C. Inc, COMSOL Multiphysics®, 2017.

R. Abraham, J. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications, vol.75, 2012.
DOI : 10.1063/1.2916250

F. Abramovici, The Shepard interpolation as the best average of a set of data, School of Math. Sci, 1984.

S. Adami, X. Hu, A. , and N. , A transport-velocity formulation for smoothed particle hydrodynamics, Journal of Computational Physics, vol.241, pp.292-307, 2013.

R. A. Adams and J. J. Fournier, Sobolev spaces, vol.140, 2003.

S. Alayrangues, G. Damiand, P. Lienhardt, and S. Peltier, A Boundary Operator for Computing the Homology of Cellular Structures, vol.71, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00683031

S. Alayrangues, X. Daragon, J. Lachaud, and P. Lienhardt, Equivalence between regular n-G-maps and n-surfaces, International Workshop on Combinatorial Image Analysis, pp.122-136, 2004.
DOI : 10.1007/978-3-540-30503-3_10

S. Alayrangues, X. Daragon, J. Lachaud, and P. Lienhardt, Equivalence between closed connected n-G-maps without multi-incidence and n-surfaces, Journal of Mathematical Imaging and Vision, vol.32, issue.1, pp.1-22, 2008.
DOI : 10.1007/s10851-008-0084-3
URL : https://hal.archives-ouvertes.fr/hal-00340945

P. S. Aleksandrov, Combinatorial topology, vol.1, 1956.

D. N. Arnold, R. S. Falk, and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica, vol.15, pp.1-155, 2006.
DOI : 10.1017/s0962492906210018

S. Atluri and T. Zhu, The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics, Computational Mechanics, vol.25, issue.2-3, pp.169-179, 2000.

I. Babu?ka, U. Banerjee, J. Osborn, and Q. Zhang, Effect of numerical integration on meshless methods, Computer Methods in Applied Mechanics and Engineering, vol.198, issue.37, pp.2886-2897, 2009.

I. Babu?ka, U. Banerjee, and J. E. Osborn, Survey of meshless and generalized finite element methods: a unified approach, Acta Numerica, vol.12, pp.1-125, 2003.

I. Babu?ka, U. Banerjee, J. E. Osborn, L. , and Q. , Quadrature for meshless methods, International Journal for Numerical Methods in Engineering, vol.76, issue.9, pp.1434-1470, 2008.

G. Backus and F. Gilbert, The resolving power of gross earth data, Geophysical Journal International, vol.16, issue.2, pp.169-205, 1968.

V. Baldoni, N. Berline, J. De-loera, M. Köppe, and M. Vergne, How to integrate a polynomial over a simplex, Mathematics of Computation, vol.80, issue.273, pp.297-325, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00320882

J. C. Barata and M. S. Hussein, The Moore-Penrose pseudoinverse: A tutorial review of the theory, Brazilian Journal of Physics, vol.42, issue.1-2, pp.146-165, 2012.

A. Barvinok, Exponential integrals and sums over convex polyhedra. Funktsional, Anal. i Prilozhen, vol.26, issue.2, pp.64-66, 1992.

B. G. Baumgart, Winged edge polyhedron representation, 1972.

B. G. Baumgart, A polyhedron representation for computer vision, Proceedings of the, pp.589-596, 1975.

Z. Ba?ant, L. Cedolin, W. Chen, and E. Lui, Stability of structures: elastic, inelastic, fracture and damage theories, 2010.

L. Beirão-da-veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini et al., Basic principles of virtual element methods, Mathematical Models and Methods in Applied Sciences, vol.23, issue.01, pp.199-214, 2013.

L. Beirão-da-veiga, F. Brezzi, L. D. Marini, and A. Russo, The hitchhiker's guide to the virtual element method, Mathematical Models and Methods in Applied Sciences, vol.24, issue.08, pp.1541-1573, 2014.

T. Belytschko and M. Fleming, Smoothing, enrichment and contact in the element-free Galerkin method, Computers & Structures, vol.71, issue.2, pp.173-195, 1999.

T. Belytschko, Y. Guo, W. Kam-liu, P. Xiao, and S. , A unified stability analysis of meshless particle methods, International Journal for Numerical Methods in Engineering, vol.48, issue.9, pp.1359-1400, 2000.

T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering, vol.139, issue.1-4, pp.3-47, 1996.

T. Belytschko, Y. Lu, and L. Gu, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering, vol.37, issue.2, pp.229-256, 1994.

T. Belytschko, Y. Y. Lu, and L. Gu, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering, vol.37, issue.2, pp.229-256, 1994.

