customizing high performance elements by fourier methods

TRENDS IN COMPUTATIONAL STRUCTURAL MECHANICS W.A. Wall, K.-U. Bletzinger and K. Schweizerhof (Eds.) c
CIMNE, Barcelona, Spain 2001

CUSTOMIZING HIGH PERFORMANCE ELEMENTS BY FOURIER METHODS

Carlos A. Felippa Department of Aerospace Engineering Sciences and Center for Aerospace Structures University of Colorado, Boulder, CO 80309-0429, USA e-mail: [email protected] web page: http://caswww.colorado.edu

Abstract. This paper illustrates, through a worked out example, the application of finite element templates to construct high performance bending elements for vibration and buckling problems. The example focuses on the improvement of the mass and geometric stiffness matrices of plane beams. Similar methods can be used for Kirchhoff plate bending elements, but this material is omitted because of space constraints. The process interweaves classical tools: Fourier analysis and orthogonal polynomials, with newer ones: finite element templates and computer algebra systems. Templates are parametrized algebraic forms that uniquely characterize an element population by a “genetic signature” defined by the set of free parameters. Specific elements are obtained by assigning numeric values to the parameters. This freedom of choice can be used to design custom elements. For the beam example, Legendre polynomials are used to construct templates for the material stiffness, geometric stiffness and mass matrices. Fourier analysis carried out through symbolic computation searches for template signatures of mass and stiffness that deliver matrices with desirable attributes for specific target situations. Three objectives are noted: high accuracy for vibration and buckling analysis, wide separation of acoustic and optical branches, and low sensitivity to mesh distortion and boundary conditions. This paper examines in some detail only the first objective. Key words: Finite element methods, templates, signatures, plane beam, mass matrix, geometric stiffness matrix, Legendre polynomials, orthogonal bases, Fourier analysis, spectral analysis, plane wave propagation, dispersion.

1 INTRODUCTION This article looks at mass, material and geometric matrices of finite elements from a purely algebraic standpoint, disconnected from the standard variationalbased derivations. The following questions are posed. Which full or lumped mass matrices give good performance in dynamic analysis if only minimal constraints are placed on its entries? Is it the consistent mass? If not, how much can be

1

Carlos A. Felippa

gained from it? Can lumped mass matrices be competitive? A related question arises in buckling and stability analysis: can one do better than merely using the consistent geometric stiffness? Conventional error analysis in FEM cannot answer these questions. It only gives worst-case asymptotic convergence rates in some norm. This is like watchmaking with a sledgehammer. A sharper toolset is needed. Basic aspects of the methodology will be illustrated on a simple but nontrivial element: a plane Bernoulli-Euler beam. The exposition intends to show a balanced synthesis of the old and the new: • Old: Orthogonal bases and Fourier analysis • New: Templates and symbolic computations by computer algebra systems Orthogonal bases and Fourier analysis are of course classical tools. Templates represent a frontier FEM subject still in the development stage. As regards the last tool, Mathematica was used to carry out the error-prone parametrized spectral computations required by templates, in minutes instead of days or months. Thus, although computer algebra systems are not exactly brand new, their increasing power on inexpensive desktops and laptops are gradually making possible the detailed analysis of 1D and 2D templates, with 3D elements likely to follow in the future. A plane beam is used because the element is well understood, fully displacement based, and exactly integrated. But it is not so trivial as, say, a bar element. Hence the basic techniques can be explained in a space-limited paper while giving some of the flavor of the far more complex algebraic manipulations that arise in plate and shell templates. 2 THE PLANE BEAM ELEMENT The element under study is the two-node, prismatic, homogeneous, thin plane beam of length
illustrated in Figure 1(a). The axes {x, y, z} are placed as indicated there. The isoparametric coordinate ξ is −1 ≤ ξ = 2x/
≤ 1 as shown in Figure 1(b). Derivatives with respect to x are denoted by a prime: (.) ≡ d(.)/dx, whereas those with respect to ξ are denoted by bullet superscripts: (.)• ≡ d(.)/dξ. The cross section properties: area A and second moment of inertia I ≡ Iyy , as well as the material properties: elastic modulus E and density ρ, are assumed to be constant along x. The kinematics of the Bernoulli-Euler beam model is fully determined by the transverse displacement field w(x). The cross-section rotation is θ(x) = w (x) and the curvature is κ(x) = w (x). In linear dynamics, the internal energy   is Π = Um − W = 12 0
EI κ2 dx − 0
q w dx whereas the kinetic energy is  T = 12 0
ρ A w2 dx. For buckling analysis the energy functional is Π = Um + Ug ,  with Ug = 12 0
P (w )2 dx, in which P is the axial force. The four degrees of freedom (DOFs) are the transverse end displacements {w1 , w2 } and the length-scaled end rotations {r1 =
θ1 , r2 =
θ2 }. Use of {r1 , r2 } renders all equations dimensionally homogeneous. Although in actual FEM analysis the rotational freedoms are {θ1 , θ2 },
-scaling is possible here because Fourier analysis is carried out on the infinite periodic-lattice structure depicted in Figure 1(c).

