GENERALIZED FINITE ELEMENT METHOD FOR VIBRATION
Blucher Mechanical Engineering ProceedingsMay 2014, vol. 1 , num. ALIZED FINITE ELEMENT METHOD FOR VIBRATION ANALYSIS OFBARSM. Arndt1, A. J. Torii1,2, R. D. Machado1, A. Scremin11Numerical Methods in Engineering Graduated Program, Federal University of Paraná(arndt@ufpr.br)2Universidade Positivo (ajtorii@up.com.br)Abstract. The vibration analysis is an important stage in the design of mechanical systemsand structures subject to dynamic loads like wind and earthquake. The Finite Element Method(FEM) is commonly used in vibration analysis and its approximated solution can be improvedusing two refinement techniques: h and p-versions. The h-version of FEM gives good resultsfor the lowest frequencies but demands great computational cost to work up the accuracy forthe higher frequencies. The accuracy of the FEM can be improved applying the polynomial prefinement. Some enriched methods based on the FEM have been developed in last 20 yearsseeking to increase the accuracy of the solutions for the higher frequencies with lowercomputational cost. The purpose of this paper is to present a formulation of the GeneralizedFinite Element Method (GFEM) to free and transient vibration analysis of bars. TheGeneralized Finite Element Method is developed by enriching the standard Finite ElementMethod space, whose basis performs a partition of unity, with knowledge about thedifferential equation being solved. The proposed method combines the best features of GFEMand enriched methods: (a) efficiency, (b) hierarchical refinements and (c) the introduction ofboundary conditions following the standard finite element procedure. In addition theenrichment functions are easily obtained. The main features of the GFEM are discussed andthe partition of unity functions and the local approximation spaces are presented. Theefficiency and convergence of the proposed method for vibration analysis of bars are checked.The results obtained by the GFEM are compared with those obtained by the analyticalsolution, some enriched methods and the h and p versions of the Finite Element Method.Keywords: Generalized finite element method, Dynamic analysis, Vibration analysis,Partition of unity.1. INTRODUCTIONThe dynamic analysis is an important stage in the design of mechanical systems andstructures subject to dynamic loads like wind and earthquake. This analysis allows obtaining
the dynamic characteristics and the time-varying response of these structures. The dynamicanalysis can be used also to identify cracks in structures.The Finite Element Method (FEM) is commonly used in vibration analysis and itsapproximated solution can be improved using two refinement techniques: h and p-versions.The h-version consists of the refinement of element mesh; the p-version may be understood asthe increase in the number of shape functions in the element domain without any change inthe mesh. The conventional p-version of FEM consists of increasing the polynomial degree inthe solution. The h-version of FEM gives good results for the lowest frequencies but demandsgreat computational cost to work up the accuracy for the higher frequencies. The accuracy ofthe FEM can be improved applying the polynomial p refinement.Some enriched methods based on the FEM have been developed in last 20 yearsseeking to increase the accuracy of the solutions for the higher frequencies with lowercomputational cost. Engels [8] and Ganesan & Engels [9] present the Assumed Mode Method(AMM) which is obtained adding to the FEM shape functions set, some interface restrainedassumed modes. The Composite Element Method (CEM) [23, 24] is obtained by enrichmentof the conventional FEM local solution space with non-polynomial functions obtained fromanalytical solutions of simple vibration problems. A modified CEM applied to analysis ofbeams is proposed by [12]. The use of products between polynomials and Fourier seriesinstead of polynomials alone in the element shape functions is recommended by [11]. Theydevelop the Fourier p-element applied to the vibration analysis of bars, beams and plates.These three methods have the same characteristics and they will be called enriched methodsin this work. The main features of the enriched methods are: (a) the introduction of boundaryconditions follows the standard finite element procedure; (b) hierarchical p refinements areeasily implemented and (c) they are more accurate than conventional h version of FEM.At the same time, the