Reduced order methods for fracture simulation and nonlinear materials

Nonlinear problems, fractures and cutting in large deformations are known to be very challenging from a computational cost point of view. In the aim of computing real time simulations of surgical operations, we cannot settle with “classical” numerical methods such as for instance finite elements. We propose to apply some Reduced Order Methods (ROM) in the field of interest, which represents an ongoing investigation subject. In general, reduced order methods are based on the idea of decreasing the number of degrees of freedom using a very low number of mesh independent shape functions. Several techniques exist in literature for the computation of these shape functions, the most largely known are for instance the Reduced-Basis method (Maday & Rønquist, 2002), the Proper Orthogonal Decomposition (POD) (Kunisch & Volkwein, 2001; Rathinam & Petzold, 2003) and the Proper Generalized Decomposition (PGD) (Chinesta et al., 2011; Ladevèze & Nouy, 2003) methods. The PGD has been successfully used by Chinesta, Cueto et al. for real-time simulation of biological soft tissues (Lafortune & Aris, 2012). Nevertheless, the method is able to deal with contact problems in non-linear hyperelasticity equations but cannot reproduce cutting problems, yet.

Partitioned Model Order Reduction for fracture simulation

Projection-based reduced-order modelling reduces greatly the number of degrees of freedom of mechanical simulations by projecting the state variables onto a space of low dimension. It typically fails when modelling phenomena of localized high nonlinearity, such as a fracture.

Recently, the application of the POD method in multiscale fracture mechanics has been the subject of a PhD dissertation (Goury, 2014) supervised by Dr. Kerfriden and Prof. Bordas. The main idea introduced in this work is the combination of a ROM with a domain decomposition technique. The domain of interest is divided in several subdomains and a POD approach is used in each subdomain except for the subdomain where the fracture is present. By partitioning the domain of study into several subdomains each represented by an individual reduced basis, we can adapt the level of accuracy by choosing an independent basis size for each region. The method allows for great time savings while keeping the accuracy in the fracture region.

 

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Reduced order methods for nonlinear materials

We are now focused on fast simulation of nonlinear hyperelastic materials. The application of ROMs is one of the possible solution. Using the same idea introduced for fracture problems, we are investigating how these methods can be combined with advanced existing techniques such as domain decompositions and Empirical Interpolation Methods (EIM) (Barrault et al., 2004).