Computational Design & Manufacturing Lab

University of Wisconsin-Madison

 

 

 

 

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Isogeometric analysis on triangulations

Acknowledgment: This research is supported in part by NSF grant #1435072.

 

We have developed a method for isogeometric analysis on the triangulation of a domain bounded by NURBS curves (or surfaces). In this method, both the geometry and the physical field are represented by bivariate (or trivariate) splines in Bernstein–Bézier form over the triangulation. We have developed a set of procedures to construct a parametric domain and its triangulation from a given physical domain, construct Cr-smooth basis functions over the domain, and establish a rational Triangular Bézier Spline (rTBS) based geometric mapping that Cr-smoothly maps the parametric domain to the physical domain and exactly recovers the NURBS boundaries at the domain boundary. As a result, this approach can achieve automated meshing of objects with complex topologies and allow highly localized refinement.

 

Figure 1Flowchart of isogeometric analysis on triangulations

Figure 1 gives a schematic illustration of the flowchart of isogeometric analysis on triangulations.

 

Recently, we have developed a smooth-refine-smooth procedure to achieve optimal convergence with such Cr rTBS basis. We show that the optimal rate of convergence can be achieved provided that the Cr geometric map remains unchanged during the refinement process. This condition can be satisfied by constructing a pre-refinement geometric map that is sufficiently smooth. Numerical experiments verify that optimal rates of convergence are achieved for Poisson and linear elasticity problems.

 

Our approach can automatically convert 3D CAD models into watertight Bezier Tetrahedra. NURBS surfaces without trimmed boundaries are converted to Bezier triangles exactly. Trimmed NURBS surfaces are converted to Bezier triangles with controlled approximation error.

Relevant publications

o   Songtao Xia and Xiaoping Qian, `` Isogeometric analysis with Bézier tetrahedra ,'' Computer Methods in Applied Mechanics and Engineering, Vol. 316, 782-816, 2017. [pdf]

o   Songtao Xia, Xilu Wang and Xiaoping Qian, ``Continuity and convergence in rational triangular Bézier spline based isogeometric analysis,'' Computer Methods in Applied Mechanics and Engineering, Vol. 297, 292-324, 2015. [pdf]

o   Jaxon, N. and Qian, X., ``Isogeometric analysis on triangulations,'' Computer-Aided Design, Special Issue on GD/SPM 2013: SIAM/ACM Joint Conference on Geometric and Physical Modeling, Vol. 46, pp. 45- 57,  2014. [pdf]

o   Wang, X. and Qian, X., ``An optimization approach for constructing trivariate B-spline solids,'' Computer-Aided Design, Special Issue on GD/SPM 2013: SIAM/ACM Joint Conference on Geometric and Physical Modeling, Vol. 46, pp. 179 - 191, 2014. [pdf]

o   Li, K. and Qian, X., "Isogeometric analysis and shape optimization via boundary integral", Proceedings of SIAM/ACM Joint Conference on Geometric and Physical Modeling (GD/SPM11) (Best Paper Award), Computer-Aided Design, Vol. 43, No. 11, pp. 1427 - 1437, 2011. [pdf]

o   Qian, X. and Sigmund, O, "Isogeometric shape optimization of photonic crystals via Coons patches," Computer Methods in Applied Mechanics and Engineering, Vol. 200, No. 25 - 28, pp. 2237 - 2255, 2011. [pdf]

o   Qian, X., "Full Analytical Sensitivities in NURBS based Isogeometric Shape Optimization," Computer Methods in Applied Mechanics and Engineering, Vol. 199, No. 29 -32, pp. 2059 - 2071, June 2010. [pdf]

 

 

 

Last updated May 15, 2017.

 

 

 

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