Finite elements in space and time for parallel computing of viscoelastic deformation

The efficient parallel computation of time dependent problems, e.g. parabolic problems of viscoelastic material deformation, underlies the “bottleneck” of the serial approach in time. The usual method of lines, also called semidiscretization, leads to an iterative calculation in time, i.e. a sequential solution of the spatial problems for all time steps. Due to that, only one spatial problem can be solved in parallel at a certain time step. For an efficient parallelization, it is necessary to compute the whole problem in a distributed way. Furthermore, both h- and p-adaptive approximation should be possible in time and space. For these purposes, in addition to the spatial FE-discretization, a continuous finite element discretization in time is used. Thus, one obtains a total algebraic equation system in space and time, whose solution has to be parallelized efficiently, and h- and p-adaptivity in time and space within the frame of the overall Galerkin-process has to be realized. The present paper treats symmetric and non-symmetric formulations of two different viscoelastic three-parameter models. The new numerical approach concerns first for the Malvern Model (generalized Maxwell Model). The numerical examples for the new non-symmetric formulation and the traditional semidiscretization show the advantage (with respect to convergence to the problem solution) of the new finite element approach with simultaneous discretizations in time and space. But the algebraic systems are bad-conditioned such that parallel iterative solvers with various preconditions are not efficient. The symmetric formulation for the Malvern Model can be obtained for the one-dimensional case only. A numerical example showed the good iterative solvability of the symmetric formulation. Therefore, in order to obtain a symmetric formulation in the 3D-case the generalized Kelvin–Voigt Model was chosen as an alternative one. It should be mentioned that the numerical examples show both the effectiveness of parallel computation and the efficiency of h- and p-adaptation (p-adaptation yields the higher rate of convergence than h-adaptation). Received 19 April 1998     

The optimal graph partitioning problem

Summary In this paper we consider the problem of partitioning the set of nodes in a graph in at mostp classes, such that the sum of node weights in any class is not greater than the class capacityb, and such that the sum of edge weights, for edges connecting nodes in the same class, is maximal. This problem can be formulated as a MILP, which turns out to be completely symmetrical with respect to thep classes, and the gap between the relaxed LP solution and the optimal solution is the largest one possible. These two properties make it very difficult to solve even smaller problems. In this paper it is shown how it is possible to preassign certain nodes to certain classes, thus reducing both the symmetric nature of the formulation, the number of variables and constraints and the gap. It is also shown how the gap can be reduced even further by introducing combinatorial cuts. Computational results based on the two formulations of the problem and combinatorial cuts are presented.     

On the discrete core of quadrilateral mesh refinement

We present a new approach to quadrilateral mesh refinement, which reduces the problem to its structural core. The resulting problem formulation belongs to a class of discrete problems, network‐flow problems, which has been thoroughly investigated and is well understood. The network‐flow model is flexible enough to allow the simultaneous incorporation of various aspects such as the control of angles and aspect ratios, local density control, and templates (meshing primitives) for the internal refinement of mesh elements. We show that many different variants of the general quadrilateral mesh‐refinement problem are covered. In particular, we present a novel strategy, which provably finds a conformal refinement unless there is none. Copyright © 1999 John Wiley & Sons, Ltd.     

Robotic U-shaped assembly line balancing using particle swarm optimization

Automation in an assembly line can be achieved using robots. In robotic U-shaped assembly line balancing (RUALB), robots are assigned to workstations to perform the assembly tasks on a U-shaped assembly line. The robots are expected to perform multiple tasks, because of their capabilities. U-shaped assembly line problems are derived from traditional assembly line problems and are relatively new. Tasks are assigned to the workstations when either all of their predecessors or all of their successors have already been assigned to workstations. The objective function considered in this article is to maximize the cycle time of the assembly line, which in turn helps to maximize the production rate of the assembly line. RUALB aims at the optimal assignment of tasks to the workstations and selection of the best fit robot to the workstations in a manner such that the cycle time is minimized. To solve this problem, a particle swarm optimization algorithm embedded with a heuristic allocation (consecutive) procedure is proposed. The consecutive heuristic is used to allocate the tasks to the workstation and to assign a best fit robot to that workstation. The proposed algorithm is evaluated using a wide variety of data sets. The results indicate that robotic U-shaped assembly lines perform better than robotic straight assembly lines in terms of cycle time.     

