Exact solution of the displacement boundary-value problem of elasticity for a torus

This paper presents an exact analytical solution to the displacement boundary-value problem of elasticity for a torus. The introduced form of the general solution of elastostatics equations allows to solve exactly a broad class of boundary-value problems in coordinate systems with incomplete separation of variables in the harmonic equation. The original boundary-value problem for a torus is reduced to infinite systems of linear algebraic equations with tridiagonal matrices. An analytical technique for solving systems of diagonal form is developed. Uniqueness of the solutions of vector boundary-value problems involving the generalized Cauchy-Riemann equations is investigated, and it is shown that the obtained solution for the displacement boundary-value problem for a torus is unique due to the specific properties of the suggested general solution. The analogy between problems of elastostatics and steady Stokes flows is demonstrated, and the developed elastic solution is used to solve the Stokes problem for a torus.     

Job shop scheduling with the option of jobs outsourcing

Incorporating outsourcing in scheduling is addressed by several researchers recently. However, this scope is not investigated thoroughly, particularly in the job shop environment. In this paper, a new job shop scheduling problem is studied with the option of jobs outsourcing. The problem objective is to minimise a weighted sum of makespan and total outsourcing cost. With the aim of solving this problem optimally, two solution approaches of combinatorial optimisation problems, i.e. mathematical programming and constraint programming are examined. Furthermore, two problem relaxation approaches are developed to obtain strong lower bounds for some large scale problems for which the optimality is not proven by the applied solution techniques. Using extensive numerical experiments, the performance of the solution approaches is evaluated. Moreover, the effect the objectives's weights in the objective function on the performance of the solution approaches is also investigated. It is concluded that constraint programming outperforms mathematical programming significantly in proving solution optimality, as it can solve small and medium size problems optimally. Moreover, by solving the relaxed problems, one can obtain good lower bounds for optimal solutions even in some large scale problems.     

General solution technique for transient thermoelasticity of transversely isotropic solids in cylindrical coordinates

Summary Goodier has proposed the thermoelastic potential function in order to analyze thermoelastic problems for isotropic solids. The thermoelastic problem can be reduced to the elastic problem by his technique. Elastic problems are in general analyzed by the generalized Boussinesq solutions and the Michell function. This paper discusses a new solution technique for thermoelastic problems of transversely isotropic solids in cylindrical coordinates. The present solution technique consists of five fundamental solutions which are developed from the Goodier's thermoelastic potential function, the generalized Boussinesq solutions and the Michell function. Considering the relations among the material constants of transverse isotropy, the present solution technique can be classified into two cases. One of them can be reduced to the three solution techniques above which are specifically for isotropic solids only. As an application of the present solution technique, a transient thermoelastic problem in a transversely isotropic cylinder with an external crack is analyzed.     

A dynamic elasto/viscoplastic solution for a thick-walled spherical container subjected to impulsive loading

A general method for the solution of dynamic elasto/viscoplastic solid problems is presented. This method is an extension of ray theory to cater for dynamic elasto/viscoplastic deformations. The method reduces the elasto/viscoplastic problem to a sequence of elastic problems with initial strains. The solution of this problem is determined by using four displacement functions. Using the foregoing method, the solution is derived for the dynamic elasto/viscoplastic behaviour of a thick-walled spherical shell subjected to internal impact load. The numerical results show how the dynamic stresses in a sphere with viscoplastic properties vary with time.     

A hybrid heuristic for the multiple choice multidimensional knapsack problem

In this article, a new solution approach for the multiple choice multidimensional knapsack problem is described. The problem is a variant of the multidimensional knapsack problem where items are divided into classes, and exactly one item per class has to be chosen. Both problems are NP-hard. However, the multiple choice multidimensional knapsack problem appears to be more difficult to solve in part because of its choice constraints. Many real applications lead to very large scale multiple choice multidimensional knapsack problems that can hardly be addressed using exact algorithms. A new hybrid heuristic is proposed that embeds several new procedures for this problem. The approach is based on the resolution of linear programming relaxations of the problem and reduced problems that are obtained by fixing some variables of the problem. The solutions of these problems are used to update the global lower and upper bounds for the optimal solution value. A new strategy for defining the reduced problems is explored, together with a new family of cuts and a reformulation procedure that is used at each iteration to improve the performance of the heuristic. An extensive set of computational experiments is reported for benchmark instances from the literature and for a large set of hard instances generated randomly. The results show that the approach outperforms other state-of-the-art methods described so far, providing the best known solution for a significant number of benchmark instances.     

