The Cell Method: An Overview on the Main Features
Curved and Layered Structures (2015)
- Volume: 2, Issue: 1
- ISSN: 2353-7396
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topElena Ferretti. "The Cell Method: An Overview on the Main Features." Curved and Layered Structures 2.1 (2015): null. <http://eudml.org/doc/276919>.
@article{ElenaFerretti2015,
abstract = {The Cell Method (CM) is a computational tool that maintains critical multi-dimensional attributes of physical phenomena in analysis. This information is neglected in the differential formulations of the classical approaches of finite element, boundary element, finite volume, and finite difference analysis, often leading to numerical instabilities and spurious results. This paper highlights the central theoretical concepts of the CM that preserve a more accurate and precise representation of the geometric and topological features of variables for practical problem solving. Important applications occur in fields such as electromagnetics, electrodynamics, solid mechanics and fluids. CM addresses non-locality in continuum mechanics, an especially important circumstance in modelling heterogeneous materials.},
author = {Elena Ferretti},
journal = {Curved and Layered Structures},
language = {eng},
number = {1},
pages = {null},
title = {The Cell Method: An Overview on the Main Features},
url = {http://eudml.org/doc/276919},
volume = {2},
year = {2015},
}
TY - JOUR
AU - Elena Ferretti
TI - The Cell Method: An Overview on the Main Features
JO - Curved and Layered Structures
PY - 2015
VL - 2
IS - 1
SP - null
AB - The Cell Method (CM) is a computational tool that maintains critical multi-dimensional attributes of physical phenomena in analysis. This information is neglected in the differential formulations of the classical approaches of finite element, boundary element, finite volume, and finite difference analysis, often leading to numerical instabilities and spurious results. This paper highlights the central theoretical concepts of the CM that preserve a more accurate and precise representation of the geometric and topological features of variables for practical problem solving. Important applications occur in fields such as electromagnetics, electrodynamics, solid mechanics and fluids. CM addresses non-locality in continuum mechanics, an especially important circumstance in modelling heterogeneous materials.
LA - eng
UR - http://eudml.org/doc/276919
ER -
References
top- [1] Alotto P., Freschi F., Repetto M., Multiphysics problems via the Cell Method: The role of Tonti diagrams, IEEE Transactions on Magnetics, 2010, 46(8), 2959-2962.
- [2] Alotto P., Freschi F., Repetto M., Rosso C., Tonti diagrams, The Cell Method for Electrical Engineering and Multiphysics Problems, Lecture Notes in Electrical Engineering, no. 230, Springer-Verlag Berlin Heidelberg, pp. 1-9, 2013. Zbl1280.78001
- [3] Alotto P., Freschi F., Repetto M., Rosso C., Topological equations, The Cell Method for Electrical Engineering and Multiphysics Problems, Lecture Notes in Electrical Engineering, no. 230, Springer-Verlag Berlin Heidelberg, pp. 11-20, 2013. Zbl1280.78001
- [4] Bellina F., Bettini P., Tonti E., Trevisan F., Finite Formulation for the Solution of a 2D Eddy-Current Problem, IEEE Transaction on Magnetics, 2002, 38(2), 561-564.
- [5] Cosmi F., Applicazione del Metodo delle Celle con Approssimazione Quadratica, Proc. AIAS 2000, Lucca, Italy, 2000, 131-140.
- [6] Cosmi F., Numerical Solution of Plane Elasticity Problems with the Cell Method, CMES: Comput. Model. Eng. Sci., 2001, 2(3), 365-372. Zbl1017.74082
- [7] Cosmi F., DiMarino F., Modelling of the Mechanical Behaviour of Porous Materials: A New Approach, Acta of Bioengineering and Biomechanics, 2001, 3(2), 55-66.
- [8] Delprete C., Freschi F., Repetto M., Rosso. C., A Proposal of Nonlinear Formulation of Cell Method for Thermo-Elastostatic Problems, CMES: Comput. Model. Eng. Sci., 2013, 94(5), 397-420.
- [9] Ferretti E., Modellazione del Comportamento del Cilindro Fasciato in Compressione, Ph.D. Thesis (in Italian), University of Lecce, Italy, 2001.
- [10] Ferretti E., Crack Propagation Modeling by Remeshing using the Cell Method (CM), CMES: Comput. Model. Eng. Sci., 2003, 4(1), 51-72. Zbl1057.74048
- [11] Ferretti E., Crack-Path Analysis for Brittle and Non-Brittle Cracks: A Cell Method Approach, CMES: Comput. Model. Eng. Sci., 2004, 6(3), 227-244. Zbl1074.74627
- [12] Ferretti E., A Cell Method (CM) Code for Modeling the Pullout Test Step-Wise, CMES: Comput. Model. Eng. Sci., 2004, 6(5), 453-476. Zbl1080.74548
- [13] Ferretti E., A Discrete Nonlocal Formulation using Local Constitutive Laws, Int. J. Fracture, 2004, 130(3), L175-L182. Zbl1196.74006
- [14] Ferretti E., Modeling of the Pullout Test through the Cell Method, In: Sih G. C., Nobile L. (Eds.), RRRTEA - International Conference of Restoration, Recycling and Rejuvenation Technology for Engineering and Architecture Application, Aracne, 180-192, 2004.
- [15] Ferretti E., A Local Strictly Nondecreasing Material Law for Modeling Softening and Size-Effect: A Discrete Approach, CMES: Comput. Model. Eng. Sci., 2005, 9(1), 19-48.
- [16] Ferretti E., On nonlocality and locality: Differential and discrete formulations, 11th International Conference on Fracture - ICF11, 2005, 3, 1728-1733.
- [17] Ferretti E., Cell Method Analysis of Crack Propagation in Tensioned Concrete Plates, CMES: Comput. Model. Eng. Sci., 2009, 54(3), 253-282.
- [18] Ferretti E., The Cell Method: An Enriched Description of Physics Starting from the Algebraic Formulation, CMC: Comput. Mater. Con., 2013, 36(1), 49-72.
- [19] Ferretti E., A Cell Method Stress Analysis in Thin Floor Tiles Subjected to Temperature Variation, CMC: Comput. Mater. Con., 2013, 36(3), 293-322.
