Numerical Benchmark Problems Research Articles

Limit analysis of 3D masonry block structures with non-associative frictional joints using cone programming

A three-dimensional limit analysis model for masonry structures is presented. In the model the masonry is discretized as an assemblage of rigid blocks, which interact via no-tension contact surfaces with Coulomb friction. A concave contact formulation is adopted and an iterative solution procedure is used to allow the underlying non-associative friction problem to be solved. Second order cone programming (SOCP) is used to allow direct modelling of the conic failure surface. The formulation is validated against various numerical benchmark problems and then successfully applied to masonry walls and a small-scale masonry building tested experimentally.

The Great Salmon Run Metaheuristic for Robust Shape and Size Design of Truss Structures with Dynamic Constraints

In this investigation, the authors intend to demonstrate the applicability of a recent spotlighted metaheuristic called the great salmon run (TGSR) algorithm for shape and size design of truss structures. The algorithmic functioning of TGSR emulates the annual migration of salmons together with dangers laid through their pathways. In a previous study by the authors, it has been proved that the method is as effective as most of the state-of-the-art metaheuristics for a wide range of numerical benchmark problems. Here, the authors utilize TGSR together with some rival metaheuristics, i.e. bee algorithm (BA), scale factor local search differential evolutionary algorithm (SFLSDEA), chaotic particle swarm optimization (CPSO) algorithm, self adaptive penalty function genetic algorithm (SAPFGA) and mutable smart bee algorithm (MSBA), for optimal design of truss structures with dynamic frequency constraints. To effectively handle the constraints, the authors take the advantage of self-adaptive penalty function (SAPF) constraint handling technique to free the user from any priori penalty coefficient tuning. Therefore, an algorithm for automation of constraint shape and size design of truss structures is proposed here. Furthermore, for more elaboration, the authors consider the results of some previous reports for same problems to find out whether TGSR is capable of yielding comparative results as compared to other metaheuristics. Through the experiments, the exploration/exploitation capabilities of TGSR for truss design are investigated. It is proved that TGSR is not only able to handle the nonlinearities and decision making difficulties associated with shape and size optimization of truss structures but also can show comparative results as compared to powerful state-of-the-art metaheuristics.

The Lagrangian relaxation for the combinatorial integral approximation problem

We are interested in methods to solve mixed-integer nonlinear optimal control problems constrained by ordinary differential equations and combinatorial constraints on some of the control functions. To solve these problems we use a first discretize, then optimize approach to get a specially structured mixed-integer nonlinear program (MINLP). We decompose this MINLP into a nonlinear program (NLP) and a mixed-integer linear program (MILP), which is called the combinatorial integral approximation problem (CIAP). Previous results guarantee an integer gap for the MINLP depending on the objective function value of the CIAP. The focus of this study is the analysis of the CIAP and of a tailored branch-and-bound method. We link the huge computational gains compared to state-of-the-art MILP solvers to an analysis of subproblems on the branching tree. To this end we study properties of the Lagrangian relaxation of the CIAP. Special focus is given to special ordered set constraints that are present due to an outer convexification of the control problem. Also subproblems that arise by the application of branch-and-bound schemes are of interest. We prove polynomial runtime of the algorithm for special cases and give numerical evidence for efficiency by means of a numerical benchmark problem.

Study on Evolutionary Algorithm Online Performance Evaluation Visualization Based on Python Programming Language

Abstract Evolutionary computations are kinds of random searching algorithms derived from natural selection and biological genetic evolution behavior. Evaluating the performance of an algorithm is a fundamental task to track and find the way to improve the algorithm, while visualization technique may play an important act during the process. Based on current existing algorithm performance evaluation criteria and methods, a Python-based programming tracking strategy, which employs 2-D graphical library of python matplotlib for online algorithm performance evaluation, is proposed in this paper. Tracking and displaying the performance of genetic algorithm (GA) and particle swarm optimization (PSO) optimizing two typical numerical benchmark problems are employed for verification and validation. Results show that the tracking strategy based on Python language for online performance evaluation of evolutionary algorithms is valid, and can be used to help researchers on algorithms’ performance evaluation and finding ways to improve it.

