S–R decomposition based numerical manifold method

• A dynamics formulation based on the S–R decomposition is deduced.
• An update scheme for the co-moving coordinate used in the S–R decomposition is built.
• A new procedure named S–R-D-based NMM is established for addressing small or large deformation together with impact/contact.
• Some issues such as the false volume expansion in the conventional small deformation based NMM are effectively overcome.

The numerical manifold method (NMM) surmounting the mesh dependence has successfully solved very complicated problems involving small deformation and large movement, but had few applications to large deformation and large rotation problems because the false volume expansion and other issues exist. In this study it is shown that the false volume expansion in NMM can be excellently resolved by using the S–R (strain–rotation) decomposition theorem which can precisely reflect complex physical behaviors occurring in the process of large rotation and large deformation. The numerical methods based on the S–R decomposition theorem have been limited to the static analysis of large deformations. To remove this limitation, a new formulation taking into account dynamical features is proposed based on the weak form of momentum conservation law. Under the framework of NMM, the generalized-αα method is employed to discretize the temporal variables. The updates of variables are described using the updated co-moving coordinate system. Thus, a new method named S–R-D-based NMM is established. The new formulation can be implemented in any other partition of unity based methods as well, so as to improve the performances of such methods in the analysis of dynamic large deformations.