Balancing of mixed-model parallel U-shaped assembly lines considering model sequences

As a consequence of increasing interests in customised products, mixed-model lines have become the most significant components of today’s manufacturing systems to meet surging consumer demand. Also, U-shaped assembly lines have been shown as the intelligent way of producing homogeneous products in large quantities by reducing the workforce need thanks to the crossover workstations. As an innovative idea, we address the mixed-model parallel U-shaped assembly line design which combines the flexibility of mixed-model lines with the efficiency of U-shaped lines and parallel lines. The multi-line stations utilised in between two adjacent lines provide extra efficiency with the opportunity of assigning tasks into workstations in different combinations. The new line configuration is defined and characterised in details and its advantages are explained. A heuristic solution approach is proposed for solving the problem. The proposed approach considers the model sequences on the lines and seeks efficient balancing solutions for their different combinations. An explanatory example is also provided to show the sophisticated structure of the studied problem and explain the running mechanism of the proposed approach. The results of the experimental tests and their statistical analysis indicated that the proposed line design requires fewer number of workstations in comparison with independently balanced mixed-model U-lines.     

Compact Mathematical Formulation for Graph Partitioning

The graph partitioning problem consists of dividing the vertices of a graph into clusters, such that the weight of the edges crossing between clusters is minimized. We present a new compact mathematical formulation of this problem, based on the use of binary representation for the index of clusters assigned to vertices. This new formulation is almost minimal in terms of the number of variables and constraints and of the density of the constraint matrix. Its linear relaxation brings a very fast computational resolution, compared with the standard one.Experiments were conducted on classical large benchmark graphs designed for comparing heuristic methods. On one hand, these experiments show that the new formulation is surprisingly less time efficient than expected on general k-partitioning problems. On the other hand, the new formulation applied on bisection problems allows to obtain the optimum solution for about ten instances, where only best upper bounds were previously known.     

High‐order Higdon‐like boundary conditions for exterior transient wave problems

Recently developed non‐reflecting boundary conditions are applied for exterior time‐dependent wave problems in unbounded domains. The linear time‐dependent wave equation, with or without a dispersive term, is considered in an infinite domain. The infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC) is imposed on ??. Then the problem is solved numerically in the finite domain bounded by ??. The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. We consider here two reformulations: one that involves high‐order derivatives with a special discretization scheme, and another that does not involve any high derivatives beyond second order. The latter formulation is made possible by introducing special auxiliary variables on ??. In both formulations the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. In contrast to previous papers using similar formulations, here the method is applied to a fully exterior two‐dimensional problem, with a rectangular boundary. Numerical examples in infinite domains are used to demonstrate the performance and advantages of the new method. In the auxiliary‐variable formulation long‐time corner instability is observed, that requires special treatment of the corners (not addressed in this paper). No such difficulties arise in the high‐derivative formulation. Published in 2005 by John Wiley & Sons, Ltd.     

Single machine multi-product capacitated lot sizing with sequence-dependent setups

In production planning in the glass container industry, machine-dependent setup times and costs are incurred for switch overs from one product to another. The resulting multi-item capacitated lot-sizing problem has sequence-dependent setup times and costs. We present two novel linear mixed-integer programming formulations for this problem, incorporating all the necessary features of setup carryovers. The compact formulation has polynomially many constraints, whereas the stronger formulation uses an exponential number of constraints that can be separated in polynomial time. We also present a five-step heuristic that is effective both in finding a feasible solution (even for tightly capacitated instances) and in producing good solutions to these problems. We report computational experiments.     

Integral methodology for distribution systems reconfiguration based on optimal power flow using Benders decomposition technique

A new and efficient methodology for distribution network reconfiguration integrated with optimal power flow (OPF) based on a Benders decomposition approach is presented. The objective minimises power losses, balancing load among feeders and subject to constraints such as capacity limit of branches, minimum and maximum power limits of substations or distributed generators, minimum deviation of bus voltages and radial optimal operation of networks. A variant of the generalised Benders decomposition algorithm is applied to solve the problem. The formulation can be embedded under two stages; the first one is the master problem and is formulated as a mixed integer non-linear programming problem. This stage determines the radial topology of the distribution network. The second stage is the slave problem and is formulated as a non-linear programming problem. This stage is used to determine the feasibility of the master problem solution by means of an OPF and provides information to formulate the linear Benders cuts that connect both problems. The model is programmed in GAMS mathematical modeling language. The effectiveness of the proposal is demonstrated through two examples extracted from the specialised literature.     

A novel enthalpy formulation for multidimensional solidification and melting of a pure substance

This paper presents a new finite-difference formulation of the multidimensional phase change problems involving unique phase change temperature. The solutions obtained with this formulation show that the problem of “waviness” of the temperature histories encountered with the conventional enthalpy formulation is now removed. The formulation derived provides a simple method for “local” tracking of the interface using the enthalpy variable in a novel way. During the solution of the finite-difference equations, the present formulation obviates the need for “book-keeping” of the phase-change nodes, and hence allows solution of the equations by tridiagonal matrix algorithm. It is argued that the benefits of enthalpy formulation can be extended to phase-change problems involving convection by solving the equations of motion on non-staggered grid.     