- [20] Ferretti E., The Cell Method: A Purely Algebraic Computational Method in Physics and Engineering, Momentum Press, New York, 2014. Zbl1312.65001
- [21] Ferretti E., The Cell Method as a Case of Bialgebra, Mathematics and Computers in Science and Engineering Series Nº 34 - Recent Advances in Applied Mathematics, Modelling and Simulation: Proceedings of the 8th International Conference on Applied Mathematics, Simulation, Modelling (ASM ‘14), WSEAS Press, Athens (Greece), 322-331, 2014.
- [22] Ferretti E., Similarities between Cell Method and Non-Standard Calculus, Mathematics and Computers in Science and Engineering Series Nº 39 - Recent Advances in Computational Mathematics: Proceedings of the 3rd International Conference on Applied and Computational Mathematics (ICACM ’14), WSEAS Press, Athens (Greece), 110-115, 2014.
- [23] Ferretti E., The Assembly Process for Enforcing Equilibrium and Compatibility with the CM: a Coboundary Process, CMES: Comput. Model. Eng. Sci., (in press).
- [24] Ferretti E., The Mathematical Foundations of the Cell Method, International Journal of Mathematical Models and Methods in Applied Sciences, (submitted). Zbl1057.74048
- [25] Ferretti E., Some new Findings on the Mathematical Structure of the Cell Method, International Journal of Mathematical Models and Methods in Applied Sciences, (submitted).
- [26] Ferretti E., The Algebraic Formulation: Why and How to Use it, Curved and Layer. Struct., 2015, 2, 106-149.
- [27] Ferretti E., Casadio E., Di Leo A., Masonry Walls under Shear Test: A CM Modeling, CMES: Comput. Model. Eng. Sci., 2008, 30(3), 163-190.
- [28] Ferretti E., Di Leo A., Modelling of Compressive Tests on FRP Wrapped Concrete Cylinders through a Novel Triaxial Concrete Constitutive Law, SITA: Scientific Israel – Technological Advantages, 2003, 5, 20-43.
- [29] Ferretti E., Di Leo A., Viola E., Computational Aspects and Numerical Simulations in the Elastic Constants Identification, CISM Courses and Lectures Nº 471: Problems in Structural Identification and Diagnostic: General Aspects and Applications, Springer, Wien – New York, 133-147, 2003.
- [30] Freschi F., Giaccone L., Repetto M., Educational value of the algebraic numerical methods in electromagnetism, COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 2008, 27(6), 1343-1357. Zbl1158.78341
- [31] Freschi F., Giaccone L., Repetto M., Magneto-Thermal Analysis of Induction Heating Processes, CMES: Comput. Model. Eng. Sci., 2013, 94(5), 371-395.
- [32] Marrone M., Convergence and Stability of the Cell Method with Non Symmetric Constitutive Matrices, Proc. 13th COMPUMAG, Evian, France, 2001.
- [33] Marrone M., Computational Aspects of Cell Method in Electrodynamics, Progress in Electromagnetics Research, PIER 32 (Special Volume on Geometrical Methods for Comp. Electromagnetics), 2001, 317-356.[Crossref]
- [34] Marrone M., Rodrìguez-Esquerre V.F., Hernàndez-Figueroa H.E., Novel Numerical Method for the Analysis of 2D Photonic Crystals: the Cell Method, Opt. Express, 2002, 10(22), 1299-1304.[Crossref]
- [35] Nappi A., Rajgelj S., Zaccaria D., Application of the Cell Method to the Elastic-Plastic Analysis, Proc. Plasticity ‘97, 1997, 14-18.
- [36] Nappi A., Rajgelj S., Zaccaria D., A Discrete Formulation Applied to Crack Growth Problem, In: Sih G. G. (Ed.), Mesomechanics 2000, Tsinghua University Press, Beijing, P. R. China, 395-406, 2000.
- [37] Nappi A., Tin-Loi F., A discrete formulation for the numerical analysis of masonry structures, In: Wang C. M., Lee K. H., Ang K. K. (Eds.), Computational Mechanics for the Next Millennium, Elsevier, Singapore, 81-86, 1999.
- [38] Nappi A., Tin-Loi F., A Numerical Model for Masonry Implemented in the Framework of a Discrete Formulation, Struct. Eng. Mech., 2001, 11(2), 171-184.[Crossref]
- [39] Pani M., Taddei F., The Cell Method: Quadratic Interpolation with Tetraedra for 3D Scalar Fields, CMES: Comput. Model. Eng. Sci., 2013, 94(4), 279-300.
- [40] Viola E., Tornabene F., Ferretti E., Fantuzzi N., Soft Core Plane State Structures Under Static Loads Using GDQFEM and Cell Method, CMES: Comput. Model. Eng. Sci., 2013, 94(4), 301-329.
- [41] Viola E., Tornabene F., Ferretti E., Fantuzzi N., GDQFEM Numerical Simulations of Continuous Media with Cracks and Discontinuities, CMES: Comput. Model. Eng. Sci., 2013, 94(4), 331-369.
- [42] Viola E., Tornabene F., Ferretti E., Fantuzzi N., On Static Analysis of Composite Plane State Structures via GDQFEM and Cell Method, CMES: Comput. Model. Eng. Sci., 2013, 94(5), 421-458.
- [43] Zovatto L., Nuovi Orizzonti per il Metodo delle Celle: Proposta per un Approccio Meshless, Proc. AIMETA – GIMC (in Italian), Taormina, Italy, 2001, 1-10.
- [44] Tonti E., On the Mathematical Structure of a Large Class of Physical Theories, Rend. Acc. Lincei, 1972, 52, 48-56.
- [45] Tonti E., The Algebraic - Topological Structure of Physical Theories, Conference on Symmetry, Similarity and Group Theoretic Methods in Mechanics, Calgary (Canada), 1974, 441-467.
- [46] Tonti E., On the formal structure of physical theories, Monograph of the Italian National Research Council, 1975.
