caltech asci technical report 080 - Semantic Scholar

CALTECH ASCI TECHNICAL REPORT 080 caltechASCI/2000.080

The Computation of the Exponential and Logarithmic Mappings and their First and Second Linearizations M. Ortiz, R.A. Radovitzky, and E.A. Repetto

The Computation of the Exponential and Logarithmic Mappings and their First and Second Linearizations M. Ortiz, R. A. Radovitzky and E. A. Repetto Graduate Aeronautical Laboratories California Institute of Technology Pasadena, CA 91125 February 15, 2001 Abstract We describe two simple methods for the evaluation of the exponential and logarithmic mappings and their first and second linearizations based on the Taylor expansion and the spectral representation. We also provide guidelines for switching between those representations on the basis of the size of the argument. The first and second linearizations of the exponential and logarithmic mappings provided here are based directly on the exponential formula for the solutions of systems of linear ordinary differential equations. This representation does not require the use of perturbation formulae for eigenvalues and eigenvectors. Our approach leads to workable and straightforward expressions for the first and second linearizations of the exponential and logarithmic mappings regardless of degeneracies in the spectral decomposition of the argument.

Keywords Exponential and Logarithmic Mappings. Linearization. Finite Element Analysis.

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1

Introduction

The need to evaluate the exponential and logarithmic mappings of square matrices, and their first and second linearizations, arises in a number of applications. For instance, some models of finite-deformation material behavior are formulated in terms of Hencky’s logarithmic strain (see [7, 8] and references therein; see also [11, 12, 13, 14, 4, 9, 10, 18, 21, 22, 2]). In addition, several update algorithms for finite deformation plasticity have been proposed which make explicit use of the exponential and logarithmic mappings [5, 20, 3]. In these examples, the computation of the stress tensor requires the evaluation of the exponential and logarithmic mappings as well as their first linearizations. If, in addition, a Newton-Raphson solution procedure is used, the computation of the tangent moduli requires a second linearization of the exponential and logarithmic mappings. The exponential and logarithmic mappings may be evaluated by a variety of means, including the direct summation of their corresponding Taylor expansions, or by recourse to an eigenvalue or spectral decomposition of the argument and the use of the spectral function theorem. A standard starting point for studies in this area is the work of Moler et al [15], who discuss nineteen different ways of computing the exponential of a matrix and argue on the computational stability and efficiency of each method. It sometimes happens that the argument of the exponential mapping is close to the null matrix or that the argument of the logarithmic mapping is close to the identity. For instance, this situation may arise during constitutive updates when the incremental deformations are small. Under these conditions, the Taylor series converge in a small number of iterations and is advantageous in comparison to the spectral decompositions. Conversely, when the argument of the exponential mapping is large or the argument of the logarithmic mapping differs greatly from the identity, the Taylor series may be expected to converge slowly or not at all, and the spectral decomposition becomes advantageous. We present numerical tests which provide an indication of where the optimal cross-over point between the Taylor and spectral representations lies. The first and second derivatives of the exponential and logarithmic mappings may be evaluated differentiating the Taylor expansion term by term. or from the spectral decomposition by recourse to perturbation formulae for the eigenvalues and eigenvectors (see, e. g., [19] and references therein). However, this latter approach is cumbersome and fraught with difficulties, 2

especially in the presence of repeated eigenvalues. Here we present a simpler approach based on the well-known exponential representation of the solutions of systems of linear ordinary differential equations which does not require the linearization of the spectral decomposition. Our approach leads to workable and straightforward expressions for the first and second linearizations of the exponential and logarithmic mappings regardless of degeneracies in the spectral decomposition of the argument. In the case of symmetric argument, our results are consistent with those of Ogden [16], who treated explicitly the computation of first and second derivatives; see also [6, 17].

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Evaluation of the exponential and logarithmic mapping and their derivatives

Let A ∈ Rn×n , B ∈ Rn×n be square real matrices, not necessarily symmetric. The exponential of A is defined as ∞ X 1 k exp(A) = A k! k=0

(1)

This series is absolutely convergent [1]. The logarithm of B is defined as log(B) =

∞ X (−1)k−1

k

k=1

(B − I)k

(2)

This series is absolutely convergent if k B − I k< 1 [1]. The problem now is to derive computationally convenient and straightforward formulae for the evaluation of the mappings exp(A) and log(B), as well as their first and second derivatives, D exp(A), D2 exp(A), and D log(B), D2 log(B).

