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The present paper focuses on the discretization and implementation of the Thick Level Set (TLS) approach for quasi-brittle failure when dealing with both local and non-local development of damage. The TLS damage model introduces a length scale and nonlocality in the Fracture Process Zone (FPZ) and it handles the continuous to discontinuous failure transition. The added nonlocality is limited to the FPZ and therefore the model is local away from the FPZ or before its emergence. It leads to concurrent development of local and nonlocal damage during simulations. This paper presents a space–time discretization that handles concurrent computations in a unified manner. It is based on an explicit time discretization written as a predictor–corrector scheme and a space discretization that uses specific approximation functions (modes) that embed nonlocality. It also uses an alternative implementation of the eXtended Finite Element Method (X-FEM) enrichment used in the TLS which is based on virtual nodes. [ABSTRACT FROM AUTHOR]

This paper presents a new isogeometric topology optimization (TO) method based on moving morphable components (MMC), where the R-functions are used to represent the topology description functions (TDF) to overcome the C 1 discontinuity problem of the overlapping regions of components. Three new ersatz material models based on uniform, Gauss and Greville abscissae collocation schemes are used to represent both the Young’s modulus of material and the density field based on the Heaviside values of collocation points. Three benchmark examples are tested to evaluate the proposed method, where the collocation schemes are compared as well as the difference between isogeometric analysis (IGA) and finite element method (FEM). The results show that the convergence rate using R-functions has been improved in a range of 17%–60% for different cases in both FEM and IGA frameworks, and the Greville collocation scheme outperforms the other two schemes in the MMC-based TO. [ABSTRACT FROM AUTHOR]

This paper presents a hybrid finite-element/boundary-element method for fluid–structure-interaction simulations of inflatable structures. The flow model consists of the steady Stokes equation, which admits a boundary-integral formulation. The structure is represented by a Kirchhoff–Love shell. The boundary-element approximation of the Stokes equation reduces the flow problem to an integral equation on the actual structure configuration, thus obviating the need for volumetric meshing of the strongly deforming fluid domain. The Stokes model moreover exhibits a lubrication effect that acts as an intrinsic mechanism to treat the ubiquitous self-contact that occurs in inflation problems. The aggregated fluid–structure-interaction problem, composed of the boundary-integral equation and the Kirchhoff–Love shell connected by dynamic and kinematic interface conditions, is approximated by means of isogeometric discretizations to accommodate the smoothness requirements on the approximation spaces imposed by the flexural rigidity in the Kirchhoff–Love shell and to provide an accurate and smooth representation of the boundary for the boundary-element method. Auxiliary results presented in this paper are: (1) a parametrization-free Kirchhoff–Love formulation; (2) establishment of a cubic relationship between distance and tractions due to the lubrication effect; and (3) the interpretation of the Lagrange multiplier pertaining to fluid incompressibility as the total excess pressure. [ABSTRACT FROM AUTHOR]

A stable basis for isogeometric analysis of trimmed models is obtained by combining extended B-splines with truncated hierarchical B-splines. While extended B-splines guarantee that the condition number of system matrices is independent of the location of a trimming curve, local refinement is used to improve the robustness of the procedure and the accuracy of the numerical results. The present extended B-spline construction works in the context of Galerkin and collocation methods. The paper focuses on the latter and introduces a new collocation scheme for truncated hierarchical B-splines. A proper transition between refinement levels is assured by a novel balancing algorithm that employs a simple criterion. The enhanced performance of the locally refined stabilization is verified by scalar Laplace and linear elasticity problems analyzed by a collocation based isogeometric boundary element method. The proposed approach yields excellent results and requires few refinement levels to improve the stabilization procedure and accuracy along trimming curves. [ABSTRACT FROM AUTHOR]

The main aim of the current paper is to propose a new truly meshless numerical technique to solve the one- and two-dimensional elliptic interface problems. The employed meshless approach is based on a new class of MLS approximation. The numerical procedure is based on the interpolating stabilized MLS (ISMLS) approximation. The new shape functions that have been made by the ISMLS technique, have the δ -Kronecker property thus the Dirichlet boundary conditions can be applied, directly. In the current investigation, we propose a meshless collocation method for solving elliptic interface problems based on the shape functions of ISMLS technique. Several test problems with sufficient complexity have been studied to check the accuracy and efficiency of the new numerical procedure. Moreover, test problems show the acceptable accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

Discontinuous Galerkin with finite difference rules (DGFD) is applied to mechanical plane stress state problem. The considered domain is discretized by polygonal mesh. The polygonal elements can be for example a hexagon, pentagon or just quadrangle or triangle. They do not have to be convex and a fish mesh, where the elements have fish shapes, is used. When the elements are rectangular then the orthogonality of Chebyshev basis functions can be utilized. In such a case very high-order approximate solution can be obtained. In this work the approximation order exceeds 10 and reaches 60, which in the latter case means 3600 numbers of degrees of freedom in a single element. The paper is illustrated by a benchmark example in which the exact solution is recovered by DGFD method for various meshes. In the other example the stress concentration is easily recovered by very high-order version of DGFD method. [ABSTRACT FROM AUTHOR]