- [47] Tonti E., The Reason for Analogies between Physical Theories, Appl. Math. Modelling, 1976, 1, 37-50.[Crossref]
- [48] Tonti E., On the Geometrical Structure of the Electromagnetism, In: Ferrarese G. (Ed.), Gravitation, Electromagnetism and Geometrical Structures, for the 80th birthday of A. Lichnerowicz, Pitagora, Bologna, 281-308, 1995.
- [49] Tonti E., Algebraic Topology and Computational Electromagnetism, Fourth International Worksop on the Electric and Magnetic Field: from Numerical Models to industrial Applications, Marseille, 1998, 284-294.
- [50] Tonti E., A Direct Discrete Formulation of Field Laws: the Cell Method, CMES: Comput. Model. Eng. Sci., 2001, 2(2), 237-258.
- [51] Tonti E., A Direct Discrete Formulation for the Wave Equation, J. Comput. Acoust., 2001, 9(4), 1355-1382.[Crossref]
- [52] Tonti, E., Finite Formulation of the Electromagnetic Field, Progress in Electromagnetics Research, PIER 32 (Special Volume on Geometrical Methods for Comp. Electromagnetics), 2001, 1-44.[Crossref]
- [53] Tonti E., Finite Formulation of the Electromagnetic Field, International COMPUMAG Society Newsletter, 2001, 8(1), 5-11.
- [54] Tonti E., Finite Formulation of the Electromagnetic Field, IEE Transactions on Magnetics, 2002, 38(2), 333-336.
- [55] Tonti E., The Mathematical Structure of Classical and Relativistic Physics, Birkhäuser, 2013. Zbl1298.00082
- [56] Tonti E., Zarantonello F., Algebraic Formulation of Elastostatics: the Cell Method, CMES: Comput. Model. Eng. Sci., 2009, 39(3), 201-236. Zbl1257.74184
- [57] Tonti E., Zarantonello F., Algebraic Formulation of Elastodynamics: the Cell Method, CMES: Comput. Model. Eng. Sci., 2010, 64(1), 37-70.
- [58] Conway J.H., Burgiel H., Goodman-Strass C., The Symmetries of Things, Peters A. K. Ltd., 2008. Zbl1173.00001
- [59] Hesse M.B., Models and Analogies in Science, revised ed., Notre Dame University Press, Notre Dame, Indiana, 1966.
- [60] Lasenby J., Lasenby A.N., Doran C.J.L., A Unified Mathematical Language for Physics and Engineering in the 21st Century, Philosophical Transactions of the Royal Society of London, 2000, A 358, 1-18. Zbl0965.01007
- [61] Chen C.Y., Atkinson C., The Stress Analysis of Thin Contact Layers: A Viscoelastic Case, CMES: Comput. Model. Eng. Sci., 2009, 48(3), 219-240. Zbl1231.74322
- [62] Chen Y., Cui J., Nie Y., Li Y., A New Algorithm for the Thermo-Mechanical Coupled Frictional Contact Problem of Polycrystalline Aggregates Based on Plastic Slip Theory, CMES: Comput. Model. Eng. Sci., 2011, 76(3), 189-206.
- [63] Hartmann S., Weyler R., Oliver J., Cante J.C., Hernández J.A., A 3D frictionless contact domain method for large deformation problems, CMES: Comput. Model. Eng. Sci., 2010, 55(3), 211-269. Zbl1228.74054
- [64] Selvadurai A.P.S., Atluri S.N., Contact Mechanics in Engineering Sciences, Tech Science Press, 2010.
- [65] Willner K., Constitutive Contact Laws in Structural Dynamics, CMES: Comput. Model. Eng. Sci., 2009, 48(3), 303-336. Zbl1231.74331
- [66] Yun C., Junzhi C., Yufeng N., Yiqiang L., A New Algorithm for the Thermo-Mechanical Coupled Frictional Contact Problem of Polycrystalline Aggregates Based on Plastic Slip Theory, CMES: Comput. Model. Eng. Sci., 2011, 76(3), 189-206.
- [67] Blázquez A., París F., Effect of numerical artificial corners appearing when using BEM on contact stresses, Engineering Analysis with Boundary Elements, 2011, 35(9), 1029-1037. Zbl1259.74036
- [68] Cai Y.C., Paik J.K., Atluri S.N., Locking-free thick-thin rod/beam element for large deformation analyses of space-frame structures, based on the Reissner variational principle and a von Karman type nonlinear theory, CMES: Comput. Model. Eng. Sci., 2010, 58(1), 75-108. Zbl1231.74402
- [69] Cai Y.C., Tian L.G., Atluri S.N., A simple locking-free discrete shear triangular plate element, CMES: Comput. Model. Eng. Sci., 2011, 77(3-4), 221-238.
- [70] Dong L., Atluri S.N., A simple procedure to develop efficient & stable hybrid/mixed elements, and Voronoi cell finite elements for macro- & micromechanics, CMC: Comput. Mater. Con., 2011, 24(1), 61-104.
- [71] Imai R., Nakagawa M., A reduction algorithm of contact problems for core seismic analysis of fast breeder reactors, CMES: Comput. Model. Eng. Sci., 2012, 84(3), 253-281.
- [72] Jarak T., Soric J., On Shear Locking in MLPG Solid-Shell Approach, CMES: Comput. Model. Eng. Sci., 2011, 81(2), 157-194.
- [73] Liu C.-S., Optimally Generalized Regularization Methods for Solving Linear Inverse Problems, CMC: Comput. Mater. Con., 2012, 29(2), 103-128.
- [74] Liu C.-S., Atluri S.N., An iterative method using an optimal descent vector, for solving an Ill-conditioned system Bx = b, better and faster than the conjugate gradient method, CMES: Comput. Model. Eng. Sci., 2011, 80(3-4), 275-298.
- [75] Liu C.S., Dai H.H., Atluri S.N., Iterative Solution of a System of Nonlinear Algebraic Equations F(x) = 0, Using dot x = λ[α R + β P] or dot x = λ[α F + β P*]R is a Normal to a Hyper-Surface Function of F, P Normal to R, and P* Normal to F, CMES: Comput. Model. Eng. Sci., 2011, 81(3), 335-363.
