The Levenberg—Marquardt non—linear least squares optimization algorithm is adapted to compute the material constants in Ogden' s strain—energy function for incompressible isotropic elastic materials.
In previous papers, three terms have been included in the strain-energy function. In the present paper, four terms are used and it is shown that the optimal values of the eight material constants, which are determined using the Levenberg—Marquardt algorithm, give a much closer fit to experimental data than the strain-energy function with three terms

Ogden, R W

Twizell, E H

English

Brunel University Research Archive

            TR/05/81             June  1981 Non-linear   optimization   of  the  material constants   in   Ogden's   strain-energy   function for   incompressible   isotropic elastic   materials E.H.  Twizell  and  R.W.  Ogden (0) 0.      AbstractThe   Levenberg—Marquardt  non—linear   least   squares   optimization   algorithm is   adapted   to   compute   the  material   constants   in  Ogden' s   strain—energy function   for   incompressible   isotropic  elastic  materials. In  previous   papers,   three   terms   have  been   included  in   the   strain-energy function.      In   the  present  paper,   four   terms   are  used  and   it   is   shown that   the   optimal   values   of   the   eight   material   constants,   which   are determined   using   the   Levenberg—Marquardt   algorithm,   give   a  much   closer fit    to   experimental    data    than    the    strain-energy  function   with    three terms. w9260174 (1) 1.      Introduction In  a  number  of  publications   (see,   for  example,   Ogden   (1972) ,    Chadwick et  at   (1977) ) ,    elementary  methods   have  been  used   to  determine   the material    constants   μi,    αi    in    the    strain-energy    function ⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛ +−+−∑==∑=≡icα1λiα1λiμM1ic),iαP(λ(iμM1iF        (0)  for   incompressible    isotropic   elastic   materials.       In   equation   (0),   F represents   the   force   per   unit   undeformed  area  corresponding   to   the principal   stretch   λ.     The   units   of   the  μi   are   those  of  F  and  the  αi  are dimensionless    (i   =   1,...,M).      Considerations    of    stability   and  physically realistic   response    lead   to    the    inequalities μiαi  >  0   for  all  i = 1,...,M.            (1) The  parameter   c  in  equation   (0)   is   related   to   the  pure  homogeneous deformations   of    simple    tension,   pure   shear  and  equibiaxial   tension,   for which   c  =  -½,   -1,   -2   respectively.     For  further  details   of   the   three deformations   and   the   derivation  of   the   corresponding  values   of   c,   the reader  is   referred  to  Ogden   (1972).  The   set  of  numerical   results   for   μi   and  αi    (i  -   1,2,3)   given  by  Ogden (1972)   and   the   two  sets given   by  Chadwick  et at   (1977)  were  obtained using   linear    least   squares   methods   to   fit  curves  to  the     experimental data   of   Treloar    (1944).      Treloar's   data   were   obtained   in  three experiments   on   samples   cut   from  a  single   sheet  of  vulcanized   natural rubber   ;   his   three   sets    of    data  are  plotted   for  simple   tension,   pure shear   and  equibiaxial   tension  in  Figures   1,   2,   3  respectively.     A  brief review of  other  experiments   and  associated   fitted  curves   by  Jones   and Treloar   (1975),  James  et al  (1975)  and Treloar  and  Riding  (1980)   is (2) contained   in   Ogden   (1981). Chadwick  et  al   (1975)   and   Ogden   (1972)   obtained  values   of   μi   and  αi  (i   =   1,2,3)   by  using  the   fact   that,   at   small   strains   (λ.   ~  1),   the computation   is   dominated  by   just   one   term  μ1   p(λ ,α1 ,c),   with μ2     p(λ. ,α2,c) and  μ3 p(λ , α3  ,c)    increasing   in  importance    as  λ   increases.      The   actual values   of  μi,   αi    determined  by  Chadwick  et  al   (1977)   and  Ogden   (1972) for   the   data   of    Treloar   (1944)   are   reproduced   for   comparison  purposes in  Table   I. In  previous   papers,   authors   using  a  strain-energy  formula  of   the   form (0),   have   taken  M  =  3,   It  was  observed  by Ogden  (1972  ;  p.  578), however,   that,   by   taking  M  =   4,    a  better  fit   could  be   obtained  for λ  >  7.0.    One   purpose   of   this   paper   is   to   report  numerical   results  which verify   this   claim,   though   it  will  be   seen   that   Ogden's    estimate   of α4   -    10 is   too   low  for  Treloar's   data.      The   other  purpose   of   this   paper is   to   show   that   superior  numerical   results   for   μi,   αi     (i  =   1,2,...,M) are   obtained   using non—linear   least    squares   optimization  techniques (section   2).      Such   techniques   obviate   the  need   to   calculate   the   μi ,   αi (i   =    1,2,...,M)    successively   by  fitting   curves    to  expanding    ranges   of data.   The  optimal  values  μi*,  αi*  (i  =  1 ,2,. . . ,M)  are  determined   as   the elements  of  a  vector.    For   the   data  of  Treloar   (1944),   the   optimal values   with  M = 3,4  are  reported  in  section  3  and,  for  M  =   4,   the curves   generated   by    (0) are  plotted   in  Figures    1,   2,   3.   Comparison with   the  values   of   Ogden   (1972)   and   Chadwick   et  al   (1977)   is presented  in  Table   I   and   comparison  of   the   accuracy  obtained  using non-linear  optimization  methods  with   the   accuracy  attained  by   Ogden (1972) and Chadwick et  al   (1977)  in  Table  II. (3) 2.      A  non-linear   least   squares   algorithm In   this   section   the   Levenberg-Marquardt   iterative   algorithm   for calculating   the   optimal   values   μi*,   αi*   (i   =  1, . . . ,M)    is  outlined.   The algorithm  was   published   in   1963  by  Marquardt   and   is   similar   to   the method  published  in   1944  by   Levenberg.     In  these  two   papers   the   L2 norm  is   used   in   the  minimization   process  ;   Shrager    and  Hill   (1980) discuss   the   implementation   of   the   Levenberg-Marquardt    algorithm  in   the L1   and   L∞   norms.    The   L1   norm  is   particularly  beneficial  when   the experimental   dat a  for  λ and  F  con tain  one  or  more   wi ld   points ,  and   the L∞   norm when  the    errors   in   the   experimental  values   of   F   are   negligible. The  L2  norm has  enjoyed  much more  use   than   the   other   two   and consequently    a    large    literature    of    successful     applications   of   the Levenberg-Marquardt  algorithm  in  the   L2   norm,   and  a  large  amount  of associated   computer  software,   has resulted.       The    Levenberg-Marquardt algorithm  in  the  L2  norm is  available  to  IBM users   as   SHARE  PROGRAM #  1428   and   f rom  the  NAG  (Numerical Algorithms  Group)   subroutine  library where   it   is   implemented   in   Fortran   as   E04GAF   and  in  Algol   as   E04GAA . The   data  of   Treloar   (1944)   used   in   this   paper   contain  no  wild  points and  so   the   outline   of   the   Levenberg-Marquardt   algorithm  which   follows is   related   to   the   L2   norm. Suppose   there   are  K   data  pairs    (λk-FK)   k   =   1,...,K  with  K  ≥  2M.     