Generalized Finite Element Method (GFEM) was independentlyproposed by Babuska and colleagues [2, 6, 13] and by Duarte & Oden [7, 14] under thefollowing names: Special Finite Element Method, Generalized Finite Element Method, FiniteElement Partition of Unity Method, hp Clouds and Cloud-Based hp Finite Element Method.Actually, several meshless methods recently proposed may be considered special cases of thismethod. Strouboulis and co-workers [19] define otherwise the subclass of methods developedfrom the Partition of Unity Method including hp Cloud Method [7, 14], the eXtended FiniteElement Method (XFEM) [20, 21], the Generalized Finite Element Method (GFEM) [17, 18],the Method of Finite Spheres [4], and the Particle-Partition of Unity Method [16]. TheGFEM, which was conceived on the basis of the Partition of Unity Method, allows theinclusion of a priori knowledge about the fundamental solution of the governing differentialequation. This approach ensures accurate local and global approximations. In structuraldynamics, the Partition of Unity Method was applied by [5] and [10] to numerical vibrationanalysis of plates and by [1] to free vibration analysis of bars and trusses. Among the mainchallenges in developing the GFEM to a specific problem are: (a) choosing the appropriatespace of functions to be used as local approximation and (b) the imposition of essentialboundary conditions, since the degrees of freedom used in GFEM generally do not correspondto the nodal ones. In most cases the imposition of boundary conditions is achieved by the
degeneration of the approximation space or applying penalty methods or Lagrangemultipliers.The purpose of this work is to present a formulation of the GFEM to free and transientvibration analysis of bars. The proposed method combines the best features of GFEM andenriched methods: (a) efficiency, (b) hierarchical refinements and (c) the introduction ofboundary conditions following the standard finite element procedure. In addition theenrichment functions are easily obtained.2. GENERALIZED FINITE ELEMENT METHODThe Generalized Finite Element Method (GFEM) is a Galerkin method whose maingoal is the construction of a finite dimensional subspace of approximating functions usinglocal knowledge about the solution that ensures accurate local and global results. The GFEMlocal enrichment in the approximation subspace is incorporated by the partition of unityapproach.2.1. Partition of unityLet u Η 1 (Ω) be the function to be approximated and {Ω i } be an open cover ofdomain Ω (Fig. 1) satisfying an overlap condition: M S Ν so that x Ωcard { i x Ω i } M S .(1)Figure 1. Open cover {Ω i } of domain Ω (see [6]).A Lipschitz partition of unity subordinate to the cover {Ω i } is the set of functions {η i }satisfying the conditions:supp (η i ) { x Ω η i ( x) 0} [Ω i ] , i , ηii 1 on Ω ,(2)(3)
where supp (η i ) denotes the support of definition of the function ηi and [Ω i ] is the closureof the patch Ω i .The partition of unity set {η i } allows obtaining an enriched set of approximatingfunctions. Let S i Η 1 (Ω i Ω ) be a set of functions that locally well represents u:{ }S i s ijmj 1.(4)Then the enriched set is formed by multiplying each partition of unity function η i by thecorresponding s ij , i.e., S : η i S i η i s ij s ij S i H 1 (Ω ) .i i (5)Accordingly, the function u can be approximated by the enriched set as:u h (x ) i η s (x ) aji iij.(6)jsi S iwhere a ij are the degrees of freedom.In the proposed GFEM, the cover {Ω i } corresponds to the finite element mesh andeach patch Ω i corresponds to the sub domain of Ω formed by the union of elements thatcontain the node xi (Fig. 2).Figure 2. Patchs and partition of unity set for one-dimensional GFEM finite elementmesh2.2. Generalized C0 elements for free vibration analysisThe generalized C0 elements use the classical linear FEM shape functions as thepartition of unity, i.e.: 1 ηi 1 x xix i x i 1x xix i 1 x iif x ( x i 1 , x i )if x (x i , x i 1 ),(7)
