# Mass Matrix Templates: General Description and 1D Examples

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## Abstract

This article is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The main focus is on adjusting the mass matrix. Two well known discretization methods, described in FEM textbooks since the late 1960s, lead to diagonally lumped and consistent mass matrices, respectively. Those models are sufficient to cover many engineering applications but for some problems they fall short. The gap can be filled with a more general approach that relies on the use of templates. These are algebraic forms that carry free parameters. Templates have the virtue of producing a set of mass matrices that satisfy certain *a priori* constraint conditions such as symmetry, nonnegativity, invariance and momentum conservation. In particular, the diagonally lumped and consistent versions can be obtained as instances. Availability of free parameters, however, allows the mass matrix to be customized to special needs, such as high precision vibration frequencies or minimally dispersive wave propagation. An attractive feature of templates for FEM programming is that only one element implementation as module with free parameters is needed, and need not be recoded when the application problem class changes. The paper provides a general overview of the topic, and illustrates it with one-dimensional structural elements: bars and beams.

## Notes

### Acknowledgments

The first version of this paper was written during the 2004-2005 summer academic recesses while the first author was a visitor at CIMNE (Centro Internacional de Métodos Numéricos en Ingenieria) at Barcelona, Spain. The visits were partly supported by fellowships awarded by the Spanish Ministerio de Educación y Cultura during May-June of those years, and partly by the National Science Foundation under grant High-Fidelity Simulations for Heterogeneous Civil and Mechanical Systems, CMS-0219422. Thanks are due to Eugenio Oñate, director of CIMNE, for encouraging expanding the material to an expository paper. Thanks are also given to Manfred Bischoff and Anton Tkachuk for calling attention to recent work at the University of Stuttgart on mass modification methods, and providing articles in press.

## References

- 1.Abramowitz M, Stegun LA (eds) (1964) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Applied Mathematics Series 55, Natl. Bur. Standards, U.S. Department of Commerce, Washington, DC (reprinted by Wiley)Google Scholar
- 2.Achenbach JD (1973) Wave propagation in elastic solids. Elsevier, AmsterdamzbMATHGoogle Scholar
- 3.Aitken AC (1939) Determinants and Matrices. Oliver and Boyd, Edinburgh and London (2nd-8th editions, 1942–56, 9th edition, reset and reprinted, 1967, Greenwood Press, Westport CN, 1983)Google Scholar
- 4.Anonymous The NASTRAN Theoretical Manual, NASA SP-221 (1970) The NASTRAN User’s Manual, NASA SP-222, 1970; The NASTRAN Programmer’s Manual, NASA SP-223, 1970; The NASTRAN Demonstration Problem Manual, NASA SP-223Google Scholar
- 5.Archer JS (1963) Consistent mass matrix for distributed mass systems. J ASCE Struct Div 89:161–178Google Scholar
- 6.Archer JS (1965) Consistent mass matrix formulation for structural analysis using finite element techniques. AIAA J 3:1910–1918CrossRefzbMATHMathSciNetGoogle Scholar
- 7.Banerjee B (2011) An introduction to metamaterials and waves in composites. Taylor and Francis, Boca RatonGoogle Scholar
- 8.Belytschko T, Mullen R (1978) On dispersive properties of finite element solutions. In: Miklowitz J, Achenbach JD (eds) Modern problems in elastic wave propagation. Wiley, New York, pp 67–82Google Scholar
- 9.Belytschko T, Hughes TJR (eds) (1983) Computational methods for transient analysis. Elsevier, AmsterdamGoogle Scholar
- 10.Bergan PG, Hanssen L (1976) A new approach for deriving ‘good’ finite elements. In: Whiteman JR (ed) The mathematics of finite elements and applications II. Academic Press, London, pp 483–497Google Scholar
- 11.Bergan PG (1980) Finite elements based on energy-orthogonal functions. Int J Numer Methods Eng 15:1141–1555MathSciNetGoogle Scholar