A. Ben-israel, A volume associated with m × n matrices, Linear Algebra and its Applications, vol.167, pp.87-111, 1992.

A. Ben-israel, Motzkin's transposition theorem, and the related theorems of Farkas, Gordan and Stiemke. Encyclopaedia of Mathematics, 2002.

B. Ben-moussa and J. Vila, Convergence of SPH method for scalar nonlinear conservation laws, SIAM Journal on Numerical Analysis, vol.37, issue.3, pp.863-887, 2000.

M. Berger, A. Tagliasacchi, L. Seversky, P. Alliez, J. Levine et al., State of the art in surface reconstruction from point clouds, EUROGRAPHICS star reports, vol.1, pp.161-185, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01017700

G. Bertrand and M. Couprie, A model for digital topology, Proceedings of 8th Discrete Geometry for Computer Imagery (DGCI'99), pp.229-241, 1999.
URL : https://hal.archives-ouvertes.fr/hal-00622003

P. Bochev and J. Hyman, Principles of mimetic discretizations of differential operators, Compatible Spatial Discretizations, pp.89-119, 2006.

J. Bonelle, Compatible Discrete Operator schemes on polyhedral meshes for elliptic and Stokes equations, 2014.
URL : https://hal.archives-ouvertes.fr/tel-01116527

J. Bonet and S. Kulasegaram, Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations, International Journal for Numerical Methods in Engineering, vol.47, issue.6, pp.1189-1214, 2000.

J. Bonet and S. Kulasegaram, Remarks on tension instability of Eulerian and Lagrangian corrected smooth particle hydrodynamics (CSPH) methods, International Journal for Numerical Methods in Engineering, vol.52, issue.11, pp.1203-1220, 2001.

J. Bonet and S. Kulasegaram, A simplified approach to enhance the performance of smooth particle hydrodynamics methods, Applied Mathematics and Computation, vol.126, issue.2-3, pp.133-155, 2002.

J. Bonet and T. Lok, Variational and momentum preservation aspects of smooth particle hydrodynamic formulations, Computer Methods in Applied Mechanics and Engineering, vol.180, issue.1, pp.97-115, 1999.

L. Bos and K. Salkauskas, Moving least-squares are Backus-Gilbert optimal, Journal of Approximation Theory, vol.59, issue.3, pp.267-275, 1989.
DOI : 10.1016/0021-9045(89)90090-7
URL : https://doi.org/10.1016/0021-9045(89)90090-7

S. Brenner and R. Scott, The mathematical theory of finite element methods, vol.15, 2007.

F. Brezzi, K. Lipnikov, and M. Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM Journal on Numerical Analysis, vol.43, issue.5, pp.1872-1896, 2005.

F. Brezzi, K. Lipnikov, and V. Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes, Mathematical Models and Methods in Applied Sciences, vol.15, issue.10, pp.1533-1551, 2005.

M. Brion, Points entiers dans les polyèdres convexes, Ann. Sci. Ecole Norm, 1988.
DOI : 10.24033/asens.1572
URL : http://www.numdam.org/article/ASENS_1988_4_21_4_653_0.pdf

. Sup, , vol.21, pp.653-663

E. Brisson, Representing geometric structures in d dimensions: topology and order, Proceedings of the fifth annual symposium on Computational Geometry, pp.218-227, 1989.
DOI : 10.1007/bf02189330
URL : https://link.springer.com/content/pdf/10.1007%2FBF02189330.pdf

E. Brisson, Representing geometric structures in d dimensions: Topology and order, Discrete & Computational Geometry, vol.9, issue.1, pp.387-426, 1993.
DOI : 10.1007/bf02189330
URL : https://link.springer.com/content/pdf/10.1007%2FBF02189330.pdf

B. Büeler, A. Enge, and K. Fukuda, Exact volume computation for polytopes: a practical study, Polytopes-Combinatorics and Computation, pp.131-154, 2000.

H. Bui and K. D. Van, Trois problèmes non linéaires de l'étude mécanique de la rupture des matériaux, Revue de Physique Appliquée, vol.9, issue.4, pp.605-610, 1974.

H. D. Bui, Mécanique de la rupture fragile, 1978.

D. E. Cardoze, G. L. Miller, and T. Phillips, Representing topological structures using cell-chains, International Conference on Geometric Modeling and Processing, pp.248-266, 2006.
DOI : 10.1007/11802914_18
URL : http://www.cs.cmu.edu/~tp517/papers/cardoze06representing.pdf

G. Carré, S. Del-pino, B. Després, L. , and E. , A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension, Journal of Computational Physics, vol.228, issue.14, pp.5160-5183, 2009.