Solutions of partial differential equations with random Dirichlet boundary conditions by multiquadric collocation method

Problems described by deterministic partial differential equations with random Dirichlet boundary conditions are considered. Formulation of the solution to such a problem by the global collocation method using multiquadrics is presented. The quality of the solution to a stochastic problem depends on both its expected value and its variance. It is proposed that the shape parameter of multiquadrics should be chosen to optimize both the accuracy and the variance of the solution. Test problems described by the Poisson, the Helmholtz, and the diffusion–convection equations with random Dirichlet boundary conditions are solved by the multiquadric collocation method. It is found that there is a trade-off between solution accuracy and solution variance for each problem.     

基于Kriging代理模型的拉压不同模量平面问题的近似求解

该文建议采用Kriging代理模型数值求解拉压不同模量平面问题。通过本构方程光滑化、有限元法及拉丁超立方采样技术,对拉压不同模量桁架与二维平面问题,给出了基于Kriging模型的近似数值解,以代理基于有限元的数值解,并探讨了样本点数目和问题规模对所建Kriging近似模型求解精度/效率的影响。数值算例表明:所提方法可为求解拉压不同模量平面问题提供精度合理的近似数值解。当问题规模较大且正问题需要多次求解时,该方法有望显著减少计算时间,这对于降低拉压不同模量反问题与优化问题的计算开销十分重要。     

Aggregate planning in hybrid flowshops

We present a linear programming based heuristic for the solution of a class of aggregate level planning problems in hybrid flowshops (flowshops with several machines per stage). First, the general planning problem is modelled as multi-level with parallel processors, multi-item, capacitated, lot-sizing with set up times. We suggest a hierarchical approach which sequentially loads the stages; each stage is constrained by the solution of its preceding stage and each stage is treated as a multi-item, capacitated, lot-sizing problem with setup times on parallel processors. We show how this latter problem may be reformulated and solved heuristically as a sequence of network problems (trans-shipment problems) in which the amount of capacity lost in setups is fixed for each period and each processor. The model is within the computing reach of a PC.     

On a rigid inclusion pressed between two elastic half spaces

A solution to the problem of a rigid cylindrical inclusion pressed between two elastic half spaces is obtained using the distributed dislocation technique. The solution is compared with previously published analytical and numerical results for a rigid cylindrical inclusion bounded by two parabolic arcs with rounded corners. A simplified solution to the problem based on the classical contact theory and well-known results for crack problems is also suggested and validated. The simplified solution agrees well with analytical results in the case when the length of the opening around inclusion is much larger than the length of the contact zone.     

Convex multilevel decomposition algorithms for non-monotone problems

A convex, multilevel decomposition algorithm is proposed in this paper for the solution of static analysis problems involving non-monotone, possibly multivalued laws. The theory is developed here for a model structure with non-monotone interface or boundary conditions. First the non-monotone laws are written in the form of a difference of two monotone functions. Under this decomposition, the non-linear elastostatic analysis problem is equivalent to a system of convex variational inequalities and to non-convex min-min problems for appropriately defined Lagrangian functions. The solution(s) of each one of the aforementioned problems describe the position(s) of static equilibrium of the considered structure. In this paper a multilevel optimization scheme, due to Auchmuty,¹ is used for the numerical solution of the problem. The most interesting feature of this method, from the computational mechanics' standpoint, is the fact that each one of the subproblems involved in the multilevel algorithm is a convex optimization problem, or, in terms of mechanics, an appropriately modified monotone ‘unilateral’ problem. Thus, existing algorithms and software can be used for the numerical solution with minor modifications. Numerical results concerning the calculation of elastic and rigid stamp problems and of material inclusion problems with delamination and non-monotone stick-slip frictional effects illustrate the theory.     

