Thep‐Version of the Finite Element and Finite Cell Methods

Abstract

In the first part of this chapter, the basic algorithmic structure and performance characteristics of the p ‐version of the finite element method (FEM) are surveyed with reference to elliptic problems in solid mechanics. For this class of problems, the theoretical basis of the p ‐version is fully established, and a very substantial amount of engineering experience covering linear and nonlinear applications is available. It is shown that p ‐extensions on properly designed meshes make realization of exponential rates of convergence in practical computations possible and provide for the estimation and control of relative errors in terms of any quantity of interest. In the second part, the p ‐version of the FEM is extended to a high‐order fictitious domain approach, the finite cell method (FCM). While the FCM inherits the advantages of the p ‐version with respect to accuracy and robustness, it relieves analysts from the necessity of generating finite element meshes. Thus, it strongly supports the analysis of problems with highly complicated geometry, for which meshing with finite elements would be very difficult. Given the growing demand for verified numerical solutions, the p ‐version of the finite element and FCMs are expected to play an increasingly important role.

Experimental and numerical investigation of single pores for identification of effective metal foams properties

“…Concerning the spatial discretization error, the FCM shows similar convergence properties like the FEM when performing a mesh refinement or increasing the polynomial degree of the shape functions . However, in order to achieve the same convergence behaviour, the numerical integration of the broken cells has to be carried out accurately enough . Finally, there was the proof of principle that the averaged pore level material parameters from the parameter identification can be used as effective material properties to simulate the mechanical behavior of any pore, regardless of the specific geometry.…”

Direct structural analysis of domains defined by point clouds

“…The description is limited only to the necessary minimum for the context of this paper. For further details, refer to [19]. Similar to standard finite elements, the FCM is derived from the principle of virtual work [20]:…”

Residual stresses in metal deposition modeling: Discretizations of higher order

“…They have, therefore, emerged as the method of choice to perform numerical simulations on bodies with a complex shape, topology or a combination of both. The finite cell method (FCM) introduced in [18,19], is a prominent representative of immersed methods whose core idea is to combine the advantages of the fictitious domain approach with the computational efficiency of discretizations of high order. Its basic idea is depicted in fig.…”