J. E. Castillo and G. F. Miranda, Mimetic discretization methods, 2013.
DOI : 10.1201/b14575

J. Chen, C. Wu, S. Yoon, and Y. You, A stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering, vol.50, issue.2, pp.435-466, 2001.
DOI : 10.1002/1097-0207(20010120)50:2<435::aid-nme32>3.0.co;2-a

J. Chen, M. Hillman, and M. Rüter, An arbitrary order variationally consistent integration for Galerkin meshfree methods, International Journal for Numerical Methods in Engineering, vol.95, issue.5, pp.387-418, 2013.
DOI : 10.1002/nme.4512

J. Chen, C. Wu, S. Yoon, and Y. You, A stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering, vol.50, issue.2, pp.435-466, 2001.

J. Chen, S. Yoon, and C. Wu, Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering, vol.53, issue.12, pp.2587-2615, 2002.

E. K. Chiu, J. , and A. , An edge-averaged semi-meshless framework for numerical solution of conservation laws, 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, p.1269, 2010.

E. K. Chiu, Q. Wang, R. Hu, J. , and A. , A conservative mesh-free scheme and generalized framework for conservation laws, SIAM Journal on Scientific Computing, vol.34, issue.6, pp.2896-2916, 2012.

E. K. Chiu, Q. Wang, J. , and A. , A conservative meshless scheme: general order formulation and application to Euler equations, 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, p.651, 2011.

K. Chiu, A conservative meshless framework for conservation laws with applications in computational fluid dynamics, 2011.

A. Cholaquidis, A. Cuevas, and R. Fraiman, On Poincaré cone property, The Annals of Statistics, vol.42, issue.1, pp.255-284, 2014.
DOI : 10.1214/13-aos1188
URL : https://doi.org/10.1214/13-aos1188

G. Clair, B. Després, L. , and E. , A new method to introduce constraints in cell-centered Lagrangian schemes, Computer Methods in Applied Mechanics and Engineering, vol.261, pp.56-65, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00789201

K. Crane, F. De-goes, M. Desbrun, and P. Schröder, Digital geometry processing with discrete exterior calculus, ACM SIGGRAPH 2013 courses, SIGGRAPH '13, 2013.

S. B. Damelin, On bounds for diffusion, discrepancy and fill distance metrics, Principal Manifolds for Data Visualization and Dimension Reduction, pp.261-270, 2008.

G. Damiand and P. Lienhardt, Combinatorial maps: efficient data structures for computer graphics and image processing, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01090890

G. Danaraj and V. Klee, A representation of 2-dimensional pseudomanifolds and its use in the design of a linear-time shelling algorithm, Annals of Discrete Mathematics, vol.2, pp.53-63, 1978.

F. De-goes, C. Wallez, J. Huang, D. Pavlov, and M. Desbrun, Power particles: an incompressible fluid solver based on power diagrams, ACM Trans. Graph, vol.34, issue.4, pp.50-51, 2015.

J. A. De-loera, Actually doing it: Polyhedral computation and its applications, 2010.

J. A. De-loera, B. Dutra, M. Köppe, S. Moreinis, G. Pinto et al., Software for exact integration of polynomials over polyhedra, Computational Geometry, vol.46, issue.3, pp.232-252, 2013.

M. Delfour and J. Zolésio, Shapes and geometries: metrics, analysis, differential calculus, and optimization, 2011.
DOI : 10.1137/1.9780898719826

M. Desbrun, A. N. Hirani, M. Leok, and J. E. Marsden, Discrete exterior calculus, 2005.

M. Desbrun, E. Kanso, and Y. Tong, Discrete differential forms for computational modeling, Discrete Differential Geometry, pp.287-324, 2008.
DOI : 10.1145/1508044.1508059

B. Després, Weak consistency of the cell-centered Lagrangian GLACE scheme on general meshes in any dimension, Computer Methods in Applied Mechanics and Engineering, vol.199, pp.2669-2679, 2010.

B. Després and E. Labourasse, Angular momentum preserving cell-centered Lagrangian and Eulerian schemes on arbitrary grids, Journal of Computational Physics, vol.290, pp.28-54, 2015.

P. Destuynder, M. Djaoua, L. , and S. , Quelques remarques sur la mécanique de la rupture élastique, Journal de mécanique théorique et appliquée, vol.2, issue.1, pp.113-135, 1983.

P. Destuynder, P. E. Djaoua, L. Chesnay, and J. Nedelec, Sur une interprétation mathématique de l'intégrale de Rice en théorie de la rupture fragile, Mathematical Methods in the Applied Sciences, vol.3, issue.1, pp.70-87, 1981.

