Numerical modelling of seismic waves using imageprocessing

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1 Southern Cross University 23rd Australasian Conference on the Mechanics of Structures and Materials 2014 Numerical modelling of seismic waves using imageprocessing and quadtree meshes C Birk University of New South Wales C Song University of New South Wales Publication details Birk, C, Song, C 2014, 'Numerical modelling of seismic waves using image-processing and quadtree meshes', in ST Smith (ed.), 23rd Australasian Conference on the Mechanics of Structures and Materials (ACMSM23), vol. II, Byron Bay, NSW, 9-12 December, Southern Cross University, Lismore, NSW, pp ISBN: epublications@scu is an electronic repository administered by Southern Cross University Library. Its goal is to capture and preserve the intellectual output of Southern Cross University authors and researchers, and to increase visibility and impact through open access to researchers around the world. For further information please contact epubs@scu.edu.au.

2 23rd Australasian Conference on the Mechanics of Structures and Materials (ACMSM23) Byron Bay, Australia, 9-12 December 2014, S.T. Smith (Ed.) NUMERICAL MODELLING OF SEISMIC WAVES USING IMAGE- PROCESSING AND QUADTREE MESHES C. Birk* School of Civil and Environmental Engineering, University of New South Wales Sydney, NSW, 2052, Australia. (Corresponding Author) C. Song School of Civil and Environmental Engineering, University of New South Wales Sydney, NSW, 2052, Australia. ABSTRACT This paper presents an automatic meshing procedure which is based on image processing techniques. It is suitable for problems with heterogeneous geometric features, which cannot be described by accurate drawings, in particular if details of very different scales are present. The scaled boundary finite element method facilitates structural analysis on the resulting quadtree mesh, as it incorporates hanging nodes in an elegant and straightforward way. The proposed method is applied to the problem of seismic waves in highly heterogeneous soils. Here, the maximum mesh size is governed by the material properties and frequency content of the excitation. A large-scale two-dimensional soilstructure interaction problem is studied to illustrate the potential of the proposed approach. KEYWORDS Seismic waves, image processing, quadtree mesh, scaled boundary finite element method. INTRODUCTION Earthquakes are natural events with potentially catastrophic consequences. For the prediction of ground motions and the resulting effects on our built environment, it is crucial to understand how seismic waves interact with geological features, such as soil layers, cracks and faults. With the rapid development of computing capacity and numerical methods, computer simulations of seismic waves are becoming increasingly important. The numerical modelling of wave fields in the ground is challenging for a number of reasons, such as the requirement to accurately model radiation damping, potential nonlinearities and the need of efficient modelling techniques to handle the large size of the problem. Recent developments in computational seismology are mainly based on finite difference and finite element methods (Aochi et al. 2013; Semblat and Brioist 2000), or on spectral element approaches (Chaljub et al. 2007; Mazzieri et al. 2011). A comprehensive review of numerical methods is given in Semblat (2011). One common feature of these methods is that they require a discretization of the region to be analysed. Realistic soil deposits, however, are highly heterogeneous, involving several materials and complex geometries. Geomechanical analyses are typically based on approximate graphical information rather than on precise engineering drawings. Moreover, seismic analysis may require the resolution of geometrical features at very different scales. The generation of high-quality meshes based on this information is a time-consuming and challenging task which requires considerable experience. This is This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit 819

further complicated by the existence of additional mesh requirements for wave propagation analyses depending on material properties and frequency content.

It is based on producing a quadtree decomposition of the domain using image processing techniques.

