# Reformulation of the interface multi-scale compatibility condition

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## Initial-boundary value problem

For the sake of simplicity, a generic initial-boundary value problem for dynamic analysis of structures is considered here, which is a case sufficient to introduce the main concepts that will be used in the thesis. This is not the case when considering contact or impact problems, in which formulation has to be adopted according for instance to [Wriggers and Simo, 1985] for nonlinear contact, to [Wriggers et al., 1990] or [Oancea and Laursen, 1997] for frictional effects, to [Wriggers, 2002] for a wide description of computational procedures.
The initial-boundary value problem for dynamics of structures, defining the spatial domain Ω ⊂ RN , in the strong form reads:
ρu¨ = div(σ) + fbody in Ω × [tin,tend].
u = over ∂Ωu × [tin,tend] u (1.1).
σ · n = fsurf over ∂Ω f × [tin,tend].
{u, u˙} = {u0, v0} in Ω|tin.
where ρ is the density of the material, σ is the stress tensor, fbody indicates the body forces, fsurf is the impact surface load, symbols · and × indicate scalar and vector prod-ucts, respectively, n is the unit normal to the boundary ∂Ω f , u0 and v0 are the initial displacement and velocity, respectively. The times tin and tend denote the initial and the final instants, respectively. The stress tensor σ is related to the strain tensor ε(u) by a material constitutive law.
The first equation of System (1.1) represents the momentum balance in terms of the displacements u(x,t ), x being the position in the domain Ω and t being the time varying in [tin,tend]. The second and the third ones denote essential and natural boundary condi-tions, respectively. The fourth one denotes initial conditions. The prescribed boundary displacements are applied to the boundary ∂Ωu and the prescribed tractions are applied to the boundary ∂Ω f . The union between ∂Ωu and ∂Ω f constitutes the entire boundary of the problem ∂Ω, so that ∂Ω f ∪ ∂Ωu ≡ ∂Ω and ∂Ω f ∩ ∂Ωu ≡ 0/.

### Alternative schemes

The parameters choice of the Central Difference scheme does not introduce any numerical dissipation. Spurious oscillations can so be detected for highly transient dynamic cases, as wave propagation problems. The introduction of a non-diagonal damping matrix should be then necessary to attenuate or avoid such spurious oscillations, requiring a matrix fac-torization to solve the system. Other explicit time integration schemes have been formu-lated for introducing numerical dissipation. For instance, the extension of the same HHT algorithm (briefly described in Remark 4) has been formulated and studied in [Chung and Lee, 1994] and [Hulbert and Chung, 1996]. The so-called Tchamwa-Wielgosz scheme represents another interesting approach, introduced in [Tchamwa et al., 1999] as purely explicit scheme (with γ = 0) and so only first-order accurate. Comparative studies in [Rio et al., 2005] have then shown that remaining dissipative explicit schemes are second-order accurate but often provide less accurate solutions than the Tchamwa-Wielgosz scheme. So, the research of accurate time integration schemes for wave propagation analysis is still ongoing. In what follows, two potential strategies are presented.

#### 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.

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Energy-Momentum integration scheme for large rotations

The initial-boundary value problem has been so far considered assuming the linear elas-tic 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], tem-poral 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 es-timate. For instance, the conservation form of the mid-point rule is an exact momentum conserving algorithm which does not conserve energy for autonomous Hamiltonian sys-tems, 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 [San-sour 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).

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
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
interpolation
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