Integration of Design and Analysis Through Boundary Integral Equations

Abstract

Abstract. 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. 6526Benjamin Marussig, Jürgen Zechner, Gernot Beer, and Thomas-Peter Fries INTRODUCTIONIsogeometric analysis aims to close the existing gap between the design process and analysis such that a simulation can be performed without generating a mesh. Consequently, the accuracy and efficiency of the overall simulation process is improved, since meshing is timeconsuming [1, 2] and introduces additional (geometrical) approximation errors. In addition, the basis functions used by design models, i.e. NURBS, provide further benefits such as high continuity [3,4].However, during the last years, it has become clear that a true integration of design and analysis is far from trivial due to several reasons: first of all, most engineering design models are based on a boundary representation (B-Rep) rather than a volume description. Secondly, three dimensional B-Rep models are usually defined by a non-conforming partition of NURBS surfaces, i.e. their mathematical parametrizations have no explicit relation to each other. Thirdly, each boundary surface is based on a tensor product structure which is a very efficient representation but has limitations due to its four sided nature. As a result, almost all NURBS based design models use trimming procedures to increase the flexibility of tensor product surfaces. This means that only a certain area of a surface is visualized while the underlying mathematical parametrization remains unchanged.In this work, a coherent framework is presented which allows a seamless integration of trimmed NURBS models into an analysis. In general, the governing equations of the problem are expressed by means of boundary integral equations which are discretized by a numerical approximation method. Here, the boundary element method (BEM) is used since it is the most versatile approach. However, it should be pointed out that other schemes like the Nyst...

Analysis‐suitable CAD Models based on Watertight Boolean Operations

“…The system of equation is set-up by collocation at the Greville abscissae, and the integration is performed as detailed in [5]. The system of equation is set-up by collocation at the Greville abscissae, and the integration is performed as detailed in [5].…”

“…Following the isogeometric paradigm, we discretize (2) using the basis functions provided by the CAD model that describes Γ. The system of equation is set-up by collocation at the Greville abscissae, and the integration is performed as detailed in [5].…”

Isogeometric Analysis with Trimmed CAD Models

“…For comparison, a model that consists of regular surfaces only is used as well. The problem is analyzed by a direct isogeometric boundary element formulation using a collocation approach, for details see e.g., [12]. Fig.…”

Stable isogeometric analysis of trimmed geometries

“…In order to apply extended B-splines for the analysis, two different approaches may be used: (i) conversion of the CAGD model to a B-spline representation or (ii) application of an independent field approximation [23,24]. In this paper, the latter is preferred since it allows to perform the analysis based on the original NURBS model without any geometrical approximations.…”

“…However, this paper is mainly dedicated to the stabilization of trimmed geometries and hence the BEM is only briefly reviewed. For an in-depth discussion of the applied BEM implementation the interested reader is referred to [23,24].…”