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The $$hp$$-$$d$$-adaptive finite cell method for geometrically nonlinear problems of solid mechanics. (English) Zbl 1242.74161
Summary: The finite cell method (FCM) combines the fictitious domain approach with the p-version of the finite element method and adaptive integration. For problems of linear elasticity, it offers high convergence rates and simple mesh generation, irrespective of the geometric complexity involved. This article presents the integration of the FCM into the framework of nonlinear finite element technology. However, the penalty parameter of the fictitious domain is restricted to a few orders of magnitude in order to maintain local uniqueness of the deformation map. As a consequence of the weak penalization, nonlinear strain measures provoke excessive stress oscillations in the cells cut by geometric boundaries, leading to a low algebraic rate of convergence. Therefore, the FCM approach is complemented by a local overlay of linear hierarchical basis functions in the sense of the $$hp$$-$$d$$ method, which synergetically uses the h-adaptivity of the integration scheme. Numerical experiments show that the $$hp$$-$$d$$ overlay effectively reduces oscillations and permits stronger penalization of the fictitious domain by stabilizing the deformation map. The $$hp$$-$$d$$-adaptive FCM is thus able to restore high convergence rates for the geometrically nonlinear case, while preserving the easy meshing property of the original FCM. Accuracy and performance of the present scheme are demonstrated by several benchmark problems in one, two, and three dimensions and the nonlinear simulation of a complex foam sample.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics
##### Software:
ParaView; Trilinos; Netgen; Matlab; HYPLAS
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