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Carlos A. Felippa

z,w

(a) w1

z,w (b)

EI = constant

x

w2

θ1 y



ξ = −1



ξ =1

x

θ2

(c)

−∞

∞ 











Figure 1. (a,b): Beam element under study; (c) periodic lattice for Fourier analysis.

3 LEGENDRE POLYNOMIAL BASIS The beam displacement basis is formed by the first four Legendre polynomials: L = [ L1 L2 L3 L4 ] ,

L = [ L1 L2 L3 L4 ] ,

L = [ L1 L2 L3 L4 ] ,

(1)

L4 (ξ) = 12 (5ξ 3 − 3ξ).

(2)

in which L1 (ξ) = 1,

L2 (ξ) = ξ,

L3 (ξ) = 12 (3ξ 2 − 1),

This is a FEM oriented nomenclature; L1 through L4 are called P0 through P3 in mathematical handbooks; e.g. Chapter 22 of Abramowitz and Stegun.1 They are normalized as per the usual conventions: Li (1) = 1, Li (−1) = (−1)i−1 . The polynomials (2) and their first two derivatives are plotted in Figure 2. The resultant displacement interpolation is w = L1 q1 + L2 q2 + L3 q3 + L4 q4 = L q.

(3)

The amplitudes qi are generalized coordinates, which are related to the element DOFs by the transformations 









w1 1 −1 1 −1 q1 r   0 2 −6 12   q   1    2     = Gq, =  w2   1 1 1 1   q3  r2 q4 0 2 6 12 Here G = H−1 . This follows the presented by Bergan and Nyg˚ ard2 product integrals over the element 

0 0 Q= L LT dx = 1/5 −
/2 0  0 0
/2 48  0 0   T S= L (L ) dx = 3 
0 0 −
/2 0 0
/2

1 0 1/3 0 0 0

0 
 0











q1 30 5 30 −5 w1 q    1   2  −36 −3 36 −3   r1   =    = Hu.  q3  60  0 −5 0 −5   w2  q4 r2 6 3 −6 3 (4) standard notation of the Free Formulation as and Bergan and Felippa.3 The function innerlength yield the covariance matrices 



0 00
/2 0 1 4 0    , R = L (L )T dx =  0 
0 0 −
/2 1/7 01  0 0 0 0   . 3 0  0 25

0 0 3 0



0 1   0 6

(5)

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Carlos A. Felippa

1

L3 -1

L4

0.5

L1

6

L2

4

L2=dL2 /dξ -0.5

0.5 -0.5

15

L4=dL 4 /dξ L3 =d 2L 3 /dξ 2

L3 =dL3 /dξ

2

-1

1

10 5

-0.5

0.5 -5

-1

L1 =dL1 /dξ

-0.5

0.5

1

-2

-10 -15

-1

L4 =d 2L4 /dξ 2

1

L1 =d 2L 1 /dξ 2 and L2 =d 2L 2 /dξ 2

Figure 2. The Legendre polynomial functions and their derivatives.

As can be observed Li and their second derivatives are orthogonal over the beam length, whereas their first derivatives are not. The representation (3) is more useful than the well known Hermite basis for developing templates for beam elements, because the Legendre functions are hierarchical in nature and have clear physical meaning. Polynomials L1 and L2 represent translational and rotational rigid motions, respectively; L3 is a constantcurvature symmetric bending mode; finally, L4 is a linear-curvature antisymmetric bending mode. For Timoshenko beam models, L4 is a constant shear mode. 4 TEMPLATES A finite element template, or simply template, is an algebraic form that represents element-level equations, and which fulfills the following four conditions: (C) consistency, (S) stability (correct rank), (I) observer invariance and (P) parametrization. These are discussed at length in other papers, for example Felippa, Haugen and Militello4 and Felippa.5 Conditions (C) and (S) are imposed to ensure convergence as the mesh size is reduced by enforcing a priori satisfaction of the Individual Element Test of Bergan and Hanssen.6 Condition (P) means that the template contains free parameters; this permits performance optimization as well as tuning elements, or combinations of elements, to fulfill specific needs. Setting the free parameters to numeric values yields specific element instances. The set of free parameters is called the template signature, a term introduced in Felippa and Militello7 and Felippa.8 Borrowing terminology from biogenetics, the signature may be viewed as an “element DNA” that uniquely characterizes it as an individual entity. Elements derived by different techniques that share the same signature are called clones. 4.1 Full Mass and Material Stiffness Templates Templates for the full mass and material stiffness are obtained by scaling the entries of the covariance matrices Q and S as follows: 





1 0 0 0 0 0  0 µ /3 48  0 0    0 0 µ β 1  , S = S(β) = 3  Q = Q(µ1 , µ2 , µ3 ) =
 0 0 µ2 /5 0 
0 0 0 0 0 µ3 /7 0 0



0 0 0 0   , 3 0  0 25β (6) under the constraints µi ≥ 0, i = 1, 2, 3, and β > 0. The recipe for this generalization is: traverse down the main diagonal of the covariance matrix, and scale the diagonal coefficients that follow the first nonzero entry.

