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919 KiB

Open AccessArticle

An Adaptive h-Refinement Algorithm for Local Damage Models

by Jonathan S. Pitt and Francesco Costanzo

Algorithms 2009, 2(4), 1281-1300; https://doi.org/10.3390/a2041281 - 6 Oct 2009

Cited by 1 | Viewed by 8969

An adaptive mesh refinement strategy is proposed for local damage models that often arise from internal state variable based continuum damage models. The proposed algorithm employs both the finite element method and the finite difference method to integrate the equations of motion of [...] Read more.

An adaptive mesh refinement strategy is proposed for local damage models that often arise from internal state variable based continuum damage models. The proposed algorithm employs both the finite element method and the finite difference method to integrate the equations of motion of a linear elastic material with simple isotropic microcracking. The challenges of this problem include the time integration of coupled partial differential equations with time-dependent coefficients, and the proper choice of solution spaces to yield a stable finite element formulation. Discontinuous elements are used for the representation of the damage field, as it is believed that this reduction in regularity is more consistent with the physical nature of evolving microcracking. The adaptive mesh refinement algorithm relies on custom refinement indicators, two of which are presented and compared. The two refinement indicators we explore are based on the time rate of change of the damage field and on the energy release rate, respectively, where the energy release rate measures the energy per unit volume available for damage to evolve. We observe the performance of the proposed algorithm and refinement indicators by comparing the predicted damage morphology on different meshes, hence judging the capability of the proposed technique to address, but not eliminate, the mesh dependency present in the solutions of the damage field. Full article

(This article belongs to the Special Issue Numerical Simulation of Discontinuities in Mechanics)

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366 KiB

Open AccessArticle

Stefan Problem through Extended Finite Elements: Review and Further Investigations

by Luca Salvatori and Niccolò Tosi

Algorithms 2009, 2(3), 1177-1220; https://doi.org/10.3390/a2031177 - 21 Sep 2009

Cited by 15 | Viewed by 11784

Abstract

A general review of the extended finite element method and its application to the simulation of first-order phase transitions is provided. Detailed numerical investigations are then performed by focusing on the one-dimensional case and studying: (i) spatial and temporal discretisations, (ii) different numerical [...] Read more.

A general review of the extended finite element method and its application to the simulation of first-order phase transitions is provided. Detailed numerical investigations are then performed by focusing on the one-dimensional case and studying: (i) spatial and temporal discretisations, (ii) different numerical techniques for the interface-condition enforcement, and (iii) different treatments for the blending elements. An embeddeddiscontinuity finite element approach is also developed and compared with the extended finite element method, so that a clearer insight of the latter can be given. Numerical examples for melting/solidification in planar, cylindrical, and spherical symmetry are presented and the results are compared with analytical solutions. Full article

(This article belongs to the Special Issue Numerical Simulation of Discontinuities in Mechanics)

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1395 KiB

Open AccessArticle

Automated Modelling of Evolving Discontinuities

by Mehdi Nikbakht and Garth N. Wells

Algorithms 2009, 2(3), 1008-1030; https://doi.org/10.3390/a2031008 - 18 Aug 2009

Cited by 5 | Viewed by 8468

Abstract

The automated approximation of solutions to differential equations which involve discontinuities across evolving surfaces is addressed. Finite element technology has developed to the point where it is now possible to model evolving discontinuities independently of the underlying mesh, which is particularly useful in [...] Read more.

The automated approximation of solutions to differential equations which involve discontinuities across evolving surfaces is addressed. Finite element technology has developed to the point where it is now possible to model evolving discontinuities independently of the underlying mesh, which is particularly useful in simulating failure of solids. However, the approach remains tedious to program, particularly in the case of coupled problems where a variety of finite element bases are employed and where a mixture of continuous and discontinuous fields may be used. We tackle this point by exploring the scope for employing automated code generation techniques for modelling discontinuities. Function spaces and variational forms are defined in a language that resembles mathematical notation, and computer code for modelling discontinuities is automatically generated. Principles underlying the approach are elucidated and a number of two- and three-dimensional examples for different equations are presented. Full article

(This article belongs to the Special Issue Numerical Simulation of Discontinuities in Mechanics)

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616 KiB

Open AccessArticle

Failure Assessment of Layered Composites Subject to Impact Loadings: a Finite Element, Sigma-Point Kalman Filter Approach

by Stefano Mariani

Algorithms 2009, 2(2), 808-827; https://doi.org/10.3390/a2020808 - 4 Jun 2009

Cited by 8 | Viewed by 10224

Abstract

We present a coupled finite element, Kalman filter approach to foresee impactinduced delamination of layered composites when mechanical properties are partially unknown. Since direct numerical simulations, which require all the constitutive parameters to be assigned, cannot be run in such cases, an inverse [...] Read more.

