Numerical Analysis of Benchmark Finite Strain Plasticity Problems using High-order Finite Elements and Finite Cells
Authors
Abstract
finite cell method, are employed to compute a series of benchmark problems in the finite strain von Mises or J2 theory of plasticity. The hierarchical (integrated Legendre) shape functions are used for the finite element approximation of incompressible plastic dominated deformations occurring in the finite strain plasticity of ductile metals. The computational examples include the necking under uniaxial tension with notched and un-notched samples and the compression of a perforated plate. These computations demonstrate that the high-order finite element methods can provide a locking-free behavior with a pure displacement-based formulation. They also provide high convergence rates and robustness against high mesh distortions. In addition, it is shown that the finite cell method, on the top of the aforementioned advantages, provides easy mesh generation capabilities for highly complex geometries. The computational results are verified in comparison with the results obtained using a standard low-order finite element method known as the F-bar method. The numerical investigations reveal that both methods are good candidates for the plasticity analysis of engineering materials and structures made up of ductile materials, particularly those involving complex geometries.
Keywords
1. Szabo, B. A., and Babuska, I., Finite Element Analysis, John Wiley & Sons, 1991.
2. Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z., The Finite Element Method -Its Basis and Fundamentals, Vol. 1. Butterworth-Heinemann, 7th Edition, 2013.
3. Szabo, B. A., Duster, A., and Rank, E., The p-version of the Finite Element Method, pp. 119-139, In: Stein, E., de Borst, R., and Hughes, T. J. R., (Eds.), Encyclopedia of Computational Mechanics, Vol. 1, Chapter. 5, John Wiley & Sons, 2004.
4. de Souza Neto, E. A., Peric, D., and Owen, D. R. J., Computational Methods for Plasticity, Theory and Applications, John Wiley & Sons, 2008.
5. Wriggers, P., Nonlinear Finite-Element-Methods, Springer-Verlag, 2008.
6. Duster, A., Hartmann, S., and Rank, E., “p-FEM Applied to Finite Isotropic Hyperelastic Bodies”, Computer Methods in Applied Mechanics and Engineering, Vol. 192, pp. 5147-5166, 2003.
7. Lipton, S., Evans, J. A., Bazilevs, Y., Elguedj, T., and Hughes, T. J. R., “Robustness of Isogeometric Structural Discretizations under Severe Mesh Distortion”, Computer Methods in Applied Mechanics and Engineering, Vol. 199, pp. 357-373, 2010.
8. Rank, E., Broker, H., Duster, A., Krause, R., and Rucker, M., The p-version of the Finite Element Method for Structural Problems, pp. 263-307, In: Stein, E., (Eds.), Error-controlled Adaptive Finite Elements in Solid Mechanics, Chapter. 8, John Wiley & Sons, 2002.
9. Kiralyfalvi, G., and Szabo, B. A., Quasi-regional Mapping for the p-version of the Finite Element Method, Finite Elements in Analysis and Design, Vol. 27, pp. 85-97, 1997.
10. Cottrell, J. A., Hughes, T. J. R., and Bazilevs, Y., Isogeometric Analysis: Towards Integration of CAD and FEM, John Wiley & Sons, 2009.
11. Szabo, B. A., and Babuska, I., Introduction to Finite Element Analysis: Formulation, Verification, and Validation, Wiley-Blackwell, 2011.
12. Krause, R., Mucke, R., and Rank, E., “hp-version Finite Elements for Geometrically Nonlinear Problems”, Communications in Numerical Methods in Engineering, Vol. 11, pp. 887-897, 1995.
13. Yosibash, Z., and Priel, E., “High-order Fems for Thermohyperelasticity at Finite Strains”, Computers & Mathematics with Applications, Vol. 67, pp. 477-496, 2014.
14. Yosibash, Z., Weissa, D., and Hartmann, S., “p-fems for Hyperelastic Anisotropic Nearly Incompressible Materials under Finite Deformations with Applications to Arteries Simulation”, International Journal for Numerical Methods in Engineering, Vol. 88, pp. 1152-1174, 2011.
15. Netz, T., Duster, A., and Hartmann, S., “High-order Finite Elements Compared to Low-order Mixed Element Formulations”, ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik, Vol. 93, No. 2-3, pp. 163-176, 2013.
