Finite element analysis on implicitly defined domains: An accurate representation based on arbitrary parametric surfaces

Abstract

a b s t r a c tIn this paper, we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis. The novel contributions of this paper are threefold: (1) describe and validate a method to represent arbitrary parametric surfaces implicitly; (2) represent arbitrary solids implicitly, including sharp features using level sets and boolean operations; (3) impose arbitrary Dirichlet and Neumann boundary conditions on the resulting implicitly defined boundaries. The methods proposed do not require local refinement of the finite element mesh in regions of high curvature, ensure the independence of the domain's volume on the mesh, do not rely on boundary regularization, and are well suited to methods based on fixed grids such as the extended finite element method (XFEM). Numerical examples are presented to demonstrate the robustness and effectiveness of the proposed approach and show that it is possible to achieve optimal convergence rates using a fully implicit representation of object boundaries. This approach is one step in the desired direction of tying numerical simulations to computer aided design (CAD), similarly to the isogeometric analysis paradigm.

Implementation of regularized isogeometric boundary element methods for gradient‐based shape optimization in two‐dimensional linear elasticity

“…The geometry in the design mesh are set as design variables. The lower bound is [45,15,15,15], and the upper bound is [70,70,70,70]. The monitoring points are chosen on GH.…”

“…However, due to the separation of the geometry and the analysis mesh, the capture of the geometry boundary for domain integration is not a trivial task. Moës [44] proposed an XFEM-based method for complex microstructures, which was generalized by Moumnassi et al [45] to treat arbitrary CAD geometries implicitly, including corners and sharp edges. Isogeometric analysis A recent trend in shape optimization is isogeometric analysis (IGA) [46], which integrates the geometry and analysis representations.…”

A hybrid smoothed extended finite element/level set method for modeling equilibrium shapes of nano-inhomogeneities

“…The level set method was originally devised for tracking a moving interface [32], and became a key ingredient of XFEM to implicitly describe complicated geometrical interfaces of microstructures without tracking them explicitly, such as cracks [33], holes and inclusions [20,34], dislocations [35] and biofilms [23,24], as well as fluid-structure interfaces [36].…”

Stabilized method of imposing Dirichlet boundary conditions using a recovered stress field

“…Integration: As the mesh does not conform to the geometry of the domain it is necessary to use special procedures to evaluate integrals. Different approaches have been considered in the bibliography to ensure that the integration in each element is only extended to the exact part of the volume (area in 2D), see for example references [3,4,5,6]. In general terms the solution to this problem consists of using two different meshes, one for interpolation and another for integration.…”