h-Adaptive mesh refinement strategy for the boundary element method based on local error analysis

An adaptive mesh refinement technique of the h type is proposed for the boundary element method (BEM). Error indicators at element level are evaluated based on a collocation scheme using ad hoc uniform norms that compare values of the field variables at successive iterations. Different approaches are applied depending on whether the boundary conditions are of the Neumann, Dirichlet, or Dirichlet–Neumann mixed type. For Neumann problems the error norms are evaluated using the standard discretised boundary integral equation (DBIE). Dirichlet problems are approached using both the standard DBIE and the hypersingular DBIE. Mixed problems are treated depending on the type of boundary condition. The technique is illustrated with examples for two-dimensional potential problems governed by the Laplace equation.     

Error estimators using global–local methods

Error estimators were derived using global-local and refined global-local finite element methods and an extrapolation technique. The proposed error estimators are to be used in estimating the global error energy norm, local error energy norm and nodal displacement error. These error estimators were applied to three numerical examples. The results clearly demonstrated that these error estimators are of high quality.     

Adaptive time stepping strategy for solidification processes based on modified local time truncation error

Adaptive time step methods provide a computationally efficient means of simulating transient problems with a high degree of numerical accuracy. However, choosing appropriate time steps to model the transient characteristics of solidification processes is difficult. The Gresho–Lee–Sani predictor–corrector strategy, one of the most commonly applied adaptive time step methods, fails to accurately model the latent heat release associated with phase change due to its exaggerated time steps while the apparent heat capacity method is applied. Accordingly, the current study develops a modified local time truncation error‐based strategy designed to adaptively adjust the size of the time step during the simulated solidification procedure in such a way that the effects of latent heat release are more accurately modeled and the precision of the computational solutions correspondingly improved. The computational accuracy and efficiency of the proposed method are demonstrated via the simulation of several one‐dimensional and two‐dimensional thermal problems characterized by different phase change phenomena and boundary conditions. The feasibility of the proposed method for the modeling of solidification processes is further verified via its applications to the enthalpy method. Copyright © 2009 John Wiley & Sons, Ltd.     

Nonlinear analysis via global–local mixed finite element approach

A computational algorithm, based on the combined use of mixed finite elements and classical Rayleigh–Ritz approximation, is presented for predicting the nonlinear static response of structures; The fundamental unknowns consist of nodal displacements and forces (or stresses) and the governing nonlinear finite element equations consist of both the constitutive relations and equilibrium equations of the discretized structure. The vector of nodal displacements and forces (or stresses) is expressed as a linear combination of a small number of global approximation functions (or basis vectors), and a Rayleigh–Ritz technique is used to approximate the finite element equations by a reduced system of nonlinear equations. The global approximation functions (or basis vectors) are chosen to be those commonly used in static perturbation technique; namely a nonlinear solution and a number of its path derivatives. These global functions are generated by using the finite element equations of the discretized structure. The potential of the global–local mixed approach and its advantages over global–local displacement finite element methods are discussed. Also, the high accuracy and effectiveness of the proposed approach are demonstrated by means of numerical examples.     

An enhanced global–local higher-order theory for the free edge effect in laminates

According to the double-superposition hypothesis proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Methods Eng 1997;40:1197–212], an enhanced global–local higher-order theory (EGLHT-mn) is developed to analyze the edge-effect problems in laminates. The in-plane displacement field consists of mth-order (9  m  3) polynomial in global coordinate z along the thickness direction and order 3 power series in local coordinate ζ within each layer whereas the transverse deflection is represented by a nth-order (9  n  3) polynomial of global coordinate z. By imposing the free surface conditions and the geometric and the stress continuity conditions at interfaces, the number of variables of the higher-order theory is independent of the number of layers of the laminates. As an improvement to the global–local theory proposed by Li and Liu (1997), the present higher-order theory takes into account all the effects of shear and normal stresses. A three-node triangular element satisfying the requirement of C¹ weak-continuity conditions between elements is also presented. Comparing to previous published results, it is found that the present higher-order theory is capable of treating free-edge problems in symmetric and unsymmetric laminates under extension, bending and thermal loading. Other characteristic of the present theory is that transverse shear stresses can be accurately computed directly from the constitutive equations without smoothing. However, to determine transverse normal stresses, the local equilibrium equation approach has to been adopted.     

