5.1. Solid Plate Numerical Example
A numerical example is proposed to validate the method: the dynamic performance of a solid plate.
The solid plate is composed of two independent parts of the same size:
. One side of the solid plate is fixed, and on the opposite side, there is a loading point and a response point, as shown in
Figure 2. The two parts are directly connected on the interface.
The external force acting on the loading point lasts for 0.1 s:
The FE model is shown in
Figure 3. The two FE substructures with coincident interface nodes are tied by a multi-point constraint (MPC).
First, to implement the interpolation function, the assembly of the IBS method is discussed. To implement the IBS method, the IRFs matrix is obtained using FEM (shown in
Figure 4). The IRF matrix of Substructure 1 is obtained under the free boundary condition, and Substructure 2 is one-side fixed.
To validate the method, the support nodes on the interface are purposely chosen differently from each grid. In Substructure 1, only 18 points are extracted on the interface, and there are 34 interface points in Substructure 2. Each point has three translational DOFs. The interface has a size of
and the distribution of the interface support nodes of the substructures is shown in
Figure 5, yellow circles represent for interface node.
In this configuration, we set the intermediate surface nodes to coincide with Interface 2.
Then,
,
and
where
, considering the conservation of energy, and
since there is only one intermediate surface in this configuration.
In order to obtain
, the moving Kriging shape function matrix is employed [
31]:
where
denotes the Euclidean distance between the given interpolation nodes
and
,
is the radius of the compact support domain, and
is the correlation coefficient to fit the model, which should be chosen properly. Using different correlation coefficients
will obtain results with different precision levels. While using the moving least squares interpolation to obtain
, we employ a weight function model with the same expression as in Equation (71) [
33].
First, we set the radius of the compact support domain
and
. On the interface, displacement compatibility is naturally satisfied in Equation (15). For both the moving Kriging and moving least squares interpolation schemes, the sum of interface forces of direction Y on the two sides is in the order of
(equal to zero, the force equilibrium satisfied), as shown in
Figure 6.
The displacement, velocity and acceleration of the response point are used to illustrate the performance of the method. The analysis results in the Z direction are compared in
Figure 7.
We used R-squared (R2) and the relative average absolute error (RAAE) [
27] to illustrate accuracy. The R2 and RAAE are computed using:
where
is the length of response data;
and
are the acceleration response results of the FEM and IBS methods, respectively;
and STD are, respectively, the mean and standard deviation of the FEM result. The accuracy of NMI-IBS is presented in
Table 1.
The results of the two interpolation schemes are slightly different. However, as shown in
Figure 6 and
Table 1, displacement and velocity achieve strong agreement with the reference. The acceleration result is not as good as that of displacement but still shows relatively high accuracy. This could be affected by the correlation function (or weight function) and the distribution of the interface support nodes.
Under the distribution of the interface support nodes in
Figure 4 and
, the R2 of the acceleration in the Z direction with the varying radius of the compact support domain
is shown in
Table 2.
As shown in
Table 2, the accuracy of the moving Kriging interpolation scheme changes slightly when
changes. Comparatively, the accuracy of the moving least squares interpolation scheme is much more sensitive with
, even though they both perform quite well when
changes in an appropriate range.
However, both interpolation schemes meet numerical problems when
varies outside the desired range. For the moving least squares interpolation, it cannot form its interpolation function when the compact support domain is too small, such as
, in this configuration of the interface. Meanwhile, the interpolation function of the moving Kriging interpolation no longer satisfies the normalization condition when a relatively large compact support domain is chosen, such as
. Once the normalization condition is not satisfied (i.e.,
), the force equilibrium cannot be achieved and the results become incorrect, as shown in
Figure 8.
The coefficient
in Equation (64) is used to fit the model. Now, we set
, the R2 of the acceleration in the Z direction, with the varying
, as shown in
Table 3.
The results in
Table 3 show that the performance of the moving Kriging interpolation scheme is more sensitive with
. However, as shown in
Table 2 and
Table 3, both interpolation schemes perform well enough when
and
remain within an appropriate range. In practice,
and
can be selected empirically by considering the configuration of the interfaces.
