Int J Adv Manuf Technol (2006) 30: 40–44 DOI 10.1007/s00170-005-0028-7
ORIGINA L ARTI CLE
Yuanyin Li . Xianlong Jin . Lijun Li . Yuan Cao
A parallel and integrated system for structural dynamic response analysis
Received: 2 July 2002 / Accepted: 11 March 2005 / Published online: 21 September 2005 # Springer-Verlag London Limited 2005
Abstract A new method, to parallel the dominating computing work of structural dynamic response analysis, is presented in this paper. Then, based on “ShenWei I” supercomputer, MSC.NASTRAN and PATRAN, a parallel and integrated system is implemented. The system tightly couples advantages of the FEA code and high performance of the supercomputer. The system can improve the structural scale and velocity of dynamic response analysis greatly. Finally two examples are presented to demonstrate the validity and efficiency of the system.
parallel FEA code is based on secondary development of commercially serial FEA code, MSC.NASTRAN. The remainder of the paper is organized as follows. Section 2 describes the dominating computing work of dynamic response analysis. Section 3 describes algorithm of the parallel dynamic response analysis. Section 4 describes the development and integration of system. In Section 5, two examples are given to demonstrate the system. Section 6 contains concluding remarks.
Keywords Dynamic response . Integration . Parallel
2 The dominating computing work of dynamic response analysis
1 Introduction
The equation of motion of a discretized structural model can be written as
The scale and complexity of structure dynamic response analysis problems are ever increasing. They request more memory and computing time than static analysis. So it is very necessary to make structural dynamic response analysis paralleled. On the other hand, the demands of analysis velocity and precision are well beyond the serial computer and finite element codes. Thus, as larger and faster parallel computers become more widely available, the use of parallel finite element codes is becoming increasingly attractive. By contrast with the quick development of massively parallel machines and parallel algorithms, the technology of commercial general-purpose parallel finite element analysis (FEA) codes lags badly. FEA and parallel technology are limited in vibration engineering field. It needs a great deal of intelligence and funds to develop parallel FEA codes from the beginning. This paper provides a new method to expand the scale and heighten velocity of dynamic response analysis. That is to parallel the dominating computing work of structural dynamic response analysis. The Y. Li (*) . X. Jin . L. Li . Y. Cao High Performance Computing Center, Shanghai JiaoTong University, Shanghai, China e-mail:
[email protected]
::
:
Mx ðt Þ þ Cxðt Þ þ KxðtÞ ¼ f ðtÞ
(1)
where M is system mass matrix, C ::is system damping matrix, :K is system stiffness matrix, xðt Þ is acceleration vector, xðt Þ is velocity vector, x(t) is displacement vector, f(t) is the known external force vector. The whole finite elements analysis process can be divided into steps as follows: (1) Generation of data files, including dissipation by FE mesh and information processing of nodes and elements. (2) Element analysis, including element stiffness matrix, damping matrix, and mass matrix. (3) System matrix assembly, including M, C, K. (4) Boundary constraints elimination, which modifies system equation. (5) System equation solution, which gets the results of all displacement of nodes. (6) Post-processing, dealing with strain and stress of each element. If the problem is static analysis, step (5) is 70% of the whole process. If dynamic analysis, for solving the equa-
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tion for many times, the computing work of step (5) is beyond 90% of the whole process [1]. So the parallel dynamic response analysis does not touch the other steps, only step (5) is paralleled to increase the scale and velocity.
The second order central difference method is one of the most widely used among explicit (numerical integration) techniques in large scale structural dynamics programs. It is based on the following difference formulas:
2ðtÞ2
(2)
:
5. Form effective mass matrix M ¼ a0 M þ a1 C ^
^
6. Triangularize M : M ¼ LDLT :
ðxtþt 2xt þ xtt Þ
(3) (2) For each time step:
:
::
Substituting the relations for xt and xt from (2) and (3) respectively, into (1), and rearranging terms, we obtain a fully discrete temporal system 1 1 2 C xtþt ¼ ft K 2 M xt Mþ t 2 2t t (4) 1 1 C xtt M t 2 2t from which we can solve for xtþt . To calculate the solution at time Δt, a : special starting :: procedure must be employed. :Since x0, x0:: and x0 are known (note that with x0 and x0 known, x0 can be calculated using (1) at time t=0), the relations in (2) and (3) can be used to obtain xtt . By substituting for xtþt in (3) from (2) and rearranging the terms, we have for xtt , :
xtt ¼ xt txt þ
t2 :: xt 2
(5)
t 2 : x0 ¼ x0 tx0 þ 2 :
1. Calculate effective loads at time t: :
f t ¼ ft ðK a2 M Þxt ða0 M a1 C Þxtt 2. Solve for displacements at time t+Δt: ^
LDLT xtþt ¼ f t : 3. If required, evaluate velocities and accelerations at time t: :
xt ¼ a1 ðxtþt xtt Þ; ::
xt ¼ a0 ðxtþt 2xt þ xtt Þ:
3.2 Parallel computing ^
For t=0, (5) becomes xtt
:
xtt ¼ x0 tx0 þ a3 x0 :
^
1 xt ¼ ðxtþt xtt Þ 2t :
1
1 1 ; ; a1 ¼ 2 t 2t
4. Calculate
3.1 Second order central difference method [2]
::
a0 ¼
a2 ¼ 2a0 ; a3 ¼ 1=a2 :
3 The parallel dynamic response analysis
xt ¼
3. Select time step Δt, Δt
