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This paper presents a new isogeometric mortar contact formulation based on an extended finite element interpolation to capture physical pressure discontinuities at the contact boundary. The so called two-half-pass algorithm is employed, which leads to an unbiased formulation and, when applied to the mortar setting, has the additional advantage that the mortar coupling term is no longer present in the contact forces. As a result, the computationally expensive segmentation at overlapping master-slave element boundaries, usually required in mortar methods (although often simplified with loss of accuracy), is not needed from the outset. For the numerical integration of general contact problems, the so-called refined boundary quadrature is employed, which is based on adaptive partitioning of contact elements along the contact boundary. The contact patch test shows that the proposed formulation passes the test without using either segmentation or refined boundary quadrature. Several numerical examples are presented to demonstrate the robustness and accuracy of the proposed formulation. [ABSTRACT FROM AUTHOR]

In this paper, a new type of hybrid finite element method (FEM), hybrid fundamental-solution-based FEM (HFS-FEM), is developed for analyzing plane piezoelectric problems by employing fundamental solutions (Green's functions) as internal interpolation functions. A modified variational functional used in the proposed model is first constructed, and then the assumed intra-element displacement fields satisfying a priori the governing equations of the problem are constructed by using a linear combination of fundamental solutions at a number of source points located outside the element domain. To ensure continuity of fields over inter-element boundaries, conventional shape functions are employed to construct the independent element frame displacement fields defined over the element boundary. The proposed methodology is assessed by several examples with different boundary conditions and is also used to investigate the phenomenon of stress concentration in infinite piezoelectric medium containing a hole under remote loading. The numerical results show that the proposed algorithm has good performance in numerical accuracy and mesh distortion insensitivity compared with analytical solutions and those from ABAQUS. In addition, some new insights on the stress concentration have been clarified and presented in the paper. [ABSTRACT FROM AUTHOR]

Many applications in computational science and engineering concern composite materials, which are characterized by large discontinuities in the material properties. Such applications require fine-scale finite-element meshes, which lead to large linear systems that are challenging to solve with current direct and iterative solutions algorithms. In this paper, we consider the simulation of asphalt concrete, which is a mixture of components with large differences in material stiffness. The discontinuities in material stiffness give rise to many small eigenvalues that negatively affect the convergence of iterative solution algorithms such as the preconditioned conjugate gradient (PCG) method. This paper considers the deflated preconditioned conjugate gradient (DPCG) method in which the rigid body modes of sets of elements with homogeneous material properties are used as deflation vectors. As preconditioner we consider several variants of the algebraic multigrid smoothed aggregation method. We evaluate the performance of the DPCG method on a parallel computer using up to 64 processors. Our test problems are derived from real asphalt core samples, obtained using CT scans. We show that the DPCG method is an efficient and robust technique for solving these challenging linear systems. [ABSTRACT FROM AUTHOR]

The immersed finite element method (IFEM) is a novel numerical approach to solve fluid-structure interaction types of problems that utilizes non-conforming meshing concept. The fluid and the solid domains are represented independently. The original algorithm of the IFEM follows the interpolation process as illustrated in the original immersed boundary method where the fluid velocity and the interaction force are explicitly coupled. However, the original approach presents many numerical difficulties when the fluid and solid physical properties have large mismatches, such as when the density difference is large and when the solid is a very stiff material. Both situations will lead to divergent or unstable solutions if not handled properly. In this paper, we develop a semi-implicit formulation of the IFEM algorithm so that several terms of the interfacial forces are implicitly evaluated without going through the force distribution process. Based on the 2-D and 3-D examples that we study in this paper, we show that the semi-implicit approach is robust and is capable of handling these highly discontinuous physical properties quite well without any numerical difficulties. [ABSTRACT FROM AUTHOR]

