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In part I of this study, a three‐dimensional finite difference iterative solver capable of handling the coupled Navier‐Stokes and energy equations for incompressible viscous flows was described and validated with two‐ and three‐dimensional benchmarks. Part II describes the results of the computational study of two distinct complex geometries: 1) two‐dimensional and three‐dimensional natural convection in cavity whose surface is cooled while two internal blocks are heated; 2) two‐dimensional and three‐dimensional natural convection in the region defined by two interconnected cavities of different sizes which are differentially heated. All computations have been performed for a Prandtl number of 1.0, and different values of the Rayleigh number ranging between 10³ and 10⁶ depending on the problem. In the first problem, three‐dimensional effects in the top region of the cavity trap fluid in vortices near the top of the heated blocks adversely affecting heat transfer in the region while enhancing it in the region between the two heated blocks. In the second problem, the sudden expansion of fluid as it leaves the top cavity and enters the bottom one generates three‐dimensional wakes in the bottom cavity that enhance the convective heat transfer across the system walls near them. These studies tend to suggest that three‐dimensional effects play a very important role in the enhancement of convective heat transfer in complex geometries, especially at higher Rayleigh numbers.

As a critical problem in sophisticated distribution systems, vehicle routing plays a pivotal role in dealing with time windows and capacities constraints. The purpose of this paper is to addresses a new integrated model to incorporate both three-dimensional and time windows aspects of the routing problem. First, capacitated vehicle routing decisions are made subject to a soft time interval to meet the customers’ demands. Afterward, these decisions are entered into the three-dimensional loading problem.

The problem is solved using generalized algebraic modeling system software in small-size problems. The problem is NP-hard and requires an efficient solution methodology. For this purpose, a hybrid algorithm has been proposed to solve the large-size problems. The efficiency of this algorithm is checked by making comparisons with exact solutions for small and medium size test problems, and with the related literature for large size problems.

The numerical experiments show that the proposed model covers more effectively the broader aspects of the transportation problem. Furthermore, the proposed algorithm supports competitive and satisfactory results by giving reasonable outputs in comparison with previous studies.

The main purpose of this integration is to achieve minimum total transportation costs, which cannot be guaranteed without applying two referred constraints, simultaneously.

The purpose of this study is to simultaneously determine the constitutive parameters and boundary conditions by solving inverse mechanical problems of power hardening elastoplastic materials in three-dimensional geometries.

The power hardening elastoplastic problem is solved by the complex variable finite element method in software ABAQUS, based on a three-dimensional complex stress element using user-defined element subroutine. The complex-variable-differentiation method is introduced and used to accurately calculate the sensitivity coefficients in the multiple parameters identification method, and the Levenberg–Marquardt algorithm is applied to carry out the inversion.

Numerical results indicate that the complex variable finite element method has good performance for solving elastoplastic problems of three-dimensional geometries. The inversion method is effective and accurate for simultaneously identifying multi-parameters of power hardening elastoplastic problems in three-dimensional geometries, which could be employed for solving inverse elastoplastic problems in engineering applications.

The constitutive parameters and boundary conditions are simultaneously identified for power hardening elastoplastic problems in three-dimensional geometries, which is much challenging in practical applications. The numerical results show that the inversion method has high accuracy, good stability, and fast convergence speed.

A variety of meshless methods have been developed in the last 20 years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The purpose of this paper is to develop an efficient and accurate algorithms based on meshless methods for the solution of problems involving both material and geometrical nonlinearities.

A parallel two-dimensional linear elastic computer code is presented for a maximum entropy basis functions based meshless method. The two-dimensional algorithm is subsequently extended to three-dimensional adaptive nonlinear and three-dimensional parallel nonlinear adaptively coupled finite element, meshless method cases. The Prandtl-Reuss constitutive model is used to model elasto-plasticity and total Lagrangian formulations are used to model finite deformation. Furthermore, Zienkiewicz and Zhu and Chung and Belytschko error estimation procedure are used in the FE and meshless regions of the problem domain, respectively. The message passing interface library and open-source software packages, METIS and MUltifrontal Massively Parallel Solver are used for the high performance computation.

