Special Issue "Numerical Simulation of Discontinuities in Mechanics"

Guest Editor
Dr. Stefano Mariani
Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy
Website: http://www.stru.polimi.it
E-mail:

Guest Editor
Dr. Paolo Maria Mariano
Università di Firenze, DICeA, Via Santa Marta 3, 50139, Firenze, Italy
Website: http://www.dicea.unifi.it/paolo.mariano/
E-mail:

Guest Editor
Dr. Paolo Venini
Università di Pavia, Dipartimento di Meccanica Strutturale, Via Ferrata 1, 27100, Pavia, Italy
Website: http://www-1.unipv.it/venini/
E-mail:

Submission

All papers should be submitted to algorithms@mdpi.org. To be published continuously until the deadline and papers will be listed together at the special issue website.

Submitted papers should not have been published nor be under consideration for publication elsewhere. All papers are refereed through a peer-review process. A guide for authors is available on the Instructions for Authors page. Algorithms is an international peer-reviewed quarterly journal published by Molecular Diversity Preservation International.

Open Access publication fees are 300 CHF per paper. English correction fees and/or formatting fees (250 CHF) will be added in certain cases (550 CHF per paper for those papers that require extensive additional formatting and/or English corrections).

Article Processing Charges (APC)

Article Processing Charges (APC) will be waived for well prepared manuscripts of invited papers. For the first two volumes of this new journal the APC are of 300 CHF (or 550 CHF per paper for those papers that require extensive additional formatting and/or English corrections).

• solid mechanics
• fluid mechanics
• weak and strong discontinuities
• complex bodies
• multi-scale methods

Title: Numerical relaxation of nonconvex functionals in solid mechanics
Author: Antonio Orlando
Affiliation: School of Engineering, Swansea University, UK
Abstract: The occurrence of microstructures in solid mechanics can be attributed to a loss of the convexity characteristics of the underlying energy potentials. Specific model problems include, for instance, phase transitions in crystalline solids, material instabilities in finite strain plasticity, and material deteroriation due to damage processes. While the material deforms macroscopically, structures in the form of shear bands, cracks or laminates can be formed on microscopic scales. The numerical simulation of these problems through energy minimization poses, how- ever a very challenging task because of the enforced high oscillatory character of the developed microstructure and of the clustering of a large number of local minimizers around some global finite element approximation. Macroscopic properties of the resulting microstructure can, nevertheless, be recovered by replacing the underlying potential with its quasiconvex hull. The latter is a nonlocal notion and extremely difficult to analyse in theory and computation. Therefore, one introduces weaker or stronger notions, like rank-one convexity and polyconvexity, which are more amenable to numerical treatment. In this paper, we illustrate the effects of nonconvexity and describe relaxation theory from calculus of variations in the scalar and vector case using two model examples: the Young-Tartar problem and the modeling of the elastoplastic behavior of single crystals in finite strain.

Title: Algorithms of the Heterogeneous Cohesive Model for Fracture Simulation
Authors: Keqiang Hu, X. Frank Xu¹; Zhenjun Yang²
Affiliations: ¹: Stevens Institute of Technology, Hoboken, NJ 07030, USA; ²: The University of Liverpool, Liverpool L69 3GH, UK
Abstract: Stochastic fracture modeling is challenging especially due to unresolved puzzles on crack resolution or fractals. Research effort in this area is needed to address emerging technologies in MEMS and biosystems. By generalizing the conventional Griffith energy-balance concept to heterogeneous materials, a heterogeneous cohesive (HC) crack model is proposed in (Yang & Xu, 2008) to predict macroscopic strength of materials based on meso-scale random fields of fracture properties. A new stress-based criterion is proposed to determine the crack growth direction by taking into account both crack-tip stress state and heterogeneity of tensile strength. In this paper the algorithms developed and employed in the HC model are presented. Particularly the algorithms for simulation of Weibull random fields using classical Monte Carlo and Sparse Grid methods are introduced in detail.

Title: Controlling Damage Localization through adaptive H-refinement
Authors: Jonathan S. Pitt and Francesco Costanzo
Affiliation: Engineering Science and Mechanics Department; The Pennsylvania State University.
Abstract: An adaptive mesh refinement strategy is proposed for controlling the damage localization typically found in internal state variable based continuum damage models. Specifically, an algorithm employing both the finite element method and finite difference method is used to integrate previously derived equations of motion of a linear elastic material with simple isotropic microcracking. Challenges of this problem include the time integration of coupled PDE with time- dependent coefficents, and the proper choice of finite elements to yield a stable finite element formulation. Discontinuous elements are used for the local damage variable, and a correlation is drawn between this method and the physical nature of the problem.
The adaptive mesh refinement algorithm relies on custom error indicators, two of which are presented and compared. The error indicators are based on both the evolution of damage in the domain, as well as the energy release rate - a precursor quantity to damage evolution. Additionally, the use of isotropic refinement is presented, and is considered a key result. It is shown that even with radically different initial meshes, local damage models can produce similar results, in terms of failure locations and final damage profiles; thus, we demonstrate that gradient theories are not always necessary for generating robust algorithms. Implementation of the overall algorithm is covered extensively, including parallelization for large simulations.
Keywords: Adaptive Mesh Refinement, Finite Element Method, Dynamic Fracture, Continuum Damage Mechanics

Title: Automated Modelling of Evolving Discontinuities
Authors: Mehdi Nikbakht and Garth N. Wells
Abstract: The automated modelling of differential equations with discontinuous solutions across evolving surfaces is addressed. Finite element technology has developed to the point where it is now possible model evolving discontinuities independently of the underlying mesh, which is particularly useful in modelling material failure. 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 reassembles mathematical notation, and efficient computer code for modelling discontinuities is automatically generated. Principles underlying the approach are elucidated and a number of two- and three-dimensional examples are presented.