Abstract: This is the second paper out of a series of papers devoted to model- and hpq-adaptive finite element methods assigned for the modeling and analysis of elastic structures of complex mechanical description. In our previous publication we investigated the issue of hierarchical models and approximations of such structures. We applied 3D or 3D-based mechanical models, hierarchical modeling, and hierarchical approximations within the proposed finite element formulation. Furthermore, we assumed that the mechanical model and discretization parameters (such as: the size h of the element, and the longitudinal and transverse approximation orders, p and q) could vary locally, i.e. they could be different in each finite element. The a posteriori error estimation discussed in the present paper is based on the generalization of the residual equilibration method on the models with internal constraints. The generalized method is applied to the assessment of the total and approximation errors, while the modeling error is calculated as the difference between the former two errors. The corresponding error-controlled adaptive procedures are based on a three-step strategy, with possible iterations of h- and p-steps. [Copyright &y& Elsevier]

In Cai and Zhang (2009), we introduced and analyzed an improved Zienkiewicz–Zhu (ZZ) estimator for the conforming linear finite element approximation to elliptic interface problems. The estimator is based on the piecewise “constant” flux recovery in the H ( div ; Ω ) conforming finite element space. This paper extends the results of Cai and Zhang (2009) to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear H ( div ) space. [ABSTRACT FROM AUTHOR]

This paper introduces a new computational methodology for determining a-posteriori multi-objective error estimates for finite-element approximations, and for constructing corresponding (quasi-)optimal adaptive refinements of finite-element spaces. As opposed to the classical goal-oriented approaches, which consider only a single objective functional, the presented methodology applies to general closed convex subsets of the dual space and constructs a worst-case error estimate of the finite-element approximation error. This worst-case multi-objective error estimate conforms to a dual-weighted residual, in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error. We regard both standard approximation errors and data-incompatibility errors associated with incompatibility of boundary data with the trace of the finite-element space. Numerical experiments are presented to demonstrate the efficacy of applying the proposed worst-case multi-objective error estimate in adaptive refinement procedures. [ABSTRACT FROM AUTHOR]

In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement–pressure) and the three-field (stress–displacement–pressure) formulations. The method presented is based on a subgrid scale concept, and depends on the approximation of the unresolvable scales of the continuous solution. In general, subgrid scale techniques consist in the addition of a residual based term to the basic Galerkin formulation. The application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient. As a distinguished feature of the present study, we take the space of the unresolved subscales orthogonal to the finite element space, which promises a remedy to the above mentioned complication. In essence, we put forward that only if the orthogonal projection is used, the residual is simplified and the use of term by term stabilization is allowed. Thus, we do not need to put the whole residual in the formulation, and the linear eigenproblem form is recovered properly. We prove that the method applied is convergent, and present the error estimates for the eigenvalues and the eigenfunctions. We report several numerical tests in order to illustrate that the theoretical results are validated. [ABSTRACT FROM AUTHOR]

Abstract: A known problem of partition of unity-based generalized finite element methods is the linear dependence of the approximation space, which leads to singular stiffness matrix. Up to now, the linear dependence problem has not been fully understood and an efficient way to alleviate it is not available. In our previous paper “Prediction of rank deficiency in partition of unity-based methods with plane triangular or quadrilateral meshes” [Comput. Methods Appl. Mech. Engrg. 200 (2011) 665–674], the origin of the linear dependence problem was first dissected and then a method was proposed to reliably predict the rank deficiency of the linearly dependent global approximations of two-dimensional partition of unity-based generalized finite element methods. This paper extends the previous work to three-dimensional cases. The linear dependence problem is first investigated at an element level and then extended to the whole mesh. Derivation of general formulations in a three-dimensional setting is undoubtedly more challenging than a two-dimensional setting because of the complicated element topology. The principle of the increase of rank deficiency is once more applied. The methodology of summing up the added rank deficiency of each element as that of the whole mesh is further proved to be valid in three-dimensional cases. This work together with the previous work is regarded as the essential step to successfully and completely solve the linear dependence problem in partition of unity-based finite element methods. [Copyright &y& Elsevier]

Abstract: This paper establishes the superconvergence and the related recovery type a posteriori error estimators based on projection method for finite element approximation of the elliptic eigenvalue problems. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares method. The results are based on some regularity assumption for the elliptic problem, and are applicable to the finite element approximations of self-adjoint elliptic eigenvalue problems with general quasi-regular partitions. Therefore, the result of this paper can be employed to provide useful a posteriori error estimators in adaptive finite element computation under unstructured meshes. [Copyright &y& Elsevier]

