曹艳平，清华大学航天航空学院工程力学系教授，博士生导师。
教育背景1998/09 – 2002/07，清华大学，工程力学系，博士
1996/09 – 1998/07，清华大学，工程力学系，硕士
1992/09 – 1996/07，清华大学，工程力学系，学士
工作履历2014/12–至今，清华大学，航天航空学院工程力学系，教授。
2008/09 – 2014/12，清华大学，航天航空学院工程力学系，副研究员。
2007/08 – 2008/09，德国杜赛尔多夫马普所，科研人员（Research Scientist）。
2005/11-2007/07, 卡尔斯鲁厄研究中心/德国杜赛尔多夫马普所，洪堡访问（Humboldt Research Fellow）。
2002/07-2005/07，法国特鲁瓦技术大学，机械与系统工程系，博士后。
研究领域固体力学，生物力学
研究概况曹艳平的研究兴趣涉及超声弹性成像、软材料和生物材料力学、接触力学、计算力学等领域。从2002年至今，在国内外期刊发表论文100余篇，其中SCI论文99篇，包括《J. Mech. Phys. Solids》11篇，《Soft Matter》8篇，《Acta Mater.》2篇。另有多篇论文发表在《Angew. Chem. Int. Ed.》、《Phys. Rev. Lett.》、《Nano Lett.》、《ACS Nano》、《Med. Image Anal.》、《Acta Biomater.》、《J. Biomech.》、《Biomech. Model. Mechanobiol.》、《Int. J. Solids Struct.》、《Int. J. Numer. Meth. Eng.》、《Proc. Roy. Soc. A》、《Langmuir》、《Appl. Phys. Lett.》等国际知名期刊。
奖励与荣誉2005-2007,洪堡基金会“Research Fellow”
2009年入选“清华大学基础研究青年人才计划”
2012年入选“教育部新世纪优秀人才支持计划”
学术成果(1). Mechanics of ultrasound elastography
[100] He Q., Li G. Y., Lee F. F., Zhang Q., Cao Y. P., *Luo J. W. (2017): A novel method for vessel cross-sectional shear wave imaging, Ultrasound in Medine and Biology, in press.
[99] Li G. Y. and *Cao Y. P. (2017): Mechanics of ultrasound elastography, Proceedings of Royal Society A. 473, ** (Invited review).
[98] Li G. Y., He Q., Mangan R., Luo J. W., Destrade M., *Cao Y. P. (2017): Guided waves in prestressed hyperelastic plates and tubes: Application to the ultrasound elastography of thin-walled soft materials, Journal of the Mechanics and Physics of Solids 102, 67-79.
[97] Li G. Y., He Q., Xu G. Q., Jia L., Liu Y. L., Luo J. W., *Cao Y. P. (2017): An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves, Journal of Biomechanics 51, 97-104.
[96] Li G. Y., He Q., Jia L., He P., Luo J. W., *Cao Y. P. (2017): An inverse method to determine the arterial stiffness with guided axial waves, Ultrasound in Medine and Biology 43, 505-516.
[95] Liu Y. L., Li G. Y., He P., Mao Z. Q., *Cao Y. P. (2017): Temperature-dependent elastic properties of brain tissues measured with the shear wave elastography method, Journal of the Mechanical Behavior of Biomedical Materials 65, 652-656.
[94] Li G. Y., Zheng Y., Liu Y. L., Destrate M., *Cao Y. P. (2016): Elastic Cherenkov effect in transversely isotropic soft materials-I: Theoretical analysis, simulations and inverse method, Journal of the Mechanics and Physics of Solids 96, 388-410.
[93] Li G. Y., He Q., Qian L. X., Liang S., Liu Y. L., Yang X. Y., Luo J. W., *Cao Y. P. (2016): Elastic Cherenkov effect in transversely isotropic soft materials-II: Ex vivo and in vivo experiments, Journal of the Mechanics and Physics of Solids 94, 181-190.
[92] Luo C. C., Qian L. X., Li G. Y., Jiang Y., Liang S., *Cao Y. P. (2015): Determining the in vivo elastic properties of dermis layer of human skin using the supersonic shear imaging technique and inverse analysis, Medical Physics 42, 4106-4115.