Hydroelastic analysis of insulation containment of LNG carrier by global–local approach

The insulation containment of liquefied natural gas (LNG) carriers is a large‐sized elastic structure made of various metallic and composite materials of complex structural composition to protect the heat invasion and to sustain the hydrodynamic pressure. The goal of the present paper is to present a global–local numerical approach to effectively and accurately compute the local hydroelastic response of a local containment region of interest. The global sloshing flow and hydrodynamic pressure fields of interior LNG are computed by assuming the flexible containment as a rigid container. On the other hand, the local hydroelastic response of the insulation containment is obtained by solving only the local hydroelastic model in which the complex and flexible insulation structure is fully considered and the global analysis results are used as the initial and boundary conditions. The interior incompressible inviscid LNG flow is solved by the first‐order Euler finite volume method, whereas the structural dynamic deformation is solved by the explicit finite element method. The LNG flow and the containment deformation are coupled by the Euler–Lagrange coupling scheme. Copyright © 2008 John Wiley & Sons, Ltd.     

An efficient block preconditioner for Jacobian‐free global–local multiscale methods

In this paper, we develop a block preconditioner for Jacobian‐free global–local multiscale methods, in which the explicit computation of the Jacobian may be circumvented at the macroscale by using a Newton–Krylov process. Effective preconditioning is necessary for the Krylov subspace iterations (e.g. GMRES) to enhance computational efficiency. This is, however, challenging since no explicit information regarding the Jacobian matrix is available. The block preconditioning technique developed in this paper circumvents this problem by effectively deflating the spectrum of the Jacobian matrix at the current Newton step using information about only the Krylov subspaces corresponding to the Jacobian matrices in the previous Newton steps and their representations on those subspaces. This approach is optimal and results in exponential convergence of the GMRES iterations within each Newton step, thus minimizing expensive microscale computations without requiring explicit Jacobian formation in any step. In terms of both computational cost and storage requirements, the action of a single block of the preconditioner per GMRES step scales linearly as the number of degrees of freedom of the macroscale problem as well as the dimension of the invariant subspace of the preconditioned Jacobian matrix. Copyright © 2011 John Wiley & Sons, Ltd.     

Refined global–local higher‐order theory and finite element for laminated plates

Based on completely three‐dimensional elasticity theory, a refined global–local higher‐order theory is presented as enhanced version of the classical global–local theory proposed by Li and Liu (Int. J. Numer. Meth. Engng. 1997; 40 :1197–1212), in which the effect of transverse normal deformation is enhanced. Compared with the previous higher‐order theory, the refined theory offers some valuable improvements these are able to predict accurately response of laminated plates subjected to thermal loading of uniform temperature. However, the previous higher‐order theory will encounter difficulty for this problem. A refined three‐noded triangular element satisfied the requirement of C¹ weak‐continuity conditions in the inter‐element is also presented. The results of numerical examples of moderately thick laminated plates and even thick plates with span/thickness ratios L/h = 2 are given to show that in‐plane stresses and transverse shear stresses can be reasonably predicted by the direct constitutive equation approach without smooth technique. In order to accurately obtain transverse normal stresses, the local equilibrium equation approach in one element is employed herein. Copyright © 2006 John Wiley & Sons, Ltd.     

