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A coupling strategy for adaptive local refinement in space and time with a fixed global model in explicit dynamics

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Abstract

In dynamics, domain decomposition methods (DDMs) enable one to use different spatial and temporal discretizations depending on the physical phenomenon being taken into account. Thus, DDMs provide the analyst with key tools for dealing with problems in which phenomena occur on different temporal and spatial scales. This paper focuses on a less intrusive variation of this type of method which enables the global (industrial) mesh to remain unchanged while the local problem is being refined in space and in time where needed. This property is particularly useful in the case of a local problem whose localization evolves rapidly with time, as is the case for delamination. The downside is that the technique is iterative. The method is presented in the context of linear explicit dynamics, but, as with domain decomposition, its extension to other integration schemes and to nonlinear problems should be possible.

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Acknowledgments

The authors wish to thank Victor Oancea Simulia in Providence for his fruitful comments and support.

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Correspondence to Omar Bettinotti.

Appendix: connectivity matrices and spatial operators

Appendix: connectivity matrices and spatial operators

With reference to Fig. 3 and numbering nodes firstly from the bottom to the top and secondly from left to right, the connectivity matrices for the DDM write:

$$\begin{aligned} \mathbf{C}_1&= \left[ \begin{array}{cccccc} \mathbf{0}_{10}&\mathbf{0}_{10}&\mathbf{0}_{10}&\mathbf{0}_{10}&\mathbf{0}_{10}&\mathbf{I}_{10} \end{array} \right] ,\nonumber \\ \mathbf{C}_2&= \left[ \begin{array}{cc} \mathbf{I}_{4}&\mathbf{0}_{4} \end{array} \right] , \end{aligned}$$
(55)

where \(\mathbf{0}_{2n}\) is a squared matrix of zeros of dimension \(2n\) and \(\mathbf{I}_{2n}\) is an identity matrix of dimension \(2n.\)

With reference to Figs. 4 and 5 and numbering nodes firstly from the bottom to the top and secondly from left to right, the connectivity matrices write:

$$\begin{aligned}&\mathbf{C}_g=\left[ \begin{array}{ccc} \mathbf{0}_{4}&\mathbf{I}_{4}&\mathbf{0}_{4} \end{array} \right] ,\nonumber \\&\mathbf{C}_l=\left[ \begin{array}{cccccc} \mathbf{0}_{10}&\mathbf{0}_{10}&\mathbf{0}_{10}&\mathbf{0}_{10}&\mathbf{0}_{10}&\mathbf{I}_{10} \end{array} \right] , \end{aligned}$$
(56)

with the same notation mentioned above in the Appendix.

Furthermore, the incompatibility between the meshes both in domain decomposition and in substitution leads to the definition of projection and interpolation operators, used to impose constraints or assign values at the nodes not in common. So, referred to position parameters \(y_i\) and \(h\) in Fig. 25, the projection operator writes:

$$\begin{aligned} \tilde{\varPi }_h^1(\mathbf{X})=\left[ \begin{array}{ccccc} \mathbf{I}_{2} &{} \ldots &{} \left( 1-\frac{y_i}{h}\right) \mathbf{I}_{2} &{} \ldots &{} \mathbf{0}_{2}\\ \mathbf{0}_{2} &{} \ldots &{} \frac{y_i}{h}\mathbf{I}_{2} &{} \ldots &{} \mathbb I _{2} \end{array} \right] \mathbf{X}, \end{aligned}$$
(57)

and the interpolation operator simply writes:

$$\begin{aligned} \varPi _h^1(\mathbf{X})=\left[ \begin{array}{ccccc} \mathbf{I}_{2} &{} \ldots &{} \left( 1-\frac{y_i}{h}\right) \mathbf{I}_{2} &{} \ldots &{} \mathbf{0}_{2}\\ \mathbf{0}_{2} &{} \ldots &{} \frac{y_i}{h}\mathbf{I}_{2} &{} \ldots &{} \mathbb I _{2} \end{array} \right] ^T\mathbf{X}. \end{aligned}$$
(58)

Considering the homogeneous structured mesh, the two Eqs. (57) and (58) simply become:

$$\begin{aligned} \tilde{\varPi }_h^1(\mathbf{X})&= \left[ \begin{array}{ccccc} \mathbf{I}_{2} &{} \frac{3}{4}\mathbf{I}_{2} &{} \frac{1}{2}\mathbf{I}_{2} &{} \frac{1}{4}\mathbf{I}_{2} &{} \mathbf{0}_{2}\\ \mathbf{0}_{2} &{} \frac{1}{4}\mathbf{I}_{2} &{} \frac{1}{2}\mathbf{I}_{2} &{} \frac{3}{4}\mathbf{I}_{2} &{} \mathbb I _{2} \end{array} \right] \mathbf{X},\nonumber \\ \varPi _h^1(\mathbf{X})&= \left[ \begin{array}{ccccc} \mathbf{I}_{2} &{} \frac{3}{4}\mathbf{I}_{2} &{} \frac{1}{2}\mathbf{I}_{2} &{} \frac{1}{4}\mathbf{I}_{2} &{} \mathbf{0}_{2}\\ \mathbf{0}_{2} &{} \frac{1}{4}\mathbf{I}_{2} &{} \frac{1}{2}\mathbf{I}_{2} &{} \frac{3}{4}\mathbf{I}_{2} &{} \mathbf{I}_{2} \end{array} \right] ^T\mathbf{X}. \end{aligned}$$
(59)
Fig. 25
figure 25

The position along the interface that determines the values in the projection and interpolation operators

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Bettinotti, O., Allix, O. & Malherbe, B. A coupling strategy for adaptive local refinement in space and time with a fixed global model in explicit dynamics. Comput Mech 53, 561–574 (2014). https://doi.org/10.1007/s00466-013-0917-9

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