- [76] Liu C.S., Hong H.K., Atluri S.N., Novel algorithms based on the conjugate gradient method for inverting ill-conditioned matrices, and a new regularization method to solve ill-posed linear systems, CMES: Comput. Model. Eng. Sci., 2010, 60(3), 279-308. Zbl1231.65071
- [77] Qian Z.Y., Han Z.D., Atluri S.N., A fast regularized boundary integral method for practical acoustic problems, CMES: Comput. Model. Eng. Sci., 2013, 91(6), 463-484.
- [78] Reaz Ahmed S., Deb Nath S.K., A Simplified Analysis of the Tire-Tread Contact Problem using Displacement Potential Based Finite-Difference Technique, CMES: Comput. Model. Eng. Sci., 2009, 44(1), 35-64.
- [79] Yeih W., Liu C.S., Kuo C.L., Atluri S.N., On solving the direct/inverse Cauchy problems of Laplace equation in a multiply connected domain, using the generalized multiple-source-point boundary-collocation Trefftz method & characteristic lengths, CMC: Comput. Mater. Con., 2010, 17(3), 275-302.
- [80] Zhang Y., Noda N.A., Takaishi K., Effects of geometry on intensity of singular stress fields at the corner of single-lap joints, World Academy of Science, Engineering and Technology, 2011, 79, 911-916.
- [81] Zheng G., Li Z.W., Finite element analysis of adjacent building response to corner effect of excavation, Tianjin Daxue Xuebao (Ziran Kexue yu Gongcheng Jishu Ban)/Journal of Tianjin University Science and Technology, 2012, 45(8), 688-699.
- [82] Zhou Y.T., Li X., Yu D.H., Lee K.Y., Coupled crack/contact analysis for composite material containing periodic cracks under periodic rigid punches action, CMES: Comput. Model. Eng. Sci., 2010, 63(2), 163-189. Zbl1231.74333
- [83] Zhu H.H., Cai Y.C., Paik J.K., Atluri S.N., Locking-free thick-thin rod/beam element based on a von Karman type nonlinear theory in rotated reference frames for large deformation analyses of space-frame structures, CMES: Comput. Model. Eng. Sci., 2010, 57(2), 175-204. Zbl1231.74451
- [84] Zhu T., Atluri S.N., A Modified Collocation Method and a Penalty Formulation for Enforcing the Essential Boundary Conditions in the Element Free Galerkin Method, Comput. Mech., 1998, 21(3), 211-222.[Crossref] Zbl0947.74080
- [85] Hallen E., Electromagnetic Theory, Chapman & Hall, 1962.
- [86] Barut A.O., Electrodynamics and Classical Theory of Fields and Particles, Courier Dover Publications, 1964.
- [87] Griflths D.J., Introduction to electrodynamics, 3rd ed., Prentice Hall, 1999.
- [88] Penfield P., Haus H., Electrodynamics of Moving Media, M.I.T. Press, Cambridge MA, 1967.
- [89] Fung Y.C., A First Course in Continuum Mechanics, 2nd ed., Prentice-Hall, Inc., 1977.
- [90] Gurtin M.E., An Introduction to Continuum Mechanics, Academic Press, New York, 1981. Zbl0559.73001
- [91] Wu H.C., Continuum Mechanics and Plasticity, CRC Press, 2005. Zbl1057.74002
- [92] Bažant Z.P., Chang T.P., Is Strain-Softening Mathematically Admissible?, Proc. 5th Engineering Mechanics Division, 1984, 2, 1377-1380.
- [93] Bažant Z.P., Belytschko T.B., Chang T.P., Continuum Model for Strain Softening, J. Eng. Mech., 1984, 110(12), 1666-1692.[Crossref]
- [94] Belytschko T., Bažant Z.P., Hyun Y.W., Chang T.P., Strain Softening Materials and Finite-Element Solutions, Comput. Struct., 1986, 23, 163-180.[Crossref]
- [95] Jirásek M., Rolshoven S., Comparison of Integral-Type Nonlocal Plasticity Models for Strain-Softening Materials, Int. J. Engineering Science, 2003, 41(13-14), 1553-1602.[Crossref] Zbl1211.74039
- [96] Nilsson C., On Local Plasticity, Strain Softening, and Localization, Rep. No. TVSM-1007, Division of Structural Mechanics, Lund Institute of Technology, Lund, Sweden, 1994.
- [97] Nilsson C., Nonlocal Strain Softening Bar Revisited, Int. J. Solids Struct., 1997, 34, 4399-4419.[Crossref] Zbl0942.74591
- [98] Vermeer P.A., Brinkgreve R.B.J., A new Effective Non-Local Strain Measure for Softening Plasticity, In: Chambon R., Desrues J., Vardoulakis I. (Eds.), Localization and Bifurcation Theory for Solis and Rocks, Balkema, Rotterdam, The Netherlands, 89-100, 1994.
- [99] Carslaw H.S., Jaeger J.C., Conduction of Heat in Solids, 2nd ed., Oxford University Press, 1959. Zbl0029.37801
- [100] Hestenes D., New foundations for classical mechanics: Fundamental Theories of Physics, 2nd ed., Springer, 1999.
- [101] Rudin W., Functional Analysis, McGraw-Hill Science/Engineering/Math, 1991.
- [102] Okada S., Onodera R., Algebraification of Field Laws of Physics by Poincaré Process, Bull. of Yamagata University – Natural Sciences, 1951, 1(4), 79-86.
- [103] Branin F.H.Jr., The Algebraic Topological Basis for Network Analogies and the Vector Calculus, Proc. Symp. on Generalized Networks, Brooklyn Polit., 1966, 453-487.
- [104] Veblen O., Whitehead J.H.C., The Foundations of Differential Geometry, Cambr. Tracts, no. 29, 1932. Zbl0005.21801
- [105] Fernandes M.C.B., Vianna J.D.M., On the generalized phase space approach to Dufln–Kemmer–Petiau particles, Foundations of Physics, 1999, 29(2), 201-219.