3 further complicated by the existence of additional mesh requirements for wave propagation analyses depending on material properties and frequency content. To overcome these challenges, an automatic meshing approach which requires only a graphical representation of the soil region is proposed in this paper. It is based on producing a quadtree decomposition of the domain using image processing techniques. Quadtree structures are highly efficient in capturing geometrical features at very different scales and in transitioning from coarser to finer mesh resolutions (Greaves and Borthwick 1999). Finite and spectral element approaches, however, cannot be used on quadtree meshes straightforwardly due to the presence of hanging nodes (Tabarraei and Sukumar 2005). The scaled boundary finite element method (SBFEM) (Song 2009) provides an elegant way of combining the automatic quadtree mesh generation with efficient dynamic analysis. Here, each quadrilateral element is represented by a rectangular scaled boundary finite element subdomain, which is modeled using a number of line elements. Arbitrary node configurations can be created, which allows hanging nodes to be easily incorporated. If a 2:1 rule is applied, each SBFE subdomain is characterized by one of only six different stiffness and mass matrices. Making use of this property, a highly efficient computational model for the time-domain anaysis of large-scale wave propagation problems is obtained. AUTOMATIC MESH GENERATION Consider a graphic representation of an example soil deposit as shown in Figure 1(a). Here, the black and grey coloured regions represent rock and soft soil with given material properties,, and,,, respectively. The white regions indicate discontinuities in the form of cracks and a cavity. (a) Image (b) Quadtree mesh Figure 1. Soil deposit with discontinuities The following algorithm is based on representing any rectangular digital image as a matrix [ ] of size, where and are the number of pixels in the vertical and horizontal direction. For a greyscale image, the value of each component of [ ] represents the colour of the corresponding pixel. This mathematical representation of the image can be used to obtain a quadtree decomposition of the region. Since this is based on recursively splitting a quadratic cell into smaller squares, [ ] is expanded to a square matrix by filling it with values corresponding to the void colour. The Matlab function qtdecomp is used to decompose the image [ ] Here, a quadratic cell is further divided if the maximum and minimum value inside a block differ by more than a specified threshold, i.e. if the colour changes. The function returns blocks that are not smaller than a specified minimum and not bigger than a specified maximum size. While the minimum size determines how smoothly geometrical features can be represented and is limited to 1 pixel, the maximum size governs the accuracy of the subsequent structural analysis. In a wave propagation analysis, the maximum allowable element size depends on the material properties, frequency content and element type. Based on the requirement of at least 10 nodes per wavelength to avoid dispersion and dissipation errors (Ham and Bathe 2012), different maximum element sizes are imposed on soil regions with different material properties. Moreover, a 2:1 rule is applied to the quadtree mesh, allowing neighbouring elements to differ in size by not more than a factor of 2. The resulting quadtree decomposition of the example soil deposit is shown in Figure 1(b). Observe that, ACMSM

4 - the overall mesh size in the top soil is smaller than in the rock according to its smaller shear wave velocity, - geometrical discontinuities (cracks, interfaces, cavities) are captured by very small elements - the element size rapidly transitions to the maximum size away from such discontinuities. STRUCTURAL ANALYSIS USING SBFEM As explained earlier, standard finite element formulations are not compatible on quadtree meshes. The scaled boundary finite element method provides an elegant link between automatic mesh generation and structural analysis. Here, each quadrilateral element is regarded as a special case of a general - sided polygon. If the 2:1 rule is applied, only the 6 configurations shown in Figure 2 are possible. Figure 2. Possible polygon configurations in a quadtree mesh abiding the 2:1 rule Each polygon is treated as a scaled boundary finite element subdomain. In the SBFEM, a local coordinate system is introduced as shown in Figure 3 for a 5-sided polygon as an example. Figure 3. Scaled boundary finite element coordinates Only the boundary is discretized using line elements. A radial coordinate is defined pointing outwards from the scaling centre, which is selected as the centroid of each quadrilateral domain. The circumferential coordinate is defined counter-clockwise along the boundary. The local Cartesian coordinates, are related to the so-called scaled boundary coordinates and by the scaled boundary transformation given in Eq. (1), [ ], [ ] (1) Here, the nodal coordinates and are interpolated by shape functions [ ] in terms of the circumferential coordinate. Formulating the governing equations of elastodynamics using the scaled boundary transformation and applying the method of weigthed residuals in the circumferential direction, the SBFE equation in dynamic stiffness (2) is obtained, [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] (2) In Eq. (2), [ ], [ ], [ ] and [ ] are coefficient matrices that depend on the geometry of the boundary and the constitutive law only. These matrices can be assembled using FE technology. For 2- node line elements and linear elasticity closed-form expressions can be found in (Wolf and Song, 1996). For, Eq. 2 yields an algebraic Riccati equation for the static stiffness matrix [ ], which can be solved using Schur decomposition (Song 2009). The mass matrix [ ] of a polygonal subdomain follows from a low-frequency expansion of Eq. (2) (Song 2009). The stiffness and mass matrices of all polygons are assembled in global equations of motion, [ ] [ ] (3) ACMSM