16. Babuska, I., and Suri, M., “On Locking and Robustness in the Finite Element Method”, SIAM Journal on Numerical Analysis, Vol. 29, pp. 1261-1293, 1992.
17. Suri, M., “Analytical and Computational Assessment of Locking in the hp Finite Element Method”, Computer Methods in Applied Mechanics and Engineering, Vol. 133, pp. 347-371, 1996.
18. Heisserer, U., Hartmann, S., Duster, A., and Yosibash. Z., “On Volumetric Locking-free Behavior of p-version Finite Elements under Finite Deformations”, Communications in Numerical Methods in Engineering, Vol. 24, pp. 1019-1032, 2008.
19. Holzer, S., and Yosibash, Z., “The p-version of the Finite Element Method in Incremental Elasto-plastic Analysis”, International Journal for Numerical Methods in Engineering, Vol. 39, pp. 1859-1878, 1996.
20. Szabo, B. A., Actis, R., and Holzer, S., Solution of Elastic-plastic Stress Analysis Problems by the p-version of the Finite Element Method, pp. 395-416, In: Babuska, I., and Flaherty, J., (Eds.), Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations, IMA Volumes in Mathematics and its Applications, Vol. 75, Springer, NewYork, 1995.
21. Duster, A., and Rank, E., “The p-version of the Finite Element Method Compared to an Adaptive h-version for the Deformation Theory of Plasticity”, Computer Methods in Applied Mechanics and Engineering, Vol. 190, pp. 1925-1935, 2001.
22. Duster, A., and Rank. E., “A p-version Finite Element Approach for Two- and Three-dimensional Problems of the J2 Flow Theory with Non-linear Isotropic Hardening”, International Journal for Numerical Methods in Engineering, Vol. 53, pp. 49-63, 2002.
23. Duster, A., Niggl, A., Nubel, V., and Rank, E., “A Numerical Investigation of High-order Finite Elements for Problems of Elastoplasticity”, Journal of Scientific Computing, Vol. 17, pp. 429-437, 2002.
24. Tin-Loi, F., and Ngo, N. S., “Performance of the p-version Finite Element Method for Limit Analysis”, International Journal of Mechanical Sciences, Vol. 45, pp. 1149-1166, 2003.
25. Tin-Loi F., and Ngo, N. S., “Performance of a p-Adaptive Finite Element Method for Shakedown Analysis”, International Journal of Mechanical Sciences, Vol. 49, pp. 1166-1178, 2007.
26. Heisserer, U., Hartmann, S., Duster, A. D., Bier, W., Yosibash, Z., and Rank, E., “p-fem for Finite Deformation Powder Compaction”, Computer Methods in Applied Mechanics and Engineering, Vol. 197, pp. 727-740, 2008.
27. Yosibash, Z., Hartmann, S., Heisserer, U., Duster, A., Rank, E., and Szanto, M., “Axisymmetric Pressure Boundary Loading for Finite Deformation Analysis using p-FEM”, Computer Methods in Applied Mechanics and Engineering, Vol. 196, pp. 1261-1277, 2007.
28. Parvizian, J., Duster, A., and Rank, E. “Finite Cell Method - h- and p-extension for Embedded Domain Problems in Solid Mechanics”, Computational Mechanics, Vol. 41, pp. 121-133, 2007.
29. Duster, A., Parvizian, J., Yang, Z., and Rank, E., “The Finite Cell Method for Three-dimensional Problems of Solid Mechanics”, Computer Methods in Applied Mechanics and Engineering, Vol. 197, pp. 3768-3782, 2008.
30. Dauge, M., Duster, A., and Rank, E., “Theoretical and Numerical Investigation of the Finite Cell Method”, Journal of Scientific Computing, doi: 10.1007/s10915-015-9997-3, 2015.
31. Zander, N., Bog, T., Elhaddad, M., Espinoza, R., Hu, H., Joly, A., Wu, C., Zerbe, P., Duster, A., Kollmannsberger, S., Parvizian, J., Ruess, M., Schillinger, D., and Rank, E., “FCMLab: a Finite Cell Research Toolbox for MATLAB”, Advances in Engineering Software, Vol. 74, pp. 49-63, 2014.
