An efficient method for generating the mass matrix inverse is presented, which can be tailored to improve the accuracy of target frequency ranges and/or wave contents. The present method bypasses the use of biorthogonal construction of a kernel inverse mass matrix that requires special procedures for boundary conditions and free edges or surfaces, and constructs the free-free inverse mass matrix employing the standard FEM procedure. The various boundary conditions are realized by the method of localized Lagrange multipliers. Numerical experiments with the proposed inverse mass matrix method are carried out to validate the effectiveness proposed technique when applied to vibration analysis of bars and beams. A perfect agreement is found between the exact inverse of the mass matrix and its direct inverse computed through biorthogonal basis functions

Cho, Sang S.

Felippa, C.A.

González Pérez, José Ángel

Kolman, Radek

Papadrakakis M., Fragiadakis M. (Coordinador)

Park, K.C.

English

Kolman, R.

idUS. Depósito de Investigación Universidad de Sevilla

COMPDYN 20176th ECCOMAS Thematic Conference onComputational Methods in Structural Dynamics and Earthquake EngineeringM. Papadrakakis, M. Fragiadakis (eds.)Rhodes Island, Greece, 15–17 June, 2017INVERSE MASS MATRIX VIA THE METHOD OF LOCALIZEDLAGRANGE MULTIPLIERSJose´ A. Gonza´lez1, R. Kolman2, S.S. Cho3, C. A. Felippa4, K.C. Park41Escuela Te´cnica Superior de Ingenierı´a, Universidad de SevillaCamino de los Descubrimientos s/n, Seville E-41092, Spaine-mail: japerez@us.es2Institute of Thermomechanics, The Czech Academy of SciencesDolejsˇkova 5, 182 00 Prague, Czech Republice-mail: kolman@it.cas.cz3Reactor Mechanical Engineering Division, Korea Atomic Energy Research Institute999-111 Daedeok-Daero, Yuseong-gu, Daejeon 305-353, Republic of Koreae-mail: sscho96@kaeri.re.kr4Department of Aerospace Engineering SciencesUniversity of Colorado at Boulder, CO 80309-429, USAe-mail: {carlos.felippa,kcpark}@colorado.eduKeywords: Inverse mass matrix, explicit time integration, localized Lagrange multipliers; par-titioned analysis.Abstract. An efficient method for generating the mass matrix inverse is presented, which can betailored to improve the accuracy of target frequency ranges and/or wave contents. The presentmethod bypasses the use of biorthogonal construction of a kernel inverse mass matrix that re-quires special procedures for boundary conditions and free edges or surfaces, and constructsthe free-free inverse mass matrix employing the standard FEM procedure. The various bound-ary conditions are realized by the the method of localized Lagrange multipliers. Numericalexperiments with the proposed inverse mass matrix method are carried out to validate the effec-tiveness proposed technique when applied to vibration analysis of bars and beams. A perfectagreement is found between the exact inverse of the mass matrix and its direct inverse computedthrough biorthogonal basis functions.1304Available online at www.eccomasproceedia.org Eccomas Proceedia COMPDYN (2017) 1304-1311 © 2017 The Authors. Published by Eccomas Proceedia.Peer-review under responsibility of the organizing committee of COMPDYN 2017. doi: 10.7712/120117.5493.17792Jose´ A. Gonza´lez, R. Kolman, S.S. Cho, C. A. Felippa and K.C. Park1 INTRODUCTIONRecently, Tkachuk and Bischoff [1] presented an innovative method for constructing an in-verse mass matrix based on Hamilton’s principle with (p,u)-variables. A key idea in their workis the discretization of the momentum-based kinetic energy in terms of a set of biorthogonalbasis functions corresponding to the displacement basis functions. The method proposed in [1]resorts to non-standard finite element construction with special tailoring for treating the Dirich-let boundary terms. Hence, considerable modifications of the existing finite element softwaremodules are required. When one contemplates element-by-element construction of the inverseof the global mass matrix, many degrees of freedom belong to the cross points and boundarynodes, which implies that one has to replace the existing mass matrix routines by speciallyconstructed mass matrix inverse.The present method, while employing the same momentum-velocity formulation as adoptedin [1] only as a formulation vehicle, utilizes standard FEM procedures, thus alleviates the specialtreatments associated the use of biorthogonal construction and mass matrix modification via apenalty procedure.2 BASIS FOR CONSTRUCTING INVERSE MASS MATRIX VIA THE LOCALIZEDLAGRANGE MULTIPLIERSThe prevailing practice for computing the mass matrixM in the finite element method is touse a bilinear form of the system kinetic