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Carlos A. Felippa

Notice that the first nonzero diagonal entry must not be parametrized. This is a consequence of the (C) requirement, which can be viewed as a conservation condition. The physical interpretation is: the element rigid-translational mass must be preserved, and the stiffness response to a constant curvature state must be exact. The (S) condition restricts the three mass parameters to be nonnegative and the stiffness parameter to be positive nonzero. The parametrized full-mass and material stiffness matrices are µ

β

= ρA HT Q H, M


= EI HT S H, K m


µ

β

(7)

distinguishes matrices in terms of the homogenized freedoms where (.) {w1 , r1 , w2 , r2 }. These have the following spectral representations as sum of rankone matrices:  

µ =ρA
  M 



36 6 1 6 1 6 144 36 −6 −1  0 0 µ2  1 0  + 0 720  0 0 −1    

 

β K m

EI = 3


   

0

0   0

0

36 6 36 −6 0 0 0 0

0 0 1 0 0 0 −1 0







−6 144 12 −144 12  µ −1  12 1 −12 1   1  +   −6  1200  −144 −12 144 −12  1 12 1 −12 1     0 4 2 4 2  µ3  −1  1 −2 1  2   +    0  2800  4 −2 4 −2   1 2 1 −2 1 (8)     0 12 6 −12 6   −1  6 3 −6 3     +β   (9)  −12 −6 0 12 −6   1 6 3 −6 3

Setting µ1 = µ2 = µ3 = β = 1 produces the well known consistent mass and material stiffness matrix, respectively, associated with the Hermite basis. 4.2 Lumped Mass Template To include block-diagonal-lumped mass matrices in this study we introduce directly the two-parameter template

ν

= M(ν

M 1 , ν2 ) = L

 1 2 ν  2 ρA
  0

0

ν2 ν1 0 0

0 0 1 2

−ν2



0 0   , −ν2  ν1

(10)

where ν1 ≥ 0 and 12 ν1 ≥ ν22 to enforce nonnegativity. Diagonal lumped matrices



are obtained if ν2 = 0. The only common instance of M(µ 1 , µ2 , µ3 ) and ML (ν1 , ν2 )

(1/36, 1/12). is M(5/3, 5, 35) ≡ M L 4.3 Geometric Stiffness Template The covariance matrix R, given by the second of (5), which appears in the geometric stiffness is nondiagonal because of the coupling of L2 and L4 . To ˜ 4 (ξ) = L4 −L2 = produce a diagonal covariance the last polynomial is changed to L 2 5ξ(1−ξ )/2. This is not a Legendre polynomial because it vanishes at ξ = ±1, but

5

Carlos A. Felippa

j

i

k





Figure 3. Three-node repeating patch for Fourier analysis.

˜ = (4/
) diag [ 0 1 3 5 ], which its derivative is. The new covariance matrix is R parametrizes to 

˜ 1 , γ2 ) = 4 ˜ γ = R(γ R


0

0   0

0

0 1 0 0

0 0 3γ1 0



0 0    0  5γ2

(11)

˜ 4 q˜4 the freedom in which γ1 ≥ 0 and γ2 ≥ 0. Using w = L1 q˜1 + L2 q˜2 + L3 q˜3 + L transformation matrices are reworked: 









w1 1 −1 1 0 q˜1 r   0 2 −6 −10   q˜   1    2 ˜     = Gq, =  w2  1 1 1 0   q˜3  r2 q˜4 0 2 6 −10











q˜1 30 5 30 −5 w1  q˜    1   2  −30 0 30 0   r1  ˜  =    = Hu.  q˜3  60  0 −5 0 −5   w2  q˜4 r2 −6 −3 6 −3 (12) γ T ˜γ ˜

˜ The spectral form of the geometric stiffness template: Kg = P H R H is 





0 −1 0 0 0 γ1  0 0 0 1  0 +  0 1 0  12  0 0 0 0 0 0 −1







0 0 12 6 −12 6  γ2  0 −1  6 3 −6 3     +   0 0  60  −12 −6 12 −6   0 1 6 3 −6 3 (13) This hierarchical form has an interesting historical interpretation. The first term is the rank-one geometric stiffness of the two-node bar element, which was used for framework buckling analysis in the early 1960; see for example Argyris, Kelsey and Kamel.9 Later Martin10 formulated the consistent geometric stiffness, which may be obtained by setting γ1 = γ2 = 1 in (13), and reported a significant increase in accuracy for the same FEM discretization. An intermediate rank-two form, not available in the literature, is obtained by setting γ1 = 1 and γ2 = 0. This illustrates the power of templates to subsume all possible elements in one expression. 1  γ P   0