- [106] Bohm D., Hiley B.J., Stuart A., On a New Mode of Description in Physics, Int. J. Theor. Phys. 1970, 3(3), 171-183.[Crossref]
- [107] Schönberg M., On the hydrodynamical model of the quantum mechanics, Il Nuovo Cimento, 1954, 12(1), 103-133. Zbl0056.21901
- [108] Schönberg M., Quantum mechanics and geometry, An. Acad. Brasil. Cien., 1958, 30, 1-20.
- [109] Rosenberg S., The Laplacian on a Riemannian manifold, Cambridge University Press, 1997. Zbl0868.58074
- [110] Sternberg S., Lectures on Differential Geometry, Prentice Hall, 1964.
- [111] Snygg J., A New Approach to Differential Geometry Using Clifford's Geometric Algebra, Birkhäuser, 2012. Zbl1232.53002
- [112] Frescura F.A.M., Hiley B.J., The algebraization of quantum mechanics and the implicate order, Foundations of Physics, 1980, 10(9-10), 705-722.
- [113] Frescura F.A.M., Hiley B.J., Algebras, quantum theory and prespace, Revista Brasileira de Fisica, Volume Especial, Los 70 anos de Mario Schonberg, 1984, 49-86.
- [114] Hiley B.J., A note on the role of idempotents in the extended Heisenberg algebra, Implications, Scientific Aspects of ANPA, Cambridge, 2001, 22, 107–121.
- [115] Hiley B.J., Algebraic quantum mechanics, algebraic spinors and Hilbert space, Boundaries, Scientific Aspects of ANPA, 2003.
- [116] Twiss R.J., Moores E.M., §2.1 The orientation of structures, In Structural geology, 2nd ed., Macmillan, 1992.
- [117] Burke W.L., Applied Differential Geometry, Cambridge University Press, Cambridge, 1985. Zbl0572.53043
- [118] Schutz B., A first course in general relativity, Cambridge University Press, Cambridge, UK, 1985.
- [119] Catoni F., Boccaletti D., Cannata R., Mathematics of Minkowski Space, Birkhäuser Verlag, Basel, 2008. Zbl1151.53001
- [120] Naber G.L., The Geometry of Minkowski Spacetime, Springer-Verlag, New York, 1992. Zbl0757.53046
- [121] Schey H.M., Div, grad, curl, and all that, W. W. Norton & Company, 1996.
- [122] Hestenes D., Space-time Algebra, Gordon and Breach, New York, 1966.
- [123] Penrose R., Rindler W., Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Field, Cambridge University Press, 1986. Zbl0602.53001
- [124] van Dantzing D., On the Relation Between Geometry and Physics and the Concept of Space-Time, Helv. Phys. Acta, 1956, Suppl. IV, 48-53.
- [125] Solomentsev E. D., Shikin E. V., Laplace–Beltrami equation, In: Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, 2001.
- [126] George P.L., Automatic Mesh Generator using the Delaunay Voronoi Principle. Surv. Math. Ind., 1995, 239-247. Zbl0824.65122
- [127] Butcher J.C., Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, New York, 2003. Zbl1040.65057
- [128] Courant R., Friedrichs K., Lewy H., On the partial difference equations of mathematical physics, AEC Research and Development Report, NYO-7689, New York, AEC Computing and Applied Mathematics Centre – Courant Institute of Mathematical Sciences, 1956 [1928]. Zbl0145.40402
- [129] Evans L.C., Partial Differential Equations, American Mathematical Society, 1998. Zbl0902.35002
- [130] Flanders H., Differential forms with applications to the physical sciences, Dover Publications, 1989.
- [131] Hairer E., Lubich C., Wanner G., Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer Series in Computational Mathematics 31, Springer-Verlag, Berlin, New York, 2006. Zbl1094.65125
- [132] Lambert J.D., The initial value problem for ordinary differential equations, In: Jacobs D. (Ed.), The State of the Art in Numerical Analysis, Academic Press, New York, 451-501, 1977.
- [133] Lambert J.D., Numerical Methods for Ordinary Differential Systems, Wiley, New York, 1992.
- [134] Gilbarg D., Trudinger N., Elliptic partial differential equations of second order, Springer, 2001. Zbl1042.35002
- [135] Ehle B.L., On Padé approximations to the exponential function and A-stable methods for the numerical solution of initial value problems, Report 2010, University of Waterloo, 1969.
- [136] Chang C.-W., A New Quasi-Boundary Scheme for Three-Dimensional Backward Heat Conduction Problems, CMC: Comput. Mater. Con., 2011, 24(3), 209-238.
- [137] Birdsall C.K., Langdon A.B., Plasma Physics via Computer Simulations, McGraw-Hill Book Company, 1985.
- [138] Fenner R.T., Finite Element Methods for Engineers, Imperial College Press, London, 1996. Zbl0929.74001
- [139] Huebner K.H., The Finite Element Method for Engineers, Wiley, 1975.
- [140] Livesley R.K., Finite Elements, an Introduction for Engineers, Cambridge University Press, 1983. Zbl0527.73067
- [141] Mavripilis D.J., Multigrid Techniques for Unstructured Meshes, Lecture Series 1995-02, Computational Fluid Dynamics, Von Karman Institute of Fluid Dynamics, 1995.
- [142] Morton K.W., Stringer S.M., Finite Volume Methods for Inviscid and Viscous Flows, Steady and Unsteady, Lecture Series 1995-02, Computational Fluid Dynamics, Von Karman Institute of Fluid Dynamics, 1995.
- [143] Mattiussi C., The Finite Volume, Finite Difference, and Finite Elements Methods as Numerical Methods for Physical Field Problems, In: P. Hawkes (Ed.), Advances in Imaging and Electron Physics, 113, 1-146, 2000.
- [144] Alireza H.M., Vijaya S., William F.H., Computation of Electromagnetic Scattering and Radiation Using a Time-Domain Finite-Volume Discretization Procedure, Computer Physics Communications, 1991, 68(1), 175-196.
- [145] Yee K., Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Transactions on Antennas and Propagation, 1966, 14(3), 302-307. Zbl1155.78304
- [146] Mattiussi C., The Geometry of Time-Stepping, Progress in Electromagnetics Research, PIER 32 (Special Volume on Geometrical Methods for Comp. Electromagnetics), 2001, 123-149.[Crossref]
- [147] Einstein A., Quanten-Mechanik Und Wirklichkeit [Quantum Mechanics and Reality], Dialectica, 1948, 2(3-4), 320-324.
