Processing math: 100%

FormLab_formfinding

1553 days ago by esamec@grad.hr


Metoda gustoća sila

(Force Density Method)


[Ćelije u kojima je prvi redak %auto ne trebate izvoditi ( evaluate ); one se izvode „same” pri otvaranju radnoga lista.  Međutim, ako radni list ponovno pokrenete—ne izlazeći iz njega—s  Restart worksheet  (u izborniku  Action... ), i njih ćete morati izvesti.]
       

Proračun metodom gustoća sila:

ncoords = FDM (nodes, elems, supports, qs)

parametri:

nodes — lista koordinata čvorova; zadane moraju biti koordinate ležajnih čvorova

elems — lista štapova određenih indeksima početnoga i krajnjeg čvora

supports — lista indeksa ležajnih čvorova

qs — lista gustoća sila u štapovima

rezultat:

ncoords — matrica tipa nˇc×3 koja sadrži koordinate čvorova upisane po redcima

Čvorovi:
       

Elementi:
       

Ležajevi:
       

Gustoće sila:
       
       


 

Pronalazak oblika:
       

Vizualizacija

pp= plot3d_mesh (ncoords, elems, node_color = 'green', elem_color = 'orange')
       
        
(5.830044826445895, 1.0276728616870547, 14.43206203658118)
(5.830044826445896, 27.97232713831294, 5.9803703053964306)
(1.0276728616870552, 23.1699551735541, 3.9931819061858764)
(27.97232713831294, 23.1699551735541, 23.594385751836512)
(1.5293849349891295, 17.3856339390173, 7.606110639561248)
(27.470615065010865, 17.3856339390173, 17.587944783480395)
(1.5293849349891293, 11.614366060982695, 11.026738299054944)
(27.47061506501086, 11.614366060982695, 11.779206277903414)
(1.0276728616870547, 5.830044826445895, 14.432062036581181)
(27.972327138312938, 5.830044826445895, 5.9803703053964306)
(23.1699551735541, 1.027672861687055, 3.993181906185875)
(23.1699551735541, 27.97232713831294, 23.594385751836512)
(17.385633939017296, 1.529384934989129, 7.606110639561246)
(17.3856339390173, 27.47061506501087, 17.587944783480395)
(11.614366060982693, 1.5293849349891289, 11.026738299054943)
(11.614366060982695, 27.47061506501087, 11.779206277903413)
(0.0, 29.0, 0.0)
(29.0, 29.0, 30.0)
(0.0, 0.0, 18.0)
(29.0, 0.0, 0.0)
(6.2872807455368624, 22.712719254463128, 7.795713634290924)
(6.546505668009872, 17.255100373995358, 9.529121378377988)
(6.546505668009871, 11.744899626004633, 11.179777518729544)
(6.287280745536862, 6.287280745536862, 12.805919777655362)
(22.712719254463128, 22.712719254463128, 19.60265295376278)
(17.255100373995358, 22.453494331990115, 15.61092015568906)
(11.744899626004635, 22.45349433199012, 11.680180947203407)
(22.453494331990115, 17.255100373995354, 15.610920155689055)
(17.177148676496365, 17.177148676496365, 13.572901938309647)
(11.822851323503626, 17.177148676496365, 11.534883720930234)
(22.453494331990115, 11.744899626004633, 11.680180947203407)
(17.177148676496362, 11.822851323503624, 11.53488372093023)
(11.822851323503622, 11.822851323503624, 11.357330619829888)
(22.712719254463124, 6.2872807455368624, 7.795713634290922)
(17.255100373995354, 6.546505668009871, 9.529121378377985)
(11.74489962600463, 6.546505668009871, 11.179777518729544)
(5.830044826445895, 1.0276728616870547, 14.43206203658118)
(5.830044826445896, 27.97232713831294, 5.9803703053964306)
(1.0276728616870552, 23.1699551735541, 3.9931819061858764)
(27.97232713831294, 23.1699551735541, 23.594385751836512)
(1.5293849349891295, 17.3856339390173, 7.606110639561248)
(27.470615065010865, 17.3856339390173, 17.587944783480395)
(1.5293849349891293, 11.614366060982695, 11.026738299054944)
(27.47061506501086, 11.614366060982695, 11.779206277903414)
(1.0276728616870547, 5.830044826445895, 14.432062036581181)
(27.972327138312938, 5.830044826445895, 5.9803703053964306)
(23.1699551735541, 1.027672861687055, 3.993181906185875)
(23.1699551735541, 27.97232713831294, 23.594385751836512)
(17.385633939017296, 1.529384934989129, 7.606110639561246)
(17.3856339390173, 27.47061506501087, 17.587944783480395)
(11.614366060982693, 1.5293849349891289, 11.026738299054943)
(11.614366060982695, 27.47061506501087, 11.779206277903413)
(0.0, 29.0, 0.0)
(29.0, 29.0, 30.0)
(0.0, 0.0, 18.0)
(29.0, 0.0, 0.0)
(6.2872807455368624, 22.712719254463128, 7.795713634290924)
(6.546505668009872, 17.255100373995358, 9.529121378377988)
(6.546505668009871, 11.744899626004633, 11.179777518729544)
(6.287280745536862, 6.287280745536862, 12.805919777655362)
(22.712719254463128, 22.712719254463128, 19.60265295376278)
(17.255100373995358, 22.453494331990115, 15.61092015568906)
(11.744899626004635, 22.45349433199012, 11.680180947203407)
(22.453494331990115, 17.255100373995354, 15.610920155689055)
(17.177148676496365, 17.177148676496365, 13.572901938309647)
(11.822851323503626, 17.177148676496365, 11.534883720930234)
(22.453494331990115, 11.744899626004633, 11.680180947203407)
(17.177148676496362, 11.822851323503624, 11.53488372093023)
(11.822851323503622, 11.822851323503624, 11.357330619829888)
(22.712719254463124, 6.2872807455368624, 7.795713634290922)
(17.255100373995354, 6.546505668009871, 9.529121378377985)
(11.74489962600463, 6.546505668009871, 11.179777518729544)