[91] Jiang Y., Li G. Y., Qian L. X., Liang S., *Destrade M., *Cao Y. P. (2015): Measuring the linear and non-linear elastic properties of brain tissue with shear waves and inverse analysis, Biomechanics and Modeling in Mechanobiology 14, 1119-1128.
[90] Jiang Y., Li G. Y., Qian L. X., Hu X. D., Liu D., Liang S., *Cao Y. P. (2015): Characterization of the nonlinear elastic properties of soft tissues using the supersonic shear imaging (SSI) technique: Inverse method, ex vivo and in vivo experiments, Medical Image Analysis 20, 97-111.
(2). Mechanics of soft materials and soft-matter composites
[89] Zheng Y., Li G. Y., *Cao Y. P., Feng X. Q. (2017): Wrinkling of a stiff-film resting on a fiber-filled soft substrate and its potential application as tunable metamaterials, Extreme Mechanics Letters 11, 121-127 (Special Issue on 3D Assembly by Cutting, Bending and Folding).
[88] Yang L., Zhao Y., Xu W. J., Shi E. Z., Wei W.J., Li X. M., Cao A. Y., *Cao Y. P., *Fang Y. (2017): Highly crumpled all-carbon transistors for brain activity recording, Nano Letters 17, 71-77.
[87] Li G. Y., Zheng Y., *Cao Y. P. (2016): Tunable defect mode in a soft wrinkled bilayer system, Extreme Mechanics Letters 9, 171-174.
[86] Liu Y. P., Zhang L. Q., Mo C., Cao Y. P., Wu W. G., *Wang W. (2016): Caulking polydimethylsiloxane molecular network by thermal chemical vapor deposition of parylene-C, Lab on a Chip 16, 4220-4229.
[85] Shao Z. C., Zhao Y., Zhang W. Y., *Cao Y. P., Feng X. Q. (2016): Curvature induced hierarchical wrinkling patterns in soft bilayers, Soft Matter 12, 7977-7982.
[84] Ji H. P., Zhao Y., Zong C. Y., Xie J. X., Han X., Wang J. J., Zhao J. X., Jiang S. C., *Cao Y. P., *Lu C. H. (2016): A Simple and versatile strategy to prevent surface wrinkling by visible light irradiation, ACS Applied Materials & Interfaces, 8 (29), 19127–19134.
[83] Li G. Y., Zheng Y., *Cao Y. P., Feng X. Q., Zhang W. Y. (2016): Controlling elastic wave propagation in a soft bilayer system via wrinkling-induced stress patterns, Soft Matter 12, 4204-4213.
[82] Zong C. Y., Zhao Y., Ji H. P., Xie J. X., Han X., Wang J. J., Cao Y. P., *Jiang S. C., *Li H. F., *Lu C. H. (2016): Patterning surfaces on Azo-based multilayer films via surface wrinkling combined with visible light irradiation, Macromol. Rapid Commun., 37, 1288-1294.
[81] Huang X., Li B., Hong W., Cao Y. P., *Feng X. Q. (2016): Effects of tension–compression asymmetry on the surface wrinkling of film–substrate systems, Journal of the Mechanics and Physics of Solids 94, 88-104.
[80] Yang X., Zhao Y., Xie J., Han X., Wang J. J., Zong C. Y., Ji H. P., Zhao J. X., Jiang S. C., *Cao Y. P., *Lu C. H. (2016): Bioinspired fabrication of free-standing conducting films with hierarchical surface wrinkling patterns, Acs Nano 10, 3801-3808.
[79] Zong C. Y., Zhao Y., Ji H. P., Han X., Xie J. X., Wang J. J., *Cao Y. P., *Jiang S. C., *Lu C. H. (2016): Tuning and erasing surface wrinkles by reversible visible-light-induced photoisomerization, Angew. Chem. Int. Ed. 55, 3931-3935 (Selected as the very important paper by the journal).
[78] Zhao Y., Li J., *Cao Y. P., Feng X. Q. (2016): Buckling of an elastic fiber with finite length in a soft matrix, Soft Matter 12, 2086-2094.
[77] Wang J. W., Li B., Cao Y. P., *Feng X. Q., Gao H. J. (2016): Wrinkling micropatterns regulated by a hard skin layer with a periodic stiffness distribution on a soft material, Applied Physics Letters 108, 021903.
