A fast, certified and "tuning free" two-field reduced basis method for the metamodelling of affinely-parametrised elasticity problems

in Computer Methods in Applied Mechanics and Engineering (2016), 298

This paper proposes a new reduced basis algorithm for the metamodelling of parametrised elliptic problems. The developments rely on the Constitutive Relation Error (CRE), and the construction of separate ... [more ▼]

This paper proposes a new reduced basis algorithm for the metamodelling of parametrised elliptic problems. The developments rely on the Constitutive Relation Error (CRE), and the construction of separate reduced order models for the primal variable (displacement) and flux (stress) fields. A two field greedy sampling strategy is proposed to construct these two fields simultaneously and in an efficient manner: at each iteration, one of the two fields is enriched by increasing the dimension of its reduced space in such a way that the CRE is minimised. This sampling strategy is then used as a basis to construct goal-oriented reduced order modelling. The resulting algorithm is certified and “tuning free”: the only requirement from the engineer is the level of accuracy that is desired for each of the outputs of the surrogate. It is also shown to be significantly more efficient in terms of computational expense than competing methodologies. [less ▲]

Numerical analysis of the inclusion-crack interaction by the extended finite element method

in International Journal for Computational Methods in Engineering Science and Mechanics (2014)

One of the partition of unity methods, the extended finite ele- ment method (XFEM), is applied to study the inclusion-crack in- teractions in an elastic medium. Both the inclusions and the crack are ... [more ▼]

One of the partition of unity methods, the extended finite ele- ment method (XFEM), is applied to study the inclusion-crack in- teractions in an elastic medium. Both the inclusions and the crack are modelled within the XFEM framework. A structured quadri- lateral mesh is used and the influence of crack length, the number of inclusions, and the geometry of the inclusions on the crack tip stress field are numerically studied. The interaction integral for non-homogeneous materials is used to compute the stress intensity factors ahead of the crack tip. The accuracy and flexibility of the XFEM is demonstrated by various numerical examples. [less ▲]

Molecular dynamics/xfem coupling by a three-dimensional extended bridging domain with applications to dynamic brittle fracture

in International Journal for Multiscale Computational Engineering (2013), 11(6), 527-541

We propose a method to couple a three-dimensional continuum domain to a molecular dynamics domain to simulate propagating cracks in dynamics. The continuum domain is treated by an extended finite element ... [more ▼]

We propose a method to couple a three-dimensional continuum domain to a molecular dynamics domain to simulate propagating cracks in dynamics. The continuum domain is treated by an extended finite element method to handle the discontinuities. The coupling is based on the bridging domain method, which blends the continuum and atomistic energies. The Lennard-Jones potential is used to model the interactions in the atomistic domain, and the Cauchy-Born rule is used to compute the material behavior in the continuum domain. To our knowledge, it is the first time that a three dimensional extended bridging domain method is reported. To show the suitability of the proposed method, a threedimensional crack problem with an atomistic region around the crack front is solved. The results show that the method is capable of handling crack propagation and dislocation nucleation. © 2013 by Begell House, Inc. [less ▲]

A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics

in Computer Methods in Applied Mechanics and Engineering (2013), 256

We propose in this paper a reduced order modelling technique based on domain partitioning for parametric problems of fracture. We show that coupling domain decomposition and projection-based model order ... [more ▼]

We propose in this paper a reduced order modelling technique based on domain partitioning for parametric problems of fracture. We show that coupling domain decomposition and projection-based model order reduction permits to focus the numerical effort where it is most needed: around the zones where damage propagates. No a priori knowledge of the damage pattern is required, the extraction of the corresponding spatial regions being based solely on algebra. The efficiency of the proposed approach is demonstrated numerically with an example relevant to engineering fracture. © 2012 Elsevier B.V. [less ▲]

Statistical extraction of process zones and representative subspaces in fracture of random composites

in International Journal for Multiscale Computational Engineering (2013), 11(3), 253-287

We propose to identify process zones in heterogeneous materials by tailored statistical tools. The process zone is redefined as the part of the structure where the random process cannot be correctly ... [more ▼]

We propose to identify process zones in heterogeneous materials by tailored statistical tools. The process zone is redefined as the part of the structure where the random process cannot be correctly approximated in a low-dimensional deterministic space. Such a low-dimensional space is obtained by a spectral analysis performed on precomputed solution samples. A greedy algorithm is proposed to identify both process zone and low-dimensional representative subspace for the solution in the complementary region. In addition to the novelty of the tools proposed in this paper for the analysis of localized phenomena, we show that the reduced space generated by the method is a valid basis for the construction of a reduced-order model. © 2013 by Begell House, Inc. [less ▲]

