Ratel: Extensible, performance-portable solid mechanics¶
Ratel is a solid mechanics library and applications based on libCEED and PETSc with support for efficient high-order elements and CUDA and ROCm GPUs.
Solid mechanics simulations provide vital information for many engineering applications, using a large amount of computational resources from workstation to supercomputing scales. The industry standard for implicit analysis uses assembled sparse matrices with low-order elements, typically \(Q_1\) hexahedral and \(P_2\) tetrahedral elements, with the linear systems solved using sparse direct solvers, algebraic multigrid, or multilevel domain decomposition. This approach has two fundamental inefficiencies: poor approximation accuracy per Degree of Freedom (DoF) and high computational and memory cost per DoF due to choice of data structures and algorithms. High-order finite elements implemented in a matrix-free fashion with appropriate preconditioning strategies can overcome these inefficiencies.
For further details on the benefits of high-order, matrix-free finite elements for solid mechanics, see our preprint on arXiv.
Fig. 1 Ratel badger moving simulation¶
Indices and tables¶
Bilen Emek Abali, Andre Klunker, Emilio Barchiesi, and Luca Placidi. A novel phase-field approach to brittle damage mechanics of gradient metamaterials combining action formalism and history variable. 2021.
Marreddy Ambati, Tymofiy Gerasimov, and Laura De Lorenzis. A review on phase-field models of brittle fracture and a new fast hybrid formulation. Computational Mechanics, 55:383–405, 2015.
Hanen Amor, Jean-Jacques Marigo, and Corrado Maurini. Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. Journal of the Mechanics and Physics of Solids, 57(8):1209–1229, 2009.
Douglas N Arnold and Marie E Rognes. Stability of lagrange elements for the mixed laplacian. Calcolo, 46(4):245–260, 2009. doi:10.1007/s10092-009-0009-6.
Halphen B. and Q. S. Nguyen. Sur les materiaux standard generalises. Journal de Mécanique, 14:39–63, 1975.
Hudobivnik B. and Korelc J. Closed-form representation of matrix functions in the formulation of nonlinear material models. Finite Elements in Analysis and Design, 111:19–32, 2016. doi:10.1016/j.finel.2015.12.002.
Michael J. Borden, Thomas Joseph Robert Hughes, Chad M. Landis, Amin Anvari, and Isaac J. Lee. A phase-field formulation for fracture in ductile materials: finite deformation balance law derivation, plastic degradation, and stress triaxiality effects. Computer Methods in Applied Mechanics and Engineering, 312:130–166, 2016. doi:10.1016/j.cma.2016.09.005.
Franco Brezzi and Michel Fortin. Mixed and hybrid finite element methods. Volume 15. Springer Science & Business Media, 2012.
Miehe C. Aspects of the formulation and finite element implementation of large strain isotropic elasticity. International Journal for Numerical Methods in Engineering, 37(12):1981–2004, 1994. doi:10.1002/nme.1620371202.
Miehe C. Comparison of two algorithms for the computation of fourth-order isotropic tensor functions. Computers & Structures, 66(1):37–43, 1998. doi:10.1016/s0045-7949(97)00073-4.
D. Chapelle and K.J. Bathe. The inf-sup test. Computers & Structures, 47(4):537–545, 1993. URL: https://www.sciencedirect.com/science/article/pii/004579499390340J, doi:10.1016/0045-7949(93)90340-J.
Alexander Cheng. Poroelasticity. Springer Cham, Chichester New York, 2016. ISBN 978-3-319-25200-1. doi:10.1007/978-3-319-25202-5.
J.D. Clayton. Nonlinear mechanics of crystals. Springer, 2011.
J.D. Clayton. Nonlinear elastic and inelastic models for shock compression of crystalline solids. Springer, 2019.
B.D. Coleman and W. Noll. The thermodynamics of elastic materials with heat conduction and viscosity. Archive for Rational Mechanics and Analysis, 13:167–178, 1963.