[76] Zhao Y., Han X., Li G. Y., *Lu C. H., *Cao Y. P., Feng X. Q., Gao H. J. (2015): Effect of lateral dimension on the surface wrinkling of a thin film on compliant substrate induced by differential growth/swelling, Journal of the Mechanics and Physics of Solids 83, 129-145.
[75] Han X., Zhao Y., *Cao Y. P., *Lu C. H. (2015): Controlling and prevention of surface wrinkling via size-dependent critical wrinkling strain, Soft Matter 11, 4444-4452.
[74] Wang J. W., Li B., Cao Y. P., *Feng X. Q. (2015): Surface wrinkling patterns of film–substrate systems with a structured interface, Journal of Applied Mechanics-ASME 82, 051009.
[73] Jia F., Li B., *Cao Y. P., *Feng X. Q. (2015): Wrinkling pattern evolution of cylindrical biological tissueswith differential growth, Physical Review E 91, 012403.
[72] Zhao Y., *Cao Y. P., Hong W., Wadee M. K., Feng X. Q. (2015): Towards a quantitative understanding of period-doubling wrinkling patterns occuring in film/substrate bilayer systems, Proceedings of Royal Society A. 471, **.
[71] Zhao Y., *Cao Y. P., Feng X. Q., Ma K. (2014): Axial compression-induced wrinkles on a core-shell soft cylinder: Theoretical analysis, simulations and experiments, Journal of the Mechanics and Physics of Solids 73, 212-227.
[70] Yin J., Han X., Cao Y. P., *Lu C. H. (2014): Surface wrinkling on polydimethylsiloxane microspheres via wet surface chemical oxidation, Scientific Report 4, 5710.
[69] Xie W. H., Huang X., Cao Y. P., Li B., *Feng X. Q. (2014): Buckling and postbuckling of stiff lamellae in a compliant matrix, Composite Science and Technology 99, 89-95.
[68] Chen K. L., *Cao Y. P., Zhang M. G., Feng X. Q. (2014): Indentation triggered pattern transformation in hyperelastic soft cellular solids. Comptes Rendus Mécanique 342, 292-298.
[67] Zhang M. G., *Cao Y. P., Li G. Y., Feng X. Q. (2014): Pipette aspiration of hyperelastic compliant materials: Theoretical analysis, simulations and experiments. Journal of the Mechanics and Physics of Solids 68, 179-196.
[66] *Cao Y. P., Li G. Y., Zhang M. G., Feng X. Q. (2014): Determination of the reduced creep function of viscoelastic compliant materials using pipette aspiration method, Journal of Applied Mechanics 81, 071006.
[65] Jia F., *Cao Y. P., Zhao Y., Feng X. Q. (2014): Buckling and surface wrinkling of an elastic graded cylinder with elastic modulus arbitrarily varying along radial direction. International Journal of Applied Mechanics 6, **.
[64] Wang J. W., Cao Y. P., *Feng X. Q. (2014): Archimedean spiral wrinkles on a film-substrate system induced by torsion, Applied Physics Letters 104, 031910.
[63] Xie W. H., Li B., Cao Y. P., *Feng X. Q. (2014): Effects of internal pressure and surface tension on the growth-induced wrinkling of mucosae. Journal of the Mechanical Behavior of Biomedical Materials 29， 594-601.
[62] Liang X. D., Zu Y., Cao Y. P., *Yang C. (2013): A dual-scale model for the caveolae-mediated vesiculation. Soft Matter 9, 7981-7987.
[61] Yang Y., Han X., Ding W. L., Jiang S. C., *Cao Y. P., *Lu C. H. (2013): Controlled free edge effects in surface wrinkling via combination of external straining and selective O2 plasma exposure. Langmuir 29, 7170-7177.
[60] Ding W. L., Yang Y., Zhao Y., Jiang S.C., *Cao Y. P., *Lu C. H. (2013): Well-defined orthogonal surface wrinkles directed by the wrinkled boundary. Soft Matter 9, 3720-3726.
[59] *Cao Y. P., Jiang Y., Li B., *Feng X. Q. (2012): Biomechanical modeling of surface wrinkling of soft tissues with growth-dependent mechanical properties. Acta Mechanica Solida Sinica 25, 483-492. (Invited paper)
[58] *Cao Y. P., Jia F., *Feng X. Q., Zhao Y., Yu S. W. (2012): Buckling and post-buckling of a stiff film resting on an elastic graded substrate. International Journal of Solids and Structures 49, 1656-1664.
