Wave propagation numerical models in damage detection based on the time domain spectral element method

Transcription

1 IOP Conference Series: Materials Science and Engineering Wave propagation numerical models in damage detection based on the time domain spectral element method To cite this article: W Ostachowicz and P Kudela 2010 IOP Conf. Ser.: Mater. Sci. Eng View the article online for updates and enhancements. Related content - Spectral element modelling of wave propagation in isotropic and anisotropic shell-structures including different types of damage R T Schulte, C-P Fritzen and J Moll - Optimizing a spectral element for modeling PZT-induced Lamb wave propagation in thinplates Sungwon Ha and Fu-Kuo Chang - Interactive Simulation and Visualization of Lamb Wave Propagation in Isotropic and Anisotropic Structures J Moll, C Rezk-Salama, R T Schulte et al. Recent citations - Damage detection sensitivity characterization of acousto-ultrasoundbased structural health monitoring techniques Vishnuvardhan Janapati et al - Numerical modeling of PZT-induced Lamb wave-based crack detection in plate-like structures Luyao Ge et al This content was downloaded from IP address on 31/01/2018 at 23:09

2 Wave Propagation Numerical Models in Damage Detection Based on the Time Domain Spectral Element Method W Ostachowicz 1,2,3 and P Kudela 1,4 1 Polish Academy of Sciences, Institute of Fluid-Flow Machinery Fiszera 14 St, Gdansk, Poland 2 Gdynia Maritime University, Faculty of Navigation, Jana Pawla II 3, Gdynia, Poland Abstract. A Spectral Element Method is used for wave propagation modelling. A 3D solid spectral element is derived with shape functions based on Lagrange interpolation and Gauss- Lobatto-Legendre points. This approach is applied for displacement approximation suited for fundamental modes of Lamb waves as well as potential distribution in piezoelectric transducers. The novelty is the model geometry extension from flat to curved elements for application in shell-like structures. Exemplary visualisations of waves excited by the piezoelectric transducers in curved shell structure made of aluminium alloy are presented. Simple signal analysis of wave interaction with crack is performed. The crack is modelled by separation of appropriate nodes between elements. An investigation of influence of the crack length on wave propagation signals is performed. Additionally, some aspects of the spectral element method implementation are discussed. 1. Introduction Guided waves induced by piezoelectric transducers are extensively used for damage detection purposes. Piezoelectric transducers are relatively cheap and are very well suited for elastic wave actuation in structures. They are brittle so recently Macro Fibre Composites (MFC) has been developed for application in structures with significant curvature. The analysis of piezoelectric composite structures such as piezoelectric laminated plates or structural elements activated by piezoactuators requires efficient and accurate electromechanical modelling of both the mechanical and electric responses such as mechanical displacements and electric potentials. Exact 3D analytical solutions have been presented for the piezoelectric response of simply supported flat panels and rectangular plates in [1 4]. Since the exact 3D analytical solutions are available only for some regular shapes with specified simple boundary conditions, it is desirable to use approximate method for more complex structures (i. e. the finite element method, the finite difference method, the boundary element method). Modelling of mechanical displacement field by the full 3D finite elements [5, 6] typically results in huge number of degrees of freedom and high computational cost. For this reason many 2D models with different assumptions made on the through-the-thickness mechanical displacement field have been proposed including layerwise approximation [7 9] or multi-layer modelling [10]. More recently method of sublayers has been proposed [11] which combines a 2D single-layer representation model 3 4 Corresponding author: wieslaw@imp.gda.pl Author would like to thank the Foundation for Polish Sciences for their finance support. c 2010 Published under licence by Ltd 1