A sub-cycling integration algorithm (or named multi-time-steps integration algorithm), which has been successfully applied to FEM dynamical analysis, was firstly presented by Belytschko et al. (Comput Methods Appl Mech Eng 17/18:259–275, 1979). However, the problem of how to apply this type of algorithm to flexible multi-body dynamics (FMD) problems still lacks investigation up to now. Similar to the region-partitioning method used in FEM, this paper presents a central-difference-based sub-cycling integral method by decomposing the variables of an FMD equation into several groups and adopting different integral step sizes to each group of the variables. Based on the condensed form of an FMD equation, a group of common update formulae and a sub-step update formula, which constitute the sub-cycling together, are established in the paper. Furthermore, an implementation flowchart of the sub-cycling is presented. Stability of the sub-cycling will be analyzed and numerical examples will be performed to verify availability and precision of the sub-cycling in part II of the paper. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to detect damage (stiffness degradation) of laminated composite plates from noisy impact response data. The combined finite element method (FEM) with five degrees of freedom (DOF) and the advanced noise filtering algorithm described in this paper may allow us not only to detect the deteriorated elements but also to find their locations and the extents. A first order shear deformation theory (FSDT) is used to predict the structural behavior and to detect damage of laminated composite plates. The filtering procedure is designed by means of a wavelet decomposition together with a selection of the measuring points, and the optimization criterion is constructed on an estimate of the probability of detection using genetic algorithms. All these techniques are applied for the first time to composites. The effects of filtered noise associated with the uncertainty of measurements due to the complex nature of composites are considered for different layup sequences, number of layers, and length–thickness ratios. Several numerical results show that the noise filtering system is computationally efficient in identifying stiffness degradation for complex structures such as laminated composites. [ABSTRACT FROM AUTHOR]

In this paper a three-dimensional Voronoi cell finite element model is developed for analyzing heterogeneous materials containing a dispersion of ellipsoidal inclusions or voids in the matrix. The paper starts with a description of 3D tessellation of a domain with ellipsoidal heterogeneities, to yield a 3D mesh of Voronoi cells containing the heterogeneities. A surface based tessellation algorithm is developed to account for the shape and size of the ellipsoids in point based tessellation methods. The 3D Voronoi cell finite element model, using the assumed stress hybrid formulation, is developed for determining stresses and displacements in a linear elastic material domain. Special stress functions that introduce classical Lamé functions in ellipsoidal coordinates are implemented to enhance solution convergence. Numerical methods for implementation of algorithms and yielding stable solutions are discussed. Numerical examples are conducted with inclusions and voids to demonstrate the effectiveness of the model. [ABSTRACT FROM AUTHOR]

This paper presents a method to find all intersections between curved lines such as structural line elements and finite element meshes with intentions to generate smaller, non-compatible, line cells (e.g. bar elements) between crossings. The intersection finding algorithm works for two and three-dimensional meshes constituted by linear, quadratic or higher order elements. Using the proposed algorithm, meshes can then be automatically prepared for finite element analyses with techniques for embedding elements within others or analyses that require lines within solids. The application of the method is demonstrated by a number of numerical examples illustrating its capabilities in handling complex geometries, relative speed and convenience. [ABSTRACT FROM AUTHOR]

In this paper, a coupled hydro-mechanical formulation is developed for deformable porous media subjected to crack interfaces in the framework of extended finite element method. Governing equations of the porous medium consist of the momentum balance of the bulk together with the momentum balance and continuity equations of the fluid phase, known as [InlineEquation not available: see fulltext.] formulation. The discontinuity in fractured porous medium is modeled for both opening and closing modes that results in the fluid flow within the fracture, and/or contact behavior at the crack edges. The fluid flow through the fracture is assumed to be viscous and is modeled by employing the Darcy law in which the permeability of fracture is obtained using the cubic law. The contact condition in fractured porous medium is handled by taking the advantage from two different algorithms of LATIN method and penalty algorithm. The effect of contact on fluid phase is employed by considering no leak-off from/into the porous medium. The nonlinearity of coupled equations produced due to opening and closing modes is carried out using an iterative algorithm in the Newton-Raphson procedure. Finally, several numerical examples are solved to illustrate the performance of proposed X-FEM method for hydro-mechanical behavior of fractured porous media with opening and closing modes. [ABSTRACT FROM AUTHOR]