32. Rank, E., Ruess, M., Kollmannsberger, S., Schillinger, D., and Duster. A., “Geometric Modeling, Isogeometric Analysis and the Finite Cell Method”, Computer Methods in Applied Mechanics and Engineering, Vol. 249-252, pp. 104-115, 2012.
33. Schillinger, D., and Rank, E., “An unfitted hp Adaptive Finite Element Method Based on Hierarchical B-splines for Interface Problems of Complex Geometry”, Computer Methods in Applied Mechanics and Engineering, Vol. 200, pp. 3358-3380, 2011.
34. Schillinger, D., Ruess, M., Zander, N., Bazilevs, Y., Duster, A., and Rank, E., “Small and Large Deformation Analysis with the Pand B-spline Versions of the Finite Cell Method”, Computational Mechanics, Vol. 50, pp. 445-478, 2012.
35. Abedian, A., Parvizian, J., Duster, A., and Rank, E., “The Finite Cell Method for the J2 Flow Theory of Plasticity”, Finite Elements in Analysis and Design, Vol. 69, pp. 37-47, 2013.
36. Abedian, A., Parvizian, J., Duster, A., and Rank. E., “Finite Cell Method Compared to h-version Finite Element Method for Elasto-plastic Problems”, Applied Mathematics and Mechanics, Vol. 35, No. 10, pp. 1239-1248, 2014.
37. Ranjbar, M., Mashayekhi, M., Parvizian, J., Duster, A., and Rank, E., “Using the Finite Cell Method to Predict Crack Initiation in Ductile Materials”, Computational Material Science, Vol. 82, pp. 427-434, 2014.
38. Ranjbar, M., Mashayekhi, M., Parvizian, J., Duster, A., and Rank, E., “The Finite Cell Method Applied to Nonlocal Damage Mechanics”, World Congress on Computational Mechanics (WCCM XI), Barcelona, Spain, 2014.
39. حدادگر، ا.، مشایخی م.، و جمشید پرویزیان، ج.، "کاربرد روش سلول محدود در پیشبینی آسیب نرم با درنظر گرفتن اثر بسته شدن ترکها"، مجله مهندسی مکانیک مدرس، دوره 14، شماره 13، صص. 118-107.
40. de Souza Neto, E. A., Peric, D., Dutko, M., and Owen, D. R. J., “Design of Simple Low Order Finite Elements for Large Strain Analysis of Nearly Incompressible Solids”, International Journal of Solids and Structures, Vol. 33, pp. 3277-3296, 1996.
41. Duster, A., Broker, H., and Rank, E., “The p-version of the Finite Element Method for Three-dimensional Curved Thin Walled Structures”, International Journal for Numerical Methods in Engineering, Vol. 52, pp. 673-703, 2001.
42. Ruess, M., Tal, D., Trabelsi, N., Yosibash, Z., and Rank, E., “The Finite Cell Method for Bone Simulations: Verification and Validation”, Biomechanics and Modeling in Mechanobiology, Vol. 11, pp. 425-437, 2012.
43. Abedian, A., Parvizian, J., Duster, A., Khademyzadeh, H., and Rank, E., “Performance of Different Integration Schemes in Facing Discontinuites in the Finite Cell Method”, International Journal of Computational Methods, Vol. 10, No. 3, 1350002/1-24, 2013.
44. Duster, A., and Kollmannsberger, S., AdhoC 4 - User's Guide. Lehrstuhl fur Computation in Engineering, TU Munchen, Numerische Strukturanalyse mit Anwendungen in der Schiffstechnik, TU Hamburg-Harburg, 2010.
45. http://www.pardiso-project.org/.
46. http://www.paraview.org/.
47. Simo J. C., and Miehe, C., “Geometrically Non-linear Enhanced Strain Mixed Methods and the Method of Incompatible Modes”, Computer Methods in Applied Mechanics and Engineering, Vol. 98, pp. 41-104, 1992.
