×

Mass lumping techniques in the spectral element method: on the equivalence of the row-sum, nodal quadrature, and diagonal scaling methods. (English) Zbl 1441.74238

Summary: In the context of wave propagation analysis, the spectral element method (SEM) in conjunction with a diagonal mass matrix is often the method of choice. Therefore, it is of high importance to investigate the influence of different mass lumping schemes on the accuracy of the numerical results. To this end, we compare the performance of three established methods including the row-sum method, the nodal quadrature method, and the diagonal scaling method. The theoretical analysis of these methods reveals a close connection between them. Under certain conditions, that are discussed in detail in this article, we are able to show a direct equivalence between these three approaches. In this regard, the attainable accuracy of the numerical integration of the mass matrix plays an important role. By means of several dynamic benchmark problems we verify the theoretical results and illustrate the convergence properties of the lumped mass matrix SEM in comparison to a formulation based on the consistent mass matrix.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Software:

RegSEM
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Delsanto, P. P.; Whitcombe, T.; Chaskelis, H. H.; Mignogna, R. B., Connection machine simulation of ultrasonic wave propagation in materials. I: The one-dimensional case, Wave Motion, 16, 65-80 (1992)
[2] Delsanto, P. P.; Schechter, R. S.; Chaskelis, H. H.; Mignogna, R. B.; Kline, R. B., Connection machine simulation of ultrasonic wave propagation in materials. II: The two-dimensional case, Wave Motion, 20 (1994) · Zbl 0918.73378
[3] Delsanto, P. P.; Schechter, R. S.; Mignogna, R. B., Connection machine simulation of ultrasonic wave propagation in materials. III: The three-dimensional case, Wave Motion, 26, 329-339 (1997) · Zbl 0929.74051
[4] Paćko, P.; Bielak, T.; Spencer, A. B.; Staszewski, W. J.; Uhl, T.; Worden, K., Lamb wave propagation modelling and simulation using parallel processing architecture and graphical cards, Smart Mater. Struct., 21, 1-13 (2012)
[5] Hopman, R. K.; Leamy, M. J., Application of cellular automata modeling to seismic elastodynamics, Int. J. Solids Struct., 45, 4835-4849 (2008) · Zbl 1169.74487
[6] Hopman, R. K.; Leamy, M. J., Triangular cellular automata for computing two-dimensional elastodynamic response on arbitrary domain, J. Appl. Mech., 78, 021020 (2011)
[7] Kluska, P.; Staszewski, W. J.; Leamy, M. J.; Uhl, T., Cellular automata for Lamb wave propagation modelling in smart structures, Smart Mater. Struct., 22, 8, 13pp. (2013)
[8] Song, C.; Wolf, J. P., The scaled boundary finite-element method - alias consistent infinitesimal finite-element cell method - for elastodynamics, Comput. Methods Appl. Mech. Engrg., 147, 329-355 (1997) · Zbl 0897.73069
[9] Gravenkamp, H.; Saputra, A. A.; Song, C.; Birk, C., Efficient wave propagation on quadtree meshes using SBFEM with reduced modal basis, Internat. J. Numer. Methods Engrg., 110, 12, 1119-1141 (2017)
[10] Krome, F.; Gravenkamp, H., A semi-analytical curved element for linear elasticity based on the scaled boundary finite element method, Internat. J. Numer. Methods Engrg., 109, 6, 790-808 (2016)
[11] Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987), Prentice-Hall · Zbl 0634.73056
[12] Bathe, K. J., Finite Element Procedures (2014), Prentice Hall
