SUMMARYA unified strategy for the higher-order accurate integration of implicitly defined geometries is proposed. The geometry is represented by a higher-order level-set function. The task is to integrate either on the zero-level set or in the sub-domains defined by the sign of the level-set function. In three dimensions, this is either an integration on a surface or inside a volume. A starting point is the identification and meshing of the zero-level set by means of higher-order interface elements. For the volume integration, special sub-elements are proposed where the element faces coincide with the identified interface elements on the zero-level set. Standard Gauss points are mapped onto the interface elements or into the volumetric sub-elements. The resulting integration points may, for example, be used in fictitious domain methods and extended finite element methods. For the case of hexahedral meshes, parts of the approach may also be seen as a higher-order marching cubes algorithm.
An accurate implicit description of geometries is enabled by the level-set method.Level-set data is given at the nodes of a higher-order background mesh and the interpolated zero-level sets imply boundaries of the domain or interfaces within. The higher-order accurate integration of elements cut by the zero-level sets is described.The proposed strategy relies on an automatic meshing of the cut elements. Firstly, the zero-level sets are identified and meshed by higher-order interface elements. Secondly, the cut elements are decomposed into conforming sub-elements on the two sides of the zero-level sets. Any quadrature rule may then be employed within the sub-elements.The approach is described in two and three dimensions without any requirements on the background meshes. Special attention is given to the consideration of corners and edges of the implicit geometries.
SUMMARYA new higher-order accurate method is proposed that combines the advantages of the classical p-version of the FEM on body-fitted meshes with embedded domain methods. A background mesh composed by higherorder Lagrange elements is used. Boundaries and interfaces are described implicitly by the level set method and are within elements. In the elements cut by the boundaries or interfaces, an automatic decomposition into higher-order accurate sub-elements is realized. Therefore, the zero level sets are detected and meshed in a first step, which is called reconstruction. Then, based on the topological situation in the cut element, higher-order sub-elements are mapped to the two sides of the boundary or interface. The quality of the reconstruction and the mapping largely determines the properties of the resulting, automatically generated conforming mesh. It is found that optimal convergence rates are possible although the resulting sub-elements are not always well-shaped.
Numerical simulations of structures using higher-order finite elements is still a challenging task, in particular for domains with curved boundaries. A new higher-order accurate approach is proposed, combining the advantages of the classical p-FEM with embedded domain methods. Boundaries and/or interfaces are described implicitly using the level set method. In the elements cut by the zero level set, an automatic decomposition into interface aligned, i.e. conforming sub-elements is realized. Transfinite mappings are utilized to construct higher-order sub-elements by mappings of reference elements to the two sides of the boundary or interface. It is shown that although the resulting sub-elements are not always well-shaped, optimal convergence rates are possible.
In this work, an approach is presented to improve global accuracy properties for physically non-linear problems in the frame of elastoplasticity. The work is motivated by the fact that convergence characteristics of a finite element solution are dominated by the regularity of the exact solution. For a material undergoing inelastic deformations, however, very few analytical solutions for the field variables are known, especially for the displacement field. Considering a simple, one-dimensional example, it is shown that the convergence rates are far from optimal. The reason is explained by establishing ties to a familiar, but more demonstrative problem. In the next section two remedies for the problem, based on the Extended Finite Element Method (XFEM) are presented and discussed, in the last section a 2D-problem is considered.
Elasticity in 1DAs shown in , it is possible to make a priori estimates on convergence rates of a finite element solution. Depending on the regularity properties of the exact solution and the polynomial degree of the elements (further referred to as m), the optimal convergence rate (k) for the displacement field using a uniform refinement in a L 2 -norm, is given by k = m + 1. This result however, is valid for a smooth distribution of the field variables only, therefore the exact solution u ex is required to be at leastConsider a one-dimensional elastic bar, consisting of two non-overlapping domains Ω = Ω 1 ∪ Ω 2 with the intersection Γ = Ω 1 ∩ Ω 2 being the interface. The properties associated with the two domains are the cross-section A and the Young's moduli E 1 and E 2 . Now, in case that E 1 = E 2 = E, E ∈ C ∞ , a finite element analysis gives optimal convergence rates, as pictured in Fig. 1 by the continuous lines. In this case, u ex ∈ C ∞ (Ω), therefore clearly fulfilling the minimum requirement of C m (Ω)-continuity. However, altering the material properties, let e.g. E 1 > E 2 , E ∈ C −1 , changes the behaviour drastically. The convergence rates drop, settling at a rate of 1, this is depicted in Fig. 1 by the dashed lines. This is explained by a kink in the analytical displacement field u ex , right at the material interface Γ. The analytical solution is therefore C 0 (Ω)-continuous, but still C ∞ in Ω 1 and Ω 2 .
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