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Based on the hybrid numerical method (HNM) combining with a reduced-basis method (RBM), the real-time transient response of a functionally graded material (FGM) plates is obtained. The large eigenvalue problem in wavenumber domain has been solved through real-time off-line/on-line calculation. At off-line stage, a reduced-basis space is constructed in sample wavenumbers according to the solved eigenvalue problems. The matrices independent of parameters are projected onto the reduced-basis spaces. At on-line stage, the reduced eigenvalue problems of the arbitrary wavenumbers are built. Subsequently, the responses in wavenumber domain are obtained by the approximated eigen-pairs. Because of the application of RBM, the computational cost of transient displacement analysis of FGM plate is decreased significantly, while the accuracy of the solution and the physics of the structure are still retained. The efficiency and validity of the proposed method are demonstrated through a numerical example.

Generally, a complex structure is modeled by a discrete multi-degree-of-freedom system, and the dynamic analysis of them often requires solving a large set of equations. Nevertheless the numerically solution of the transient structural response for such a large system is expensive; thus methods that can not only reduce significantly the size of the problem and the computational cost but also retain the accuracy of the solution and the physics of the structures are very desirable.

Many methods on model-order reduction, such as Guyan reduction [

Hybrid numerical method (HNM) which combines the finite element method with the Fourier transform, a very efficient method to perform the transient analysis of laminated structures, is proposed by Liu et al [

In this paper we apply the reduced-basis method in wavenumber domain to the transient analysis of structures based on the modified HNM. The truncated eigenvectors corresponding to the carefully selected sample parameter points are extracted to construct the reduced-basis space onto which the original large system problem is projected. In this manner, a reduced system is obtained and the eigenvalue problem can be solved more effectively. And then the eigenvectors of the full problem are obtained by the inverse projection; the response in wavenumber domain can be obtained by a real-time manner. Eventually the transient response in space-time domain is obtained through performing the inverse Fourier transform.

A functionally graded material (FGM) plate with varying material properties in the thickness direction as shown in

where

where

The superscripts l, m and u denote the lower, middle and upper surfaces of the nth element respectively.

The initial conditions of the plate are given by:

d is the transient displacement responses vector of the nodal planes and “.” denotes the differentiation with respect to time t.

With the principle of virtual work, a set of approximate partial differential equations for an element is obtained. The dynamic equilibrium equation of the whole plate can be formed by assembling the matrices of all the adjacent elements. The Fourier transform from space to wavenumber is used to deduce a set of system equations of the FGM plate in wavenumber domain [

where

which is constant matrix for the given wavenumbers k_{x} and k_{y}. The expressions of the matrices

An alternative expression of Equation (8) can be obtained through rearranging columns and rows in the matrices by degrees of freedom rather than by interface. The resulting representation is given by

where

In the particular case where the elastic material possesses a symmetric plane, multiplying the third row of Equation (10) by

where

The above equation can be converted to standard form by performing Cholesky decomposition of mass matrix.

where

It requires the repeated analysis of eigenvalue problem in wavenumber domain. When the inverse Fourier transform is performed, the range of integration is theoretically from negative infinite to positive infinite wavenumbers. For a practical calculation, the ranges of wavenumbers and the number of sampling points should be chosen properly according to the required accuracy before Fourier transform. Generally, hundreds of sample points in wavenumber domain are required to guarantee the accuracy of results. Nevertheless, the eigenvalue problem for individual wavenumber should be solved to perform the modal superposition before the inverse Fourier transform, this is computationally expensive. Furthermore, for higher accuracy of results, the computational cost will increase exponentially when layered element is adopted.

The expensive computational cost of the repeated analysis for the large eigenvalue problems can be decreased by reduced-basis method through an off-line/on-line decomposition [

At off-line stage, we introduce a sample set

where G is the number of the sample point,

A reduced-basis space can be introduced by extracting the first m eigenvectors of each point in the sample set as follows [

To simplify the notation, we rewrite Equation (16) as follows:

The bases of above equation are orthonormalized to guarantee their independence, and then the approximate eigenvectors of the standard eigenvalue problem for a new

Galerkin projection:

where the subscript j denotes the j^{th} mode, ‘^’ denotes the approximated variable. It can also be rewritten in the form of a matrix.

where

The matrices independent of wavenumber are projected onto the reduced-basis space:

At on-line stage, because of the wavenumber independence of

The resultant reduced eigenvalue problem can be expressed as:

where

Combining the method of modal analysis, the initial condition Equation (7) with the addition of the Duhamel integral, the displacements in Fourier transform domain can be obtained.

Finally, the approximate displacement response in the space-time domain can be obtained by performing the inverse Fourier transform.

It is considerable for the reduced standard eigenvalue problem that the reduced-basis space which is constructed by the truncated modes can only approximate the same modes of the eigenvalue problem for a new

According to the classical theory of matrix projection, the following equation holds if Z is an invariant subspace for matrix Q.

Here

The Euclid Norm is defined as:

It is easy to guarantee the error within bound of eigenvalue.

The error of eigenvectors is also bounded.

