Numerical simulation of arbitrary holes in orthotropic media by an efficient computational method based on adaptive XIGA

Abstract We present in this paper an efficient computational approach based on an adaptive extended isogeometric analysis (XIGA) for simulation of arbitrary holes in orthotropic materials. The interfaces of holes are described by multiple level set functions so that the developed XIGA can capture the hole geometry without considering its interfaces. The approach is further enhanced by using the locally refined (LR) B-spline basis functions, which dominate over B-spline or non-uniform rational B-spline functions (NURBS) due to the local refinement. This study only deals with the structured mesh strategy for local refinement. The implementation of the local refinement is guided by posteriori error estimator based on stress recovery. Accuracy study is performed for isotropic media due to the availability of analytical solutions. For numerical experiments, different types of holes in orthotropic media are studied, and the computed numerical results are compared with reference solutions derived from ABAQUS (FEM). We also compare the convergence rate obtained by our adaptive local refinement with that derived from uniform global refinement to indicate the greater advantage of the developed adaptive XIGA.

[1]  Tinh Quoc Bui,et al.  Numerical simulation of 2-D weak and strong discontinuities by a novel approach based on XFEM with local mesh refinement , 2018 .

[2]  Zhen Wang,et al.  Numerical modeling of 3-D inclusions and voids by a novel adaptive XFEM , 2016, Adv. Eng. Softw..

[3]  Trond Kvamsdal,et al.  Divergence-conforming discretization for Stokes problem on locally refined meshes using LR B-splines , 2015 .

[4]  Hendrik Speleers,et al.  THB-splines: The truncated basis for hierarchical splines , 2012, Comput. Aided Geom. Des..

[5]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[6]  V. G. Ukadgaonker,et al.  A general solution for stresses around holes in symmetric laminates under inplane loading , 2000 .

[7]  Trond Kvamsdal,et al.  Adaptive isogeometric finite element analysis of steady‐state groundwater flow , 2016 .

[8]  E. Dill,et al.  Theory of Elasticity of an Anisotropic Elastic Body , 1964 .

[9]  Tinh Quoc Bui,et al.  Multi-inclusions modeling by adaptive XIGA based on LR B-splines and multiple level sets , 2018, Finite Elements in Analysis and Design.

[10]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[11]  Guriĭ Nikolaevich Savin,et al.  Stress concentration around holes , 1961 .

[12]  László Szabó,et al.  On the configurational-force-based r-adaptive mesh refinement in isogeometric analysis , 2017 .

[13]  T. Hughes,et al.  Local refinement of analysis-suitable T-splines , 2012 .

[14]  N. Nguyen‐Thanh,et al.  Extended isogeometric analysis based on PHT‐splines for crack propagation near inclusions , 2017 .

[15]  T. Q. Bui,et al.  A simple FSDT-based isogeometric analysis for geometrically nonlinear analysis of functionally graded plates , 2015 .

[16]  Trond Kvamsdal,et al.  Superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis , 2017 .

[17]  Le Van Lich,et al.  Fracture modeling with the adaptive XIGA based on locally refined B-splines , 2019, Computer Methods in Applied Mechanics and Engineering.

[18]  Ernst Rank,et al.  Finite cell method , 2007 .

[19]  Tinh Quoc Bui,et al.  Adaptive multi-patch isogeometric analysis based on locally refined B-splines , 2018, Computer Methods in Applied Mechanics and Engineering.

[20]  Kapil Khandelwal,et al.  On the locking free isogeometric formulations for 3-D curved Timoshenko beams , 2018 .

[21]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[22]  D. Sharma,et al.  Stress Concentration around Circular/Elliptical/Triangular Cutouts in Infinite Composite Plate , 2011 .

[23]  T. K. Paul,et al.  Finite element stress analysis of laminated composite plates containing two circular holes under transverse loading , 1995 .

[24]  T. Belytschko,et al.  Arbitrary branched and intersecting cracks with the eXtended Finite Element Method , 2000 .

[25]  Ahmad H. Nasri,et al.  T-splines and T-NURCCs , 2003, ACM Trans. Graph..

[26]  Soheil Mohammadi,et al.  XFEM Fracture Analysis of Composites , 2012 .

[27]  Sohichi Hirose,et al.  NURBS-based isogeometric analysis of buckling and free vibration problems for laminated composites plates with complicated cutouts using a new simple FSDT theory and level set method , 2016 .

[28]  Pilseong Kang,et al.  Isogeometric topology optimization of shell structures using trimmed NURBS surfaces , 2016 .

[29]  T. Belytschko,et al.  MODELING HOLES AND INCLUSIONS BY LEVEL SETS IN THE EXTENDED FINITE-ELEMENT METHOD , 2001 .

[30]  Tom Lyche,et al.  Polynomial splines over locally refined box-partitions , 2013, Comput. Aided Geom. Des..

[31]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[32]  Andrea Bressan,et al.  Some properties of LR-splines , 2013, Comput. Aided Geom. Des..

[33]  Yingjun Wang,et al.  Structural Design Optimization Using Isogeometric Analysis: A Comprehensive Review , 2018, Computer Modeling in Engineering & Sciences.

[34]  Trond Kvamsdal,et al.  Simple a posteriori error estimators in adaptive isogeometric analysis , 2015, Comput. Math. Appl..

[35]  S. M. Chern,et al.  On Displacement Fields in Orthotropic Laminates Containing an Elliptical Hole , 2000 .

[36]  Moussa Karama,et al.  Stress concentration in a circular hole in composite plate , 2005 .

[37]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[38]  T. Strouboulis,et al.  The generalized finite element method: an example of its implementation and illustration of its performance , 2000 .

[39]  Dan Givoli,et al.  Stress Concentration at a Nearly Circular Hole With Uncertain Irregularities , 1992 .

[40]  Jun Wang,et al.  Parallel and adaptive surface reconstruction based on implicit PHT-splines , 2011, Comput. Aided Geom. Des..

[41]  T. Enab.  Stress concentration analysis in functionally graded plates with elliptic holes under biaxial loadings , 2014 .

[42]  Grégory Legrain,et al.  High order X-FEM and levelsets for complex microstructures: Uncoupling geometry and approximation , 2012 .

[43]  N. Muskhelishvili.  Some basic problems of the mathematical theory of elasticity , 1953 .

[44]  Deshin Liu,et al.  Eigenvalue Analysis of Thin Plate with Complicated Shapes By a Novel Infinite Element Method , 2019, Computer Modeling in Engineering & Sciences.

[45]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[46]  Rakesh K. Kapania,et al.  Global/local analysis of composite plates with cutouts , 1997 .

[47]  V. G. Ukadgaonker,et al.  Stress distribution around triangular holes in anisotropic plates , 1999 .

[48]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[49]  John A. Evans,et al.  Isogeometric analysis using T-splines , 2010 .

[50]  Trond Kvamsdal,et al.  Isogeometric analysis using LR B-splines , 2014 .

[51]  N. Valizadeh,et al.  Extended isogeometric analysis for simulation of stationary and propagating cracks , 2012 .

[52]  J. Niiranen,et al.  Analytical solution with validity analysis for an elliptical void and a rigid inclusion under uniform or nonuniform anti-plane loading , 2018, Theoretical and Applied Fracture Mechanics.

[53]  Roland Wüchner,et al.  Analysis in computer aided design: Nonlinear isogeometric B-Rep analysis of shell structures , 2015 .