Equilibrated patch recovery for accurate evaluation of upper error bounds in quantities of interest

in ECCOMAS Thematic Conference - ADMOS 2011: International Conference on Adaptive Modeling and Simulation, An IACM Special Interest Conference (2012)

There is an increasing interest on the use of goal-oriented error estimates which help to measure and control the local error on a linear or non-linear quantity of interest (QoI) that might result ... [more ▼]

There is an increasing interest on the use of goal-oriented error estimates which help to measure and control the local error on a linear or non-linear quantity of interest (QoI) that might result relevant for design purposes (e.g. the mean stress value in a particular area, displacements, the stress intensity factor for fracture problems,⋯). In general, residual-based error estimators have been used to obtain upper and lower bounds of the error in quantities of interest for finite element approximations. In this work, we propose a novel a posteriori recovery technique to obtain an upper error bound of the QoI. We use a recovery procedure based on the superconvergent patch recovery (SPR) technique to obtain nearly statically admissible recovered stress fields for the primal and dual problems. This recovery technique was previously used to obtain upper bounds of the error in energy norm and has been used in this paper to obtain a computable version of the upper bound for the quantity of interest. [less ▲]

Error estimation and error bounding in quantities of interest based on equilibrated recovered displacement fields

in ECCOMAS 2012 - European Congress on Computational Methods in Applied Sciences and Engineering, e-Book Full Papers (2012)

Over the last ten years there has been an increase on the use of goal-oriented error estimates aimed to quantify the local error on a (non)linear quantity of interest (QoI) that might result relevant for ... [more ▼]

Over the last ten years there has been an increase on the use of goal-oriented error estimates aimed to quantify the local error on a (non)linear quantity of interest (QoI) that might result relevant for design purposes. Residual-based error estimators have been used recursively to obtain upper and lower bounds of the error in quantities of interest for finite element approximations. In this paper, we present a recovery technique for 2D linear elasticity problems, based on the superconvergent patch recovery (SPR), which provides recovered displacement and stress fields that are then utilised to evaluate practical upper and lower error bounds in QoI. [less ▲]

Extended finite element method with edge-based strain smoothing (ESm-XFEM) for linear elastic crack growth

in Computer Methods in Applied Mechanics and Engineering (2012), 209-212

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 ... [more ▼]

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. [less ▲]

Natural frequencies of cracked isotropic & specially orthotropic plates using the extended finite element method

Scientific Conference (2011, April)

In this paper, the linear free flexural vibration of cracked isotropic and specially orthotropic plates is studied using the extended finite element method. The mixed interpolation technique of the well ... [more ▼]

In this paper, the linear free flexural vibration of cracked isotropic and specially orthotropic plates is studied using the extended finite element method. The mixed interpolation technique of the well- established MITC4 [1] quadrilateral finite element with 12 standard degrees of freedom per element is used for this study. The natural frequencies of simply supported square plates are computed as a function of crack length and crack location. [less ▲]

On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM)

in International Journal for Numerical Methods in Engineering (2011), 86(4-5), 637-666

By using the strain smoothing technique proposed by Chen et al. (Comput. Mech. 2000; 25:137-156) for meshless methods in the context of the finite element method (FEM), Liu et al. (Comput. Mech. 2007; 39 ... [more ▼]

By using the strain smoothing technique proposed by Chen et al. (Comput. Mech. 2000; 25:137-156) for meshless methods in the context of the finite element method (FEM), Liu et al. (Comput. Mech. 2007; 39(6):859-877) developed the Smoothed FEM (SFEM). Although the SFEM is not yet well understood mathematically, numerical experiments point to potentially useful features of this particularly simple modification of the FEM. To date, the SFEM has only been investigated for bilinear and Wachspress approximations and is limited to linear reproducing conditions. The goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations. We focus on three widely used enrichment schemes, namely: (a) weak discontinuities; (b) strong discontinuities; (c) near-tip linear elastic fracture mechanics functions. The main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial (cases (a) and (b)), but that non-polynomial enrichment of type (c) lead to inferior methods compared to the standard enriched FEM (e.g. XFEM). © 2011 John Wiley & Sons, Ltd. [less ▲]