Authors:Christian Steinke; Imadeddin Zreid; Michael Kaliske Pages: 717 - 735 Abstract: Abstract The finite element implementation of a gradient enhanced microplane damage model is compared to a phase-field model for brittle fracture. Phase-field models and implicit gradient damage models share many similarities despite being conceived from very different standpoints. In both approaches, an additional differential equation and a length scale are introduced. However, while the phase-field method is formulated starting from the description of a crack in fracture mechanics, the gradient method starts from a continuum mechanics point of view. At first, the scope of application for both models is discussed to point out intersections. Then, the analysis of the employed mathematical methods and their rigorous comparison are presented. Finally, numerical examples are introduced to illustrate the findings of the comparison which are summarized in a conclusion at the end of the paper. PubDate: 2017-05-01 DOI: 10.1007/s00466-016-1369-9 Issue No:Vol. 59, No. 5 (2017)

Authors:Xue Zhang; Chet Vignes; Scott W. Sloan; Daichao Sheng Pages: 737 - 752 Abstract: Abstract The phase-field model has been attracting considerable attention due to its capability of capturing complex crack propagations without mesh dependence. However, its validation studies have primarily focused on the ability to predict reasonable, sharply defined crack paths. Very limited works have so far been contributed to estimate its accuracy in predicting force responses, which is majorly attributed to the difficulty in the determination of the length scale. Indeed, accurate crack path simulation can be achieved by setting the length scale to be sufficiently small, whereas a very small length scale may lead to unrealistic force-displacement responses and overestimate critical structural loads. This paper aims to provide a critical numerical investigation of the accuracy of phase-field modelling of brittle fracture with special emphasis on a possible formula for the length scale estimation. Phase-field simulations of a number of classical fracture experiments for brittle fracture in concretes are performed with simulated results compared with experimental data qualitatively and quantitatively to achieve this goal. Furthermore, discussions are conducted with the aim to provide guidelines for the application of the phase-field model. PubDate: 2017-05-01 DOI: 10.1007/s00466-017-1373-8 Issue No:Vol. 59, No. 5 (2017)

Authors:Martin Horák; David Ryckelynck; Samuel Forest Pages: 753 - 778 Abstract: Abstract This paper deals with the reduced order modeling of micromorphic continua. The reduced basis model relies on the proper orthogonal decomposition and the hyper-reduction. Two variants of creation of reduced bases using the proper orthogonal decomposition are explored from the perspective of additional micromorphic degrees of freedom. In the first approach, one snapshot matrix including displacement as well as micromorphic degrees of freedom is assembled. In the second approach, snapshots matrices are assembled separately for displacement and micromorphic fields and the singular value decomposition is performed on each system separately. Thereafter, the formulation is extended to the hyper-reduction method. It is shown that the formulation has the same structure as for the classical continua. The relation of higher order stresses introduced in micromorphic balance equations to creation of the reduced integration domain is examined. Finally, the method is applied to examples of microdilatation extension and clamped tension and to a size-dependent stress concentration in Cosserat elasticity. It is shown that the proposed approach leads to a good level of accuracy with significant reduction of computational time. PubDate: 2017-05-01 DOI: 10.1007/s00466-016-1371-2 Issue No:Vol. 59, No. 5 (2017)

Authors:Yan Gu; Hongwei Gao; Wen Chen; Huijuan Wang; Chuanzeng Zhang Pages: 779 - 793 Abstract: Abstract One of the typical and most significant issues of almost all boundary element analyses is the accurate evaluation of nearly singular boundary element integrals. In this study, we review some numerical techniques used currently to calculate nearly singular integrals and propose an improved algorithm to calculate integrals with nearly strong (and beyond) singularities. This new method has full generally and can be easily included in any existing computer code. The method is tested in general three-dimensional boundary element analysis. Comparison of this method with some of the existing methods is also presented. It is shown that several orders of magnitude improvement in relative errors can be obtained using the proposed method when compared to a straightforward implementation of Gaussian quadrature. PubDate: 2017-05-01 DOI: 10.1007/s00466-016-1372-1 Issue No:Vol. 59, No. 5 (2017)

