×

Adjoint a posteriori error measures for anisotropic mesh optimisation. (English) Zbl 1206.65250

Summary: An adjoint- (or sensitivity-) based error measure is formulated which measures the error contribution of each solution variable to an overall goal The goal is typically embodied in an integral functional, e.g., the solution in a small region of the domain of interest. The resulting a posteriori error measures involve the solution of both primal and adjoint problems. A comparison of a number of important a posteriori error measures is made in this work. There is a focus on developing relatively simple methods that refer to information from the discretised equation sets (often readily accessible in simulation codes) and do not explicitly use equation residuals. This method is subsequently used to guide anisotropic mesh adaptivity of tetrahedral finite elements. Mesh adaptivity is achieved here with a series of optimisation heuristics of the landscape defined by mesh quality. Mesh quality is gauged with respect to a Riemann metric tensor embodying an a posteriori error measure, such that an ideal element has sides of unit length when measured with respect to this metric tensor. This results in meshes in which each finite-element node has approximately equal (subject to certain boundary-conforming constraints and the performance of the mesh optimisation heuristics) error contribution to the functional (goal).

MSC:

65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Rannacher, R.; Suttmeier, F. T., A-posteriori error control in finite element methods via duality techniques: Application to perfect plasticity, Comput. Mech., 21, 123-133 (1998) · Zbl 0910.73064
[2] Paraschivoiu, M.; Patera, A., A hierarchical duality approach to bounds for the outputs of partial differential equations, Comput. Meth. Appl. Mech. Eng., 158, 389-407 (1998) · Zbl 0953.76054
[3] Peraire, J.; Patera, A. T., Advances in Adaptive Computational Methods in Mechanics (1998), Elsevier, pp. 199-215
[4] Peraire, J.; Vahdati, M.; Morgan, K.; Zienkiewicz, O. C., Adaptive remeshing for compressible flow computations, J. Comput. Phys., 72, 449-466 (1987) · Zbl 0631.76085
[5] Wu, J.; Zhu, J. Z.; Szmelter, J.; Zienkiewicz, O. C., Error estimation and adaptivity in Navier-Stokes incompressible flows, Comput. Mech., 6, 259-270 (1990) · Zbl 0699.76035
[6] Lohner, R.; Morgan, K.; Zienkiewicz, O. C., An adaptive finite element procedure for compressible high speed flows, Comput. Meth. Appl. Mech. Eng., 51, 441-465 (1985) · Zbl 0568.76074
[7] Piggott, M. D.; Pain, C. C.; Gorman, G. J.; Power, P. W.; Goddard, A. J.H., \(h, r\) and hr adaptivity with applications in numerical ocean modelling, Ocean Modelling, 10, 95-113 (2005)
[8] Strouboulis, T.; Oden, J. T., A-posteriori error estimation of finite element approximations in fluid mechanics, Comput. Meth. Appl. Mech. Eng., 78, 201-242 (1990) · Zbl 0711.76061
[9] Ainsworth, M.; Oden, J. T., A-posteriori error estimation in finite element analysis, Comput. Meth. Appl. Mech. Eng., 142, 1-88 (1997) · Zbl 0895.76040
[10] Ainsworth, M.; Oden, J. T., A unified approach to a-posteriori error estimation using element residual methods, Numer. Math., 65, 23-50 (1993) · Zbl 0797.65080
