Abstract
We present an open-source software framework called PERMIX for multiscale modeling and simulation of fracture in solids. The framework is an object oriented open-source effort written primarily in Fortran 2003 standard with Fortran/C++ interfaces to a number of other libraries such as LAMMPS, ABAQUS, LS-DYNA and GMSH. Fracture on the continuum level is modeled by the extended finite element method (XFEM). Using several novel or state of the art methods, the piece software handles semi-concurrent multiscale methods as well as concurrent multiscale methods for fracture, coupling two continuum domains or atomistic domains to continuum domains, respectively. The efficiency of our open-source software is shown through several simulations including a 3D crack modeling in clay nanocomposites, a semi-concurrent FE-FE coupling, a 3D Arlequin multiscale example and an MD-XFEM coupling for dynamic crack propagation.
Similar content being viewed by others
References
Ju JW, Lee HK (2001) A micromechanical damage model for effective elastoplastic behavior of partially debonded ductile matrix composites. Int J Solids Struct 38:6307–6332
Abaqus 6.11 standard user’s manual, DASSAULT SYSTEMES, (2011)
Ahrens J, Geveci B, Law C (2005) Paraview: an end-user tool for large data visualization. Vis Handb 717:731
Akin JE (1999) Object oriented programming via fortran 90. Eng Comput 16(1):26–48
Alder BJ, Wainwright TE (1959) Studies in molecular dynamics. I. general method. J Chem Phys 31:459
Cid Alfaro MV, Suiker ASJ, Verhoosel CV, de Borst R (2010) Numerical homogenization of cracking processes in thin fibre-epoxy layers. Eur J Mech 29(2):119–131
Allen MP, Tildesley DJ (1989) Comput Simul Liq, vol 18. Oxford University Press, Oxford
Amestoy P, Duff I, LExcellent J-Y, Koster J (2001) Mumps: a general purpose distributed memory sparse solver. Appl Parallel Comput 1947:121–130
Aubertin P, Réthoré J, de Borst R (2010) A coupled molecular dynamics and extended finite element method for dynamic crack propagation. Int J Numer Methods Eng 81(1):72–88
Aubertin P, Rthor J, de Borst R (2009) Energy conservation of atomistic/continuum coupling. Int J Numer Methods Eng 78(11):1365–1386
Aurenhammer F (1991) Voronoi diagramsa survey of a fundamental geometric data structure. ACM Comput Surv 23(3):345–405
Badia S, Parks M, Bochev P, Gunzburger M, Lehoucq R (2008) On atomistic-to-continuum coupling by blending. Multiscale Model Simul 7(1):381–406
Bangerth W, Hartmann R, Kanschat G (2007) General-purpose object-oriented finite element library. ACM Trans Math Softw 33(4):24
Bazant ZP (2010) Can multiscale-multiphysics methods predict softening damage and structural failure. Int J Multiscale Comput Eng 8(1):61–67
Beazley DM et al. (1996) Swig: an easy to use tool for integrating scripting languages with c and c++, Proceedings of the 4th USENIX Tcl/Tk workshop, pp 129–139.