- 12.Bergan PG, Nygård MK (1984) Finite elements with increased freedom in choosing shape functions. Int J Numer Methods Eng 20:643–664CrossRefzbMATHGoogle Scholar
- 13.Bergan PG, Felippa CA (1985) A triangular membrane element with rotational degrees of freedom. Comput Methods Appl Mech Eng 50:25–69CrossRefzbMATHGoogle Scholar
- 14.Bhat SP, Bernstein DS (1996) Second-order systems with singular mass matrix and an extension of Guyan reduction. SIAM J Matrix Anal Appl 17:649–657CrossRefzbMATHMathSciNetGoogle Scholar
- 15.Born M, Huang K (1954) Dynamical theory of crystal Lattices. Oxford, LondonzbMATHGoogle Scholar
- 16.Brillouin L (1946) Wave propagation in periodic structures. Dover, New YorkzbMATHGoogle Scholar
- 17.Chandrasekhar S (1960) Radiative transfer. Dover, New YorkGoogle Scholar
- 18.Clough RW, Penzien J (1993) Dynamics of structures, 2nd edn. McGraw-Hill, New YorkGoogle Scholar
- 19.Cook RD, Malkus DS, Plesha ME (1989) Concepts and application of finite element methods, 3rd edn. Wiley, New YorkGoogle Scholar
- 20.Craig RR, Bampton MCC (1968) Coupling of substructures for dynamic analyses. AIAA J 6:1313–1319CrossRefzbMATHGoogle Scholar
- 21.Craig RR, Kurdila AJ (2006) Fundamentals of structural dynamics. Wiley, New York zbMATHGoogle Scholar
- 22.Deymier PA (ed) (2013) Acoustic metamaterials and phononic crystals. Springer, New YorkGoogle Scholar
- 23.Duncan WJ, Collar AR (1934) A method for the solution of oscillations problems by matrices. Philos Mag Ser 7 17:865–885CrossRefGoogle Scholar
- 24.Duncan WJ, Collar AR (1935) Matrices applied to the motions of damped systems. Philos Mag Ser 7 19:197–214CrossRefGoogle Scholar
- 25.Elias ZM (1973) On the reciprocal form of Hamilton’s principle. J Appl Mech 40:93–100CrossRefzbMATHGoogle Scholar
- 26.Ergatoudis J, Irons BM, Zienkiewicz OC (1968) Curved, isoparametric, “quadrilateral” elements for finite element analysis. Int J Solids Struct 4:31–42CrossRefzbMATHGoogle Scholar
- 27.Ewing WM, Jardetsky WS, Press F (1957) Wave propagation in layered media. McGraw-Hill, New YorkGoogle Scholar
- 28.Felippa CA (1966) Refined finite element analysis of linear and nonlinear two-dimensional structures, Ph.D. Dissertation, Department of Civil Engineering, University of California at Berkeley, BerkeleyGoogle Scholar
- 29.Felippa CA, Bergan PG (1987) A triangular plate bending element based on an energy-orthogonal free formulation. Comput Methods Appl Mech Eng 61:129–160CrossRefzbMATHGoogle Scholar
- 30.Felippa CA (1994) A survey of parametrized variational principles and applications to computational mechanics. Comput Methods Appl Mech Eng 113:109–139CrossRefMathSciNetGoogle Scholar
- 31.Felippa CA (2000) Recent advances in finite element templates. In: Topping BHV (ed) Computational Mechanics for the Twenty-First Century. Saxe-Coburn Publications, Edinburgh, Chapter 4, pp 71–98Google Scholar
- 32.Felippa CA (2001) A historical outline of matrix structural analysis: a play in three acts. Comput Struct 79:1313–1324CrossRefGoogle Scholar
- 33.Felippa CA (2001) Customizing high performance elements by Fourier methods. In: Wall WA et al. (eds). Trends in Computational Mechanics, CIMNE, Barcelona, pp 283–296Google Scholar
- 34.Felippa CA (2001) Customizing the mass and geometric stiffness of plane thin beam elements by Fourier methods. Eng Comput 18:286–303CrossRefzbMATHGoogle Scholar
- 35.Felippa CA (2003) A study of optimal membrane triangles with drilling freedoms. Comput Methods Appl Mech Eng 192:2125–2168CrossRefzbMATHGoogle Scholar
- 36.Felippa CA (2004) A template tutorial. In: Mathisen KM Kvamsdal T, Okstad KM (eds) Computational mechanics: theory and practic, 29th edn. CIMNE, Barcelona, pp 29–66Google Scholar
- 37.Felippa CA (2004) A compendium of FEM integration rules for finite element work. Eng Comput 21:867–890CrossRefzbMATHGoogle Scholar
- 38.Felippa CA (2005) The amusing history of shear flexible beam elements. IACM Expr 17:15–19Google Scholar