energy T , viz.,T =Z⌦12⇢v(x, t) · v(x, t) d⌦ ⇡ 12 u˙(t)TM u˙(t) (1)where v(x, t) and u˙(t) are the continuum velocity and discrete approximate velocity, respec-tively. Since our objective is to generate the inverse mass matrix directly, instead of employingmatrix inversion, we employ a dual form of the kinetic energy:T =Z⌦12⇢p(x, t) · p(x, t) d⌦ ⇡ 12p(t)TCp(t) (2)where p(x, t) and p(t) are respectively the continuum and discrete approximate momentumvectors, and C is defined as the Reciprocal Mass Matrix (RMM).It will be shown that an inverse of the mass matrix can be obtained via:M 1 = A TCA 1 (3)whereA is the dual-base projection matrix, that can be transformed into a diagonal area/volume-like matrix. Hence, the computational simplicity of obtaining an inverse of the mass-matrix viathis new route.To this end, we invoke Hamilton’s principle for constrained elastodynamic problems. Thisprinciple states that the path followed by a dynamic system is the one which minimizes theaction integral of the Lagrangian; condition that can be expressed using the following three-field variational form: H(u,p, `) =Z t2t1  {T (u˙,p)  U(u, `) +W (u)} dt = 0 (4)1305Jose´ A. Gonza´lez, R. Kolman, S.S. Cho, C. A. Felippa and K.C. Parkwere  T is the virtual kinetic energy,  U the virtual elastic energy and  W the virtual workdone by the external loads, magnitudes that can be expressed: T =Z⌦ (12p · u˙) d⌦ (5) U =Z⌦ " :   d⌦+Z   {` · (u  ub)} d  (6) W =Z⌦ u · f d⌦ (7)with known displacement boundary conditions ub imposed at interface   by using a field oflocalized Lagrange multipliers ` and external body forces per unit volume f acting on ⌦.By making use of the momentum-velocity relation (p = ⇢u˙) in the following identity:12p · u˙ = p · u˙ 12⇢p · p (8)we can obtain an equivalent expression for the virtual kinetic energy per unit volume:  12p · u˙ =  u˙ · p+  p ·✓u˙  1⇢p◆(9)and finally performing an integration by parts of the second term of the last equation yields:Z t2t1 u˙ · p dt =  Z t2t1 u · p˙ dt (10)relation that can be used to eliminate the virtual velocity field from the formulation.Introducing previous identities, (9) and (10), into the principle of stationary action (4), weobtain the final the three-field variational form of the Hamilton’s principle for constrained elas-todynamics: H(u,p, `) =Z t2t1⇢Z⌦ p ·✓u˙  1⇢p◆d⌦ Z⌦( u · p˙+  " :  ) d⌦+Z⌦ u · f d⌦ Z  u · ` d  Z  ` · (u  ub) d  dt = 0 (11)expression that will be used to derive the equations of motion.Discretization in space of this mixed form is performed by using independent shape functionsfor displacements, momenta and Lagrangian multipliers. We carry out then a standard mixedFEM discretization with independent shape functions for the three fields:u = Nuu, p = Npp, ` = N   (12)and introducing these approximations in (11) one obtains the following set of semi-discreteequations:ATu˙ Cp = 0 Momentum equation (13)Ap˙+B  = r Equilibrium equation (14)BTu  Lbub = 0 Boundary (and interface) constraints (15) LTb   = 0 Newton’s 3rd law on the boundaries (16)1306Jose´ A. Gonza´lez, R. Kolman, S.S. Cho, C. A. Felippa and K.C. Parkwhere vector r = f   f int is the external-internal forces residual, the internal forces for linearproblems are simply given by f int = Ku and the matrix components are expressed:A =Z⌦NTu Np d⌦ (17)C =Z⌦1⇢NTp Np d⌦ (18)B =Z NTu N  d  (19)Lb =Z NT  Nub d  (20)where A is the global projection matrix, C the global reciprocal mass matrix, B the boundaryassembly operator and Lb is the Localized multipliers assembly matrix.Eliminating symbolically the momentum variable (p) from (13) and (14), one obtains theclassical equation of motion expressed in terms of displacements: AC 1AT u¨+B  = r (21)with the mass matrix approximated as:M = AC 1AT (22)and observe that there must exist an inverse mass matrix (denoted asM 1) given by:M 1 = A TCA 1 (23)assuming that the global projection matrix is invertible.Since the objective of the present work is to obtain inverse mass-matrices in efficient and ac-curate ways, one must insist on easily to invert diagonal or narrowly banded projection matrices.This can be accomplished by diagonalizing the projection A-matrix given in (17), leaving thetask of constructing the reciprocal C-matrix efficiently and accurately.3 CONSTRUCTION OF THE MASS INVERSEFor the calculation of the inverse mass-matrix, biorthogonal basis functions must be firstbuild for each particular finite element. However, basis functions are known for a limited num-ber of elements, such as line, triangular, quadrilateral and tetrahedral elements. Therefore, wepropose a different route for the evaluation of the mass-matrix inverse that