[EN]We present a new strategy for construction spline spaces over hierarchical T-meshes with quad- and
octree subdivision scheme.MINECO y FEDER; CONACYT-SENER Project, Fondo Sectorial, contract: 16372

Brovka, Marina

Escobar, José María

López González, José Iván

Montenegro Armas, Rafael

English

Gestion del Repositorio Documental de la Universidad de Salamanca

Construction of polynomial spline spaces over quadtree andoctree T-meshes for its application in isogeometric analysisJ.I. López1, M. Brovka1, J. M. Escobar1 and R. Montenegro1,∗1 University Institute for Intelligent Systems and Numerical Applications in Engineering,University of Las Palmas de Gran Canaria, Spain{jilopez,mbrovka,jmescobar,rmontenegro}@siani.esABSTRACTWe present a new strategy for construction spline spaces over hierarchical T-meshes with quad- andoctree subdivision scheme [1]. The proposed method is based on some simple rules for inferring,from a given T-mesh, local knot vectors to define tensor product spline blending functions. A set ofcubic spline functions defined by means of this technique span a space with nice properties: it canreproduce cubic polynomials, the functions are C2-continuous, linearly independent, and spacesspanned by nested T-meshes are also nested. In order to define spline spaces with desirable prop-erties applying the proposed rules, the T-mesh should fulfill a mild restriction of being a stronglybalanced quadtree or octree. A T-mesh with a quadtree (octree) structure is said to be stronglybalanced if any cell has contact (through vertex, edge or face) only with cells that differ at mosttwice in depth. Balanced tree condition is commonly used in FEM to guarantee a good qualityof the approximation space constructed over the mesh. To obtain a strongly balanced quadtree,a standard balancing procedure is applied. The straightforward implementation of the proposedstrategy (both in 2D and 3D) and the simplicity of tree structures can make it attractive for itsuse in geometric design and isogeometric analysis. We give a detailed description of our techniqueand illustrate some examples of its application in isogeometric analysis performing adaptive re-finement for 2D and 3D problems. Optimal rates of convergence are obtained during adaptiverefinement for all test problems. Parameterization of computational domains is obtained using thealgorithm described in our previous works [2, 3]. This technique, based on a T-mesh untangling andoptimization procedure, allows us to obtain a good quality parameterization from the boundaryrepresentation of the geometry. The procedure is an extension of the ideas presented in our works[4, 5].REFERENCES[1] Brovka M., López J.I., Escobar J.M., Montenegro R., Cascón J.M. (2016) A simple strategyfor defining polynomial spline spaces over hierarchical T-meshes. Computer-Aided Design, 72,pp. 140–156.[2] Brovka M., López J.I., Escobar J.M., Cascón J.M., Montenegro R. (2014) A new method forT-spline parameterization of complex 2D geometries. Engineering with Computers 30: 457-473.[3] López J.I., Brovka M., Escobar J.M., Montenegro R., Socorro G.V. (2016) Strategies foroptimization of hexahedral meshes and their comparative study. Engineering with Computers,available online May 2016, DOI: 10.1007/s00366-016-0454-1.[4] Escobar JM, Cascón JM, Rodŕıguez E, Montenegro R (2011), A new approach to solid modelingwith trivariate T-splines based on mesh optimization.Comput. Meth. Appl. Mech. Eng 200:3210-3222.[5] Escobar JM, Montenegro R, Rodŕıguez E, Cascón JM (2014) The meccano method for isoge-ometric solid modeling and applications. Engineering with Computers 30: 331-343.

Construction of polynomial spline spaces over quadtree and octree T-meshes for its application in isogeometric analysis

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Construction of polynomial spline spaces over quadtree and octree T-meshes for its application in isogeometric analysis

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