The Finite Cell Method for Transient, Non-linear Heat Conduction

Since the early years of computational engineering, the classical Finite Element Method (FEM) has become the state-of-the-art approach to solve initial boundary value problems numerically. Although major enhancements allowed for highly sophisticated simulations, the idea to geometrically resolve the physical domain on the discretization level remained unchanged. In cases of complex geometries, this intrinsic need for a conform mesh is still a burden in today’s engineering practice. The recently introduced Finite Cell Method (FCM) overcomes this problem by combining the benefits of high-order Finite Elements (p-FEM) with the idea of fictitious domains. The approach embeds the possibly complex physical domain phys in a fictitious domain ?fict and solves the problem on their simply shaped union. The localization factor ? allows to recover the original geometry on the integration level. Recent research results confirm the excellent applicability of this new method in the fields of non-linear continuum mechanics, topology optimization, thin-walled structures, bone mechanics and advection-diffusion problems. Also the method’s potential for multi-physical problems has been demonstrated in the context of linear thermoelasticity. In the present work, the Finite Cell Method is employed to solve the transient, non-linear heat equation on non-conforming meshes. In particular, the weak enforcement of Dirichlet boundary conditions is addressed. The results of a conventional p-FEM simulation serve as reference and the convergence characteristics of both approaches are compared.     «

Since the early years of computational engineering, the classical Finite Element Method (FEM) has become the state-of-the-art approach to solve initial boundary value problems numerically. Although major enhancements allowed for highly sophisticated simulations, the idea to geometrically resolve the physical domain on the discretization level remained unchanged. In cases of complex geometries, this intrinsic need for a conform mesh is still a burden in today’s engineering practice. The recently...     »