2.1

Spectral representation

Let A have eigenvalues {λα , α = 1, . . . n}, right eigenvectors {uα , α = 1, . . . n}, and left eigenvectors {v α , α = 1, . . . n}. Then, Auα = λα uα , A T v α = λα v α ,

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α = 1, . . . , n, α = 1, . . . , n,

(3) (4)

and A=

n X

λα uα ⊗ v α

(5)

α=1

The exponential of A then admits the spectral representation exp(A) =

n X

eλα uα ⊗ v α

(6)

α=1

Next we wish to linearize the exponential mapping twice. To this end, we begin by recalling that the solution to the problem ˙ x(t) = Ax(t) + f (t), x(0) = x0

t≥0

(7) (8)

in Rn is x(t) = exp(tA)x0 +

Z

t

exp((t − τ )A)f (τ ) dτ,

t≥0

(9)

0

It therefore follows that exp(A)ij is the ith component of the solution of the initial-value problem (7 - 8) at t = 1 with f (t) = 0 and x0 = ej ≡ jth standard basis vector in Rn . Imagine now perturbing the matrix A to A + δA in (7 - 8), resulting in a perturbed solution x(t) + δx(t). To first order, δx(t) is the solution of the problem ˙ δ x(t) = Aδx(t) + δAx(t), δx(0) = 0

t≥0

(10) (11)

From (9), δx(t) =

Z

t

exp((t − τ )A)δA exp(τ A) x0 dτ

(12)

0

But we also have x(t) + δx(t) = exp(t(A + δA))x0 = [exp(tA) + Dexp(tA)δA]x0 + hot (13) Comparison of (12) and (13) yields Z 1 exp((1 − τ )A)δA exp(τ A) dτ Dexp(A)δA = 0

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(14)

or, in components Dexp(A)ijkl =

1

Z

exp((1 − τ )A)ik exp(τ A)lj dτ

(15)

0

We may further make use of representation (5) to write (15) in the form Dexp(A)ijkl =

n X

e

λα

α=1

n Z X

1

e

τ (λβ −λα )



dτ uαi vαk uβl vβj

(16)

0

β=1

which evaluates to Dexp(A)ijkl =

n X n X

f (λα , λβ )uαi vαk uβl vβj

(17)

α=1 β=1

or, in invariant notation, Dexp(A) =

n X n X

f (λα , λβ )uα ⊗ v β ⊗ v α ⊗ uβ

(18)

α=1 β=1

In these expressions we have written eλβ − eλα if λβ 6= λα λ β − λα f (λα , λβ ) = eλα otherwise f (λα , λβ ) =

(19) (20)

Note that the above expressions are valid in the presence of repeated eigenvalues. In order to determine the second derivative of the exponential mapping we may differentiate (15) to obtain Z 1 2 D exp(A)ijklmn = (1 − τ )Dexp((1 − τ )A)ikmn exp(τ A)lj dτ 0 Z 1 + τ exp((1 − τ )A)ik Dexp(τ A)ljmn dτ (21) 0

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Inserting (5) and (17) into this expression gives D2 exp(A)ijklmn =  n X n X n Z 1 X τ λγ (1 − τ )f ((1 − τ )λα , (1 − τ )λβ )e dτ uαi vαm uβn vβk uγl vγj 0 α=1 β=1 γ=1 n X n X n Z 1 X

+

α=1 β=1 γ=1

(1−τ )λγ

τ f (τ λα , τ λβ )e



dτ uαl vαm uβn vβj uγi vγk (22)

0

Defining F (λα , λβ , λγ ) =

Z

1

τ f (τ λα , τ λβ )e(1−τ )λγ dτ

(23)

0

(22) simplifies to D2 exp(A)ijklmn = n X n X n X

F (λα , λβ , λγ )vαm uβn [uαl vβj uγi vγk + uαi vβk uγl vγj ] (24)

α=1 β=1 γ=1

which is the sought expression. Finally, a straightforward evaluation of (23) leads to the result:

F (λα , λβ , λγ ) =  λ eλα −λ eλα −λ eλβ +λ eλβ +λ eλγ −λ eλγ γ α γ α β β  , if λα = 6 λβ , λα = 6 λγ , λβ = 6 λγ  (λα −λβ )(λα −λγ )(λβ −λγ ) λγ λ λ λ α α α −e +λα e −λγ e +e , if λα = λβ , λα = 6 λγ , λβ 6= λγ (25)  (λα −λγ )2  1 λα e , if λα = λβ , λα = λγ , λβ = λγ 2 where the remaining cases follow by a cyclic permutation of the labels {α, β, γ}. The derivatives of the logarithmic mapping can be obtained directly from the previous expressions by recognizing that log = exp−1 and using the properties of inverse functions [1]. This leads to [1] Dlog(B)ijkl = Dexp(A)−1 ijkl 2 D log(B)ijklmn = −1 −1 2 −Dexp(A)ijpq Dexp(A)−1 rskl Dexp(A)tumn D exp(A)pqrstu 6