Bellevue W (2005) Tecplot user’s manual. Amtec Engineering, Inc., Bellevue
Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45(5):601–620
Belytschko T, Chen H, Xu J, Zi G (2003) Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int J Numer Methods Eng 58(12):1873–1905
Belytschko T, Liu WK, Moran B (2000) Nonlinear finite elements for continua and structures. Wiley, Chichester
Belytschko T, Lu YY, Gu L (1995) Crack propagation by element-free galerkin methods. Eng Fract Mech 51(2):295–315
Belytschko T, Loehnert S, Song J-H (2008) Multiscale aggregating discontinuities: a method for circumventing loss of material stability. Int J Numer Methods Eng 73(6):869–894
Belytschko T, Song J-H (2010) Coarse-graining of multiscale crack propagation. Int J Numer Methods Eng 81(5):537–563
Ben RG, Dhia H (2005) The Arlequin method as a flexible engineering design tool. Int J Numer Methods Eng 62:1442–1462
Bonet J, Wood RD (1997) Nonlinear continuum mechanics for finite element analysis. Cambridge University Press, New York
Bordas S, Nguyen VP, Dunant C, Nguyen-Dang H, Guidoum A (2007) An extended finite element library. Int J Numer Methods Eng 71(6):703–732. doi:10.1002/nme.1966
Bordas S, Rabczuk T, Zi G (2008) Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment. Eng Fract Mech 75(5):943–960
Bouchard PO, Bay F, Chastel Y (2003) Numerical modelling of crack propagation: automatic remeshing and comparison of different criteria. Comput Methods Appl Mech Eng 192(35):3887–3908
Chivers I, Sleightholme J (2010) Compiler support for the fortran 2003 and 2008 standards revision 6 ACM SIGPLAN Fortran Forum. ACM 29:26–34
Christman T, Needleman A, Suresh S (1989) An experimental and numerical study of deformation in metal-ceramic composites. Acta Metall 37(11):3029–3050
Crisfield MA (1983) An arc-length method including line searches and accelerations. Int J Numer Methods Eng 19(9):1269–1289
Dagum L, Menon R (1998) Openmp: an industry standard api for shared-memory programming. Computl Sci Eng IEEE 5(1):46–55
Dascalu C, Bilbie G, Agiasofitou EK (2008) Damage and size effects in elastic solids: a homogenization approach. Int J Solids Struct 45(2):409–430
Datta DK, Picu RC, Shephard MS (2004) Composite grid atomistic continuum method: an adaptive approach to bridge continuum with atomistic analysis. Int J Multiscale Comput Eng 2:3
Decyk VK, Norton CD, Szymanski BK (1997) Expressing object-oriented concepts in fortran 90. ACM SIGPLAN Fortran Forum 16:13–18
Eckardt S, Könke C (2008) Adaptive damage simulation of concrete using heterogeneous multiscale models. J Algorithms Comput Technol 2(2):275–297
Epperly TGW, Gary K, Tamar D, Dietmar E, Jim L, Adrian P, Scott K (2012) High-performance language interoperability for scientific computing through babel. Int J High Perform Comput Appl 26(3):260–274
Feyel F, Chaboche J-L (2000) \({FE}^2\) multiscale approach for modeling the elastoviscoplastic behavior of long fiber SiC/Ti composite materials. Comput Methods Appl Mech Eng 183:309–330
Fish J, Yuan Z (2005) Multiscale enrichment based on partition of unity. Int J Numer Methods Eng 62(10):1341–1359
Fish J, Nuggehally MA, Shephard MS, Picu CR, Badia S, Parks ML, Gunzburger M (2007) Concurrent atc coupling based on a blend of the continuum stress and the atomistic force. Comput Methods Appl Mech Eng 196(45–48):4548–4560
Fish J, Shek K, Pandheeradi M, Shephard MS (1997) Computational plasticity for composite structures based on mathematical homogenization: theory and practice. Comput Methods Appl Mech Eng 148(1–2):53–73
Fish J, Qing Y (1999) Computational damage mechanics for composite materials based on mathematical homogenization. Int J Numer Methods Eng 45(11):1657–1679
Forde BWR, Stiemer SF (1987) Improved arc length orthogonality methods for nonlinear finite element analysis. Comput Struct 27(5):625–630