In the current paper we present a fast, reliable technique for simulating wave propagation in complex structures made of heterogeneous materials. The proposed approach, the spectral cell method, is a combination of the finite cell method and the spectral element method that significantly lowers preprocessing and computational expenditure. The spectral cell method takes advantage of explicit time-integration schemes coupled with a diagonal mass matrix to reduce the time spent on solving the equation system. By employing a fictitious domain approach, this method also helps to eliminate some of the difficulties associated with mesh generation. Besides introducing a proper, specific mass lumping technique, we also study the performance of the low-order and high-order versions of this approach based on several numerical examples. Our results show that the high-order version of the spectral cell method together requires less memory storage and less CPU time than other possible versions, when combined simultaneously with explicit time-integration algorithms. Moreover, as the implementation of the proposed method in available finite element programs is straightforward, these properties turn the method into a viable tool for practical applications such as structural health monitoring [-], quantitative ultrasound applications [], or the active control of vibrations and noise [, ]. [ABSTRACT FROM AUTHOR]

The present paper focuses on the development of a fully implicit, incrementally objective integration algorithm for a hypoelastic formulation of $$J_{2}$$ -viscoplasticity, which employs the mechanical threshold strength model to compute the material's flow stress, taking into account its dependence on strain rate and temperature. Heat generation due to high-rate viscoplastic deformation is accounted for, assuming adiabatic conditions. The implementation of the algorithm is discussed, and its performance is assessed in the contexts of implicit and explicit dynamic finite element analysis, with the aid of example problems involving a wide range of loading rates. Computational results are compared to experimental data, showing very good agreement. [ABSTRACT FROM AUTHOR]

Finite element based formulations for flexible multibody systems are becoming increasingly popular and as the complexity of the configurations to be treated increases, so does the computational cost. It seems natural to investigate the applicability of parallel processing to this type of problems; domain decomposition techniques have been used extensively for this purpose. In this approach, the computational domain is divided into non-overlapping sub-domains, and the continuity of the displacement field across sub-domain boundaries is enforced via the Lagrange multiplier technique. In the finite element literature, this approach is presented as a mathematical algorithm that enables parallel processing. In this paper, the divided system is viewed as a flexible multibody system, and the sub-domains are connected by kinematic constraints. Consequently, all the techniques applicable to the enforcement of constraints in multibody systems become applicable to the present problem. In particular, it is shown that a combination of the localized Lagrange multiplier technique with the augmented Lagrange formulation leads to interesting solution strategies. The proposed algorithm is compared with the well-known FETI approach with regards to convergence and efficiency characteristics. The present algorithm is relatively simple and leads to improved convergence and efficiency characteristics. Finally, implementation on a parallel computer was conducted for the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, heat transfer problems with sharp spatial gradients are analyzed using the Generalized Finite Element Method with global-local enrichment functions ( GFEM). With this approach, scale-bridging enrichment functions are generated on the fly, providing specially-tailored enrichment functions for the problem to be analyzed with no a-priori knowledge of the exact solution. In this work, a decomposition of the linear system of equations is formulated for both steady-state and transient heat transfer problems, allowing for a much more computationally efficient analysis of the problems of interest. With this algorithm, only a small portion of the global system of equations, i.e., the hierarchically added enrichments, need to be re-computed for each loading configuration or time-step. Numerical studies confirm that the condensation scheme does not impact the solution quality, while allowing for more computationally efficient simulations when large problems are considered. We also extend the GFEM to allow for the use of hexahedral elements in the global domain, while still using tetrahedral elements in the local domain, to allow for automatic localized mesh refinement without the use of constrained approximations. Simulations are run with the use of linear and quadratic hexahedral and tetrahedral elements in the global domain. Convergence studies indicate that the use of a different partition of unity (PoU) in the global (hexahedral elements) and local (tetrahedral elements) domains does not adversely impact the solution quality. [ABSTRACT FROM AUTHOR]

The rotational quaternions represent a unique four dimensional parametrization of rotations in the three dimensional Euclidean space. In the present paper they are used as the basic rotational parameters in formulating the finite-element approach of geometrically exact beam-like structures. The classical concept of parameterizing the rotation matrix by the rotational vector is completely abandoned so that the only rotational parameters are the rotational quaternions representing both rotations and rotational strains in the beam. The space discretization based on the collocation method is used and the adjustment of the Newmark time-integration algorithm to the quaternion parameterizations of rotation is presented. [ABSTRACT FROM AUTHOR]

