Robust eXtended finite elements for complex cutting of deformables

Abstract

In this paper we present a robust remeshing-free cutting algorithm on the basis of the eXtended Finite Element Method (XFEM) and fully implicit time integration. One of the most crucial points of the XFEM is that integrals over discontinuous polynomials have to be computed on subdomains of the polyhedral elements. Most existing approaches construct a cut-aligned auxiliary mesh for integration. In contrast, we propose a cutting algorithm that includes the construction of specialized quadrature rules for each dissected element without the requirement to explicitly represent the arising subdomains. Moreover, we solve the problem of ill-conditioned or even numerically singular solver matrices during time integration using a novel algorithm that constrains non-contributing degrees of freedom (DOFs) and introduce a preconditioner that efficiently reuses the constructed quadrature weights. Our method is particularly suitable for fine structural cutting as it decouples the added number of DOFs from the cut's geometry and correctly preserves geometry and physical properties by accurate integration. Due to the implicit time integration these fine features can still be simulated robustly using large time steps. As opposed to this, the vast majority of existing approaches either use remeshing or element duplication. Remeshing based methods are able to correctly preserve physical quantities but strongly couple cut geometry and mesh resolution leading to an unnecessary large number of additional DOFs. Element duplication based approaches keep the number of additional DOFs small but fail at correct conservation of mass and stiffness properties. We verify consistency and robustness of our approach on simple and reproducible academic examples while stability and applicability are demonstrated in large scenarios with complex and fine structural cutting.

Anisotropic elasticity for inversion-safety and element rehabilitation

“…Degenerate elements are also encountered during fracture simulation. Remeshing alternatives such as virtual node [Molino et al 2004] and XFEM [Belytschko et al 2013;Koschier et al 2017] methods have been developed, but these algorithms assume that an initially well-conditioned element has fractured into badly-conditioned components. We instead examine the case where high-quality elements are ruined via warping.…”

“…We accomplish this by replacing the numerically troublesome direction of the (arbitrary) isotropic energy with a well-behaved anisotropic energy. In contrast to accuracy-centered p-refinement [Schneider et al 2018] or XFEM [Koschier et al 2017] techniques, our robustness-centered approach tends to decrease the condition number of the final Hessian.…”

Cd-MPM

“…These physics-based approaches further deviate in their choice of discretization of the governing equations: finite element methods (FEMs) were among the first to be successful [O'Brien et al 2002;O'Brien and Hodgins 1999], boundary element methods (BEMs) are recently gaining traction [Da et al 2016;Wojtan 2015, 2016], and meshless methods showed success for large topology change of solids early on [Pauly et al 2005] and are on the rise with the growing versatility and popularity of the Material Point Method (MPM) [Sulsky et al 1995]. Though they show great success, mesh-based methods require computationally intensive re-meshing routines or complex algorithms like VNA and XFEM [Koschier et al 2017;Molino et al 2005] and BEMs in particular have not yet been successfully applied to ductile fracture. Conversely, meshless methods such as MPM show great promise due to automatic support for arbitrarily large topological changes, natural collision handling, and a wide variety of successfully simulated continuum materials.…”

Accelerated Simulation of Brittle Objects for Interactive Applications

“…Some methods propose to virtually cut volumes while keeping well-shaped elements but the used mesh is no longer manifold (Molino et al, 2004). XFEM approaches consider discontinuous element shape functions to represent and simulate fractures (Koschier et al, 2017). Some interactive methods tend to use predefined crack patterns that locally modify the mesh (Müller et al, 2013;Chen et al, 2014;Parker and O'Brien, 2009), but cracks are not "physicallymanaged" in such cases.…”