in the patch Ω i (x i 1 , x i 1 ) .The proposed local approximation space in the patch Ω i (x i 1 , x i 1 ) takes the form:S i span{ 1 γ 1 jγ 2jϕ1 jϕ2 jK}, j 1,2, K , nl ,0 if x ( x i 1 , x i ) , sin β Rj (x x i ) if x (x i , x i 1 )γ1j [][] sin β Lj ( x x i ) if x ( x i 1 , x i ),0 if x ( x i , x i 1 ) γ 2j ϕ1 j cos[β 0 if x ( x i 1 , x i ),x ( x i , x i 1 )Rj ( x x i ) 1 if][] cos β Lj (x x i ) 1 if x (x i 1 , x i ),0 if x (x i , x i 1 ) ϕ2 j β Rj β Lj ρRERρLEL(8)(9)(10)(11)(12)µj,(13)µj,(14)where ER and ρR are the Young modulus and specific mass on sub domain ( x i , x i 1 ) , EL andρL are the Young modulus and specific mass on sub domain ( x i 1 , x i ) , and µ j is a frequencyrelated to the enrichment level j.The enriched set S, so proposed, vanishes at element nodes, which allows theimposition of boundary conditions in the same fashion of the finite element procedure.This C0 element can be applied in the free vibration analysis of shafts, bars andtrusses. Different frequencies µ j produce different enriched elements. The increase in thenumber of elements in the mesh with only one level of enrichment (j 1) and a fixedparameter β 1 β R1 β L1 , for example β 1 π , produces an h refinement. Otherwise theincrease in the number of levels of enrichment, with a different parameter β j β Rj β Ljeach, for example, β j jπ , produces a hierarchical p refinement. Another refinementpossible in the proposed GFEM is the adaptive refinement, which is presented below.The adaptive GFEM is an iterative approach presented first by [1] whose main goal isto increase the accuracy of the frequency (eigenvalue) related to a chosen vibration mode withorder denoted by “target order”. The flowchart with blocks A to H presented in Figure 3represents the adaptive process.
Choice of the targetvibration modetarget chosen mode order(A)(B)Solution by FEM (GFEM nl 0 )mesh ndof targetObtain ωtarget,FEM(C)i 1ωtarget,i ωtarget,FEM(D)(E)i i 1j 1Solution by GFEMnl j and µj ωtarget,i-1Obtain ωtarget,GFEMNO(F)ωtarget,i ωtarget,GFEMConvergence test ωtarget,i - ωtarget,i-1 tolerance(G)YES(H)EndShow resultsFigure 3. Flowchart of the adaptive GFEM.In this flowchart, ωtarget corresponds to the frequency related to the target mode. Thefirst step of the adaptive GFEM process (blocks A to C) consists in obtaining anapproximation of the target frequency by the standard FEM (GFEM with nl 0) with a coarsemesh. The finite element mesh used in the analysis has to be as coarse as is necessary tocapture a first approximation of the target frequency. The subsequent steps (blocks D to G)consist in applying the GFEM with only one enrichment level (nl 1) to the same finiteelement mesh assuming the frequency µ j (j 1, blocks D and E) of the enrichment functions(Eqs. 9-14) as the target frequency obtained in the last step. Thus, no mesh refinement isnecessary along the iterative process.Both the standard FEM and the adaptive GFEM allow as many frequencies as the totalnumber of degrees of freedom to be obtained. However, in the latter, only the precision of thetarget frequency is effectively improved by the iterative process. The other frequencies
present errors similar to those obtained by the standard FEM with the same mesh. In order toimprove the precision of another frequency, it is necessary to perform a new adaptive GFEManalysis, taking this new one as the target frequency.2.3. Generalized C0 elements for time response analysisThe C0 elements described in the previous section can also be used for time responseanalysis. When damping is not considered, time response analysis can be made by solving thefollowing system of equations [3]:&& F ,Ku Mu(15)&& is the vector of accelerations, K is the stiffnesswhere u is the vector of displacements, umatrix, M is the mass matrix and F is the vector of applied forces.The system of equations from Eq. (15) can be solved by some time integration schemefound in the literature. In this paper, Modal Superposition is used to obtain a set ofindependent equations from Eq. (15) and then each equation is solved separately byNewmark’s Method [3]. The advantage of using this approach is that one is able to choosewhich vibration modes to include in the Modal Superposition procedure. This can be aninteresting tool to investigate the accuracy of the individual approximated vibration modes ofthe structure.The error between a reference solution and an approximate solution in a fixed positioninside the structure can be approximated bynte t u (i ) u h(i )i 1(16),where nt is the number of time steps used, t is the time step used to obtain the approximatesolution, u(i) is the analytical solution at time step (i) and uh(i) is the approximate solution attime step (i) in a fixed position inside the structure. Error evaluation according to Eq. (16) isillustrated in Fig. 4. The error in a given time interval is approximated by the product between t and u(i).uu(i) u(i)uh(i) tt(i-1) t(i)tFigure 4: Error evaluation according to Eq. (16).
