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Keywords: finite cell method, elasto-plasticity, numerical integration scheme, Voronoi polygons.

Summary

The finite cell method (FCM) [1,2] can be considered as an embedding or fictitious domain method combined with high-order finite elements [3]. The FCM, as a high order finite element method for which only an integration mesh is necessary, is implemented in this paper to elasto-plastic problems. The Newton-Raphson iterative algorithm has been adapted by a scalar integration parameter suitable for the integration scheme. The potential of the FCM to solve materially nonlinear elasto-plastic problems is demonstrated in a simple plane strain example. In this embedding method, the mesh is not necessarily conforming to the boundaries. The boundary is extended to a simple domain which can be discretized using a simple mesh. Thus, the problem of discretization is replaced by a problem of integration. The classic Gauss quadrature integration scheme is modified to improve the convergence behavior of the method to produce more accurate results at a lower cost. In the integration procedure of the FCM, a test is normally needed to find out if the integration point is in the physical domain or in the extended domain. The efficiency of the method has been improved when replacing this test by checking if the Voronoi Polygon, associated with each integration point, is in the physical (material) domain, totally or partially. Then, the corresponding weight of each integration point is modified according to the ratio of the material area to the total area of the polygon. In a further attempt, the position of the integration point for the weak discontinuity problems is changed to the centroid of the physical part of the Voronoi polygon. These two modifications have improved the convergence behavior of the method. Converging to acceptable results, even for singular problems, when the mesh does not conform to the boundaries, and the shape functions are standard high order polynomials, is the key advantage of the finite cell method. Effort to enrich the approximation space is not necessary. This paper shows that the new integration method can provide accurate results for elasto-plastic problems too.

References

1: J. Parvizian, A. Düster, E. Rank, "Finite cell method, h- and p-extension for embedded domain problems in solid mechanics", Comput. Mech., 41, 121-133, 2007. doi:10.1007/s00466-007-0173-y
2: A. Düster, J. Parvizian, Z. Yang, E. Rank, "The finite cell method for three-dimensional problems of solid mechanics", Comput. Methods Appl. Mech. Engrg., 197, 3768-3782, 2008. doi:10.1016/j.cma.2008.02.036
3: B. Szabo, A. Düster, E. Rank, "The p-version of the Finite Element Method", in E. Stein, R. de Borst, T.J.R. Hughes, (Editors), Encyclopedia of Computational Mechanics, John Wiley & Sons, Volume 1, Chapter 5, 119-139, 2004. doi:10.1002/0470091355.ecm003g

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 93 PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: Paper 310 The Finite Cell Method for Elasto-Plastic Problems A. Abedian¹, J. Parvizian², A. Düster³, H. Khademyzadeh¹ and E. Rank⁴ ¹Department of Mechanical Engineering, ²Department of Industrial Engineering, Isfahan University of Technology, Iran ³Numerical Structural Analysis with Application in Ship Technology, Technische Universität Hamburg-Harburg, Germany ⁴Chair for Computation in Engineering, Technische Universität München, Germany doi:10.4203/ccp.93.310 purchase the full-text of this paper Full Bibliographic Reference for this paper , "The Finite Cell Method for Elasto-Plastic Problems", in , (Editors), "Proceedings of the Tenth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 310, 2010. doi:10.4203/ccp.93.310 Keywords: finite cell method, elasto-plasticity, numerical integration scheme, Voronoi polygons. Summary The finite cell method (FCM) [1,2] can be considered as an embedding or fictitious domain method combined with high-order finite elements [3]. The FCM, as a high order finite element method for which only an integration mesh is necessary, is implemented in this paper to elasto-plastic problems. The Newton-Raphson iterative algorithm has been adapted by a scalar integration parameter suitable for the integration scheme. The potential of the FCM to solve materially nonlinear elasto-plastic problems is demonstrated in a simple plane strain example. In this embedding method, the mesh is not necessarily conforming to the boundaries. The boundary is extended to a simple domain which can be discretized using a simple mesh. Thus, the problem of discretization is replaced by a problem of integration. The classic Gauss quadrature integration scheme is modified to improve the convergence behavior of the method to produce more accurate results at a lower cost. In the integration procedure of the FCM, a test is normally needed to find out if the integration point is in the physical domain or in the extended domain. The efficiency of the method has been improved when replacing this test by checking if the Voronoi Polygon, associated with each integration point, is in the physical (material) domain, totally or partially. Then, the corresponding weight of each integration point is modified according to the ratio of the material area to the total area of the polygon. In a further attempt, the position of the integration point for the weak discontinuity problems is changed to the centroid of the physical part of the Voronoi polygon. These two modifications have improved the convergence behavior of the method. Converging to acceptable results, even for singular problems, when the mesh does not conform to the boundaries, and the shape functions are standard high order polynomials, is the key advantage of the finite cell method. Effort to enrich the approximation space is not necessary. This paper shows that the new integration method can provide accurate results for elasto-plastic problems too. References 1 J. Parvizian, A. Düster, E. Rank, "Finite cell method, h- and p-extension for embedded domain problems in solid mechanics", Comput. Mech., 41, 121-133, 2007. doi:10.1007/s00466-007-0173-y 2 A. Düster, J. Parvizian, Z. Yang, E. Rank, "The finite cell method for three-dimensional problems of solid mechanics", Comput. Methods Appl. Mech. Engrg., 197, 3768-3782, 2008. doi:10.1016/j.cma.2008.02.036 3 B. Szabo, A. Düster, E. Rank, "The p-version of the Finite Element Method", in E. Stein, R. de Borst, T.J.R. Hughes, (Editors), Encyclopedia of Computational Mechanics, John Wiley & Sons, Volume 1, Chapter 5, 119-139, 2004. doi:10.1002/0470091355.ecm003g purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £145 +P&P)
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