DepartementPhysique et Mecanique des

Materiaux

Finite Element solver FoXtroTM. Gueguena

amikael.gueguen@ensma.fr

Introduction

I Context : Development of a finite element solverI devoted to material modeling and multiphysic coupling (thermo-mechanical ;

diffuso-mechanical)I overcome limitations of industrial code for dealing complex multiphysic problemsI simulation on representative microstructure on different kind of materials (metallic

polycristalline aggregate ; textile composite material)

Current Implementation

I reading of well-known abaqus fileformat, with same kind of functionality(load and boundary conditionsfonction of time, stacking step solver)

I Usage of complex behavior law(crystalline plasticity ; viscoelasticity)with many internal variables storedupon finite element integration point

I Classical non linear resolution withincremental Newton Raphsonalgorithm

→ Need of different linear solver, IOformat and material integrationmethod to deal with complex materialmicrostructure

code

linearsolver

IO

finiteele-

ment

materiallaw

Nonlin-ear

solver

Implemented finite element

I 3D Mechanical, thermal and diffusion isoparametric element (tetrahedron/hexahedron))I Geometric non linearity with total lagrangian formulation ; Hughes and Winget algorithmImplemented behavior law : abaqus UMAT formalism on each integrationpoint

I strain ( ε) , or transformationgradient (F ) on input

I stress σ, and tangent matrix∂σ∂ε on output

I update of internal variablesν i

finite element

εt+∆t, νit

behavior law

σt+∆t, νit+∆t,

∂σ∂ε

Types

Isoelasticity hypoelastic

Isotropic or kinematic plasticity

crystalline plasticity: HCP/FCC system

viscoelasticity

UMAT

Current parallel implementation

→ Linear system for mesh domain Ω assembled andsolved for each non linear iteration [K].u = f

Domain decomposition partitioningI sequential mesh readingI all material data are copied on each processI partitioning into n MPI process with classical

partitioner (METIS1,SCOTCH2)1 http://glaros.dtc.umn.edu/gkhome/views/metis2 http://www.labri.fr/perso/pelegrin/scotch/scotch_fr.html#resources

I elemental phase, assembling et boundaryconditions insertion on n domains.

I linear system solved by direct MUMPS solver (http://mumps.enseeiht.fr ) on distributed tangentmatrix and global vector

Ω 1Ω 2Ω 3

Ω 4

FEM domain Ω

subdomain

interface Γ

Work in progress : Petsc parallel improvement

DMPlex→ usage of DMPlex object to handle unstructured grids using the generic DM interface for

hierarchy and multi-physics→ capacity to store and manipulate material data and internal variables for complex behavior

law→ direct transfer to SNES object to evaluate residual and tangent matrix→ usage of iterative linear solver : parallel efficiency for large model created for

representative microstructure (Millions of dofs)

Mesh ReaderI Modification of DMPlexCreateGmsh to get physical entities tags from element reading ;I Insertion of element parameters using PetscSectionI Usage of Moab/DMPlex interface for better efficiency

Input mesh example

Computed tomography image based meshing on textilecomposite materialY. Sinchuk, M. Gigliotti, Y. Pannier, M. Gueguen and D. Tandiang.

2

half of the nodes are hanging-type nodes [2]. The hanging nodes are handled by linear and bilinear multipoint constrains [3], which can be easy included within the ABAQUS® model.

Fig. 1. Voxel mesh of the textile composite material

Fig. 2. Adaptive voxel mesh approach

REFERENCES [1] T. Belytschko, R. Gracie, G. Ventura, A review of Extended/Generalized finite element

methods for material modeling. Model. Simul. Mater. Sci. Eng., Vol. 17(4), 2009. [2] T.P. Fries, A. Byfut, A. Alizada, K.W. Cheng, A. Schröder, Hanging nodes and XFEM,

Int. J. Numer. Meth. Engng. Vol 86, pp, 404-430, 2011. [3] J.H. Kim, C.C. Swan, Voxel-based meshing and unit-cell analysis of textile composites,

Int. J. Numer. Meth. Engng, Vol 56, pp. 977–1006, 2003.

3D reconstruction from stack of 2D EBSD maps andvoxel based mesh of polycrystalline aggregate

21

(a) (b)

(c) (d)

Fig. 16: 3D reconstruction and mesh of the polycrystalline aggregate containing the studied

crack: (a) Illustration of the stack of 2D EBSD maps, (b) 3D reconstructed aggregate, (c)

detailed view of the grain where the studied crack has initiated, (d) other examples of

reconstructed grains. (Approximate location and size of the studied crack are depicted in (b)

and (c) by white lines).

Specific development

implementation of specific loading and boundary conditions for gassaturation in elastomerobjective effect due to gas decompression inside cavity on bulk stress field :

first stage of damage? mutiphysic coupling ; mechanical and diffusion gas solving ; degrees of freedom : partial

pression for diffusion p and displacement uk ; diffusion expansionI material exchange between bulk and cavityI inside cavity, perfect gas law : p(t) = nm(t)RT

V (t)

I cavity volume depend of mechanical solution :

V (t) = f (u∂Ω(t))

I moles nm(t) inside cavity is function of flux −→q onsurface ∂Ω :

nm(t) = nm(0) +

∫t

∫∂Ω

−→q .−→n dS

∂ΩContinuum Ω

cavity

I computation on material exchange and loading update on cavity interface ∂Ω for eachiteration ;

nm(t + ∆t) = nm(t) +nbel∂Ω∑

e

(

nbpti∑i

−→q i.−→n i wi)∆tS∂Ω

−→q∂Ω = −D∂c∂Xi

= −Dcr∂φr

∂ηj

∂ηj

∂Xi

Figure: gas saturation/decompression on t = 20s

Simulation examples

Crystalline plasticity modeling : RVE with 630 virtual grains simulations

Figure: Equivalent Young Modulus Figure: schmid factor for RVE location occuringwith 80%εmax

eq < εeq < εmaxeq

Figure: transverse displacement field

morphological simplification with 2 phase materials : RVE simulation onbulk with inclusion clusterDetermination of equivalent inclusion based on covariogram