h- and p-adaptivity driven by recovery and residual-based error estimators for PHT-splines applied to time-harmonic acoustics

Abstract In this work, we demonstrate the application of PHT-splines for time-harmonic acoustic problems, modeled by the Helmholtz equation. Solutions of the Helmholtz equation have two features: global oscillations associated with the wave number and local gradients caused by geometrical irregularities. We show that after a sufficient number of degrees of freedom is used to approximate global oscillations, adaptive refinement can capture local features of the solution. We compare residual-based and recovery-based error estimators and investigate the performance of p -refinement. The simulations are done in the context of recently introduced Geometry Independent Field approximaTion (GIFT), where PHT-splines are only used to approximate the solution, while the computational domain is parameterized with NURBS. This approach builds on the natural adaptation ability of PHT-splines and avoids the re-parameterization of the NURBS geometry during the solution refinement process.

[1]  Jiansong Deng,et al.  Polynomial splines over hierarchical T-meshes , 2008, Graph. Model..

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

[3]  Wim Desmet,et al.  A performance study of NURBS-based isogeometric analysis for interior two-dimensional time-harmonic acoustics , 2016 .

[4]  Pascal Bouvry,et al.  Management of an academic HPC cluster: The UL experience , 2014, 2014 International Conference on High Performance Computing & Simulation (HPCS).

[5]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[6]  Haijun Wu,et al.  Isogeometric finite element analysis of interior acoustic problems , 2015 .

[7]  Marc Duflot,et al.  Derivative recovery and a posteriori error estimate for extended finite elements , 2007 .

[8]  I. Babuska,et al.  Finite Element Solution of the Helmholtz Equation with High Wave Number Part II: The h - p Version of the FEM , 1997 .

[9]  Philippe Lasaygues,et al.  Ultrasonic Computed Tomography , 2011 .

[10]  J. Scruggs,et al.  Harvesting Ocean Wave Energy , 2009, Science.

[11]  S. Nicaise,et al.  High-frequency behaviour of corner singularities in Helmholtz problems , 2018, ESAIM: Mathematical Modelling and Numerical Analysis.

[12]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

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

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

[15]  Jon Trevelyan,et al.  High-order finite elements for the solution of Helmholtz problems , 2017 .

[16]  Mark Ainsworth,et al.  Discrete Dispersion Relation for hp-Version Finite Element Approximation at High Wave Number , 2004, SIAM J. Numer. Anal..

[17]  Isaac Harari,et al.  A survey of finite element methods for time-harmonic acoustics , 2006 .

[18]  S. Bordas,et al.  A simple error estimator for extended finite elements , 2007 .

[19]  N. Booij,et al.  A third-generation wave model for coastal regions-1 , 1999 .

[20]  J. Tinsley Oden,et al.  A posteriori error estimation for acoustic wave propagation problems , 2005 .

[21]  I. Babuska,et al.  Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM☆ , 1995 .

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

[23]  Christophe Geuzaine,et al.  Phase reduction models for improving the accuracy of the finite element solution of time-harmonic scattering problems I: General approach and low-order models , 2009, J. Comput. Phys..

[24]  D. Givoli.  High-order local non-reflecting boundary conditions: a review☆ , 2004 .

[25]  Joao Cruz,et al.  Ocean Wave Energy: Current Status and Future Prespectives , 2008 .

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

[27]  N. Nguyen-Thanh,et al.  An adaptive three-dimensional RHT-splines formulation in linear elasto-statics and elasto-dynamics , 2014 .

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

[29]  H. Nguyen-Xuan,et al.  Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling , 2017 .

[30]  G. M. L. Gladwell,et al.  A variational formulation of damped acousto structural vibration problems , 1966 .

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

[32]  Nicholas S. North,et al.  T-spline simplification and local refinement , 2004, SIGGRAPH 2004.

[33]  Thomas-Peter Fries,et al.  Fast Isogeometric Boundary Element Method based on Independent Field Approximation , 2014, ArXiv.

[34]  L. Thompson.  A review of finite-element methods for time-harmonic acoustics , 2006 .

[35]  Thomas J. R. Hughes,et al.  Explicit residual-based a posteriori error estimation for finite element discretizations of the Helmholtz equation: Computation of the constant and new measures of error estimator quality , 1996 .

[36]  Xavier Antoine,et al.  Bayliss-Turkel-like radiation conditions on surfaces of arbitrary shape , 1999 .

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

[38]  Alessandro Gardi,et al.  Acoustic Sensors for Air and Surface Navigation Applications , 2018, Sensors.

[39]  Timon Rabczuk,et al.  An isogeometric collocation method using superconvergent points , 2015 .

[40]  Ivo Babuška,et al.  Computer‐based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's, and the elasticity equations , 1996 .

[41]  Elena Atroshchenko,et al.  Weakening the tight coupling between geometry and simulation in isogeometric analysis: From sub‐ and super‐geometric analysis to Geometry‐Independent Field approximaTion (GIFT) , 2016, 1706.06371.

[42]  Joseph B. Keller,et al.  A hybrid numerical asymptotic method for scattering problems , 2001 .

[43]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[44]  Naim Hossain,et al.  Recovery-based error estimation and adaptivity using high-order splines over hierarchical T-meshes , 2018 .

[45]  Stefan A. Sauter,et al.  Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers? , 1997, SIAM Rev..

[46]  Ari Ben-Menahem,et al.  Seismic waves and sources , 1981 .

[47]  M. Gunzburger,et al.  Boundary conditions for the numerical solution of elliptic equations in exterior regions , 1982 .

[48]  Gulshan,et al.  Theoretical analysis of extreme wave oscillation in Paradip Port using a 3-D boundary element method , 2018, Ocean Engineering.

[49]  C. Geuzaine,et al.  An Amplitude Formulation to Reduce the Pollution Error in the Finite Element Solution of Time-Harmonic Scattering Problems , 2008, IEEE Transactions on Magnetics.

[50]  T. Rabczuk,et al.  T-spline based XIGA for fracture analysis of orthotropic media , 2015 .

[51]  A. Bayliss,et al.  Radiation boundary conditions for wave-like equations , 1980 .

[52]  S. Bordas,et al.  A posteriori error estimation for extended finite elements by an extended global recovery , 2008 .

[53]  M. Dinachandra,et al.  Plane wave enriched Partition of Unity Isogeometric Analysis (PUIGA) for 2D-Helmholtz problems , 2018, Computer Methods in Applied Mechanics and Engineering.

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

[55]  Jiansong Deng,et al.  Dimensions of spline spaces over T-meshes , 2006 .

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

[57]  Vinh Phu Nguyen,et al.  Isogeometric analysis: An overview and computer implementation aspects , 2012, Math. Comput. Simul..

[58]  Ping Wang,et al.  Adaptive isogeometric analysis using rational PHT-splines , 2011, Comput. Aided Des..

[59]  Eli Turkel,et al.  On surface radiation conditions for an ellipse , 2010, J. Comput. Appl. Math..

[60]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[61]  J. Trevelyan,et al.  Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems , 2013 .

[62]  Pablo Gamallo,et al.  Comparison of two wave element methods for the Helmholtz problem , 2006 .

[63]  Ivo Babuška,et al.  The generalized finite element method for Helmholtz equation: Theory, computation, and open problems , 2006 .

[64]  O. C. Zienkiewicz,et al.  Recovery procedures in error estimation and adaptivity Part I: Adaptivity in linear problems , 1999 .

[65]  Jens Markus Melenk,et al.  Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions , 2010, Math. Comput..

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