Let Fk    be   the   value   of   Fk   yielded  by ⎟⎟⎠⎞⎜⎜⎝⎛ +−−+−∑== icα1λkiα1kλiμM1ikF  and   let  Ek   =   Fk   -  Fˆ k   be  the  error  in  Fk  .     The   least   squares   criterion requires   that (4)  (2)2icα1λkiα1kλiμM1ikFK1k2kEK1k)MαMμ,,1α,1(μSS⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛ +−−+−∑==∑==∑==− K be  minimized,   this  minimum  of   S   being  reached  by obtaining  optimal   values  μi*,   αi*  of   the  parameters   μi,   αi.   (i  =  1,. . . ,M).  In  order  to  implement  the Levenberg-Marquardt   algorithm  to  minimize   S,   it   is   convenient   to  introduce a vector  x  of  order  2M  defined  by ,T)Mα,Mμ,,1α,1(μT)2Mx,12Mx,,2x,1(xx KK =−= where  T  denotes   transpose. The   Levenberg-Marquardt   algorithm  calculates   iteratively   a  sequence   of points   x(r) (r  =   0 ,1 ,2 , - . . )    with  x(0) some   initial  point   chosen   so   that   the  sequence  {x(r)}  will  converge  to  a  point T)*Mα,*Mμ,,*1α,*1(μ*x K=   that  minimizes  S   (the  superscript  r  denoting   the  rth  iterate).     The algorithm  calculates   the  vector  x (r + 1) from  the  vector  x (r) using  the equation    x ( r +1 ) = x (r) - [ ( p (r)) T p (r) = γ (r)  I ]  -1  (p (r) ) T  E (r) ; r = 0, 1,2,. . . . (3) where   γ(r) (r  =  0 ,1 ,2 , . . . )    is   an  arbitrary parameter  and  E  =   (E 1, E2,...,Ek)Tis   the  vector  of   errors   (see  equation   (2)).  The  matrix I   is  the  identity matrix of  order 2M and P  is   the matrix  of   first   derivatives  of  order k  x 2M  whose  element  pk i .  at  the  rt h  iterate  is  given  by (r)xxixkE(r)kiP =∂∂=     (k  = 1,…,K  ;  I  =  1,…,2M  ;  r  =  0,1,2,…). (5) Thus ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ +−−+−−=−(r)cα1kλ(r)α1kλ(r)1k,2P lll  and knλ(r)cα1kcλ(r)α1kλ(r)μ(r)k,2P lllll ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ +−−+−−= for  k   =   1,.. . ,K   ;  ℓ  =   1,. ..,M   ;   r  =  0,1,2,...    . Marquardt   (1963)   has   shown  that  a  sufficiently  large  γ(r) always   exists such   that S(r+l)   <  S(r) (4) (unless  x (r) =  x*),  where  S(r) denotes  the  value  of  S  at  the  rthiteration   (r  =  0 , 1 , 2 , . . . ) .       It  is   clear  therefore   that  the  method converges   from  poor   starting  values  μi(0) ,  αi(0)  (i  =  1,...,M)   and convergence  proceeds   as   follows: (i) arbitrarily  choose   γ(0)       and a parameter  u >  1  ;  say  γ(0) = 0.01 and  u  =   10   ; (ii)        let  T( γ(r)),   T( γ(r)/u)   be   the  values   of   S(r)   when   γ(r)   and   γ(r)/u, respectively,   are   used   in  equation   (3)    ; (iii)       calculate   S(r+1),    T ( γ(r))    and  T( γ(r)/u)   ; (iv)        then   (a)   if   T( γ(r)/u)   ≤   S(r+1),   let   γ(r+1)   =   γ(r)/u   ; (b) if   T( γ(r)/u)   >  S( r + 1 ) .and  T( γ(r))   <   S(r+1),   let  γ(r + 1) =  γ(r)      ; (c) if   T( γ(r)/u)   >  S(r+1)   and  T( γ(r))   >  S(r+1) ,   increase  γ(r)  by  successive  multiplication  by  u  until   the positive   integer  n  is  reached  such   that  T( γ(r) un) ≤   S(r+1).      Let  γ(r+1)   =   γ(r)un   ; (v) test   for   convergence  of  all   the  material   constants μi, αi. (6) (i  = 1,...,M)   to  the  required  accuracy.     If   the   accuracy criterion   is   met   the   iterations  cease,   otherwise   r  is   incremented by  unity   and   control  returns  to  (ii). The   convergence   tests   described   in   steps   (iv)   and   (v)   of   the   strategy do   lead   to   increased  computer  time   and   storage   in   comparison  with   less sophisticated  methods.      With  γ(r)        ≡  0,   for   instance,   the   Levenberg- Marquardt   algorithm   (3)   becomes   the   Gauss-Newton  algorithm  which,   for some  problems,   may  well   converge  faster,   from   good   initial   values, than   the   Levenberg-Marquardt   algorithm.      