[13] Szabó, B.; Babuška, I., Finite Element Analysis (1991), John Wiley and Sons · Zbl 0792.73003
[14] Solin, P.; Segeth, K.; Dolezel, I., Higher-Order Finite Element Methods (2004), Chapman and Hall · Zbl 1032.65132
[15] Demkowicz, L., Computing with hp-Adaptive Finite Elements: Volume 1: One and Two Dimensional Elliptic and Maxwell Problems (2006), Chapman and Hall
[16] Demkowicz, L.; Kurtz, J.; Pardo, D.; Paszyński, M.; Rachowicz, W.; Zdunek, A., Computing with hp-Adaptive Finite Elements: Volume 2 Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications (2008), Chapman and Hall · Zbl 1148.65001
[17] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194, 4135-4195 (2005) · Zbl 1151.74419
[18] Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA (2009), John Wiley & Sons · Zbl 1378.65009
[19] Schillinger, D.; Dede, L.; Scott, M. A.; Evans, J. A.; Borden, M. J.; Rank, E.; Hughes, T. J.R., An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces, Comput. Methods Appl. Mech. Engrg., 249-252, 116-150 (2012) · Zbl 1348.65055
[20] Zienkiewicz, O. C.; Taylor, R. L., The Finite Element Method: Volume 1, The Basis (2000), Butterworth Heinemann · Zbl 0991.74002
[21] Anitescu, C.; Nguyen, C.; Rabczuk, T.; Zhuang, X., Isogeometric analysis for explicit elastodynamics using a dual-basis diagonal mass formulation, Comput. Methods Appl. Mech. Engrg., 346, 574-591 (2019) · Zbl 1440.74371
[22] Tkachuk, A.; Bischoff, M., Direct and sparse construction of consistent inverse mass matrices: General variational formulation and application to selective mass scaling, Internat. J. Numer. Methods Engrg., 101, 6, 435-469 (2015) · Zbl 1352.74441
[23] Schaeuble, A.-K.; Tkachuk, A.; Bischoff, M., Variationally consistent inertia templates for B-spline- and NURBS-based FEM: Inertia scaling and customization, Comput. Methods Appl. Mech. Engrg., 326, 596-621 (2017) · Zbl 1439.65120
[24] González, J. A.; Kopačka, J.; Kolman, R.; Cho, S. S.; Park, K. C., Inverse mass matrix for isogeometric explicit transient analysis via the method of localized Lagrange multipliers, Internat. J. Numer. Methods Engrg., 117, 9, 939-966 (2018)
[25] Noh, G.; Bathe, K. J., An explicit time integration scheme for the analysis of wave propagations, Comput. Struct., 129, 178-193 (2013)
[26] Soares, D., A simple and effective single-step time marching technique based on adaptive time integrators, Int. J. Numer. Methods Biomed. Eng., 109, 1344-1368 (2017)
[27] Soares, D., A novel family of explicit time marching techniques for structural dynamics and wave propagation models, Comput. Methods Appl. Mech. Engrg., 311, 838-855 (2016) · Zbl 1439.65078
[28] Soares, D., A simple and effective new family of time marching procedures for dynamics, Comput. Methods Appl. Mech. Engrg., 283, 1138-1166 (2015) · Zbl 1425.65077
[29] Deville, M. O.; Fischer, P. F.; Mund, E. H., (High-Order Methods for Incompressible Fluid Flow. High-Order Methods for Incompressible Fluid Flow, Cambridge Monographs on Applied and Computational Mathematics, vol. 9 (2002), Cambridge University Press) · Zbl 1007.76001
[30] Karniadakis, G. E.; Sherwin, S. J., Spectral/hp Element Methods for Computational Fluid Dynamics (2005), Oxford Science Publications · Zbl 1116.76002
[31] Pozrikidis, C., Introduction to Finite and Spectral Element Methods using MATLAB, 830 (2014), Chapman and Hall/CRC · Zbl 1337.65001
[32] Patera, A. T., A spectral element method for fluid dynamics: Laminar flow in a channel expansion, J. Comput. Phys., 54, 468-488 (1984) · Zbl 0535.76035
