Bardell’s Hierarchical Functions¶

Introduction¶

Bardell (1991) [bardell1991] proposed a very convinient set of approximation functions based on hierarchical polynomials. The convenience comes from the fast convergence and from the easiness to simulate practically any type of boundary conditions.

The boundary condition is controlled by the first 4 terms of the approximation function, herein defined as:

t1: the translation at extremity 1 ( $\xi = -1$ )
r1: the rotation at extremity 1
t2: the translation at extremity 2 ( $\xi = +1$ )
r2: the rotation at extremity 2

Generating Bardell’s functions¶

The following code can be used to generate the Bardell functions for a given number of terms nmax. The substitution replace('**', '^') aims to create an input to Mathematica.

../../../../theory/func/bardell/bardell.py¶

from __future__ import division
import numpy as np
from sympy import var, factorial, factorial2, sympify

nmax = 40

xi = var('xi')
var('t1, r1, t2, r2')

u = list(map(sympify, ['1/2 - 3/4*xi + 1/4*xi**3',
                  '1/8 - 1/8*xi - 1/8*xi**2 + 1/8*xi**3',
                  '1/2 + 3/4*xi - 1/4*xi**3',
                  '-1/8 - 1/8*xi + 1/8*xi**2 + 1/8*xi**3']))

for r in range(5, nmax+1):
    utmp = []
    for n in range(0, r//2+1):
        den = 2**n*factorial(n)*factorial(r-2*n-1)
        utmp.append((-1)**n*factorial2(2*r - 2*n - 7)/den * xi**(r-2*n-1))
    u.append(sum(utmp))

u[0] = t1*u[0]
u[1] = r1*u[1]
u[2] = t2*u[2]
u[3] = r2*u[3]

with open('bardell.txt', 'w') as f:
    f.write("Bardell's hierarchical functions\n\n")
    f.write('Number of terms: {0}\n\n'.format(len(u)))
    f.write(',\n'.join(list(map(str, u))).replace('**', '^') + '\n\n')

The output of the code above is:

../../../../theory/func/bardell/bardell.txt¶

Bardell's hierarchical functions

Number of terms: 40

t1*(xi^3/4 - 3*xi/4 + 1/2),
r1*(xi^3/8 - xi^2/8 - xi/8 + 1/8),
t2*(-xi^3/4 + 3*xi/4 + 1/2),
r2*(xi^3/8 + xi^2/8 - xi/8 - 1/8),
xi^4/8 - xi^2/4 + 1/8,
xi^5/8 - xi^3/4 + xi/8,
7*xi^6/48 - 5*xi^4/16 + 3*xi^2/16 - 1/48,
3*xi^7/16 - 7*xi^5/16 + 5*xi^3/16 - xi/16,
33*xi^8/128 - 21*xi^6/32 + 35*xi^4/64 - 5*xi^2/32 + 1/128,
143*xi^9/384 - 33*xi^7/32 + 63*xi^5/64 - 35*xi^3/96 + 5*xi/128,
143*xi^10/256 - 429*xi^8/256 + 231*xi^6/128 - 105*xi^4/128 + 35*xi^2/256 - 1/256,
221*xi^11/256 - 715*xi^9/256 + 429*xi^7/128 - 231*xi^5/128 + 105*xi^3/256 - 7*xi/256,
4199*xi^12/3072 - 2431*xi^10/512 + 6435*xi^8/1024 - 1001*xi^6/256 + 1155*xi^4/1024 - 63*xi^2/512 + 7/3072,
2261*xi^13/1024 - 4199*xi^11/512 + 12155*xi^9/1024 - 2145*xi^7/256 + 3003*xi^5/1024 - 231*xi^3/512 + 21*xi/1024,
7429*xi^14/2048 - 29393*xi^12/2048 + 46189*xi^10/2048 - 36465*xi^8/2048 + 15015*xi^6/2048 - 3003*xi^4/2048 + 231*xi^2/2048 - 3/2048,
37145*xi^15/6144 - 52003*xi^13/2048 + 88179*xi^11/2048 - 230945*xi^9/6144 + 36465*xi^7/2048 - 9009*xi^5/2048 + 1001*xi^3/2048 - 33*xi/2048,
334305*xi^16/32768 - 185725*xi^14/4096 + 676039*xi^12/8192 - 323323*xi^10/4096 + 692835*xi^8/16384 - 51051*xi^6/4096 + 15015*xi^4/8192 - 429*xi^2/4096 + 33/32768,
570285*xi^17/32768 - 334305*xi^15/4096 + 1300075*xi^13/8192 - 676039*xi^11/4096 + 1616615*xi^9/16384 - 138567*xi^7/4096 + 51051*xi^5/8192 - 2145*xi^3/4096 + 429*xi/32768,
1964315*xi^18/65536 - 9694845*xi^16/65536 + 5014575*xi^14/16384 - 16900975*xi^12/49152 + 7436429*xi^10/32768 - 2909907*xi^8/32768 + 323323*xi^6/16384 - 36465*xi^4/16384 + 6435*xi^2/65536 - 143/196608,
3411705*xi^19/65536 - 17678835*xi^17/65536 + 9694845*xi^15/16384 - 11700675*xi^13/16384 + 16900975*xi^11/32768 - 7436429*xi^9/32768 + 969969*xi^7/16384 - 138567*xi^5/16384 + 36465*xi^3/65536 - 715*xi/65536,
23881935*xi^20/262144 - 64822395*xi^18/131072 + 300540195*xi^16/262144 - 48474225*xi^14/32768 + 152108775*xi^12/131072 - 37182145*xi^10/65536 + 22309287*xi^8/131072 - 969969*xi^6/32768 + 692835*xi^4/262144 - 12155*xi^2/131072 + 143/262144,
42077695*xi^21/262144 - 119409675*xi^19/131072 + 583401555*xi^17/262144 - 100180065*xi^15/32768 + 339319575*xi^13/131072 - 91265265*xi^11/65536 + 185910725*xi^9/393216 - 3187041*xi^7/32768 + 2909907*xi^5/262144 - 230945*xi^3/393216 + 2431*xi/262144,
149184555*xi^22/524288 - 883631595*xi^20/524288 + 2268783825*xi^18/524288 - 3305942145*xi^16/524288 + 1502700975*xi^14/262144 - 882230895*xi^12/262144 + 334639305*xi^10/262144 - 79676025*xi^8/262144 + 22309287*xi^6/524288 - 1616615*xi^4/524288 + 46189*xi^2/524288 - 221/524288,
265937685*xi^23/524288 - 1641030105*xi^21/524288 + 4418157975*xi^19/524288 - 6806351475*xi^17/524288 + 3305942145*xi^15/262144 - 2103781365*xi^13/262144 + 882230895*xi^11/262144 - 239028075*xi^9/262144 + 79676025*xi^7/524288 - 7436429*xi^5/524288 + 323323*xi^3/524288 - 4199*xi/524288,
3811773485*xi^24/4194304 - 6116566755*xi^22/1048576 + 34461632205*xi^20/2097152 - 27981667175*xi^18/1048576 + 115707975075*xi^16/4194304 - 9917826435*xi^14/524288 + 9116385915*xi^12/1048576 - 1386362835*xi^10/524288 + 2151252675*xi^8/4194304 - 185910725*xi^6/3145728 + 7436429*xi^4/2097152 - 88179*xi^2/1048576 + 4199/12582912,
6861192273*xi^25/4194304 - 11435320455*xi^23/1048576 + 67282234305*xi^21/2097152 - 57436053675*xi^19/1048576 + 251835004575*xi^17/4194304 - 23141595015*xi^15/524288 + 23141595015*xi^13/1048576 - 3907022535*xi^11/524288 + 6931814175*xi^9/4194304 - 239028075*xi^7/1048576 + 37182145*xi^5/2097152 - 676039*xi^3/1048576 + 29393*xi/4194304,