The calculation method for SP3 discontinuity factor and its application

The simplified-P3 (SP3) method is considered a promising approximate spherical harmonics method with comparable accuracy to low order discrete ordinate method, but with about the same order of speed as that of the diffusion method. However the difficulty in calculating the SP3 discontinuity factors (DF) could significantly reduce its advantage over the diffusion method for practical applications. Recently Chao and Yamamoto proposed a new SP3 derivation that overcomes this difficulty with the explicit derivation of how to calculate DF in terms of the volume average flux, surface average flux, surface average partial currents, and the surface average flux gradient from the heterogeneous transport solution of a single reflective assembly. This paper follows that formulation and approach, but with an error correction and some improvements, to work out a specific methodology of calculating DF for SP3 and validates the method with a set of challenging numerical benchmark problems. It is concluded that although SP3 is significantly more accurate than diffusion, diffusion with DF actually gives more accurate results than SP3 without DF. However, when SP3 also uses DF its superiority over diffusion is restored. But in case of energy group collapsing or large nodal meshes, the advantage of using SP3 over diffusion diminishes.

Finite element simulation of large-strain single-crystal viscoplasticity: An investigation of various hardening relations

Hardening relations describe the increase in resistance to deformation during plastic flow. Three hardening relations are compared here in the context of conventional large-strain single-crystal viscoplasticity. The first is an isotropic hardening relation. The second is a hardening relation that is expressed as an ordinary differential equation in the slip resistance. The third is a new relation, originally developed in the context of gradient crystal plasticity, in which the slip resistance is expressed explicitly in terms of the accumulated slip on each slip system. The numerical solution of the governing equations is found using the finite element method coupled with a predictor–corrector type algorithm. The features of the hardening relations are elucidated using a series of numerical benchmark problems. The parameters for the hardening relations are calibrated using a model problem. Various crystal structures are investigated, including single- and double slip, and face-centred cubic crystals. The hardening relations are compared and their relative features discussed.

An Imbricate Finite Element Method (I-FEM) using full, reduced, and smoothed integration

A method to design finite elements that imbricate with each other while being assembled, denoted as imbricate finite element method, is proposed to improve the smoothness and the accuracy of the approximation based upon low order elements. Although these imbricate elements rely on triangular meshes, the approximation stems from the shape functions of bilinear quadrilateral elements. These elements satisfy the standard requirements of the finite element method: continuity, delta function property, and partition of unity. The convergence of the proposed approximation is investigated by means of two numerical benchmark problems comparing three different schemes for the numerical integration including a cell-based smoothed FEM based on a quadratic shape of the elements edges. The method is compared to related existing methods.

An edge‐based smoothed finite element method for 3D analysis of solid mechanics problems

SUMMARYThe edge‐based smoothed finite element method (ES‐FEM) was proposed recently in Liu, Nguyen‐Thoi, and Lam to improve the accuracy of the FEM for 2D problems. This method belongs to the wider family of the smoothed FEM for which smoothing cells are defined to perform the numerical integration over the domain. Later, the face‐based smoothed FEM (FS‐FEM) was proposed to generalize the ES‐FEM to 3D problems. According to this method, the smoothing cells are centered along the faces of the tetrahedrons of the mesh. In the present paper, an alternative method for the extension of the ES‐FEM to 3D is investigated. This method is based on an underlying mesh composed of tetrahedrons, and the approximation of the field variables is associated with the tetrahedral elements; however, in contrast to the FS‐FEM, the smoothing cells of the proposed ES‐FEM are centered along the edges of the tetrahedrons of the mesh. From selected numerical benchmark problems, it is observed that the ES‐FEM is characterized by a higher accuracy and improved computational efficiency as compared with linear tetrahedral elements and to the FS‐FEM for a given number of degrees of freedom. Copyright © 2013 John Wiley & Sons, Ltd.

Optimal design of classic Atkinson engine with dynamic specific heat using adaptive neuro-fuzzy inference system and mutable smart bee algorithm

In this article, an improved version of Artificial Bee Colony (ABC) algorithm is developed to optimize a multi-modal thermodynamic power system with dynamic specific heat. Since original Karaboga's ABC for constraint problems does not consider the initial population to be feasible, a modified method called “Mutable Smart Bee Algorithm” (MSBA) is used which utilizes mutable heuristic agents. These mutable agents are also capable to maintain their historical memory for the location and quality of food sources. These features have been found as strong elements for mining data in constraint areas. In additions, our implementations reveal that MSBA is faster than Karaboga's method. To elaborate on authenticity of MSBA, several state-of-the-art techniques are used as rival methods to optimize well-known benchmark problems. Then, two main steps are made to optimize Atkinson engine. Firstly, an Adaptive Neuro-Fuzzy Inference System (ANFIS) is developed to identify the dynamic behavior of specific heat. Then, MSBA is hired to design the optimum features of the engine. It is observed that the proposed method is capable to successfully handle the real-life engineering problem as well as the numerical benchmark problems.