Geuzaine C, Remacle JF (2009) Gmsh: a 3-D finite element mesh generator with built-in pre-and post-processing facilities. Int J Numer Methods Eng 79(11):1309–1331
Ghosh S, Lee K, Moorthy S (1996) Two scale analysis of heterogeneous elastic-plastic materials with asymptotic homogenization and voronoi cell finite element model. Comput Methods Appl Mech Eng 132(1–2):63–116
Ghosh S, Lee K, Raghavan P (2001) A multi-level computational model for multi-scale damage analysis in composite and porous materials. Int J Solids Struct 38(14):2335–2385
Gitman IM, Askes H, Sluys LJ (2007) Representative volume: existence and size determination. Eng Fract Mech 74(16):2518–2534
Gitman IM, Askes H, Sluys LJ (2008) Coupled-volume multi-scale modelling of quasi-brittle material. Eur J Mech A 27(3):302–327
Gracie R, Belytschko T (2009) Concurrently coupled atomistic and xfem models for dislocations and cracks. Int J Numer Methods Eng 78(3):354–378
Gracie R, Belytschko T (2011) An adaptive concurrent multiscale method for the dynamic simulation of dislocations. Int J Numer Methods Eng 86(4–5):575–597
Guedes J, Kikuchi N (1990) Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput Methods Appl Mech Eng 83(2):143–198
Guidault PA, Allix O, Champaney L, Navarro JP (2007) A two-scale approach with homogenization for the computation of cracked structures. Comput Struct 85(17–18):1360–1371
Hashin Z (1983) Analysis of composite materials. J Appl Mech 50(2):481–505
Hettich T, Hund A, Ramm E (2008) Modeling of failure in composites by x-fem and level sets within a multiscale framework. Comput Methods Appl Mech Eng 197(5):414–424
Hill R (1967) The essential structure of constitutive laws for metal composites and polycrystals. J Mech Phys Solids 15:79–95
Hirschberger CB, Sukumar N, Steinman P (2008) Computational homogenization of material layers with micromorphic mesostructure. Philos Mag 88(30–32):3603–3631
Hoover WG (2006) Smooth particle applied mechanics: the state of the art. World Scientific Publishing Co. Inc., Hackensack
Horstemeyer MF (2010) Multiscale modeling: a review. In: Leszczynski J, Shukla MK (eds) Practical aspects of computational, chemistry. Springer, Dordrecht, pp 87–135
Hughes Thomas JR, Cottrell John A (2005) Isogeometric analysis: cad, finite elements, nurbs, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39):4135–4195
Ibrahimbegovi A, Markovi D (2003) Strong coupling methods in multi-phase and multi-scale modeling of inelastic behavior of heterogeneous structures. Comput Methods Appl Mech Eng 192(28–30):3089–3107
Intel, Math kernel library. http://software.intel.com/en-us/intel-mkl. Accessed 03 Dec 2013
Jain JR, Ghosh S (2009) Damage evolution in composites with a homogenization-based continuum damage mechanics model. Int J Damage Mech 18(6):533–568
Kelchner CL, Plimpton SJ, Hamilton JC (1998) Dislocation nucleation and defect structure during surface indentation. Phys Rev B 58(17):11085
Kohlhoff S, Gumbsch P, FischmeisteR HF (1991) Crack propagation A in. b.c.c. crystals studied with a combined finite-element and atomistic model. Philos Mag A 64:851–878
Kouznetsova V. (2002) Computational homogenization for the multi-scale analysis of multi-phase materials, PhD Thesis, Netherlands Institute for Metals Research, The Netherlands
Kouznetsova V, Brekelmans WAM, Baaijens FPT (2001) An approach to micro-macro modeling of heterogeneous materials. Comput Mech 27(1):37–48
Kouznetsova VG, Geers MGD, Brekelmans WAM (2004) Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput Methods Appl Mech Eng 193(48):5525–5550
Kulkarni MG, Geubelle PH, Matouš K (2009) Multi-scale modeling of heterogeneous adhesives: effect of particle decohesion. Mech Mater 41(5):573–583
Larsson Fredrik, Runesson Kenneth (2011) On two-scale adaptive fe analysis of micro-heterogeneous media with seamless scale-bridging. Comput Methods Appl Mech Eng 200(37–40):2662–2674