J.J. Cross. Mixtures of fluids and isotropic solids. Archive of Mechanics, 6:1025–1039, 1973.
L. Davison. Fundamentals of Shock Wave Propagation in Solids. Springer, 2008.
Denis Davydov, Jean-Paul Pelteret, Daniel Arndt, Martin Kronbichler, and Paul Steinmann. A matrix-free approach for finite-strain hyperelastic problems using geometric multigrid. International Journal for Numerical Methods in Engineering, 121(13):2874–2895, 2020. doi:10.1002/nme.6336.
R. de Boer. Trends in Continuum Mechanics of Porous Media: Theory and Applications of Transport in Porous Media. Springer, 2005.
R. de Boer and W. Ehlers. A Historical Review of the Formulation of Porous Media Theories. Acta Mechanica, 74:1–8, 1988.
Perić D. de Souza Neto E.A. and Owen R.J. Continuum modelling and numerical simulation of material damage at finite strains. Archives of Computational Methods in Engineering, 5(311):311–384, 1998. doi:10.1007/BF02905910.
Boyang Ding, Alexander H.-D. Cheng, and Zhanglong Chen. Fundamental solutions of poroelastodynamics in frequency domain based on wave decomposition. Journal of Applied Mechanics, 80(6):061021, 08 2013. doi:10.1115/1.4023692.
Lee E.H. Elastic-plastic deformation at finite strains. Journal of Applied Mechanics, 36(1):65–76, 1969. doi:10.21236/ad0678483.
W. Ehlers. Foundations of multiphasic and porous materials. In W. Ehlers and J. Bluhm, editors, Porous Media: Theory, Experiments and Numerical Applications, pages 3–86. Springer Berlin Heidelberg, Berlin, Heidelberg, 2002. doi:10.1007/978-3-662-04999-0_1.
Adrian Luis Eterovic and Klaus-Jurgen Bathe. A hyperelastic-based large strain elasto-plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures. International Journal for Numerical Methods in Engineering, 30(6):1099–1114, 1990.
Eduardo Fancello, Jean-Philippe Ponthot, and Laurent Stainier. A variational formulation of constitutive models and updates in non-linear finite viscoelasticity. International Journal for Numerical Methods in Engineering, 65(11):1831–1864, 2006.
Gilles A Francfort and J-J Marigo. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 46(8):1319–1342, 1998.
Johannes Friedlein, Julia Mergheim, and Paul Steinmann. Observations on additive plasticity in the logarithmic strain space at excessive strains. International Journal of Solids and Structures, 239:111416, 2022.
T. Christian Gasser, Ray W Ogden, and Gerhard A Holzapfel. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. Journal of The Royal Society Interface, 3(6):15–35, September 2005. doi:10.1098/rsif.2005.0073.
M.E. Gurtin, E. Fried, and L. Anand. The mechanics and thermodynamics of continua. Cambridge University Press, 2010. doi:10.1017/CBO9780511762956.
Gerhard Holzapfel. Nonlinear solid mechanics: a continuum approach for engineering. Wiley, Chichester New York, 2000. ISBN 978-0-471-82319-3.
Z.T. Irwin. A Multiphase Continuum Mechanics Model for Shock Loading of Soft Porous Materials. PhD thesis, University of Colorado Boulder, 2024.
Zachariah T. Irwin, John D. Clayton, and Richard A. Regueiro. A large deformation multiphase continuum mechanics model for shock loading of soft porous materials. International Journal for Numerical Methods in Engineering, 125(6):e7411, 2024. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.7411, arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.7411, doi:https://doi.org/10.1002/nme.7411.
Simo J.C. and Taylor R.L. Consistent tangent operators for rate-independent elastoplasticity. Computer Methods in Applied Mechanics and Engineering, 48(1):101–118, 1985. doi:10.1016/0045-7825(85)90070-2.
Charlotte Kuhn, Alexander Schlüter, and Ralf Müller. On degradation functions in phase field fracture models. Computational Materials Science, 108:374–384, 2015.