Two-phase approach to GT cell formation using efficient p-median formulations

The purpose of this paper is to develop an efficient p-median approach applicable to large cell formation (CF) problems. A two-phase methodology that seeks to minimize the number of exceptional elements is proposed. In phase I, two efficient p-median formulations which contain fewer binary variables than existing p-median formulations are constructed. For a CF problem with m machines existing p-median formulations contains m² or more binary variables, whereas the new formulation proposed in phase I contains not more than 5m binary variables at the expense of a slightly increased number of continuous variables and constraints for practical values of p less than 32. This makes it possible to implement large CF problem within reasonable computer runtime with commercially available linear integer programming codes. Given the initial cell configuration found with the new p-median formulation, in phase II bottleneck machines and parts are reassigned to reduce the number of exceptional elements. This procedure has the flexibility of providing the cell designer with alternative solutions. Test results on large CF problems show a substantial increase in the efficiency of the new p-median formulations compared with existing p-median formulations.     

Simulated annealing algorithms for the multi-manned assembly line balancing problem: minimising cycle time

Multi-manned assembly lines are often designed to produce big-sized products, such as automobiles and trucks. In this type of production lines, there are multi-manned workstations where a group of workers simultaneously performs different operations on the same individual product. One of the problems, that managers of such production lines usually encounter, is to produce the optimal number of items using a fixed number of workstations, without adding new ones. In this paper, such a class of problems, namely, the multi-manned assembly line balancing problem is addressed, with the objective of minimising the cycle time. A mixed-integer mathematical programming formulation is proposed for the considered problem. This model has the primary objective of minimising the cycle time for a given number of workstations and the secondary objective of minimising the total number of workers. Since the addressed problem is NP-hard, two meta-heuristic approaches based on the simulated annealing algorithm have been developed: ISA and DSA. ISA solves the problem indirectly while DSA solves it directly. The performance of the two algorithms are tested and compared on a set of test problems taken from the literature. The results show that DSA outperforms ISA in term of solution quality and computational time.     

SPH with the multiple boundary tangent method

In this article, we present an improved solid boundary treatment formulation for the smoothed particle hydrodynamics (SPH) method. Benchmark simulations using previously reported boundary treatments can suffer from particle penetration and may produce results that numerically blow up near solid boundaries. As well, current SPH boundary approaches do not properly treat curved boundaries in complicated flow domains. These drawbacks have been remedied in a new boundary treatment method presented in this article, called the multiple boundary tangent (MBT) approach. In this article we present two important benchmark problems to validate the developed algorithm and show that the multiple boundary tangent treatment produces results that agree with known numerical and experimental solutions. The two benchmark problems chosen are the lid‐driven cavity problem, and flow over a cylinder. The SPH solutions using the MBT approach and the results from literature are in very good agreement. These solutions involved solid boundaries, but the approach presented herein should be extendable to time‐evolving, free‐surface boundaries. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive multigrid method using duality in plane elasticity

This work presents an adaptive multigrid method for the mixed formulation of plane elasticity problems. First, a mixed‐hybrid formulation is introduced where the continuity of the normal components of the stress tensor is indirectly imposed using a Lagrange multiplier. Two different numerical approximations, naturally associated with the primal problem and the dual problem, are then proposed. The Complementary Energy Principle provides an a posteriori error estimate. For the effective solving of both systems of equations, a non‐standard multigrid algorithm has been designed that allows us to solve the two problems, dual and primal, with reasonable cost and in an integrated way. Finally, a significant numerical application is presented to check the efficiency of the error estimator and the good performance of the algorithm. Copyright © 2001 John Wiley & Sons, Ltd.     

Increasing production rate in U-type assembly lines with sequence-dependent set-up times

U-shaped assembly lines are commonly used in just-in-time production systems as they have some advantages over straight lines. Although maximizing production rates on these lines by assigning tasks to stations is common practice in industrial environments, studies on the stated assembly line balancing problem are limited. This article deals with maximizing the production rate on U-shaped assembly lines under sequence-dependent set-up times. Sequence-dependent set-up times mean that after a task is performed, a set-up time, the duration of which depends on adjacent tasks, is required to start the next task operation. These set-ups are considered by dividing them into two groups, named forward and backward set-ups, to make the problem more practical. Two heuristics based on simulated annealing and genetic algorithms are improved beside the mathematical model. Experimental results show that solving the stated problem using the mathematical model is nearly impossible, while heuristics may obtain solutions that have acceptable deviations from the lower bounds.     