Solution of a low Prandtl number natural convection benchmark by a local meshless method

PurposeThe purpose of this paper is to present the solution of a highly nonlinear fluid dynamics in a low Prandtl number regime, typical for metal‐like materials, as defined in the call for contributions to a numerical benchmark problem for 2D columnar solidification of binary alloys. The solution of such a numerical situation represents the first step towards understanding the instabilities in a more complex case of macrosegregation.Design/methodology/approachThe involved temperature, velocity and pressure fields are represented through the local approximation functions which are used to evaluate the partial differential operators. The temporal discretization is performed through explicit time stepping.FindingsThe performance of the method is assessed on the natural convection in a closed rectangular cavity filled with a low Prandtl fluid. Two cases are considered, one with steady state and another with oscillatory solution. It is shown that the proposed solution procedure, despite its simplicity, provides stable and convergent results with excellent computational performance. The results show good agreement with the results of the classical finite volume method and spectral finite element method.Originality/valueThe solution procedure is formulated completely through local computational operations. Besides local numerical method, the pressure velocity is performed locally also, retaining the correct temporal transient.

Finite element methods for variable density flow and solute transport

Saltwater intrusion into coastal freshwater aquifers is an ongoing problem that will continue to impact coastal freshwater resources as coastal populations increase. To effectively model saltwater intrusion, the impacts of increased salt content on fluid density must be accounted for to properly model saltwater/freshwater transition zones and sharp interfaces. We present a model for variable density fluid flow and solute transport where a conforming finite element method discretization with a locally conservative velocity post-processing method is used for the flow model and the transport equation is discretized using a variational multiscale stabilized conforming finite element method. This formulation provides a consistent velocity and performs well even in advection-dominated problems that can occur in saltwater intrusion modeling. The physical model is presented as well as the formulation of the numerical model and solution methods. The model is tested against several 2-D and 3-D numerical and experimental benchmark problems, and the results are presented to verify the code.

Enhancing Differential Evolution Performance with Self-Adaptive Population Topologies on Numerical Benchmark Problems

Enhancing Differential Evolution Performance with Self-Adaptive Population Topologies on Numerical Benchmark Problems

A complementary-energy based criterion for the stability analysis of geometrically exact framed structures

Stability analysis is a crucial aspect of structural design, particularly for beam type structures. This paper addresses the development of a complementary-energy based (dual) stability criterion for the quasi-static analysis of three-dimensional framed structures modeled using the so-called geometrically exact (Reissner–Simo) beam theory. The criterion is derived from the well known primal stability criterion based on the condition of minimum total potential energy. This is accomplished by resorting to the Lagrangian multiplier method together with the Legendre transformation. Several numerical benchmark problems are analyzed. The analyses are carried out using two dual finite element models and their corresponding stability criteria, namely, the traditional two-node displacement/rotation-based model with the primal stability criterion, and a recently introduced equilibrium-based model with the proposed dual stability criterion.

One-point quadrature ANS solid-shell element based on a displacement variational formulation Part I – Geometrically linear assessment

A demonstration of a computationally efficient, physically stabilised one point-quadrature solid-shell finite element in pure displacement variational formulation and within the context of hierarchical displacement modes is presented in this paper. The displacement field of the 24 degree of freedom 8-node solid finite element is enhanced by a hierarchical, thickness-wise, quadratic displacement field and associated single displacement degree of freedom. The displacement enhancements expand the covariant element strain space resolving thickness locking and in combination with the assumed natural strain approach, attenuates shear and trapezoidal locking in thin shell simulations. Formulated within the recent framework of the reduced integration element technology and hourglass physical stabilisation, a locking free, 8-node solid-shell element for general three dimensional shell analysis is obtained. The theoretical relevance and applicability of the proposed displacement-kinematics 8-node solid-shell element offering comparable accuracy and performance flexibility as physically stabilised reduced integration solid-shell elements based on mixed and EAS variational statements is demonstrated in selected linear numerical benchmark problems.

Stochastic Collocation Methods on Unstructured Grids in High Dimensions via Interpolation

In this paper we propose a method for conducting stochastic collocation on arbitrary sets of nodes. To accomplish this, we present the framework of least orthogonal interpolation, which allows one to construct interpolation polynomials based on arbitrarily located grids in arbitrary dimensions. These interpolation polynomials are constructed as a subspace of the family of orthogonal polynomials corresponding to the probability distribution function on stochastic space. This feature enables one to conduct stochastic collocation simulations in practical problems where one cannot adopt some popular node selections such as sparse grids or cubature nodes. We present in detail both the mathematical properties of the least orthogonal interpolation and its practical implementation algorithm. Numerical benchmark problems are also presented to demonstrate the efficacy of the method.