Anders Logg, Kent-Andre Mardal, and Garth Wells. Automated solution of differential equations by the finite element method: The FEniCS book. Volume 84. Springer Science & Business Media, 2012. doi:10.1007/978-3-642-23099-8.
Filipe Marques, Paulo Flores, J. C. Pimenta Claro, and Hamid M. Lankarani. A survey and comparison of several friction force models for dynamic analysis of multibody mechanical systems. Nonlinear Dynamics, 86(3):1407–1443, November 2016. doi:10.1007/s11071-016-2999-3.
Christian Miehe, N Apel, and Matthias Lambrecht. Anisotropic additive plasticity in the logarithmic strain space: modular kinematic formulation and implementation based on incremental minimization principles for standard materials. Computer methods in applied mechanics and engineering, 191(47-48):5383–5425, 2002. doi:10.1016/S0045-7825(02)00438-3.
Christian Miehe, Martina Hofacker, and Fabian Welschinger. A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering, 199(45-48):2765–2778, 2010.
Christian Miehe, Fabian Welschinger, and Martina Hofacker. Thermodynamically consistent phase-field models of fracture: variational principles and multi-field fe implementations. International journal for numerical methods in engineering, 83(10):1273–1311, 2010.
Rabii Mlika. Nitsche method for frictional contact and self-contact: Mathematical and numerical study. phdthesis, Université de Lyon, Jan 2018. URL: https://theses.hal.science/tel-02067118.
Patrizio Neff, Bernhard Eidel, and Robert J Martin. Geometry of logarithmic strain measures in solid mechanics. Archive for Rational Mechanics and Analysis, 222:507–572, 2016.
Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer-Verlag, New York, 1999.
D. R. Nolan, A. L. Gower, M. Destrade, R. W. Ogden, and J. P. McGarry. A robust anisotropic hyperelastic formulation for the modelling of soft tissue. Journal of the Mechanical Behavior of Biomedical Materials, 39:48–60, November 2014. doi:10.1016/j.jmbbm.2014.06.016.
Raymond W Ogden. Non-linear elastic deformations. Courier Dover Publications, 1997.
D Peric and W Dettmer. A computational model for generalized inelastic materials at finite strains combining elastic, viscoelastic and plastic material behaviour. Engineering Computations, 20(5/6):768–787, 2003.
Owen D.R.J. Peric D. and Honnor M.E. A model for finite strain elasto-plasticity based on logarithmic strains: computational issues. Computer Methods in Applied Mechanics and Engineering, 94(61):35–61, 1992. doi:10.1016/0045-7825(92)90156-e.
Stefanie Reese and Sanjay Govindjee. A theory of finite viscoelasticity and numerical aspects. International journal of solids and structures, 35(26-27):3455–3482, 1998.
M. Schanz and S. Diebels. A comparative study of biot's theory and the linear theory of porous media for wave propagation problems. Acta Mechanica, 161:213–235, April 2003. doi:10.1007/s00707-002-0999-5.
Rezgar Shakeri, Leila Ghaffari, Jeremy L Thompson, and Jed Brown. Stable numerics for finite-strain elasticity. International Journal for Numerical Methods in Engineering, 125(21):e7563, 2024.
Juan C Simo. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Computer Methods in Applied Mechanics and Engineering, 99(1):61–112, 1992.
Erwan Tanné, Tianyi Li, Blaise Bourdin, J-J Marigo, and Corrado Maurini. Crack nucleation in variational phase-field models of brittle fracture. Journal of the Mechanics and Physics of Solids, 110:80–99, 2018.
Luigi Vergori, Michel Destrade, Patrick McGarry, and Ray W. Ogden. On anisotropic elasticity and questions concerning its finite element implementation. Computational Mechanics, 52(5):1185–1197, November 2013. doi:10.1007/s00466-013-0871-6.
Gustavo Weber and Lallit Anand. Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids. Computer Methods in Applied Mechanics and Engineering, 79(2):173–202, 1990.