Showing papers in "Engineering Computations in 2004"
TL;DR: Particle scale simulation of industrial particle flows using discrete element method (DEM) offers the opportunity for better understanding the flow dynamics leading to improvements in equipment design and operation that can potentially lead to large increases in equipment and process efficiency, throughput and/or product quality as mentioned in this paper.
Abstract: Particle scale simulation of industrial particle flows using discrete element method (DEM) offers the opportunity for better understanding the flow dynamics leading to improvements in equipment design and operation that can potentially lead to large increases in equipment and process efficiency, throughput and/or product quality. Industrial applications can be characterized as large, involving complex particulate behaviour in typically complex geometries. In this paper, with a series of examples, we will explore the breadth of large scale modelling of industrial processes that is currently possible. Few of these applications will be examined in more detail to show how insights into the fundamentals of these processes can be gained through DEM modelling. Some examples of our collaborative validation efforts will also be described.
323 citations
TL;DR: In this article, a new approach for calculating the critical time increment in explicit discrete element simulations is proposed, which is a function of the current contact conditions, and it is shown that the proposed refined estimates of the critical step indicate that the earlier recommendations contained in the literature can often overestimate the actual critical time step.
Abstract: The distinct element method as proposed by Cundall and Strack uses the computationally efficient, explicit, central difference time integration scheme. A limitation of this scheme is that it is only conditionally stable, so small time steps must be used. Some researchers have proposed using an implicit time integration scheme to avoid the stability issues arising from the explicit time integrator typically used in these simulations. However, these schemes are computationally expensive and can require a significant number of iterations to form the stiffness matrix that is compatible with the contact state at the end of each time step. In this paper, a new, simple approach for calculating the critical time increment in explicit discrete element simulations is proposed. Using this approach, it is shown that the critical time increment is a function of the current contact conditions. Considering both two‐ and three‐dimensional scenarios, the proposed refined estimates of the critical time step indicate that the earlier recommendations contained in the literature can be unconservative, in that they often overestimate the actual critical time step. A three‐dimensional simulation of a problem with a known analytical solution illustrates the potential for erroneous results to be obtained from discrete element simulations, if the time‐increment exceeds the critical time step for stable analysis.
250 citations
TL;DR: In this article, a coupled numerical method for direct simulation of particlefluid systems is formulated and implemented, where the Navier-Stokes equations governing fluid flow are solved using the lattice Boltzmann method, while the equations of motion governing particles are solved with the discrete element method.
Abstract: A coupled numerical method for the direct simulation of particle‐fluid systems is formulated and implemented. The Navier‐Stokes equations governing fluid flow are solved using the lattice Boltzmann method, while the equations of motion governing particles are solved with the discrete element method. Particle‐fluid coupling is realized through an immersed moving boundary condition. Particle forcing mechanisms represented in the model to at least the first‐order include static and dynamic fluid‐induced forces, and intergranular forces including particle collisions, static contacts, and cementation. The coupling scheme is validated through a comparison of simulation results with the analytical solution of cylindrical Couette flow. Simulation results for the fluid‐induced erosive failure of a cemented particulate constriction are presented to demonstrate the capability of the method.
161 citations
TL;DR: In this article, a 3D DEM program TRUBAL is employed to study the evolution of internal structure of particle assemblies during the consolidation process, and the results in particle scale (microscopic) are obtained and correlated to the statistical bulk response of the assembly.
Abstract: In this paper, a 3D DEM program TRUBAL, which is capable of calculating the contact between particles considering friction and local plastic deformation, is employed to study the evolution of internal structure of particle assemblies during the consolidation process. Uniaxial powder compaction process is simulated in a cubic periodic unit cell by applying the strain rate to the individual particles. The selection of the proper time steps in DEM for quasi‐static case is discussed. Results in particle scale (microscopic) are obtained and correlated to the statistical bulk response of the assembly. The effects of the microscopic properties of particles (such as friction, plastic contact) on the bulk mechanical response are examined by numerical tests. Correlations between the microscopic properties of particles and the macroscopic continuum behaviours of compacts are discussed. These discussions make it possible to fit DEM results at a macroscopic scale to the experimental measurements by adjusting the particle properties in DEM calculation. An example test is carried out to demonstrate that DEM results could be fitted properly to the experimental results, in the mean time, also provide some microscopic results which are hard to be measured. DEM has the potential to incorporate the microscopic properties of particles into a proper continuum model to perform combined macro and micro study of the powder compaction process.
