An Uncoupling Strategy in the Newmark Method for Dynamic Problems

Abstract

When the semidiscrete formulation of the finite element method (FEM) is employed in traditional elastodynamic problems, a system of ordinary differential equations (ODEs) is obtained. The present paper focuses on the development of a numerical strategy to decouple the resulting system by means of the implicit unconditionally stable Newmark method, allowing the parts to be solved independently, and through an iterative procedure, managing to preserve the stability and accuracy properties of the original method. It is observed that only one iteration is sufficient to achieve the same level of accuracy of the solution of the fully coupled system, rendering a very efficient algorithm. The accuracy and potentialities of the proposed decoupling strategy will be studied through the solution of two 2D structural dynamic problems that present materials with functionally graded properties.

Appendix: Coupling Parameter

This appendix presents the necessary assumptions to obtain the expressions employed in Sect. 4. Thus, supposing a regular uniform mesh (leading to $N_{i,1}=N_{i,2}$) and same nodal displacement constraints in each direction (leading to $nq_1=nq_2$) with material properties governed by functions in the spatial domain, as well as FGM models, and recalling that

$$\begin{aligned} c_{1}=\lambda +2\mu = \frac{E(1-\nu )}{(1+\nu )(1-2\nu )},\;\; c_{2}=\lambda =\nu c_{1},\;\; c_{3}=\mu =c_{1}\frac{(1-2\nu )}{2(1-\nu )} \end{aligned}$$

(23)

from (13) and (15) we have

$$\begin{aligned} K^{11}_{ij}=\int _{\varOmega }hN_{i,1}N_{j,1}(c_{1}+c_{3})dxdy; K^{22}_{ij}=\int _{\varOmega }hN_{i,1}N_{j,1}(c_{3}+c_{1})dxdy=K^{11}_{ij} \end{aligned}$$

(24)

Furthermore, (14) can be rewritten (after proper use of the Integral Mean Value Theorem) as follows:

$$\begin{aligned} K^{12}_{ij}=K^{21}_{ji}=&\int _{\varOmega }h(\nu c_{1}N_{i,1}N_{j,2}+c_{3}N_{i,2}N_{j,1})dxdy\end{aligned}$$

(25)

$$\begin{aligned} =&\int _{\varOmega }hN_{i,1}N_{j,1}(\nu c_{1}+c_{3})dxdy\end{aligned}$$

(26)

$$\begin{aligned} =&\int _{\varOmega }hN_{i,1}N_{j,1}(\nu c_{1}+c_{3})\frac{(c_{1}+c_{3})}{(c_{1}+c_{3})}dxdy\end{aligned}$$

(27)

$$\begin{aligned} =&\sum \limits _e \left. \frac{( vc_1 + c_3 )}{( c_1 + c_3 )}\right| _{\mathbf {\xi } \in \varOmega _e}\int _{\varOmega _e} hN_{i,1}N_{j,1}( c_1 + c_3 )dxdy \end{aligned}$$

(28)

$$\begin{aligned} =&\sum \limits _e \alpha _e\int _{\varOmega _e} hN_{i,1}N_{j,1}( c_1 + c_3 )dxdy \end{aligned}$$

(29)

where $\alpha _e=\frac{(\nu c_{1}+c_{3})}{(c_{1}+c_{3})}=\frac{1-2\nu ^{2}}{3-4\nu }$ plays a role of a parameter which varies as a function of the Poisson’s ratio with $\nu \in [0,0.5]$ in common cases. Since we are interested in the variation of such a parameter over the whole range, one can assume $K^{12}_{ij}=K^{21}_{ji}=\alpha K^{11}_{ij}$.