Kg =  
  −1 0

5 MASS-STIFFNESS FOURIER ANALYSIS To assess the performance of mass-stiffness combinations for dynamics we perform Fourier analysis of the infinite beam lattice of Figure 1(c), taking the typical three-node {i, j, k} patch shown in Figure 3. The operational techniques introduced by Park and Flaggs11,12 for element analysis are followed. The unforced semidiscrete dynamical equations of the patch are µ ¨ P + KβP uP = 0, MP u

6

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Carlos A. Felippa

in which a superposed dot denotes differentiation with respect to time t, and uP = [ wi

ri 

µ

MP = ρA


wj

rj

rk ]T ,

wk

36(175 − 84µ1 − µ3 ) 6(175 − 42µ1 − 3µ3 ) 72(175 + 84µ1 + µ3 ) −6(175 − 42µ1 − 3µ3 ) 175 + 21µ1 − 35µ2 + 9µ3 0 

−6(175 − 42µ1 − 3µ3 ) 0 36(175 − 84µ1 − µ3 ) , 2(175 + 21µ1 + 35µ2 + 9µ3 ) 6(175 − 42µ1 − 3µ3 ) −175 + 21µ1 − 35µ2 + 9µ3 β



KP =

EI −12β 6β
3

−6β −1 + 3β

24β 0

0 2 + 6β

−12β −6β



6β . −1 + 3β (15)

5.1 Plane Wave Propagation Analysis We study the propagation over the lattice of plane waves of wavelength L, wavenumber kp = 2π/L, and circular frequency ωp : w(x, t) = A ei(kp x−ωp t) ,

r(x, t) =
θ(x, t) = B ei(kp x−ωp t) .

(16)

in which subscript p means “physical.” To simplify the subsequent analysis we will work with the dimensionless frequency ω 2 = ωp2 /(EI/ρA
3 ) and wavenumber k = kp
. The dimensionless wavespeed is c = ω/k. A curve plotting solutions ω = ω(k) of (16) is called a dispersion relation. Evaluating (16) at the nodes {i, j, k}, inserting into (14), dropping the time dependency and requiring plane wave solutions with nonzero A, B yields a frequency equation biquadratic in ω that has nonnegative real roots. For convenience define ψ1 = 60 + 5µ1 + µ2 , ψ2 = 15µ1 − µ2 − 20, ψ3 = 28µ2 − 5µ3 + 35, ψ4 = µ2 + 20, ψ5 = µ2 − 40, ψ6 = 36µ2 + 285, ψ7 = 84µ1 + µ3 , ψ8 = 4µ2 + 15, ψ9 = 36µ1 +5βψ1 +25, ψ10 = 5βψ2 −36µ1 +75, ψ11 = 140βψ5 +12ψ7 , ψ12 = 7ψ10 −3µ3 , ψ13 = −21ψ9 − 9µ3 , ψ14 = 84µ1 ψ8 + 4ψ5 µ3 , ψ15 = 3µ1 ψ3 − 175µ2 + ψ4 µ3 , ψ16 = 175µ2 + 7µ1 ψ6 + 3ψ1 µ3 , φ1 = −ψ13 − ψ11 cos k − ψ12 cos 2k and φ2 = ψ16 −ψ14 cos k +ψ15 cos 2k. Then the dispersion relation obtained by Mathematica can be succintly written 

 ωa2 120 = φ ∓ φ21 − 3360 βφ2 sin4 21 k 1 ωo2 φ2



(17)

Frequencies ωa2 and ωo2 , which depend on the four free parameters {β, µ1 , µ2 , µ3 }, pertain to the so-called acoustic and optical branches, respectively; a nomenclature explained by Brillouin.13 Only the acoustic branch appears in the continuum Bernoulli-Euler model, for which ω 2 = k 4 ; see for example Graff.14 The optical branch is spurious. It is introduced by the FEM discretization and corresponds to high-frequency lattice oscillations or “mesh modes,” as discussed by Belytschko and Mullen15 for bar elements. The Taylor expansion about k = 0, which is of interest in low-frequency accuracy studies, contains only even powers of k: ωa2 = k 4 −

1 + β(µ1 − 2) 6 5 + β (3µ1 − 10 + β φ3 ) 8 k + O(k 10 ) k + 12β 720β 2 



ωo2

2100 7(µ1 − 10βµ1 + βµ21 − 5βµ2 ) + 3µ3 2 25200β = + k + O(k 4 ) 7µ1 + 3µ3 (7µ1 + 3µ3 )2

7

(18)

(a) CFMCK: Consistent Full Mass with Consistent Stiffness β =µ1 = µ2 = µ3 = 1 50

(b) BFMCK: Best Full Mass with Consistent Stiffness β =1, µ1 = 1, µ2 = 2, µ3 =26/3 50