The solution of a mixed boundary value problem in the theory of diffraction

Summary An exact solution is obtained for the problem of the diffraction of a cylindrical sound wave by an absorbent semi-infinite plane. The two faces of the half-plane have different impedance boundary conditions. The problem which is solved is a mathematical model for a noise barrier whose surface is treated with two different acoustically absorbent materials.The usual Wiener-Hopf method (which is the standard technique for solving half-plane problems) has to be modified to give a solution to the present mixed boundary value problem.     

Iterative computational identification of a space-wise dependent source in parabolic equation

Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems, tasks of recovering the dependence of the right-hand side on time and spatial variables can be treated as independent. These tasks relate to a class of linear inverse problems, which sufficiently simplifies their study. This work is devoted to finding the dependence of right-hand side of multidimensional parabolic equation on spatial variables using additional observations of the solution at the final point of time – the final overdetermination. More general problems are associated with some integral observation of the solution in time – the integral overdetermination. The first method of numerical solution of inverse problems is based on iterative solution of boundary value problem for time derivative with non-local acceleration. The second method is based on the known approach with iterative refinement of desired dependence of the right-hand side on spatial variables. Capabilities of proposed methods are illustrated by numerical examples for a two-dimensional problem of identifying the right-hand side of a parabolic equation. The standard finite-element approximation in space is used, whilst the time discretization is based on fully implicit two-level schemes.     

Stress intensity solutions for the interaction between a hole edge crack and a line crack

The principle of superposition is used to solve the problem and the original problem is converted into two particular hole edge crack problems. The remote stresses are applied at infinity in the first problem. Meantime, a dislocation distribution is assumed outside the hole contour in the second problem. Singular integral equation is proposed for the solution of the second problem, in which the right hand side of the integral equation is obtained from the solution of the first problem. The first problem as well as the elementary solution of the second problem are solved by means of the rational mapping approach. Finally, numerical examples with the calculated results of stress intensity factors are presented.     

The solution of steady-state axtsymmetric thermoelastic problems by means of complex variables

In this paper is introduced a method of solution of steady-state axisymmetrie thermoelastic problems by means of functions of complex variable. There the equations of the problem are obtained by a rotation of the plane state about an axis of symmetry or by a linear translation of the axisymmetrie state. The equations are utilized for the solution of steady-state thermoelastic problem in a sphere. The stresses in a sphere are given in cylindrical coordinates and coincide with the known solution.     

An adaptive multigrid technique for three-dimensional elasticity

A program for finite element analysis of 3D linear elasticity problems is described. The program uses quadratic hexahedral elements. The solution process starts on an initial coarse mesh; here error estimators are determined by the standard Babu?ka-Rheinboldt method and local refinement is performed by partitioning of indicated elements, each hexahedron into eight new elements. Then the discrete problem is solved on the second mesh and the refinement process proceeds in the following way-on the ith mesh only the elements caused by refinement on the (i-1)th mesh can be refined. The control of refinement is the task of the user because the dimension of the discrete problem grows very rapidly in 3D. The discrete problem is being solved by the frontal solution method on the initial mesh and by a newly developed and very efficient local multigrid method on the refined meshes. The program can be successfully used for solving problems with structural singularities, such as re-entrant corners and moving boundary conditions. A numerical example shows that such problems are solved with the same efficiency as regular problems.     