Finally, to implement the assembly of multi-domain substructure methods, the results of the assembly of IBS-IBS, IBS-FEM and IBS-CMS are compared in Configuration 1 with moving Kriging interpolation functions. Under the configuration of the interface support nodes with
and
, the analysis results in the Z direction are compared in
Figure 9.
The result shows that the proposed method performs excellently for the dynamics of multi-domain substructures with non-matching interfaces and that there are only slight differences between IBS and IBS-FEM. The R2 and RAAE of the three methods are listed in
Table 4.
Ultimately, both of the interpolation schemes satisfy the basic rule of the normalization condition and show good performance in dynamic response prediction. The examples illustrate that the proposed method is accurate and efficient in the dynamic response prediction of substructure IRFs with non-matching interfaces.
5.2. Application to Lunar Lander with a Rover
We analyzed the dynamic response of the lunar lander landing process to validate the method; the FEM model is shown in
Figure 9. The main components of the lander include: (1) the center body, (2) the payloads and auxiliary equipment and (3) the landing mechanism.
In this example, we only retained the main body, the solar wings and the rover. Other payloads are simplified as the distributed inertia, which adheres to the original position.
The rover attaches to the main body of the lunar lander when it is transported to the moon. When arriving at the destination, it separates from the main body and starts cruising, becoming an independent system.
Usually, the rover and the center body of the lander are developed by different branches separately during the prototype design process. Moreover, the development process of the rover is an iterative optimization process, and during every iteration, the modified rover should be assembled with the center body for coupling analysis. So, we used a modal substructure to model it.
However, the center body and the rover usually have non-matched node distributions on the interface since their FE models are established by different design groups, as shown in
Figure 10 (right). Moreover, there are some complex connecting relations inside the center body and the rover that are commonly based on nodes. Consequently, remeshing will lead to an enormous cost for rebuilding those connecting relations. Therefore, the non-matching interface method is suitable in this case.
In this example, the lunar lander contains two types of substructures (the center body with solar wings is an IBS substructure and the rover is an FE substructure), and the method of initial applied forces with the Newmark- scheme is used to compute the approximate IRFs.
There are 40 support nodes selected on the rover interface and 34 support nodes on the center body interface. The intermediate surface coincides with the rover interface, shown in
Figure 11, and with the coefficient
and
(referring to the size of the interface), the interpolation functions are generated.
According to Chen [
15], we obtain the buffer loads acting on the center body. Take the longitudinal (X direction) buffer load generated from the Z-direction main strut, for instance [
13]; its time history is shown in
Figure 12.
There are two response points (see
Figure 10) selected on the lunar lander: Point 1 is on the rover and Point 2 is on the left solar wing. Using the proposed NMI-IBS, NMI-IBS-FEM and NMI-IBS-CB (all modes retained), the acceleration responses of the response points are computed. To illustrate the accuracy of these methods, an FE model with the coinciding interface is given as the reference, and the dynamic responses are obtained under the same conditions: gravity (
) and initial vertical landing velocity (
). The results are shown in
Figure 13.
To make a clear comparison of the accuracy, R2 is listed in
Table 5.
These results show that displacement and velocity have strong agreement with the reference at both of the two response points. Meanwhile, the acceleration result has a relatively high accuracy performance.
Moreover, when the IRFs of the substructures are known, the total CPU times of NMI-IBS, NMI-IBS-FEM and full Newmark-
β time integration are 38.63 s, 173.96 s and 2685 s, respectively, under the same computer hardware conditions (CPU: Intel Core i5-2300, RAM: 8 GB), shown in
Table 6. The most time-consuming process for the NMI-IBS is the calculation of IRFs. However, in practice, this preparation time for the IRFs can be reduced because the columns of the IRF matrix can be calculated simultaneously and the calculation is applicable to all. When facing design optimization or multiple load cases, by using NMI-IBS, efficiency can be drastically improved. NMI-IBS-FEM takes more time than NMI-IBS, but it could still improve the calculation efficiency. Moreover, these two methods do not need to calculate IRFs in advance, which might be more efficient than NMIS-IBS when the number of cases is not too large.
This example shows that the multi-domain substructure methods can manage the practical assembly of substructures with non-matching interfaces conveniently and efficiently, with high accuracy.