[33] Rønquist, E. M.; Patera, A. T., A Legendre spectral element method for the Stefan problem, Internat. J. Numer. Methods Engrg., 24, 2273-2299 (1987) · Zbl 0632.65126
[34] Chung, E. T.; Yu, T. F., Staggered-grid spectral element methods for elastic wave simulations, J. Comput. Appl. Math., 285, 132-150 (2015) · Zbl 1315.65089
[35] Cohen, G. C., Higher-Order Numerical Methods for Transient Wave Equations (2002), Springer-Verlag Berlin Heidelberg · Zbl 0985.65096
[36] Cohen, G.; Joly, P.; Tordjman, N., Higher-order finite elements with mass-lumping for the 1D wave equation, Finite Elem. Anal. Des., 16, 3-4, 329-336 (1994) · Zbl 0865.65072
[37] Cupillard, P.; Delavaud, E.; Burgos, G.; Festa, G.; Vilotte, J.-P.; Capdeville, Y.; Montagner, J.-P., RegSEM: A versatile code based on the spectral element method to compute seismic wave propagation at the regional scale, Geophys. J. Int., 188, 1203-1220 (2012)
[38] Dauksher, W.; Emery, A. F., An evaluation of the cost effectiveness of Chebyshev spectral and \(p\)-finite element solutions to the scalar wave equation, Internat. J. Numer. Methods Engrg., 45, 1099-1113 (1999) · Zbl 0942.74070
[39] Dauksher, W.; Emery, A. F., Accuracy in modeling the acoustic wave equation with Chebyshev spectral finite elements, Finite Elem. Anal. Des., 26, 115-128 (1997) · Zbl 0896.65067
[40] Duczek, S., Higher Order Finite Elements and the Fictitious Domain Concept for Wave Propagation Analysis (2014), VDI Fortschritt-Berichte Reihe 20 Nr. 458, URL http://edoc2.bibliothek.uni-halle.de/urn/urn:nbn:de:gbv:ma9:1-5434
[41] Seriani, G.; Priolo, E., Spectral element method for acoustic wave simulation in heterogeneous media, Finite Elem. Anal. Des., 16, 337-348 (1994) · Zbl 0810.73079
[42] Zimmermann, E.; Eremin, A.; Lammering, R., Analysis of the continuous mode conversion of Lamb waves in fibre composites by a stochastic material model and laser vibrometer experiments, GAMM - Mitteilungen, 41, Article e201800001 pp. (2018), (15pp.)
[43] Ha, S.; Chang, F.-K., Optimizing a spectral element for modeling PZT-induced Lamb wave propagation in thin plates, Smart Mater. Struct., 19, 1-11 (2010)
[44] Kudela, P.; Radzieṅski, M.; Ostachowicz, W., Impact induced damage assessment by means of Lamb wave image processing, Mech. Syst. Signal Process., 102, 23-36 (2018)
[45] Kudela, P.; Radzieṅski, M.; Ostachowicz, W., Wave propagation modeling in composites reinforced by randomly oriented fibers, J. Sound Vib., 414, 110-125 (2018)
[46] (Lammering, R.; Gabbert, U.; Sinapius, M.; Schuster, T.; Wierach, P., Lamb-Wave Based Structural Health Monitoring in Polymer Composites. Lamb-Wave Based Structural Health Monitoring in Polymer Composites, Research Topics in Aerospace, vol. 5 (2018), Springer International Publishing)
[47] Lonkar, K.; Chang, F.-K., Modeling of piezo-induced ultrasonic wave propagation in composite structures using layered solid spectral element, Struct. Health Monit., 13, 50-67 (2014)
[48] Xu, H.; Cantwell, C. D.; Monteserin, C.; Eskilsson, C.; Engsig-Karup, A. P.; Sherwin, S. J., Spectral/hp element methods: Recent developments, applications, and perspectives, J. Hydrol., 30, 1-22 (2018)
[49] Dauksher, W.; Emery, A. F., The solution of elastostatic and elastodynamic problems with Chebyshev spectral finite elements, Comput. Methods Appl. Mech. Engrg., 188, 217-233 (2000) · Zbl 0963.74059