- [148] Tegmark M., Wheeler J.A., 100 Years of the Quantum. Scientific American, 2001, 284, 68-75.
- [149] Horodecki R., Horodecki P., Horodecki M., Horodecki K., Quantum entanglement, Rev. Mod. Phys., 2007, 81(2), 865-942.
- [150] Plenio M.B., Virmani S., An introduction to entanglement measures, Quant. Inf. Comp., 2007, 1, 1-51. Zbl1152.81798
- [151] Hesse M.B., Action at a Distance in Classical Physics, Isis, 1955, 46(4), 337-353. Zbl0065.39103
- [152] Misner C.W., Thorne K.S., Wheeler J.A., Gravitation, Freeman W. H., 1973.
- [153] Wheeler J.A., Misner C., Thorne K.S., Gravitation, Freeman W.H. & Co, 1973.
- [154] Einstein A., Podolsky B., Rosen N., Can Quantum-Mechanical Description of Physical Reality be Considered Complete?, Physical Review, 1935, 47(10), 777-780. Zbl0012.04201
- [155] Aspect A., Grangier P., Roger G., Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities, Physical Review Letters, 1982, 49(2), 91-94.
- [156] Penrose R., The Road to Reality, Vintage books, 2007.
- [157] Clauser J.F., Horne M.A., Shimony A., Holt R.A., Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett., 1969, 23(15), 880-884.[Crossref]
- [158] Kakuda K., Obara S., Toyotani J., Meguro M., Furuichi M., Fluid Flow Simulation Using Particle Method and Its Physics-based Computer Graphics, CMES: Comput. Model. Eng. Sci., 2012, 83(1), 57-72.
- [159] Kakuda K., Nagashima T., Hayashi Y., Obara S., Toyotani J., Katsurada N., Higuchi S., Matsuda S., Particle-based Fluid Flow Simulations on GPGPU Using CUDA, CMES: Comput. Model. Eng. Sci., 2012, 88(1), 17-28.
- [160] Rogula D., Introduction to Nonlocal Theory of Material Media, In: Rogula D. (Ed.), Nonlocal theory of material media, CISM courses and lectures, Springer, Wien, 268, 125-222, 1982.
- [161] Truesdell C., Noll W., The non-linear field theories of mechanics, 3rd ed., Springer, 2004.
- [162] Noll W., A new Mathematical Theory of Simple Materials, Arch. Ration. Mech. Anal., 1972, 48, 1-50.
- [163] Duhem P., Le Potentiel Thermodynamique et la Pression Hydrostatique, Ann. Sci. Ecole Norm. S., 1893, 10, 183-230.
- [164] Rayleigh O.M., Notes on the Theory of Lubrication, Philos. Mag., 1918, 35, 1-12.[Crossref]
- [165] Oseen C.W., The Theory of Liquid Crystals, Trans. Faraday Soc., 1933, 29, 883-899.[Crossref] Zbl0008.04203
- [166] Chandrasekhar S., Radiation Transfer, Oxford University Press, London, 1950.
- [167] Hodgkin A.L., The Conduction of Nervous Impulses, Thomas, Springfield, Ill., 1964.
- [168] Edelen D.G.B., Green A.E., Laws N., Nonlocal Continuum Mechanics, Arch. Ration. Mech. Anal., 1971, 43, 36-44. Zbl0225.73006
- [169] Eringen A.C., Linear Theory of Nonlocal Elasticity and Dispersion of Plane Waves, Int. J Eng. Sci., 1972, 10, 425-435.[Crossref] Zbl0241.73005
- [170] Eringen A.C., Edelen D.G.B., On Nonlocal Elasticity, Int. J Eng. Sci., 1972, 10, 233-248.[Crossref] Zbl0247.73005
- [171] Eringen A.C., Kim B.S., Stress Concentration at the Tip of a Crack, Mech. Res. Commun., 1974, 1, 233-237.[Crossref]
- [172] Eringen A.C., Speziale C.G., Kim B.S., Crack-Tip Problem in Nonlocal Elasticity, J. Mech. Phys. Solids, 1977, 25, 339-355.[Crossref] Zbl0375.73083
- [173] Eringen A.C., On Nonlocal Plasticity, Int. J Eng. Sci., 1981, 19, 1461-1474.[Crossref] Zbl0474.73028
- [174] Eringen A. C., Theories of Nonlocal Plasticity, Int. J Eng. Sci., 1983, 21, 741-751.[Crossref] Zbl0519.73024
- [175] Pijaudier-Cabot G., Bažant Z. P., Nonlocal Damage Theory, J. Eng. Mech., 1987, 113, 1512-1533.[Crossref]
- [176] Bažant Z.P., Lin F.–B., Nonlocal Smeared Cracking Model for Concrete Fracture, J. Struct. Eng., 1988, 114(11), 2493-2510.[Crossref]
- [177] Bažant Z.P., Lin F.–B., Nonlocal Yield-Limit Degradation, Int. J. Numer. Methods Eng., 1988, 26, 1805-1823.[Crossref] Zbl0661.73041
- [178] Bažant Z.P., Pijaudier-Cabot G., Nonlocal Continuum Damage, Localization Instability and Convergence, J. Appl. Mech, 1988, 55, 287-293.[Crossref] Zbl0663.73075
- [179] Saouridis C., Identification et Numérisation Objectives des Comportements Adoucissants: une Approche Multiéchelle de l'Endommagement du Béton, Ph.D. Thesis, Univ. Paris VI, France, 1988.
- [180] Bažant Z.P., Pijaudier-Cabot G., Measurement of the Characteristic Length of Nonlocal Continuum, J. Eng. Mech., 1989, 113, 2333-2347.