24805848987*xi^26/8388608 - 171529806825*xi^24/8388608 + 263012370465*xi^22/4194304 - 470975640135*xi^20/4194304 + 1091285019825*xi^18/8388608 - 856239015555*xi^16/8388608 + 115707975075*xi^14/2097152 - 42977247885*xi^12/2097152 + 42977247885*xi^10/8388608 - 6931814175*xi^8/8388608 + 334639305*xi^6/4194304 - 16900975*xi^4/4194304 + 676039*xi^2/8388608 - 2261/8388608,
135054066707*xi^27/25165824 - 322476036831*xi^25/8388608 + 514589420475*xi^23/4194304 - 964378691705*xi^21/4194304 + 2354878200675*xi^19/8388608 - 1964313035685*xi^17/8388608 + 285413005185*xi^15/2097152 - 115707975075*xi^13/2097152 + 128931743655*xi^11/8388608 - 23876248825*xi^9/8388608 + 1386362835*xi^7/4194304 - 91265265*xi^5/4194304 + 16900975*xi^3/25165824 - 52003*xi/8388608,
327988447717*xi^28/33554432 - 1215486600363*xi^26/16777216 + 8061900920775*xi^24/33554432 - 3945185556975*xi^22/8388608 + 20251952525805*xi^20/33554432 - 8948537162565*xi^18/16777216 + 11131107202215*xi^16/33554432 - 611599296825*xi^14/4194304 + 1504203675975*xi^12/33554432 - 157583242245*xi^10/16777216 + 42977247885*xi^8/33554432 - 882230895*xi^6/8388608 + 152108775*xi^4/33554432 - 1300075*xi^2/16777216 + 7429/33554432,
599427163069*xi^29/33554432 - 2295919134019*xi^27/16777216 + 15801325804719*xi^25/33554432 - 8061900920775*xi^23/8388608 + 43397041126725*xi^21/33554432 - 20251952525805*xi^19/16777216 + 26845611487695*xi^17/33554432 - 1590158171745*xi^15/4194304 + 4281195077775*xi^13/33554432 - 501401225325*xi^11/16777216 + 157583242245*xi^9/33554432 - 3907022535*xi^7/8388608 + 882230895*xi^5/33554432 - 11700675*xi^3/16777216 + 185725*xi/33554432,
6593698793759*xi^30/201326592 - 17383387729001*xi^28/67108864 + 61989816618513*xi^26/67108864 - 131677715039325*xi^24/67108864 + 185423721177825*xi^22/67108864 - 182267572732245*xi^20/67108864 + 128262365996765*xi^18/67108864 - 65196485041545*xi^16/67108864 + 23852372576175*xi^14/67108864 - 6183948445675*xi^12/67108864 + 1103082695715*xi^10/67108864 - 128931743655*xi^8/67108864 + 9116385915*xi^6/67108864 - 339319575*xi^4/67108864 + 5014575*xi^2/67108864 - 37145/201326592,
4041299260691*xi^31/67108864 - 32968493968795*xi^29/67108864 + 121683714103007*xi^27/67108864 - 268622538680223*xi^25/67108864 + 395033145117975*xi^23/67108864 - 407932186591215*xi^21/67108864 + 303779287887075*xi^19/67108864 - 164908756281555*xi^17/67108864 + 65196485041545*xi^15/67108864 - 18551845337025*xi^13/67108864 + 3710369067405*xi^11/67108864 - 501401225325*xi^9/67108864 + 42977247885*xi^7/67108864 - 2103781365*xi^5/67108864 + 48474225*xi^3/67108864 - 334305*xi/67108864,
238436656380769*xi^32/2147483648 - 125280277081421*xi^30/134217728 + 956086325095055*xi^28/268435456 - 1095153426927063*xi^26/134217728 + 6715563467005575*xi^24/536870912 - 1817152467542685*xi^22/134217728 + 2855525306138505*xi^20/268435456 - 824543781407775*xi^18/134217728 + 2803448856786435*xi^16/1073741824 - 108660808402575*xi^14/134217728 + 48234797876265*xi^12/268435456 - 3710369067405*xi^10/134217728 + 1504203675975*xi^8/536870912 - 23141595015*xi^6/134217728 + 1502700975*xi^4/268435456 - 9694845*xi^2/134217728 + 334305/2147483648,
1322239639929719*xi^33/6442450944 - 238436656380769*xi^31/134217728 + 1879204156221315*xi^29/268435456 - 6692604275665385*xi^27/402653184 + 14236994550051819*xi^25/536870912 - 4029338080203345*xi^23/134217728 + 6662892380989845*xi^21/268435456 - 2039660932956075*xi^19/134217728 + 7420894032669975*xi^17/1073741824 - 311494317420715*xi^15/134217728 + 152125131763605*xi^13/268435456 - 13154944875345*xi^11/134217728 + 6183948445675*xi^9/536870912 - 115707975075*xi^7/134217728 + 9917826435*xi^5/268435456 - 100180065*xi^3/134217728 + 9694845*xi/2147483648,