The present method is applied to an actual stainless steel-silicon nitride (SS-SN) FGM plate, in which the silicon nitride is considered as the inclusion material. The material constants for SS-SN are listed in

The material property of the FGM can be obtained by using the rule-of-mixture [

In the computational procedure, the following dimensionless parameters are used [

For the SS-SN FGM plate, the SS material is taken as the referenced material,

Consider two-dimensional case, a vertical line sin load is acted on the upper surface at

where

Equation (26) implies that the time history of the load is only one cycle of the sin function,

In this case where

Material | E (GPa) | ν | ρ (kg/m^{3}) |
---|---|---|---|

Stainless steel | 207.82 | 0.3177 | 8166 |

Silicon nitride | 322.4 | 0.24 | 2370 |

The eigen-pairs analysis in six wavenumbers, namely

Modal | |||||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | ||

1 | E_{ori } E_{RBM} deviation | 0.0015476 0.0015584 (0.69706%) | 0.053833 0.053833 (0%) | 0.136998 0.137009 (0.00823%) | 3.994015 3.994015 (0%) | 4.123671 4.123691 (0.00049%) | 10.856721 10.857459 (0.00680%) |

2 | E_{ori } E_{RBM} Deviation | 0.0110826 0.0110911 (0.07657%) | 0.148887 0.148887 (0%) | 0.376779 0.376789 (0.00263%) | 4.107236 4.107236 (0%) | 4.461153 4.461169 (0.000367%) | 10.659006 10.659572 (0.005308%) |

3 | E_{ori } E_{RBM} Deviation | 0.445148 0.445160 (0.00276%) | 1.194321 1.194321 (0%) | 2.840223 2.840235 (0.00041%) | 5.451391 5.451391 (0%) | 7.8919523 7.8920416 (0.00113%) | 10.320135 10.320669 (0.00518%) |

4 | E_{ori } E_{RBM} Deviation | 1.136406 1.136437 (0.00529%) | 2.178978 2.178978 (0%) | 4.884445 4.884461 (0.00068%) | 6.871678 6.871678 (0%) | 9.501102 9.501483 (0.008%) | 12.827217 12.827740 (0.00764%) |

5 | E_{ori } E_{RBM} Deviation | 4.512785 4.512792 (0.00276%) | 5.954376 5.934376 (0%) | 10.541770 10.541834 (0.00034%) | 13.050998 13.050998 (0%) | 15.398164 15.398204 (0.00401%) | 23.799351 23.799351 (0%) |

6 | E_{ori } E_{RBM} Deviation | 9.046870 9.046886 (0.00019%) | 11.155957 11.155957 (0%) | 18.462705 18.462744 (0.00022%) | 21.648815 21.648816 (0%) | 25.946279 25.946399 (0.00046%) | 33.442165 33.442174 (2.7e-005%) |

k | Modal | ||||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | ||

1 | F_{ori } F_{RBM} Deviation | 9.451689 9.451740 (0.00054%) | 9.413008 9.413008 (1.94e-006%) | 9.376146 9.375485 (0.00705%) | 11.131472 11.131472 (3.61e-006%) | 11.135540 11.133934 (0.01443%) | 11.197055 11.200604 (0.03170%) |

2 | F_{ori } F_{RBM} Deviation | 9.456450 9.456501 (0.00054%) | 9.352517 9.352518 (3.23e-006%) | 9.253267 9.252640 (0.00678%) | 11.167351 11.167352 (6.09e-006%) | 11.175687 11.174335 (0.01210%) | 11.226883 11.231277 (0.03914%) |

3 | F_{ori } F_{RBM} Deviation | 9.409621 9.410042 (0.00447%) | 8.751445 8.751449 (3.52e-005%) | 8.236840 8.236911 (0.00086%) | 11.478927 11.478937 (8.07e-005%) | 10.931823 10.939108 (0.06664%) | 12.054381 12.053443 (0.00778%) |

4 | F_{ori } F_{RBM} Deviation | 9.271804 9.273026 (0.01318%) | 8.314886 8.314896 (0.00012%) | 7.872469 7.872762 (0.00373%) | 11.611885 11.611933 (0.00041%) | 10.6239002 10.644070 (0.18985%) | 12.367178 12.371036 (0.03120%) |

5 | F_{ori } F_{RBM} Deviation | 8.334712 8.335180 (0.00561%) | 7.554797 7.554795 (2.13e-005%) | 10.118705 10.114811 (0.03848%) | 11.051769 11.051787 (0.000164%) | 9.256165 9.258184 (0.02181%) | 11.525035 11.525118 (0.00073%) |

6 | F_{ori } F_{RBM} Deviation | 7.589872 7.590152 (0.00369%) | 7.325107 7.325104 (4.54e-005%) | 11.147883 11.144348 (0.03171%) | 9.731358 9.731460 (0.00105%) | 9.223409 9.215928 (0.08111%) | 11.767512 11.768749 (0.01052%) |

Either too many or too few sample points selected in the parameter domain is unfeasible, the former leads to computational inefficiency, while the latter leads to unacceptable error. There is often a tradeoff between the computational cost and the accuracy of the simulated result. The transient displacements from RBM with 6, 8 and 15 sample points compared with that from HNM without the introducing of RBM are drawn in

CPU time (s) | With RBM | Without RBM |
---|---|---|

1.625 | 4.188 |

The Fortran codes for the calculation of transient displacements are executed on a PC with a CPU clock speed of 2.8 GHz. The CPU time for the analysis of eigenvalue problems with and without the introducing of RBM are compared in

A reduced-basis method in wavenumber domain is proposed to obtain the real-time transient response of FGM plate based on the modified HNM in this paper. The repeated and expensive numerical analyzes of the large eigenvalue problems have been simplified by RBM. Through a real-time off-line/on-line decomposition technique, high accuracy and less cost of the simulation are achieved. Because of the outstanding performance of the RBM, it is a promising numerical method which can be extended to the dynamic analysis of other complex structures.

This project is supported by National Natural Science Foundation of China (Grant No. 51305045), and by China Postdoctoral Science Foundation (No. 2014M562099).

YonghuiHuang,YiHuang, (2015) A Real-Time Transient Analysis of a Functionally Graded Material Plate Using Reduced-Basis Methods. Advances in Linear Algebra & Matrix Theory,05,98-108. doi: 10.4236/alamt.2015.53010