We review the treatment of trimmed geometries in the context of design, data exchange, and computational simulation. Such models are omnipresent in current engineering modeling and play a key role for the integration of design and analysis. The problems induced by trimming are often underestimated due to the conceptional simplicity of the procedure. In this work, several challenges and pitfalls are described.
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.
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 Bsplines 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.
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.Comment: 20 pages and 16 figure
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.
Isogeometric methods have become the latest trend in numerical simulation. Their attractiveness stems from the fact that the description of the geometry can be taken directly from CAD programs, avoiding the need for mesh generation. Since NURBS (non-uniform rational B-splines) functions that exhibit desirable properties such as high continuity and efficient refinement algorithms are also used for the description of the unknowns, a higher quality of results can be expected. This paper presents some results from the research project 'Fast isogeometric boundary element methods for tunneling'. The aim of the project is to implement a fast and accurate method for the simulation of underground excavations without the need for the generation and subsequent refinement of a mesh.
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