Thomas-Peter Fries

Fast isogeometric boundary element method based on independent field approximation

Abstract An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology are demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.

The extended finite element method for two-phase and free-surface flows: A systematic study

In immiscible two-phase flows, jumps or kinks are present in the velocity and pressure fields across the interfaces of the two fluids. The extended finite element method (XFEM) is able to reproduce such discontinuities within elements. Robust and accurate interface capturing schemes with no restrictions on the interface topology are thereby enabled. This paper investigates different enrichment schemes and time-integration schemes within the XFEM. Test cases with and without surface tension on moving or stationary meshes are studied and compared to interface tracking results when possible. A particularly useful setting is extracted which is recommended for two-phase flows. An extension of this formulation for the simulation of free-surface flows and of floating objects is proposed.

Crack propagation criteria in three dimensions using the XFEM and an explicit–implicit crack description

This paper studies propagation criteria in three-dimensional fracture mechanics within the extended finite element framework (XFEM). The crack in this paper is described by a hybrid explicit–implicit approach as proposed in Fries and Baydoun (Int J Numer Methods Eng, 2011). In this approach, the crack update is realized based on an explicit crack surface mesh which allows an investigation of different propagation criteria. In contrast, for the computation of the displacements, stresses and strains by means of the XFEM, an implicit description by level set functions is employed. The maximum circumferential stress criterion, the maximum strain energy release rate criterion, the minimal strain energy density criterion and the material forces criterion are realized. The propagation paths from different criteria are studied and compared for asymmetric bending, torsion, and combined bending and torsion test cases. It is found that the maximum strain energy release rate and maximum circumferential stress criterion show the most favorable results.

Higher-order meshing of implicit geometries—Part I: Integration and interpolation in cut elements

Abstract 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 surface 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.

The XFEM with an Explicit-Implicit Crack Description for Hydraulic Fracture Problems

The Extended Finite Element Method (XFEM) approach is applied to the coupled problem of fluid flow, solid deformation, and fracture propagation. The XFEM model description of hydraulic fracture propagation is part of a joint project in which the developed numerical model will be verified against large-scale laboratory experiments. XFEM forms an important basis towards future combination with heat and mass transport simulators and extension to more complex fracture systems. The crack is described implicitly using three level-sets to evaluate enrichment functions. Additionally, an explicit crack representation is used to up‐ date the crack during propagation. The level-set functions are computed exactly from the ex‐ plicit representation. This explicit/implicit representation is applied to a fluid-filled crack in an impermeable, elastic solid and compared to the early-time solution of a plane-strain hy‐ draulic fracture problem with a fluid lag.

Stable isogeometric analysis of trimmed geometries

Abstract We explore extended B-splines as a stable basis for isogeometric analysis with trimmed parameter spaces. The stabilization is accomplished by an appropriate substitution of B-splines that may lead to ill-conditioned system matrices. The construction for non-uniform knot vectors is presented. The properties of extended B-splines are examined in the context of interpolation, potential, and linear elasticity problems and excellent results are attained. The analysis is performed by an isogeometric boundary element formulation using collocation. It is argued that extended B-splines provide a flexible and simple stabilization scheme which ideally suits the isogeometric paradigm.

Meshfree Petrov-Galerkin Methods for the Incompressible Navier-Stokes Equations

Meshfree stabilised methods are employed and compared for the solution of the incompressible Navier-Stokes equations in Eulerian formulation. These Petrov-Galerkin methods are standard tools in the FEM context, and can be used for meshfree methods as well. However, the choice of the stabilisation parameter has to be reconsidered. We find that reliable and successful approximation with standard formulas for the stabilisation parameter can only be expected for shape functions with small supports or dilatation parameters.

Higher-order meshing of implicit geometries, Part II: Approximations on manifolds

Abstract A new concept for the higher-order accurate approximation of partial differential equations on manifolds is proposed where a surface mesh composed by higher-order elements is automatically generated based on level-set data. Thereby, it enables a completely automatic workflow from the geometric description to the numerical analysis without any user-intervention. A master level-set function defines the shape of the manifold through its zero-isosurface which is then restricted to a finite domain by additional level-set functions. It is ensured that the surface elements are sufficiently continuous and shape regular which is achieved by manipulating the background mesh. The numerical results show that optimal convergence rates are obtained with a moderate increase in the condition number compared to handcrafted surface meshes.

Higher-Order Accurate Integration for Cut Elements with Chen-Babuška Nodes

The higher-order accurate numerical integration of geometries that are implicitly defined by level-set functions is considered. A higher-order background mesh is employed providing an interpolation of the level-set function by Lagrangian shape functions. The integration may take place on the zero-level set or in the domains defined by the sign of the level-set function. This work is a follow-up of Fries and Omerovic (Int J Numer Methods Eng, doi:10.1002/nme.5121). Herein, it is shown that special distributions of the element nodes, which are optimized for integration, yield significantly better results than equally-spaced nodes. Different error norms are proposed which allow to investigate the accuracy of general implicit geometries in two and three dimensions.

Stress Intensity Factors Through Crack Opening Displacements in the XFEM

The computation of stress intensity factors (SIFs) for two- and three-dimensional cracks based on crack opening displacements (CODs) is presented in linear elastic fracture mechanics. For the evaluation, two different states are involved. An approximated state represents the computed displacements in the solid, which is obtained by an extended finite element method (XFEM) simulation based on a hybrid explicit-implicit crack description. On the other hand, a reference state is defined which represents the expected openings for a pure mode I, I​I and I​I​I. This reference state is aligned with the (curved) crack surface and extracted from the level-set functions, no matter whether the crack is planar or not. Furthermore, as only displacements are fitted, no additional considerations for pressurized crack surfaces are required. The proposed method offers an intuitive, robust and computationally cheap technique for the computation of SIFs where two- and three-dimensional crack configurations are treated in the same manner.