1633354849324947*xi^34/4294967296 - 14544636039226909*xi^32/4294967296 + 7391536347803839*xi^30/536870912 - 18165640176806045*xi^28/536870912 + 60233438480988465*xi^26/1073741824 - 71184972750259095*xi^24/1073741824 + 30891591948225645*xi^22/536870912 - 19988677142969535*xi^20/536870912 + 38753557726165425*xi^18/2147483648 - 14017244283932175*xi^16/2147483648 + 934482952262145*xi^14/536870912 - 179784246629715*xi^12/536870912 + 48234797876265*xi^10/1073741824 - 4281195077775*xi^8/1073741824 + 115707975075*xi^6/536870912 - 3305942145*xi^4/536870912 + 300540195*xi^2/4294967296 - 570285/4294967296,
3033373291603473*xi^35/4294967296 - 27767032438524099*xi^33/4294967296 + 14544636039226909*xi^31/536870912 - 36957681739019195*xi^29/536870912 + 127159481237642315*xi^27/1073741824 - 156606940050570009*xi^25/1073741824 + 71184972750259095*xi^23/536870912 - 48543930204354585*xi^21/536870912 + 99943385714847675*xi^19/2147483648 - 38753557726165425*xi^17/2147483648 + 2803448856786435*xi^15/536870912 - 594670969621365*xi^13/536870912 + 179784246629715*xi^11/1073741824 - 18551845337025*xi^9/1073741824 + 611599296825*xi^7/536870912 - 23141595015*xi^5/536870912 + 3305942145*xi^3/4294967296 - 17678835*xi/4294967296,
67745336845810897*xi^36/51539607552 - 106168065206121555*xi^34/8589934592 + 916312070471295267*xi^32/17179869184 - 450883717216034179*xi^30/3221225472 + 1071772770431556655*xi^28/4294967296 - 686661198683268501*xi^26/2147483648 + 1305057833754750075*xi^24/4294967296 - 233893481893708455*xi^22/1073741824 + 1019422534291446285*xi^20/8589934592 - 210991592064678425*xi^18/4294967296 + 131762096268962445*xi^16/8589934592 - 3822884804708775*xi^14/1073741824 + 2576907535025915*xi^12/4294967296 - 152125131763605*xi^10/2147483648 + 23852372576175*xi^8/4294967296 - 285413005185*xi^6/1073741824 + 115707975075*xi^4/17179869184 - 583401555*xi^2/8589934592 + 1964315/17179869184,
42111966147395963*xi^37/17179869184 - 203236010537432691*xi^35/8589934592 + 1804857108504066435*xi^33/17179869184 - 305437356823765089*xi^31/1073741824 + 2254418586080170895*xi^29/4294967296 - 1500481878604179317*xi^27/2147483648 + 2975531860960830171*xi^25/4294967296 - 559310500180607175*xi^23/1073741824 + 2572828300830793005*xi^21/8589934592 - 566345852384136825*xi^19/4294967296 + 379784865716421165*xi^17/8589934592 - 11978372388087495*xi^15/1073741824 + 8920064544320475*xi^13/4294967296 - 594670969621365*xi^11/2147483648 + 108660808402575*xi^9/4294967296 - 1590158171745*xi^7/1073741824 + 856239015555*xi^5/17179869184 - 6806351475*xi^3/8589934592 + 64822395*xi/17179869184,
157365768235005967*xi^38/34359738368 - 1558142747453650631*xi^36/34359738368 + 7113260368810144185*xi^34/34359738368 - 19853428193544730785*xi^32/34359738368 + 9468558061536717759*xi^30/8589934592 - 13075627799264991191*xi^28/8589934592 + 13504336907437613853*xi^26/8589934592 - 10626899503431536325*xi^24/8589934592 + 12864141504153965025*xi^22/17179869184 - 6003266035271850345*xi^20/17179869184 + 2152114239059719935*xi^18/17179869184 - 586940247016287255*xi^16/17179869184 + 59891861940437475*xi^14/8589934592 - 8920064544320475*xi^12/8589934592 + 934482952262145*xi^10/8589934592 - 65196485041545*xi^8/8589934592 + 11131107202215*xi^6/34359738368 - 251835004575*xi^4/34359738368 + 2268783825*xi^2/34359738368 - 3411705/34359738368,
883669313935033507*xi^39/103079215104 - 2989949596465113373*xi^37/34359738368 + 14023284727082855679*xi^35/34359738368 - 40308475423257483715*xi^33/34359738368 + 19853428193544730785*xi^31/8589934592 - 28405674184610153277*xi^29/8589934592 + 91529394594854938337*xi^27/25769803776 - 25079482828098425727*xi^25/8589934592 + 31880698510294608975*xi^23/17179869184 - 15722839616188179475*xi^21/17179869184 + 6003266035271850345*xi^19/17179869184 - 1760820741048861765*xi^17/17179869184 + 195646749005429085*xi^15/8589934592 - 32249464121774025*xi^13/8589934592 + 3822884804708775*xi^11/8589934592 - 311494317420715*xi^9/8589934592 + 65196485041545*xi^7/34359738368 - 1964313035685*xi^5/34359738368 + 27981667175*xi^3/34359738368 - 119409675*xi/34359738368