(26) (27)

where B = exp(A). The first of these expressions evaluates to Dlog(B)ijkl =

n X n X

g(µα , µβ )uαi vαk uβl vβj

(28)

α=1 β=1

where {µα , α = 1, . . . n} are the eigenvalues of B, {uα , α = 1, . . . n} are its right eigenvectors, and {v α , α = 1, . . . n} its left eigenvectors. In invariant notation n X n X Dlog(B) = g(µα , µβ )uα ⊗ v β ⊗ v α ⊗ uβ (29) α=1 β=1

where we write logµβ − logµα if µβ 6= µα µβ − µα 1 g(µα , µβ ) = otherwise µα g(µα , µβ ) =

(30) (31)

Note again that the above expressions are valid in the presence of repeated eigenvalues. Using (26) and (29), (27) may be rewritten explicitly in the form: D2 log(B)ijklmn = −Dlog(B)ijpq Dlog(B)rskl Dlog(B)tumn D2 exp(A)pqrstu

(32)

which does not require matrix inversion.

2.2

Taylor series expansion

The evaluation of the formulae in the preceding section requires the computation of eigenvalues and eigenvectors, which entails a non-negligible computational overhead. Therefore, for small A it may be cheaper to evaluate the exponential mapping and its derivatives directly from its Taylor series expansion (1). Likewise, for B close to the identity it may be cost-effective to use (2) directly. To this end, it proves convenient to express (1) in the form ∞ X exp(A) = exp(k) (A) (33) k=0

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where the terms exp(k) (A) in the expansion follow from the recurrence relation exp(0) (A) = I exp(k+1) (A) =

(34)

1 exp(k) (A)A, k+1

k = 0, . . .

(35)

Here I is the identity matrix. It follows from this representation that Dexp(A) =

∞ X

Dexp(k) (A)

(36)

k=1

where Dexp(1) (A) = DA Dexp(k+1) (A) = 1 [Dexp(k) (A)A + exp(k) (A)DA], k+1

(37) k = 1, . . .

(38)

In components Dexp(1) (A)ijkl = δik δjl Dexp(k+1) (A)ijkl = 1 [Dexp(k) (A)ipkl Apj + exp(k) (A)ik δjl ], k+1 Likewise 2

D exp(A) =

∞ X

(39) k = 1, . . .

D2 exp(k) (A)

(40)

(41)

k=2

where D2 exp(2) (A) = 21 D2 A2 D2 exp(k+1) (A) = 1 {D2 exp(k) (A)A + 2sym[Dexp(k) (A)DA]}, k+1

(42) k = 2, . . .

(43)

where sym denotes the symmetrization algorithm. In components D2 exp(2) (A)ijklmn = 12 (δik δlm δjn + δim δkn δjl ) 1 D2 exp(k+1) (A)ijklmn = k+1 [D2 exp(k) (A)ipklmn Apj + Dexp(k) (A)imkl δjn + Dexp(k) (A)ikmn δjl ], 8

k = 2, . . .

(44) (45)

The logarithmic mapping can be given a similar treatment. Begin by expressing (2) in the form log(B) =

∞ X

log(k) (B)

(46)

k=1

where the terms log(k) (B) in the expansion follow from the recurrence relation log(1) (B) = B − I k log(k+1) (B) = − log(k) (B)(B − I), k+1

(47) k = 1, . . .

(48)

It follows from this representation that Dlog(B) =

∞ X

Dlog(k) (B)

(49)

k=1

where Dlog(1) (B) = DB Dlog(k+1) (B) = k − k+1 [Dlog(k) (B)(B − I) + log(k) (B)DB],

(50) k = 1, . . .

(51)

In components Dlog(1) (B)ijkl = δik δjl (k+1)

Dlog

(B)ijkl =

k − k+1 [Dlog(k) (B)ipkl (Bpj − δpj ) + log(k) (B)ik δjl ],

Likewise 2

D log(B) =

(52)

∞ X

D2 log(k) (B)

k = 1, . . .

(53)

(54)

k=2

where D2 log(2) (B) = 12 D2 B 2 D2 log(k+1) (B) = k − k+1 {D2 log(k) (B)(A − I) + 2sym[Dlog(k) (B)DB]}, 9

(55) k = 2, . . . (56)

In components D2 log(2) (B)ijklmn = 21 (δik δlm δjn + δim δkn δjl ) (k+1)

D2 log

(57)

(k)

k (B)ijklmn = − k+1 [D2 log (B)ipklmn (Apj − δpj )+

Dlog(k) (B)imkl δjn + Dlog(k) (B)ikmn δjl ],

k = 2, . . .

