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

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1 Southern Cross University 3rd Australasian Conference on the Mechanics of Structures and Materials 14 Automatic image-based stress analysis by the scaled boundary polytope elements Chongmin Song University of New South Wales Publication details Song, C 14, 'Automatic image-based stress analysis by the scaled boundary polytope elements', in ST Smith (ed.), 3rd Australasian Conference on the Mechanics of Structures and Materials (ACMSM3), vol. II, Byron Bay, NSW, 9-1 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 3rd Australasian Conference on the Mechanics of Structures and Materials (ACMSM3) Byron Bay, Australia, 9-1 December 14, S.T. Smith (Ed.) AUTOMATIC IMAGE-BASED STRESS ANALYSIS BY THE SCALED BOUNDARY POLYTOPE ELEMENTS Chongmin Song* Centre for Infrastructure Engineering & Safety, School of Civil & Environmental Engineering, University of New South Wales, Sydney, NSW 5, Australia. (Corresponding Author) ABSTRACT Digital imaging technology is increasingly being applied in material, biomedical and other disciplines of engineering and science. Image-based stress analysis provides an attracting way to perform virtual testing of materials and structural analysis. Existing computational methods for stress analysis are developed with computer-aided design (CAD) models as input and are faced with difficulties, e.g. in mesh generation, when handing digital images. The paper presents a numerical technique to perform stress analysis directly from a D digital image segmented to its constituent materials. The quadtree algorithm is employed for mesh generation. The quadtree cells are modelled by scaled boundary polytope elements, which eliminate the issue of hanging nodes faced by standard finite elements. The whole analysis is fully automated. A numerical example is presented to demonstrate the high efficiency of the proposed technique for the analysis of high-definition images. KEYWORDS Scaled boundary finite element method, quadtree algorithm, octree algorithm, image-based analysis. INTRODUCTION The advances in the area of computational mechanics have transformed how analysis and design of engineering structures are carried out in practice. Nowadays, commercial computer software based on the finite element method (FEM) is routinely used in engineering practice. One of the key steps in the finite element analysis (FEA) of a complex structure is the mesh generation starting from, typically, a computer-aided design (CAD) model. This task is performed by numerical analysts with the aid of mesh generation software. Digital imaging technology has undergone rapid progress and is increasingly used in structural and material engineering. For example, digital images of microstructure of concrete specimens obtained by X-ray computed tomography (XCT) offers in-depth understanding of its material behaviour that would otherwise be difficult to observe (Ren, 13). In digital images, the internal structure of a material is segmented into different phases represented by the gray level of the image. Owing to the difference with CAD models in data format, existing technology for automatic mesh generation becomes often inadequate for high-fidelity analysis. The simplest approach to generate a finite element mesh from a digital image is to model each voxel as a finite element (Hollister and Kikuchi, 1994; Huang and Li, 13). However, this results in a huge number of elements for high-resolution images (a 3D image of 14 voxels in each direction has 14 3, i.e. more than 1 billion, voxels) and is not feasible for practical engineering applications. Several more sophisticated approaches exist. One approach is to generate a boundary model (similar to a CAD model) of various phases of the segmented image and use it as an input to a finite element mesh generator (Frey, 4). Another approach is to directly covering the image using triangular elements without explicitly generating a boundary model first This work is licensed under the Creative Commons Attribution 4. International License. To view a copy of this license, visit 81

3 (Langer et al., 1). These approaches necessitate significant user interaction to mesh multi-material images and often involve appreciable simplification of the images. We present the use of scaled boundary polytope elements (Ooi et al. 14) for two-dimensional image-based analysis. The polytope elements can assume the shapes of any star convex polygons in two dimensions and offers greater flexibility in mesh generation than conventional triangular and quadrilateral elements. This feature allows the use of quadtree algorithm for mesh generation without causing displacement incompatibility between quadtree cells of different size. SCALED BOUNDARY POLYTOPE ELEMENTS The scaled boundary polytope elements are constructed using the scaled boundary finite element method (Song and Wolf, 1997). As in the standard finite element method, a complex two-dimensional domain is divided into elements of simple geometry. In the present approach, these elements are starconvex polygons. Only the key concepts and equations for a polytope element are summarised in this paper. The algebraic equations of the global system are obtained by following the finite element procedure of assemblage. A scaled boundary polytope element is shown in Figure 1. It is a star-convex polygon. A scaling centre O is chosen within its kernel from which the entire polygon boundary is visible. The boundary of the polygon is discretized with one-dimensional (line) elements. The larger dots in Figure 1 show the end nodes of the elements. Any type of displacement-based elements can be used. One edge can also be divided into more than one element. The nodal coordinates of a line element are denoted as { x },{ y} and the shape functions are defined in the local coordinate as [ N] [ N( )]. At a point on the boundary, the Cartesian coordinates x x( ) and y y( ) are expressed as x [ N( )]{ x}; y [ N( )]{ y} The polygonal domain is described by scaling the boundary continuously with respect to the scaling centre O. A radial coordinate pointing from the scaling centre to a point on the boundary is introduced (Figure 1) for this purpose. is chosen as at the scaling centre and as 1 on the boundary. The coordinates of a point in the domain xˆ xˆ(, ), yˆ yˆ(, ) is expressed by using Eq. (1) as xˆ x [ N( )]{ x}; yˆ y [ N( )]{ y} where, are called the scaled boundary coordinates. (1) () Figure 1. A polytope element in scaled boundary coordinates Introducing the scaled boundary transformation defined in Eq.(), the governing differential equations for elastostatics with vanishing body force are expressed as (Song and Wolf, 1997) ACMSM3 14 8