[50] Ainsworth, M.; Wajid, H. A., Dispersive and dissipative behavior of the spectral element method, SIAM J. Numer. Anal., 47, 5, 3910-3937 (2009) · Zbl 1204.65137
[51] Ainsworth, M.; Wajid, H. A., Optimally blended spectral-finite element scheme for wave propagation and nonstandard reduced integration, SIAM J. Numer. Anal., 48, 346-371 (2010) · Zbl 1213.65129
[52] Seriani, G., 3-d large-scale wave propagation modeling by spectral element method on Cray T3E multiprocessor, Comput. Methods Appl. Mech. Engrg., 164, 235-247 (1998) · Zbl 0962.76072
[53] Seriani, G.; Oliveira, S. P., DFT modal analysis of spectral element methods for acoustic wave propagation, J. Comput. Acoust., 16, 531-561 (2008) · Zbl 1257.76078
[54] Seriani, G.; Oliveira, S. P., Dispersion analysis of spectral element methods for elastic wave propagation, Wave Motion, 45, 729-744 (2008) · Zbl 1231.74185
[55] Komatitsch, D.; Tromp, J., Spectral-element simulations of global seismic wave propagation I. - validation, Int. J. Geophys., 149, 390-412 (2002)
[56] Jensen, M. S., High convergence order finite elements with lumped mass matrix, Internat. J. Numer. Methods Engrg., 39, 1879-1888 (1996) · Zbl 0880.76044
[57] Duczek, S.; Joulaian, M.; Düster, A.; Gabbert, U., Numerical analysis of Lamb waves using the finite and spectral cell methods, Internat. J. Numer. Methods Engrg., 99, 1, 26-53 (2014) · Zbl 1352.74144
[58] Joulaian, M.; Duczek, S.; Gabbert, U.; Düster, A., Finite and spectral cell method for wave propagation in heterogeneous materials, Comput. Mech., 54, 3, 661-675 (2014) · Zbl 1311.74056
[59] Giraldo, D.; Restrepo, D., The spectral cell method in nonlinear earthquake modeling, Comput. Mech., 60, 6, 883-903 (2017) · Zbl 1398.74331
[60] Düster, A.; Parvizian, J.; Yang, Z.; Rank, E., The finite cell method for three-dimensional problems of solid mechanics, Comput. Methods Appl. Mech. Engrg., 197, 3768-3782 (2008) · Zbl 1194.74517
[61] Willberg, C.; Duczek, S.; Vivar Perez, J. M.; Schmicker, D.; Gabbert, U., Comparison of different higher order finite element schemes for the simulation of Lamb waves, Comput. Methods Appl. Mech. Engrg., 241-244, 246-261 (2012) · Zbl 1353.74077
[62] Willberg, C.; Duczek, S.; Vivar Perez, J. M.; Ahmad, Z. A.B., Simulation methods for guided-wave based structural health monitoring: A review, Appl. Mech. Rev., 67, 1, 1-20 (2015)
[63] Tschöke, K.; Gravenkamp, H., On the numerical convergence and performance of different spatial discretization techniques for transient elastodynamic wave propagation problems, Wave Motion, 82, 62-85 (2018) · Zbl 1524.74446
[64] Kopriva, D. A., (Implementing Spectral Methods for Partial Differential Equations. Implementing Spectral Methods for Partial Differential Equations, Scientific Computation (2009), Springer Verlag) · Zbl 1172.65001
[65] Kopriva, D. A.; Gassner, G., On the quadrature and weak form choices in collocation type discontinuous Galerkin spectral element methods, J. Sci. Comput., 44, 2, 136-155 (2010) · Zbl 1203.65199
[66] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., (Spectral Methods. Spectral Methods, Scientific Computations (2006), Springer-Verlag Berlin Heidelberg) · Zbl 1093.76002
[67] Teukolsky, S. A., Short note on the mass matrix for Gauss-Lobatto grid points, J. Comput. Phys., 283, 408-413 (2015) · Zbl 1352.65578
[68] Durufle, M.; Grob, P.; Joly, P., Influence of Gauss and Gauss-Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain, Numer. Methods Partial Differential Equations, 25, 3, 526-551 (2009) · Zbl 1167.65057