G. A. Dilts, Moving-least-squares-particle hydrodynamics-I. consistency and stability, International Journal for Numerical Methods in Engineering, vol.44, issue.8, pp.1115-1155, 1999.

G. A. Dilts, Moving least-squares particle hydrodynamics II: conservation and boundaries, International Journal for Numerical Methods in Engineering, vol.48, issue.10, pp.1503-1524, 2000.

M. Dorff and L. Hall, Solids in Rn whose area is the derivative of the volume, College Mathematics Journal, vol.34, issue.5, pp.350-358, 2003.

T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces, Mathematics of Computation, vol.34, issue.150, pp.441-463, 1980.

M. E. Dyer and A. M. Frieze, On the complexity of computing the volume of a polyhedron, SIAM Journal on Computing, vol.17, issue.5, pp.967-974, 1988.

C. Dyka and R. Ingel, An approach for tension instability in smoothed particle hydrodynamics (SPH), Computers & Structures, vol.57, issue.4, pp.573-580, 1995.

C. Dyka, P. Randles, and R. Ingel, Stress points for tension instability in SPH, International Journal for Numerical Methods in Engineering, vol.40, issue.13, pp.2325-2341, 1997.

N. Dyn, Interpolation of scattered data by radial functions, Topics in Multivariate Approximation, pp.47-61, 1987.

E. Oliveira, The patch test and the general convergence criteria of the finite element method, International Journal of Solids and Structures, vol.13, issue.3, pp.159-178, 1977.

M. A. Eisenberg and L. E. Malvern, On finite element integration in natural co-ordinates, International Journal for Numerical Methods in Engineering, vol.7, issue.4, pp.574-575, 1973.

M. Ellero, M. Serrano, and P. Espanol, Incompressible smoothed particle hydrodynamics, Journal of Computational Physics, vol.226, issue.2, pp.1731-1752, 2007.

L. Evans, Partial Differential Equations, 1996.

J. Farkas, Über die Theorie der einfachen Ungleichungen, Journal für die Reine und Angewandte Mathematik, vol.124, pp.1-27, 1902.

G. Fasshauer, Lecture notes in multivariate meshfree approximation, 2003.

G. E. Fasshauer, Solving partial differential equations by collocation with radial basis functions, Proceedings of Chamonix, vol.1997, pp.1-8, 1996.

R. Fattal, Blue-noise point sampling using kernel density model, In ACM Transactions on Graphics, vol.30, p.48, 2011.

H. Faure, P. Kritzer, and F. Pillichshammer, From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules, Indagationes Mathematicae, vol.26, issue.5, pp.760-822, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01256774

R. L. Fernandes, University of Illinois course on differentiable manifolds, 2014.

S. Fernández-méndez, P. Díez, and A. Huerta, Convergence of finite elements enriched with mesh-less methods, Numerische Mathematik, vol.96, issue.1, pp.43-59, 2003.

S. Fernández-méndez and A. Huerta, Coupling finite elements and particles for adaptivity: An application to consistently stabilized convection-diffusion, Meshfree Methods for Partial Differential Equations, pp.117-129, 2003.

S. Fernández-méndez and A. Huerta, Imposing essential boundary conditions in mesh-free methods, Computer Methods in Applied Mechanics and Engineering, vol.193, issue.12, pp.1257-1275, 2004.

M. Fleming, Y. Chu, B. Moran, and T. Belytschko, Enriched element-free Galerkin methods for crack tip fields, International Journal for Numerical Methods in Engineering, vol.40, issue.8, pp.1483-1504, 1997.

M. Floater, Generalized barycentric coordinates and applications, Acta Numerica, vol.24, pp.161-214, 2015.
DOI : 10.1017/s0962492914000129

T. Frankel, The geometry of physics: an introduction, 2011.

R. V. Garimella, J. Kim, and M. Berndt, Polyhedral mesh generation and optimization for non-manifold domains, Proceedings of the 22nd International Meshing Roundtable, pp.313-330, 2014.
DOI : 10.1007/978-3-319-02335-9_18

E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden, and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D: Nonlinear Phenomena, vol.240, issue.21, pp.1724-1760, 2011.
DOI : 10.1016/j.physd.2011.07.011
URL : http://arxiv.org/pdf/1010.4851

D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, 2015.

R. Gingold and J. Monaghan, Kernel estimates as a basis for general particle methods in hydrodynamics, Journal of Computational Physics, vol.46, issue.3, pp.429-453, 1982.