bypasses this neces-sity through the following steps:1. Consider all the structures as free-floating.2. Approximate the element projection matrixAe by a lumping of the element mass-matrix.3. Compute the element reciprocal mass-matrix Ce from a direct inversion of the elementmass-matrix.4. Assemble the global projection and reciprocal matrices and compute the global mass-matrix inverse.1307Jose´ A. Gonza´lez, R. Kolman, S.S. Cho, C. A. Felippa and K.C. Park5. Impose boundary conditions using localized Lagrange multipliers[2]-[3].The first four steps of this process are described next and the last step, that is a little bit moreinvolved, will be analyzed in the next Section.First, we consider an element at a time by utilizing the element mass matrices existing instandard FEM software systems. Second, we use the element mass-matrix to approximate theelement projection matrix by diagonalization, viz.:[Ae]ii =1⇢eXj[Me]ij (24)assuming that the density is constant inside the element. Third, the elemental reciprocal mass-matrix is evaluated inverting numerically the element mass-matrix:Ce = ATeM 1e Ae (25)where (Ae,Ce,Me) are elemental matrices, and particularlyAe is a diagonalized matrix.Finally, we proceed with the assembly of the global reciprocal mass-matrix:C =neAe=1Ce (26)and the global projection matrix:A =neAe=1Ae (27)observing that as long as the elemental Ae is diagonal, so is the assembled matrix A and itsinverse. These steps permit to evaluate the inverse of the global mass-matrix through expression(23) in an efficient way.4 NUMERICAL RESULTSWe have assessed the performance of the present mass-matrix inverse for bar and beamstructural elements, in both free-free states and cantilevered conditions. In so doing, for thederivation of elemental reciprocal mass-matrix Ce, we have utilized a parametrized elementalmass-matrix expressed in the form:Me = (1   )Mce +  Mle (28)in which a parameter   2 [0, 1] is used to balance between the consistent element mass-matrixMce and the lumped element mass-matrixMle. Using   = {0, 0.5, 1} the element mass-matrixis then easily reduced respectively to the consistent mass-matrix (CMM), averaged mass-matrix(AMM) and lumped mass-matrix (LMM). Unless otherwise stated, in the following numericalexperiments we employ AMM as a reference and compute the mass-matrix inverse using RMMwith   = 0.5 to test the proposed methodology.4.1 Frequencies of a free-free and a fixed-free barIn this first example, we consider a long bar of length L = 1m with a rectangular sectionof width b = 0.05 m and thickness h = 0.01 m. The material of the bar is linear elastic,with Young’s modulus E = 69 GPa and a constant density ⇢ = 2700kg/m3. We discretizethe bar using a uniform mesh of 40 two-node linear bar elements and compute the vibration1308Jose´ A. Gonza´lez, R. Kolman, S.S. Cho, C. A. Felippa and K.C. Park5 10 15 20 25 30 35 4000.511.522.533.544.55 1055 10 15 20 25 30 35 4000.511.522.533.544.55 105Figure 1: Bar vibration problem. Frequencies for the free-free (left) and fixed-free (right) cases. Comparison ofexact frequencies and numerical frequencies obtained with: lumped mass-matrix (LMM), average mass-matrix(AMM) and the proposed inverse mass-matrix (RMM   = 0.5).modes of the structure with different approximations for the mass-matrix inverse. In particular,we investigate differences in the use of the lumped mass-matrix LMM, averaged mass-matrixAMM and the proposed mass-matrix inverse computed through RMM with   = 0.5.The results of the eigenvalue analysis are presented in Figure 1 for two different boundaryconditions, free-free case (left) and fixed-free case (right). We start with the free-free case, bycomparing the numerical frequencies with the exact solution obtained from continuum theory.The exact frequencies of this problem are !exacti =pE/L(⇡/L)(i   1) rad/sec for modenumber i = 1, . . . n and the first eigenfrequency is zero, corresponding to the horizontal rigidbody motion. We observe that the present method, utilizing the RMM, yields far better accuracythan the LMM and approximates very well to the AMM results for all frequencies.For the fixed-free case, we obtain very similar results. The only difference now is in the re-striction of the rigid body motion and the shift of exact frequencies to !exacti =pE/L(⇡/L)(i 1/2) rad/sec for mode number i = 1, . . . n. We corroborate that the AMM and its approximationthrough the RMM with   = 0.5 provides exactly the same result. This means that the presentmethod utilizing the inverse mass-matrix yields far better accuracy than the lumped mass-matrixwith similar computational effort.It can also be observed that LMM produces