4 T { } [ L]{ u} ; { } [ E]{ } ; [ L ] { } (3) with the strains { }, displacements { u} and stresses { }. The differential operator [ L ] is written as 1 1 [ L] [ b ] [ b ] where [ b 1 ] and [ b ] depend on the location of the scaling centre and the boundary geometry only 1 1 [ b ] x, ; J x, y, y 1 x J x y [ b ] y, with the determinant of the Jacobian matrix on the boundary J xy, yx, (6) Along the radial lines passing through the scaling centre O and a node on the boundary, nodal displacement functions { u ( )} are introduced. The displacements at a point (, ) inside the domain are obtained by interpolating the nodal functions { u(, )} [ N( )]{ u( )} (7) Applying the weighted residual technique (Deeks and Wolf, ) in leads to the scaled boundary finite element equation 1 [ E ] { u( )}, ([ E 1 1 T ] [ E ] [ E ] ) { u( )}, [ E ]{ u( )} where [ E ], [ E ] and [ E ] are coefficient matrices assembled from the element matrices with 1 1 T 1 [ E ] [ B ] [ D][ B ] J d; T [ E ] [ B ] [ D][ B ] J d; 1 1 T [ E ] [ B ] [ D][ B ] J d [ B ] [ b ][ N] [ B ] [ b ][ N] The nodal force functions along the radial lines are expressed as 1 { q( )} [ E ] { u( )}, [ E ]{ u( )} (4) (5) (8) (9) (1) (11) Equations (8) and (11) can be solved by using the eigenvalue decomposition 1 1 T 1 [ E ] [ E ] [ E ] [ u] [ u] T 1 1 [ E ] [ E ][ E ] [ E ] [ E ][ E ] [ q] [ q] where only the eigenvalues with the negative real part are retained in, [ u] and [ q ] constitute the eigenvectors. The solutions are expressed as (1) ACMSM

5 { u( )} [ ] { c}; { q( )} [ ] { c} u q where { c } are the integration constants. Eliminating { c } from Eq. (13) leads to the stiffness matrix of the polytope element [ K] [ ][ ] For elasto-plasticity analysis, the reader is referred to Ooi et al. (14). Quadtree Algorithm for Mesh Generation (13) 1 q u (14) The quadtree algorithm for mesh generation is based on the recursive subdivision of cells into four smaller ones of equal size (Samet, 199). A part of a balanced (also called regularized) quadtree mesh, where the ratio between two adjacent cells is limited to, is shown in Figure a as an example. The quadtree data structure allows fast data retrieval and efficient storage. The quadtree mesh provides a simple and efficient technique for nonuniform and adaptive mesh refinement. However, a hanging node appears when two adjacent quadtree cells are at different levels as indicated by the solid squares. When standard finite elements are used, the displacements become incompatible across the edges with a hanging node, which greatly hinders the application of quadtree mesh in the finite element method. a) b) Figure. a) Quadtree mesh; b) Quadtree cells modeled as scaled boundary polytope elements where the numbers in circle are the line element numbers on the edges The scaled boundary polytope elements overcome the issue of displacement incompatibility by modeling an edge with a hanging node as two line elements as illustrated in Figure b. No special treatments are necessary. Owing to the structure of a quadtree mesh, only the six types of cells in Figure b exist. The quadtree algorithm is ideally suited to process digital images. As a part of its Image Processing Toolbox, MATLAB provides a function qtdecomp. This function performs a quadtree decomposition of an image according to a criterion of homogeneity. To reduce the number of unique elements to the six shown in Figure b, the quadtree decomposition is balanced, i.e., enforcing the :1 ratio between adjacent quadtree cells. A mesh of scaled boundary polytope elements can be easily generated from the balanced quadtree decomposition. The whole mesh generation process is fully automatic. NUMERICAL EXAMPLE A digital image (Figure 3) reported in Ren et al. (13) is analysed as an example to illustrate the digital based analysis by the scaled boundary polytope element. The image is obtained from an in-situ microscale X-ray Computed Tomography test. The image size of the concrete specimen is 37 by 37. Each pixel represents.1mm by.1mm. The black phase represents aggregates, the gray phase mortar ACMSM

and the white phase voids. The Young s modulus of the aggregates and mortar are 7GPa and 5 GPa, respectively. The Poisson s ratio is equal to. for both the aggregates and mortar. Figure 3.