[11] Babuska, I.; Rheinboldt, W. C., A-posteriori error estimates for the finite element method, Int. J. Num. Meth. Eng., 12, 1597-1615 (1978) · Zbl 0396.65068
[12] Bank, R. E.; Smith, R. K., A-posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal., 30, 921-935 (1993) · Zbl 0787.65078
[13] Bank, R. E.; Weiser, A., Some a-posteriori error estimators for elliptic partial differential equations, Math. Comp., 44, 283-301 (1985) · Zbl 0569.65079
[14] Verfurth, R., A Review of a-Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (1996), Wiley- Teubner · Zbl 0853.65108
[15] Zienkiewicz, O. C.; Zhu, J. Z., The superconvergent patch recovery and a-posteriori error estimates. Part 1: The recovery technique, Int. J. Num. Meth. Eng., 33, 1331-1364 (1992) · Zbl 0769.73084
[16] Zienkiewicz, O. C.; Zhu, J. Z., The superconvergent patch recovery and a-posteriori error estimates. Part 2: Error estimates and adaptivity, Int. J. Num. Meth. Eng., 33, 1365-1382 (1992) · Zbl 0769.73085
[17] Preitag, L. A.; Ollivier-Gooch, C., Tetrahedral mesh improvement using swapping and smoothing, Int. J. Num. Meth. Eng., 40, 3979-4002 (1997) · Zbl 0897.65075
[18] Buscaglia, G. C.; Dari, E. A., Anisotropic mesh optimization and its application in adaptivity, Int. J. Num. Meth. Eng., 40, 4119-4136 (1997) · Zbl 0899.76264
[19] P.L. George, Delaunay Triangulation and Meshing: Application to Finite Elements; P.L. George, Delaunay Triangulation and Meshing: Application to Finite Elements · Zbl 0908.65143
[20] George, P. L.; Hecht, F.; Saltel, E., Automatic mesh generator with a specified boundary, Comput. Meth. Appl. Mech. Eng., 92, 269-288 (1991) · Zbl 0756.65133
[21] Pain, C. C.; Umpleby, A. P.; de Oliveira, C. R.E.; Goddard, A. J.H., Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations, Comput. Meth. Appl. Mech. Eng., 190, 3771-3796 (2001) · Zbl 1008.76041
[22] Ford, R.; Pain, C. C.; Piggott, M. D.; Goddard, A. J.H., C.R.E. de Oliveira and A.P. Umpleby, A non-hydrostatic finite element model for three-dimensional stratified ocean flows. Part I: Model formulation, Monthly Weather Rev., 132, 2816-2831 (2004)
[23] Ford, R.; Pain, C. C.; Piggott, M. D.; Goddard, A. J.H.; de Oliveira, C. R.E.; Umpleby, A. P., A non-hydrostatic finite element model for three-dimensional stratified ocean flows. Part II: Model validation, Monthly Weather Rev., 132, 2832-2844 (2004)
[24] Pain, C. C.; Piggott, M. D.; Goddard, A. J.H.; Fang, F.; Gorman, G. J.; Marshall, D. P.; Eaton, M. D.; Power, P. W.; de Oliveira, C. R.E., Three-dimensional unstructured mesh ocean modelling, Ocean Modelling, 10, 5-33 (2005)
[25] Kallinderis, Y.; Vijayan, P., Adaptive refinement-coarsening scheme for three-dimensional unstructured meshes, AIAA J., 31, 1440-1447 (1993)
[26] Lohner, R.; Parikh, P., Generation of three-dimensional unstructured grids by the advancing-front method, Int. J. Num. Meth. Fluids, 8, 1135-1149 (1988) · Zbl 0668.76035
[27] Moller, P.; Hansbo, P., On advancing front mesh generation in three dimensions, Int. J. Num. Meth. Eng., 38, 3551-3569 (1995) · Zbl 0835.73093
[28] Xu, X.; Pain, C. C.; Goddard, A. J.H.; de Oliveria, C. R.E., An automatic adaptive meshing technique for delaunay triangulations, Comput. Meth. Appl. Mech. Eng., 161, 297-303 (1998) · Zbl 0935.65134
[29] Borouchaki, H.; Frey, P. J., Adaptive triangular-quadrilateral mesh generation, Int. J. Num. Meth. Eng., 41, 915-934 (1998) · Zbl 0905.65111