a maximum frequency around 20% lower thanAMM, so the critical time-step for explicit time-integration can be reduced by the same amountusing a diagonal mass-matrix. Even higher gains can be obtained with selective mass scalingtechniques, maintaining at the same time the advantages in terms of accuracy of using a non-diagonal mass-matrix.4.2 Frequencies of a free-free and a fixed-free beamIn this second case, we consider the same beam of the previous example but now it is dis-cretized using 20 Euler-Bernoulli beam elements with two DOFs per node, i.e., deflection androtation. We know that the shape functions for the Euler-Bernoulli beam require C1-continuityfor deflections and rotations, hence cubic Hermite polynomials are used as interpolation func-tions. This means that we would need to derive biorthogonal basis functions for the Hermitecubic shape functions in order to compute the reciprocal mass-matrix Ce from its definition(18). As previously described, we bypass this complex derivation obtaining the projection ma-trixAe from the diagonalization of the element mass-matrix.1309Jose´ A. Gonza´lez, R. Kolman, S.S. Cho, C. A. Felippa and K.C. Park5 10 15 20 25 30 35 40012345678910 1045 10 15 20 25 30 35 40012345678910 104Figure 2: Beam vibration problem. Frequencies for the free-free (left) and fixed-free (right) cases. Comparisonof exact frequencies and numerical frequencies obtained with: lumped mass-matrix (LMM), average mass-matrix(AMM) and the proposed inverse mass-matrix (RMM   = 0.5).In Figure 2 (left) we analyze the frequencies obtained for the free-free beam case. The exactfrequencies of this problem obtained from continuum theory are !exacti =pEI/⇢A(ci/L)2rad/sec with c = {0, 0, 4.73, 7.853, 10.996, 14.137, 17.279, . . . } for mode number i = 1, . . . nand two zero frequencies corresponding to the displacement and rotation rigid-body motions.We solve again the problem with exact LMM and AMM inverse mass-matrices and then usingthe proposed RMM with   = 0.5. Results illustrate, first, that the consistent AMM gives farbetter accuracy than diagonal LMM for this particular case, specially in the high-frequencyrange. Secondly, that the present RMM method utilizing the inverse mass-matrix yields thesame accuracy than the consistent AMM.The results for the fixed-free alternative of this problem are represented in Figure 2 (right).The exact frequencies of this case are !exacti =pEI/⇢A(ci/L)2 rad/sec with c = {1.875, . . . }.Exactly the same precision is observed in the free-free and fixed-free solutions. This convincesus to the conclusion that the proposed method for generating the inverse of mass matricesM 1b ,yields almost the same resonant frequencies as those obtained by the consistent mass matrices,as evidenced from Figure 2, thus confirming the validity of the present formulation.5 CONCLUSIONSA new methodology for direct generation of the inverse mass-matrix for discrete finite ele-ment equations of elastodynamics is presented, which is completely element-independent anddoes not require any special treatments of the elements adjacent to the Dirichlet boundaries. Inthe present method, the Dirichlet boundary conditions are handled by the method of localizedLagrange multipliers. This process is found to be computationally efficient, because matricesare sparse and only the factorization of a small matrix associated with the constrained DOFs isrequired.The present inverse mass-matrices have been tested for rod and beam elements, demonstrat-ing excellent accuracy in terms of frequencies and time evolution of the DOFs obtained withexplicit time-integration methods.REFERENCES[1] Tkachuk A, Bischoff, M. Direct and sparse construction of consistent inverse mass ma-trices: general variational formulation and application to selective mass scaling. Interna-1310Jose´ A. Gonza´lez, R. Kolman, S.S. Cho, C. A. Felippa and K.C. Parktional Journal for Numerical Methods in Engineering 2015; 101:435-469.[2] Park KC, Felippa CA. A variational framework for solution method developments in struc-tural mechanics. Journal of Applied Mechanics 1998; 65(1):242–249.[3] Gonzalez JA, Park KC, Felippa CA, Abascal R. A formulation based on localized La-grange multipliers for BEM-FEM coupling in contact problems Computer Methods inApplied Mechanics and Engineering 2007, Volume: 197 Issue: 6-8 Pages: 623-640.1311

Inverse mass matrix via the method of localized lagrange multipliers

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Inverse mass matrix via the method of localized lagrange multipliers

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idUS. Depósito de Investigación Universidad de Sevilla