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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]

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]

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]

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]

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 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]