- [181] Bažant Z.P., Ožbolt J., Nonlocal Microplane Model for Fracture, Damage, and Size Effect in Structures, J. Eng. Mech., 1990, 116(11), 2485-2505.[Crossref]
- [182] Bažant Z.P., Tabbara M.R., Kazemi M.T., Pijaudier-Cabot G., Random Particle Model for Fracture of Aggregate or Fiber Composites, J. Eng. Mech., 1990, 116(8), 1686-1705.[Crossref]
- [183] Bažant Z.P., Why Continuum Damage is Nonlocal: Micromechanics Arguments, J. Eng. Mech., 1991, 117(5), 1070-1087.[Crossref]
- [184] Saouridis C., Mazars J., Prediction of the Failure and Size Effect in Concrete via a Biscale Damage Approach, Eng. Comput., 1992, 9, 329-344.[Crossref]
- [185] Schlangen E., van Mier J.G.M., Simple Lattice Model for Numerical Simulation of Fracture of Concrete Materials and Structures, Mater. Struct., 1992, 25, 534-542.[Crossref]
- [186] Planas J., Elices M., Guinea G.V., Cohesive Cracks versus Nonlocal Models: Closing the Gap, Int J. Fract., 1993, 63, 173-187.[Crossref]
- [187] Schlangen E., Experimental and Numerical Analysis of Fracture Processes in Concrete, Ph.D. Thesis, Delft Univ. of Technology, Delft, The Netherlands, 1993.
- [188] Bažant Z.P., Nonlocal Damage Theory based on Micromechanics of Crack Interactions, J. Eng. Mech., 1994, 120(3), 593-617.[Crossref]
- [189] Huerta A., Pijaudier-Cabot G., Discretization Influence on Regularization by two Localization Limiters, J. Eng. Mech., 1994, 120(6), 1198-1218.[Crossref]
- [190] Leblond J.B., Perrin G., Devaux J., Bifurcation Effects in Ductile Metals Incorporating Void Nucleation, Growth and Interaction, J. Appl. Mech., 1994, 236-242.[Crossref] Zbl0807.73004
- [191] Jirásek M., Bažant Z.P., Macroscopic Fracture Characteristics of Random Particle Systems, Int. J. Fract., 1995, 69, 201-228.
- [192] Tvergaard V., Needleman A., Effects of Nonlocal Damage in Porous Plastic Solids, Int. J. Solids Struct., 1995, 32, 1063-1077.[Crossref] Zbl0866.73060
- [193] Drugan W.J., Willis J.R., A Micromechanics-Based Nonlocal Constitutive Equation and Estimates of Representative Volume Element Size for Elastic Composites, J. Mech. Phys. Solids, 1996, 44, 497-524.[Crossref] Zbl1054.74704
- [194] Ožbolt J., Bažant Z.P., Numerical Smeared Fracture Analysis: Nonlocal Microcrack Interaction Approach, Int. J. Numer. Methods Eng., 1996, 39, 635-661.[Crossref] Zbl0868.73066
- [195] Planas J., Guinea G.V., Elices M., Basic Issues on Nonlocal Models: Uniaxial Modeling, Tech. Rep., No. 96-jp03, Departamento de Ciencia de Materiales, ETS de Ingenieros de Caminos, Univ. Politécnica de Madrid, Ciudad Univ. sn., 28040 Madrid, Spain, 1996.
- [196] Strömberg L., Ristinmaa M., FE Formulation of a Nonlocal Plasticity Theory, Comput. Methods Appl. Mech. Eng., 1996, 136, 127-144.[Crossref] Zbl0918.73118
- [197] van Mier J.G.M., Fracture Processes of Concrete, CRC, Boca Raton, Fla., 1997.
- [198] Jirásek M., Embedded Crack Models for Concrete Fracture, In: de Borst R., Bićanić N., Mang H., Meschke G. (Eds.), Computational Modelling of Concrete Structures, Balkema, Rotterdam, 291-300, 1998.
- [199] Jirásek M., Nonlocal Models for Damage and Fracture: Comparison of Approaches, Int. J. Solids Struct., 1998, 35, 4133-4145.[Crossref] Zbl0930.74054
- [200] Jirásek M., Zimmermann T., Rotating Crack Model with Transition to Scalar Damage, J. Eng. Mech., 1998, 124(3), 277-284.[Crossref]
- [201] Needleman A., Tvergaard V., Dynamic Crack Growth in a Nonlocal Progressively Cavitating Solid, Eur. J. Mech. A–Solid, 1998, 17, 421-438.[Crossref] Zbl0933.74059
- [202] Borino G., Fuschi P., Polizzotto C., A Thermodynamic Approach to Nonlocal Plasticity and Related Variational Approaches, J. Appl. Mech., 1999, 66, 952-963.[Crossref]
- [203] Jirásek M., Computational Aspects of Nonlocal Models, Proc. ECCM ’99, München, Germany, 1999, 1-10.
- [204] Chen J.S., Wu C.T., Belytschko T., Regularization of Material Instabilities by Meshfree Approximations with Intrinsic Length Scales, Int. J. Numer. Meth. Engng., 2000, 47, 1303-1322.[Crossref] Zbl0987.74079
- [205] Hu X.Z., Wittmann F.H., Size Effect on Toughness Induced by Cracks Close to Free Surface, Eng. Fract. Mech., 2000, 65, 209-211.[Crossref]
- [206] Jirásek M., Bažant Z.P., Inelastic Analysis of Structures, John Wiley and Sons, 2001.
- [207] Gao H., Huang Y., Taylor-Based Nonlocal Theory of Plasticity, Int. J. Solids Struct., 2001, 38, 2615-2637.[Crossref] Zbl0977.74009
- [208] Luciano R., Willis J.R., Nonlocal Constitutive Response of a Random Laminate Subjected to Configuration-Dependent Body Force, J. Mech. Phys. Solids, 2001, 49, 431-444.[Crossref] Zbl0983.74006
- [209] Bažant Z.P., Jirásek M., Nonlocal Integral Formulations of Plasticity and Damage: Survey of Progress, J. Eng. Mech., 2002, 128(11), 1119-1149.[Crossref]
- [210] Jirásek M., Patzák B., Consistent Tangent Stiffness for Nonlocal Damage Models, Comput. Struct., 2002, 80(14-15), 1279-1293.[Crossref]
- [211] Krumhansl J.A., Generalized Continuum Field Representation for Lattice Vibrations, In: Wallis R. F. (Ed.), Lattice dynamics, Pergamon, London, 627-634, 1965.