[57] Cao Y. P. and *Hutchinson J. W. (2012): Wrinkling phenomena in neo-Hookean film/substrate bilayers. Journal of Applied Mechanics 79, 031019. (Dedicated to Prof. James R Rice for his 70th birthday)
[56] Li B., Cao Y. P., *Feng X. Q., Gao H. J. (2012): Mechanics of morphological instabilities and surface wrinkling in soft materials: A review. Soft Matter 8, 5728. (Invited review).
[55] Zang J. F., Zhao X. H., *Cao Y. P., *Hutchinson J. W. (2012): Localized ridge wrinkling of stiff films on compliant substrates. Journal of the Mechanics and Physics of Solids 60, 1265-1279.
[54] Jia F., *Cao Y. P., LiuT. S., Jiang Y., Feng X. Q., Yu S. W. (2012): Wrinkling of a bilayer resting on a soft substrate under in-plane compression. Philosophical Magazine 92, 1554-1568.
[53] Cao Y. P., Li B., *Feng X. Q. (2012): Surface wrinkling and folding of core–shell soft cylinders. Soft Matter 8, 556-562.
[52] Cao Y. P. and *Hutchinson J. W. (2012): From wrinkles to creases in elastomers: The instability and imperfection-sensitivity of wrinkling. Proceedings of Royal Society A. 468, 94-115.
[51] *Cao Y. P., Zheng X. P., Jia F., *Feng X. Q. (2012): Wrinkling and creasing of a compressed elastoplastic film resting on a soft substrate. Computational Materials Science 57, 111-117.
[50] Li B., Jia F., Cao Y. P., *Feng X. Q., Gao H. J. (2011): Surface wrinkling patterns on a core–shell soft sphere. Physical Review Letters 106, 234301.
[49] Li B., Cao Y. P., *Feng X. Q., Yu S. W. (2011): Mucosal wrinkling in animal antra induced by volumetric growth. Applied Physics Letters 98, 153701.
[48] Li B., Cao Y. P., *Feng X. Q., Gao H. J. (2011): Surface wrinkling of mucosa induced by volumetric growth: Theory, simulation and experiment. Journal of the Mechanics and Physics of Solids 59, 758-774.
[47] Li B., Cao Y. P., *Feng X. Q. (2011): Growth and surface folding of esophageal mucosa: A biomechanical model. Journal of Biomechanics 44, 182–188.
[46] Jia F., Zheng X. P., *Cao Y. P., Feng X. Q. (2010): Theoretical study on the bilayer buckling technique for thin film metrology. CMC-Computers, Materials & Continua 18, 105.
[45] Zheng X. P., *Cao Y. P., Li B., Feng X. Q.,Yu S. W. (2010): Surface wrinkling of nanostructured thin films on a compliant substrate. Computational Materials Science 49, 767.
[44] Zheng X. P., Cao Y. P., Li B., *Feng X. Q., Wang G. F. (2010): Surface effects in various bending-based test methods for measuring the elastic property of nanowires. Nanotechnology 21, 205702.
[43] Cao Y. P., Zheng X. P., Li B., *Feng X. Q. (2009): Determination of the elastic modulus of micro- and nanowires/tubes using a buckling-based metrology, Scripta Materialia 61, 1044-1047.
[42] Zheng X. P., *Cao Y. P., Li B., Feng X. Q., Jiang H. Q., Huang Y. G. (2009): Determining the elastic modulus of thin films using a buckling-based method: Computational study. Journal of Physics D: Applied Physics 42, 175506.
(3). Mechanics of indentation
[41] Zhang M. G., Chen J. J, Feng X. Q., *Cao Y. P. (2014): On the applicability of Sneddon’s solution for interpreting the indentation of nonlinear elastic biopolymers, Journal of Applied Mechanics-ASME 81(9), 091011.
[40] Zhang M. G., *Cao Y. P., Li G. Y., Feng X. Q. (2014): Spherical indentation method for determining the constitutive parameters of hyperelastic soft materials, Biomechanics and Modeling in Mechanobiology 13 (1), 1-11.