Abstract: This paper considers the formulation of finite elements for thin Kirchhoff rods in the three-dimensional case, extending our work on plane rods presented in . The new issue to be addressed, besides the basic consideration of the three-dimensional geometry, is the approximation of the rod’s torsion, still in the geometrically linear setting of infinitesimal strains. In particular, it is the goal of this paper to identify approximations that preserve the basic invariance properties of the underlying structural theory, in the sense that the fundamental rigid-body modes of translations and infinitesimal rotations are represented exactly without straining, accounting fully for the rod’s curvature and associated Kirchhoff’s kinematics in contrast to common straight (framework) element approximations based on the simplified Euler–Bernoulli beam kinematics. To this purpose, different basic and new interpolations of the angle of twist rotation as well as interpolations of directors increments in the setting provided by a Cosserat treatment of the rod’s kinematics are explored and studied in detail, including rigorous analytical characterizations of the aforementioned rigid-body modes. These different interpolations are combined with the new C 1-continuous Hermitian interpolation developed in that reference for the axis displacements, extended to general three-dimensionally curved rods, as well as with an assumed strain mixed treatment to improve the overall performance of the finite elements. Several representative numerical simulations are presented to illustrate the properties and the numerical performance of the different new finite elements. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with meshfree particle methods for the solutions of the classical plate model. The vertical displacement of a thin plate is governed by a fourth order elliptic equation and thus the approximation functions for numerical solutions are required to have continuous partial derivatives. Hence, the conventional finite element method has difficulties to solve the fourth order problems. Meshfree methods have the advantage of constructing smooth approximation functions, however, most of the earlier works on meshfree methods for plate problems used either moving least squares method with penalty method or coupling FEM with meshfree method to deal with essential boundary conditions. In this paper, by using generalized product partition of unity, introduced by Oh et al., we introduce meshfree particle methods in which approximation functions have high order polynomial reproducing property and the Kronecker delta property. We also prove error estimates for the proposed meshfree methods. Moreover, to demonstrate the effectiveness of our method, results of the proposed method are compared with existing results for various shapes of plates with variety of boundary conditions and loads. [Copyright &y& Elsevier]

Abstract: This paper presents a residual-based turbulence model for the incompressible Navier–Stokes equations. The method is derived employing the variational multiscale (VMS) framework. A multiscale decomposition of the continuous solution and a priori unique decomposition of the admissible spaces of functions lead to two coupled nonlinear problems termed as the coarse-scale and the fine-scale sub-problems. The fine-scale velocity field is assumed to be nonlinear and time-dependent and is modeled via the bubble functions approach applied directly to the fine-scale sub-problem. A significant contribution in this paper is a systematic and consistent derivation of the fine-scale variational operator, commonly termed as the stabilization tensor that possesses the right order in the advective and diffusive limits, and variationally projects the fine-scale solution onto the coarse-scale space. A direct treatment of the fine-scale problem via bubble functions offers several fine-scale approximation options with varying degrees of mathematical sophistication that are investigated via benchmark problems. Numerical accuracy of the proposed method is shown on a forced-isotropic turbulence problem, statistically stationary turbulent channel flow problems at Re T =395 and 590, and non-equilibrium turbulent flow around a cylinder at Re =3,900. [Copyright &y& Elsevier]

Abstract: This is a paper out of a series of three papers devoted to model- and hpq-adaptive finite element methods assigned for modeling and analysis of complex structures. In this paper we focus our attention on hierarchical models and hierarchical approximations, while the issues of error estimation and adaptivity control will be presented in the next papers of the series. In all the papers we consider the elastic structures of complex mechanical description. We apply 3D or 3D-based mechanical models, hierarchical modeling, and hierarchical approximations within our finite element formulation. In the proposed approach, the mechanical model and discretization parameters, such as the size h of the element, and the longitudinal and transverse approximation orders, p and q, can vary locally, i.e. they can be different in each finite element. The a posteriori error estimation is based on the residual equilibration methods applied to the assessment of the modeling and approximation errors. The error-controlled adaptive procedures are derived from a four-step strategy, with possible iterations on h- and p-steps. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, heat transfer problems exhibiting sharp thermal gradients are analyzed using the classical and generalized finite element methods. The effect of solution roughness on the ability of the methods to obtain accurate approximations is investigated. Convergence studies show that low order (linear and quadratic) elements require strongly refined meshes for acceptable accuracy. We propose a generalized FEM with global–local enrichments for the class of problems investigated in the paper. In this procedure, a global solution space defined on a coarse mesh is enriched through the partition of unity framework of the generalized FEM with solutions of local boundary value problems. The local problems are defined using the same procedure as in the global–local FEM, where boundary conditions are provided by a coarse scale global solution. Coarse, uniform, global meshes are acceptable even at regions with thermal spikes that are orders of magnitude smaller than the element size. Convergence on these discretizations was achieved even when no or limited convergence was observed in the local problems. Two approaches are proposed to improve the boundary conditions prescribed on local problems and their convergence. The use of the corresponding improved local solutions as enrichments for the global problem extends the range of target error level for the enriched global problem. The two-way information transfer provided by the proposed generalized FEM is appealing to several classes of problems, especially those involving multiple spatial scales. The proposed methodology brings the benefits of generalized FEM to problems where limited or no information about the solution is known a priori. [Copyright &y& Elsevier]