Larsson Fredrik, Runesson Kenneth (2010) Variationally consistent computational homogenization of transient heat flow. Int J Numer Methos Eng 81(13):1659–1686
Lemaitre J, Desmorat R, Sauzay M (2000) Anisotropic damage law of evolution. Eur J Mech A 19(2):187–208
Lene F, Leguillon D (1982) Homogenized constitutive law for a partially cohesive composite material. Int J Solids Struct 18(5):443–458
Lim JH, Sohn D, Lee JH, Im S (2010) Variable-node finite elements with smoothed integration techniques and their applications for multiscale mechanics problems. Comput Struct 88(7–8):413–425
Liu GR, Liu MB (2003) Smoothed particle hydrodynamics: a meshfree particle method. World Scientific Publishing Co. Inc., Dordrecht
Lloberas-Valls O, Rixen DJ, Simone A, Sluys LJ (2012) Multiscale domain decomposition analysis of quasi-brittle heterogeneous materials. Int J Numer Methods Eng 89(11):1337–1366
Luan BQ, Hyun S, Molinari JF, Bernstein N (2006) Multiscale modeling of two-dimensional contacts. Phys Rev E 74:046710
Markus A (2008) Design patterns and fortran 2003. ACM SIGPLAN Fortran Forum 27:2–15
Massart TJ, Peerlings RHJ, Geers MGD (2007) An enhanced multi-scale approach for masonry wall computations with localization of damage. Int J Numer Methods Eng 69(5):1022– 1059
Massart TJ, Peerlings RHJ, Geers MGD (2007) Structural damage analysis of masonry walls using computational homogenization. Int J Damage Mech 16(2):199–226
Matou K, Kulkarni MG, Geubelle PH (2008) Multiscale cohesive failure modeling of heterogeneous adhesives. J Mech Phys Solids 56(4):1511–1533
Mei Xu TB (2008) Conservation properties of the bridging domain method for coupled molecular/continuum dynamics. Int J Numer Methods Eng 76:278–294
Menouillard T, Réthoré J, Combescure A, Bung H (2006) Efficient explicit time stepping for the extended finite element method (x-fem). Int J Numer Methods Eng 68(9):911–939
Mesarovic SD, Padbidri J (2005) Minimal kinematic boundary conditions for simulations of disordered microstructures. Philos Mag A 85(1):65–78
Metcalf M, Reid JK, Cohen M (2004) Fortran 95/2003 explained, vol 416. Oxford University Press, NewYork
Miehe C, Schrder J, Schotte J (1999) Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials. Comput Methods Appl Mech Eng 171(3–4):387–418
Miller RE, Tadmor EB (2009) A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Modell Simul Mater Sci Eng 17:053001
Moes N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):133–150
Monteiro E, Yvonnet J, He QC (2008) Computational homogenization for nonlinear conduction in heterogeneous materials using model reduction. Comput Mater Sci 42(4):704–712
Nakamura T, Suresh S (1993) Effects of thermal residual stresses and fiber packing on deformation of metal-matrix composites. Acta Metall Mater 41(6):1665–1681
Nemat-Nasser S, Hori M (1993) Micromechanics: overall properties of heterogeneous materials. Elsevier, Amsterdam
Nguyen VP, Lloberas-Valls O, Stroeven M, Sluys JL (2012) Computational homogenization for multiscale crack modeling implementational and computational aspects. Int J Numer Methods Eng 89(2):192–226
Nguyen VP, Lloberas-Valls O, Stroeven M, Sluys LJ (2010) On the existence of representative volumes for softening quasi-brittle materials—a failure zone averaging scheme. Comput Methods Appl Mech Eng 199(45–48):3028–3038
Nguyen VP, Lloberas-Valls O, Stroeven M, Sluys LJ (2011) Homogenization-based multiscale crack modelling: from micro-diffusive damage to macro-cracks. Comput Methods Appl Mech Eng 200(9):1220–1236
Nguyen VP, Lloberas-Valls O, Stroeven M, Sluys LJ (2011) Multiscale continuous and discontinuous modeling of heterogeneous materials: a review on recent developments. J Multiscale Modell 3(4):229–270
Nguyen VP, Stroeven M, Sluys LJ (2012) An enhanced continuous-discontinuous multiscale method for modeling mode-i cohesive failure in random heterogeneous quasi-brittle materials. Eng Fract Mech 79:78–102
