arXiv subjects

Gernot Beer

Publications and source records attributed to Gernot Beer.

12 recordsLinked to original sources

arXiv2025

Efficient isogeometric Boundary Element simulation of elastic domains containing thin inclusions

This paper is concerned with the Boundary Element simulation of elastic domains that contain thin inclusions that have elastic material properties, which are different to the domain. With thin inclusions we mean inclusions with extreme aspect ratios, i.e. where one dimension is much smaller than the other ones. Examples of this are reinforcements in civil/mechanical engineering or concrete linings in underground construction. The fact that an inclusion has an extreme aspect ratio poses a challenge to the numerical integration of the arising singular integrals and novel approaches are presented to deal with it. Several examples demonstrate the efficiency and accuracy of the proposed methods and show that the results are in good agreement with analytical and other numerical solutions.

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arXiv2022

Immersed isogeometric Boundary Elements: A user friendly method for the 3-D elasto-plastic simulation of underground excavations

The immersed isogeometric Boundary Element Method is presented and applied to the simulation of underground excavations. Nonuniform rational B-splines (NURBS) are used for the accurate definition of complex geometries with few parameters. Immersed technology is applied to automatically generate cell meshes. This allows heterogeneous and anisotropic ground conditions as well as nonlinear material behaviour to be considered. On a practical example the user friendliness and accuracy is demonstrated.

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arXiv2022

Algorithms for geometrical operations with NURBS surfaces

The aim of the paper is to show algorithms for geometrical manipulations on NURBS surfaces. These include generating NURBS surfaces that pass through given points, calculating the minimum distance to a point and include line to surface and surface to surface intersections.

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arXiv2020

Efficient and realistic 3-D Boundary Element simulations of underground construction using isogeometric analysis

The paper outlines some recent developments of the boundary element method (BEM) that makes it more user friendly and suitable for a realistic simulation in geomechanics, especially for underground excavations and tunnelling. The innovations refer to the introduction of isogeometric concepts, elasto-plastic analysis and the simulation of ground support. The introduction of isogeometric concepts for the description of the excavation boundaries results in less user and analysis effort, since complex geometries can be modelled with few parameters and degrees of freedom. No mesh generation is necessary. In order to consider heterogeneous and inelastic ground conditions a domain discretisation is used but the use of isogeometric methods results in a minimum of user effort. Efficient methods are also presented for modelling rock bolts. A comparison of results of test examples with other numerical methods and analytical solutions confirm the efficiency and accuracy of the proposed implementation. A practical example with a complex geometry is presented.

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arXiv2020

Efficient simulation of inclusions and reinforcement bars with the isogeometric Boundary Element method

The paper is concerned with the development of efficient and accurate solution procedures for the isogeometric boundary element method (BEM) when applied to problems that contain inclusions that have elastic properties different to the computed domain. This topic has been addressed in previous papers but the approach presented here is a considerable improvement in terms of efficiency and accuracy. One innovation is that initial stresses instead of body forces are used. This then allows a one step solution without iteration. In addition, a novel approach is used for the computation of strains, that avoids the use of highly singular fundamental solutions. Finally, a new type of inclusion is presented that can be used to model reinforcement bars or rock bolts and where analytical integration can be used. Test examples, where results are compared with Finite Element simulations, show that the proposed approach is sound.

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arXiv2016

Stable Isogeometric Analysis of Trimmed Geometries

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.

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arXiv2015

Isogeometric Boundary Element Analysis with elasto-plastic inclusions. Part 1: Plane problems

In this work a novel approach is presented for the isogeometric Boundary Element analysis of domains that contain inclusions with different elastic properties than the ones used for computing the fundamental solutions. In addition the inclusion may exhibit inelastic material behavior. In this paper only plane stress/strain problems are considered. In our approach the geometry of the inclusion is described using NURBS basis functions. The advantage over currently used methods is that no discretization into cells is required in order to evaluate the arising volume integrals. The other difference to current approaches is that Kernels of lower singularity are used in the domain term. The implementation is verified on simple finite and infinite domain examples with various boundary conditions. Finally a practical application in geomechanics is presented.

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arXiv2015

The Isogeometric Nystr\"om Method

In this paper the isogeometric Nystr\"om method is presented. It's outstanding features are: it allows the analysis of domains described by many different geometrical mapping methods in computer aided geometric design and it requires only pointwise function evaluations just like isogeometric collocation methods. The analysis of the computational domain is carried out by means of boundary integral equations, therefor only the boundary representation is required. The method is thoroughly integrated into the isogeometric framework. For example, the regularization of the arising singular integrals performed with local correction as well as the interpolation of the pointwise existing results are carried out by means of Bezier elements. The presented isogeometric Nystr\"om method is applied to practical problems solved by the Laplace and the Lame-Navier equation. Numerical tests show higher order convergence in two and three dimensions. It is concluded that the presented approach provides a simple and flexible alternative to currently used methods for solving boundary integral equations, but has some limitations.

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arXiv2015

A simple approach to the numerical simulation with trimmed CAD surfaces

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 sim- plicity 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.

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arXiv2014

Boundary Element Analysis with trimmed NURBS and a generalized IGA approach

A novel approach to the simulation with the boundary element method using trimmed NURBS patches is presented. The advantage of this approach is its efficiency and easy implementation. The analysis with trimmed NURBS is achieved by double mapping. The variation of the unknowns on the boundary is specified in a local coordinate system and is completely independent of the description of the geometry. The method is tested on a branched tunnel and the results compared with those obtained from a conventional analysis. The conclusion is that the proposed approach is superior in terms of number of unknowns and effort required.

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arXiv2014

Isogeometric Boundary Element Method with Hierarchical Matrices

In this work we address the complexity problem of the isogeometric Boundary Element Method by proposing a collocation scheme for practical problems in linear elasticity and the application of hierarchical matrices. For mixed boundary value problems, a block system of matrices similar to Galerkin formulations is constructed allowing an effective application of that matrix format. We introduce a strategy for the geometric bisection of surfaces based on NURBS patches. The approximation of system matrices is carried out by means of kernel interpolation. Numerical results are shown that prove the success of the formulation.

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arXiv2014

Fast Isogeometric Boundary Element Method based on Independent Field Approximation

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 is demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.

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