Dimensionless frequency ω

Dimensionless frequency ω

Carlos A. Felippa

40 30

Optical branch

20 10 0

Acoustic branch 0

1 2 3 4 5 Dimensionless wavenumber k

Dimensionless frequency ω

Dimensionless frequency ω

30 Optical branch 20

Acoustic branch 1 2 3 4 5 Dimensionless wavenumber k

10

Acoustic branch 0

1 2 3 4 5 Dimensionless wavenumber k

6

50

40

0

Optical branch 20

(d) DFMBK: Dark Full Mass with Best Stiffness β =1/2, µ1 = 0, µ2 = 9, µ 3 = 0

(c) BFMBK: Best Full Mass with Best Stiffness β =4/3, µ1 = 5/4, µ2 = 39/16, µ3 =1055/108

0

30

0

6

50

10

40

Optical branch 40 30 20 10 0

6

Acoustic branch 0

1 2 3 4 5 Dimensionless wavenumber k

6

Figure 4. Dispersion relation for four full-mass-matrix template instances.

where φ3 = 9 + 5µ21 − 10µ1 − µ2 . Only a few leading terms are shown in (18) since succeeding ones, found by Mathematica through O(k 14 ), get increasingly complicated. If one selects µ1 =

2β − 1 , β

µ2 =

7 − 14β + 9β 2 , β2

µ3 =

245 − 546β + 427β 2 − 100β 3 , (19) 3β

then the coefficients of k 6 , k 8 and k 10 in the first of (18) vanish, and ωa2 = k 4 + O(k 12 ). The coefficient of k 12 cannot be zeroed out for any positive real value of β, but is very approximately minimized by taking β = 4/3. A similar analysis of the lumped mass template (10) leads to a dispersion relation that depends on two parameters: {β, ν1 } because the effect of ν2 cancels out on a regular lattice: 

[1 + βχ1 + (βχ2 − 1) cos k]2 − 384βν1 sin4 12 k , 2ν1 (20) in which χ1 = 3 + 24 ν1 and χ2 = 3 − 24 ν1 . The Taylor expansions about k = 0 are 1 + β χ1 + (β χ2 − 1) cos k ∓ ωa2 = ωo2

β − 1 − 24βν1 6 5 + 3β (120ν1 − 5 + 3βχ3 ) 8 k + O(k 10 ), k + 12β 720β 2 6β 1 − 3β + 24βν1 2 + k + O(k 4 ), ωo2 = ν1 2ν1 ωa2 = k 4 +

8

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Carlos A. Felippa

(f) HRZCK: HRZ Lumped Mass with Consistent Stiffness β = 1, ν1 = 1/78, ν2 = 0 50 Dimensionless frequency ω

Dimensionless frequency ω

50

(e) UFMBK: Uncoupled Full Mass with Best Stiffness β =3, µ1 =5/3, µ2 = 5, µ3 = 35

40 30 Optical branch 20 10 0

Acoustic branch 0

1 2 3 4 5 Dimensionless wavenumber k

(g) BLMBK:√Lumped Mass √ with Best Stiffness β =(5- 5)/2, ν1 = (5- 5)/240, ν2 = 0 Dimensionless frequency ω

Dimensionless frequency ω

Optical branch 20

Acoustic branch 1 2 3 4 5 Dimensionless wavenumber k

10

Acoustic branch 0

1 2 3 4 5 Dimensionless wavenumber k

6

(h) TLMCK: Translational Lumped Mass with Consistent Stiffness β = 1, ν1 = ν2 = 0

30

0

Optical branch

20

50

40

0

30

0

6

50

10

40

40 30 20 10 0

6

Acoustic branch 0

1 2 3 4 5 Dimensionless wavenumber k

6

Figure 5. Dispersion relations for four lumped-mass-matrix template instances.

1 (β − 1)/β the coefficient of k 6 vanishes. in which χ3 = 1 − 40 ν1 + 320 ν12 . If ν1 = 24 √ √ If β = (5 − 5)/2 and ν1 = (5 − 5)/240 the coefficients of k 6 and k 8 vanish. Figures 4 and 5 plot the acoustic and optical branches showing |ωa | and |ωo |, respectively, as functions of k, of the eight mass-stiffness combinations listed in Table 1. These combinations possess historical or practical interest. (The meaning of the acronyms is discussed below.). Branches are shown for k ∈ [0, 2π] and repeat with period 2π. The interval −π ≤ k ≤ π is called the first Brillouin zone. The vertical distance between the maximum of ωa , which usually occurs at k = π, and the minimum of ωo , is called the stopping band or forbidden band, a term derived from filter technology. 5.2 Multiobjective Parameter Selection The four parameters of the full-mass plus stiffness template or the three parameters of the lumped-mass plus stiffness template may be used to achieve various performance goals. Three important ones are: Objective A. Low frequency accuracy in the sense that ωa2 agrees well with k 4 as k → 0. This is of interest for structural dynamics, in which low frequency behavior dominates. Objective B. Wide forbidden band between acoustic and optical branches. This is of interest in wave propagation dynamics to reduce spurious noise, particularly at discontinuous wavefronts.