48. de Souza Neto, E. A., Peric, D., Dutko, M., and Owen, D. R. J., “Design of Simple Low Order Finite Elements for Large Strain Analysis of Nearly Incompressible Solids”, International Journal of Solids and Structures, Vol. 33, pp. 3277-3296, 1996.
49. Elguedj, T., and Hughes, T. J. R., Isogeometric Analysis of Nearly Incompressible Large Strain Plasticity”, Computer Methods in Applied Mechanics and Engineering, Vol. 268, pp. 388-416, 2014.
50. Crisfield, M. A., “A Fast Incremental/iterative Solution Procedure that Handles snap-through”, Computers & Structures, Vol. 13, pp. 55-62, 1981.
2. Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z., The Finite Element Method -Its Basis and Fundamentals, Vol. 1. Butterworth-Heinemann, 7th Edition, 2013.
3. Szabo, B. A., Duster, A., and Rank, E., The p-version of the Finite Element Method, pp. 119-139, In: Stein, E., de Borst, R., and Hughes, T. J. R., (Eds.), Encyclopedia of Computational Mechanics, Vol. 1, Chapter. 5, John Wiley & Sons, 2004.
4. de Souza Neto, E. A., Peric, D., and Owen, D. R. J., Computational Methods for Plasticity, Theory and Applications, John Wiley & Sons, 2008.
5. Wriggers, P., Nonlinear Finite-Element-Methods, Springer-Verlag, 2008.
6. Duster, A., Hartmann, S., and Rank, E., “p-FEM Applied to Finite Isotropic Hyperelastic Bodies”, Computer Methods in Applied Mechanics and Engineering, Vol. 192, pp. 5147-5166, 2003.
7. Lipton, S., Evans, J. A., Bazilevs, Y., Elguedj, T., and Hughes, T. J. R., “Robustness of Isogeometric Structural Discretizations under Severe Mesh Distortion”, Computer Methods in Applied Mechanics and Engineering, Vol. 199, pp. 357-373, 2010.
8. Rank, E., Broker, H., Duster, A., Krause, R., and Rucker, M., The p-version of the Finite Element Method for Structural Problems, pp. 263-307, In: Stein, E., (Eds.), Error-controlled Adaptive Finite Elements in Solid Mechanics, Chapter. 8, John Wiley & Sons, 2002.
9. Kiralyfalvi, G., and Szabo, B. A., Quasi-regional Mapping for the p-version of the Finite Element Method, Finite Elements in Analysis and Design, Vol. 27, pp. 85-97, 1997.
10. Cottrell, J. A., Hughes, T. J. R., and Bazilevs, Y., Isogeometric Analysis: Towards Integration of CAD and FEM, John Wiley & Sons, 2009.
11. Szabo, B. A., and Babuska, I., Introduction to Finite Element Analysis: Formulation, Verification, and Validation, Wiley-Blackwell, 2011.
12. Krause, R., Mucke, R., and Rank, E., “hp-version Finite Elements for Geometrically Nonlinear Problems”, Communications in Numerical Methods in Engineering, Vol. 11, pp. 887-897, 1995.
13. Yosibash, Z., and Priel, E., “High-order Fems for Thermohyperelasticity at Finite Strains”, Computers & Mathematics with Applications, Vol. 67, pp. 477-496, 2014.
14. Yosibash, Z., Weissa, D., and Hartmann, S., “p-fems for Hyperelastic Anisotropic Nearly Incompressible Materials under Finite Deformations with Applications to Arteries Simulation”, International Journal for Numerical Methods in Engineering, Vol. 88, pp. 1152-1174, 2011.
15. Netz, T., Duster, A., and Hartmann, S., “High-order Finite Elements Compared to Low-order Mixed Element Formulations”, ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik, Vol. 93, No. 2-3, pp. 163-176, 2013.
16. Babuska, I., and Suri, M., “On Locking and Robustness in the Finite Element Method”, SIAM Journal on Numerical Analysis, Vol. 29, pp. 1261-1293, 1992.
17. Suri, M., “Analytical and Computational Assessment of Locking in the hp Finite Element Method”, Computer Methods in Applied Mechanics and Engineering, Vol. 133, pp. 347-371, 1996.