In order to calculate the displacement, strain of stress fields using Cython, the above output is not adequate due to very long integer numbers that will cause precision overflows. The code below should be used to create an input to Cython:

../../../../theory/func/bardell/bardell_floating_point.py¶

from __future__ import division
import numpy as np
from sympy import var, factorial, factorial2, sympify, diff

nmax = 30

xi = var('xi')

u = map(sympify, ['1./2. - 3./4.*xi + 1./4.*xi**3',
                  '1./8. - 1./8.*xi - 1./8.*xi**2 + 1./8.*xi**3',
                  '1./2. + 3./4.*xi - 1./4.*xi**3',
                  '-1./8. - 1./8.*xi + 1./8.*xi**2 + 1./8.*xi**3'])

for r in range(5, nmax+1):
    utmp = []
    for n in range(0, r//2+1):
        den = 2**n*factorial(n)*factorial(r-2*n-1)
        utmp.append((-1)**n*factorial2(2*r - 2*n - 7)/den * xi**(r-2*n-1)/1.)
    u.append(sum(utmp))

with open('bardell_floating_point.txt', 'w') as f:
    f.write("Bardell's hierarchical functions\n\n")
    f.write('Number of terms: {0}\n\n'.format(len(u)))
    f.write('cdef void calc_vec_f(double *f, double xi, double xi1t=1., double xi1r=1.,\n' +
            '                 double xi2t=1., double xi2r=1.) nogil:\n')
    consts = {0:'xi1t', 1:'xi1r', 2:'xi2t', 3:'xi2r'}
    for i in range(len(u)):
        const = consts.get(i)
        if const is None:
            f.write('    f[%d] = %s\n' % (i, str(u[i])))
        else:
            f.write('    f[%d] = %s*(%s)\n' % (i, const, str(u[i])))

    f.write('\n\n')
    f.write('cdef void calc_vec_fxi(double *fxi, double xi, double xi1t=1., double xi1r=1.,\n' +
            '                       double xi2t=1., double xi2r=1.) nogil:\n')
    for i in range(len(u)):
        const = consts.get(i)
        if const is None:
            f.write('    fxi[%d] = %s\n' % (i, str(diff(u[i], xi))))
        else:
            f.write('    fxi[%d] = %s*(%s)\n' % (i, const, str(diff(u[i], xi))))

    f.write('\n\n')
    f.write('cdef double calc_f(int i, double xi, double xi1t, double xi1r,\n' +
            '                   double xi2t, double xi2r) nogil:\n')
    for i in range(len(u)):
        const = consts.get(i)
        f.write('    if i == %d:\n' % i)
        if const is None:
            f.write('        return %s\n' % str(u[i]))
        else:
            f.write('        return %s*(%s)\n' % (const, str(u[i])))

    f.write('\n\n')
    f.write('cdef double calc_fxi(int i, double xi, double xi1t, double xi1r,\n' +
            '                     double xi2t, double xi2r) nogil:\n')
    for i in range(len(u)):
        const = consts.get(i)
        f.write('    if i == %d:\n' % i)
        if const is None:
            f.write('        return %s\n' % str(diff(u[i], xi)))
        else:
            f.write('        return %s*(%s)\n' % (const, str(diff(u[i], xi))))