Goal-oriented <Emphasis Type=Italic>h</Emphasis>-adaptivity for the Helmholtz equation: error estimates,local indicators and refinement strategies

This paper introduces a new goal-oriented adaptive technique based on a simple and effective post-process of the finite element approximations. The goal-oriented character of the estimate is achieved by analyzing both the direct problem and an auxiliary problem, denoted as adjoint or dual problem, which is related to the quantity of interest. Thus, the error estimation technique proposed in this paper would fall into the category of recovery-type explicit residual a posteriori error estimates. The procedure is valid for general linear quantities of interest and it is also extended to non-linear ones. The numerical examples demonstrate the efficiency of the proposed approach and discuss: (1) different error representations, (2) assessment of the dispersion error, and (3) different remeshing criteria.     

An efficient coarse‐grained parallel algorithm for global–local multiscale computations on massively parallel systems

The existing global–local multiscale computational methods, using finite element discretization at both the macro‐scale and micro‐scale, are intensive both in terms of computational time and memory requirements and their parallelization using domain decomposition methods incur substantial communication overhead, limiting their application. We are interested in a class of explicit global–local multiscale methods whose architecture significantly reduces this communication overhead on massively parallel machines. However, a naïve task decomposition based on distributing individual macro‐scale integration points to a single group of processors is not optimal and leads to communication overheads and idling of processors. To overcome this problem, we have developed a novel coarse‐grained parallel algorithm in which groups of macro‐scale integration points are distributed to a layer of processors. Each processor in this layer communicates locally with a group of processors that are responsible for the micro‐scale computations. The overlapping groups of processors are shown to achieve optimal concurrency at significantly reduced communication overhead. Several example problems are presented to demonstrate the efficiency of the proposed algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

An improved nonintrusive global–local approach for sharp thermal gradients in a standard FEA platform

Several classes of important engineering problems—in this case, problems exhibiting sharp thermal gradients—have solution features spanning multiple spatial scales and, therefore, necessitate advanced hp finite element discretizations. Although hp‐FEM is unavailable off‐the‐shelf in many predominant commercial analysis software packages, the authors herein propose a novel method to introduce these capabilities via a generalized FEM nonintrusively in a standard finite element analysis (FEA) platform. The methodology is demonstrated on two verification problems as well as a representative, industrial‐scale problem. Numerical results show that the techniques utilized allow for accurate resolution of localized thermal features on structural‐scale meshes without hp‐adaptivity or the ability to account for complex and very localized loads in the FEA code itself. This methodology enables the user to take advantage of all the benefits of both hp‐FEM discretizations and the appealing features of many available computer‐aided engineering /FEA software packages to obtain optimal convergence for challenging multiscale problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Stabilized global–local X‐FEM for 3D non‐planar frictional crack using relevant meshes

A stabilized global–local quasi‐static contact algorithm for 3D non‐planar frictional crack is presented in the X‐FEM/level set framework. A three‐field weak formulation is considered and allows an independent discretization of the bulk and the crack interface. Then, a fine discretization of the interface can be defined according to the possible complex contact state along the crack faces independently from the mesh in the bulk. Furthermore, an efficient stabilized non‐linear LATIN solver dedicated to contact and friction is proposed. It allows solving in a unified framework frictionless and frictional contact at the crack interface with a symmetric formulation, no iterations on the local stage (unilateral contact law with/without friction), no calculation of any global tangent operator, and improved convergence rate. 2D and 3D patch tests are presented to illustrate the relevance of the proposed model and an actual 3D frictional crack problem under cyclic fretting loading is modeled. Copyright © 2011 John Wiley & Sons, Ltd.     

Refined global–local higher-order theory for angle-ply laminated plates under thermo-mechanical loads and finite element model

To analyze angle-ply laminated composite and sandwich plates coupled bending and extension under thermo-mechanical loading, a refined global–local higher-order theory considering transverse normal strain is presented in this work. Hitherto, present theory for angle-ply laminates has never been reported in the literature, and this theory can satisfy continuity of transverse shear stresses at interfaces. In addition, the number of unknowns in present model is independent of layer numbers of the laminate. Based on this theory as well as methodology of the refined triangular discrete Kirchhoff plate element, a triangular laminated plate element satisfying the requirement of C¹ continuity is presented. Numerical results show that the present refined theory can accurately analyze the bending problems of angle-ply composite and sandwich plates as well as thermal expansion problem of cross-ply plates, and the present refined theory is obviously superior to the existing global–local higher-order theory proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212]. After ascertaining the accuracy of present model, the distributions of displacements and stresses for angle-ply laminated plates under temperature loads are also given in present work. These results can serve as a reference for future investigations.     

