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# Hybrid Galerkin-collocation methods for surface-oriented modeling of nonlinear problems in solid mechanics

Project members: Prof. Dr. Bernd Simeon and M. Sc. Clarissa Arioli

**Project poster:** YASON poster

Project partners:

- Prof. Dr. Bernd Simeon (TU Kaiserslautern, Felix-Klein-Zentrum)
- Prof. Dr.-Ing. Sven Klinkel (RWTH Aachen)

Project duration: 3 years, 2016 - 2018

# Project contents and goals

The geometric model employed for the design process in standard Computer Aided Design (CAD) software differs completely from the geometric description used in the well-established Finite Element Method (FEM) for structural analysis. A common method to define solids in CAD is the boundary representation modeling technique that defines the solid in terms of bounded NURBS surfaces (Non-Uniform Rational B-Splines). The interior geometry needs to be remodeled by finite element meshing or by deriving a tri-variate NURBS parametrization for Isogeometric Analysis (IGA). In case of the FEM, it results in an approximation of the geometry and in an additional error in the response analysis. The Scaled Boundary Finite Element Method (SB-FEM) parametrizes the structure by a radial scaling parameter that emanates from a scaling center and a parameter in circumferential direction along the boundary. While the weak form of equilibrium is enforced in circumferential direction and treated by standard projection onto the space of piecewise Lagrange polynomials, the strong form is enforced in scaling direction. For linear problems, it results in an ordinary differential equation (ODE) in terms of the scaling parameter. Similar to the properties of the FEM, this boundary-oriented approach does not represent the geometry exactly but converges to the exact geometry under mesh refinement. The main objective of this project is to develop a computational method that combines the features of isogeometric analysis and of the scaled-boundary approach in order to make direct use of the surface modeling technique that dominates in CAD today. Moreover, we seek for methods that apply to a wide class of nonlinear continuum mechanics problems. While a NURBS-based Galerkin projection will be used to treat the surface integrals, the remaining ODE problem with its weak singularity in the scaling center demands for novel discretization methods. Finally, inter-patch connections for 3D surface configurations are addressed to cover general engineering shapes.