3 for the mechanical displacement field with a 3D layerwise-like approximation for the electric potential field. However, these models are rarely used in elastic wave propagation problems. Furthermore, the through-the-thickness electric potential field distribution can be approximated by various models: linear [12] for thin actuators or nonlinear [10, 13] or trigonometric [7]. Kapuria [14] even proposed an electromechanical model that combines the displacement field approximations of the third order zigzag theory with a layerwise approximation for the electric potential. Each electric potential representation model has its own advantages and disadvantages in terms of accuracy and ease of use and computational cost. Moreover, in the case of elastic wave propagation, the major numerical challenge is to minimize numerical dispersion error [15] (error of phase and group velocities). Minimisation of the error requires sufficient discrete spatial resolution per minimum wavelength. In order to obtain solution with a less than 1% dispersive error, explicit finite element method requires more than 20 elements per minimum wavelength [15]. Thus, application of 3D finite elemtnts for wave propagation modelling is impractical. The spectral element method (SEM) in the time domain proposed by Patera [16] is an alternative to the existing numerical methods in the field of elastic wave modelling. This method originates from the use of spectral series for the solution of partial differential equations [17]. The idea of SEM is very similar to FEM except for the specific approximation functions it uses. Elemental interpolation nodes are located at points corresponding to zeros of an appropriate family of orthogonal polynomials (Legendre or Chebyshev). A set of local shape functions consisting of Lagrange polynomials, which are spanned on these points, are built and used. As a consequence of this, as well as the use of the Gauss-Lobatto-Legendre integration rule, a diagonal form of the mass matrix is obtained. In this way the cost of numerical calculations is much less expensive than in the case of any classic FE approach. Moreover, the numerical errors decrease faster than any power of 1/p (so called spectral convergence), where p is the order of the applied polynomial. The main fields of application of SEM nowadays include: fluid dynamics [18], heat transfer [19], acoustics [20, 21], seismology [22, 23], etc. The application of SEM for problems of propagating waves in anisotropic crystals has been shown in [24]. The first attempt to the use of SEM for problems of propagation of elastic waves in 2D structural elements with crack has been done by Zak et al. [25]. 36-node spectral membrane element with two degrees of freedom per node has been developed. The crack has been modelled by simple splitting of the nodes between appropriate spectral elements. This approach has been extended to isotropic and composite plates [25 27]. The simplified actuator-induced Lamb wave propagation analysis using 3D SEM has been performed in [28]. The aim of this paper is to extend spectral element approach presented in [29] to curvilinear coordinate system. Higher order 3D spectral element, which assures fast convergence of mechanical displacement field as well as electric potential field has been developed. Particular case of node distribution in element has been chosen specifically for fundamental Lamb wave modelling. Both actuation and sensing is taken into account. 2. 3D spectral elements The spectral elements can be derived in the same manner as finite elements. The main difference is construction of shape functions. In this paper Langrange s interpolants going through Gauss Lobatto Legendre points are considered. 3D spectral elements outperform spectral elements based on Mindlin s theory in terms of accuracy and prediction of Lamb wave behaviour especially in higher frequency regimes. Moreover, 3D spectral elements may be better suited for PZT modelling. On the other hand, calculations are very time and memory consuming. A good compromise is an application of 3D spectral elements for wave propagation modelling with higher order approximation of in-plane displacements and lower order approximation of out of plane displacements. In the current approach 36 nodes are used for in-plane approximation and 3 nodes through the thickness of the solid element. Such approach guarantees accurate modelling of fundamental Lamb wave modes which in fact have parabolic displacement distribution through the thickness. At the same time algorithm is very efficient. 2

4 In case of higher frequencies in which higher Lamb wave modes propagate more through the thickness nodes should be used Spectral element formulation Spectral element is derived in the same manner as finite element based on weight residual method. Considering three-dimensional theory of elasticity, displacement field within the element can be expressed as: (1) where, and denote displacements in x, y and z direction, respectively while t denotes particular moment in time. The strain field can be expressed as follows: (2) The SEM approximates the field variables in elements using higher-order one-dimensional Lagrange polynomial and its tensor product. The displacement field can be approximated as follows: (3) where is the one-dimensional shape function (nth-order 1D Lagrange interpolation function at n+1 Gauss-Lobatto-Legendre points) specified in a curvilinear coordinate system, - nodal degrees of freedom, - unit matrix of the size 3x3. The curvilinear coordinates are normalised and they vary between -1 and +1. Approximation of potential and geometry of the element can be assumed similarly: (4) (5) 3

5 The linear piezoelectric constitutive equations in Voigt notation can be expressed as: where is the strain vector, the electric displacement vector, the stress vector, the elasticity matrix under constant electric field, the piezoelectric constant matrix, the electric field vector, the dielectric constant matrix. The electric field vector is related to the electric potential field by using a gradient vector : (8) Taking into account small deformations of any point in the solid linear strains can be approximated as: (9) (6) (7) where is the strain-nodal displacement matrix calculated as: (10) where is the Jacobi matrix and is the matrix of the directional cosines. Using Eq. (8) and the electric field approximation in Eq. (4) the electric field vector is given by the equation: (11) where is the electric field-nodal potential matrix calculated as: (12) The governing equations of motion of piezoelectric solid element can be derived by using Hamilton's variational principle in which the total work done by the external mechanical and electrical forces is taken into account. Finally equations of motion of piezoelectric element can be written in the matrix form: (13) where denotes the elementary mass matrix, the elementary stiffness matrix, and the piezoelectric coupling matrices, the dielectric permittivity matrix, the elementary electric (14) 4

