A flow‐aligning algorithm for convection‐dominated problems

This paper proposes and studies an algorithm for aligning a triangulation with a given convection field. Approximate solutions of convection‐dominated problems on flow‐aligned meshes typically have sharper internal layers, less over and undershooting and higher accuracy. The algorithm we present can be imported easily into any 2D finite element solver, does not change the number of meshpoints, and can improve solution quality quite dramatically. This improvement in solution quality on the flow‐aligned triangulation is illustrated for both the usual Galerkin method and the streamline‐diffusion method. Copyright © 1999 John Wiley & Sons, Ltd.     

Local RBF‐FD solutions for steady convection–diffusion problems

This paper describes the application of radial basis function (RBF) based finite difference type scheme (RBF‐FD) for solving steady convection–diffusion equations. Numerical studies are made using multiquadric (MQ) RBF. By varying the shape parameter in MQ, the accuracy of the solution is seen to be highly improved for large values of Reynolds' numbers. The developed scheme has been compared with the corresponding finite difference scheme and found that the solutions obtained using the former are non‐oscillatory. Copyright © 2007 John Wiley & Sons, Ltd.     

A boundary element numerical scheme for the two‐dimensional convection–diffusion equation

In this study, the theoretical and numerical fundamentals of BIEM techniques for the two‐dimensional convection–diffusion problem are presented. After an extended presentation of the basic integral formulation, the discretizing and iterative processes for its resolution are introduced. Interesting remarks on general expressions versus previously published particularized results are worth mentioning. A numerical solution scheme is provided, which has been completely developed and designed to the physical problem posed. A novel scheme based in the simultaneous solving of the potential and the gradient of the potential boundary integral equations is included. A diversity of problems is tested to prove the possibilities of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

An alternative TLM method for steady‐state convection–diffusion

Recent papers have introduced a novel and efficient scheme, based on the transmission line modelling (TLM) method, for solving one‐dimensional steady‐state convection–diffusion problems. This paper introduces an alternative method. It presents results obtained using both techniques, which suggest that the new scheme outlined in this paper is the more accurate and efficient of the two. Copyright © 2009 John Wiley & Sons, Ltd.     

A TLM method for steady‐state convection–diffusion: Some additions and refinements

A recent paper introduced a novel and efficient scheme, based on the transmission line modelling (TLM) method, for solving steady‐state convection–diffusion problems. This paper shows how this one‐dimensional scheme can be adapted to include reaction and source terms and how it can be implemented with non‐equidistant nodes. It introduces new ways of calculating the necessary model parameters which can improve the accuracy of the scheme, shows how steady‐state solutions can be obtained directly, and compares results with those from two finite difference (FD) methods. While the cost of implementation is higher than for the FD schemes, the new TLM scheme can be significantly more accurate, especially when convection dominates. Copyright © 2008 John Wiley & Sons, Ltd.     

Time accurate consistently stabilized mesh‐free methods for convection dominated problems

The behaviour of high‐order time stepping methods combined with mesh‐free methods is studied for the transient convection–diffusion equation. Particle methods, such as the element‐free Galerkin (EFG) method, allow to easily increase the order of consistency and, thus, to formulate high‐order schemes in space and time. Moreover, second derivatives of the EFG shape functions can be constructed with a low extra cost and are well defined, even for linear interpolation. Thus, consistent stabilization schemes can be considered without loss in the convergence rates. Copyright © 2003 John Wiley & Sons, Ltd.     

An optimal weighted upwinding covolume method on non‐standard grids for convection–diffusion problems in 2D

In this paper we develop an optimal weighted upwinding covolume method on non‐standard covolume grids for convection–diffusion problems in two dimensions. The novel feature of our method is that we construct the non‐standard covolume grid in which the nodes of covolumes vary in the interior of different volumes of primary grid depending on the local weighted factors and further on the local Peclet's numbers. A simple method of finding the local optimal weighted factors is also derived from a non‐linear function of local Peclet's numbers. The developed method leads to a totally new scheme for convection–diffusion problems, which overcomes numerical oscillation, avoids numerical dispersion, and has high‐order accuracy. Some theoretical analyses are given and numerical experiments are presented to illustrate the performance of the method. Copyright © 2006 John Wiley & Sons, Ltd.     