Abstract: We begin the mathematical study of the k-method utilizing the theory of Kolmogorov n-widths. The k-method is a finite element technique where spline basis functions of higher-order continuity are employed. It is a fundamental feature of the new field of isogeometric analysis. In previous works, it has been shown that using the k-method has many advantages over the classical finite element method in application areas such as structural dynamics, wave propagation, and turbulence. The Kolmogorov n-width and sup–inf were introduced as tools to assess the effectiveness of approximating functions. In this paper, we investigate the approximation properties of the k-method with these tools. Following a review of theoretical results, we conduct a numerical study in which we compute the n-width and sup–inf for a number of one-dimensional cases. This study sheds further light on the approximation properties of the k-method. We finish this paper with a comparison study of the k-method and the classical finite element method and an analysis of the robustness of polynomial approximation. [Copyright &y& Elsevier]

Abstract: A spatial stabilization via bubble functions of linear finite element methods for nonlinear evolutionary convection–diffusion equations is discussed. The method of lines with SUPG discretization in space leads to numerical schemes that are not only difficult to implement, when considering nonlinear evolutionary equations, but also do not produce satisfactory results. The method we propose can be seen as an alternative to this kind of methods. Once the numerical approximation belonging to a linear finite element space enriched with bubble functions is obtained, the bubble part is discarded. The linear part is shown to give a stabilized approximation to the solution being approached. The bubble functions are deduced using a linear steady convection–diffusion model in such a way that the linear part of the approximation to the linear steady model (after static condensation of bubbles) reproduces the SUPG method in the convection-dominated regime. However, for the nonlinear evolutionary equations we consider in the paper the method we propose is not equivalent to the SUPG method. Some numerical experiments are provided in the paper to show the efficiency of the procedure. [Copyright &y& Elsevier]

The paper presents investigations on the performance of the finite element numerical integration algorithm for first order approximations and three processor architectures, popular in scientific computing, classical x86_64 CPU, Intel Xeon Phi and NVIDIA Kepler GPU. We base the discussion on theoretical performance models and our own implementations for which we perform a range of computational experiments. For the latter, we consider a unifying programming model and portable OpenCL implementation for all architectures. Variations of the algorithm due to different problems solved and different element types are investigated and several optimizations aimed at proper optimization and mapping of the algorithm to computer architectures are demonstrated. The experimental results show the varying levels of performance for different architectures, but indicate that the algorithm can be effectively ported to all of them. The conclusions indicate the factors that limit the performance for different problems and types of approximation and the performance ranges that can be expected for FEM numerical integration on different processor architectures. [ABSTRACT FROM AUTHOR]

This paper introduces a new recovery-based a posteriori error estimator for the lowest order Nédélec finite element approximation to the H ( curl ) interface problem. The error estimator is analyzed by establishing both the reliability and the efficiency bounds and is supported by numerical results. Under certain assumptions, it is proved that the reliability and efficiency constants are independent of the jumps of the coefficients. [ABSTRACT FROM AUTHOR]

In this paper we analyze time marching schemes for the wave equation in mixed form. The problem is discretized in space using stabilized finite elements. On the one hand, stability and convergence analyses of the fully discrete numerical schemes are presented using different time integration schemes and appropriate functional settings. On the other hand, we use Fourier techniques (also known as von Neumann analysis) in order to analyze stability, dispersion and dissipation. Numerical convergence tests are presented for various time integration schemes, polynomial interpolations (for the spatial discretization), stabilization methods, and variational forms. To analyze the behavior of the different schemes considered, a 1D wave propagation problem is solved. [ABSTRACT FROM AUTHOR]

In the present paper we study a finite element method for the incompressible Stokes problem with a boundary immersed in the domain on which essential constraints are imposed. Such type of methods may be useful to tackle problems with complex geometries, interfaces such as multiphase flow and fluid–structure interaction. The method we study herein consists in locally refining elements crossed by the immersed boundary such that newly added elements, called subelements, fit the immersed boundary. In this sense, this approach is of a fitted type, but with an original mesh given independently of the location of the immersed boundary. We use such a subdivision technique to build a new finite element basis, which enables us to represent accurately the immersed boundary and to impose strongly Dirichlet boundary conditions on it. However, the subdivision process may imply the generation of anisotropic elements, which, for the incompressible Stokes problem, may result in the loss of inf–sup stability even for well-known stable element schemes. We therefore use a finite element approximation, which appears stable also on anisotropic elements. We perform numerical tests to check stability of the chosen finite elements. Several numerical experiments are finally presented to illustrate the capabilities of the method. The method is presented for two-dimensional problems. [ABSTRACT FROM AUTHOR]