6 potential vector, the nodal external force vector, the nodal externally applied charge vector. The definition of characteristic elemental matrices can be found in [29]. After the global matrices are assembled, the vectors of unknown displacements and potentials would be directly solved for. However, for the piezoelectric materials, typical element values of are of the order 10 8, while typical element values of are of the order This huge difference in magnitudes would make the global matrices too ill conditioned if the governing equations of motion are taken as a whole. To overcome this problem, the general method of static condensation by matrix algebra can be adopted. The governing equation of motions (13 14) can be further condensed in such a way that the unknown potentials are sacrificed in favour of the unknown displacements: where denotes stiffness matrix induced by electromechanical coupling which depends on electric boundary condition (open circuit, closed circuit, actuator) and is obtained as: It should be noticed that the elementary dielectric permittivity matrix is not positive definite. To make static condensation (15) possible, electric boundary conditions must be applied. Imposed electrical boundary conditions are applied to matrices and. (15) (16) Actuation is accomplished by the equivalent mechanical force vector piezoelectric actuator which can be expressed as follows: of the applied voltage of the In case of excitation applied at the top surface of the piezoelectric electrode and assuming that the electric potential is zero at the bottom surface of the actuator, the induced potential distribution can be calculated using formula: in which and are the submatrices of the matrix corresponding to electrical boundary conditions. For example in case of nth-order 3D spectral element matrix corresponds to set of node numbers whereas matrix corresponds to set of node numbers, where colon denotes range of natural numbers. Sensing is performed according to the equation: where and are the corresponding submatrices of and, respectively (open circuit is considered here). (17) (18) (19) 3. Implementation aspects Procedure for solving elastic wave propagation equations of motion (15 19) has been implemented in MATLAB using the explicit scheme. Element by element approach along with central difference 5

7 scheme has been applied in which characteristic spectral element matrices are calculated at each time step [29]. A multiplication of stiffness matrix of an element by actual displacement vector is most costly operation. In order to speed up calculations this part of procedure has been written as mex file with support of lapack and blas libraries. Moreover, PGI compiler has been used for compilation which enables automatic loop unrolling and vectorisation (without any code modification). As a result efficient concurrent calculations have been performed on workstation equipped with 4 AMD Quad- Core Opterons with very good scalability. It should be noticed that due to discrete orthogonality of shape functions, mass matrix of 3D spectral element is diagonal. Moreover, this property can be utilised for reduction of operations necessary for calculation of Jacobians as well as stiffness matrices by appropriate loop splitting and avoiding operations on zero matrix elements. 4. Results Numerical calculations have been carried out for a half-pipe structural element made out of aluminium alloy (Young s modulus 71 GPa, Poisson ratio 0.33, mass density 2700 kg/m 3 ). The radius of the element was R=0.2 m, length L=0.5 m and thickness 2 mm (figure 1). 34 piezoelectric transducers were used in which 17 for wave actuation. The placement of piezoelectric transducers together with the mesh of spectral elements is presented in figure 2. Each piezoelectric transducer is modelled by four spectral elements with 108 nodes each (3 nodes through the thickness). The detail of the mesh near the transducer is presented in figure 3. The thickness of transducers is 1 mm and it is assumed that it is made of PZT material type 4 [30]. The excitation was applied simultaneously to 17 actuators on the one side of half-pipe in the form of sine pulse of frequency 100 khz modulated by Hanning window (3 cycles). To be more precise, the voltage was applied at the upper surface of piezoelectric transducers whereas the bottom of each transducer was assumed to be perfectly bonded to the surface of the half-pipe and grounded. Damage in the form of crack was modelled by splitting appropriate nodes in neighbouring elements. The crack is located at distance d=0.153 m from the end of the halfpipe (figure 1). Simulations have been carried out for the crack about 24.8 mm long. Figure 1. Geometry of half-pipe with crack. Figure 2. Mesh of spectral elements with piezoelectric transducers. 6

8 Figure 3. Detail of the mesh showing piezoelectric transducer composed of 4 spectral elements. Figure 4. Energy comparison calculated for signals registered by sensors. Piezoelectric actuators due to the applied voltage deform. The deformation in the form of extension/compression and vertical shear is transferred to the surface of the half-pipe. Complex wave mechanism consisting of shear vertical waves in connection with longitudinal P waves is formed in thin shell element. This mechanism results in Lamb wave generation at some distance from actuators. Results of numerical simulations are presented in figures 5 7. It is easy to notice that in case of inplane displacements elastic waves start propagating at circumference of transducers (figure 6) than the front of symmetric mode is created and next slower antisymmetric mode propagates. Reflections between piezoelectric transducers as well as reflections from the crack are clearly visible. Reflections from the crack cause that energy transmitted to sensors drops significantly in comparison to signals for undamaged half-pipe (see figure 4 and 5). This energy drops correlate with the position and the length of the crack. Such a simple feature can be used for damage detection purposes. The bars presented in figure 4 are calculated as: (20) where denotes j-th sampling point in i-th sensor. Next amplitudes of energy are normalized in respect to maximum amplitude for the case of reference signals. It can be noticed that energy drop occurs only in sensors which are located on direct path from the opposite actuator through damage to this particular sensor. Figure 5. Comparison of signals registered by sensor located near the edge of the half pipe along the path going through the crack. 7