R. A. Gingold and J. Monaghan, Smoothed particle hydrodynamics: theory and application to non-spherical stars, Monthly Notices of the Royal Astronomical Society, vol.181, issue.3, pp.375-389, 1977.
DOI : 10.1093/mnras/181.3.375
URL : https://academic.oup.com/mnras/article-pdf/181/3/375/3104055/mnras181-0375.pdf

G. H. Golub and V. Pereyra, The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate, SIAM Journal on Numerical Analysis, vol.10, issue.2, pp.413-432, 1973.

P. Gordan, Über die Auflösung linearer Gleichungen mit reellen Coefficienten, Mathematische Annalen, vol.6, issue.1, pp.23-28, 1873.

C. Grasset-simon, G. Damiand, and P. Lienhardt, nD generalized map pyramids: three equivalent representations, 2005.

N. Gravin, D. V. Pasechnik, B. Shapiro, and M. Shapiro, On moments of a polytope, 2012.

M. Griebel and M. A. Schweitzer, A particle-partition of unity method part V: boundary conditions, Geometric Analysis and Nonlinear Partial Differential Equations, pp.519-542, 2003.

P. Griffiths and J. Harris, Principles of algebraic geometry, 2014.

B. Grünbaum, Are your polyhedra the same as my polyhedra?, Discrete and Computational Geometry, pp.461-488, 2003.

G. Guinea, J. Planas, and M. Elices, K I evaluation by the displacement extrapolation technique, Engineering Fracture Mechanics, vol.66, issue.3, pp.243-255, 2000.

F. C. Günther and W. K. Liu, Implementation of boundary conditions for meshless methods, Computer Methods in Applied Mechanics and Engineering, vol.163, issue.14, pp.205-230, 1998.

A. Hatcher, Algebraic topology, p.606, 2002.

R. Hiptmair, Discrete Hodge operators, Numerische Mathematik, vol.90, issue.2, pp.265-289, 2001.
DOI : 10.1007/s002110100295

A. N. Hirani, Discrete exterior calculus, 2003.

H. Hiyoshi and K. Sugihara, Improving continuity of Voronoi-based interpolation over Delaunay spheres, Computational Geometry, vol.22, issue.1-3, pp.167-183, 2002.

A. Huerta and S. Fernandez-mendez, Enrichment and coupling of the finite element and meshless methods, International Journal for Numerical Methods in Engineering, vol.48, issue.11, pp.1615-1636, 2000.
URL : https://hal.archives-ouvertes.fr/hal-01428629

A. Huerta, S. Fernández-méndez, and P. Díez, Enrichissement des interpolations d'éléments finis en utilisant des méthodes sans maillage, ESAIM: Mathematical Modelling and Numerical Analysis, vol.36, issue.6, pp.1027-1042, 2002.
DOI : 10.1051/m2an:2003004
URL : https://www.esaim-m2an.org/articles/m2an/pdf/2002/06/m2an0188.pdf

J. M. Hyman and M. Shashkov, Mimetic finite difference methods for Maxwell's equations and the equations of magnetic diffusion, Progress in Electromagnetics Research, vol.32, pp.89-121, 2001.

T. Iwaniec, J. Kauhanen, A. Kravetz, and C. Scott, The Hadamard-Schwarz inequality, Journal of Function Spaces and Applications, vol.2, issue.2, pp.191-215, 2004.

G. R. Johnson and S. R. Beissel, Normalized smoothing functions for SPH impact computations, International Journal for Numerical Methods in Engineering, vol.39, issue.16, pp.2725-2741, 1996.

A. Kamoulakos and P. Groenenboom, Moving from FE to SPH for space debris impact simulations-experience with PAM-SHOCK, European Conference on Spacecraft Structures, Materials and Mechanical Testing, 1998.

A. Kamoulakos, M. Przybylowicz, G. , and P. , Smoothed particle hydrodynamics for space debris impact simulations: An approach with the PAM-SHOCK transient dynamics code, International Workshop in New Models and Numerical Codes for Shock Wave Processes in Condensed Media, 1997.

A. Kamoulakos, M. Przybylowicz, L. , and M. , Virtual design for space debris impact (validation of the PAM-SHOCK transient dynamics code), Second European Conference on Space Debris, pp.455-458, 1997.

L. Khachiyan, Complexity of polytope volume computation, New trends in Discrete and Computational Geometry, pp.91-101, 1993.

M. Khosravifard, M. Esmaeili, and H. Saidi, Extension of the LasserreAvrachenkov theorem on the integral of multilinear forms over simplices, Applied Mathematics and Computation, vol.212, issue.1, pp.94-99, 2009.