Mechanically based models: Adaptive refinement for B‐spline finite element

This article presents two new methods for adaptive refinement of a B‐spline finite element solution within an integrated mechanically based computer aided engineering system. The proposed techniques for adaptively refining a B‐spline finite element solution are a local variant of np‐refinement and a local variant of h‐refinement. The key component in the np‐refinement is the linear co‐ordinate transformation introduced into the refined element. The transformation is constructed in such a way that the transformed nodal configuration of the refined element is identical to the nodal configuration of the neighbour elements. Therefore, the assembly proceeds as with classic finite elements, while the solution approximation conforms exactly along the inter‐element boundaries. For the h‐refinement, this transformation is introduced into a construction that merges the super element from the finite element world with the hierarchical B‐spline representation from the computational geometry. In the scope of developing sculptured surfaces, the proposed approach supports C⁰ as well as the Hermite B‐spline C¹ continuous shapes. For sculptured solids, C⁰ continuity only is considered in this article. The feasibility of the proposed methods in the scope of the geometric design is demonstrated by several examples of creating sculptured surfaces and volumetric solids. Numerical performance of the methods is demonstrated for a test case of the two‐dimensional Poisson equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Adaptive POD‐based low‐dimensional modeling supported by residual estimates

An adaptive low‐dimensional model is considered to simulate time‐dependent dynamics in nonlinear dissipative systems governed by PDEs. The method combines an inexpensive POD‐based Galerkin system with short runs of a standard numerical solver that provides the snapshots necessary to first construct and then update the POD modes. Switching between the numerical solver and the Galerkin system is decided ‘on the fly’ by monitoring (i) a truncation error estimate and (ii) a residual estimate. The latter estimate is used to control the mode truncation instability and highly improves former adaptive strategies that detected this instability by monitoring consistency with a second instrumental Galerkin system based on a larger number of POD modes. The most computationally expensive run of the numerical solver occurs at the outset, when the whole set of POD modes is calculated. This step is improved by using mode libraries, which may either be generic or result from former applications of the method. The outcome is a flexible, robust, computationally inexpensive procedure that adapts itself to the local dynamics by using the faster Galerkin system for the majority of the time and few, on demand, short runs of a numerical solver. The method is illustrated considering the complex Ginzburg–Landau equation in one and two space dimensions. Copyright © 2015 John Wiley & Sons, Ltd.     

An error estimator for adaptive mesh refinement analysis based on strain energy equalisation

The two most widely used error estimators for adaptive mesh refinement are discussed and developed in the context of non-linear elliptic problems. The first is based on the work of Babuska and Rheinboldt (1978) where the error norm is a function of the residual and the inter-element discontinuity of the stress field. The discontinuous stress field arises in the Finite Element formulation where C ₀ continuity of the velocity field is assumed. The second error estimator is based on the work of Zienkiewicz and Zhu (1987). This method also uses the discontinuous stress field to measure the error, but results in a more simplified expression for the error norm. In fact, the equivalence between the two error norms has been shown by Zienkiewicz. Finally, an error estimator which is based on the approximation velocity space only is proposed. This estimator has the advantage in that it does not require the a posteriori calculation of the pressure (or stress) field. The method is applied to non-Newtonian Stokes flow which has a similar formulation to non-linear elasticity problems.     