 

Iteracijska primjena metode gustoća sila odabrani broj puta:

ncoords, forces, f_dens = multistepFDM (nodes, elems, supports, qs, fcs = [], lcs = [], l0cs = [], steps = 250)

parametri:

nodes — lista koordinata čvorova; zadane moraju biti koordinate ležajnih čvorova

elems — lista štapova određenih indeksima početnoga i krajnjeg čvora

supports — lista indeksa ležajnih čvorova

qs — lista gustoća sila u štapovima za prvi korak

fcs — lista zadanih vrijednosti sila u štapovima

komponente liste: parovi (idx, f); idx – indeks štapa, f – vrijednost sile

lcs — lista zadanih duljina štapova

komponente liste: parovi (idx, l); idx – indeks štapa, l – duljina

l0cs — lista zadanih nerastegnutih duljina štapova

komponente liste: parovi (idx, (l0, AE)); idx – indeks štapa, l0 – duljina, AE – uzdužna krutost 

steps — broj koraka

rezultati:

ncoords — matrica tipa nˇc×3 koja sadrži po redcima svrstane koordinate čvorova

forces — lista konačnih vrijednosti sila u štapovima

f_dens — lista gustoća sila u štapovima izračunanih u posljednjem koraku
       
       
        
[5.502632679658823,
 4.685732556345492,
 4.504939966798522,
 5.0364098068922205,
 6.358285140174391,
 6.080666229751414,
 4.860255799450472,
 4.5789595701377355,
 5.250511175564079,
 7.165581559780133,
 4.979892874392657,
 4.720524269753153,
 4.687127674981986,
 5.064447005778335,
 5.641271603873372,
 4.828079272088791,
 4.655096968499492,
 4.647375523738498,
 4.978809778768222,
 5.42500297326895,
 5.502632679658829,
 4.685732556345496,
 4.504939966798528,
 5.036409806892217,
 6.358285140174388,
 6.080666229751416,
 4.860255799450471,
 4.578959570137731,
 5.250511175564079,
 7.165581559780131,
 4.979892874392657,
 4.72052426975315,
 4.687127674981986,
 5.064447005778336,
 5.641271603873369,
 4.828079272088788,
 4.655096968499486,
 4.647375523738491,
 4.978809778768224,
 5.42500297326895,
 70.20490026780273,
 69.65217511245262,
 69.6258018277385,
 69.55889895380508,
 70.17359141822546,
 70.1832643371487,
 69.74903345110548,
 69.7228636665183,
 69.68826771771762,
 70.17754428626347,
 70.18326433714871,
 69.74903345110549,
 69.7228636665183,
 69.68826771771764,
 70.17754428626344,
 70.20490026780276,
 69.6521751124527,
 69.62580182773843,
 69.55889895380504,
 70.17359141822541]
[5.502632679658823,
 4.685732556345492,
 4.504939966798522,
 5.0364098068922205,
 6.358285140174391,
 6.080666229751414,
 4.860255799450472,
 4.5789595701377355,
 5.250511175564079,
 7.165581559780133,
 4.979892874392657,
 4.720524269753153,
 4.687127674981986,
 5.064447005778335,
 5.641271603873372,
 4.828079272088791,
 4.655096968499492,
 4.647375523738498,
 4.978809778768222,
 5.42500297326895,
 5.502632679658829,
 4.685732556345496,
 4.504939966798528,
 5.036409806892217,
 6.358285140174388,
 6.080666229751416,
 4.860255799450471,
 4.578959570137731,
 5.250511175564079,
 7.165581559780131,
 4.979892874392657,
 4.72052426975315,
 4.687127674981986,
 5.064447005778336,
 5.641271603873369,
 4.828079272088788,
 4.655096968499486,
 4.647375523738491,
 4.978809778768224,
 5.42500297326895,
 70.20490026780273,
 69.65217511245262,
 69.6258018277385,
 69.55889895380508,
 70.17359141822546,
 70.1832643371487,
 69.74903345110548,
 69.7228636665183,
 69.68826771771762,
 70.17754428626347,
 70.18326433714871,
 69.74903345110549,
 69.7228636665183,
 69.68826771771764,
 70.17754428626344,
 70.20490026780276,
 69.6521751124527,
 69.62580182773843,
 69.55889895380504,
 70.17359141822541]
       

Klizanje kabela!
       