In this paper we present a new approach for solving elastoplastic problems as second order cone complementarity problems (SOCCPs). Specially, two classes of elastoplastic problems, i.e. the J plasticity problems with combined linear kinematic and isotropic hardening laws and the Drucker-Prager plasticity problems with associative or non-associative flow rules, are taken as the examples to illustrate the main idea of our new approach. In the new approach, firstly, the classical elastoplastic constitutive equations are equivalently reformulated as second order cone complementarity conditions. Secondly, by employing the finite element method and treating the nodal displacements and the plasticity multiplier vectors of Gaussian integration points as the unknown variables, we obtain a standard SOCCP formulation for the elastoplasticity analysis, which enables the using of general SOCCP solvers developed in the field of mathematical programming be directly available in the field of computational plasticity. Finally, a semi-smooth Newton algorithm is suggested to solve the obtained SOCCPs. Numerical results of several classical plasticity benchmark problems confirm the effectiveness and robustness of the SOCCP approach. [ABSTRACT FROM AUTHOR]

In this paper we propose a finite element approach which simulates the mechanical behaviour of beam assemblies that are subject to large deformations and that develop contact-friction interactions. We focus on detecting and modeling contact-friction interactions within the assembly of beams. Contact between beams-or between parts of the same beam in the case of self-contact, is detected from intermediate geometries defined within proximity zones associating close parts of beam axes. The discretization of contact-friction interactions is performed on these intermediate geometries by means of contact elements, constituted of pairs of material particles which are predicted to enter into contact. A 3D finite strain beam model is used to represent the behaviour of each beam. This model describes the kinematics of each beam cross-section using nine degrees of freedom, and is therefore able to represent plane deformations of these cross-sections. Algorithms are proposed to solve the global nonlinear problem using an implicit scheme, under quasi-static assumptions. Simulation results of the tightening and releasing of knots made on monofilament and multifilament yarns are shown as an application. Straight fibers are first twisted together to make a yarn, before suitable conditions are applied to their ends to form and tighten the knot. Tightening forces are finally released to obtain an equilibrium configuration of the knot without external forces. [ABSTRACT FROM AUTHOR]

A contact algorithm in the context of the combined discrete element (DE) and finite element (FE) method is proposed. The algorithm, which is based on the node-to-surface method used in finite element method, treats each spherical discrete element as a slave node and the surfaces of the finite element domain as the master surfaces. The contact force on the contact interface is processed by using a penalty function method. Afterward, a modification of the combined DE/FE method is proposed. Following that, the corresponding numerical code is implemented into the in-house developed code. To test the accuracy of the proposed algorithm, the impact between two identical bars and the vibration process of a laminated glass plate under impact of elastic sphere are simulated in elastic range. By comparing the results with the analytical solution and/or that calculated by using LS-DYNA, it is found that they agree with each other very well. The accuracy of the algorithm proposed in this paper is proved. [ABSTRACT FROM AUTHOR]

Following the approach developed for rods in Part 1 of this paper (Pimenta et al. in Comput. Mech. 42:715-732, ), this work presents a fully conserving algorithm for the integration of the equations of motion in nonlinear shell dynamics. We begin with a re-parameterization of the rotation field in terms of the so-called Rodrigues rotation vector, allowing for an extremely simple update of the rotational variables within the scheme. The weak form is constructed via non-orthogonal projection, the time-collocation of which ensures exact conservation of momentum and total energy in the absence of external forces. Appealing is the fact that general hyperelastic materials (and not only materials with quadratic potentials) are permitted in a totally consistent way. Spatial discretization is performed using the finite element method and the robust performance of the scheme is demonstrated by means of numerical examples. [ABSTRACT FROM AUTHOR]

An algorithm combining three-dimensional (3D) discrete and finite element methods is proposed. This new approach is conducted by decomposing the calculation domain into a finite element (FE) calculation domain and a discrete element (DE) calculation domain; the interaction between the two sub-domains is processed by using a penalty function method. Following the established model that combines spherical DEs and FEs, the corresponding numerical code is developed. The vibration process of two cantilever beams under dynamic force is simulated. By comparing the results calculated with different penalty factors set and also with that calculated by the finite element code LS-DYNA, it is found that the calculated results are unanimous and the precision is almost the same as LS-DYNA, as long as the penalty factor is large enough. Moreover, the vibration processes of two plates under impact of rigid spheres are simulated and the accuracy of the model proposed in this paper is further proved in the field of contact mechanics by comparing the simulating results with that calculated by using LS-DYNA. Finally, the impact fracture behavior of a laminated glass plate is simulated, with the influence of model parameters taken into consideration. And the numerical experiments show that the combined model can be used to predict some macroscopical physical quantities, such as the impact force of impactor. [ABSTRACT FROM AUTHOR]