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Carlos A. Felippa

Table 1. Eight Interesting Mass-Stiffness Combinations Acronym β CFMCK

Template Signature µ 1 µ2 µ3 ν1 ν2

1

Low frequency (k → 0) Taylor expansion of ωa2

1 1

1

k4 +

k8 720

26 3

k −

41 k12 18144000

k −

41 k12 185794560

k4 +

53 k10 30240

4

−

BFMCK

1

1 2

BFMBK

4 3

5 4

DFMBK

1 2

0 9

UFMBK

3

5 3

UFMBK

3

1 36

1 12

HRZCK

1

0

k4 −

k6 39

BLMBK

5− 5 2

1 78 √ 5− 5 240

0

k −

k10 10080

0

k −

k8 720

TLMCK

√

1

4

39 1055 16 108

5

0

4

35

k +

0

k8 6480

k10 25200

−

−

+

k12 604800

+ O(k14 )

Low

14

+ O(k )

Medium

14

+ O(k ) 13 k12 34560

k10 13608

Very high

+ O(k14 )

−

Distortion sensitivity

107 k12 20995200

High 14

+ O(k )

Medium

[same as above, UFMBK has two signatures] 4 4

89 k8 121680

−

−

−

−

k12 129600

k10 3024

−

151 k8 1845480

+

239329 k12 51821078400

14

+ O(k )

7 k12 259200

+ O(k14 )

Low High

14

+ O(k )

Low

Objective S. Low sensitivity of frequency predictions to mesh distortion and boundary conditions. This goal is self-explanatory: it tries to lessen effects of the deviation from the infinite regular lattice assumed in classical Fourier analysis. Ideally the parameters should be chosen to improve performance in the three objectives. Unfortunately they are not compatible, and should be prioritized depending on the intended application. For all-around use in general purpose codes, a compromise may be necessary. 5.3 Existing Mass-Stiffness Combinations Before discussing parameter selection it is of interest to note how existing elements fare in Objectives A and B. Table 1 includes three well known combinations: CFMCK, TLMCK, and HRZCK. The combination of consistent full mass and consistent stiffness, called CFMCK, is obtained if β = µ1 = µ2 = µ3 = 1. The simplest lumped mass matrix corresponds to ν1 = ν2 = 0, which has only translational point masses and zero rotational masses; when used in conjunction with the consistent stiffness β = 1 this combination is called TLMCK. The HRZ lumped mass matrix obtained through the widely used scheme of Hinton, Rock and Zienkiewicz,16 defined by ν1 = 1/78 and ν2 = 0, is denoted as HRZCK when used together with the consistent stiffness β = 1. The Taylor expansions of ωa2 given in Table 1 show that CFMCK and TLMCK achieve low frequency accuracy of O(h4 ), h =
/L, in |ωa |. The sign of the first truncation term indicates that CFMCK overestimates |ωa | whereas TLMCK underestimates it, which is a well known property. 5.4 New Mass-Stiffness Combinations Achieving Objective A (high low-frequency accuracy) is easy if one forgets about the other two. Applying the selection rule (19) with β = 1 and β = 4/3 provides combinations called BFMCK (Best Full Mass and Consistent Stiffness) and BFMBK (Best Full Mass and Best Stiffness), respectively. These possess accuracy of O(h6 ) for |ωa | on an infinite regular lattice. BFMBK displays very nearly the lowest truncation error in ωa2 and practically represents the most that can be “squeezed out” of the full-mass-stiffness template if Objectives B and S are ignored. Combination BLMBK (Best Lumped Mass and Best Stiffness) is

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Carlos A. Felippa

12

12

d 10

Digits of accuracy in first fundamental frequency ω1

BFMBK

d

BFMCK

10

Digits of accuracy in second fundamental frequency ω2 BFMBK

8

8 BLMBK

BFMCK

DFMBK UFMBK TLMCK CFMCK

6

6 BLMBK UFMBK DFMBK CFMCK TLMCK

4

4 HRZCK

2

2 HRZCK

0 1

2

4

8

0 1

16 N

2

4

8

16 N

Figure 6. Convergence of first two natural frequencies of SS uniform beam for the eight models of Table 1, using a regular mesh of N elements.