We propose an approach for goal-oriented error estimation in finite element approximations of second-order elliptic problems that combines the dual-weighted residual method and equilibrated-flux reconstruction methods for the primal and dual problems. The objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution. It is only assumed here that the discretization methods come with a priori error estimates and an equilibrated-flux reconstruction algorithm. A high-order discontinuous Galerkin (dG) method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes. One contribution of the paper is to show how the order of the dG method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method, a dG method, or a mixed Raviart–Thomas method is used for the solution of the primal problem. Numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size. [ABSTRACT FROM AUTHOR]

This article is a review of our work towards a parameter-free method for simulation of turbulent flow at high Reynolds numbers. In a series of papers we have developed a model for turbulent flow in the form of weak solutions of the Navier–Stokes equations, approximated by an adaptive finite element method, where: (i) viscous dissipation is assumed to be dominated by turbulent dissipation proportional to the residual of the equations, and (ii) skin friction at solid walls is assumed to be negligible compared to inertial effects. The result is a computational model without empirical data, where the only model parameter is the local size of the finite element mesh. Under adaptive refinement of the mesh based on a posteriori error estimation, output quantities of interest in the form of functionals of the finite element solution converge to become independent of the mesh resolution, and thus the resulting method has no adjustable parameters. No ad hoc design of the mesh is needed, instead the mesh is optimized based on solution features, in particular no boundary layer mesh is needed. We connect the computational method to the mathematical concept of a dissipative weak solution of the Euler equations, as a model of high Reynolds number turbulent flow, and we highlight a number of benchmark problems for which the method is validated. The purpose of the article is to present the computational framework in a concise form, to report on recent progress, and to discuss open problems that are subject to ongoing research. [ABSTRACT FROM AUTHOR]

We present a fully computable a posteriori error estimator for piecewise linear finite element approximations of reaction–diffusion problems with mixed boundary conditions and piecewise constant reaction coefficient formulated in arbitrary dimension. The estimator provides a guaranteed upper bound on the energy norm of the error and it is robust for all values of the reaction coefficient, including the singularly perturbed case. The approach is based on robustly equilibrated boundary flux functions of Ainsworth and Oden (2000) and on subsequent robust and explicit flux reconstruction. This paper simplifies and extends the applicability of the previous result of Ainsworth and Vejchodský (2011) in three aspects: (i) arbitrary dimension, (ii) mixed boundary conditions, and (iii) non-constant reaction coefficient. It is the first robust upper bound on the error with these properties. An auxiliary result that is of independent interest is the derivation of new explicit constants for two types of trace inequalities on simplices. [ABSTRACT FROM AUTHOR]

In this paper, we present a new multiscale method which is capable of coupling atomistic and continuum domains for high frequency wave propagation analysis. The problem of non-physical wave reflection, which occurs due to the change in system description across the interface between two scales, can be satisfactorily overcome by the proposed method. We propose an efficient spectral domain decomposition of the total fine scale displacement along with a potent macroscale equation in the Laplace domain to eliminate the spurious interfacial reflection. We use Laplace transform based spectral finite element method to model the macroscale, which provides the optimum approximations for required dynamic responses of the outer atoms of the simulated microscale region very accurately. This new method shows excellent agreement between the proposed multiscale model and the full molecular dynamics (MD) results. Numerical experiments of wave propagation in a 1D harmonic lattice, a 1D lattice with Lennard-Jones potential, a 2D square Bravais lattice, and a 2D triangular lattice with microcrack demonstrate the accuracy and the robustness of the method. In addition, under certain conditions, this method can simulate complex dynamics of crystalline solids involving different spatial and/or temporal scales with sufficient accuracy and efficiency. [ABSTRACT FROM AUTHOR]

In this paper a finite difference scheme is proposed for multidimensional convection-diffusion-reaction equations, particularly designed to treat the most interesting case of small diffusion. It is based closely on the work Şendur and Neslitürk (2011). Application of the method to multidimensional convection-diffusion-reaction equation is based on a simple splitting of the convection-diffusion-reaction equation and then joining their approximations obtained with Şendur and Neslitürk (2011). The method adapts very well to all regimes with continuous transitions from one regime to another. Numerical tests show good performance of the method and superiority with respect to well known stabilized finite element methods. [ABSTRACT FROM AUTHOR]

