An Adaptive Frequency Domain Finite Element Model ...

AN ADAPTIVE FREQUENCY DOMAIN FINITE ELEMENT MODEL FOR SURFACE WAVE TESTING OF PAVEMENTS N. Ryden1 and M. Castaings2 1 2

Engineering Geology, Faculty of Engineering, Lund University, 22100 Lund, Sweden Université Bordeaux 1, Laboratoire de Mécanique Physique, UMR CNRS 5469, France

ABSTRACT. The proposed adaptive frequency domain finite element (FE) model makes it possible to simulate wave propagation in a pavement structure over a wide frequency range using a normal size computer. The geometry and mesh size are optimized and regenerated for each frequency using a simple script in a commercial FE software package. Results from the FE model agree well with normal mode dispersion curves for a 3-layer reference pavement model. The proposed approach is fast and computationally efficient and intended to be used as a forward model for iterative surface wave inversion. Keywords: Surface Waves, Finite Element Modeling, Pavement, Dispersion PACS: 43.35.Cg, 43.20.Ks, 43.20.Mv

INTRODUCTION Aging infrastructure and increasing loads fuel the need for efficient non-destructive quality control/assurance of civil infrastructures. Surface wave testing of pavements is a promising non-destructive technique for obtaining both thickness and velocity (stiffness) of all the layers in a pavement construction [1]. Surface wave techniques are based on the dispersive properties of surface waves in a layered medium. Dominating phase velocities are measured along the surface over a wide frequency range in order to estimate the elastic structural properties of the unknown layered medium. Recent developments have been focused on data processing [2,3] along with optimization techniques for iteratively finding the best theoretical match to the measured response [4,5]. However, the forward model used to predict the theoretical response of a layered structure during the iterative inversion process, is still based on normal mode dispersion curves for a layered half-space [6]. These traditional matrix based techniques are fast and efficient for normal mode dispersion curve calculations, but are only valid for far field propagation of surface waves in horizontally infinite homogeneous layers. The traditional matrix techniques cannot be used to accurately predict the effect of 2D and 3D effects from the actual geometry, variable material properties, scattering from cracks and/or other anomalies. For future research and development of surface wave techniques it is important to move towards more realistic 2D and 3D theoretical models. Finite Element (FE) models have been used to study 2D effects on surface wave propagation [7,8]. However, these

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studies have been based on time-marching finite element or finite difference modeling using only a few elastic layers without large velocity contrasts and without frequency dependent material properties. The computational load increases immensely for problems with large velocity contrasts and an increasing frequency range since the range of wavelengths increases with both of these factors. More over it is difficult to model realistic frequency dependent material properties in time domain. An efficient alternative based on frequency domain modeling has been presented in [9]. The problem is decomposed into a few stationary frequencies and by satisfying infinite boundary conditions the solution can be transformed back to time domain using an inverse Fourier transform. With this approach it is straight forward to implement user defined frequency dependent velocity and damping properties, as well as properties varying in any direction of space, even within any layer. It is the purpose of this paper to demonstrate an efficient frequency domain finite element model for surface wave testing of pavements. The viscoelastic frequency domain approach [9] has been extended using an adaptive geometry and mesh model to accommodate the extremely wide range of wavelengths used in surface wave testing of pavements. However, the same approach can also be used for normal soil site applications. The model is intended to be fast enough to be used as a forward model during the iterative inversion procedure of surface wave data. FINITE ELEMENT MODEL The equations of dynamic equilibrium can be solved in either space-frequency or spacetime domain using finite elements. The accuracy of wave propagation simulations using FE models depends heavily on the space-time discretization of the model [10,11]. Each wavelength (λ) is approximated with a couple of finite elements and it is generally recommended to use at least 10-12 degrees of freedom (DOF) per wavelength. In a 2D model with axial symmetry the total number of DOF’s then becomes a function of the length (R), depth (Z) and the minimum wavelength (λmin) present in the model. Meshing a typical 3-layer pavement construction in 2D using a fixed geometry and mesh size can easily generate several million DOF’s. If this type of model is solved using a time-marching technique (typically from 0 to 0.030 s) several thousand time steps may be necessary. However, if the same model is solved in frequency domain it may be enough to solve for 50 discrete frequencies [9]. By using this method the total time for wave propagation simulations can be drastically reduced turning hours into seconds. This approach is especially appealing for surface wave inversion since the inversion procedure is performed in frequency domain. With an adaptive mesh it also becomes easier to optimize the mesh and absorbing boundaries regions for each frequency, avoiding artifacts like mesh dispersion typically encountered at the high frequency limit using time marching techniques [10]. Absorbing Regions Energy absorbing regions are used to avoid reflections at the boundaries in FE wave propagation simulations. For frequency domain models it is important to use excellent absorbing boundaries to simulate an infinite space avoiding potential artifacts from reflections at the boundaries. Viscoelastic absorbing regions have been successfully used by [9,11,12]. In this approach the loss factor (η) of the complex viscoelastic modulus is gradually increased from 0 to 1 over the absorbing region. Compared to perfectly matched layers this approach requires more elements but are more robust and easy to use since the absorption over a fixed distance is not dependent on the direction of wave-fronts