From  poor   initial   values, however,   the  Gauss—Newton method may diverge while  the  Levenberg- Marquardt   algorithm  will  converge.       It   is   this    factor   which   highlights the    superior    reliability   of    the  Levenberg-Marquardt    algorithm. (7) 3.     Numerical   results The  optimal  values  of  μi,  αi.  (i = 1,.. . ,M) for  M  =  3  and  M  =  4  were determined  using   the  Levenberg-Marquardt  algorithm  for  the  data  of Treloar   (1944)   relating   to  his   simple   tension  experiment   (c  =  -½). These  values   are   contained   in  Table   I.     Also   contained  in  Table  I  are the   set  of  values   of μi,  αi    (i  =   1,2,3)  obtained by Ogden   (1972)   and the   two  sets   obtained  by  Chadwick et at   (1977  :   pp.   74,75).   The  minimum sums   of   squares,   defined  by  equation   (2) ,    were   also  determined  for  all five   sets  of   values   for  the   simple   tension  experiment   ;    the    five  values of   S   are   contained  in  Table  II. The   two   optimal  sets   of  material   constants   obtained  by   the  non-linear optimization   algorithm,   and   those  obtained  by  Chadwick et  at   (1977) and  Ogden   (1972) ,   were  also  used  to determine   the  value  of  S  for the data   of   Treloar   relating   to   his   pure  shear   (c  =  -1)   and   equibiaxial tension   (c = -2)   experiments.     These   ten   values  of  S  are  also contained  in  Table   II.  The   sets   of   optimal  values  of  μi,  αi   (i  =   1 , . . . ,M)   determined  for  both M  =   3  and  M = 4  by  the  Levenberg-Marquardt  algorithm  are  seen  to satisfy   the   inequality   ( 1 ) .       In  addition  each  optimum  value  α*   for iM  =   3  is   seen  to  satisfy   the   condition αi   ≤  -1  or   αi  ≥  2 (5) (Chadwick et al   (1977 ; p.63).    This  is  not  so  for  M  =  4   ;   he re   the optimal  value  of  α1   (α1*  =   1.23)   violates   (5).     In  their  paper  Chadwick et  at  reject  Ogden's   (1972)   value  of  α1   -   1.3  because  it,   too, violates   (5)   ;   it  is  noted   that  for  M  =  4  the   Levenberg-Marquardt algorithm  yields  optimal  values   μ*i   and  α*i   which  are  very  close  to those  of  Ogden   (1972).     Other  grounds  for  accepting  values  of  αi(8) between   1   and  2  have  been   summarized  by  Ogden   (1981). For   Treloar's   simple   tension   data,   the   minimum  value   of   S   obtained  by the   Levenberg-Marquardt   algorithm  with  M  =   3   is   smaller   than   the   three values   of   S   yielded  by   the   values   of   μi,  αi    (i   =   1 ,2 ,3) of   Chadwick   et al †(1977)   and  Ogden   (1972).      For  pure   shear   and  equibiaxial   tension, however,   the  minimum  value  of  S  with  M  =  3   is  superior  only   to  that   of Chadwick  et  at   (i)   (1977   ;   p.74). It   is   seen   that   the   computed  minimum  values   of   S   relating  to   Treloar's equibiaxial   ten sion  data   are   very   large,   indicating   that  while   the optimal   values  of  μi,  αi   (i  = 1 , . . . ,M ; M = 3 or  4)   computed  in   the present  paper  fit   the  simple   tension  and  pure  shear   data  closely.     The same  observation  may  be  made  of   the  values   of  μi,  αi.  (i = 1,2,3) given in  Ogden   (1972)  and Chadwick  et al  (1977). Figures  1,2,3 contain the curves  of  best   and  worst   fit   for  simple   tension,   pure  shear   and   equibiaxial tension,   respectively,   as  well   as   the  data  points   of   Treloar   (1944).     