Authors:B. Nedjar Pages: 795 - 807 Abstract: Abstract The first objective of this contribution is the formulation of nonlinear problems in magneto-elasticity involving finite geometry of the surrounding free space. More specifically for the magnetic part of the problem, the surrounding free space is described by means of a boundary integral equation for which boundary elements are used that are appropriately coupled with the finite element discretization used inside the material. The second objective is to develop a numerical strategy to solve the strongly coupled magneto-mechanics problem at hand. Herein we provide a staggered scheme consisting of a magnetostatic resolution employing the above coupled BEM-FEM procedure at fixed deformation, followed by a mechanical resolution at fixed magnetic fields. This decoupled method renders the whole solution strategy very appealing since, among others, the first BEM-FEM resolution is linear for some prototype models, and the remaining mechanical resolution is analogous to nowadays classical nonlinear elastostatic problems in the finite strain range. Some nonlinear boundary-value problems are simulated to demonstrate the applicability of the proposed framework. PubDate: 2017-05-01 DOI: 10.1007/s00466-016-1370-3 Issue No:Vol. 59, No. 5 (2017)

Authors:A. K. Gaonkar; S. S. Kulkarni Pages: 809 - 829 Abstract: Abstract In this paper a model order reduction technique for the dynamic simulation of beams undergoing large rotations is presented. The finite element model for the motion of such beams is based on the corotational formulation. The trajectory piecewise linear model order reduction (TPWLMOR) method with second order Krylov subspace is used to obtain the reduced order model from the finite element model. Improvements are suggested to improve the accuracy and the computational efficiency of the TPWLMOR model. Several numerical examples which include forced undamped and damped beams are presented to validate the proposed method. PubDate: 2017-05-01 DOI: 10.1007/s00466-017-1374-7 Issue No:Vol. 59, No. 5 (2017)

Authors:E. Martínez-Pañeda; S. Natarajan; S. Bordas Pages: 831 - 842 Abstract: Abstract Strain gradient plasticity theories are being widely used for fracture assessment, as they provide a richer description of crack tip fields by incorporating the influence of geometrically necessary dislocations. Characterizing the behavior at the small scales involved in crack tip deformation requires, however, the use of a very refined mesh within microns to the crack. In this work a novel and efficient gradient-enhanced numerical framework is developed by means of the extended finite element method (X-FEM). A mechanism-based gradient plasticity model is employed and the approximation of the displacement field is enriched with the stress singularity of the gradient-dominated solution. Results reveal that the proposed numerical methodology largely outperforms the standard finite element approach. The present work could have important implications on the use of microstructurally-motivated models in large scale applications. The non-linear X-FEM code developed in MATLAB can be downloaded from www.empaneda.com/codes. PubDate: 2017-05-01 DOI: 10.1007/s00466-017-1375-6 Issue No:Vol. 59, No. 5 (2017)

Authors:Baiyili Liu; Shaoqiang Tang; Jun Chen Pages: 843 - 859 Abstract: Abstract In this paper, we propose a heat jet approach for atomic simulations at finite temperature of a triangular lattice. First we design a matching boundary condition by carefully examining a residual function based on the lattice dispersion relation. It leads to a two-way boundary condition, where prescribed incoming waves are included with a source term. Meanwhile, we adopt a phonon representation to determine Fourier mode amplitudes. The heat jet approach is then formulated by combining the two-way boundary condition and the phonon representation of heat source. Numerical tests of a tube-shaped computational domain illustrate the accuracy and effectiveness in simultaneously resolving thermal fluctuations and non-thermal motion at a given temperature. PubDate: 2017-05-01 DOI: 10.1007/s00466-017-1376-5 Issue No:Vol. 59, No. 5 (2017)

