# Investigation and implementation of the Finite Element Tearing and Interconnecting (FETI) approach

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## Two-steps explicit time integration

A two-steps explicit time integration scheme is formulated and shown to be stable and accurate in [Noh and Bathe, 2013]. Such integration scheme is based on the splitting of the time step in two sub-steps in order to keep a non-diagonal damping matrix always on the right-hand side of the solving systems. Five parameters are then introduced to optimize the stability and accuracy of calculation.
Introducing the non-diagonal viscous damping matrix C and the first parameter p ∈ (0,1) as marker of the position within the time step, the solution in terms of accelerations, velocities and displacements in the first sub-step reads:
1. update of mid-step displacements: n+pU = nU+nVpDt +nAp2 Dt2 2.
2. update of mid-step velocities: n+p ˜V = 􀀀nV+nApDt 2 (1−s)+nVs.

### Multi-field space-time integration

In contrast to explicit time integration schemes that aim at minimizing the computational effort for the systems resolution, a multi-fields space-time monolithic discretization was proposed in [Hulbert and Hughes, 1990], exploiting a Time-Discontinuous Galerkin method (TDG) in order to consider jumps in the velocity field, as in wave-propagation problems.
The formulation is based on the integration of the weak formulation of the initialboundary value problem in System (1.1) both in space and time as: Z tend tin ZW (r¨u·d˙u+(u) : « (d˙u)) dWdt = Z tend tin ZW fbody ·d˙u dWdt.

#### Energy-Momentum integration scheme for large rotations

The initial-boundary value problem has been so far considered assuming the linear elastic formulation presented in Equations (1.1) involving only translational displacements u(x, t). Out of a more accurate formulation, as stated in [Simo and Tarnow, 1992], temporal and spatial Finite Element discretizations of the continuum dynamics need not, and in general will not, inherit the conservation of momentum properties and the a-priori estimate. For instance, the conservation form of the mid-point rule is an exact momentum conserving algorithm which does not conserve energy for autonomous Hamiltonian systems, except for the linear regime.
In [Simo and Tarnow, 1992], a so-called Energy-Momentum (EM) integration scheme was introduced for geometrical non-linearities of quadratic nature, aiming at preserving specific features of the continuous system such as conservation of momentum, angular momentum and energy when the system and the applied forces allow to. The extensions to nonlinear dynamics of shells and rods were later proposed in [Simo and Tarnow, 1994] and [Simo et al., 1995], where displacement-based discretizations are used, and in [Sansour et al., 2002], where a multi-field discretization is treated involving enhanced-strain Finite Elements (see the theoretical basis of the Enhanced Assumed Strain method for linear elastic problems in Appendix B).

Multi-scale coupling strategies in space and time

The need to couple different models and discretizations arises when treating large structures with localized phenomena, e.g. small details, micro-cracks, defects, highly localized loading conditions, structured material and so on. In this case, a refined and specialized model should be focused on a restricted zone of the overall domain. In the rest of the domain, not involved in particular phenomena, such as in the case of regular geometry, smoothly distributed loading and homogeneous material, the model should be as coarse as possible, in order to save computing time. Hereafter, the problem in Figure 1.3 will be referred to as the reference problem. It consists of a sample structure in a domain W composed of two parts. The part on the left, characterized by fine features, requires a fine resolution both in space and time. The part on the right requires only a coarse resolution both in space and time.

Partitioning in time for a given non-uniform mesh

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A potential methodology to solve the problem in Figure 1.3 consists in discretizing the domain with a non-uniform mesh Wh, for instance pulling together two different uniform meshes Wh1 and Wh2, so that Wh1 ∪Wh2 ≡ Wh and treating the embedded meshes over the interface Gh ≡ Wh1 ∩Wh2 using a tie constraint. Two different time integration schemes or time steps are then applied in the two partitions Wh1 and Wh2 according to the two different mesh sizes.
For instance, Figure 1.4 shows the partitioning of the overall domain in a fine mesh Wh1 composed of 20 quadrilateral elements and a coarse mesh Wh2 composed of just 1 quadrilateral element. The common nodes between the two regions constitute the interface Gh. In literature, several works have been devoted to algorithms for coupling implicit and explicit time integration schemes using the same time scale everywhere. These approaches are usually referred to as Mixed-Methods in time. On the other hand, several other works have been exploring the possibility of associating different time scales, developing a multi-scale approach in time, usually called Subcycling technique.

R´esum´e
Contents
List of Figures
List of Tables
Introduction
I State of the art and focus on non-overlapping Domain Decomposition methods
1 Transient dynamics and existing Finite Element methodologies
1 Introduction
2 Initial-boundary value problem
3 Time integration
3.1 Central Difference scheme
3.2 Alternative schemes
3.3 Energy-Momentum integration scheme for large rotations
4 Multi-scale coupling strategies in space and time
4.1 Partitioning in time for a given non-uniform mesh
4.2 Domain Decomposition method
4.3 Arlequin framework
4.4 Global-local approaches and Multi-grid algorithms
5 Existing coupling techniques inside Abaqus
5.1 Impact test example
5.2 Submodeling technique
5.3 Subcycling technique
5.4 Co-Simulation technique
5.5 Comparison between the techniques in [Heimbs, 2011]
6 Conclusions
A weakly-intrusive multi-scale Substitution method in explicit dynamics
2 Investigation and implementation of the Finite Element Tearing and Interconnecting (FETI) approach
1 Introduction
2 Multi-time-step FETI approach
2.1 Formulation
2.2 Application to assess the coupling properties
2.3 Stability assessment with the energy method
3 Energy-preserving multi-time-step FETI approach
3.1 Formulation
3.2 Application to assess the coupling properties
3.3 Conservation assessment with the energy method
4 Solution comparison between GC and GCbis algorithms
5 Conclusions
II Proposal and development of a weakly-intrusive multi-scale Substitution method
3 A weakly-intrusive substitution-based coupling technique
1 Introduction
2 Reference problem
2.1 Interface compatibility and momentum balance
3 Substitution method
3.1 Interface multi-scale compatibility condition
3.2 Global time integration with the correction forces
3.3 Iterative scheme: fixed-point algorithm
3.4 Time down-scaling operator
3.5 Substitution strategy
3.6 Application and results analysis
4 Energy-preserving Substitution method
4.1 Definition of the new interface constraint
4.2 Application and results analysis
4.3 Conclusions
5 Example of simplified impact
5.1 Acceleration of the iterative scheme: direct substitution
6 Conclusions
4 Enhancement of the iterative scheme
1 Introduction
2 Time down-scaling operator property
3 Reformulation of the interface multi-scale compatibility condition
4 Improved iterative scheme
5 Applications
A weakly-intrusive multi-scale Substitution method in explicit dynamics
5.1 Local mesh refinement
5.2 Local defects and heterogeneities
5.3 Composite structure with damageable interface
6 Conclusions
6.1 Investigation towards weakening procedures
6.2 Investigation towards de-refinement procedures
6.3 Investigation towards the displacements continuity: quintic Hermitian
Conclusions and prospects
A Meso-scale model for the Matlab prototype
1 Cohesive element technology
2 Interface contact
3 Isotropic damage model
4 Application to mode-2 mechanism
B Macro-scale model for the Matlab prototype