Abstract: This paper introduces a first-order viscosity method for the explicit approximation of scalar conservation equations with Lipschitz fluxes using continuous finite elements on arbitrary grids in any space dimension. Provided the lumped mass matrix is positive definite, the method is shown to satisfy the local maximum principle under a usual CFL condition. The method is independent of the cell type; for instance, the mesh can be a combination of tetrahedra, hexahedra, and prisms in three space dimensions. [Copyright &y& Elsevier]

Abstract: In this paper, some new generalized Newton’s methods for the resolution of elastostatic frictional contact problem approximated by finite elements are presented and compared to existing ones. A numerical experimentation is performed to compare the different methods, especially with respect to the sensitivity to the method parameter. Two different strategies to approximate the contact and friction condition are considered: a nodal and an integral one. Existence and uniqueness results of the solution to the discretized problem are also discussed. [Copyright &y& Elsevier]

Abstract: In the present paper, a high-speed adaptive meshing using a hierarchical mesh is studied. We use the level-of-detail approximations of the mesh data structure used often in the computer graphics field. To achieve the correct level of detail, we employ the hierarchical regional partitions. Meshes using this data structure can be realized in a very quick manner. Finally, we apply the level of detail approximation to the adaptive analyses of crack propagation, demonstrating the efficiency of the present method. [Copyright &y& Elsevier]

Abstract: In this paper, we propose and analyze a new stabilized finite element method using continuous piecewise linear (or bilinear) elements for solving 2D reaction–convection–diffusion equations. The equation under consideration involves a small diffusivity and a large reaction coefficient , leading to high Péclet number and high Damköhler number. In addition to giving error estimates of the approximations in and norms, we explicitly establish the dependence of error bounds on the diffusivity, the norm of convection field, the reaction coefficient and the mesh size. Our analysis shows that the proposed method is particularly suitable for problems with a small diffusivity and a large reaction coefficient, or more precisely, with a large mesh Péclet number and a large mesh Damköhler number. Several numerical examples exhibiting boundary or interior layers are given to illustrate the high accuracy and stability of the proposed method. The results obtained are also compared with those of existing stabilization methods. [Copyright &y& Elsevier]

Abstract: We study the performance of direct solvers on linear systems of equations resulting from isogeometric analysis. The problem of choice is the canonical Laplace equation in three dimensions. From this study we conclude that for a fixed number of unknowns and polynomial degree of approximation, a higher degree of continuity k drastically increases the CPU time and RAM needed to solve the problem when using a direct solver. This paper presents numerical results detailing the phenomenon as well as a theoretical analysis that explains the underlying cause. [Copyright &y& Elsevier]

Abstract: In this paper we formulate and analyze two non-conforming high order strategies for the approximation of elastic wave problems in heterogeneous media, namely the Mortar Spectral Element Method and the Discontinuous Galerkin Spectral Element Method. Starting from a common variational formulation we make a full comparison of the two techniques from the points of view of accuracy, convergence, grid dispersion and stability. [Copyright &y& Elsevier]

Abstract: This paper presents a strain smoothing procedure for the extended finite element method (XFEM). The resulting “edge-based” smoothed extended finite element method (ESm-XFEM) is tailored to linear elastic fracture mechanics and, in this context, to outperform the standard XFEM. In the XFEM, the displacement-based approximation is enriched by the Heaviside and asymptotic crack tip functions using the framework of partition of unity. This eliminates the need for the mesh alignment with the crack and re-meshing, as the crack evolves. Edge-based smoothing (ES) relies on a generalized smoothing operation over smoothing domains associated with edges of simplex meshes, and produces a softening effect leading to a close-to-exact stiffness, “super-convergence” and “ultra-accurate” solutions. The present method takes advantage of both the ES-FEM and the XFEM. Thanks to the use of strain smoothing, the subdivision of elements intersected by discontinuities and of integrating the (singular) derivatives of the approximation functions is suppressed via transforming interior integration into boundary integration. Numerical examples show that the proposed method improves significantly the accuracy of stress intensity factors and achieves a near optimal convergence rate in the energy norm even without geometrical enrichment or blending correction. [Copyright &y& Elsevier]

Abstract: In this paper we address the numerical approximation of the incompressible Navier–Stokes equations in a moving domain by the spectral element method and high order time integrators. We present the Arbitrary Lagrangian Eulerian (ALE) formulation of the incompressible Navier–Stokes equations and propose a numerical method based on the following kernels: a Lagrange basis associated with Fekete points in the spectral element method context, BDF time integrators, an ALE map of high degree, and an algebraic linear solver. In particular, the high degree ALE map is appropriate to deal with a computational domain whose boundary is described with curved elements. Finally, we apply the proposed strategy to a test case. [Copyright &y& Elsevier]