Authors:Xuechuan Wang; Satya N. Atluri Pages: 861 - 876 Abstract: Abstract A new class of time-integrators is presented for strongly nonlinear dynamical systems. These algorithms are far superior to the currently common time integrators in computational efficiency and accuracy. These three algorithms are based on a local variational iteration method applied over a finite interval of time. By using Chebyshev polynomials as trial functions and Dirac–Delta functions as the test functions over the finite time interval, the three algorithms are developed into three different discrete time-integrators through the collocation method. These time integrators are labeled as Chebyshev local iterative collocation methods. Through examples of the forced Duffing oscillator, the Lorenz system, and the multiple coupled Duffing equations (which arise as semi-discrete equations for beams, plates and shells undergoing large deformations), it is shown that the new algorithms are far superior to the 4th order Runge–Kutta and ODE45 of MATLAB, in predicting the chaotic responses of strongly nonlinear dynamical systems. PubDate: 2017-05-01 DOI: 10.1007/s00466-017-1377-4 Issue No:Vol. 59, No. 5 (2017)

Authors:Alireza Abedian; Alexander Düster Pages: 877 - 886 Abstract: Abstract In the finite cell method, the fictitious domain approach is combined with high-order finite elements. The geometry of the problem is taken into account by integrating the finite cell formulation over the physical domain to obtain the corresponding stiffness matrix and load vector. In this contribution, an extension of the FCM is presented wherein both the physical and fictitious domain of an element are simultaneously evaluated during the integration. In the proposed extension of the finite cell method, the contribution of the stiffness matrix over the fictitious domain is subtracted from the cell, resulting in the desired stiffness matrix which reflects the contribution of the physical domain only. This method results in an exponential rate of convergence for porous domain problems with a smooth solution and accurate integration. In addition, it reduces the computational cost, especially when applying adaptive integration schemes based on the quadtree/octree. Based on 2D and 3D problems of linear elastostatics, numerical examples serve to demonstrate the efficiency and accuracy of the proposed method. PubDate: 2017-05-01 DOI: 10.1007/s00466-017-1378-3 Issue No:Vol. 59, No. 5 (2017)

Authors:Haemin Jeon; Jaesang Yu; Hunsu Lee; G. M. Kim; Jae Woo Kim; Yong Chae Jung; Cheol-Min Yang; B. J. Yang Abstract: Abstract Continuous fiber-reinforced composites are important materials that have the highest commercialized potential in the upcoming future among existing advanced materials. Despite their wide use and value, their theoretical mechanisms have not been fully established due to the complexity of the compositions and their unrevealed failure mechanisms. This study proposes an effective three-dimensional damage modeling of a fibrous composite by combining analytical micromechanics and evolutionary computation. The interface characteristics, debonding damage, and micro-cracks are considered to be the most influential factors on the toughness and failure behaviors of composites, and a constitutive equation considering these factors was explicitly derived in accordance with the micromechanics-based ensemble volume averaged method. The optimal set of various model parameters in the analytical model were found using modified evolutionary computation that considers human-induced error. The effectiveness of the proposed formulation was validated by comparing a series of numerical simulations with experimental data from available studies. PubDate: 2017-04-25 DOI: 10.1007/s00466-017-1398-z

Authors:S. Huang; Y. J. Liu Abstract: Abstract A new fast direct linear equation solver for the boundary element method (BEM) is presented in this paper. The idea of the new fast direct solver stems from the concept of the hierarchical off-diagonal low-rank matrix. The hierarchical off-diagonal low-rank matrix can be decomposed into the multiplication of several diagonal block matrices. The inverse of the hierarchical off-diagonal low-rank matrix can be calculated efficiently with the Sherman–Morrison–Woodbury formula. In this paper, a more general and efficient approach to approximate the coefficient matrix of the BEM with the hierarchical off-diagonal low-rank matrix is proposed. Compared to the current fast direct solver based on the hierarchical off-diagonal low-rank matrix, the proposed method is suitable for solving general 3-D boundary element models. Several numerical examples of 3-D potential problems with the total number of unknowns up to above 200,000 are presented. The results show that the new fast direct solver can be applied to solve large 3-D BEM models accurately and with better efficiency compared with the conventional BEM. PubDate: 2017-04-18 DOI: 10.1007/s00466-017-1407-2