High‐order global–regional model interaction: Extension of Carpenter's scheme

A two‐dimensional global–regional model interaction problem for linear time‐dependent waves is considered. The setup, which is sometimes called ‘one‐way nesting,’ arises in numerical weather prediction as well as in other fields concerning waves in very large domains. It involves the interaction of a coarse global model and a fine limited‐area (regional) model through an ‘open boundary.’ The multiscale nature of this general problem is described. The Carpenter scheme, originally proposed in a note by K. M. Carpenter in 1982 for this type of problem, is then revisited, in the context of the linear scalar wave equation. The original Carpenter scheme is based on the Sommerfeld radiation operator and thus is associated with low‐order accuracy. By replacing the Sommerfeld operator with the high‐order Hagstrom–Warburton absorbing operator, a modified Carpenter open‐boundary condition emerges, which possesses high‐order accuracy. This boundary condition is incorporated in a computational scheme, which uses finite element discretization in space and Newmark time‐stepping. Error analysis and numerical tests for wave guides demonstrate the performance of the modified scheme for combinations of incoming and outgoing waves. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive mesh refinement of the boundary element method for potential problems by using mesh sensitivities as error indicators

This paper presents a novel method for error estimation and h-version adaptive mesh refinement for potential problems which are solved by the boundary element method (BEM). Special sensitivities, denoted as mesh sensitivities, are used to evaluate a posteriori error indicators for each element, and a global error estimator. A mesh sensitivity is the sensitivity of a physical quantity at a boundary node with respect to perturbation of the mesh. The element error indicators for all the elements can be evaluated from these mesh sensitivities. Mesh refinement can then be performed by using these element error indicators as guides.The method presented here is suitable for both potential and elastostatics problems, and can be applied for adaptive mesh refinement with either linear or quadratic boundary elements. For potential problems, the physical quantities are potential and/or flux; for elastostatics problems, the physical quantities are tractions/displacements (or tangential derivatives of displacements). In this paper, the focus is on potential problems with linear elements, and the proposed method is validated with two illustrative examples. However, it is easy to extend these ideas to elastostatics problems and to quadratic elements.The computing for this research has been supported by the Cornell National Supercomputer Facility.     

Adaptive isogeometric analysis based on a combined r-h strategy

In the present work, an r-h adaptive isogeometric analysis is proposed for plane elasticity problems. For performing the r-adaption, the control net is considered to be a network of springs with the individual spring stiffness values being proportional to the error estimated at the control points. While preserving the boundary control points, relocation of only the interior control points is made by adopting a successive relaxation approach to achieve the equilibrium of spring system. To suit the noninterpolatory nature of the isogeometric approximation, a new point-wise error estimate for the h-refinement is proposed. To evaluate the point-wise error, hierarchical B-spline functions in Sobolev spaces are considered. The proposed adaptive h-refinement strategy is based on using De-Casteljau’s algorithm for obtaining the new control points. The subsequent control meshes are thus obtained by using a recursive subdivision of reference control mesh. Such a strategy ensures that the control points lie in the physical domain in subsequent refinements, thus making the physical mesh to exactly interpolate the control mesh and thereby allowing the exact imposition of essential boundary conditions in the classical isogeometric analysis (IGA). The combined r-h adaptive refinement strategy results in better convergence characteristics with reduced errors than r- or h-refinement. Several numerical examples are presented to illustrate the efficiency of the proposed approach.     

Adaptive training of local reduced bases for unsteady incompressible Navier–Stokes flows

This report presents a numerical study of reduced‐order representations for simulating incompressible Navier–Stokes flows over a range of physical parameters. The reduced‐order representations combine ideas of approximation for nonlinear terms, of local bases, and of least‐squares residual minimization. To construct the local bases, temporal snapshots for different physical configurations are collected automatically until an error indicator is reduced below a user‐specified tolerance. An adaptive time‐integration scheme is also employed to accelerate the generation of snapshots as well as the simulations with the reduced‐order representations. The accuracy and efficiency of the different representations is compared with examples with parameter sweeps. Copyright © 2015 John Wiley & Sons, Ltd.