[39] *Cao Y. P., Zhang M. G., Feng X. Q. (2013): Indentation method for measuring the viscoelastic kernel function of nonlinear viscoelastic soft materials, Journal of Materials Research 28, 806-816.
[38] *Cao Y. P., Chen K. L. (2012): Theoretical and computational modeling of instrumented indentation of viscoelastic composites. Mechanics of Time-Dependent Materials 16, 1-18.
[37] *Cao Y. P., Ji X. Y., Feng X. Q. (2011): On determination of the damping factor of linear viscoelastic materials using dynamic indentation: A theoretical study. Science China G. 54, 598-605.
[36] *Qian X. Q., Cao Y. P., Liu Z. C., Lu J. (2010): Influence of hardening behaviour of elastoplastic materials on the determination of elastic modulus using indentation tests, Material Science and Engineering: A. 527, 4751-4575.
[35] *Cao Y. P., Ji X. Y., Feng X. Q. (2010): Geometry independence of the normalized relaxation functions of viscoelastic materials in indentation. Philosophical Magazine 90, 1639-1655.
[34] *Keerthika B., Cao Y. P., Raabe D. (2009): Mechanical characterization of viscoelastic-plastic soft matter using spherical indentation. CMC-Computers, Materials & Continua 10, 243-258.
[33] *Cao Y. P., Ma D., Raabe D. (2009): The use of flat punch indentation to determine the viscoelastic properties in the time and frequency domains of a soft layer bonded to a rigid substrate. Acta Biomaterialia 5, 240-248.
[32] *Cao Y. P., Xue Z. Y., *Chen X., Raabe D. (2008): Correlation between the flow stress and the nominal indentation hardness of soft metals. Scripta Materialia. 59, 518-521.
[31] Qian X. Q., *Cao Y. P., Zhang J. Y., Raabe D., Yao Z. H., Fei B. J. (2008): An inverse approach to determine the mechanical properties of elastoplastic materials using indentation tests. CMC-Computers, Materials & Continua 7, 33-42.
[30] *Cao Y. P. (2007): Determination of the creep exponent of a power-law creep solid using indentation tests. Mechanics of Time-dependent Materials 11, 159-173.
[29] Qian X. Q., *Cao Y. P., Lu J. (2007): Dependence of the representative strain on the hardening functions of metallic materials in indentation. Scripta Materialia 57, 57-60.
[28] Cao Y. P., Dao M., *Lu J. (2007): A precise correcting method for the study of the superhard material using nanoindentation tests. Journal of Materials Research 22, 1255-1264.
[27] *Cao Y. P., Qian X. Q., Huber N. (2007): Spherical indentation into elastoplastic materials: indentation-response based definitions of the representative strain. Material Science and Engineering: A. 454-455, 1-13.
[26] *Cao Y. P. and Huber N. (2006): Further investigation on the definition of the representative strain in conical indentation. Journal of Materials Research 21, 1810-1821.
[25] *Cao Y. P., Qian X. Q., Lu J. (2006): On the determination of reduced Young’s modulus and hardness of elastoplastic materials using a single sharp indenter. Journal of Materials Research 21, 215-224.
[24] Cao Y. P., Qian X. Q., *Lu J., Yao Z. H. (2005): An energy-based method to extract plastic properties of metal materials from conical indentation tests. Journal of Materials Research 20, 1194-1206.
[23] Cao Y. P. and *Lu J. (2005): Size-dependent sharp indentation-I: A closed-form expression of the indentation loading curve. Journal of the Mechanics and Physics of Solids 53, 33-48.
[22] Cao Y. P. and *Lu J. (2005): Size-dependent sharp indentation-II: A reverse algorithm to identify plastic properties of metallic materials. Journal of the Mechanics and Physics of Solids 53, 49-62.
[21] Cao Y. P. and *Lu J. (2004): A new method to extract plastic properties of metal materials from an instrumented spherical indentation loading curve. Acta Materialia 52, 4023-4032.
[20] Cao Y. P. and *Lu J. (2004): Depth-sensing instrumented indentation with dual sharp indenters: stability analysis and corresponding regularization schemes. Acta Materialia 52, 1143-1153.
[19] Cao Y. P. and *Lu J. (2004): A new scheme for computational modelling of conical indentation in plasticallygraded materials. Journal of Materials Research 19, 1704-1716.