[30] Borouchaki, H.; Lo, S. H., Fast Delaunay triangulation in three dimensions, Comput. Meth. Appl. Mech. Eng., 128, 153-167 (1995) · Zbl 0859.65149
[31] Borouchaki, H.; George, P. L., Optimal Delaunay point insertion, Int. J. Num. Meth. Eng., 39, 3407-3437 (1996) · Zbl 0883.73086
[32] Castro-Diaz, M. J.; Hecht, F.; Mohammadi, B.; Pironneau, O., Anisotropic unstructured mesh adaption for flow simulations, Int. J. Num. Meth. Fluids, 25, 475-491 (1997) · Zbl 0902.76057
[33] Weatherill, N. P., Efficient three-dimensional Delaunay triangulation with automatic point creation and im-posed boundary constraints, Int. J. Num. Meth. Eng., 37, 2005-2039 (1994) · Zbl 0806.76073
[34] Joe, B., Three-dimensional triangulations from local transformations, SIAM J. Sci. Stat. Comput., 10, 718-741 (1989) · Zbl 0681.65087
[35] Cacuci, D. G., Sensitivity theory for non-linear systems. I: Nonlinear function analysis approach, J. Math. Phys., 22, 2794-2802 (1981)
[36] Sanders, B. F.; Katopodes, N. D., Control of canal flow by adjoint sensitivity method, J. Irrigation Drainage Eng., 125, 287-297 (1999)
[37] Piasecki, M.; Katopodes, N., Control of contaminant releases in rivers. I: Adjoint sensitivity analysis, J. Hydr. Engrg., 123, 488-492 (1997)
[38] Sanders, B. F.; Katopodes, N. D., Adjoint sensitivity analysis for shallow water wave control, J. Eng. Mech., 126, 909-919 (2000)
[39] Zhang, S.; Zou, X.; Ahlquist, J.; Navon, I. M.; Sela, J. G., Use of differentiable and non-differentiable op-timization algorithms for variational data assimilation with discontinuous cost functions, Monthly Weather Rev., 128, 4031-4044 (2000)
[40] Gunzburger, M., Sensitivities, adjoints and flow optimization, Int. J. Num. Meth. Fluids, 31, 53-78 (1999) · Zbl 0962.76030
[41] Langland, R. H.; Elsberry, R. L.; Errico, R. M., Evaluation of physical processes in an idealized extratropical cyclone using adjoint sensitivity, Q.J.R. Meteorol. Soc., 121, 1349-1386 (1995)
[42] Homescu, C.; Navon, I. M., Numerical and theoretical considerations for sensitivity calculation of discontinuous flow, Systems and Control Letters, 48, 253-260 (2003) · Zbl 1157.93361
[43] Cacuci, D. G., The Forward and Adjoint Methods of Sensitivity Analysis (1988), CRC Press, Chapter 3 pp. 71-144
[44] Alekseev, A. K.; Navon, I. M., A-posteriori pointwise error estimation for compressible fluid flows using adjoint parameters and Lagrange remainder, Int. J. Num. Meth. Fluids, 47, 45-74 (2005) · Zbl 1063.76067
[45] Le Dimet, F.-X.; Navon, I. M.; Daescu, D. N., Second-order information in data assimilation, Monthly Weather Rev., 130, 629-648 (2002)
[46] Alekseev, A. K.; Navon, I. M., The analysis of an ill-posed problem using multi-scale resolution and second- order adjoint techniques, Comput. Meth. Appl. Mech. Eng., 190, 1937-1953 (2001) · Zbl 1031.76040
[47] Navon, I. M., Variational data assimilation with an adiabatic version of the NMC spectral model, Monthly Weather Rev., 120, 1433-1446 (1992)
[48] F-X. Le Dimet and I.M. Navon, Technical report: Early review on variational data assimilation, SCRI Technical Report, (1988).; F-X. Le Dimet and I.M. Navon, Technical report: Early review on variational data assimilation, SCRI Technical Report, (1988).