18. Heisserer, U., Hartmann, S., Duster, A., and Yosibash. Z., “On Volumetric Locking-free Behavior of p-version Finite Elements under Finite Deformations”, Communications in Numerical Methods in Engineering, Vol. 24, pp. 1019-1032, 2008.
19. Holzer, S., and Yosibash, Z., “The p-version of the Finite Element Method in Incremental Elasto-plastic Analysis”, International Journal for Numerical Methods in Engineering, Vol. 39, pp. 1859-1878, 1996.
20. Szabo, B. A., Actis, R., and Holzer, S., Solution of Elastic-plastic Stress Analysis Problems by the p-version of the Finite Element Method, pp. 395-416, In: Babuska, I., and Flaherty, J., (Eds.), Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations, IMA Volumes in Mathematics and its Applications, Vol. 75, Springer, NewYork, 1995.
21. Duster, A., and Rank, E., “The p-version of the Finite Element Method Compared to an Adaptive h-version for the Deformation Theory of Plasticity”, Computer Methods in Applied Mechanics and Engineering, Vol. 190, pp. 1925-1935, 2001.
22. Duster, A., and Rank. E., “A p-version Finite Element Approach for Two- and Three-dimensional Problems of the J2 Flow Theory with Non-linear Isotropic Hardening”, International Journal for Numerical Methods in Engineering, Vol. 53, pp. 49-63, 2002.
23. Duster, A., Niggl, A., Nubel, V., and Rank, E., “A Numerical Investigation of High-order Finite Elements for Problems of Elastoplasticity”, Journal of Scientific Computing, Vol. 17, pp. 429-437, 2002.
24. Tin-Loi, F., and Ngo, N. S., “Performance of the p-version Finite Element Method for Limit Analysis”, International Journal of Mechanical Sciences, Vol. 45, pp. 1149-1166, 2003.
25. Tin-Loi F., and Ngo, N. S., “Performance of a p-Adaptive Finite Element Method for Shakedown Analysis”, International Journal of Mechanical Sciences, Vol. 49, pp. 1166-1178, 2007.
26. Heisserer, U., Hartmann, S., Duster, A. D., Bier, W., Yosibash, Z., and Rank, E., “p-fem for Finite Deformation Powder Compaction”, Computer Methods in Applied Mechanics and Engineering, Vol. 197, pp. 727-740, 2008.
27. Yosibash, Z., Hartmann, S., Heisserer, U., Duster, A., Rank, E., and Szanto, M., “Axisymmetric Pressure Boundary Loading for Finite Deformation Analysis using p-FEM”, Computer Methods in Applied Mechanics and Engineering, Vol. 196, pp. 1261-1277, 2007.
28. Parvizian, J., Duster, A., and Rank, E. “Finite Cell Method - h- and p-extension for Embedded Domain Problems in Solid Mechanics”, Computational Mechanics, Vol. 41, pp. 121-133, 2007.
29. Duster, A., Parvizian, J., Yang, Z., and Rank, E., “The Finite Cell Method for Three-dimensional Problems of Solid Mechanics”, Computer Methods in Applied Mechanics and Engineering, Vol. 197, pp. 3768-3782, 2008.
30. Dauge, M., Duster, A., and Rank, E., “Theoretical and Numerical Investigation of the Finite Cell Method”, Journal of Scientific Computing, doi: 10.1007/s10915-015-9997-3, 2015.
31. Zander, N., Bog, T., Elhaddad, M., Espinoza, R., Hu, H., Joly, A., Wu, C., Zerbe, P., Duster, A., Kollmannsberger, S., Parvizian, J., Ruess, M., Schillinger, D., and Rank, E., “FCMLab: a Finite Cell Research Toolbox for MATLAB”, Advances in Engineering Software, Vol. 74, pp. 49-63, 2014.
32. Rank, E., Ruess, M., Kollmannsberger, S., Schillinger, D., and Duster. A., “Geometric Modeling, Isogeometric Analysis and the Finite Cell Method”, Computer Methods in Applied Mechanics and Engineering, Vol. 249-252, pp. 104-115, 2012.
33. Schillinger, D., and Rank, E., “An unfitted hp Adaptive Finite Element Method Based on Hierarchical B-splines for Interface Problems of Complex Geometry”, Computer Methods in Applied Mechanics and Engineering, Vol. 200, pp. 3358-3380, 2011.