Abstract: Generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries. The paper at hand introduces a B-spline finite element method, which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved. Its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement, which adds several levels of local basis functions in the close vicinity of interfaces, but unfitted to their exact location, so that a simple regular grid of knot span elements can be maintained. Numerical experiments show that an hp-refinement strategy, which simultaneously increases the polynomial degree of the B-spline basis and the levels of refinement around interfaces, achieves exponential rates of convergence despite the presence of discontinuities. It is also demonstrated that the hierarchical B-spline FEM can be used to transfer the recently introduced Finite Cell concept to geometrically nonlinear problems. Its computational performance, imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one, two and three dimensions, and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite. [Copyright &y& Elsevier]

Abstract: The paper presents a study of solution of inverse medium scattering problems for 3D time-harmonic electromagnetics. The solution is based on minimization of the misfit function between the observed scattered fields and trial solutions of direct problems for the sought approximate distribution of electric permittivity. The sources of illuminating waves and the observation points of the scattered fields are located in the nearfield the scatterer. The simulations of the direct problems are performed with the Finite Element Method. Application of adaptive mesh refinement to improve accuracy of solutions is illustrated. Tests with real- and complex-valued distributions of electric permittivity, and with complete and incomplete measurement data are presented. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the phase separation on general surfaces by solving the nonlinear Cahn–Hilliard equation using a finite element method. A fully discrete approximation scheme is introduced, and we establish a priori estimates for the discrete solution that does not rely on any knowledge of the exact solution beyond the initial time. This in turn leads to convergence and optimal error estimates of the discretization scheme. Numerical examples are also provided to substantiate the theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we study finite element approximations of a class of nonlinear eigenvalue problems arising from quantum physics. We derive both a priori and a posteriori finite element error estimates and obtain optimal convergence rates for both linear and quadratic finite element approximations. In particular, we analyze the convergence and complexity of an adaptive finite element method. In our analysis, we utilize certain relationship between the finite element eigenvalue problem and the associated finite element boundary value approximations. We also present several numerical examples in quantum physics that support our theory. [Copyright &y& Elsevier]

Abstract: In this paper we develop an a posteriori error analysis of a new fully mixed finite element method for the coupling of fluid flow with porous media flow in 2D. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We consider dual-mixed formulations in both media, which yields the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium, and the traces of the porous media pressure and the fluid velocity on the interface, as the resulting unknowns. The set of feasible finite element subspaces includes Raviart–Thomas elements of lowest order and piecewise constants for the velocities and pressures, respectively, in both domains, together with continuous piecewise linear elements for the traces. We derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. The proof of reliability makes use of the global inf–sup condition, Helmholtz decompositions in both media, and local approximation properties of the Clément interpolant and Raviart–Thomas operator. On the other hand, inverse inequalities, the localization technique based on element-bubble and edge-bubble functions, and known results from previous works, are the main tools for proving the efficiency of the estimator. Finally, some numerical results confirming the theoretical properties of this estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities of the solution, are reported. [Copyright &y& Elsevier]

Abstract: In this paper we construct an approximation that uses midpoints of edges on tetrahedra in three dimensions. The construction is based on the three-dimensional version of the rotated Q 1-approximation proposed by Rannacher and Turek (1992) . We prove a priori error estimates for finite element solutions of the elasticity equations using the new element. Since it contains (rotated) bilinear terms it performs substantially better than the standard constant strain element in bending. It also allows for under-integration (in the form of one point Gauss integration of volumetric terms) in near incompressible situations. Numerical examples are included. [Copyright &y& Elsevier]

Abstract: A formulation for the bending analysis of composite laminated plates with piezoelectric layers is implemented using the generalized finite element method. This formulation is derived from a mechanical description based on Higher-Order Shear Deformation Theory which allows for the use of C 0 continuous approximation functions on the domain. On the other hand, a Layerwise Theory is employed for interpolation of electric potential across the thickness of piezoelectric layers, in such a way that the kinematical hypotheses result in a mixed model. The paper presents an analysis of the approximation capability of the proposed numerical model for static analysis, using C 0 continuous Partition of Unity and polynomial enrichments to span the approximation spaces, by assessment of convergence. Analytical solutions obtained from the same kinematical hypotheses are used as references. Results for relative error in the energy norm considering p- and h-refinements for regular and distorted meshes, in addition to a pointwise evaluation of the stresses and electric field, are presented. The evaluations show that the numerical methodology is a very effective tool for improving the solution through the enrichment, even for pointwise values across the thickness, and is robust to mesh distortions. Moreover, the results furnish insight about the physical modeling for both active and sensory modes, for thick and thin plates. [ABSTRACT FROM AUTHOR]