For the time response analysis, higher order polynomial finite elements were obtainedusing Lobatto’s polynomials as shape functions, as described in details by [22]. These polynomials are different from Lagrange’s polynomials that are commonly used in p-version ofFEM. However, both families of polynomials form a basis for the subspace Pn of polynomialsup to order n when n 1 shape functions are used. Consequently, the approximations given byboth schemes are the same.3. APPLICATIONNumerical solutions for a uniform fixed-free bar (Fig. 5) are given below to check theefficiency of the proposed formulation of GFEM.Figure 5. Uniform fixed-free bar.The number of degrees of freedom (ndof) considered in each analysis is the totalnumber of effective degrees of freedom after introduction of boundary conditions.3.1. Free vibration analysisThe free axial vibration of a fixed-free bar (Fig. 5 with F(t) 0) with length L,elasticity modulus E, mass density ρ and uniform cross section area A, has exact naturalfrequencies ( ω r ) given by:ωr (2r 1)πE2Lρ,r 1,2, K(17)In order to compare the exact solution with the approximated ones, in this example anon-dimensional eigenvalue χ r given by:χr ρL2ω r 2E(18)will be used.To check the efficiency of the p refinement of GFEM the results were compared tothose obtained by AMM, by CEM, by Fourier, by linear and cubic h-versions of FEM and byconventional p-version of FEM. The shape functions of the conventional p-version of FEMare Lagrangian polynomials. The p-version of GFEM consists in a progressive increase oflevels of enrichment with parameter β j jπ . In the analyses by p-version of FEM and by all
the enriched methods, the bar was described geometrically by one element and the successiverefinements were obtained increasing the number of shape functions. Figures 6 to 8 presentthe behavior of relative error for the six earliest eigenvalues in logarithmic scale.Analyzing the results obtained for the fixed-free bar, one observes that the resultsobtained by GFEM and all enriched methods show convergence rates greater than the linear hrefinement of the FEM. The cubic h-version of FEM shows better results than CEM / Fourierand AMM just for three earliest eigenvalues and it shows worst results than GFEM for alleigenvalues. The conventional hierarchical p refinement of the FEM has greater accuracy thanCEM / Fourier and AMM. Otherwise, the GFEM showed worst precision than the p versionof the FEM only for the first eigenvalue.Figure 6. Relative error (%) for the 1st and 2nd bar eigenvalues.Figure 7. Relative error (%) for the 3rd and 4th bar eigenvalues.
Figure 8. Relative error (%) for the 5th and 6th bar eigenvalues.Four different adaptive GFEM analyses are performed in order to obtain the first fourfrequencies. In order to capture an initial approximation of the target vibration frequency, forthe first frequency, the finite element mesh must have at least one bar element (one effectivedegree of freedom), for the second frequency, it must have at least two bar elements (twoeffective degrees of freedom), and so on.Table 1 presents the relative errors obtained by the numerical methods. The linearFEM solution is obtained with 100 elements, that is, 100 effective degrees of freedom (dof).The cubic FEM solution is obtained with 20 elements, that is, 60 effective degrees offreedom. The CEM solution is obtained with one element and 15 enrichment functionscorresponding to one nodal degree of freedom and 15 field degrees of freedom resulting in 16effective degrees of freedom. The conventional hierarchical p FEM solution is obtained with a17-node element corresponding to 16 effective degrees of freedom. The analyses by theadaptive GFEM have no more than 20 degrees of freedom in each iteration. For example, thefourth frequency is obtained taking 4 degrees of freedom in the first iteration and 20 degreesof freedom in the two subsequent ones.Table 1. Results to free vibration of uniform fixed-free barEigenvalue1234linear hFEM(100e)ndof 100error (%)2,056 e-31,851 e-25,141 e-21,008 e-1cubic hFEM(20e)ndof 60error (%)8,564 e-101,694 e-73,619 e-62,711 e-5CEMp FEM(1e 17n) (1e 15c)ndof 16 ndof 16error (%)3,780 e-132,560 e-131,382 e-131,602 e-11error (%)8,936 e-48,188 e-32,299 e-24,579 e-2Adaptive GFEM(after 3 iterations)error (%)3,780 e-132,560 e-132,304 e-145,289 e-13ndof in iterations1x 1 dof 2x 5 dof1x 2 dof 2x 10 dof1x 3 dof 2x 15 dof1x 4 dof 2x 20 dofThe adaptive process converges rapidly, requiring three iterations in order to achieveeach target frequency with precision of the 10-13 order. For the uniform fixed-free bar, onenotes that the adaptive GFEM reaches greater precision than the h versions of FEM and theCEM. The p-version of FEM is as precise as the adaptive GFEM only for the first twoeigenvalues. After this, the precision of the adaptive GFEM prevails among the others.