- [212] Rogula D., Influence of Spatial Acoustic Dispersion on Dynamical Properties of Dislocations, I. Bulletin de l'Académie Polonaise des Sciences, Séries des Sciences Techniques, 1965, 13, 337-343.
- [213] Eringen A.C., A Unified Theory of Thermomechanical Materials, Int. J Eng. Sci., 1966, 4, 179-202.[Crossref] Zbl0139.20204
- [214] Kunin I.A., Theory of Elasticity with Spatial Dispersion, Prikl. Mat. Mekh. (in Russian), 1966, 30, 866.
- [215] Kröner E., Elasticity Theory of Materials with Long-Range Cohesive Forces, Int. J. Solids Struct., 1968, 3, 731-742. Zbl0163.19402
- [216] Belytschko T., Krongauz Y., Organ D., Fleming M., Krysl P., Meshless Methods: An Overview and Recent Developments, Computer Methods Appl. Mech Engrg., 1996, 139, 3-47. Zbl0891.73075
- [217] Han Z.D., Liu H.T., Rajendran A.M, Atluri S.N., The Applications of Meshless Local Petrov-Galerkin (MLPG) Approaches in High-Speed Impact, Penetration and Perforation Problems, CMES: Comput. Model. Eng. Sci., 2006, 14(2), 119-128.
- [218] Pimprikar N., Teresa J., Roy D., Vasu RM., Rajan K., An approximately H1-optimal Petrov-Galerkin meshfree method: application to computation of scattered light for optical tomography, CMES: Comput. Model. Eng. Sci., 2013, 92(1), 33-61.
- [219] Soares D.Jr., A Time-Domain Meshless Local Petrov-Galerkin Formulation for the Dynamic Analysis of Nonlinear Porous Media, CMES: Comput. Model. Eng. Sci., 2010, 66(3), 227-248. Zbl1231.76164
- [220] Stevens D., Power W., A Scalable Meshless Formulation Based on RBF Hermitian Interpolation for 3D Nonlinear Heat Conduction Problems, CMES: Comput. Model. Eng. Sci., 2010, 55(2), 111-146. Zbl1231.65182
- [221] Schnack E., Weber W., Zhu Y., Discussion of experimental data for 3D crack propagation on the basis of three dimensional singularities, CMES: Comput. Model. Eng. Sci., 2011, 74(1), 1-38. Zbl1231.74396
- [222] Theilig H., Eflcient fracture analysis of 2D crack problems by the MVCCI method, Structural Durability and Health Monitoring, 2010, 6(3-4), 239-271.
- [223] Taddei F., Pani M., Zovatto L., Tonti E., Viceconti M., A New Meshless Approach for Subject-Specific Strain Prediction in Long Bones: Evaluation of Accuracy, Clinical Biomechanics, 2008, 23(9), 1192-1199.
- [224] Ferretti E., A Discussion of Strain-Softening in Concrete, Int. J. Fracture (Letters section), 2004, 126(1), L3–L10. Zbl1187.74227
- [225] Ferretti E., Experimental Procedure for Verifying Strain-Softening in Concrete, Int. J. Fracture (Letters section), 2004, 126(2), L27–L34.[Crossref]
- [226] Ferretti E., On Poisson's Ratio and Volumetric Strain in Concrete, Int. J. Fracture (Letters section), 2004, 126(3), L49-L55.
- [227] Ferretti E., On Strain-Softening in Dynamics, Int. J. Fracture (Letters section), 2004, 126(4), L75-L82. Zbl1187.74039
- [228] Ferretti E., Shape-Effect in the Effective Law of Plain and Rubberized Concrete, CMC: Comput. Mater. Con., 2012, 30(3), 237-284.
- [229] Ferretti E., Di Leo A., Cracking and Creep Role in Displacement at Constant Load: Concrete Solids in Compression, CMC: Comput. Mater. Con., 2008, 7(2), 59-80.
- [230] Ferretti E., Di Leo A., Viola E., A novel approach for the identification of material elastic constants, CISM Courses and Lectures Nº 471: Problems in Structural Identification and Diagnostic: General Aspects and Applications, Springer, Wien – New York, 117-131, 2003.
- [231] Anderson T.L., Fracture Mechanics, Fundamentals and Applications, 3rd ed., Taylor & Francis Group, 2004. Zbl0999.74001
- [232] Reinhardt H.W., Cornelissen H.A.W., Post-Peak Cyclic Behavior of Concrete in Uniaxial Tensile and Alternating Tensile and Compressive Loading, Cement Concrete Res., 1984, 14, 263-270.
- [233] Hadamard J., Leçons sur la Propagation des Ondes – Chapter VI, Paris, France, 1903.
- [234] Hudson J.A., Brown E.T., Fairhurst C., Shape of the Complete Stress-Strain Curve for Rock, Proc. 13th Symposiumon Rock Mechanics, University of Illinois, Urbana, Ill., 1971.
- [235] Dresher A., Vardoulakis I., Geometric Softening in Triaxial Tests on Granular Material, Geotechnique, 1982, 32(4), 291-303.
- [236] Bergan P.G., Record of the Discussion on Numerical Modeling, Proc. IUTAM W. Prager Symposium, Northwestern University, Evanston, Ill., 1983.
- [237] Hegemier G.A., Read H.E., Some Comments on Strain-Softening, Proc. DARPA-NSF Workshop, Northwestern University, Evanston, Ill., 1983.
- [238] Sandler I., Wright J.P., Summary of Strain-Softening, DARPANSF Workshop, Northwestern University, Evanston, Ill., 1983.
- [239] Wu F.H., Freud L.B., Report MRL-E-145, Brown University, Providence, RI, USA, 1983.
- [240] Brace W.F., Paulding B.W., Scholz C., Dilatancy in the Fracture of Crystalline Rocks, J. Geophys. Res., 1966,71(16), 3939-3953.[Crossref]
- [241] Schutz B., Geometrical methods of mathematical physics, Cambridge University Press, 1980. Zbl0462.58001
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