Abstract: In the partition of unity (PU)-based methods, the global approximation is built by multiplying a partition of unity by local approximations. Within this framework, high-order approximations are achieved by directly adopting high-order polynomials as local approximations, and therefore nodes along sides or inside elements, which are usually adopted in the conventional finite element methods, are no more required. However, the PU-based approximation constructed in this way may suffer from rank deficiency due to the linear dependence of the global degrees of freedom. In this paper, the origin of the rank deficiency in the PU-based approximation space is first dissected at an element level, and then an approach to predict the rank deficiency for a mesh is proposed together with the principle of the increase of rank deficiency. Finally, examples are investigated to validate the present approach. The current work indicates such a fact that the rank deficiency is an unrelated issue to the nullity of the global matrix. It can be resolved in its own manner. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation. These methods are based on the decomposition of the unknowns into resolvable and subgrid scales, with an approximation for the latter that yields a problem to be solved for the former. This approach allows us to design the algorithmic parameters on which the method depends, which we do here when the discrete ordinates method is used for the directional approximation. We concentrate on two stabilized methods, namely, the classical SUPG technique and the orthogonal subscale stabilization. A numerical analysis of the spatial approximation for both formulations is performed, which shows that they have a similar behavior: they are both stable and optimally convergent in the same mesh-dependent norm. A comparison with the behavior of the Galerkin method, for which a non-standard numerical analysis is done, is also presented. [ABSTRACT FROM AUTHOR]

Abstract: We propose a new second-order cone linear complementarity problem (SOCLCP) formulation for the numerical finite element analysis of three-dimensional (3D) frictional contact problems by the parametric variational principle. Specifically, we develop a regularization technique to resolve the multi-valued difficulty involved in the frictional contact law, and use a second-order cone complementarity condition to handle the regularized Coulomb friction law in contact analysis. The governing equations of the 3D frictional contact problem is represented by an SOCLCP via the parametric variational principle and the finite element method, which avoids the polyhedral approximation to the Coulomb friction cone so that the problem to be solved has much smaller size and the solution has better accuracy. In this paper, we reformulate the SOCLCP as a semi-smooth system of equations via a one-parametric class of second-order cone complementarity functions, and then apply the non-smooth Newton method for solving this system. Numerical results confirm the effectiveness and robustness of the SOCLCP approach developed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we derive and analyze some a posteriori error estimators for the stabilized P1 nonconforming approximation of the Stokes problem involving the strain tensor. This will be done by decomposing the numerical error in a proper way into conforming and nonconforming contributions. The error estimator for the nonconforming error is obtained in the standard way, and the implicit error estimator for the conforming error is derived by applying the equilibrated residual method. A crucial part of this work is construction of approximate normal stresses on interelement boundaries which will serve as equilibrated Neumann data for local Stokes problems. It turns out that such normal stresses can be simply computed by local weak residuals of the discrete system plus jumps of the velocity solution and that a stronger equilibration condition is satisfied to ensure solvability of local Stokes problems. We also derive a simple explicit error estimator based on the nonsymmetric tensor recovery of the normal stress error. Numerical results are provided to illustrate the performance of our error estimators. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a simple idea based on midpoint integration rule is utilized to solve a particular class of mechanics problems; namely static problems defined on unbounded domains where the solution is required to be accurate only in an interior (and not in the far field). By developing a finite element mesh that approximates the stiffness of an unbounded domain directly (without approximating the far-field displacement profile first), the current formulation provides a superior alternative to infinite elements (IEs) that have long been used to incorporate unbounded domains into the finite element method (FEM). In contrast to most IEs, the present formulation (a) requires no new shape functions or special integration rules, (b) is proved to be both accurate and efficient, and (c) is versatile enough to handle a large variety of domains including those with anisotropic, stratified media and convex polygonal corners. In addition to this, the proposed model leads to the derivation of a simple error expression that provides an explicit correlation between the mesh parameters and the accuracy achieved. This error expression can be used to calculate the accuracy of a given mesh a-priori. This in-turn, allows one to generate the most efficient mesh capable of achieving a desired accuracy by solving a mesh optimization problem. We formulate such an optimization problem, solve it and use the results to develop a practical mesh generation methodology. This methodology does not require any additional computation on the part of the user, and can hence be used in practical situations to quickly generate an efficient and near optimal finite element mesh that models an unbounded domain to the required accuracy. Numerical examples involving practical problems are presented at the end to illustrate the effectiveness of this method. [Copyright &y& Elsevier]

Abstract: In this paper we study from the numerical point of view a strain-adaptive bone remodelling that couples the displacements and the apparent density (the porosity) of the bone. The rate of this density at a particular location is described as an objective function, which depends on a particular stimulus at that location. The variational problem is written as a coupled system of a nonlinear variational equation for the displacement field and a nonlinear parabolic variational inequality for the apparent density. Then, fully discrete approximations are provided by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are proved, from which, under adequate regularity conditions, the linear convergence of the algorithm is deduced. Finally, some numerical simulations involving one- and two-dimensional test examples are presented to demonstrate the accuracy of the approximation and the behaviour of the solution. [Copyright &y& Elsevier]

Abstract: In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. By applying some error estimates techniques of mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present a numerical example which confirms our theoretical results. [Copyright &y& Elsevier]