It is   seen   that  using  four  terms  in   (0)   gives  a  very   close   fit   for   large strains   for   all   three  experiments. The  numerical  results   reported  in  the  present  paper  verify  that   the  use of   non-linear   least   squares   optimization   methods   is   justified  when   fitting curves   of   the   form   (0)   to  experimental  data  ; the  wide  availability  of relevant   software   enforces   this   point.      It  has   further  been  verified   that the   use  of  four  terms  in  the  strain  energy  function produces  a much  closer fit   than   the  use of  three  terms.   This  was  suggested  in 1972 by Ogden   (1972) but  his  estimated value  of  α4,   for  Treloar's  data has  been  seen   to  be   too low  by  a  factor  of  about  2, though  it  is  almost equal  to α   for   the   case *3M  =   3.  Not  one of   the   five   sets   of   μi,  αi    (i = 1,...,M ; M = 3 or 4) contained  in Table  I  yields  a  close  fit   to   the  data  of   Treloar   (1944) simultaneously  for  each  of   the   simple  tension,   pure  shear  or  equibiaxial tension   data  sets. (9) Table   I   :  Numerical  values  of  μi,  αi   (i  = 1, . . . ,M ;  M = 3  or  4)   μ1 α1 μ2 α2 μ3 α3 μ4 α4Twizell   and Ogden   (i) 6.27 1.23 -0.054 -1.99 0.036 4.44 0,8(-15) 19.49Twizell   and Ogden   (ii) 2.22 2.26 -0.45 -2.01 3.88(-7) 10.01    - - Ogden   (1972) 6.3 1 ,3   -0.1  -2.0   0.012  5.0    -  - Chadwick  et al (i)    eqn.    (5.4) 3-0 2.0  -0, 1  -2.0 3.7(-5) 7.82        -      -  Chadwick  et al  (ii)   eqn.   (5.5) 3.24 2.0  -0.  1  -2.0 6.2(-6)  8.7          -       - Table  II     :     Sums  of  squares   (S).  Method  s    Simple tension Pure  shear Eqtubiaxial tension Twizell   and Ogden   (i) 6.3 1.40 3.77 Twizell   and Ogden  (ii) 12.8 2.20 9.81 Ogden  (1972) 302.9 1.60 3.91 Chadvi ck et al   (i) 20.4 2.83 10.32 Chadwick et at   (ii) 16. 1 1.44 4.42  Figure  1:  Best- and  worst-fitting  curves  for  Treloar's  simple  tension  data. (11) Figure  2:    Best-  and  worst-fitting  curves   for  Treloar's  pure  shear  data.  Figure  3:    Best-  and worst-fitting  curves  for  Treloar's  equibiaxial  tension  data. (13) References Chadwick  P.,  Creasy C.F.M.   and  Hart V. G. ,  1977,  J.  Austral.  Math,   Soc. (Series   B)  20,   62  -   96. James  A. G.,  Green  A.  and  Simpson  G.M. ,  1975,  J. Appl. Polymer  Sci. 19, 2033   -   2058. Jones   D.F. and  Treloar  L. R. G. , 1975,  J. Phys. D.  :  Appl.   Phys.    8, 1285   -   1304. Levenberg   K. ,    1944,   Q.   Appl.   Math.    2,    164  -   168. Marquardt   D.W. ,    1963,   SI AM J.   11 ( 2) ,   431   -   441. Ogden   R.W.,    1972,   Proa.   R.    Soo.  Lond.   A   326,  565   -  584. Ogden   R. W. ,  1981.  In  Hopkins  H.G.  and   Sewell   M.J.  (eds.)   Mechanics   of    Solids,  The   Rodney  Hill  60th  Anniversary   Volume   (Oxford,   Pergamon Press),   499   -   537. Shrager  R.I.   and  Hill  E.   Jr . ,     1980,   Math.    Comp.   34(150),   529 - 541. Treloar  L.R.G.,    1944,   Trans.    Faraday Soc.   40,   59   -   70. Treloar  L.R.G.    and   Riding  G. ,    1980,   Proc.   R.  Soc.  Lond.  A 369,  261 -   280. AcknowledgementThe   authors   acknowledge   the   financial   support  of   the  Science  and Engineering  Research  Council   to   attend   the   Strathclyde   Biomechanics Workshop  in  July  1980   at  the   University  of  Strathclyde,   Glasgow,  where the   research   for   this   paper   began.            NOT TO BE REMOVED FROM  THE  LIBRARY XB  2356413 X  

Non-linear optimization of the material constants in Ogden's strain-energy function for incompressible isotropic elastic materials

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