## 3 Numerical Examples and Results
### Validation Study  
In order to show the accuracy of the present layerwise finite element model with a homogenization technique, a validation study is performed by comparing the dynamic characteristics of the HCS plates from the present method with those from an MSC.Nastran simulation of a detailed 3D finite element model under different boundary conditions and temperature change. The HCS plates all made of Ti-6 Al-4V with Rohacell 31 foam filled are taken into consideration. The material properties of the foam and base structures that are listed in Tables 1 and 2 refer to the values given by [PERSON] [19]. The number of the unit cells with triangular corrugated-core shape is \(n_{c}\) = 25 and other dimensions are \(a/h\) = 50, \(b/a\) = 1, \(h_{1}\) = \(h_{3}\) = \(h_{f}\) = 0.1 \(h\). In the MSC.Nastran software, the corrugated sandwich plate and the filling foam are both simulated by CHEXA solid elements with totally 142,500 elements and 193,431 nodes as shown in Figure 3. This mesh refinement is considered to be sufficient enough that the results of the 3D finite element model can be the exact solutions. Modal analysis of the HCS plates under different boundary conditions with no temperature change is firstly performed. The boundary conditions include: (1) simply supported at all sides (SSSS); (2) clamped at all sides (CCCC); (3) clamped in \(MD\) and simply supported in \(CD\) (CSCS); (4) simply supported in \(MD\) and clamped in \(CD\) (SCSC); (5) clamped in \(MD\) and free in \(CD\) (CFCF); (6) free in \(MD\) and clamped in \(CD\) (FCFC). Figure 4 shows the convergence study of the presented layerwise model. The dimensionless fundamental frequency of the HCS plates under SSSS and CCCC boundary conditions with different mesh sizes is illustrated in Figure 4. It is observed that the solution converges with the mesh size of 10\(\times\)10. Therefore, the following examples are calculated with 10 \(\times\) 10 mesh size. The first six dimensionless natural frequencies under different boundary conditions obtained from the present method and MSC.Nastran simulation are shown in Figure 5. It is noted that the results calculated by the present method are in good agreement with those by MSC.Nastran, and the relative error is less than 5% for all cases within the frequency range of interest. Figure 4: Dimensionless fundamental frequency of the HCS plate with different mesh sizes under (**a**) SSSS; (**b**) CCCC boundary conditions. Figure 5: _Cont._  
Pre-stressed modal analysis in thermal environments of the clamped HCS plates is performed then. The first six dimensionless natural frequencies against temperature change \(\Delta T\) are shown in Figure 6. It is found that for the first three modes, the results from the present method fit well with those from the MSC.Nastran simulation; meanwhile, for higher-order modes, the relative error of the dimensionless frequencies is a little larger as temperature change increases, but still within an acceptable range. Figure 7 shows the first six mode shapes of the clamped HCS plate with temperature change \(\Delta T=100\). The mode shapes from the present method are as same as those from the MSC.Nastran simulation. The present method not only has enough calculation accuracy but also has less computational cost compared to commercial software. All the calculations are performed using an Intel Core i7-4790K CPU with a 4 GHz main frequency and a 16G memory. It takes about 240 seconds for MSC.Nastran to finish the calculation for one thermal condition; however, the average calculating time for the present method is only 1.5 seconds. Therefore, within an acceptable error range, the present method can save an enormous amount of computing time, much more efficient than MSC.Nastran simulation. Figure 5: First six dimensionless natural frequencies of the HCS plate with no temperature change under different boundary conditions: (**a**) SSSS; (**b**) CCCC; (**c**) CSCS; (**d**) SCSC; (**e**) CFCF; (**f**) FCFC.