Nie JH, Hopkins DA, Chen YT, Hsieh HT (2010) Development of an object-oriented finite element program with adaptive mesh refinement for multi-physics applications. Adv Eng Softw 41(4):569–579
Norton CD, Decyk VK, Szymanski BK (1997) High performance object-oriented scientific programming in fortran 90. MIT Press, Cambridge
Özdemir I, Brekelmans WAM, Geers MGD (2008) Computational homogenization for heat conduction in heterogeneous solids. Int J Numer Methods Eng 73(2):185–204
Özdemir I, Brekelmans WAM, Geers MGD (2008) Computational homogenization for the thermo-mechanical analysis of heterogeneous solids. Comput Methods Appl Mech Eng 198(3–4):602–613
Parks ML, Lehoucq RB, Plimpton SJ, Silling SA (2008) Implementing peridynamics within a molecular dynamics code. Comput Phys Commun 179(11):777–783
Patzák B, Bittnar Z (2001) Design of object oriented finite element code. Adv Eng Softw 32(10):759–767
Pettermann HE, Suresh S (2000) A comprehensive unit cell model: a study of coupled effects in piezoelectric composites. Int J Solids Struct 37(39):5447–5464
Plimpton S (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117(1):1–19
Plimpton S (2001) Atomistic stress simulator (warp)
Press WH (1992) Numerical recipes in fortran: the art of scientific computing, vol 1. Cambridge University Press, New York
Qian D, Wagner GJ, Liu WK (2004) A multiscale projection method for the analysis of carbon nanotubes. Comput Methods Appl Mech Eng 193(17–20):1603–1632
Rabczuk T, Belytschko T (2004) Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int J Numer Methods Eng 61(13):2316–2343
Raghavan P, Ghosh S (2005) A continuum damage mechanics model for unidirectional composites undergoing interfacial debonding. Mech Mater 37(9):955–979
Rahman A (1964) Correlations in the motion of atoms in liquid argon. Phys Rev 136(2A):405–411
Robert JS (1983) Comments on virial theorems for bounded systems. Am J Phys 51:940–942
Rouson DWI, Xia J, Xu X (2010) Object construction and destruction design patterns in fortran 2003. Procedia Comput Sci 1(1):1495–1504
Rudd RE, Broughton JQ (2000) Concurrent coupling of length scales in solid state systems. Phys Status Solid B 217(1):251– 291
Schenk O, Gärtner K, Fichtner W, Fichtner A (2001) Pardiso: a high-performance serial and parallel sparse linear solver in semiconductor device simulation. Future Gener Comput Syst 18(1):69–78
Schröder J, Keip M (2010) A framework for the two-scale homogenization of electro-mechanically coupled boundary value problems. Comput Methods Mech 1:311–329
Shao-Qiang TANG, Liu Wing K, Karpov Eduard G, Hou Thomas Y (2007) Bridging atomistic/continuum scales in solids with moving dislocations. Chin Phys Lett 24(1):161
Shenoy VB, Miller R, Tadmor EB, Phillips R, Ortiz M (1998) Quasicontinuum models of interfacial structure and deformation. Phys Rev Lett 80:742–745
Shephard MS, Nuggehally MA, Dale BF, Picu CR, Fish J, Klaas O, Beall MW (2009) Component software for multiscale simulation. In: Fish J (ed) Multiscale methods: bridging the scales in science and engineering. Oxford University Press, New York
Shewchuk J (1996) Triangle: engineering a 2D quality mesh generator and delaunay triangulator. Appl Comput Geom 1148:203–222
Shilkrot LE, Miller RE, Curtin WA (2002) Coupled atomistic and discrete dislocation plasticity. Phys Rev Lett 89:025501
Shilkrot LE, Miller Ronald E, Curtin William A (2004) Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics. J Mech Phys Solids 52(4):755–787
Si H (2007) Tetgen: a quality tetrahedral mesh generator and three-dimensional delaunay triangulator, research group: numerical mathematics and scientific computing, Weierstrass Institute for Applied Analysis and Stochastics. http://wias-berlin.de/software/tetgen/. Accessed 03 Dec 2013
Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1): 175–209
Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88(2): 151–184
Smit RJM, Brekelmans WAM, Meijer HEH (1998) Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Comput Methods Appl Mech Eng 155(1–2):181–192
Souza FV, Allen DH (2010) Multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks. Int J Numer Methods Eng 82(4):464–504
Souza FV, Allen DH (2011) Modeling the transition of microcracks into macrocracks in heterogeneous viscoelastic media using a two-way coupled multiscale model. Int J Solids Struct 48 (22–23):3160–3175
Strouboulis T, Copps K, Babuška I (2000) The generalized finite element method: an example of its implementation and illustration of its performance. Int J Numer Methods Eng 47(8): 1401–1417
Subramaniyan AK, Sun CT (2008) Continuum interpretation of virial stress in molecular simulations. Int J Solids Struct 45 (14–15):4340–4346
Swope WC, Andersen HC, Berens PH, Wilson KR (1982) A computer simulation method for the calculation of equilibrium constants for the formation of physical clusters of molecules: application to small water clusters. J Chem Phys 76:637
Tadmor EB, Ortiz M, Phillips R (1996) Quasicontinuum analysis of defects in solids. Philos Mag A 73(6):1529–1563
Talebi H, Samaniego C, Samaniego E, Rabczuk T (2012) On the numerical stability and mass-lumping schemes for explicit enriched meshfree methods. Int J Numer Methods Eng 89(8):1009–1027
Talebi H, Gi Z, Silani M, Samaniego E, Rabczuk T (2012) A simple circular cell method for multilevel finite element analysis, J Appl Math 2012:526846. doi:10.1155/2012/526846
Temizer Ä, Wriggers P (2011) An adaptive multiscale resolution strategy for the finite deformation analysis of microheterogeneous structures. Comput Methods Appl Mech Eng 200(37–40):2639–2661
van der Sluis O, Schreurs PJG, Brekelmans WAM, Meijer HEH (2000) Overall behaviour of heterogeneous elastoviscoplastic materials: effect of microstructural modelling. Mech Mater 32(8):449–462
Verhoosel CV, Remmers JJC, Gutirrez MA, de Borst R (2010) Computational homogenization for adhesive and cohesive failure in quasi-brittle solids. Int J Numer Meth Eng 83(8–9): 1155–1179
Wagner GJ, Liu WK (2003) Coupling of atomistic and continuum simulations. J Comput Phys 190(1):249–274
Wells GN, Sluys LJ (2001) A new method for modelling cohesive cracks using finite elements. Int J Numer Methods Eng 50(12):2667–2682
Xiao SP, Belytschko T (2003) Coupling methods for continuum model with molecular model. Int J Multiscale Comput Eng 1:115–126
Xiao SP, Belytschko T (2004) A bridging domain method for coupling continua with molecular dynamics. Comput Methods Appl Mech Eng 193(17–20):1645–1669
Xu X-P, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42:1397–1434
Yuan Z, Fish J (2008) Toward realization of computational homogenization in practice. Int J Numer Methods Eng 73(3):361–380
Zi G, Rabczuk T, Wall W (2007) Extended meshfree methods without the branch enrichment for the cohesive crack model. Comput Mech 40(2):367–382
Acknowledgments
The authors thank the support of the German Research Foundation (DFG). Also, the authors gratefully acknowledge the help of Dr. Amitava Moitra from Pennsylvania State University for providing the WARP code. Stéphane Bordas and Pierre Kerfriden also thank partial funding for their time provided by the EPSRC under grant EP/G042705/1 Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended finite element method. The European Research Council Starting Independent Research Grant (ERC Stg Grant Agreement No. 279578) entitled “RealTCut – Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery.”
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Talebi, H., Silani, M., Bordas, S.P.A. et al. A computational library for multiscale modeling of material failure. Comput Mech 53, 1047–1071 (2014). https://doi.org/10.1007/s00466-013-0948-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-013-0948-2