Shell Finite Cell Method: A high order fictitious domain approach for thin-walled structures

This article presents a generalization of the recently proposed Finite Cell Method to thin-walled structures. This approach uses a combination of well known Fictitious Domain Methods with high order hierarchical Ansatz spaces known from the p-version of the Finite Element Method. Whereas the original concept embeds a three-dimensional structure in a simple domain being meshed into a grid of cube shaped cells, the extension presented in this paper applies the fictitious domain approach to a two-dimensional master domain defined in the parameter plane of the geometric model. Implementation details are discussed and numerical benchmark problems show the high accuracy and computational efficiency of the new approach. It is also remarked, that the present approach can easily be carried over to isogeometric analysis, opening an attractive possibility to simulate trimmed NURBS-surfaces.

Simplified two and three dimensional HTTR benchmark problems

To assess the accuracy of diffusion or transport methods for reactor calculations, it is desirable to create heterogeneous benchmark problems that are typical of whole core configurations. In this paper we have created two and three dimensional numerical benchmark problems typical of high temperature gas cooled prismatic cores. Additionally, a single cell and single block benchmark problems are also included. These problems were derived from the HTTR start-up experiment. Since the primary utility of the benchmark problems is in code-to-code verification, minor details regarding geometry and material specification of the original experiment have been simplified while retaining the heterogeneity and the major physics properties of the core from a neutronics viewpoint. A six-group material (macroscopic) cross section library has been generated for the benchmark problems using the lattice depletion code HELIOS. Using this library, Monte Carlo solutions are presented for three configurations (all-rods-in, partially-controlled and all-rods-out) for both the 2D and 3D problems. These solutions include the core eigenvalues, the block (assembly) averaged fission densities, local peaking factors, the absorption densities in the burnable poison and control rods, and pin fission density distribution for selected blocks. Also included are the solutions for the single cell and single block problems.

A comparative study of common and self-adaptive differential evolution strategies on numerical benchmark problems

Abstract Differential Evolution (DE) is a population-based stochastic global optimization technique that requires the adjustment of a very few parameters in order to produce results. However, the control parameters involved in DE are highly dependent on the optimization problem; in practice, their fine-tuning is not always an easy task. The self-adaptive differential evolution (SADE) variants are those that do not require the pre-specified choice of control parameters. On the contrary, control parameters are selfadapted by using the previous learning experience. In this paper, we discuss and evaluate popular common and self-adaptive differential evolution (DE) algorithms. In particular, we present an empirical comparison between two self-adaptive DE variants and common DE methods. In order to assure a fair comparison, we test the methods by using a number of well-known unimodal and multimodal, separable and non-separable, benchmark optimization problems for different dimensions and population size. The results show that SADE variants outperform, or at least produce similar results, to common differential evolution algorithms in terms of solution accuracy and convergence speed. The advantage of using the self-adaptive methods is that the user does not need to adjust control parameters. Therefore, the total computational effort is significantly reduced.

Single machine scheduling for minimising earliness and tardiness penalties by scatter search approach

This paper proposes a new evolutionary technique named scatter search for scheduling a number of jobs on a single machine against a restrictive common due date. Individual earliness and tardiness penalties for the jobs are under consideration and the objective is to find an optimal schedule, which jointly minimises the sum of the earliness and tardiness costs. In this paper, the results are obtained for numerous benchmark problems generated by Biskup and Feldmann (2001) for common due date (CDD) problems. The best results of each of the benchmark problems of the three meta-heuristic techniques used by Feldmann and Biskup (2003) are selected and they are compared with the results of SS. This methodology provides substantially better results than the benchmark problems. At the same time, this method holds almost the same result as the best results (best-FB) of three meta-heuristic approaches of Feldmann and Biskup (2003).

A stylized three dimensional PWR whole-core benchmark problem with Gadolinium

Recent advancements in computer technology have led to an increased ability to solve complicated reactor problems, and as a result, there has been an interest in codes and methods that utilize whole-core transport in three-dimensional, large-scale, highly-heterogeneous configurations. Most current reactor benchmark problems are either small-scale or involve spatial homogenization (pin cell or full-assembly) for diffusion codes; therefore, new whole-core benchmark problems must be designed in order to validate whole-core transport methods. This study focuses on an PWR case, with a low leakage loading pattern and Gadolinium as a burnable absorber. In this paper, a detailed description of a new 3D whole-core numerical benchmark problem based on a 4-loop layout is presented with geometric, material and cross section specifications. A set of 2-group and 4-group region-wise neutron cross section libraries is generated using the lattice depletion code HELIOS, and an MCNP model is described for three configurations (all rods in, all rods out, power-shaping rods inserted). Eigenvalues are presented for the 2-group calculations of all three configurations, and detailed pin fission density results are presented for the power-shaping configuration.