Nonlinear thermal analysis with a boundary element zone condensation technique

This paper presents a substantially more economical technique for the boundary element analysis (BEA) of a large class of nonlinear heat transfer problems including those with temperature dependent conductivity, temperature dependent convection coefficients, and radiation boundary conditions. The technique involves an exact static condensation of boundary element zones in a multi-zone boundary element model. The condensed boundary element zone contributions to be overall sparse blocked boundary element system matrices are formed once in the first step of the iterative nonlinear solution process and subsequently reused as the nonlinear parts of the overall problem are evolved to a convergent solution. Through a series of example problems it is demonstrated that the zone condensation technique facilitates the use of highly convergent iterative strategies for the solution of the nonlinear heat transfer problem involving modification and subsequent factorization of the overall boundary element system left had side matrix. For heat transfer problems with localized nonlinear effects, the condensation technique is shown to allow for the solution of nonlinear problems in less than half the CPU time required by methods that do not employ condensation.     

HEURISTIC NONSERIAL DYNAMIC PROGRAMMING FOR LARGE PROBLEMS

Dynamic programming is an extremely powerful optimization approach used for the solution of problems which can be formulated to exhibit a serial stage-state structure. However, many design problems are not serial but have highly connected interdependent structures. Existing methods, for the solution of nonserial problems require the problem to possess a certain structure or limit the size of the problem due to storage and computational time requirements. The aim of this paper is to show that nonserial problems can be solved by the use of dynamic programming incorporating algorithms based on heuristics. Two such algorithms are developed using artificial intelligence concepts of estimating the likelihood of future results on present decisions. The algorithms are explained in detail, A small problem is solved and the results of testing them on large scale problems are given. The method is then used to solve a problem drawn from the literature.     

Second-order cone programming with warm start for elastoplastic analysis with von Mises yield criterion

The incremental problem for quasistatic elastoplastic analysis with the von?Mises yield criterion is discussed within the framework of the second-order cone programming (SOCP). We show that the associated flow rule under the von?Mises yield criterion with the linear isotropic/kinematic hardening is equivalently rewritten as a second-order cone complementarity problem. The minimization problems of the potential energy and the complementary energy for incremental analysis are then formulated as the primal-dual pair of SOCP problems, which can be solved with a primal-dual interior-point method. To enhance numerical performance of tracing an equilibrium path, we propose a warm-start strategy for a primal-dual interior-point method based on the primal-dual penalty method. In this warm-start strategy, we solve a penalized SOCP problem to find the equilibrium solution at the current loading step. An advanced initial point for solving this penalized SOCP problem is defined by using information of the solution at the previous loading step.     

Regularized solutions with a singular point for the inverse biharmonic boundary value problem by the method of fundamental solutions

Two numerical methods for the Cauchy problem of the biharmonic equation are proposed. The solution of the problem does not continuously depend on given Cauchy data since the problem is ill-posed. A small noise contained in the Cauchy data sensitively affects on the accuracy of the solution. Our problem is directly discretized by the method of fundamental solutions (MFS) to derive an ill-conditioned matrix equation. As another method, our problem is decomposed into two Cauchy problems of the Laplace and the Poisson equations, which are discretized by the MFS and the method of particular solutions (MPS), respectively. The Tikhonov regularization and the truncated singular value decomposition are applied to the matrix equation to stabilize a numerical solution of the problem for the given Cauchy data with high noises. The L-curve and the generalized cross-validation determine a suitable regularization parameter for obtaining an accurate solution. Based on numerical experiments, it is concluded that the numerical method proposed in this paper is effective for the problem that has an irregular domain and the Cauchy data with high noises. Furthermore, our latter method can successfully solve the problem whose solution has a singular point outside the computational domain.     

Direct solution of ill‐posed boundary value problems by radial basis function collocation method

Numerical solution of ill‐posed boundary value problems normally requires iterative procedures. In a typical solution, the ill‐posed problem is first converted to a well‐posed one by assuming the missing boundary values. The new problem is solved by a conventional numerical technique and the solution is checked against the unused data. The problem is solved iteratively using optimization schemes until convergence is achieved. The present paper offers a different procedure. Using the radial basis function collocation method, we demonstrate that the solution of certain ill‐posed problems can be accomplished without iteration. This method not only is efficient and accurate, but also circumvents the stability problem that can exist in the iterative method. Copyright © 2005 John Wiley & Sons, Ltd.