122 citations
TL;DR: This paper introduces a new hybrid evolutionary algorithm for continuous global optimization problems, called estimation of distribution algorithm with local search (EDA/L), and demonstrates that EDA/L is better than four other recent EAs in terms of the solution quality and the computational cost.
Abstract: This paper introduces a new hybrid evolutionary algorithm (EA) for continuous global optimization problems, called estimation of distribution algorithm with local search (EDA/L). Like other EAs, EDA/L maintains and improves a population of solutions in the feasible region. Initial candidate solutions are generated by uniform design, these solutions evenly scatter over the feasible solution region. To generate a new population, a marginal histogram model is built based on the global statistical information extracted from the current population and then new solutions are sampled from the model thus built. The incomplete simplex method applies to every new solution generated by uniform design or sampled from the histogram model. Unconstrained optimization by diagonal quadratic approximation applies to several selected resultant solutions of the incomplete simplex method at each generation. We study the effectiveness of main components of EDA/L. The experimental results demonstrate that EDA/L is better than four other recent EAs in terms of the solution quality and the computational cost.
106 citations
TL;DR: The algorithm (CGRID) is a binning algorithm that extends traditional binning algorithms so that the arbitrary sizes and shapes can be handled efficiently.
Abstract: A new spatial reasoning algorithm that can be used in multi‐body contact detection is presented. The algorithm achieves the partitioning of N bodies of arbitrary shape and size into N lists in order O(N) operations, where each list consists of bodies spatially near to the target object. The algorithm has been tested for objects of arbitrary shape and size, in two and three dimensions. However, we believe that it can be extended to dimensions of four and higher. The algorithm (CGRID) is a binning algorithm that extends traditional binning algorithms so that the arbitrary sizes and shapes can be handled efficiently. The algorithm has applications in discrete element, finite element, molecular dynamics, meshless methods, and lattice‐Boltzmann codes and also in domains such as path planning, target acquisition and general clustering problems.
106 citations
TL;DR: In this article, the authors reviewed the development of simulation techniques and their application to the study of particle packing and flow, transport properties and constitutive relationships of typical static or dynamic particulate systems, and concluded, through representative comparison between simulated and measured results under different conditions, that DEM is an effective method for particle scale research of particulate matter.
Abstract: Discrete element method (DEM) has been extensively used in the laboratory of particulate and multiphase processing at the University of New South Wales (UNSW) to study the fundamentals of particulate matter at a particle scale. This paper briefly reviews the work in the laboratory, which covers the development of simulation techniques and their application to the study of particle packing and flow, transport properties and constitutive relationships of typical static or dynamic particulate systems. It is concluded, through representative comparison between simulated and measured results under different conditions, that DEM, as a major technique for discrete particle simulation, is an effective method for particle scale research of particulate matter.
86 citations
TL;DR: An energy‐based general polygon to polygon normal contact model in which the normal and tangential directions, magnitude and reference contact position of the normal contact force are uniquely defined.
Abstract: This paper proposes an energy‐based general polygon to polygon normal contact model in which the normal and tangential directions, magnitude and reference contact position of the normal contact force are uniquely defined. The model in its final form is simple and elegant with a clear geometric perspective, and also possesses some advanced features. Furthermore, it can be extended to a more complex situations and in particular, it may also provide a sound theoretical foundation to possibly unifying existing contact models for all types of (convex) objects.
