Dynamic relaxation method based on Lanczos algorithm

Abstract

This paper tries to accelerate the convergence rate of the general viscous dynamic relaxation method. For this purpose, a new automated procedure for estimating the critical damping factor is developed by employing a simple variant of the Lanczos algorithm, which does not require any re-orthogonalization process. All of the computational operations are performed by simple vector-matrix multiplication without requiring any matrix factorization or inversion. Some numerical examples with geometric nonlinear behavior are analyzed by the proposed algorithm. Results show that the suggested procedure could effectively decrease the total number of convergence iterations compared with the conventional dynamic relaxation algorithms.A. H. NAMADCHI AND J. ALAMATIAN time integration process. Large amount of research has been directed toward both improving the DR parameters and application of the DR method in various engineering problems. Otter employed DR to analyze axially symmetric pressure vessels and a three-dimensional elastic arch dam using central finite differences [5,6]. Cassel investigated the application of DR technique in analyze of cylindrical shells [7]. Rushton proposed a systematic procedure to tackle rectangular plate problems including small and large deflection of elastic plates as well as the buckling behavior [8][9][10]. In order to damp out the artificial oscillations as quick as possible, Rushton suggested that the critical damping of fundamental mode of vibration is selected as the fictitious damping factor, and an automated process for estimating this factor was also introduced [8]. Considering DR as a fictitious time-marching scheme, Brew and Brotton investigated DR characteristics based on the concept of amplification matrix and employed DR to analyze frame structures [11]. Pseudo-transient analysis of linear and geometric nonlinear Mindlin plates using finite element method was successfully carried out by adopting a specific form of DR. Some empirically based suggestions regarding time-step selection and fictitious mass matrix were also reported [12]. By assuming a load vector proportional to diagonal entries of the artificial mass matrix, which is formulated using Gershgorin's circle theorem, Bunce proposed a new technique to determine critical damping in DR algorithm based on the Rayleigh's principle [13]. In another study, Papadrakakis evaluated DR properties by transforming the fundamental difference equations into the error space, which leads to an eigenvalue problem. By examining the resultant characteristic equation, a new automated procedure for estimating DR fictitious parameters was proposed that improves the convergence rate of the method [3]. Zienkiewicz and Löhner studied the concept of viscous relaxation in which the inertial term is removed from the virtual dynamic system. They then utilized a line search technique to accelerate the convergence [14]. Instead of applying the external load in an incremental form, which is conventional in nonlinear quasi-static analyses, a novel techniq...

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رابطه‌سازی نوین برای روش رهایی پویای جنبشی بر پایه ی درون‌یابی لاگرانژ