For certain continuum problems, it is desirable and beneficial to combine two different methods together in order to exploit their advantages while evading their disadvantages. In this paper, a bridging transition algorithm is developed for the combination of the meshfree method (MM) with the finite element method (FEM). In this coupled method, the MM is used in the sub-domain where the MM is required to obtain high accuracy, and the FEM is employed in other sub-domains where FEM is required to improve the computational efficiency. The MM domain and the FEM domain are connected by a transition (bridging) region. A modified variational formulation and the Lagrange multiplier method are used to ensure the compatibility of displacements and their gradients. To improve the computational efficiency and reduce the meshing cost in the transition region, regularly distributed transition particles, which are independent of either the meshfree nodes or the FE nodes, can be inserted into the transition region. The newly developed coupled method is applied to the stress analysis of 2D solids and structures in order to investigate its’ performance and study parameters. Numerical results show that the present coupled method is convergent, accurate and stable. The coupled method has a promising potential for practical applications, because it can take advantages of both the MM and FEM when overcome their shortcomings. [ABSTRACT FROM AUTHOR]

Fluid–structure interaction computations based on interface-tracking (moving-mesh) techniques are often hindered if the structural surfaces come in contact with each other. As the distance between two structural surfaces tends to zero, the fluid mesh in between distorts severely and eventually becomes invalid. Our objective is to develop a technique for modeling problems where the contacting structural surfaces would otherwise inhibit flow modeling or even fluid-mesh update. In this paper, we present our contact tracking technique that detects impending contact and maintains a minimum distance between the contacting structural surfaces. Our Surface-Edge-Node Contact Tracking (SENCT) technique conducts a topologically hierarchical search to detect contact between each node and the elements (“surfaces”), edges and other nodes. To keep the contacting surfaces apart by a small distance, we apply to the contacted nodes penalty forces in SENCT-Force (SENCT-F) and displacement restrictions in SENCT-Displacement (SENCT-D). By keeping a minimum distance between the contacting surfaces, we are able to update the fluid mesh in between and model the flow accurately. [ABSTRACT FROM AUTHOR]

A fully conserving algorithm is developed in this paper for the integration of the equations of motion in nonlinear rod dynamics. The starting point is a re-parameterization of the rotation field in terms of the so-called Rodrigues rotation vector, which results in an extremely simple update of the rotational variables. The weak form is constructed with a non-orthogonal projection corresponding to the application of the virtual power theorem. Together with an appropriate time-collocation, it ensures exact conservation of momentum and total energy in the absence of external forces. Appealing is the fact that nonlinear hyperelastic materials (and not only materials with quadratic potentials) are permitted without any prejudice on the conservation properties. Spatial discretization is performed via the finite element method and the performance of the scheme is assessed by means of several numerical simulations. [ABSTRACT FROM AUTHOR]

Imposing rigidity constraints of an immersed elastic body in a transient flow field is not trivial. It requires solution stability and accuracy. In this paper, we present an efficient and accurate algorithm implemented to enforce fluid–structure interface constraints used in the immersed finite element method (IFEM). This interface treatment is a constraint applied onto the rigid bodies based on the fluid structure interaction force evaluated from the immersed solid object. It requires no ad hoc constants or adjustments, thus providing numerical stability and avoiding unnecessary trial-and-error procedures in defining the stiffness of the elastic body. This force term can be evaluated for both uniform and nonuniform fluid grids based on the higher order interpolation function adopted in the IFEM. The ability in handling nonuniform interpolations offers the convenience in modeling arbitrary geometrical shapes and provides solution refinements around interfaces. The results we obtained from flow past a rigid cylinder demonstrate that this convenient way of constraining the interface is a reliable and robust numerical approach to solve unsteady fluid flow interacting with immersed rigid bodies. [ABSTRACT FROM AUTHOR]