Authors:Przemysław Litewka; Roman Lewandowski Abstract: Abstract The paper is devoted to non-linear vibrations of plates, made of the Zener viscoelastic material modelled with the Caputo fractional derivative, and in particular to their response to harmonic excitation. The plate geometric non-linearity is of the von Kármán type. In the formulation shear effects and rotary inertia are considered, too. The problem is solved in the frequency domain. A one-harmonic form of the solution for plate displacements corresponding to the plate formulation is assumed. The amplitude equation is obtained from the time averaged principle of virtual work. The time averaging precedes the use of the harmonic balance method. In the space discretization the finite element method is used involving 8-noded rectangular plate elements with selective-reduced integration. Several numerical examples are analyzed and the response curves are found using a path-following method. The purpose of these analyses is to identify material features of the adopted model of viscoelasticity with the fractional derivative. PubDate: 2017-04-11 DOI: 10.1007/s00466-017-1408-1

Authors:Rossana Dimitri; Giorgio Zavarise Abstract: Abstract Nowadays the isogeometric analysis (IGA) represents an innovative method that merges design and numerical computations into a unified formulation. In such a context we apply the isogeometric concept based on T-splines and Non Uniform Rational B-Splines (NURBS) discretizations to study the interfacial contact and debonding problems between deformable bodies in large deformations. More in detail, we develop and test a generalized large deformation contact algorithm which accounts for both frictional contact and mixed-mode cohesive debonding in a unified setting. Some numerical examples are provided for varying resolutions of the contact and/or cohesive zone, which show the accuracy of the solutions and demonstrate the potential of the method to solve challenging 2D contact and debonding problems. The superior accuracy of T-splines with respect to NURBS interpolations for a given number of degrees of freedom (Dofs) is always proved. PubDate: 2017-04-10 DOI: 10.1007/s00466-017-1410-7

Authors:Thanh-Tung Nguyen; Julien Réthoré; Julien Yvonnet; Marie-Christine Baietto Abstract: Abstract A new multi-phase-field method is developed for modeling the fracture of polycrystals at the microstructural level. Inter and transgranular cracking, as well as anisotropic effects of both elasticity and preferential cleavage directions within each randomly oriented crystal are taken into account. For this purpose, the proposed phase field formulation includes: (a) a smeared description of grain boundaries as cohesive zones avoiding defining an additional phase for grains; (b) an anisotropic phase field model; (c) a multi-phase field formulation where each preferential cleavage direction is associated with a damage (phase field) variable. The obtained framework allows modeling interactions and competition between grains and grain boundary cracks, as well as their effects on the effective response of the material. The proposed model is illustrated through several numerical examples involving a full description of complex crack initiation and propagation within 2D and 3D models of polycrystals. PubDate: 2017-04-08 DOI: 10.1007/s00466-017-1409-0

Authors:P. Wriggers; B. D. Reddy; W. Rust; B. Hudobivnik Abstract: Abstract The virtual element method has been developed over the last decade and applied to problems in elasticity and other areas. The successful application of the method to linear problems leads naturally to the question of its effectiveness in the nonlinear regime. This work is concerned with extensions of the virtual element method to problems of compressible and incompressible nonlinear elasticity. Low-order formulations for problems in two dimensions, with elements being arbitrary polygons, are considered: for these, the ansatz functions are linear along element edges. The various formulations considered are based on minimization of energy, with a novel construction of the stabilization energy. The formulations are investigated through a series of numerical examples, which demonstrate their efficiency, convergence properties, and for the case of nearly incompressible and incompressible materials, locking-free behaviour. PubDate: 2017-04-06 DOI: 10.1007/s00466-017-1405-4