34. Schillinger, D., Ruess, M., Zander, N., Bazilevs, Y., Duster, A., and Rank, E., “Small and Large Deformation Analysis with the Pand B-spline Versions of the Finite Cell Method”, Computational Mechanics, Vol. 50, pp. 445-478, 2012.
35. Abedian, A., Parvizian, J., Duster, A., and Rank, E., “The Finite Cell Method for the J2 Flow Theory of Plasticity”, Finite Elements in Analysis and Design, Vol. 69, pp. 37-47, 2013.
36. Abedian, A., Parvizian, J., Duster, A., and Rank. E., “Finite Cell Method Compared to h-version Finite Element Method for Elasto-plastic Problems”, Applied Mathematics and Mechanics, Vol. 35, No. 10, pp. 1239-1248, 2014.
37. Ranjbar, M., Mashayekhi, M., Parvizian, J., Duster, A., and Rank, E., “Using the Finite Cell Method to Predict Crack Initiation in Ductile Materials”, Computational Material Science, Vol. 82, pp. 427-434, 2014.
38. Ranjbar, M., Mashayekhi, M., Parvizian, J., Duster, A., and Rank, E., “The Finite Cell Method Applied to Nonlocal Damage Mechanics”, World Congress on Computational Mechanics (WCCM XI), Barcelona, Spain, 2014.
39. حدادگر، ا.، مشایخی م.، و جمشید پرویزیان، ج.، "کاربرد روش سلول محدود در پیشبینی آسیب نرم با درنظر گرفتن اثر بسته شدن ترکها"، مجله مهندسی مکانیک مدرس، دوره 14، شماره 13، صص. 118-107.
40. de Souza Neto, E. A., Peric, D., Dutko, M., and Owen, D. R. J., “Design of Simple Low Order Finite Elements for Large Strain Analysis of Nearly Incompressible Solids”, International Journal of Solids and Structures, Vol. 33, pp. 3277-3296, 1996.
41. Duster, A., Broker, H., and Rank, E., “The p-version of the Finite Element Method for Three-dimensional Curved Thin Walled Structures”, International Journal for Numerical Methods in Engineering, Vol. 52, pp. 673-703, 2001.
42. Ruess, M., Tal, D., Trabelsi, N., Yosibash, Z., and Rank, E., “The Finite Cell Method for Bone Simulations: Verification and Validation”, Biomechanics and Modeling in Mechanobiology, Vol. 11, pp. 425-437, 2012.
43. Abedian, A., Parvizian, J., Duster, A., Khademyzadeh, H., and Rank, E., “Performance of Different Integration Schemes in Facing Discontinuites in the Finite Cell Method”, International Journal of Computational Methods, Vol. 10, No. 3, 1350002/1-24, 2013.
44. Duster, A., and Kollmannsberger, S., AdhoC 4 - User's Guide. Lehrstuhl fur Computation in Engineering, TU Munchen, Numerische Strukturanalyse mit Anwendungen in der Schiffstechnik, TU Hamburg-Harburg, 2010.
45. http://www.pardiso-project.org/.
46. http://www.paraview.org/.
47. Simo J. C., and Miehe, C., “Geometrically Non-linear Enhanced Strain Mixed Methods and the Method of Incompatible Modes”, Computer Methods in Applied Mechanics and Engineering, Vol. 98, pp. 41-104, 1992.
48. de Souza Neto, E. A., Peric, D., Dutko, M., and Owen, D. R. J., “Design of Simple Low Order Finite Elements for Large Strain Analysis of Nearly Incompressible Solids”, International Journal of Solids and Structures, Vol. 33, pp. 3277-3296, 1996.
49. Elguedj, T., and Hughes, T. J. R., Isogeometric Analysis of Nearly Incompressible Large Strain Plasticity”, Computer Methods in Applied Mechanics and Engineering, Vol. 268, pp. 388-416, 2014.
50. Crisfield, M. A., “A Fast Incremental/iterative Solution Procedure that Handles snap-through”, Computers & Structures, Vol. 13, pp. 55-62, 1981.
)