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propagation, i.e. they also absorb waves having energy and phase velocities of opposite signs. In a 2D axial symmetric case, Equation 1 is used to define the complex modulus in the r-direction inside the absorbing region and is given as: 3 ⎡ ⎛ r − ra ⎞ ⎤ ⎟⎟ ⎥ E = Ereal ⎢1 + i⎜⎜ ⎝ Rabsr ⎠ ⎦⎥ ⎣⎢

(1)

where r is the position along the r-direction and ra and Rabsr are the starting position and the length respectively. A similar equation is used for the depth (z) direction. The length of the absorbing region should be 3 times the longest wavelength in the material [9], see Figure 1. In this specific pavement application the velocity decreases from layer to layer in the z-direction making λmax and hence Rabsr shorter and shorter with depth. Adaptive Geometry and Mesh Size The fundamental difficulties with FE modeling of surface waves in a pavement structure originates from the large velocity contrast between layers and the wide frequency range utilized. The top pavement layer can have a maximum compression (P) wave velocity of more than 4000 m/s while the bottom subgrade material can have a low shear (S) wave velocity of less than 100 m/s. To resolve the top one or two meters of a complete pavement construction a sensor array of 4 to 5 meters along the surface and a frequency range of 50 to 20 000 Hz is usually necessary. This velocity and frequency range corresponds to minimum and maximum wavelengths of 0.005 to 80.0 m. A fixed model geometry and mesh size that could accommodate this complete range of wavelengths simultaneously would require almost 3 billion DOF’s and cannot be solved for using a normal size computer. Even if the mesh in each layer is adjusted according to the layer velocities it is still almost impossible to solve for the complete frequency range using one fixed model and a normal size computer. To solve the problems above and develop an efficient FE model for surface wave simulations we propose to use an adaptive geometry and mesh size for each frequency. The modeled space is divided into an analyzed region and an absorbing region, Figure 1. The analyzed region is the space of interest and the absorbing region is only used to simulate infinite boundary conditions. The analyzed region is kept fixed but the size of

FIGURE 1. Schematic of the adaptive FE model with analyzed and absorbing regions.

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each element and the absorbing regions are optimized for each modeled frequency. This means that the total model and element size is big when we solve for low frequencies and small when we solve for high frequencies. Since the analyzed region is kept fixed we can still extract the response at the surface over the same fixed distance of interest for all frequencies. Using a commercial FE package [13] an adaptive frequency domain model is setup with a 2D axial symmetric geometry. The mesh is made of 2nd order quadratic elements using 4 elements per wavelength (λmin) in each layer. The adaptive geometry is based on equally sized elements and a fully fixed analyzed region in the r-direction (R=5.0 m). A semi fixed depth of the analyzed region (Z) is defined as: Z = h1 + h2 + 2λ3max

(2)

where h1 and h2 represents the thickness of layer 1 and 2 respectively and the depth of the bottom layer (half-space) is calculated as 2 times the maximum wavelength in the 3rd layer (λ3max), see Figure 1. The element size in the r-direction is determined from the minimum wavelength in the complete model. In the z-direction the element size is optimized based on λmin in each layer. This approach saves a lot of computational power since the total geometry is reduced at higher frequencies when wavelengths are comparatively small. A vertical unit amplitude force is applied at the free surface at r