√ √ obtained with β = (5 − 5)/2 ≈ 1.382 and ν1 = (5 − 5)/240 ≈ 1/87, which gives the most accurate lumped-mass-stiffness template. Combination DFMBK ([Optically] Dark Full Mass with Best Stiffness) attempts to concurrently fulfill objectives A and B by moving the optical branch upward while maintaining reasonable accuracy in low frequencies. This is done by enforcing ωo2 → ∞ as k → 0. A glance at the second of (18) shows that this requires 7µ1 + 3µ3 = 0, but since both parameters must be nonnegative the only possibility is µ1 = µ3 = 0. Taking β = 3 and µ2 = 9 makes the coefficients of k 6 and k 8 in ωa2 vanish, although the coefficient of k 10 remains quite large compared to other elements. This mass matrix has one blemish: it is rank deficient. The combination UFMBK (Uncoupled Full Mass with Best Stiffness) adopts µ1 = 5/3, µ2 = 5 and µ3 = 35, which makes Mµ block diagonal. Taking β = 1/2 makes the coefficient of k 6 vanish. This case is exhibited as a curiosity because its mass is the only one that is a member of both the full-mass and lumped-mass templates. For the latter it is obtained by setting ν1 = 1/36 and ν2 = 1/12. 5.5 Vibration Analysis Example The performance of the eight combinations of Table 1 in vibration analysis of a simply-supported (SS) prismatic beam of length L divided into N equal elements is shown in Figure 6. This is a log-log plot that displays the error in the first two dimensionless fundamental frequencies |ω1 | = π 2 and |ω2 | = 4π 2 as functions  of N = 1, 2, . . . 32. [The physical frequencies are ω1p = π 2 EI/(ρAL3 ) and 

ω2p = 4π 2 EI/(ρAL3 ).] The error is plotted as d = log10 (|ωcomputed − ωexact |), which gives at a glance the number of correct digits. The computed results correlate well with the truncation analysis of ωa2 in Table 1. This can be expected not only because the mesh is uniform, but (most importantly) the eigenfunctions of a SS beam are sinusoidal, which agrees with the plane wave spatial distribution (16). The high accuracy of BFMCK and

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Carlos A. Felippa

6 ∆d

6 ∆d

BFMBK

5

5 Digits lost in ω1 with respect to regular discretization

4 3

Digits lost in ω2 with respect to regular discretization

BFMBK

4 3

BLMBK

BLMBK 2 1 0

2

DFMBK

0.1

0.2

0.3

0.4

BFMCK UFMBK TLMCK CFMCK HRZCK 0.5 δ

1 0

0.1

0.2

0.3

0.4

DFMBK BFMCK UFMBK TLMCK CFMCK HRZCK 0.5 δ

Figure 7. Loss of accuracy in SS beam vibration analysis due to use of irregular mesh.

BFMBK should be noted. For example, just four BFMBK elements give |ω1 | to seven figures: 9.86960424 . . . versus π 2 = 9.86960440 . . ., whereas the widely used lumped-mass combination HRZCK gives only one: 9.78975. Deviation from this ideal test problem, however, show these results to be overly optimistic. Running uniform meshes for other end conditions deliver lower accuracies and asymptotic rates, with BFMCK and BFMBK still winning but not by much. And mesh distortion is the great equalizer, as shown next. 5.6 Distortion Sensitivity To get a quick idea of the sensitivity of mass-stiffness combinations to mesh distortion, the SS beam with 16 elements is chosen. Every other interior node is displaced by
δ so elements have alternate lengths
(1 ± δ). The loss of accuracy ∆d in decimal places of the first two lowest frequencies is recorded as δ is increased from 0 to 0.50. Plots of ∆d versus δ are shown in Figure 7. A steep ∆d(δ) flags a combination with high distortion sensitivity. By far the most sensitive is BFMBK which loses close to 6 digits in ω1 and 5 in ω2 for δ ≈ 0.5. This is followed by BLMBK, which loses 2 to 3 digits, and then DFMBK, BFMCK and UFMBK, which lose 1 to 2. Combinations TLMCK, CFMCK and HRZCK lose less than one digit, and can be viewed as fairly insensitive to distortion. Note that these last three combinations pertain to existing elements. These ratings are summarized in the last column of Table 1. Rather than relying on numerical experiments it would be useful to have an explicit analytical expression of this kind of mesh sensitivity in terms of the parameters. This would simplify the design of a mass-stiffness combination attempting to meet both objectives A and S. Fourier analysis can be used for this task by examining plane wave propagation over an infinite periodic but irregular lattice. This is more laborious than regular lattice analysis because it involves a frequency equation of higher order,13 and remains to be done in the future.