A discontinuous Galerkin method for a hyperbolic model for convection–diffusion problems in CFD

This paper proposes a hyperbolic model for convection–diffusion transport problems in computational fluid dynamics (CFD). The hyperbolic model is based on the so‐called Cattaneo's law. This is a time‐dependent generalization of Fick's and Fourier's laws that was originally proposed to solve pure‐diffusive heat transfer problems. We show that the proposed model avoids the infinite speed paradox that is inherent in the standard parabolic model. A high‐order upwind discontinuous Galerkin (DG) method is developed and applied to classic convection‐dominated test problems. The quality of the numerical results is remarkable, since the discontinuities are very well captured without the appearance of spurious oscillations. These results are compared with those obtained by using the standard parabolic model and the local DG (LDG) method and with those given by the parabolic model and the Bassi–Rebay scheme. Finally, the applicability of the proposed methodology is demonstrated by solving a practical case in engineering. We simulate the evolution of pollutant being spilled in the harbour of A Coruña (northwest of Spain, EU). Copyright © 2007 John Wiley & Sons, Ltd.     

Implicit characteristic Galerkin method for convection–diffusion equations

This paper presents a characteristic Galerkin finite element method with an implicit algorithm for solving multidimensional, time‐dependent convection–diffusion equations. The method is formulated on the basis of the combination of both the precise and the implicit numerical integration procedures aiming to reference particles. The precise integration procedure with a 2^N algorithm is taken as a tool to determine the material (Lagrangian) derivative of the convective function in the operator splitting procedure. The stability analysis of the algorithm and numerical results illustrate good performance of the present method in stability and accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

An element‐wise,locally conservative Galerkin (LCG) method for solving diffusion and convection–diffusion problems

An element‐wise locally conservative Galerkin (LCG) method is employed to solve the conservation equations of diffusion and convection–diffusion. This approach allows the system of simultaneous equations to be solved over each element. Thus, the traditional assembly of elemental contributions into a global matrix system is avoided. This simplifies the calculation procedure over the standard global (continuous) Galerkin method, in addition to explicitly establishing element‐wise flux conservation. In the LCG method, elements are treated as sub‐domains with weakly imposed Neumann boundary conditions. The LCG method obtains a continuous and unique nodal solution from the surrounding element contributions via averaging. It is also shown in this paper that the proposed LCG method is identical to the standard global Galerkin (GG) method, at both steady and unsteady states, for an inside node. Thus, the method, has all the advantages of the standard GG method while explicitly conserving fluxes over each element. Several problems of diffusion and convection–diffusion are solved on both structured and unstructured grids to demonstrate the accuracy and robustness of the LCG method. Both linear and quadratic elements are used in the calculations. For convection‐dominated problems, Petrov–Galerkin weighting and high‐order characteristic‐based temporal schemes have been implemented into the LCG formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

Three‐dimensional numerical modelling by XFEM of spring‐layer imperfect curved interfaces with applications to linearly elastic composite materials

The spring‐layer interface model is widely used in describing some imperfect interfaces frequently involved in materials and structures. Typically, it is appropriate for modelling a thin soft interphase layer between two relatively stiff bulk media. According to the spring‐layer interface model, the displacement vector suffers a jump across an interface whereas the traction vector is continuous across the same interface and is, in the linear case, proportional to the displacement vector jump. In the present work, an efficient three‐dimensional numerical approach based on the extended finite element method is first proposed to model linear spring‐layer curved imperfect interfaces and then applied to predict the effective elastic moduli of composites in which such imperfect interfaces intervene. In particular, a rigorous derivation of the linear spring‐layer interface model is provided to clarify its domain of validity. The accuracy and convergence rate of the elaborated numerical approach are assessed via benchmark tests for which exact analytical solutions are available. The computated effective elastic moduli of composites are compared with the relevant analytical lower and upper bounds. Copyright © 2011 John Wiley & Sons, Ltd.     