80 citations
TL;DR: In this article, a contact detection technique for discrete element modeling is described, which is suitable for a large family of particle shapes that are based on the dilation process from mathematical morphology and demonstrated with a series of biaxial deformation simulations using a range of ellipsoidal particle shapes.
Abstract: A new contact detection technique for discrete element modeling is described. This technique is suitable for a large family of particle shapes that are based on the dilation process from mathematical morphology. In the dilation process an arbitrary shape is dilated by placing the center of a sphere of fixed diameter at every point in the basic shape. Defining a contact between two objects in this class is equivalent to determining which spheres amongst the infinite number that compose each object is in contact. The algorithm is derived for general ellipsoidal particles and demonstrated with a series of biaxial deformation simulations using a range of ellipsoidal particle shapes.
62 citations
TL;DR: A set of Mathematica modules that organizes numerical integration rules considered useful for finite element work, and may be used "as is" in support of symbolic FEM work thus avoiding contamination with floating arithmetic that precludes simplification.
Abstract: This paper presents a set of Mathematica modules that organizes numerical integration rules considered useful for finite element work Seven regions are considered: line segments, triangles, quadrilaterals, tetrahedral, wedges, pyramids and hexahedra Information can be returned in floating‐point (numerical) form, or in exact symbolic form The latter is useful for computer‐algebra aided FEM work that carries along symbolic variables A few quadrature rules were extracted from sources in the FEM and computational mathematics literature, and placed in symbolic form using Mathematica to generate own code A larger class of formulas, previously known only numerically, were directly obtained through symbolic computations Some unpublished non‐product rules for pyramid regions were found and included in the collection For certain regions: quadrilaterals, wedges and hexahedra, only product rules were included to economize programming The collection embodies most FEM‐useful formulas of low and moderate order for the seven regions noted above Some gaps as regard region geometries and omission of non‐product rules are noted in the conclusions The collection may be used “as is” in support of symbolic FEM work thus avoiding contamination with floating arithmetic that precludes simplification It can also be used as generator for low‐level floating‐point code modules in Fortran or C Floating point accuracy can be selected arbitrarily No similar modular collection applicable to a range of FEM work, whether symbolic or numeric, has been published before
52 citations
TL;DR: In this paper, the authors present the preliminary results from a parameter study investigating the stability of underground structures in response to explosion-induced strong ground motions, and demonstrate the suitability of the distinct element method (DEM) for this application.
Abstract: We present the preliminary results from a parameter study investigating the stability of underground structures in response to explosion‐induced strong ground motions. In practice, even the most sophisticated site characterization may lack key details regarding precise joint properties and orientations within the rock mass. Thus, in order to place bounds upon the predicted behavior of a given facility, an extensive series of simulations representing different realizations may be required. The influence of both construction parameters (reinforcement, rock bolts, liners) and geological parameters (joint stiffness, joint spacing and orientation, and tunnel diameter to block size ratio) must be considered. We discuss the distinct element method (DEM) with particular emphasis on techniques for achieving improved computational efficiency, including the handling of contact detection and approaches to parallelization. We introduce a new approach for simulating deformation of the discrete blocks using the theory of a Cosserat point, which does not require internal discretization of the blocks. We also outline the continuum techniques we employ to obtain boundary conditions for the distinct element simulations. We present results from simulations of dynamic loading of several generic subterranean facilities in hard rock, demonstrating the suitability of the DEM for this application. These results demonstrate the significant role that joint geometry plays in determining the response of a given facility.
TL;DR: A comprehensive review of the available literature on applications of the rough set theory is presented in this paper, where concepts of rough sets theory are discussed for approximation, dependence and reduction of attributes, decision tables and decision rules.
Abstract: This paper presents a comprehensive review of the available literature on applications of the rough set theory. Concepts of the rough set theory are discussed for approximation, dependence and reduction of attributes, decision tables and decision rules. The applications of rough sets are discussed in pattern recognition, information processing, business and finance, industry, environment engineering, medical diagnosis and medical data analysis, system fault diagnosis and monitoring and intelligent control systems. Development trends and future efforts are outlined. An extensive list of references is also provided to encourage interested readers to pursue further investigations.