This paper presents our new development of parallel finite element algorithms for elastic–plastic problems. The proposed method is based on dividing the original structure under consideration into a number of substructures which are treated as isolated finite element models via the interface conditions. Throughout the analysis, each processor stores only the information relevant to its substructure and generates the local stiffness matrix. A parallel substructure oriented preconditioned conjugate gradient method, which is combined with MR smoothing and diagonal storage scheme are employed to solve linear systems of equations. After having obtained the displacements of the problem under consideration, a substepping scheme is used to integrate elastic–plastic stress–strain relations. The procedure outlined controls the error of the computed stress by choosing each substep size automatically according to a prescribed tolerance. The combination of these algorithms shows a good speedup when increasing the number of processors and the effective solution of 3D elastic–plastic problems whose size is much too large for a single workstation becomes possible. [ABSTRACT FROM AUTHOR]

This paper presents a new contact searching algorithm for large deformation mortar-based contact formulations. In this algorithm, a bounding volume hierarchy, defined in the context of a binary tree, is built for each contact surface based on the geometry of the surface. A global contact searching procedure based on these bounding volume trees is first performed to find all candidate contact element pairs, and then a local searching procedure is done to find all the mortar segments having contributions to the mortar integrals that define the contact formulation. The searching algorithm is shown to be very efficient and readily applicable to a variety of large sliding contact problems. [ABSTRACT FROM AUTHOR]

Within the framework of the finite element method, this paper presents new algorithms implementing implicit stress integration and consistent tangent matrix calculations for an elasto-plastic model with rotational hardening. The sub-stepping technique is used for both the numerical integration of the constitutive relations and determination of the consistent tangent matrix in order to overcome the convergence difficulty arising from the complexity of the elasto-plastic model with rotational hardening. The integration of the constitutive relations and the computation of the consistent tangent matrix are incorporated into a unique procedure. Numerical tests are carried out and discussed to demonstrate the global accuracy and stability of the presented algorithms. [ABSTRACT FROM AUTHOR]

In this study, we investigate a method to detect damage in a laminated composite structure by analyzing its dynamic response to impact loads. The combined finite element method (FEM) with seven degrees of freedom (DOF) and the advanced microgenetic algorithm described in this paper may allow us not only to detect the damaged elements but also to find their locations and the extent of damage. A high order shear deformation theory (HSDT) is used to predict the structural behavior and to detect damage of laminated composite plates. The effects of noise associated with the uncertainty of measurements due to the complex nature of composites are considered for [0/90] s and [±45] s layup sequences. The results indicate that the new method is computationally efficient in characterizing damage for complex structures such as laminated composites. [ABSTRACT FROM AUTHOR]

This paper presents a new algorithm to identify the occurrence of impact (contact) among structures. Structures are modeled here by finite elements and the resulting formulation is employed to solve two-dimensional frame impact problems by the Lagrange Multiplier technique. The proposed algorithm is based on potential theory, largely employed in various fields of physics. The potential theory is used together with integral equations making possible to compare relative positions among nodes of different structures involved in a collision process. The present technique may be used to model the contact prevision among two or three-dimensional structures or between structures and rigid obstacles. In order to show the applicability of the proposed technique, numerical answers for 2D problems are compared with both analytical solutions and numerical results given by other authors. [ABSTRACT FROM AUTHOR]

This paper presents a nonlinear implicit transient formulation for parallel contact analysis. A new ‘lumping’ algorithm based on the penalty method is developed to enforce contact constraints. This algorithm has the effect of incorporating contact generated connectivity and eliminating ill conditioning in the system stiffness matrix caused by the use of a large penalty parameter. Communication schemes are also developed to facilitate contact searching and enforcement in a parallel environment. Numerical analyses were conducted to verify the accuracy of the proposed algorithm. In addition, several contact simulations are also performed to study the performance of this new algorithm under different circumstances. Efficiency plots are also presented for one of the simulations to evaluate the performance of this parallel implicit contact formulation. [ABSTRACT FROM AUTHOR]

3D elastoplastic frictional contact problems with orthotropic friction law belong to the unspecified boundary problems with nonlinearities in both material and geometric forms. One of the difficulties in solving the problem lies in the determination of the tangential slip states at the contact points. A great amount of computational efforts is needed so as to obtain high accuracy numerical results. Based on a combination of the well known mathematical programming method and iterative method, a finite element model is put forward in this paper. The problems are finally reduced to linear complementarity problems. A specially designed smoothing algorithm based on NCP-function is then applied for solving the problems. Numerical results are given to demonstrate the validity of the model and the algorithm proposed. [ABSTRACT FROM AUTHOR]