The contribution presents a new numerical method to solve nonlinear problems of solids in boundary representation. A formulation for material nonlinearities is derived. The proposed method introduces an approach where the geometrical description of the boundary is sufficient to define the complete solid. While the interior of the domain is described by a radial scaling parameter, the scaling of the boundary with respect to the specified scaling center leads to the complete solid. This idea fits perfectly to the boundary representation modeling technique commonly employed in CAD. The approach exploits the tensor‐product structure of the solid to parameterize the physical domain, i.e., two‐dimensional surfaces are represented by NURBS objects, which parameterize the boundary surfaces. Following the isogeometric paradigm, the NURBS functions that describe the boundary of the geometry form also the basis for the approximation of the displacement at the boundary. The displacement response in the circumferential and radial scaling direction is approximated by one‐dimensional NURBS. The present formulation accounts for material nonlinearity with elasto‐plastic behavior, where small strain theory is assumed. Similar to the Scaled Boundary Finite Element Method (SB‐FEM), the structure is parameterized by a radial scaling parameter that emanates from a scaling center and a parameter in circumferential direction along the boundary. The Galerkin projection of the weak form yields a system of equilibrium equations whose solution gives rise to the displacement response. Due to the nonlinear relation between the stress and the strain, the linear equilibrium equation is not applicable anymore. Applying the weak form in the circumferential and radial direction leads to a nonlinear equation with respect to the unknown displacement response, which is solved with a linearization and the Newton‐Raphson scheme. The applicability of the proposed formulation is shown by means of numerical examples. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Proceedings in Applied Mathematics & Mechanics – Wiley
Published: Jan 1, 2017
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