An interface‐enriched generalized FEM for problems with discontinuous gradient fields

A new generalized FEM is introduced for solving problems with discontinuous gradient fields. The method relies on enrichment functions associated with generalized degrees of freedom at the nodes generated from the intersection of the phase interface with element edges. The proposed approach has several advantages over conventional generalized FEM formulations, such as a lower computational cost, easier implementation, and straightforward handling of Dirichlet boundary conditions. A detailed convergence study of the proposed method and a comparison with the standard FEM are presented for heat transfer problems. The method achieves the optimal rate of convergence using meshes that do not conform to the interfaces present in the domain while achieving a level of accuracy comparable to that of the standard FEM with conforming meshes. Various application problems are presented, including the conjugate heat transfer problem encountered in microvascular materials. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical solution of convection–diffusion problems at high Péclet number using boundary elements

This paper describes a boundary element scheme for solving steady-state convection–diffusion problems at high Péclet numbers. A special treatment of the singular integrals is included which uses discontinuous elements and a regularization procedure. Transformations are performed to avoid directly evaluating Bessel functions for Cauchy principal value and hypersingular integrals. Test examples are solved with values of Péclet number up to 10⁷ to assess the numerical scheme. © 1998 John Wiley & Sons, Ltd.     

Enhanced mixed interpolation XFEM formulations for discontinuous Timoshenko beam and Mindlin‐Reissner plate

Shear locking is a major issue emerging in the computational formulation of beam and plate finite elements of minimal number of degrees of freedom as it leads to artificial overstiffening. In this paper, discontinuous Timoshenko beam and Mindlin‐Reissner plate elements are developed by adopting the Hellinger‐Reissner functional with the displacements and through‐thickness shear strains as degrees of freedom. Heterogeneous beams and plates with weak discontinuity are considered, and the mixed formulation has been combined with the extended finite element method (FEM); thus, mixed enrichment functions are used. Both the displacement and the shear strain fields are enriched as opposed to the traditional extended FEM where only the displacement functions are enriched. The enrichment type is restricted to extrinsic mesh‐based topological local enrichment. The results from the proposed formulation correlate well with analytical solution in the case of the beam and in the case of the Mindlin‐Reissner plate with those of a finite element package (ABAQUS) and classical FEM and show higher rates of convergence. In all cases, the proposed method captures strain discontinuity accurately. Thus, the proposed method provides an accurate and a computationally more efficient way for the formulation of beam and plate finite elements of minimal number of degrees of freedom.     

Incremental‐secant modulus iteration scheme and stress recovery for simulating cracking process in quasi‐brittle materials using XFEM

In this paper, an incremental‐secant modulus iteration scheme using the extended/generalized finite element method (XFEM) is proposed for the simulation of cracking process in quasi‐brittle materials described by cohesive crack models whose softening law is composed of linear segments. The leading term of the displacement asymptotic field at the tip of a cohesive crack (which ensures a displacement discontinuity normal to the cohesive crack face) is used as the enrichment function in the XFEM. The opening component of the same field is also used as the initial guess opening profile of a newly extended cohesive segment in the simulation of cohesive crack propagation. A statically admissible stress recovery (SAR) technique is extended to cohesive cracks with special treatment of non‐homogeneous boundary tractions. The application of locally normalized co‐ordinates to eliminate possible ill‐conditioning of SAR, and the influence of different weight functions on SAR are also studied. Several mode I cracking problems in quasi‐brittle materials with linear and bilinear softening laws are analysed to demonstrate the usefulness of the proposed scheme, as well as the characteristics of global responses and local fields obtained numerically by the XFEM. Copyright © 2006 John Wiley & Sons, Ltd.     