[49] Hughes, T. J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems, Comput. Meth. Appl. Mech. Eng., 58, 329 (1986) · Zbl 0587.76120
[50] Leonard, B. P., The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection, Comput. Meth. Appl. Mech. Eng., 88, 17 (1991) · Zbl 0746.76067
[51] F. Fang, M.D. Piggott, C.C. Pain, G.J. Gorman and A.J.H. Goddard, An adaptive mesh adjoint data assimilation method for coastal flows, Ocean Modelling15doi:10.1016/j.ocemod.2006.02.002http://dx.doi.org/10.1016/j.ocemod.2006.02.002; F. Fang, M.D. Piggott, C.C. Pain, G.J. Gorman and A.J.H. Goddard, An adaptive mesh adjoint data assimilation method for coastal flows, Ocean Modelling15doi:10.1016/j.ocemod.2006.02.002http://dx.doi.org/10.1016/j.ocemod.2006.02.002
[52] Griewank, A., Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation, Optimization Methods and Software, 1, 35-54 (1992)
[53] S.K. Nadarajah and A. Jameson, A comparison of the continuous and discrete adjoint approach to automatic aerodynamic optimization, \(In AIAA 38^{ th } \)Aerospace Sciences Meeting; S.K. Nadarajah and A. Jameson, A comparison of the continuous and discrete adjoint approach to automatic aerodynamic optimization, \(In AIAA 38^{ th } \)Aerospace Sciences Meeting
[54] Pierce, N. A.; Giles, M. B., Adjoint recovery of superconvergent functionals from PDE approximations, SIAM Review, 42, 247-264 (2000) · Zbl 0948.65119
[55] Müller, J.-D.; Giles, M. B., Solution adaptive mesh refinement using adjoint error analysis, \(15^{th}\) Computational Fluid Dynamics Conference (2001), American Institute of Aeronautics and Astronautics
[56] Hide, R., A note on helicity, Geophys. Astrophys. Fluid Dyn., 7, 157-161 (1976) · Zbl 0357.76078
[57] Hide, R., Superhelicity, helicity and potential vorticity, Geophys. Astrophys. Fluid Dyn., 48, 69-79 (1989) · Zbl 0683.76096
[58] Hide, R., Helicity, superhelicity and weighted relative potential vorticity: Useful diagnostic pseudoscalars?, Q. J. R. Meteorol. Soc., 128, 1759-1762 (2002)
[59] Oden, J. T.; Prudhomme, S., Goal-oriented error estimation and adaptivity for the finite element method, Computers Math. Applic., 41, 735-756 (2001) · Zbl 0987.65110
[60] Prudhomme, S.; Oden, J. T., On goal-oriented error estimation for elliptic problems, Comput. Meth. Appl. Mech. Eng., 179, 313-331 (1999) · Zbl 0945.65123
[61] Cirak, F.; Ramm, E., A-posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem, Comput. Meth. Appl. Mech. Eng., 156, 351-362 (1998) · Zbl 0947.74062
[62] Venditti, D. A.; Darmofal, D. L., Adjoint error estimation and grid adaption for functional outputs: Application to quasi-one-dimensional flow, J. Comput. Phys., 164, 204 (2000) · Zbl 0995.76057
[63] Knupp, P. M., Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II—A framework for volume mesh optimisation and the condition number of the Jacobian matrix, Int. J. Num. Meth. Eng., 48, 1165-1185 (2000) · Zbl 0990.74069
[64] Venditti, D. A.; Darmofal, D. L., Grid adaption for functional outputs: Application to two-dimensional inviscid flows, J. Comput. Phys., 176, 40-69 (2002) · Zbl 1120.76342
[65] Venditti, D. A.; Darmofal, D. L., Anisotropic grid adaption for functional outputs: Application to two- dimensional viscous flows, J. Comput. Phys., 187, 22-46 (2003) · Zbl 1047.76541
[66] O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method\(^{th} \); O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method\(^{th} \) · Zbl 0991.74002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.