5 which can be solved using standard time integration algorithms, such as Newmark s method. Note that in a two-dimensional analysis the stiffness matrix of a quadtree element only depends on the polygon configuration and material data, but not on the element size. Therefore, only a few element stiffness and mass matrices have to be calculated. The mass matrices are scaled according to their size when assembling contributions of individual polygons into the global mass matrix NUMERICAL EXAMPLES The regular three-layered domain shown in Figure 4 is analysed to verify the proposed method. A deep bedrock is overlain by two softer layers of thickness each.,,,,,, Figure 4. Three-layered soil deposit: material data and mesh The quadtree mesh obtained automatically from an image of resolution is also shown. Here, one pixel corresponds to a physical dimension of. It can be seen that smaller elements are automatically obtained in the softer two top layers, corresponding to their lower shear wave velocity. The mesh has been designed to transmit frequencies up to. Elements of the smallest possible size of 1 pixel are obtained in the transition regions between soil layers, since the layer thickness is not a power of 2. The first 30 natural frequencies of the soil deposit under plane stress, fixed at the bottom are calculated and compared to reference solutions obtained using the commercial software ANSYS. Excellent agreement is observed in Figure 5. Figure 5. Natural frequencies of a three-layered soil deposit In a second example, a large-scale soil-structure interaction problem is addressed to illustrate the potential of the proposed method. A three-layered soil deposit of total thickness is considered, which extends 8192 in the horizontal direction. The individual layers are, and thick, from bottom to top, respectively. The same soil properties as in Figure 4 are assumed. The soil-deposit is represented by an image of size pixel. Thus, the smallest possible element size of 1 pixel corresponds to a physical dimension of. A structural detail consisting of two concrete high-rise buildings and a concrete tunnel with,, is added at the centre of the image, as shown in Figure 6. A quadtree mesh is created automatically using the image shown in Figure 6(a). It is designed to transmit frequencies up to and comprises elements and nodes. The maximum mesh size in the soil layers 1-3 is 64, 64 and 32 pixel, corresponding to, and, respectively. ACMSM

The maximum mesh size in the concrete is to material boundaries and geometrical features., however, much smaller elements are obtained close (a) (b) (c) Figure 6.

(a) full image, (b) detail, (c) mesh detail Waves in the soil are caused by a Ricker pulse point load,, which is applied at point S,,, (4) The resulting displacements at points A, B and C are shown

Points A and B are located at the soil surface, approximately to the left and between the two high-rise buildings, respectively.

6 The maximum mesh size in the concrete is to material boundaries and geometrical features., however, much smaller elements are obtained close (a) (b) (c) Figure 6. Three-layered soil deposit with structural detail. (a) full image, (b) detail, (c) mesh detail Waves in the soil are caused by a Ricker pulse point load,, which is applied at point S,,, (4) The resulting displacements at points A, B and C are shown in Figure 7. All coordinates are measured from the bottom left corner. Figure 7. Displacement response due to Ricker pulse acting below the surface Point S is approximately to the left of the structures and below the surface. Points A and B are located at the soil surface, approximately to the left and between the two high-rise buildings, respectively. Points A and C are located directly above the source, with C being in the middle soil layer. Figure 7 indicates the vertical displacements are amplified when travelling upwards through increasingly softer material. It can also be seen that the maximum displacement recorded at point B, i.e. between the two buildings, is approximately 50% of the maximum displacement obtained at point A. Snapshots of the displacement field at different times are shown in Figure 8. These further illustrate the propagation of the seismic wave, reflection at the rigid bottom and refraction at interfaces. ACMSM

(a) (b) (c) (d) Figure 8. Horizontal displacement due to vertical Ricker pulse at point S The CPU times recorded for the mesh generation and the time-domain analysis (1000 steps) are 91.3 s and 431.