- 39.Felippa CA (2006) Supernatural QUAD4: a template formulation, invited contribution to J. H. Argyris Memorial Issue. Comput Methods Appl Mech Eng 195:5316–5342CrossRefzbMATHMathSciNetGoogle Scholar
- 40.Felippa CA (2006) Construction of customized mass-stiffness pairs using templates, invited contribution to special Issue in honor of A. K. Noor. ASCE J Aerosp 19:4:241–258CrossRefGoogle Scholar
- 41.Felippa CA Web-posted Lectures on Advanced Finite Element Methods. http://caswww.colorado.edu/courses.d/AFEM.d/Home.html. Accessed 2013
- 42.Felippa CA Web-posted Lectures on Introduction to Finite Element Methods http://caswww.colorado.edu/courses.d/IFEM.d/Home.html. Accessed 2013
- 43.Flaggs DL (1988) Symbolic analysis of the finite element method in structural mechanics, Ph.D. Dissertation, Dept of Aeronautics and Astronautics, Stanford UniversityGoogle Scholar
- 44.Fraeijs de Veubeke BM (1971) Dual principles of elastodynamics: finite element applications, NATO Advanced Studies Institute Lecture Series in Finite Element Methods in Continuum Mechanics, Lisbon (reprinted in M. Geradin (ed.), B. M. Fraeijs de Veubeke Memorial Volume of Selected Papers, Sitthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, pp 295–319)Google Scholar
- 45.Frazer RA, Duncan WJ, Collar AR (1938) Elementary matrices, and some applications to dynamics and differential equations, 1st edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- 46.Fried I, Malkus DS (1975) Finite element mass lumping by numerical integration with no convergence rate loss. Int J Solids Struct 11:461–466CrossRefzbMATHGoogle Scholar
- 47.Geradin M (1973) Computational efficiency of equilibrium models for eigenvalue analysis. In: Fraeijs de Veubeke B (ed) High speed computing of elastic structures. Université de Liège, Liège, pp 589–624Google Scholar
- 48.Geradin M, Rixen D (1997) Mechanical vibrations: theory and applications to structural dynamics. Wiley, New YorkGoogle Scholar
- 49.Graff KF (1991) Wave motion in elastic solids. Dover, New YorkGoogle Scholar
- 50.Guo Q (2012) Developing an optimal mass for membrane triangles with corner drilling freedoms, M.S. Dissertation, Department of Aerospace Engineering Sciences, University of Colorado at BoulderGoogle Scholar
- 51.Gurtin M (1972) The Linear Theory of Elasticity. In: Truesdell C (ed) Encyclopedia of Physics VIa, Vol II, Springer-Verlag, Berlin, pp 1–295 (reprinted as Mechanics of Solids, vol II. Springer, Berlin)Google Scholar
- 52.Guyan RJ (1965) Reduction of stiffness and mass matrices. AIAA J 3:380CrossRefGoogle Scholar
- 53.Hamming RW (1986) Numerical methods for scientists and engineers, 2nd edn. Dover, New YorkzbMATHGoogle Scholar
- 54.Hamming RW (1998) Digital filters, 3rd edn. Dover, New YorkGoogle Scholar
- 55.Harris JD (2001) Linear elastic waves. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
- 56.Higham NJ (2008) Functions of matrices: theory and computation. SIAM, PhiladelphiaCrossRefGoogle Scholar
- 57.Hinton E, Rock T, Zienkiewicz OC (1976) A note on mass lumping and related processes in the finite element method. Earthq Eng Struct Dyn 4:245–249CrossRefGoogle Scholar
- 58.Hurty WC (1960) Vibrations of structural systems by component mode synthesis. J Eng Mech 86:51–69Google Scholar
- 59.Hurty WC (1965) Dynamic analysis of structural systems using component modes. AIAA J 3:678–685CrossRefGoogle Scholar
- 60.Irons BM, Draper K (1965) Inadequacy of nodal connections in a stiffness solution for plate bending. AIAA J 3:965–966CrossRefGoogle Scholar
- 61.Irons BM (1966) Engineering application of numerical integration in stiffness methods. AIAA J 4:2035–2037CrossRefzbMATHGoogle Scholar
- 62.Irons BM, Ahmad S (1980) Techniques of finite elements. Ellis Horwood Ltd., ChichesterGoogle Scholar
- 63.Jones H (1960) The theory of Brillouin zones and electronic states in crystals. North Holland, AmsterdamGoogle Scholar
- 64.Khenous HB, Laborde P, Renard Y (2008) Mass redistribution method for finite element contact problems in elastodynamics. Eur J Mech A 27:918–932CrossRefzbMATHMathSciNetGoogle Scholar