TL;DR: In this paper, the combined use of damage criteria, genetic algorithms and advanced CFD solvers provides an effective strategy to identify locations of releases that produce maximum damage, and the implementation is simple and does not require any change to flow solvers.
Abstract: The combined use of damage criteria, genetic algorithms and advanced CFD solvers provides an effective strategy to identify locations of releases that produce maximum damage. The implementation is simple and does not require any change to flow solvers. A rather general criterion has been formulated to determine the damage inflicted by the intentional or unintentional release of contaminants. Results of two typical cases show that damage can vary considerably as a function of release location, implying that genetic algorithms are perhaps the only techniques suited for this type of optimization problem.
TL;DR: In this paper, a new homogenization technique for the determination of dynamic and kinematic quantities of representative elementary volumes (REVs) in granular assemblies is presented based on the definition of volume averages, expressions for macroscopic stress, couple stress, strain and curvature tensors are derived for an arbitrary REV.
Abstract: In this paper, a new homogenization technique for the determination of dynamic and kinematic quantities of representative elementary volumes (REVs) in granular assemblies is presented. Based on the definition of volume averages, expressions for macroscopic stress, couple stress, strain and curvature tensors are derived for an arbitrary REV. Discrete element model simulations of two different test set‐ups including cohesionless and cohesive granular assemblies are used as a validation of the proposed homogenization technique. A non‐symmetric macroscopic stress tensor, as well as couple stresses are obtained following the proposed procedure, even if a single particle is described as a standard continuum on the microscopic scale.
TL;DR: It is shown that the proposed mechanics‐based approach for model validation is very effective in filtering noise in the experimental data.
Abstract: In this paper, we discuss the application of the constitutive relation error (CRE) to model updating and validation in the context of uncertain measurements. First, a parallel is drawn between the CRE method and a general theory for inverse problems proposed by Tarantola. Then, an extension of the classical CRE method considering uncertain measurements is proposed. It is shown that the proposed mechanics‐based approach for model validation is very effective in filtering noise in the experimental data. The method is applied to an industrial structure, the SYLDA5, which is a satellite support for Ariane5. The results demonstrate the robustness of the method in actual industrial situations.
TL;DR: A new contact resolution algorithm based on the four‐arc approximation for an ellipsoid is presented, which takes advantage of the properties of the geometry to provide favorable empirical convergence properties compared with the method proposed earlier.
Abstract: The efficiency of a discrete element implementation relies on several factors, including the particle representation, neighbor‐sorting algorithm, contact resolution, and force generation. The focus of this paper is on the four‐arc approximation for an ellipsoid – a geometrical representation useful in simulations of large numbers of smoothly shaped particles. A new contact resolution algorithm based on the four‐arc approximation is presented, which takes advantage of the properties of the geometry to provide favorable empirical convergence properties compared with the method proposed earlier. Special attention is given to the software implementation of the algorithm, and a discussion of the computational efficiency of the algorithm is provided.
TL;DR: The Lagrangian Sea Ice Model (LIM) as discussed by the authors is based on a discrete element model and explicitly simulates individual ice parcels and the interactions between them, and is used to model the behavior of the sea ice in the Arctic basin.
Abstract: The ice pack covering the Arctic basin is composed of a multitude of ice parcels of different areas, ages, thicknesses, and deformation histories that are frozen together into larger plates that combine and break apart in response to the demands of ever changing boundary conditions and forcing. Current Arctic sea ice models are Eulerian continuum models that use a plastic yield surface to characterize the constitutive behavior of the pack. An alternative is to adopt a discontinuous Lagrangian approach, based on a discrete element model and explicitly simulate individual ice parcels and the interactions between them. The mechanics of the Lagrangian sea ice model are outlined in detail along with the methods that will be used for validation.