Numerical examples are given to demonstrate the correct implementation and performance of the parallel algorithms. The agreement between the numerical and analytical results in the case of linear elastic example is excellent. For the nonlinear problems load-displacement curve are compared with the reference FEM and found in a very good agreement. As compared to the FEM, no volumetric locking was observed in the case of meshless method. Furthermore, it is shown that increasing the number of processors up to a given number improve the performance of parallel algorithms in term of simulation time, speedup and efficiency.

Problems involving both material and geometrical nonlinearities are of practical importance in many engineering applications, e.g. geomechanics, metal forming and biomechanics. A family of parallel algorithms has been developed in this paper for these problems using adaptively coupled finite element, meshless method (based on maximum entropy basis functions) for distributed memory computer architectures.

The boundary integral equation (BIE) numerical technique is applied to several practical pressure vessels and piping problems. Axisymmetric and three‐dimensional formulations of the BIE method for linear elastic stress analysis are reviewed. Isoparametric quadratic elements which exhibit excellent modelling capabilities are used to discretize the surfaces. Several three‐dimensional and axisymmetric structures are analysed.

The purpose of this paper is to develop a meshless numerical method for three‐dimensional isotropic thermoelastic problems with arbitrary body forces.

This paper combines the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method (MFS‐DRM) to solve three‐dimensional isotropic thermoelastic problems with arbitrary body forces. In the DRM, the arbitrarily distributed temperature and body force are approximated by polyharmonic splines with augmented polynomial basis, whose particular solutions and the corresponding tractions are reviewed and given explicitly. The MFS is then applied to solve the complementary solution. Numerical experiments of Dirchlet, Robin, and peanut‐shaped‐domain problems are carried out to validate the method.

In literature, it is commented that the Gaussian elimination can be used reliably to solve the MFS equations for non‐noisy boundary conditions. For noisy boundary conditions, the truncated singular value decomposition (TSVD) is more accurate than the Gaussian elimination. In this paper, it was found that the particular solutions obtained by the DRM act like noises and the use of TSVD improves the accuracy.

It is the first time that the MFS‐DRM is derived to solve three‐dimensional isotropic thermoelastic problems with arbitrary body forces.

This paper gives a bibliographical review of the finite element and boundary element parallel processing techniques from the theoretical and application points of view. Topics include: theory – domain decomposition/partitioning, load balancing, parallel solvers/algorithms, parallel mesh generation, adaptive methods, and visualization/graphics; applications – structural mechanics problems, dynamic problems, material/geometrical non‐linear problems, contact problems, fracture mechanics, field problems, coupled problems, sensitivity and optimization, and other problems; hardware and software environments – hardware environments, programming techniques, and software development and presentations. The bibliography at the end of this paper contains 850 references to papers, conference proceedings and theses/dissertations dealing with presented subjects that were published between 1996 and 2002.

A numerical solution to the free boundary problem of three‐dimensional transient seepage through an earth dam with accretion is presented. The numerical model uses a Baiocchi type transformation to extend the unknown solution region to a fixed known region. The initial value problem is then solved by an iterative method of successive over‐relaxation type. A seepage situation of sudden rise of water level on one side of a dam is presented as an example problem. The effects of variation of accretion, effective porosity and hydraulic conductivity are studied.

In this article we report on progress in the development of software tools for fluid flow prediction in the polymer processing industry. This involves state‐of‐the‐art numerical techniques and the study of a number of non‐trivial model flow problems, in an effort to investigate realistic transient problems relevant to industrial processes. Here we study particularly the effects of variations in non‐Newtonian and heat transfer properties of the flowing materials in the flows, both throughout the transient development period and at steady‐state.

A numerical formulation for solving homogeneous anisotropic heat conduction problems based on the use of an isotropic fundamental solution is presented in detail. The analysis is carried out assuming a generic position of the coordinate axes, which may not coincide with the principal directions of orthotropy of the material. The two primary integral equations of the method are derived from the governing differential equation of the problem. Then, the numerical procedure is developed by rewriting the internal degrees of freedom that arise from the domain discretization in terms of the boundary nodes and solving the resulting system of linear equations for the boundary unknowns only. Special attention is given to the differentiation of singular integrals which yields additional terms as well as to the evaluation of the resulting Cauchy principal value integral. The main feature of the proposed formulation is its generality, which makes possible its direct extension to solve the problem of three‐dimensional heat conduction in anisotropic media and, foremost, to three‐dimensional orthotropic and anisotropic elasticity or elastoplasticity.