A. Kitson, R. I. Mclachlan, R. , and N. , Skew-adjoint finite difference methods on nonuniform grids, New Zealand J. Math, vol.32, issue.2, pp.139-159, 2003.

N. Konerth, Exact multivariate integration on simplices: an explanation of the Lasserre-Avrachenkov theorem, 2014.

Y. Krongauz and T. Belytschko, Enforcement of essential boundary conditions in meshless approximations using finite elements, Computer Methods in Applied Mechanics and Engineering, vol.131, issue.1-2, pp.133-145, 1996.

Y. Krongauz and T. Belytschko, Consistent pseudo-derivatives in meshless methods, Computer Methods in Applied Mechanics and Engineering, vol.146, issue.3-4, pp.371-386, 1997.
DOI : 10.1016/s0045-7825(96)01234-0

P. Lancaster and K. Salkauskas, Surfaces generated by moving least squares methods, Mathematics of Computation, vol.37, issue.155, pp.141-158, 1981.
DOI : 10.2307/2007507
URL : https://www.ams.org/mcom/1981-37-155/S0025-5718-1981-0616367-1/S0025-5718-1981-0616367-1.pdf

N. Lanson, Etude des méthodes particulaires renormalisées: applications aux problemes de dynamique rapide, 2001.

N. Lanson and J. Vila, Renormalized meshfree schemes I: consistency, stability, and hybrid methods for conservation laws, SIAM Journal on Numerical Analysis, vol.46, issue.4, pp.1912-1934, 2008.

J. Lasserre, Integration on a convex polytope, Proceedings of the American Mathematical Society, vol.126, issue.8, pp.2433-2441, 1998.

J. B. Lasserre, An analytical expression and an algorithm for the volume of a convex polyhedron in Rn, Journal of optimization theory and applications, vol.39, issue.3, pp.363-377, 1983.

J. B. Lasserre and K. E. Avrachenkov, The multi-dimensional version of´b of´of´b a x p dx, The American Mathematical Monthly, vol.108, issue.2, pp.151-154, 2001.

J. Lawrence, Polytope volume computation, Mathematics of Computation, vol.57, pp.259-271, 0195.
DOI : 10.2307/2938672
URL : https://www.ams.org/mcom/1991-57-195/S0025-5718-1991-1079024-2/S0025-5718-1991-1079024-2.pdf

R. J. Leveque, Finite volume methods for hyperbolic problems, vol.31, 2002.

S. Li and W. K. Liu, Meshfree particle methods, 2007.

P. Lienhardt, Subdivisions of n-dimensional spaces and n-dimensional generalized maps, Proceedings of the fifth annual symposium on Computational Geometry, pp.228-236, 1989.

S. Lind, R. Xu, P. Stansby, R. , and B. D. , Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves, Journal of Computational Physics, vol.231, issue.4, pp.1499-1523, 2012.

K. Lipnikov, G. Manzini, and M. Shashkov, Mimetic finite difference method, Journal of Computational Physics, vol.257, pp.1163-1227, 2014.
DOI : 10.1016/j.jcp.2013.07.031

K. Lipnikov, M. Shashkov, and D. Svyatskiy, The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes, Journal of Computational Physics, vol.211, issue.2, pp.473-491, 2006.

S. Litvinov, X. Hu, A. , and N. , Towards consistence and convergence of conservative SPH approximations, Journal of Computational Physics, vol.301, pp.394-401, 2015.

G. Liu, A G-space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory, International Journal for Numerical Methods in Engineering, vol.81, issue.9, pp.1093-1126, 2010.

G. Liu, A G-space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part II applications to solid mechanics problems, International Journal for Numerical Methods in Engineering, vol.81, issue.9, pp.1127-1156, 2010.

G. Liu, T. Nguyen-thoi, H. Nguyen-xuan, and K. Lam, A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems, Computers & Structures, vol.87, issue.1, pp.14-26, 2009.

G. Liu and N. T. Trung, Smoothed finite element methods, 2016.

W. K. Liu, S. Jun, S. Li, J. Adee, and T. Belytschko, Reproducing kernel particle methods for structural dynamics, International Journal for Numerical Methods in Engineering, vol.38, issue.10, pp.1655-1679, 1995.
DOI : 10.1002/nme.1620381005

T. Long, D. Hu, D. Wan, C. Zhuang, Y. et al., An arbitrary boundary with ghost particles incorporated in coupled FEM-SPH model for FSI problems, Journal of Computational Physics, vol.350, pp.166-183, 2017.