3.2. Time response analysisTime response analysis is made for the structure from Figure 5. This is a fixed-freeuniform bar subject to a time dependent force F(t) at the right end. Only axial displacementsare considered here.The properties of the material for this example were chosen to give a wave velocityequal to c E / ρ 1 m/s, in order to simplify the analysis. Besides, the length of the bar istaken equal to 1 meter. The bar is at rest and then both initial displacements and velocities arenull.The natural vibration frequencies for this example can be found using Eq. (17). For thetime response we assume that the time dependent force is given byF (t ) f sin(ωt ) ,(19)where f is the force magnitude and ω is the excitation frequency.The analytical solution for the time response in this case requires the application oftechniques described by [15]. The displacements inside the bar are given by u ( x, t ) fx sin(ωt ) f {sin( k n x)[C n sin( k n ct ) Bn (t )]},(20)n 1whereAnω,knc(21)Anω 2 sin(ωt ) Anω 3 sin(k n ct ) 3 3,c 2 k n2 ω 2c k n ck nω 2(22)2[k n cos( k n ) sin( k n )],k n2(23)1 kn π n .2 (24)Cn Bn (t ) An This problem is solved numerically for ω 20 rad/s and f 1 N/m2. The analysis ismade using the Modal Superposition Method for a time interval of 20 s and the resultingequations are solved using the Newmark method (with α 0,5
boundary conditions following the standard finite element procedure. In addition the enrichment functions are easily obtained. 2. GENERALIZED FINITE ELEMENT METHOD The Generalized Finite Element Method (GFEM) is a Galerkin method whose main goal is the construction of a fin
The Generalized Finite Element Method (GFEM) presented in this paper combines and extends the best features of the finite element method with the help of meshless formulations based on the Partition of Unity Method. Although an input finite element mesh is used by the pro- . Probl
Finite Element Method Partial Differential Equations arise in the mathematical modelling of many engineering problems Analytical solution or exact solution is very complicated Alternative: Numerical Solution – Finite element method, finite difference method, finite volume method, boundary element method, discrete element method, etc. 9
Finite element analysis DNV GL AS 1.7 Finite element types All calculation methods described in this class guideline are based on linear finite element analysis of three dimensional structural models. The general types of finite elements to be used in the finite element analysis are given in Table 2. Table 2 Types of finite element Type of .
1 Overview of Finite Element Method 3 1.1 Basic Concept 3 1.2 Historical Background 3 1.3 General Applicability of the Method 7 1.4 Engineering Applications of the Finite Element Method 10 1.5 General Description of the Finite Element Method 10 1.6 Comparison of Finite Element Method with Other Methods of Analysis
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The Finite Element Method: Linear Static and Dynamic Finite Element Analysis by T. J. R. Hughes, Dover Publications, 2000 The Finite Element Method Vol. 2 Solid Mechanics by O.C. Zienkiewicz and R.L. Taylor, Oxford : Butterworth Heinemann, 2000 Institute of Structural Engineering Method of Finite Elements II 2
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