TL;DR: In this article, the authors used 3D experimental particle trajectory data, acquired from a laboratory tumbling mill using bi-planar X-ray filming, to validate the discrete element method (DEM).
Abstract: Accurate 3D experimental particle trajectory data, acquired from a laboratory tumbling mill using bi‐planar X‐ray filming, are used to validate the discrete element method (DEM). Novel numerical characterisation techniques are presented that provide a basis for comparing the experimental and simulated charge behaviour. These techniques are based on fundamental conservation principles, and provide robust, new interpretations of charge behaviour that are free of operator bias. Two‐ and three‐dimensional DEM simulations of the experimental tumbling mill are performed, and the relative merits of each discussed. The results indicate that in its current form DEM can simulate some of the salient features of the tumbling mill charge, however, comparison with the experiment indicate that the technique requires refinement to adequately simulate all aspects of the system.
TL;DR: In this article, the authors present a bibliography of sheet metal forming applications using finite element methods, including the use of numerical techniques to analyze physical phenomena in the field of structural, solid and fluid mechanics as well as to simulate various processes in engineering.
Abstract: Sheet metal forming is a process of shaping thin sheets of metal by applying pressure through male or female dies or both. In most of used sheet‐formating processes the metal is subjected to primarily tensile or compressive stresses or both. During the last three decades considerable advances have been made in the applications of numerical techniques, especially the finite element methods, to analyze physical phenomena in the field of structural, solid and fluid mechanics as well as to simulate various processes in engineering. These methods are useful because one can use them to find out facts or study the processes in a way that no other tool can accomplish. Finite element methods applied to sheet metal forming are the subjects of this paper. The reason for writing this bibliography is to save time for readers looking for information dealing with sheet metal forming, not having an access to large databases or willingness to spend own time with uncertain information retrieval. This paper is organized into two parts. In the first one, each topic is handled and current trends in the application of finite element techniques are briefly mentioned. The second part, an Appendix, lists papers published in the open literature. More than 900 references to papers, conference proceedings and theses/dissertations dealing with subjects that were published in 1995‐2003 are listed.
TL;DR: In this paper, a combined finite-discrete element method was used to simulate the gravitational depositions of packs containing particles of cubical shape, and the results of such an approach were compared to the experimental results to evaluate both feasibility and accuracy of the combined finite discrete element simulation of packing problems.
Abstract: The combined finite‐discrete element method has been used to simulate the gravitational depositions of packs containing particles of cubical shape. This approach to the generation of particle packs is based on the simulation of the dynamics of pack formation including interaction among individual particles, inertia and gravitational forces. The results of such an approach are compared to the experimental results to evaluate both feasibility and accuracy of the combined finite‐discrete element simulation of packing problems.
TL;DR: An updated Lagrangian (UL) finite element (FE) formulation and an explicit time integration scheme are used together with some simplifying assumptions to linearize this highly nonlinear contact problem and obtain solutions with realistic computational cost and sufficiently good accuracy.
Abstract: The discrete element methods (DEM) are numerical techniques that have been specifically developed to enable simulations of systems of multiple distinct, typically infinitely rigid, bodies that interact with each other through contact forces. However, there are multibody systems for which it is useful to consider the deformability of the simulated bodies and enable the evaluation of their stress and strain distributions. This paper focuses on the simulation of deformable multibody systems using a combination of DEM and finite element methods (FEM). In particular, an updated Lagrangian (UL) finite element (FE) formulation and an explicit time integration scheme are used together with some simplifying assumptions to linearize this highly nonlinear contact problem and obtain solutions with realistic computational cost and sufficiently good accuracy. In addition, this paper describes a software implementation of this formulation, which utilizes the Java programming language and the Java3D graphics application programming interface (API), as well as database technology.
TL;DR: The study of a finite element model and the comparison of results with a combinatorial approach, based on Zadeh's extension principle, show the efficiency of this methodology to calculate fuzzy eigenvalues and eigenvectors of finite element structures defined by imprecise parameters.
