One-dimensional fully automatic h-adaptive isogeometric finite element method package

Paweł Piotr Lipski; Maciej Paszyński

doi:10.7494/csci.2016.17.4.439

Authors

Paweł Piotr Lipski AGH University of Science and Technology
Maciej Paszyński AGH University of Science and Technology

DOI:

https://doi.org/10.7494/csci.2016.17.4.439

Keywords:

finite element method, isogeometric analysis, parallel computing, h-adaptivity, B-splines

Abstract

This paper deals with an adaptive finite element method originally developed
by Prof. Leszek Demkowicz for hierarchical basis functions. In this paper, we
investigate the extension of the adaptive algorithm for isogeometric analysis
performed with B-spline basis functions. We restrict ourselves to h-adaptivity,
since the polynomial order of approximation must be fixed in the isogeometric
case. The classical variant of the adaptive FEM algorithm, as delivered by the
group of Prof. Demkowicz, is based on a two-grid paradigm, with coarse and
fine grids (the latter utilized as a reference solution). The problem is solved independently
over a coarse mesh and a fine mesh. The fine-mesh solution is then
utilized as a reference to estimate the relative error of the coarse-mesh solution
and to decide which elements to refine. Prof. Demkowicz uses hierarchical
basis functions, which (though locally providing C p−1 continuity) ensure only
C 0 on the interfaces between elements. The CUDA C library described in this
paper switches the basis to B-spline functions and proposes a one-dimensional
isogeometric version of the h-adaptive FEM algorithm to achieve global C p−1
continuity of the solution.

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Author Biography

Maciej Paszyński, AGH University of Science and Technology

Department of Computer Science

References

Babuška I., Guo B.: The hp-version of the finite element method, Part I: The basic approximation results. Computational Mechanics, pp. 21–41, 1986.

Babuška I., Guo B.: The hp-version of the finite element method, Part II: General results and applications. Computational Mechanics, pp. 203–220, 1986.

Babuška I., Rheinboldt W.C.: Error Estimates for Adaptive Finite Element Computations. SIAM Journal of Numerical Analysis, vol. 15(4), pp. 736–754, 1978.

Bazilevs Y., Beirão Da Veiga L., Cottrell J.A., Hughes T.J.R., Sangalli G.: Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes. Mathematical Models and Methods in Applied Sciences, vol. 16(7), pp. 1031– 1090, 2006.

Becker R., Hartmut K., R. R.: Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM Journal on Control and Optimisation, vol. 39(1), pp. 113–132, 2000.

Belytschko T., Tabbara M.: H-Adaptive finite element methods for dynamic problems, with emphasis on localization. International Journal for Numerical Methods in Engineering, vol. 36(24), pp. 4245–4265, 1993.

Cottrell J., Hughes T.J.R., Bazilevs Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley Publishing, 1st ed., 2009, ISBN 0470748737, 9780470748732.

Demkowicz L.: Computing with hp adaptive finite element methods. Part I. Elliptic and Maxwell problems with applications. Taylor & Francis, CRC Press, 2006.

Eriksson K., Johnson C.: Adaptive Finite Element Methods for Parabolic Problems I: A Linear Model Problem. SIAM Journal on Numerical Analysis, vol. 28(1), pp. 43–77, 1991.

Gang B., Guanghui H., Di L.: An h-adaptive finite element solver for the calculations of the electronic structures. Journal of Computational Physics, vol. 231(14), pp. 4967–4979, 2012.

Kardani M., Nazem M., Abbo A.J., Sheng D., Sloan S.W.: Refined h-adaptive finite element procedure for large deformation geotechnical problems. Computational Mechanics, vol. 49(1), pp. 21–33, 2012.

Krawezik G., Poole G.: Accelerating the ANSYS Direct Sparse Solver with GPUs. Symposium on Application Accelerators in High Performance Computing, SAAHPC, 2009.

Kuźnik K., Paszyński M., Calo V.: Grammar-Based Multi-Frontal Solver for One Dimensional Isogeometric Analysis with Multiple Right-Hand-Sides. Procedia Computer Science, vol. 18(0), pp. 1574 – 1583, 2013.

Lucas R.F., Wagenbreth G., Davis D.M., Grimes R.: Multifrontal Computations on GPUs and Their Multi-core Hosts. VECPAR, Lecture Notes in Computer Science, vol. 6449, pp. 71–82, Springer, 2010.

Niemi A.H., Babuška I., Pitkäranta J., Demkowicz L.: Finite element analysis of the Girkmann problem using the modern hp-version and the classical h-version. Engineering with Computers, vol. 28(2), pp. 123–134, 2012.

Nochetto R.H., Siebert K.G., Veeser A.: Multiscale, Nonlinear and Adaptive Approximation. Springer, 2009.

Piegl L., Tiller W.: The NURBS Book (Second Edition). Springer-Verlag New York, Inc., 1997.

Woźniak M., Kuźnik K., Paszyński M., Calo V., Pardo D.: Computational cost estimates for parallel shared memory isogeometric multi-frontal solvers. Computers and Mathematics with Applications, vol. 67(10), pp. 1864–1883, 2014.

Yu C.D., Wang W., Pierce D.: A CPU-GPU Hybrid Approach for the Unsymmetric Multifrontal Method. Parallel Comput., vol. 37(12), pp. 759–770, 2011.