R. Loubère, M. Shashkov, and B. Wendroff, Volume consistency in a staggered grid Lagrangian hydrodynamics scheme, Journal of Computational Physics, vol.227, issue.8, pp.3731-3737, 2008.

L. Lucy, A numerical approach to the testing of the fission hypothesis, The Astronomical Journal, vol.82, pp.1013-1024, 1977.

P. Madhavan, Numerical integration in meshfree methods, 2010.

L. G. Margolin, M. Shashkov, and P. K. Smolarkiewicz, A discrete operator calculus for finite difference approximations, Computer Methods in Applied Mechanics and Engineering, vol.187, issue.3-4, pp.365-383, 2000.

J. Marshall, University of Iowa course of computer-aided engineering, 2005.

N. Mikhailova, V. Tishkin, N. Tyurina, A. Favorskii, and M. Shashkov, Numerical modelling of two-dimensional gas-dynamic flows on a variable-structure mesh, 1986.

, USSR Computational Mathematics and Mathematical Physics, vol.26, issue.5, pp.74-84

D. Mirzaei, R. Schaback, and M. Dehghan, On generalized moving least squares and diffuse derivatives, IMA Journal of Numerical Analysis, vol.32, issue.3, pp.983-1000, 2011.

J. Monaghan, On the problem of penetration in particle methods, Journal of Computational Physics, vol.82, issue.1, pp.1-15, 1989.

J. Monaghan, Smoothed particle hydrodynamics, Annual Review of Astronomy and Astrophysics, vol.30, issue.1, pp.543-574, 1992.

J. Monaghan, Smoothed particle hydrodynamics, Reports on Progress in Physics, vol.68, issue.8, p.1703, 2005.

J. Monaghan and D. Price, Variational principles for relativistic smoothed particle hydrodynamics, Monthly Notices of the Royal Astronomical Society, vol.328, issue.2, pp.381-392, 2001.

J. J. Monaghan, SPH without a tensile instability, Journal of Computational Physics, vol.159, issue.2, pp.290-311, 2000.

J. J. Monaghan and J. B. Kajtar, SPH particle boundary forces for arbitrary boundaries, Computer Physics Communications, vol.180, issue.10, pp.1811-1820, 2009.

A. Mondino and E. Spadaro, On an isoperimetric-isodiametric inequality, Analysis & PDE, vol.10, issue.1, pp.95-126, 2017.

T. S. Motzkin, Beiträge zur Theorie der linearen Ungleichungen, 1936.

T. S. Motzkin, Two consequences of the transposition theorem on linear inequalities. Econometrica (pre-1986), vol.19, p.184, 1951.

B. B. Moussa, N. Lanson, and J. P. Vila, Convergence of meshless methods for conservation laws applications to Euler equations, Hyperbolic Problems: Theory, Numerics, Applications, pp.31-40, 1999.

Y. Mukherjee and S. Mukherjee, On boundary conditions in the element-free Galerkin method, Computational Mechanics, vol.19, issue.4, pp.264-270, 1997.

P. Mullen, F. De-goes, M. Desbrun, D. Cohen-steiner, A. et al., Signing the unsigned: Robust surface reconstruction from raw pointsets, Computer Graphics Forum, vol.29, pp.1733-1741, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00502473

P. Mullen, A. Mckenzie, D. Pavlov, L. Durant, Y. Tong et al., Discrete Lie advection of differential forms, Foundations of Computational Mathematics, vol.11, issue.2, pp.131-149, 2011.

P. Mullen, P. Memari, F. De-goes, and M. Desbrun, HOT: Hodge-optimized triangulations, ACM Transactions on Graphics (TOG), vol.30, issue.4, p.103, 2011.

B. Nayroles, G. Touzot, and P. Villon, Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics, vol.10, issue.5, pp.307-318, 1992.

J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, vol.36, pp.9-15, 1971.

W. Noll, Proof of the maximality of the orthogonal group in the unimodular group. Archive for Rational Mechanics and, Analysis, vol.18, issue.2, pp.100-102, 1965.

E. Parzen, On estimation of a probability density function and mode, The Annals of Mathematical Statistics, vol.33, issue.3, pp.1065-1076, 1962.

D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden et al., Structure-preserving discretization of incompressible fluids, Physica D: Nonlinear Phenomena, vol.240, issue.6, pp.443-458, 2011.

B. Perot, Conservation properties of unstructured staggered mesh schemes, Journal of Computational Physics, vol.159, issue.1, pp.58-89, 2000.

