code uses quasi-steady state conditions or slow power changes for time-independent problems; thus, time-dependent cases can be represented by quasi-static “snap shots.” The development and testing of the FRANCO program for deformation and heat transfer problems is described. The fuel performance models and solution procedures are provided in detail. The accuracy of FRANCO is examined through selected benchmark problems with other nuclear industry finite difference and finite element fuel analysis codes and actual Halden experimental measurements. These benchmarks show that FRANCO performs well except for the fuel-clad contact cases when compared to results from other fuel performance codes, such as FREY and ESCORE. The FRANCO program developed in this research effort is easy to run, fast, and yields comparable results.