Abstract: This paper presents an efficient methodology to calculate fuzzy eigenvalues and eigenvectors of finite element structures defined by imprecise parameters. The material and geometric parameters are then described by fuzzy numbers. The proposed methodology, based on α‐cut discretization of fuzzy numbers and Taylor's expansion, determines the extreme eigensolutions for each α‐cut. The study of a finite element model and the comparison of results with a combinatorial approach, based on Zadeh's extension principle, show the efficiency of this methodology.
TL;DR: In this article, simple discrete element models using PFC2D models with bonded assemblies of particles were used to numerically simulate direct tension and block punching tests on thin spray-on tunnel liner materials to gain insight about the liner support mechanisms.
Abstract: Simple discrete element models using PFC2D models with bonded assemblies of particles were used to numerically simulate direct tension and block punching tests on thin spray‐on tunnel liner materials to gain insight about the liner support mechanisms. PFC2D input parameters were calibrated such that the rupture load and elongation at rupture were similar to the laboratory test data. The calibrated model of the liner material was then used to simulate a liner around a highly stressed tunnel in rock where stresses caused extensive fracturing near the top of the tunnel. The effect of the liner was analysed by modelling the tunnel with and without the liner and showed that the liner had minimal impact on fracture propagation in the rock because of the liner's highly deformable nature. However, the liner was able to retain the fractured rock in place.
TL;DR: In this article, a density-stiffness interpolation scheme for topology optimization of continuum structures is proposed, which can eliminate the so-called checkerboard pattern from the final optimal topology and overcome the boundary-smooth effect associated with the traditional sensitivity averaging approach.
Abstract: In this paper, a new density‐stiffness interpolation scheme for topology optimization of continuum structures is proposed. Based on this new scheme, not only the so‐called checkerboard pattern can be eliminated from the final optimal topology, but also the boundary‐smooth effect associated with the traditional sensitivity averaging approach can also be overcome. A proof of the existence of the solution of the optimization problem is also given, therefore mesh independent optimization results can be obtained. Numerical examples illustrate the effectiveness and the advantage of the proposed interpolation scheme.
TL;DR: The major aim of the KBS of IJETSS is to generate rapid and precise engine fault diagnosis that can simulate the work of experienced aircraft maintenance mechanics and engineers.
Abstract: In this paper, an intelligent knowledge‐based system (KBS) capable of assisting aircraft mechanics and engineers to deal with fault diagnosis of the turbo‐prop aircraft engine is presented. The KBS intelligent jet engine trouble‐shooting system (IJETSS) employs expert knowledge to act in a way similar to that of a human expert in an aircraft maintenance field by using if‐then rule‐based system. The major aim of the KBS of IJETSS is to generate rapid and precise engine fault diagnosis that can simulate the work of experienced aircraft maintenance mechanics and engineers. The developed system can also be useful for the inexperienced aircraft mechanics and engineers and can be used for training module for them.
TL;DR: In this article, a bibliographical review of the finite element modelling and simulation of indentation testing from the theoretical as well as practical points of view is given, with references to papers, conference proceedings and theses/dissertations that were published between 1990 and 2002.
Abstract: This paper gives a bibliographical review of the finite element modelling and simulation of indentation testing from the theoretical as well as practical points of view. The bibliography lists references to papers, conference proceedings and theses/dissertations that were published between 1990 and 2002. At the end of this paper, 509 references are listed dealing with subjects such as, fundamental relations and modelling in indentation testing, identification of mechanical properties for specific materials, fracture mechanics problems in indentation, scaling relationship for indentation, indenter geometry and indentation testing.
TL;DR: In this article, an optimal design method of number and placements of piezoelectric patch actuators in active vibration control of a plate was presented, where the eigenvalue distribution of energy correlative matrix of control input force was applied to determine optimal number of the required actuators.