Efficient formulation and heuristics for multi-item single source ordering problem with transportation cost

Integrated inventory and transportation decisions are critical in the supply chain, providing significant gains for all parties. In this paper, we present a mathematical formulation for the dynamic demand multi-item single source replenishment problem with a piecewise linear transportation cost. Through an extensive experimental study, we find that the new formulation provides a tighter LP relaxation of the problem, while requiring fewer computational resources to optimally solve the problem when compared with existing model in the literature. We also present a new metaheuristic for this general class of coordinated capacitated replenishment problems. On average, the solutions from heuristics are within 1.23% of the optimal solution for the comprehensive set of test problems.     

A novel contact interaction formulation for voxel‐based micro‐finite‐element models of bone

Voxel‐based micro‐finite‐element (μFE) models are used extensively in bone mechanics research. A major disadvantage of voxel‐based μFE models is that voxel surface jaggedness causes distortion of contact‐induced stresses. Past efforts in resolving this problem have only been partially successful, ie, mesh smoothing failed to preserve uniformity of the stiffness matrix, resulting in (excessively) larger solution times, whereas reducing contact to a bonded interface introduced spurious tensile stresses at the contact surface. This paper introduces a novel “smooth” contact formulation that defines gap distances based on an artificial smooth surface representation while using the conventional penalty contact framework. Detailed analyses of a sphere under compression demonstrated that the smooth formulation predicts contact‐induced stresses more accurately than the bonded contact formulation. When applied to a realistic bone contact problem, errors in the smooth contact result were under 2%, whereas errors in the bonded contact result were up to 42.2%. We conclude that the novel smooth contact formulation presents a memory‐efficient method for contact problems in voxel‐based μFE models. It presents the first method that allows modeling finite slip in large‐scale voxel meshes common to high‐resolution image‐based models of bone while keeping the benefits of a fast and efficient voxel‐based solution scheme.     

Scheduling continuous aluminium casting lines

This study considers the problem of scheduling casting lines of an aluminium casting and processing plant. In aluminium processing plants, continuous casting lines are the bottleneck resources, i.e. factory throughput is limited by the amount of aluminium that can be cast. The throughput of a casting line might be increased by minimizing total setup time between jobs. The objective is to minimize setup time on production lines for a given time period while balancing workload between production lines to accommodate potential new orders. A mathematical formulation for scheduling jobs to minimize the total setup time while achieving workload balance between the production lines is presented. Since the casting scheduling problem is an NP-hard problem, even with only one casting line, a four-step algorithm to find good solutions in a reasonable amount of time is proposed. In this process, a set of asymmetric travelling salesman problems is followed by a pairwise exchange heuristic. The proposed procedure is applied to a case study using real casting data.     

A new decomposition algorithm for a liquefied natural gas inventory routing problem

We consider an inventory routing problem (IRP) in the liquefied natural gas (LNG) supply chain, called the LNG-IRP. Here, an actor is responsible for the LNG production and inventory management at the liquefaction plants, the routing and scheduling of a heterogeneous fleet of LNG ships, as well as the inventories and sales at the regasification terminals. Furthermore, all ports have a limited number of berths available for loading and unloading. The LNG-IRP is more complicated than many other maritime inventory routing problems because a constant rate of the cargo evaporates in the tanks each day and is used as fuel during transportation. In addition, a variable number of tanks are unloaded at the regasification terminals. We introduce a new path flow formulation for this problem arising from a novel decomposition scheme based on parts of a ship schedule, called duties. A ship schedule for the entire planning horizon can be divided into duties consisting of a visit to a liquefaction plant, then one or two visits to a regasification terminal before ending in a liquefaction plant. The solution method suggested is based on a priori generation of duties, and the formulation is strengthened by valid inequalities. The same problem was previously solved by a branch-price-and-cut algorithm for a schedule-based formulation. Computational results show that the new formulation provides tighter bounds than the previous schedule-based formulation. Furthermore, on a set of 27 benchmark instances, the proposed algorithm clearly outperforms the previous branch-price-and-cut algorithm both with regard to computational time and the number of problems solved within a 10-h time limit.     

Three-dimensional noise-immune phase unwrapping algorithm

The classical problem of phase unwrapping in two dimensions, that of how to create a path-independent unwrapped map, is extended to the case of a three-dimensional phase distribution. Whereas in two dimensions the path dependence problem arises from isolated phase singularity points, in three dimensions the phase singularities are shown to form closed loops in space. A closed path that links one such loop will cross a nonzero number of phase discontinuities. In two dimensions, path independence is achieved when branch-cut lines are placed between singular points of opposite sign; an equivalent path-independent algorithm for three dimensions is developed that places branch-cut surfaces so as to prevent unwrapping through the phase singularity loops. The placing of the cuts is determined uniquely by the phase data, which contrasts with the two-dimensional case for which there are many possible ways in which to pair up the singular points. The performance of the new algorithm is demonstrated on three-dimensional phase data from a high-speed phase-shifting speckle pattern interferometer.