[69] Maday, Y.; Rønquist, M., Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries, Comput. Methods Appl. Mech. Engrg., 80, 91-115 (1990) · Zbl 0728.65078
[70] Düster, A.; Bröker, H.; Rank, E., The \(p\)-version of the finite element method for three-dimensional curved thin walled structures, Internat. J. Numer. Methods Engrg., 52, 673-703 (2001) · Zbl 1058.74079
[71] Höllig, K.; Reif, U.; Wipper, J., Weighted extended B-spline approximation of Dirichlet problems, SIAM J. Numer. Anal., 39, 442-462 (2001) · Zbl 0996.65119
[72] Priolo, E.; Carcione, J. M.; Seriani, G., Numerical simulation of interface waves by high-order spectral modeling techniques, J. Acoust. Soc. Am., 95, 681-693 (1994)
[73] Komatitsch, D.; Tromp, J., Introduction to the spectral element method for three-dimensional seismic wave propagation, Geophys. J. Int., 139, 806-822 (1999)
[74] Sprague, M. A.; Geers, T. L., Legendre spectral finite elements for structural dynamics analysis, Commun. Numer. Methods. Eng., 24, 1953-1965 (2008) · Zbl 1152.74050
[75] Komatitsch, D.; Tromp, J., Spectral-element simulations of global seismic wave propagation II. - three-dimensional models, oceans, rotation and self-gravitation, Int. J. Geophys., 150, 303-318 (2002)
[76] Kudela, P.; Ostachowicz, W., 3d time-domain spectral elements for stress waves modelling, J. Phys. Conf. Ser., 181, 1-8 (2009)
[77] Ostachowicz, W.; Kudela, P.; Krawczuk, M.; Żak, A., Guided Waves in Structures for SHM: The Time-Domain Spectral Element Method (2011), John Wiley & Sons
[78] Düster, A., (High Order Finite Elements for Three-Dimensional, Thin-Walled Nonlinear Continua. High Order Finite Elements for Three-Dimensional, Thin-Walled Nonlinear Continua, Berichte aus dem Bauwesen (2002), Shaker, Technical University Munich)
[79] Becker, C., Finite Elemente Methoden zur räumlichen Diskretisierung von Mehrfeldproblemen der Strukturmechanik unter Berücksichtigung diskreter Risse (2007), Ruhr-University Bochum, (Ph.D. thesis)
[80] Bronstein, I. N.; Semendjajew, K. A.; Musiol, G.; Mühlig, H., Taschenbuch der Mathematik (2008), Harri Deutsch · Zbl 1205.00020
[81] Newmark, N. M., A method of computation for structural dynamics, ASCE J. Eng. Mech. Div., 85, 2067-2094 (1959)
[82] (Belytschko, T.; Hughes, T. J.R., Computational Methods for Transient Analysis (1983), North-Holland Publishing Company) · Zbl 0521.00025
[83] Duczek, S.; Gravenkamp, H., Critical assessment of different mass lumping schemes for higher order serendipity finite elements, Comput. Methods Appl. Mech. Engrg., 350, 836-897 (2019) · Zbl 1441.74239
[84] Malkus, D. S.; Plesha, M. E., Zero and negative masses in finite element vibration and transient analysis, Comput. Methods Appl. Mech. Engrg., 59, 281-306 (1986) · Zbl 0587.73111
[85] Fried, I.; Malkus, D. S., Finite element mass matrix lumping by numerical integration with no convergence rate loss, Int. J. Solids Struct., 11, 461-466 (1975) · Zbl 0301.65010
[86] Cook, R. D.; Malkus, D. S.; Plesha, M. E., Concepts and Applications of Finite Element Analysis (1989), John Wiley & Sons · Zbl 0696.73039
[87] Murlikrishna, R.; Prathap, G., Studies on Variational Correctness of Finite Element Elastodynamics of Some Plate Elements. Tech. Rep. (2003), CSIR Centre for Mathematical Modelling and Computer Simulation
[88] Hinton, E.; Rock, T.; Zienkiewicz, O. C., A note on mass lumping and related processes in the finite element method, Earthq. Eng. Struct. Dyn., 4, 245-249 (1976)