9 Figure 6. Displacements in y direction (along the length of the half pipe) illustrate propagating waves at selected time instances. It is worthy to notice that S0 mode which propagates faster and has greater wavelength than A0 mode also gives almost continuous wave front except some part along the edges of the half-pipe (figure 6). A0 mode is much more noisy due to reflections between transducers (figure 7). It can be explained by the fact that the distance between centres of neighbouring actuators is greater than the half of the A0 mode wavelength (18.6 mm > 6.5 mm). 8

10 Figure 7. Displacements in radial direction illustrate propagating waves at selected time instances. 5. Conclusions In case of 3D spectral elements it is possible to take advantage of discrete orthogonality of shape functions during implementation of characteristic matrices. As a result mass matrix is diagonal and stiffness matrix can be obtained by very fast computation of nonzero elements. Diagonal property of the mass matrix enables to implement efficient solver for equation of motion (central difference scheme). 9

11 The advantage of 3D spectral elements in comparison with shell elements is that piezoelectric transducers can be easily modelled in terms of geometry which is defined at mesh level as well as in terms of potential distribution through the thickness of the element. Developed model is very helpful in designing effective Structural Health Monitoring systems. Exemplary simulations show capability of modelling and testing damage detection algorithms. Experimental verification regarding obtained results will be performed on Mi-2 helicopter blade in the future. Also procedure for quantification of the crack growing will be the subject of future work. Acknowledgments The authors of this work would like to gratefully acknowledge the support for this research provided by the Foundation for Polish Science. References [1] Ray M, Bhattacharya R and Samanta B 1993 AIAA Journal [2] Tzou H and Tiersten H 1994 Smart Mater. Struct [3] Bisegna P and Maceri F 1996 ASME J. Appl. Mech [4] Ray M, Bhattacharya R and Samanta B 1998 Comput. Struct [5] Tzou H and Tseng C 1990 J. Sound Vib [6] Yao L and Lu L 2003 Int. J. Numer. Methods Eng [7] Fernandes A and Pouget J 2002 Eur. J. Mech. Solids [8] Saravanos D, Heyliger P and Hopkins D 1997 Int. J. Solids Struct [9] Reddy J 1997 Mechanics of Laminated Composite Plates: Theory and Analysis (CRC Press) [10] Bisegna P and Caruso G 2001 J. Solids Struct [11] Wang S 2004 Int. J. Solids Struct [12] Mindlin R 1972 Int. J. Solids Struct [13] Yang J 1999 Smart Mater. Struct [14] Kapuria S 2001 Int. J. Solids Struct [15] Mullen R and Belytschko T 1982 Int. J. Numer. Methods Eng [16] Patera A 1984 J. Comput. Phys [17] Boyd J 1989 Chebyshev and Fourier Spectral Methods (Springer) [18] Canuto C, Hussaini M, Quarteroni A and Zang T 1991 Spectral Methods in Fluid Dynamics 3rd ed (Springer) [19] Spall R 1995 Int. J. Heat Mass Transfer [20] Dauksher W and Emery A 1997 Finite Elem. Anal. Des [21] Seriani G 1997 J. Comput. Acoust [22] Komatitsch D and Vilotte J 1998 Bull. Seismol. Soc. Amer [23] Seriani G 1999 Phys. Chem. Earth [24] Komatitsch D, Barnes C and Tromp J 2000 Geophisics [25] Zak A, Krawczuk M, Ostachowicz W, Kudela P and Palacz M 2006 Proceedings of the Third European Workshop Structural Health Monitoring 2006 pp [26] Kudela P, Zak A, Krawczuk M and Ostachowicz W 2007 J. Sound Vib [27] Kudela P, Ostachowicz W and Zak A 2007 Key Engineering Materials 47 (Switzerland: Trans Tech Publications) pp [28] Kim Y, Ha S and Chang F K 2008 AIAA J [29] Kudela P and Ostachowicz W 2009 Journal of Physics: Conference Series 181 [30] 10