(4). Other topics
[18] *Ning H. M., Li Y., *Hu N., Cao Y. P., Yan C., Azuma T., Peng X. H., Wu L. K., Li J. H. (2014): Improvement of the mode II interface fracture toughness of glass fiber reinforced plastics/aluminum laminates through vapor grown carbon fiber interleaves, Science and Technology of Advanced Materials, 15, 035004.
[17] Zhang L. Y., Li Y., Cao Y. P., *Feng X. Q., Gao H. J. (2014): Stiffness matrix based form-finding method of tensegrity structures. Engineering Structures 58, 36-48.
[16] Zhang L. Y., Li Y., Cao Y. P., *Feng X. Q., Gao H. J. (2013): A numerical method for simulating nonlinear mechanical responses of tensegrity structures under large deformations. Journal of Applied Mechanics-ASME 80, 061018.
[15] Zhang L. Y., Li Y., Cao Y. P., *Feng X. Q. (2013): A unified solution for self-equilibrium and super-stability of rhombic truncated regular polyhedral tensegrities. International Journal of Solids Structures 50, 234-245.
[14] Zhang L. Y., Li Y., Cao Y. P., *Feng X. Q., Gao H. J. (2012): Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: A unified analytical solution. Proceedings of Royal Society A. 468, 3323-3347.
[13] Nie A., Liu J. B., Wang P., *Wang H. T., Yang W., Gao Y., Lin C., Cao Y .P., Zhuang Z. (2012): Nanofingers pulled from bulk silver. Scripta Materialia 66, 247-249.
[12] *Hu N., Cai Y. D., Zhu G. J., Tsuji C., Liu Y. L., Alamusi, Cao Y. P. (2012): Characterization of damage size in metallic plates using Lamb waves. Structural Health Monitoring 11, 125-137.
[11] Liu Y. L., *Hu N., Alamusi, Watanabe T., Koshin Y., Cao Y. P., Fukunaga H. (2011): Relative reflection intensity of lamb waves from elliptically-shaped damages in metallic plates. Smart Materials and Structures 20, 075010.
[10] *Feng X. Q., Li Y., Cao Y. P., Gu Y. T. (2010): Design methods of rhombic tensegrity structures. Acta Mechanica Sinica 26, 559.
[9] Li Y., *Feng X. Q., Cao Y. P., Gao H. J. (2010): A Monte Carlo form-finding method for large scale regular and irregular tensegrity structures. International Journal of Solids Structures 47, 1888-1898.
[8] Ji X. Y., Cao Y. P., *Feng X. Q. (2010): Micromechanics prediction of the effective elastic moduli of graphene sheet-reinforced polymer nanocomposites. Modelling and Simulations in Material Science and Engineering. 18, 045005.
[7] Li Y, *Feng X. Q., Cao Y. P., Gao H. J.(2010): Constructing tensegrity structures from one-bar elementary cells. Proceedings of Royal Society A. 466, 45–61.
[6] Qian X. Q., *Yao Z. H., Cao Y. P., Lu J. (2005): An inverse approach to construct residual stresses existing in axisymmetric structures using BEM. Engineering Analysis with Boundary Elements 29, 986-999.
[5] Cao Y. P., *Hu N., Lu J., Fukunaga H., Yao Z. H. (2004): A new scheme for designing the penalty factor in 3-D penalty-equilibrating mixed elements. Communications in Numerical Methods in Engineering 20, 455-464.
[4] Qian X. Q., *Yao Z. H., Cao Y. P., Lu J. (2004): An inverse approach for constructing residual stress using BEM. Engineering Analysis with Boundary Elements 28, 205-211.
[3] Cao Y. P., *Hu N., Lu J., Yao Z. H. (2003): A highly accurate brick element based on a three-field variational principle for elasto-plastic analysis. Finite Elements in Analysis and Design 39, 1155-1171.
[2] Cao Y. P., *Hu N., Lu J., Yao Z. H. (2002): An inverse approach for constructing the residual stresses field induced by welding. Journal of Strain Analysis for Engineering Design 37, 345-359.
[1] Cao Y. P., *Hu N., Lu J., Fukunaga H., Yao Z. H. (2002): A 3-D brick element based on Hu-Washizu variational principle for mesh distortion. International Journal for Numerical Methods in Engineering 53, 2529-2548.