(58)

The above recursive formulae are in a form which lends itself to a convenient numerical implementation. In practice, the recursive evaluation terminates when the norm of the next term is less then a prespecified tolerance, or when convergence fails to be attained within a fixed maximum number of iterations.

3

Comparison between Taylor and spectral representations

In the foregoing, we have described two different representations of the exponential and logarithmic mappings and their first and second derivatives. The Taylor expansion of the exponential (resp. logarithm) may be expected to converge rapidly, and therefore be computationally advantageous, for small matrices (resp. matrices close to the identity), whereas the spectral representation may be expected to be faster for large matrices (resp. matrices differing greatly from the identity). Our benchmark tests bear these expectations out and additionally provide a threshold for switching from a Taylor to a spectral representation. Figs. 1 and 2 collect the obtained execution times for the Taylor-expansion and spectral representations of the exponential (resp. logarithm) mapping as a function of matrix size (resp. distance to the identity matrix). The calculation of the exponential is carried out for matrices of the form   A 0 0 A =  0 0 0 (59) 0 0 0 whereas the calculation of the logarithm form  1+B B= 0 0 10

is carried out for matrices of the  0 0 1 0 (60) 0 1

Computation of exponential mapping

1.2 1.1 1 0.9

Time [sec]

0.8 0.7 0.6 taylor expansion eigenvalue expansion combined

0.5 0.4 0.3

Threshold

0.2 0.1 0

0.1

0.2

0.3

0.4

Matrix norm

Figure 1: Execution times for the Taylor-expansion and spectral representations of the exponential mapping as a function of matrix size.

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Computation of logarithmic mapping

3

Time [sec]

2

taylor expansion eigenvalue expansion combined

1

Threshold 0

0.1

0.2

0.3

0.4

Matrix norm

Figure 2: Execution times for the Taylor-expansion and spectral representations of the logarithmic mapping as a function of the distance to the identity matrix.

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The rate of convergence of the Taylor expansion of the exponential (resp. logarithm) is dictated by the magnitude of the maximum eigenvalue of A (resp. B − I). Consequently, we choose the abscissae of Figs. 1 and 2 to correspond to |A| and |B|, respectively. The execution times are computed in C language as follows:

time_0 = clock(); for (repetitions=0; repetitions < number_of_repetitions; repetitions++) { compute exponential/logarithmic mapping including 1st and 2nd derivatives } time_1 = clock(); printf("Execution time= %f ", (double) (time_1 time_0)/CLOCKS_PER_SECOND);

The calculations were carried out on a PC computer with a Pentium II processor (400 MHz) and 128 Mb of memory. The variable number of repetitions was set to 1000. All eigenvalue calculations were carried out using the rg subroutine from the fortran library EISPACK (http://www.netlib.org/eispack). This subroutine computes the eigenvalues and eigenvectors of a general matrix using Jacobi rotations. It is evident from these data that the cross over from a Taylor expansion to a spectral representation should be effected when the magnitude of the maximum eigenvalue of A or B − I is of the order of 0.15-0.20. A practical cross-over criterion which avoids the computation √ of eigenvalues may be p based simply on the Euclidean norms k A k= trAT A and k B − I k= tr(B − I)T (B − I).

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4

Summary and conclusions

We have analyzed two methods for the evaluations of the exponential and logarithmic mappings based on the Taylor expansion and the spectral function theorem. Based on the results of selected numerical tests, we have provided guidelines for switching between those representations on the basis of the size of the argument. In addition, we have presented a representation of the first and second linearizations of the exponential and logarithmic mappings based on the well-known exponential formula for the solutions of systems of linear ordinary differential equations. This representation does not require the use of perturbation formulae for eigenvalues and eigenvectors. Our approach leads to workable and straightforward expressions for the first and second linearizations of the exponential and logarithmic mappings regardless of degeneracies in the spectral decomposition of the argument. Our numerical tests suggest switching from the Taylor expansion to the spectral representation roughly when the Euclidean norms k A k or k B −I k become of the order of 0.15-0.20. The implications of this empirical criterion as regards applications to constitutive updates are noteworthy. Thus, in explicit-dynamics calculations, which are characterized by small incremental deformations, the Taylor-expansion representation may be expected to be always advantageous. Contrariwise, the spectral representation may be expected to become cost-effective in implicit calculations involving large incremental deformations, e. g., in the vicinity of stress-risers such as crack tips. In this regime, the Taylor expansion may become unacceptably slow, or simply fail to converge. It is clear, therefore, that a general and efficient implementation of the exponential and logarithmic mappings requires the use of both expansions and spectral representations.

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