We present a coupled finite element, Kalman filter approach to foresee impactinduced delamination of layered composites when mechanical properties are partially unknown. Since direct numerical simulations, which require all the constitutive parameters to be assigned, cannot be run in such cases, an inverse problem is formulated to allow for modeling as well as constitutive uncertainties. Upon space discretization through finite elements and time integration through the explicit ®¡method, the resulting nonlinear stochastic state model, wherein nonlinearities are due to delamination growth, is attacked with sigma-point Kalman filtering. Comparison with experimental data available in the literature and concerning inter-laminar failure of layered composites subject to low-velocity impacts, shows that the proposed procedure leads to: an accurate description of the failure mode; converged estimates of inter-laminar strength and toughness in good agreement with experimental data. Full article

(This article belongs to the Special Issue Numerical Simulation of Discontinuities in Mechanics)

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400 KiB

Open AccessArticle

Probabilistic Upscaling of Material Failure Using Random Field Models – A Preliminary Investigation

by Keqiang Hu and X. Frank Xu

Algorithms 2009, 2(2), 750-763; https://doi.org/10.3390/a2020750 - 30 Apr 2009

Cited by 4 | Viewed by 8938

Abstract

Complexity of failure is reflected from sensitivity of strength to small defects and wide scatter of macroscopic behaviors. In engineering practices, spatial information of materials at fine scales can only be partially measurable. Random field (RF) models are important to address the uncertainty [...] Read more.

Complexity of failure is reflected from sensitivity of strength to small defects and wide scatter of macroscopic behaviors. In engineering practices, spatial information of materials at fine scales can only be partially measurable. Random field (RF) models are important to address the uncertainty in spatial distribution. To transform a RF of micro-cracks into failure probability at full structural-scale crossing a number of length scales, the operator representing physics laws need be implemented in a multiscale framework, and to be realized in a stochastic setting. Multiscale stochastic modeling of materials is emerging as a new methodology at this research frontier, which provides a new multiscale thinking by upscaling fine-scale RFs. In this study, a preliminary framework of probabilistic upscaling is presented for bottom-up hierarchical modeling of failure propagation across micro-meso-macro scales. In the micro-to-meso process, the strength of stochastic representative volume element (SRVE) is probabilistically assessed by using a lattice model. A mixed Weibull-Gaussian distribution is proposed to characterize the statistical strength of SRVE, which can be used as input for the subsequent meso-to-macro upscaling process using smeared crack finite element analysis. Full article

(This article belongs to the Special Issue Numerical Simulation of Discontinuities in Mechanics)

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313 KiB

Open AccessArticle

ALE-PSO: An Adaptive Swarm Algorithm to Solve Design Problems of Laminates

by Paolo Vannucci

Algorithms 2009, 2(2), 710-734; https://doi.org/10.3390/a2020710 - 21 Apr 2009

Cited by 14 | Viewed by 9002

Abstract

This paper presents an adaptive PSO algorithm whose numerical parameters can be updated following a scheduled protocol respecting some known criteria of convergence in order to enhance the chances to reach the global optimum of a hard combinatorial optimization problem, such those encountered [...] Read more.

This paper presents an adaptive PSO algorithm whose numerical parameters can be updated following a scheduled protocol respecting some known criteria of convergence in order to enhance the chances to reach the global optimum of a hard combinatorial optimization problem, such those encountered in global optimization problems of composite laminates. Some examples concerning hard design problems are provided, showing the effectiveness of the approach. Full article

(This article belongs to the Special Issue Numerical Simulation of Discontinuities in Mechanics)

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417 KiB

Open AccessArticle

Mixed Variational Formulations for Micro-cracked Continua in the Multifield Framework

by Matteo Bruggi and Paolo Venini

Algorithms 2009, 2(1), 606-622; https://doi.org/10.3390/a2010606 - 27 Mar 2009

Viewed by 7837

Abstract

Within the framework of multifield continua, we move from the model of elastic microcracked body introduced in (Mariano, P.M. and Stazi, F.L., Strain localization in elastic microcracked bodies, Comp. Methods Appl. Mech. Engrg. 2001, 190, 5657–5677) and propose a few novel variational formulations [...] Read more.

Within the framework of multifield continua, we move from the model of elastic microcracked body introduced in (Mariano, P.M. and Stazi, F.L., Strain localization in elastic microcracked bodies, Comp. Methods Appl. Mech. Engrg. 2001, 190, 5657–5677) and propose a few novel variational formulations of mixed type along with relevant mixed FEM discretizations. To this goal, suitably extended Hellinger-Reissner principles of primal and dual type are derived. A few numerical studies are presented that include an investigation on the interaction between a single cohesive macrocrack and diffuse microcracks (Mariano, P.M. and Stazi, F.L., Strain localization due to crack–microcrack interactions: X–FEM for a multifield approach, Comp. Methods Appl. Mech. Engrg. 2004, 193, 5035–5062). Full article

(This article belongs to the Special Issue Numerical Simulation of Discontinuities in Mechanics)

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