- 65.Kolsky H (1953) Stress waves in solids. Oxford University Press, Oxford reprinted by Dover, 1963zbMATHGoogle Scholar
- 66.Krieg RD, Key SW (1972) Transient shell analysis by numerical time integration. In: Oden JT, Clough RW, Yamamoto Y (eds) Advances in computational methods for structural mechanics and design. UAH Press, Huntsville, pp 237–258Google Scholar
- 67.Lagrange JL (1788) Méchanique Analytique, Paris (Reprinted by J. Gabay, Paris, 1989; downloadable as Google eBook)Google Scholar
- 68.Lancaster P, Rodman L (1995) Algebraic Riccati equations. Oxford University Press, OxfordzbMATHGoogle Scholar
- 69.Lee U (2009) Spectral element method in structural dynamics. Wiley, SingaporeCrossRefzbMATHGoogle Scholar
- 70.Leung AYT (1993) Dynamic stiffness and substructures. Springer, LondonCrossRefGoogle Scholar
- 71.MacNeal RH (ed) (1970) The NASTRAN Theoretical Manual, NASA SP-221Google Scholar
- 72.MacNeal RH (1971) A hybrid method of component mode synthesis. Comput Struct 1:581–601CrossRefGoogle Scholar
- 73.MacNeal RH (1994) Finite elements: their design and performance. Marcel Dekker, New YorkGoogle Scholar
- 74.Melosh RJ (1962) Development of the stiffness method to define bounds on the elastic behavior of structures, Ph.D. Dissertation, University of Washington, SeattleGoogle Scholar
- 75.Melosh RJ (1963) Bases for the derivation of matrices for the direct stiffness method. AIAA J 1:1631–1637CrossRefGoogle Scholar
- 76.Malkus DS, Plesha ME (1986) Zero and negative masses in finite element vibration and transient analysis. Comput Methods Appl Mech Eng 59:281–306CrossRefzbMATHMathSciNetGoogle Scholar
- 77.Malkus DS, Plesha ME, Liu MR (1988) Reversed stability conditions in transient finite element analysis. Comput Methods Appl Mech Eng 68:97–114CrossRefzbMATHMathSciNetGoogle Scholar
- 78.Olovsson L, Unosson M, Simonsson K (2004) Selective mass scaling for thin wall structures modeled with trilinear solid elements. Comput Mech 34:134–136CrossRefzbMATHGoogle Scholar
- 79.Olovsson L, Simonsson K, Unosson M (2005) Selective mass scaling for explicit finite element analysis. Int J Numer Methods Eng 63:1436–1445CrossRefzbMATHGoogle Scholar
- 80.Parlett BN (1980) The symmetric eigenvalue problem. Prentice-Hall, Englewood Cliffs (Reprinted by SIAM Publications, 1980)Google Scholar
- 81.Park KC, Underwood PG (1980) A variable step central difference method for structural dynamics analysis: theoretical aspects. Comput Methods Appl Mech Eng 22:241–258CrossRefzbMATHMathSciNetGoogle Scholar
- 82.Park KC (1984) Symbolic Fourier analysis procedures for \(C^0\) finite elements. In: Liu WK, Belytschko T, Park KC (eds) Innovative methods for nonlinear problems. Pineridge Press, Swansea, pp 269–293Google Scholar
- 83.Park KC, Flaggs DL (1984) An operational procedure for the symbolic analysis of the finite element method. Comput Methods Appl Mech Eng 42:37–46CrossRefzbMATHMathSciNetGoogle Scholar
- 84.Park KC, Flaggs DL (1984) A Fourier analysis of spurious modes and element locking in the finite element method. Comput Methods Appl Mech Eng 46:65–81CrossRefzbMATHMathSciNetGoogle Scholar
- 85.Pestel EC, Leckie FA (1963) Matrix methods in elastomechanics. McGraw-Hill, New YorkGoogle Scholar
- 86.Pilkey WD, Wunderlich W (1993) Mechanics of structures: variational and computational methods. CRC Press, Boca RatonGoogle Scholar
- 87.Pilkey WD (2002) Analysis and design of elastic beams. Wiley, New YorkCrossRefGoogle Scholar
- 88.Przemieniecki JS (1968) Theory of matrix structural analysis. McGraw-Hill, New York (Dover edition 1986)Google Scholar
- 89.Raimes S (1967) The wave mechanics of electrons in metals. North-Holland, AmsterdamGoogle Scholar
- 90.Renard Y (2010) The singular dynamic method for constrained second order hyperbolic equations. J Comput Appl Methods 234:906–923CrossRefzbMATHMathSciNetGoogle Scholar
- 91.Rubin S (1975) Improved component-mode representation for structural dynamic analysis. AIAA J 13:995–1006CrossRefzbMATHGoogle Scholar
- 92.Song C (2009) The scaled boundary finite element method in structural dynamics. Int J Numer Methods Eng 77:1139–1171CrossRefzbMATHGoogle Scholar
