Method for the explicit insertion of microstructure in Cellular Automata Finite Element (CAFE) models based on an irregular tetrahedral Finite Element mesh: Application in a multi-scale Finite Element Microstructure MEshfree framework (FEMME) Luis Saucedo-Mora a,n , Thomas James Marrow a,b a University of Oxford, Department of Materials, Parks Road, Oxford OX1 3PH, United Kingdom b University of Oxford, Oxford Martin School, Parks Road, Oxford OX1 3PH, United Kingdom article info Article history: Received 23 December 2014 Received in revised form 27 April 2015 Accepted 6 July 2015 Keywords: Cellular-automata Finite elements Multiscale model Microstructure Tetrahedron Meshfree abstract Multiscale models are needed in simulations of mechanical and thermal properties that consider the microstructural heterogeneity of materials, in which a coupling is essential between the numerical models that deal with the different scales. In the class of cellular automata-finite element (CA-FE) or CAFE models, the use of a regular FE mesh restricts the model versatility due to the limited variety of engineering problems that can be simulated with such meshes. A novel methodology is proposed to create the homogeneous cells of the CA model from a mesh that is formed by irregular tetrahedrons. The result is more versatile and capable of modelling complex geometries, improving its applicability. The problem is solved through a subdivision algorithm that creates homogeneous tetrahedral cells, using a methodology to insert microstructures that can be described by particles, pores or fibres. Its use is demonstrated in a multi-scale Finite Element Microstructure MEshfree (FEMME) framework to calculate the elastic strains caused by microstructural features (pores); the method is shown to have a significantly lower computational cost than finite element simulations of equivalent levels of discretization. & 2015 Elsevier B.V. All rights reserved. 1. Introduction The importance of multi-scale models has grown in recent years, achieving increasing complexity in the problems addressed; these aim, for instance, to consider mechanical or thermal effects on the macrostructure that arise from the microstructural features. Prior to this, the effect of the microstructure has generally been neglected, or treated in a very limited manner, due to the high computational cost of introducing microstructure fidelity into large scale models; consequently, continuum material properties have been commonly used [1]. Multi-scale models address this problem by compensating for the use of a coarse discretization in the macrostructural model via the insertion of detailed micro- structure only in those zones that require this refinement; for example, the damaged regions within fracture models. Various methods have been proposed, such as the stochastic multiscale model for fracture [2] that involves a statistical description of particles; the multiscale cohesive zone model [3], in which the bulk material is modelled as a quasi-continuum and the adaptive concurrent multiscale model [4], which utilizes an explicit descrip- tion of microstructural features near the crack tip. Recently, a cellular automata (CA) finite element (FE) model with a micro- structural adaptive meshfree framework [5] has inserted a high fidelity description of the microstructure into a Finite Element Microstructure MEshfree (FEMME) multiscale model of damage development, using the methodology that is explained here. Cellular Automata Finite Element (CAFE) [6] models insert a detailed description of the microstructural material properties locally using CA within a larger scale FE model, coupling the macro and microstructural problems efficiently. The complexity of this approach lies in the achievement of continuity between the CA and FE layers, where the geometry of the cells in the CA model is embedded in the FE mesh. Traditionally, the solution to this problem is to split the regular three-dimensional finite elements into cubes of equal size representing the CA layer; this is very convenient for CA methods that are based on images [7–9]. Each cube of the discretization can be related to an image pixel, thus inserting the microstructure in the model from a description of the microstructure, which might be derived from a 2- dimensional or 3-dimensional image. The problem for CAFE is the limited versatility of regular FE meshes in the solution of engineering problems. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ finel Finite Elements in Analysis and Design http://dx.doi.org/10.1016/j.finel.2015.07.001 0168-874X/& 2015 Elsevier B.V. All rights reserved. n Corresponding author. E-mail address: luis.saucedomora@materials.ox.ac.uk (L. Saucedo-Mora). Finite Elements in Analysis and Design 105 (2015) 56–62