Gernot Beer

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.

A simple approach to the numerical simulation with trimmed CAD surfaces

Abstract In this work a novel method for the analysis with trimmed CAD surfaces is presented. The method involves an additional mapping step and the attraction stems from its simplicity and ease of implementation into existing Finite Element (FEM) or Boundary Element (BEM) software. The method is first verified with classical test examples in structural mechanics. Then two practical applications are presented one using the FEM, the other the BEM, that show the applicability of the method.

Editorial: Special Issue on Advances in Modeling Rock Engineering Problems

This special issue of the International Journal of Geomechanics contains contributions on advances in rock mechanics, rock slope stability, and tunneling. The broad topic selected for this special issue needed additional focus. This was done by selecting authors within the worldwide rockmechanics community and asking them to provide their latest advancement in modeling engineering problems. The guest editors believe that this goal was achieved, and that some insights and discussion may be stimulated after reading the authors’ work. This special issue includes nine papers and can be divided into three sections from the content point of view. The first section includes three papers devoted to the fundamental aspects in rock mechanics, such as the calibration of advanced constitutive models and the impact of individual textural features on rock strength and fracture mechanics. The topic of geomechanical characterization of shear zones and cataclastic rocks, represented as a bimrock (block-in-matrix rock), is also covered in a specific paper. The attention thenmoves directly to numerical modeling and to twomethods that are currently receiving growing interest: the new advances of the boundary element method (BEM) and the potential of the combined finite-discrete element method (FDEM). While the BEM is described in view of possible applications to underground excavations, the FDEM is shown to play an important role in the study of processes that involve fracture propagation in the rock mass. These aspects are broadly covered in the second section. Examples in numerical modeling of engineering problems constitute the third group of papers. The interest moves from tunneling to rock slope stability. This includes time dependency and damage aspects in tunneling, whereas the potential of the FDEM is shown with reference to the stability of a high rock cut. In closing, the guest editors wish to thank all the authors for their effort and their patience in accepting the time consuming review process, and ASCE for publishing this stimulating special issue.

Isogeometric Boundary Element Method for the Simulation in Tunneling

Isogeometric finite element methods and more recently boundary element methods have been successfully applied to problems in mechanical engineering and have led to an increased accuracy and a reduction in simulation effort. Isogeometric boundary element methods have great potential for the simulation of problems in geomechanics, especially tunneling because an infinite domain can be considered without truncation. In this paper we discuss the implementation of the method in the research software BEFE++. Based on an example of a spherical excavation we show that a significant reduction in the number of parameters for describing the excavation boundary as well as an improved quality of the results can be obtained.

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.

New Developments of the Boundary Element Method for Underground Constructions

AbstractIn this paper, an innovative modeling approach is presented for the simulation of underground construction with the emphasis on tunneling. The boundary element method (BEM) is used, and the theoretical background is discussed first. This is followed by test examples, where the results are compared with results from available software, showing the efficiency and accuracy of the new method. Finally, a practical example considering nonlinear material, sequential excavation, and installation of ground support is presented.

Elimination the influence of the mesh geometry to the quality of global smoothing

In the last years computer performance has become very high, solution methods have been developed, so more and more numerical calculations can be done in an acceptable time. That is one reason why such methods are being used very frequently. However, not only the calculation itself, but also the visualisation of the data places a challenge on the techniques used. Simulation results have very special properties which have to be considered for the visualisation. Ignoring this fact produces an output that makes it impossible to produce confidence in the results. A high quality display of the results in real time is necessary, in order to sell the numerical simulation to a large audience. But the basic characteristics of the data must not be changed in order to keep the basic message. The techniques described here fulfil this very important requirement of scientific information visualisation.

Fluid-Structure Interaction by a Duhamel-BEM / FEM Coupling

A pure interface coupling formulation is developed for time domain analysis of coupled fluid-structure systems. Finite elements are applied to model the structure as an elastic continuum, while the fluid region is modeled as an acoustic media by the Boundary Element Method. The coefficient matrices for the fluid-structure interface are determined by applying unit impulses at the boundary of the fluid regions using the concept of Duhamel integrals, which are numerically approximated by means of the Convolution Quadrature Method. The proposed approach, greatly simplifies the assembly of sub-regions and the coupling to finite elements. The stability and accuracy of the proposed method are verified on some selected numerical examples.

A NURBS-BEM Application in Continuum Damage Mechanics

In this paper the integral equation approach is used to describe the propagation of continuumdamage in three dimensional solids. The governing equation is of integral type and contains bothboundary and domain integrals. Such integrals are computed with the aid of the NURBS functions.The subvolume involved by the damage is modelled by a special mapping procedure that avoids theuse of the internal cells. The implementation is verified on a test case for which an analytical solutionis available.

INTEGRATION OF DESIGN AND ANALYSIS THROUGH BOUNDARY INTEGRAL EQUATIONS

The direct integration of Computer Aided Geometric Design (CAGD) models into a numerical simulation improves the accuracy of the geometrical representation of the problem as well as the efficiency of the overall analysis process. In this work, the complementary features of isogeometric analysis and boundary integral equations are combined to obtain a coalescence of design and analysis which is based on a boundary-only discretization. Following the isogeometric concept, the functions used by CAGD are employed for the simulation. An independent field approximation is applied to obtain a more flexible and efficient formulation. In addition, a procedure is presented which allows a stable analysis of trimmed geometries and a straightforward positioning of collocation points. Several numerical examples demonstrate the characteristics and benefits of the proposed approach. In particular, the independent field approximation improves the computational efficiency and reduces the storage requirements without any loss of accuracy. The proposed methodology permits a seamless integration of the most common design models into an analysis of linear elasticity problems.