Abstract: This paper presents an optimal design method of number and placements of piezoelectric patch actuators in active vibration control of a plate. Eigenvalue distribution of energy correlative matrix of control input force is applied to determine optimal number of the required actuators. Genetic algorithms (GAs) using active vibration control effects, which are taken as the objective function, are adopted to search optimal placements of actuators. The results show that disturbance exerted on a plate is a key factor of determining optimal number and placements of actuators in active structural vibration control, and a global and efficient optimization solution of multiple actuator placements can be obtained using GAs.
TL;DR: In this article, a numerical method for shape optimisation in forging is presented, where the goal of the optimisation is to eliminate work-piece defects that may arise during the forging process.
Abstract: A numerical method for shape optimisation in forging is presented. The goal of the optimisation is to eliminate work‐piece defects that may arise during the forging process. A two‐dimensional finite element code has been developed for the simulation of the mechanical process. The material is incompressible and it follows the Norton‐Hoff law. To deal with contact constraint the velocity projection algorithm is used. The optimisation process is conducted using a genetic algorithm supported by an elitist strategy. A new genetic operator called adaptive mutation has been developed to increase the efficiency of the search. The developed scheme is used to design optimal preform shapes for several axisymmetric examples. Continuous and discrete design variables are considered. The objective function of the optimisation problem is associated with the quality of the final product. Comparing the obtained optimal results with the literature validates the proposed optimisation method.
TL;DR: A second‐order accurate time‐marching pressure‐correction algorithm to accommodate weakly‐compressible highly‐viscous liquid flows at low Mach number and the piecewise‐constant interpolation scheme is advocated as a viable method of choice, with its advantages of order retention, yet efficiency in implementation.
Abstract: We introduce a second‐order accurate time‐marching pressure‐correction algorithm to accommodate weakly‐compressible highly‐viscous liquid flows at low Mach number. As the incompressible limit is approached (Ma ≈ 0), the consistency of the compressible scheme is highlighted in recovering equivalent incompressible solutions. In the viscous‐dominated regime of low Reynolds number (zone of interest), the algorithm treats the viscous part of the equations in a semi‐implicit form. Two discrete representations are proposed to interpolate density: a piecewise‐constant form with gradient recovery and a linear interpolation form, akin to that on pressure. Numerical performance is considered on a number of classical benchmark problems for highly viscous liquid flows to highlight consistency, accuracy and stability properties. Validation bears out the high quality of performance of both compressible flow implementations, at low to vanishing Mach number. Neither linear nor constant density interpolations schemes degrade the second‐order accuracy of the original incompressible fractional‐staged pressure‐correction scheme. The piecewise‐constant interpolation scheme is advocated as a viable method of choice, with its advantages of order retention, yet efficiency in implementation.
TL;DR: The aim of this paper is to obtain the analytical solution for the optimal design of reinforced concrete sections under ultimate design with the use of Heaviside functions in the definition of the ultimate strains in the reinforced concrete section.
Abstract: The aim of this paper is to obtain the analytical solution for the optimal design of reinforced concrete sections under ultimate design. The equilibrium equations of the section under bending moment and axial force in rupture are derived. The ultimate conditions are considered either in the steel or in the concrete according to the concrete design codes. The definition of the strains and stresses in the materials is based on the use of Heaviside functions. With this definition the equilibrium equations are described by unique equations. The optimization can then be developed with any design variables in the geometric definition, as area of the reinforcement and location. The optimization is developed with yielding of tensile steel and crushing of concrete. Although this is the current situation in reinforced concrete design, future developments of the model can include other steel and concrete conditions. Cost optimization and variable materials strength ratio are possible applications of the model. The interest of the model is the use of closed form unique equilibrium equations in the optimization of reinforced concrete sections. Numerical examples of the optimization of a rectangular section with minimum reinforcing steel area and economic bending moment are presented. The originality of the paper is the use of Heaviside functions in the definition of the ultimate strains in the reinforced concrete section. Unique equations for the objective function and restrictions are derived. The paper is useful for the design of reinforced concrete. The equations derived can be implemented into computer programs.