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6 GEOMETRIC STIFFNESS ANALYSIS Combinations of the geometric and material stiffness templates can be studied by similar Fourier techniques. These are simpler in that time does not enter the analysis. The main objective is accurate prediction of buckling loads. The results are not reported here because of space constraints. 7 CONCLUDING REMARKS Templates forms of finite element matrices offer the opportunity of customizing FEM discretizations to achieve good performance for specific objectives. For use in a general-purpose FEM program, which typically caters to a wide range of objectives, it is recommended that elements be programmed from the start using free parameters. These parameters can then be supplied through the argument list, and set by intermediate levels of the program. An interesting extension not considered here is the Timoshenko beam model. Using templates this can be done by adjusting some of the free parameters to account for the shear properties. For instance, the material stiffness may be corrected by taking β as function of GAs /(12EI), in which GAs is the cross sectional shear rigidity, as described in Felippa.17 In general only parameters associated with the L4 cubic mode need to be adjusted. This is a simple and effective method to account for transverse shear. It bypasses all of the complexities of the C 0 formulation that clutters the FEM literature. Rotational inertia can be ˜ similarly included in the mass matrix through the covariance matrix R. In passing to 2D and 3D finite element models the following complications arise: • The number of free parameters can be much higher. Incorporation of observer invariance and distortion sensitivity constraints from the start, as done for the plate bending triangle by Felippa5 using energy methods, is recommended because this formulation sequence reduces the dimension of the parameter space. • The stiffness matrix of high performance elements is constructed through assumed stresses or strains, and a displacement field is generally available only on the boundaries. So far only three-node flat triangle geometries for plane stress and bending have been studied in some detail at the material stiffness level. This is roughly the practical limit of symbolic manipulations with present hardware and CAS software. As desktop computers gain further processing power and storage address space, the approach should gradually become feasible for quadrilaterals and tetrahedral solid elements. ACKNOWLEDGEMENTS Preparation of the present paper has been supported by the National Science Foundation under Grant ECS-9725504, and by Sandia National Laboratories under the Advanced Strategic Computational Initiative (ASCI) Contract BF-3574. REFERENCES [1] M. Abramowitz and L.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Applied Mathematics Series 55, Natl. Bur. Standards, U.S. Department of Commerce, Washington, D.C. (1964). [2] P.G. Bergan and M.K. Nyg˚ ard, Finite elements with increased freedom in choosing shape functions, Int. J. Numer. Meth. Engrg., 20, 643–664 (1984).

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[3] P.G. Bergan and C.A. Felippa, A triangular membrane element with rotational degrees of freedom, Comp. Meths. Appl. Mech. Engrg., 50, 25–69 (1985). [4] C.A. Felippa, B. Haugen and C. Militello, From the individual element test to finite element templates: evolution of the patch test, Int. J. Numer. Meth. Engrg., 38, 199–222 (1995). [5] C.A. Felippa, Recent advances in finite element templates, in Computational Mechanics for the Twenty-First Century, ed. by B.J. Topping, Saxe-Coburn Publications, Edinburgh, U.K., 71–98 (2000). [6] P.G. Bergan and L. Hanssen, A new approach for deriving ‘good’ finite elements, in The Mathematics of Finite Elements and Applications – Volume II, ed. by J. R. Whiteman, Academic Press, London, 483–497 (1975). [7] C.A. Felippa and C. Militello, Construction of optimal 3-node plate bending elements by templates, Comput. Mech. J., 24/1, 1–13 (1999). [8] C.A. Felippa, Recent developments in basic finite element technologies, in Computational Mechanics in Structural Engineering - Recent Developments, ed. by F.Y. Cheng and Y. Gu, Elsevier, Amsterdam, 141–156 (1999). [9] J.H. Argyris, S. Kelsey and H. Kamel, Matrix methods of structural analysis — a precis of recent developments, in AGARDograph 72: Matrix Methods of Structural Analysis, ed. by B.M. Fraeijs de Veubeke, Pergamon Press, Oxford, 1–164 (1964). [10] H.C. Martin, On the derivation of stiffness matrices for the analysis of large deflection and stability problems, in Proc. Conf. on Matrix Methods in Structural Mechanics, ed. by J.S. Przemieniecki et al, AFFDL-TR-66-80, Air Force Institute of Technology, 697–716 (1966). [11] K.C. Park and D.L. Flaggs, An operational procedure for the symbolic analysis of the finite element method, Comp. Meths. Appl. Mech. Engrg., 46, 65–81 (1984). [12] K.C. Park and D.L. Flaggs, A Fourier analysis of spurious modes and element locking in the finite element method, Comp. Meths. Appl. Mech. Engrg., 42, 37– 46 (1984). [13] L. Brillouin, Wave Propagation in Periodic Structures, Dover, New York (1946). [14] K.F. Graff, Wave Motion in Elastic Solids, Dover, New York (1991). [15] T. Belytschko and R. Mullen, On dispersive properties of finite element solutions, in Modern Problems in Elastic Wave Propagation, ed. by J. Miklowitz and J.D. Achenbach, Wiley, New York, 67–82 (1978). [16] E. Hinton, T. Rock and O.C. Zienkiewicz, A note on mass lumping and related processes in the finite element method, Earthquake Engrg. Struct. Dynamics, 4, 245–249 (1976). [17] C.A. Felippa, A survey of parametrized variational principles and applications to computational mechanics, Comp. Meths. Appl. Mech. Engrg., 113, 109–139 (1994).

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