Isogeometric structural shape optimization

Wall, Wolfgang A.; Frenzel, Moritz; Cyron, Christian J.

doi:10.1016/j.cma.2008.01.025

Cited by 455 publications

(228 citation statements)

References 25 publications

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“…IGA methodologies have been summarized in the recent book [18] and studied in, e.g., [2,4,9,10,19,23,28,29,11,5,8]. IGA methods are having a growing impact on fields as diverse as fluid dynamics [6,7,40,15,26], structural mechanics [3,1,12,20,30,39], and electromagnetics [17,16].…”

mentioning

confidence: 99%

Overlapping Schwarz Methods for Isogeometric Analysis

Veiga¹,

Cho²,

Pavarino³

et al. 2012

SIAM J. Numer. Anal.

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Abstract. We construct and analyze an overlapping Schwarz preconditioner for elliptic problems discretized with isogeometric analysis. The preconditioner is based on partitioning the domain of the problem into overlapping subdomains, solving local isogeometric problems on these subdomains, and solving an additional coarse isogeometric problem associated with the subdomain mesh. We develop an h-analysis of the preconditioner, showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and "overlap sizes" for fixed polynomial degree p and regularity k of the basis functions. Numerical results in two-and three-dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters h, p, k, number of subdomains N , overlap size, and also jumps in the coefficients of the elliptic operator.Key words. domain decomposition methods, overlapping Schwarz, scalable preconditioners, isogeometric analysis, finite elements, NURBS AMS subject classifications. 65N55, 65N30, 65F10 DOI. 10.1137/110833476 1. Introduction. Isogeometric analysis (IGA) based on NURBS (nonuniform rational B-splines) was introduced by Hughes, Cottrell, and Bazilevs in [27] as an innovative numerical methodology for the analysis of PDE problems, allowing for an exact description of CAD-type geometries. NURBS are a standard in the computeraided design (CAD) community mainly because such spline functions allow excellent representations of free-form surfaces, and there are very efficient algorithms for their evaluation, refinement, and derefinement. In the isogeometric framework, the NURBS basis functions representing the CAD geometry are also used as the basis for the discrete solution space of PDEs, following an isoparametric paradigm. In addition to exact representation of CAD geometries, another advantage of using NURBS basis functions is the higher control on the regularity of the discrete space. For instance, spaces of global C k regularity are easily built, thus allowing for fewer degrees of freedom, better performance in case of vibrations, easier approximation of higher order problems, and other advantages. IGA methodologies have been summarized in the recent book [18] and studied in, e.g., [2,4,9,10,19,23,28,29,11,5,8]. IGA methods are having a growing impact on fields as diverse as fluid dynamics [6,7,40,15,26], structural mechanics [3,1,12,20,30,39], and electromagnetics [17,16].The discrete systems produced by isogeometric methods are better conditioned than the systems produced by standard finite elements or finite differences, but their conditioning can still degenerates rapidly for decreasing mesh size h, increasing poly-

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mentioning

confidence: 99%

Overlapping Schwarz Methods for Isogeometric Analysis

Veiga¹,

Cho²,

Pavarino³

et al. 2012

SIAM J. Numer. Anal.

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“…[31]. In addition, most realistic industrial problems can only be realized based on multiple patches, and the problems are most often three-dimensional and not planar.…”

Section: Discussionmentioning

confidence: 99%

Planar Parametrization in Isogeometric Analysis

Gravesen

Evgrafov

Nguyen

et al. 2014

Mathematical Methods for Curves and Surfaces

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Abstract. Before isogeometric analysis can be applied to solving a partial differential equation posed over some physical domain, one needs to construct a valid parametrization of the geometry. The accuracy of the analysis is affected by the quality of the parametrization. The challenge of computing and maintaining a valid geometry parametrization is particularly relevant in applications of isogemetric analysis to shape optimization, where the geometry varies from one optimization iteration to another. We propose a general framework for handling the geometry parametrization in isogeometric analysis and shape optimization. It utilizes an expensive non-linear method for constructing/updating a high quality reference parametrization, and an inexpensive linear method for maintaining the parametrization in the vicinity of the reference one. We describe several linear and non-linear parametrization methods, which are suitable for our framework. The non-linear methods we consider are based on solving a constrained optimization problem numerically, and are divided into two classes, geometry-oriented methods and analysisoriented methods. Their performance is illustrated through a few numerical examples.

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“…However, FE based discretization of the design model, which differs much from the analysis model from the geometrical representation point of view, leads to the unrealistic optimal design model [6].…”

Section: Introduction State Of the Art In Igamentioning

confidence: 99%

“…• In the case of optimization problems, a tight coupling between the design model and the analysis model is achieved as the design model is directly utilized in the analysis with IGA technique [6].…”

Section: Introduction State Of the Art In Igamentioning

confidence: 99%

IGA: A Simplified Introduction and Implementation Details for Finite Element Users

Agrawal

Gautam

2018

J. Inst. Eng. India Ser. C

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Isogeometric analysis (IGA) is a recently introduced technique that employs the Computer Aided Design (CAD) concept of Non-uniform Rational B-splines (NURBS) tool to bridge the substantial bottleneck between the CAD and finite element analysis (FEA) fields. The simplified transition of exact CAD models into the analysis alleviates the issues originating from geometrical discontinuities and thus, significantly reduces the design-toanalysis time in comparison to traditional FEA technique. Since its origination, the research in the field of IGA is accelerating and has been applied to various problems. However, the employment of CAD tools in the area of FEA invokes the need of adapting the existing implementation procedure for the framework of IGA. Also, the usage of IGA requires the in-depth knowledge of both the CAD and FEA fields. This can be overwhelming for a beginner in IGA. Hence, in this paper, a simplified introduction and implementation details for the incorporation of NURBS based IGA technique within the existing FEA code is presented. It is shown that with little modifications, the available standard code structure of FEA can be adapted for IGA. For the clear and concise explanation of these modifications, step-by-step implementation of a benchmark plate with a circular hole under the action of in-plane tension is included.

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Isogeometric structural shape optimization

Cited by 455 publications

References 25 publications

Overlapping Schwarz Methods for Isogeometric Analysis

Overlapping Schwarz Methods for Isogeometric Analysis

Planar Parametrization in Isogeometric Analysis

IGA: A Simplified Introduction and Implementation Details for Finite Element Users

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