A stabilized meshfree reproducing kernel‐based method for convection–diffusion problems

In this paper, we develop a meshfree particle‐based method for convection–diffusion problems. Discretization is performed by using piecewise constant kernels. The stabilized scheme is based on a new upwind kernel. We show that accurate and stable scheme can be obtained by using purpose‐built kernels. It also shown that under some conditions the classical optimal finite difference scheme can be derived by the new method. Several numerical tests validate the method. Copyright © 2011 John Wiley & Sons, Ltd.     

Anisotropic mesh refinement for problems with internal and boundary layers

The character of convection‐dominated, singularly perturbed boundary value problems requires their special numerical treatment in order to guarantee stability and resolve existing layers with acceptable accuracy. In addition to discretization methods particularly developed for this aim, recently more and more attention has been directed towards adapted triangulations of the computational domain. In this paper, an adaptive strategy based on an anisotropic refinement is developed for finite element methods. Starting from some a priori information about the location of layers, the so‐called hybrid meshes are constructed. By these meshes, the flexibility of unstructured meshes, good approximation properties in layers, and relatively simple rules for a posteriori anisotropic refinement are combined with each other. The efficiency of this procedure is demonstrated by selected numerical examples. Copyright © 1999 John Wiley & Sons, Ltd.     

A finite volume method for the approximation of convection–diffusion equations on general meshes

A new finite volume method is presented for approximating convection–diffusion equations. This method allows general (unstructured, non‐matching, distorted) meshes to be used without the numerical results being too much altered. The method has been tested for some well‐known benchmarks involving convection and convection–diffusion operators in two space dimensions. These numerical experiments show that it is between first and second‐order accurate, according to the type of the underlying mesh. Further numerical experiments regarding the striation equations have been carried out successfully. Copyright © 2012 John Wiley & Sons, Ltd.     

An object‐oriented framework for the implementation of adjoint techniques in the design and control of complex continuum systems

Specific object‐oriented software design concepts are elaborated for a novel implementation of a class of adjoint optimization problems typical of the infinite‐dimensional design and control of continuum systems. For clarity, the design steps and ideas are elucidated using an inverse natural convection design problem. Effective application of software design concepts such as inheritance, data encapsulation, information hiding, etc., is demonstrated through instances from the example considered. Two test numerical examples are considered and the CPU statistics for one of these problems are compared with those corresponding to a procedural implementation of the same problem. The numerical examples include a three‐dimensional inverse design problem that demonstrates the effectiveness of the present object‐oriented approach in developing dimension‐independent robust design codes. Copyright © 2000 John Wiley & Sons, Ltd.     

Interface handling for three‐dimensional higher‐order XFEM‐computations in fluid–structure interaction

Three‐dimensional higher‐order eXtended finite element method (XFEM)‐computations still pose challenging computational geometry problems especially for moving interfaces. This paper provides a method for the localization of a higher‐order interface finite element (FE) mesh in an underlying three‐dimensional higher‐order FE mesh. Additionally, it demonstrates, how a subtetrahedralization of an intersected element can be obtained, which preserves the possibly curved interface and allows therefore exact numerical integration. The proposed interface algorithm collects initially a set of possibly intersecting elements by comparing their ‘eXtended axis‐aligned bounding boxes’. The intersection method is applied to a highly reduced number of intersection candidates. The resulting linearized interface is used as input for an elementwise constrained Delaunay tetrahedralization, which computes an appropriate subdivision for each intersected element. The curved interface is recovered from the linearized interface in the last step. The output comprises triangular integration cells representing the interface and tetrahedral integration cells for each intersected element. Application of the interface algorithm currently concentrates on fluid–structure interaction problems on low‐order and higher‐order FE meshes, which may be composed of any arbitrary element types such as hexahedra, tetrahedra, wedges, etc. Nevertheless, other XFEM‐problems with explicitly given interfaces or discontinuities may be tackled in addition. Multiple structures and interfaces per intersected element can be handled without any additional difficulties. Several parallelization strategies exist depending on the desired domain decomposition approach. Numerical test cases including various geometrical exceptions demonstrate the accuracy, robustness and efficiency of the interface handling. Copyright © 2009 John Wiley & Sons, Ltd.