6 and the white phase voids. The Young s modulus of the aggregates and mortar are 7GPa and 5 GPa, respectively. The Poisson s ratio is equal to. for both the aggregates and mortar. Figure 3. Digital image of concrete specimen (37x37 pixels) A quadtree mesh is shown in Figure 4a. The materials are assigned to the elements based on the gray level. The size of the smallest polytope elements is chosen as 1 pixel and, thus, the full resolution of the image is perserved. The enlarged view around a void is illustrated in Figure 4b. The smallest elements concentrate around the material interfaces. Within the aggregates and mortar, larger elements present. This reduces the number of elements and degrees of freedom in the system. a) b) Figure 4. Quadtree mesh of concrete specimen. The size of the smallest elements is 1 pixel: a) Global mesh; b) Enlarged view around a void In this preliminary study, an elastic analysis of the concrete specimen under uniaxial tension is performed. The vertical displacement is constrained at the bottom of the specimen. A vertical displacement leading to an average strain of 1 5 is prescribed at the top of the specimen. The tensile principal stresses developed in the aggregates and mortar are depicted in Figure 5. Overall, the stresses in aggregates are higher than the stresses in mortar as the former is stiffer. The highest stresses occur mostly in slender aggregates and in mortar and aggregates surrounding the voids. The equivalent Young s modulus of the concrete specimen is determined as 39.6GPa. The proposed image based analysis technique is not only automatic, eliminating the human effort in mesh generation, but also computationally very efficient. The computer time is measured on a laptop computer with an i7-35m CPU. The computer code is written in MATLAB. It takes about 1.6s to create the numerical model (generating mesh, enforcing boundary conditions, etc.), 1.6s to perform the analysis to obtain nodal displacements and 3s for stress calculation and graphical output. ACMSM

7 a) b) Figure 5. Tensile principal stresses (MPa) in: a) Aggregates; b) Mortar.1.1 CONCLUSIONS An automatic procedure for D image-based analysis is presented. This technique combines the quadtree algorithm for mesh generation, which is fully automatic and highly efficient, and the scaled boundary polytope elements for stress analysis, which satisfies compatibility across edges connecting quadtree cells of different size. A numerical example is presented to demonstrate its simplicity and efficiency. Further research on extending this technique to 3D analysis, elasto-plasticity and damage mechanics is in progress. REFERENCES Deeks, A. J. and Wolf, J. P. (). A virtual work derivation of the scaled boundary finite-element method for elastostatics. Computation Mechanics, Vol. 8, pp Frey, P. J. (4). Generation and adaptation of computational surface meshes from discrete anatomical data, International Journal for Numerical Methods in Engineering, Vol. 6, pp Hollister, S. J. and Kikuchi, N. (1994). Homogenization theory and digital imaging: a basis for studying the mechanics and design principles of bone tissue, Biotechnology and Bioengineering, Vol. 43, pp Huang, M. and Li, Y. M. (13). X-ray tomography image-based reconstruction of microstructural finite element mesh models for heterogeneous materials, Computational Materials Science, Vol. 67, pp Langer, S. A., Fuller Jr., E. R. and Carter, W. C. (1). OOF: an image-based finite-element analysis of material microstructures. Computing in Science & Engineering, Vol. 3, pp Ooi, E. T., Song, C. and Tin-Loi, F. (14). A scaled boundary polygon formulation for elasto-plastic analyses, Computer Methods in Applied Mechanics and Engineering, Vol. 68, pp Ren, W., Yang, Z.J. and Withers, P. (13) Meso-scale fracture modelling of concrete based on X-ray computed tomography images, Proceedings of the 5th Asia Pacific Congress on Computational Mechanics, Singapore, 11-14th December 13. Samet, H. (199). Application of Spatial Data Structure, Addison-Wesley, New York, NY. Song, C. and Wolf, J. P. (1997). The scaled boundary finite-element method alias consistent infinitesimal finite-element cell method for elastodynamics. Computer Methods in Applied Mechanics and Engineering, Vol. 147, pp ACMSM

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