“…@U |]NQ}e p=Ut pL |= Q@ QFwt |}xw}W x@ = Q u; 'QwmPt VwQ QO G}=Dv |Q=O}=B w "CU= xOQm p}O@D u}vJty w nQR@ 'lJwm |=ypmWQ}}eD |= Q=O |r}]DUt |=yxLiY p}rLD |}=yQ VwQ 'xt=O= QO [7 w 6] "CU=}wB |}=yQ VwQ |=yOQ@Q=m Qo}O R= |Wv=tm Q=DiQ =]N |@=} RQ= [8] "CU= xOW |iQat '|}= Q}t OwHw sOa |=v@t Q@ '|W@vH |=}wB |}x]@=Q u; pL =@ xm 'OW QHvt xS}w Q=Okt |xrUt l} x@ =} wB |}=yQ VwQ QO '|W@vH |x}=tQ=m C= Q}}eD |UQQ@ =@ 'u}vJty [9] "Ot; CUO x@ |oDN=U sQH |= Q@ =@ u= QoWywSB 'Qo}O |wU R= [10] "Ot; CUO x@ |oDN=U |v=tR s=o |= Q@ |}x]@=Q |wk Q= R@= l} '=}wB |}=yQ VwQ xm OvOQm C@=F =yuwDU Vv=tm |xrUt |UQQ@ u}vJty [11] "CU= |]NQ}e COW x@ swO xHQO |r}Uv=Qi}O |xrO=at pL |= Q@ |=yVwQ \UwD 'nQR@ |=ypmWQ}}eD =@ |WtN C=LiY p}rLD |wQ Q@ |}xU}=kt [12] "CiQo CQwY =} wB |}=yQ hrDNt VwQ 'u}vJty [13] "OW K}LYD |oDN=U |v=tR s=o '|}= Qosy MQv V}=Ri= |= Q@ |wU R= [14] "CU= xDiQ Q=m x@ K} QY |OOa |m}t=v}O |=yp}rLD |= Q@ =} wB |}=yQ C=@U=Lt =@ =} wB |}=yQ VwQ =@ 'uQ=kDt=v w uQ=kDt |=yxR=U |Wv=tm Q=@ 'Qo}O R= xR=U |}=DU}= Q}Ut QO |DWoR=@ |=yx}L=v V}=t}B [15] "O};|t CUO x@ |Q=OQ@ w |}=Q}t T} QD=t x@ R=}v uwO@ u} wv |WwQ [16] "CU= QwmPt VwQ |=y|}=v=wD Qo}O Q=DiQ w Q}eDt |=yQ=@ QO VwQ u}= xm 'OW `=O@= pwtat =} wB |}=yQ |x}=B Q@ CaQU u=wD|t '=yu}= Q@ uwRi= [17] "CW=Po V}=tv x@ |QFwt OQmrta 'xv=t=U O=}R |]NQ}e xDmv u}= xm 'OQm = QH= |m}i=Qo |=yxOvR=OQB |wQ Q@ = Q =} wB |}=yQ sD} Qwor= |xDUy [18] "OwW|t T=}ktnQR@ |=ypOt |= Q@ QwmPt VwQ QD`} QU OQmrta ? @U uwUi= Q-uDw}v VwQ '2017 u= Q=mty w 10 TDw@twQ VywSB QO 'Qo}O |wU R= R}v '2017 u= Q=mty w |ra VywSB QO [19] "CU= xOW xU}=kt =} wB |}=yQ ui =@ K}LYD =} wB |}=yQ ltm x@ |r@=m |=yu=tr= l=m]Y= uwO@ CmQL |ak=wQ}e ZQi 11 RwUmvq sD} Qwor= R= xO=iDU= =@ =} wB |}=yQ VwQ |}=Qosy MQv 'u}vJty [20] "OW Vy=m Qw_vt x@ |WwQ '|r=wDt s=o wO QO =]N MQv |R=Uxv}tm =@ [21] "Ci=} Ow@y@ O}=@ [22] "OW |UQQ@ GRr |=}wB |}=yQ VwQ |= Q@ '|OOa |Q=O}=B w p}rLD u=tR…”

“…2 min ?U=vt u}tND =@ |v=tR s=o C@Uv 'CU= xOW xDiQo Q_v QO 2 n ! 2 min "O};|t CUO x@ 21 |xrO=at pL |Q=mx= Q xm [25] 'OQ}PB|t s=Hv= 14 xDi=}p=kDv= |v=wD |=yQ= QmD VwQ =@ Q=m u}= "CU= |m}t=v}O x=oDUO |xS}w |=yQ=OQ@ w =yQ=Okt |x@U=Lt |= Q@ Ovtv=wD w xO=U ?Q[ |xO=U C=}rta w x=wNrO u}DUNv Q=OQ@ l} ?=NDv= =@ 'Qw@Rt sD} Qwor= [21] "OyO|t x=Q= = Q xv=t=U |xS}w Q=Okt u} QDnQR@ R= |@U=vt Q=}U@ u}tND '|Q=OQ@ xO=iDU= |oDN=U sQH T} QD=t |}=B Q@ |= Q@ u} QwoWQo |xQov R= xmu}= x@ xHwD =@ O}=@ "OQm Oy=wNv Rw=HD 2(1+ p ) 2 (1+ ) R= xv=t=U |xS}w Q=Okt u} QDnQR@ 'CU= xOW xS}w |=yQ=Okt u}tND |= Q@ |@} QkD |xw}W 'u}QwoWQo |=yxQ}=O |xQov xm CUv=O |xS}w Q=Okt u} QDlJwm w u} QDnQR@ u}tND 'QwmPt VwQ R= xO=iDU= =@ "CU= uO=O Q= Qk Q@=Q@ =@ 'xt=O= QO "O};|t CUO x@ |oO=U x@ '|R=Ht |m}t=v}O |xv=t=U |a}@] Ot=U@ |xv}W}@ Q=Okt =@ xDi=}p=kDv= |v=wD |=yQ= QmD VwQ QO p=kDv= pt=a Q=Okt u} QDtm |wUx@ |}=Qosy x@ = Q |v=wD |=yQ= QmD u=wD|t '|oDN=U |xv=t=U x} RHD |oDN=U sQH T} QD=t CUNv 'Q=m u}= |= Q@ [21] "O=O jwU xv=t=U |xS}w %OwW|t…”