# Department of Mechanics: Seminar: Abstract Beex 2016

## Similarities and differences between the Quasicontinuum method and POD-based reduction methods

__L.A.A. Beex__, E. Schenone, J.S. Hale

Research Unit of Engineering Science, University of Luxembourg, LU

L.A.A.Beex@gmail.com

Large computations of non-linear mechanical models are inefficient due to

- the large numbers of degrees of freedom involved (and hence, the solution of large systems), and
- the large numbers of integration points that need to be visited to construct these large systems.

Numerical strategies can be employed to overcome these two issues. In this presentation, two types of such strategies will be discussed, including their resemblance and differences. The first category is the multiscale quasicontinuum (QC) method. The second consists of reduced order modelling approaches based on proper-orthogonal-decomposition (POD). Both approaches reduce the number of degrees of freedom by interpolation (avoiding issue (1)). Second, they select just a few integration points to sample the contributions of all integration points (avoiding issue (2)).

POD methods are broadly applicable, but require several full-scale computations to be performed, before they can be applied. Furthermore, relatively few procedures to select the reduced integration points exist [1-2], although we have recently made some contributions in this field [3]. The multiscale QC method on the other hand does not require full-scale computations to be performed a-priori. Dirichlet boundary conditions can also be applied more straightforwardly and the selection of reduced integration points is studied more extensively [4-7]. A major disadvantage of the QC method on the other hand is that it can only be used for regular lattice computations up till now (see e.g. [8]).

#### References

[1] Barault et al., 2004, *C. R. Math. Acad. Sci.* 339, 6667-672.

[2] Chaturantabut & Sorensen, 2010, *SIAM J. Sci. Com.* 32(5), 2737-2764.

[3] Schenone et al., in preparation.

[4] Knap & Ortiz, 2001, *JMPS* 49, 1899-1932.

[5] Beex et al., 2011, *IJNME* 87, 701-718.

[6] Beex et al., 2014, *JMPS* 70, 242-261.

[7] Amelang et al., 2015, *JMPS* 82, 378-413.

[8] Rokoš et al., *IJSS*, in press.