        
[5.657102469521567,
 4.477805027422682,
 4.2798351540031305,
 4.776679970774914,
 6.52508380386636,
 6.267404886191404,
 4.592656406115291,
 4.299406435991288,
 4.91622747301532,
 7.347709677431302,
 5.022425240797385,
 4.6160106013205775,
 4.538266785790156,
 4.941749333406757,
 5.679167184307712,
 4.865148056340795,
 4.563210125272514,
 4.515234841403726,
 4.871541773700753,
 5.463033587180715,
 5.657102469521566,
 4.47780502742268,
 4.27983515400314,
 4.776679970774913,
 6.525083803866353,
 6.267404886191401,
 4.5926564061152915,
 4.299406435991294,
 4.916227473015328,
 7.347709677431301,
 5.022425240797388,
 4.6160106013205775,
 4.53826678579016,
 4.941749333406767,
 5.679167184307712,
 4.865148056340796,
 4.563210125272515,
 4.515234841403725,
 4.871541773700755,
 5.463033587180718,
 70.0026138228571,
 69.9839768945595,
 70.01700300644225,
 69.98133999447415,
 70.00218587187477,
 70.00301334962715,
 69.98700837851014,
 70.01230820807545,
 69.9854608666972,
 70.00261278274425,
 70.00301334962708,
 69.98700837851011,
 70.01230820807547,
 69.98546086669725,
 70.00261278274427,
 70.00261382285707,
 69.9839768945595,
 70.01700300644218,
 69.98133999447418,
 70.00218587187484]
[5.657102469521567,
 4.477805027422682,
 4.2798351540031305,
 4.776679970774914,
 6.52508380386636,
 6.267404886191404,
 4.592656406115291,
 4.299406435991288,
 4.91622747301532,
 7.347709677431302,
 5.022425240797385,
 4.6160106013205775,
 4.538266785790156,
 4.941749333406757,
 5.679167184307712,
 4.865148056340795,
 4.563210125272514,
 4.515234841403726,
 4.871541773700753,
 5.463033587180715,
 5.657102469521566,
 4.47780502742268,
 4.27983515400314,
 4.776679970774913,
 6.525083803866353,
 6.267404886191401,
 4.5926564061152915,
 4.299406435991294,
 4.916227473015328,
 7.347709677431301,
 5.022425240797388,
 4.6160106013205775,
 4.53826678579016,
 4.941749333406767,
 5.679167184307712,
 4.865148056340796,
 4.563210125272515,
 4.515234841403725,
 4.871541773700755,
 5.463033587180718,
 70.0026138228571,
 69.9839768945595,
 70.01700300644225,
 69.98133999447415,
 70.00218587187477,
 70.00301334962715,
 69.98700837851014,
 70.01230820807545,
 69.9854608666972,
 70.00261278274425,
 70.00301334962708,
 69.98700837851011,
 70.01230820807547,
 69.98546086669725,
 70.00261278274427,
 70.00261382285707,
 69.9839768945595,
 70.01700300644218,
 69.98133999447418,
 70.00218587187484]
       

 

• iteracijska primjena metode gustoća sila do postizanja zadane točnosti:

ncoords, forces = multistepFDM_wtol (nodes, elems, supports, qs, fcs = [], lcs = [], l0cs = [], tol_f = 1e-3, tol_l = 1e-3, abs_error = True, steps = 250)

parametri:

nodes — lista koordinata čvorova; zadane moraju biti koordinate ležajnih čvorova

elems — lista štapova određenih indeksima početnoga i krajnjeg čvora

supports — lista indeksa ležajnih čvorova

qs — lista gustoća sila u štapovima za prvi korak

fcs — lista zadanih vrijednosti sila u štapovima

komponente liste: parovi (idx, f); idx – indeks štapa, f – vrijednost sile

lcs — lista zadanih duljina štapova

komponente liste: parovi (idx, l); idx – indeks štapa, l – duljina

l0cs — lista zadanih nerastegnutih duljina štapova

komponente liste: parovi (idx, (l0, AE)); idx – indeks štapa, l0 – duljina, AE – uzdužna krutost 

tol_f — točnost za vrijednosti sila

tol_l — točnost za duljine

abs_error — ako je True, izračunava se apsolutna, a ako je False, relativna pogreška

primjerice, za vrijednosti sila:  eabs=‖S(k){i,j}−ˉS{i,j}‖max  i  erel=‖S(k){i,j}−ˉS{i,j}‖maxˉS{i,j}

steps — broj koraka

rezultati:

ncoords — matrica tipa nˇc×3 koja sadrži po redcima svrstane koordinate čvorova

forces — lista konačnih vrijednosti sila u štapovima
        
WARNING: maximal number of steps (250) reached !
maximal force error:  0.010641154709929879
WARNING: maximal number of steps (250) reached !
maximal force error:  0.010641154709929879


 

 

 
        
steps: 450
maximal force error:  0.0009986706496647457
steps: 450
maximal force error:  0.0009986706496647457
        
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