[89] Gordon, W. J., Blending-function methods of bivariate and multivariate interpolation and approximation, SIAM J. Numer. Anal., 8, 158-177 (1971) · Zbl 0237.41008
[90] Gordon, W. J.; Hall, C. A., Construction of curvilinear co-ordinate systems and applications to mesh generation, Internat. J. Numer. Methods Engrg., 7, 461-477 (1973) · Zbl 0271.65062
[91] Gordon, W. J.; Hall, C. A., Transfinite element methods: Blending-function interpolation over arbitrary curved element domains, Numer. Math., 21, 109-129 (1973) · Zbl 0254.65072
[92] Királyfalvi, G.; Szabó, B., Quasi-regional mapping for the \(p\)-version of the finite element method, Finite Elem. Anal. Des., 27, 85-97 (1997) · Zbl 0916.73056
[93] Bröker, H., Integration Von Geometrischer Modellierung und Berechnung nach der \(p\)-Version der FEM (2001), Technical University Munich, (Ph.D. thesis)
[94] Hedayatrasa, S.; Bui, T. Q.; Zhang, C., Modeling wave propagation in functionally graded materials using time-domain spectral Chebyshev elements, J. Comput. Phys., 258, 381-404 (2014) · Zbl 1349.74199
[95] Gravenkamp, H.; Song, C.; Prager, J., A numerical approach for the computation of dispersion relations for plate structures using the scaled boundary finite element method, J. Sound Vib., 331, 2543-2557 (2012)
[96] J.V. Lambers, Numerical Analysis, online (2016). URL http://www.math.usm.edu/lambers/mat772/; J.V. Lambers, Numerical Analysis, online (2016). URL http://www.math.usm.edu/lambers/mat772/
[97] Abramowitz, M.; Stegun, I. A., (Handbook of Mathematical Functions. Handbook of Mathematical Functions, Applied Mathematics Series, vol. 55 (1972), Natinal Bureau of Standards), 1046 · Zbl 0543.33001
[98] Cottrell, J. A.; Reali, A.; Bazilevs, Y.; Hughes, T. J.R., Isogeometric analysis of structural vibrations, Comput. Methods Appl. Mech. Engrg., 195, 5257-5296 (2006) · Zbl 1119.74024
[99] Eisenberger, M., Exact longitudinal vibration frequencies of a variable cross-section rod, Appl. Acoust., 34, 123-130 (1991)
[100] Kumar, B. M.; Sujith, R. I., Exact solutions for the longitudinal vibration of non-uniform rods, J. Sound Vib., 207, 721-729 (1997) · Zbl 1235.74172
[101] Li, Q. S., Exact solutions for free longitudinal vibration of stepped non-uniform rods, Appl. Acoust., 60, 13-28 (2000)
[102] Raj, A.; Sujith, R. I., Closed-form solutions for the free longitudinal vibration of inhomogeneous rods, J. Sound Vib., 283, 1015-1030 (2005) · Zbl 1237.74083
[103] Cottereau, R.; Sevilla, R., Stability of an explicit high-order spectral element method for acoustics in heterogeneous media based on local element stability criteria, Internat. J. Numer. Methods Engrg., 116, 4, 223-245 (2018)
[104] Sevilla, R.; Cottereau, R., Influence of periodically fluctuating material parameters on the stability of explicit high-order spectral element methods, J. Comput. Phys., 373, 304-323 (2018) · Zbl 1416.65306
[105] Fries, T.-P.; Belytschko, T., The extended/generalized finite element method: An overview of the method and its applications, Internat. J. Numer. Methods Engrg., 84, 253-304 (2010) · Zbl 1202.74169
[106] Hesthaven, J. S.; Warburton, T., (Marsden, J. E.; Sirovich, L.; Antmann, S. S., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis and Applications. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis and Applications, Texts in Applied Mathematics, vol. 54 (2008), Springer-Verlag New York) · Zbl 1134.65068
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.