Based on square scaled boundary polygons, the ubiquitous hanging node problem of classical FE approaches is overcome. The stability of the proposed scheme is not limited by the size of the problem.

The structural analysis is based on the SBFEM and standard time-stepping techniques. First promising results for the analysis of seismic wave propagation in two-dimensional soil have been presented.

(2013) Finite difference simulations of seismic wave propagation for understanding earthquake physics and predicting ground motions: Advances and challenges, Journal of Physics: Conference Series,

7 (a) (b) (c) (d) Figure 8. Horizontal displacement due to vertical Ricker pulse at point S The CPU times recorded for the mesh generation and the time-domain analysis (1000 steps) are 91.3 s and s, respectively, on a desktop computer with Intel Core i CPU@3.30 GHz and 8GB RAM. CONCLUSIONS This paper has presented a versatile strategy for automatic mesh generation and dynamic analysis. Based on square scaled boundary polygons, the ubiquitous hanging node problem of classical FE approaches is overcome. The stability of the proposed scheme is not limited by the size of the problem. Using techniques based on image-processing, quadtree meshes can be created automatically in a highly efficient manner. This has been demonstrated for a high-resolution image of size pixel. The structural analysis is based on the SBFEM and standard time-stepping techniques. First promising results for the analysis of seismic wave propagation in two-dimensional soil have been presented. The extension to 3D is the subject of current research. REFERENCES Aochi, H., Ulrich, T., Ducellier, A., Dupros, F., and Michea, D. (2013) Finite difference simulations of seismic wave propagation for understanding earthquake physics and predicting ground motions: Advances and challenges, Journal of Physics: Conference Series, Vol. 454, pp Chaljub, E., Komatitsch, D., Vilotte, J.-P., Capdeville, Y., Valette, B. and Festa, G. (2007) Spectralelement analysis in seismology, Advances in Geophysics, Vol. 48, pp Greaves, D.M. and Borthwick, A. (1999) Hierarchical tree-based finite element mesh generation, International Journal for Numerical Methods in Engineering, Vol. 45, pp Ham, S. and Bathe, K.J. (2012) A finite element method enriched for wave propagation problems, Computers and Structures, Vol , pp Mazzieri, I., Smerzini, C., Antonietti, P.F., Rapetti, F., Stupazzini, M., Paolucci, R. and Quarteroni, A. (2011) Non-conforming spectral approximations for the elastic wave equation in heterogeneous media, Proceedings, ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, COMPDYN 2011, Corfu, Greece, May Semblat, J.F. and Brioist, J.J. (2000) Efficiency of higher order finite elements for the analysis of seismic wave propagation, Journal of Sound and Vibration, Vol. 231, No. 2, pp Semblat, J.F. (2011) Modeling seismic wave propagation and amplification in 1D/2D/3D linear and nonlinear unbounded media, International Journal of Geomechanics, ASCE, Vol. 11, No. 6, pp Song, C. (2009) The scaled boundary finite element method in structural dynamics, International Journal for Numerical Methods in Engineering, Vol. 77, pp Tabarraei, A. and Sukumar, N. (2005) Adaptive computations on conforming quadtree meshes, Finite Elements in Analysis and Design, Vol. 41, pp Wolf, J.P. and Song, C. (1996) Finite-element modelling of unbounded media, John Wiley & Sons, Chichester, U.K. ACMSM

Automatic image-based stress analysis by the scaled boundary polytope elements

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