Authors:R. C. Hurley; O. Y. Vorobiev; S. M. Ezzedine Abstract: Abstract We present a numerical method for modeling the mechanical effects of nonlinearly-compliant joints in elasto-plastic media. The method uses a series of strain-rate and stress update algorithms to determine joint closure, slip, and solid stress within computational cells containing multiple “embedded” joints. This work facilitates efficient modeling of nonlinear wave propagation in large spatial domains containing a large number of joints that affect bulk mechanical properties. We implement the method within the massively parallel Lagrangian code GEODYN-L and provide verification and examples. We highlight the ability of our algorithms to capture joint interactions and multiple weakness planes within individual computational cells, as well as its computational efficiency. We also discuss the motivation for developing the proposed technique: to simulate large-scale wave propagation during the Source Physics Experiments (SPE), a series of underground explosions conducted at the Nevada National Security Site (NNSS). PubDate: 2017-04-06 DOI: 10.1007/s00466-017-1403-6

Authors:J. D. R. Bordón; J. J. Aznárez; O. Maeso Abstract: Abstract This paper is concerned with a three-dimensional time harmonic model of open shell structures buried in poroelastic soils. It combines the dual boundary element method (DBEM) for treating the soil and shell finite elements for modelling the structure, leading to a simple and efficient representation of buried open shell structures. A new fully regularised hypersingular boundary integral equation (HBIE) has been developed to this aim, which is then used to build the pair of dual BIEs necessary to formulate the DBEM for Biot poroelasticity. The new regularised HBIE is validated against a problem with analytical solution. The model is used in a wave diffraction problem in order to show its effectiveness. It offers excellent agreement for length to thickness ratios greater than 10, and relatively coarse meshes. The model is also applied to the calculation of impedances of bucket foundations. It is found that all impedances except the torsional one depend considerably on hydraulic conductivity within the typical frequency range of interest of offshore wind turbines. PubDate: 2017-04-06 DOI: 10.1007/s00466-017-1406-3

Abstract: Abstract The present work deals with the formulation of a virtual element method for two dimensional structural problems. The contribution is split in two parts: in part I, the elastic problem is discussed, while in part II (Artioli et al. in Comput Mech, 2017) the method is extended to material nonlinearity, considering different inelastic responses of the material. In particular, in part I a standardized procedure for the construction of all the terms required for the implementation of the method in a computer code is explained. The procedure is initially illustrated for the simplest case of quadrilateral virtual elements with linear approximation of displacement variables on the boundary of the element. Then, the case of polygonal elements with quadratic and, even, higher order interpolation is considered. The construction of the method is detailed, deriving the approximation of the consistent term, the required stabilization term and the loading term for all the considered virtual elements. A wide numerical investigation is performed to assess the performances of the developed virtual elements, considering different number of edges describing the elements and different order of approximations of the unknown field. Numerical results are also compared with the one recovered using the classical finite element method. PubDate: 2017-04-04 DOI: 10.1007/s00466-017-1404-5

Authors:Enrique Ortega; Roberto Flores; Eugenio Oñate; Sergio Idelsohn Abstract: Abstract An a-posteriori error estimate with application to inviscid compressible flow problems is presented. The estimate is a surrogate measure of the discretization error, obtained from an approximation to the truncation terms of the governing equations. This approximation is calculated from the discrete nodal differential residuals using a reconstructed solution field on a modified stencil of points. Both the error estimation methodology and the flow solution scheme are implemented using the Finite Point Method, a meshless technique enabling higher-order approximations and reconstruction procedures on general unstructured discretizations. The performance of the proposed error indicator is studied and applications to adaptive grid refinement are presented. PubDate: 2017-04-03 DOI: 10.1007/s00466-017-1402-7