Initial solution estimation for one-step inverse isogeometric analysis in sheet metal stamping

Abstract Recently, Isogeometric Analysis (IGA) based on incremental methods for simulating the stamping process has been researched. To the best of our knowledge, however, few studies have combined IGA and One-step inverse approach which is based on total deformation theory of plasticity. A key step for One-step inverse IGA is to estimate a good initial solution. Traditional mesh-based initial solution algorithms for One-step inverse approach are not suitable for One-step inverse NURBS-based IGA. In this paper, we presented a method which can rapidly unfold the undevelopable NURBS surface onto a planar domain and obtain a good initial solution estimation for One-step inverse IGA. The key idea of the presented method is unfolding the control net of a NURBS surface for isogeometric analysis by energy-based initial solution estimation algorithm. In addition, we developed a “cutting-stitching” algorithm which can separate a complex control net into several parts with simple shapes. Numerical examples illustrate the initial solutions using the presented method are approaching the final results by One-step inverse finite element method. This implies that the iterative steps and computational time of One-step inverse IGA will be reduced significantly compared with that of One-step inverse finite element method.

[1]  W. Wall,et al.  Isogeometric structural shape optimization , 2008 .

[2]  Ping Hu,et al.  A new surface parameterization method based on one-step inverse forming for isogeometric analysis-suited geometry , 2013 .

[3]  G. Sangalli,et al.  IsoGeometric analysis using T-splines on two-patch geometries , 2011 .

[4]  T. Hughes,et al.  Converting an unstructured quadrilateral/hexahedral mesh to a rational T-spline , 2012 .

[5]  Hoon Huh,et al.  Blank Design and Strain Estimates for Sheet Metal Forming Processes by a Finite Element Inverse Approach with Initial Guess of Linear Deformation , 1998 .

[6]  Ulrich Reif,et al.  Weighted Extended B-Spline Approximation of Dirichlet Problems , 2001, SIAM J. Numer. Anal..

[7]  H. Nguyen-Xuan,et al.  Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids , 2011 .

[8]  Seung-Hyun Ha,et al.  Isogeometric shape design optimization: exact geometry and enhanced sensitivity , 2009 .

[9]  Alessandro Reali,et al.  Isogeometric Analysis of Structural Vibrations , 2006 .

[10]  Ying Huang,et al.  A new approach to solve key issues in multi-step inverse finite-element method in sheet metal stamping , 2006 .

[11]  Yuri Bazilevs,et al.  Rotation free isogeometric thin shell analysis using PHT-splines , 2011 .

[12]  Xiaoping Qian,et al.  Full analytical sensitivities in NURBS based isogeometric shape optimization , 2010 .

[13]  B. Simeon,et al.  A hierarchical approach to adaptive local refinement in isogeometric analysis , 2011 .

[14]  Hoon Huh,et al.  Blank design and strain prediction of automobile stamping parts by an inverse finite element approach , 1997 .

[15]  T. Hughes,et al.  Solid T-spline construction from boundary representations for genus-zero geometry , 2012 .

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

[17]  T. Belytschko,et al.  X‐FEM in isogeometric analysis for linear fracture mechanics , 2011 .

[18]  C. D. Boor,et al.  On Calculating B-splines , 1972 .

[19]  K. Chung,et al.  A deformation theory of plasticity based on minimum work paths , 1993 .

[20]  Peter Wriggers,et al.  A large deformation frictional contact formulation using NURBS‐based isogeometric analysis , 2011 .

[21]  D. Schillinger,et al.  An unfitted hp-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry , 2011 .

[22]  Fredrik Orderud,et al.  Isoparametric finite element analysis for Doo-Sabin subdivision models , 2012, Graphics Interface.

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

[24]  J. M. Detraux,et al.  Finite element procedures for strain estimations of sheet metal forming parts , 1990 .

[25]  Ulrich Reif,et al.  Nonuniform web-splines , 2003, Comput. Aided Geom. Des..

[26]  Hoon Huh,et al.  Construction of sliding constraint surfaces and initial guess shapes for intermediate steps in multi-step finite element inverse analysis , 2002 .

[27]  Sung-Kie Youn,et al.  Isogeometric topology optimization using trimmed spline surfaces , 2010 .

[28]  Thomas J. R. Hughes,et al.  Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device , 2009 .

[29]  Zheng-Dong Ma,et al.  Nonconforming isogeometric analysis for trimmed CAD geometries using finite-element tearing and interconnecting algorithm , 2017 .

[30]  John A. Evans,et al.  An Isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces , 2012 .

[31]  B. Simeon,et al.  Adaptive isogeometric analysis by local h-refinement with T-splines , 2010 .

[32]  Peter Wriggers,et al.  Contact treatment in isogeometric analysis with NURBS , 2011 .

[33]  J. L. Duncan,et al.  The modelling of sheet metal stampings , 1986 .

[34]  Hoon Huh,et al.  Three dimensional multi-step inverse analysis for the optimum blank design in sheet metal forming processes , 1998 .

[35]  Giancarlo Sangalli,et al.  Analysis-Suitable T-splines are Dual-Compatible , 2012 .

[36]  Ping Hu,et al.  Energy-Based Initial Guess Estimation Method for One-Step Simulation , 2007 .

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

[38]  Ping Hu,et al.  Isogeometric finite element method for buckling analysis of generally laminated composite beams with different boundary conditions , 2015 .

[39]  T. Hughes,et al.  Isogeometric fluid-structure interaction: theory, algorithms, and computations , 2008 .

[40]  Thomas J. R. Hughes,et al.  Isogeometric shell analysis: The Reissner-Mindlin shell , 2010 .

[41]  Zheng-Dong Ma,et al.  B++ splines with applications to isogeometric analysis , 2016 .

[42]  Klaus Höllig,et al.  Introduction to the Web-method and its applications , 2005, Adv. Comput. Math..

[43]  S. A. Majlessi,et al.  Development of multistage sheet metal forming analysis method , 1988 .

[44]  Jens Gravesen,et al.  Isogeometric Shape Optimization of Vibrating Membranes , 2011 .

[45]  Thomas J. R. Hughes,et al.  Isogeometric Failure Analysis , 2011 .

[46]  Hoon Huh,et al.  Finite element inverse analysis for the design of intermediate dies in multi-stage deep-drawing processes with large aspect ratio , 2001 .

[47]  Bernd Hamann,et al.  Iso‐geometric Finite Element Analysis Based on Catmull‐Clark : ubdivision Solids , 2010, Comput. Graph. Forum.

[48]  M. Cox.  The Numerical Evaluation of B-Splines , 1972 .

[49]  Klaus Höllig,et al.  Finite Element Analysis with B-Splines: Weighted and Isogeometric Methods , 2010, Curves and Surfaces.

[50]  Sung-Kie Youn,et al.  T‐spline finite element method for the analysis of shell structures , 2009 .

[51]  Thomas J. R. Hughes,et al.  A large deformation, rotation-free, isogeometric shell , 2011 .

[52]  Fehmi Cirak,et al.  Subdivision shells with exact boundary control and non‐manifold geometry , 2011 .

[53]  Hakim Naceur,et al.  Initial solution estimation to speed up inverse approach in stamping modeling , 2003 .

[54]  T. Hughes,et al.  ISOGEOMETRIC ANALYSIS: APPROXIMATION, STABILITY AND ERROR ESTIMATES FOR h-REFINED MESHES , 2006 .

[55]  Dixiong Yang,et al.  Isogeometric method based in-plane and out-of-plane free vibration analysis for Timoshenko curved beams , 2016 .