- 93.Soutas-Little RW, Inman DJ (1998) Engineering mechanics: dynamics. Prentice-Hall, Upper Saddle RiverGoogle Scholar
- 94.Sprague MA, Geers TL (2007) Legendre spectral finite elements for structural dynamics analysis. Commun Numer Methods Eng 24:1953–1965CrossRefMathSciNetGoogle Scholar
- 95.Stroud AH, Secrest D (1966) Gaussian quadrature formulas. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
- 96.Stroud AH (1971) Approximate calculation of multiple integrals. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
- 97.Tabarrok B (1973) Complementary energy methods in elastodynamics. In: Fraeijs de Veubeke B (ed) High speed computing of elastic structures. Université de Liège, Liège, pp 625–662Google Scholar
- 98.Tabarrok B, Rimrott FPJ (1994) Variational methods and complementary formulations in dynamics. Kluwer, BostonCrossRefzbMATHGoogle Scholar
- 99.Timoshenko SP (1921) On the correction for shear of the differential equation for transverse vibration of prismatic bars, Phil. Mag., XLI, pp 744–46 (Reprinted in The Collected Papers of Stephen P. Timoshenko, McGraw-Hill, London, 1953. See also S. P. Timoshenko and D. H. Young, Vibration Problems in Engineering, 3rd edition, Van Nostrand, pp 329–331, 1954)Google Scholar
- 100.Timoshenko SP, Young DH (1955) Vibration problems in engineering. Van Nostrand, PrincetonGoogle Scholar
- 101.Tkachuk A, Bischoff M (2013) Variational methods for selective mass scaling. Comput Mech 52:563–576CrossRefzbMATHMathSciNetGoogle Scholar
- 102.Tkachuk A, Wolfmuth B, Bischoff M (2013) Hybrid-mixed discretization of elastodynamic contact problems using consistent singular matrices. Int J Numer Methods Eng 94:473–493CrossRefGoogle Scholar
- 103.Toupin RA (1952) A variational principle for the mesh-type analysis of a mechanical system. Trans ASME 74:151–152MathSciNetGoogle Scholar
- 104.Turner MJ, Clough RW, Martin HC, Topp LJ (1956) Stiffness and deflection analysis of complex structures. J Acoust Soc Am 23:805–824zbMATHGoogle Scholar
- 105.Turner MJ (1959) The direct stiffness method of structural analysis. Structural and Materials Panel Paper, AachenGoogle Scholar
- 106.Turner MJ, Martin HC, Weikel BC (1964) Further developments and applications of the stiffness method. In: Fraeijs de Veubeke BM (ed) Matrix methods of structural analysis, AGARDograph 72. Pergamon Press, Oxford, pp 203–266Google Scholar
- 107.Udwadia FE, Kalaba RE (1996) Analytical dynamics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- 108.Udwadia FE, Phohomshiri P (2006) Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics. Proc R Soc A 462:2097–2117 Google Scholar
- 109.Udwadia FE, Schutte AD (2010) Equations of motion for general constrained systems in Lagrangian mechanics. Acta Mech 213:111–129CrossRefzbMATHGoogle Scholar
- 110.Udwadia FE, Wanichanon T (2012) Explicit equation of motion of constrained systems. In: Dai L, Jazar RN (eds) Nonlinear approaches in engineering applications. Springer, New YorkGoogle Scholar
- 111.Underwood PG, Park KC (1980) A variable-step central difference method for structural dynamics analysis, Part 2: implementation and performance evaluation. Comput Methods Appl Mech Eng 23:259–279CrossRefzbMATHMathSciNetGoogle Scholar
- 112.Warming RF, Hyett BJ (1974) The modified equation approach to the stability and accuracy analysis of finite difference methods. J Comput Phys 14:159–179CrossRefzbMATHMathSciNetGoogle Scholar
- 113.Wilkinson JH, Reinsch CH (eds) (1971) Handbook for automatic computation. Linear Algebra, vol 2, Springer, BerlinGoogle Scholar
- 114.Wimp J (1981) Sequence transformations and their applications. Academic Press, New YorkzbMATHGoogle Scholar
- 115.Wolf A (2003) The scaled boundary finite element method. Wiley, ChichesterGoogle Scholar
- 116.Wood A (1940) Acoustics. Blackie and Sons, London (Reprinted by Dover, 1966)Google Scholar
- 117.Zhang F (ed) (2005) The schur complement and its applications. Springer, New YorkGoogle Scholar
- 118.Ziman JM (1967) Principles of the theory of solids. North-Holland, AmsterdamGoogle Scholar