G. Pierrot, Discrete Ostrogradsky property recovery: symmetrisation of MLS coefficients, 2008.

G. Pierrot, A gentle introduction to ESI i-adjoint library, BIBLIOGRAPHY, vol.217, 2010.

G. Pierrot, Meshless methods: a unified framework and analysis, 2011.

G. Pierrot and G. Fougeron, Enforcing differentiation/integration compatibility in meshless methods with nodal integration, IV International Conference on Particle-Based Methods-Fundamentals and Applications, 2015.

G. Pierrot and G. Fougeron, Discrete gradient theorem and element-based integration in meshless methods. Presentation at the ECCOMAS Congress, 2016.

W. Pilkey, Formulas for stress, strain, and structural matrices, 1994.

H. Power and V. Barraco, A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations, Computers & Mathematics with Applications, vol.43, issue.3-5, pp.551-583, 2002.

D. Price, Smoothed particle hydrodynamics and magnetohydrodynamics, Journal of Computational Physics, vol.231, issue.3, pp.759-794, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01983030

A. V. Pukhlikov and A. G. Khovanskii, The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes, Algebra i analiz, vol.4, issue.4, pp.188-216, 1992.

J. Qian, Discrete gradient method in solid mechanics, 2009.

J. Qian, J. Li, W. , and S. , The generalized Riemann problems for hyperbolic balance laws: A unified formulation towards high order, 2013.

T. Rabczuk, S. P. Xiao, and M. Sauer, Coupling of mesh-free methods with finite elements: basic concepts and test results, Communications in Numerical Methods in Engineering, vol.22, issue.10, pp.1031-1065, 2006.

P. Randles and L. Libersky, Smoothed particle hydrodynamics: some recent improvements and applications, Computer Methods in Applied Mechanics and Engineering, vol.139, issue.1-4, pp.375-408, 1996.

P. Randles and L. Libersky, Smoothed particle hydrodynamics: some recent improvements and applications, Computer Methods in Applied Mechanics and Engineering, vol.139, issue.1-4, pp.375-408, 1996.
DOI : 10.1016/s0045-7825(96)01090-0

D. Ranganathan, A gentle introduction to grassmannians, 2010.

J. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks, ASME, 1968.

M. Rigo, Perron-Frobenius theory. Advanced Graph Theory and Combinatorics, pp.183-232, 2016.

D. P. Rooke and D. J. Cartwright, Compendium of stress intensity factors. Procurement Executive, Ministry of Defence, vol.330, p.p, 1976.

M. Rosenblatt, Remarks on some nonparametric estimates of a density function, The Annals of Mathematical Statistics, pp.832-837, 1956.

A. Samarskii, V. Tishkin, A. Favorskii, and M. Shashkov, Operational finitedifference schemes. Differential Equations, vol.17, pp.854-862, 1981.

W. Schaap, The Delaunay tessellation field estimator, 2007.

R. Schaback and H. Wendland, Kernel techniques: from machine learning to meshless methods, Acta Numerica, vol.15, pp.543-639, 2006.
DOI : 10.1017/s0962492906270016
URL : http://www.num.math.uni-goettingen.de/schaback/research/papers/KTfMLtMM.pdf

H. Schechter and R. Bridson, Ghost SPH for animating water, ACM Transactions on Graphics (TOG), vol.31, issue.4, p.61, 2012.
DOI : 10.1145/2185520.2185557
URL : http://www.cs.ubc.ca/~rbridson/docs/schechter-siggraph2012-ghostsph.pdf

I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, IJ Schoenberg Selected Papers, pp.3-57, 1988.

M. Serrano and P. Español, Thermodynamically consistent mesoscopic fluid particle models for a Van der Waals fluid, 2000.
DOI : 10.1103/physreve.64.046115

M. Serrano and P. Espanol, Thermodynamically consistent mesoscopic fluid particle model, Physical Review E, vol.64, issue.4, p.46115, 2001.
DOI : 10.1103/physreve.64.046115

M. Serrano, P. Español, and I. Zúñiga, Voronoi fluid particle model for Euler equations, Journal of Statistical Physics, vol.121, issue.1-2, pp.133-147, 2005.
DOI : 10.1007/s10955-005-8414-y

D. Shepard, A two-dimensional interpolation function for irregularly-spaced data, Proceedings of the 1968 23rd ACM National Conference, pp.517-524, 1968.
DOI : 10.1145/800186.810616
URL : http://revistaseug.ugr.es/index.php/cuadgeo/article/download/5914/6786

R. Sibson, A vector identity for the Dirichlet tessellation, Mathematical Proceedings of the Cambridge Philosophical Society, vol.87, pp.151-155, 1980.
DOI : 10.1017/s0305004100056589

G. Smith, On a fundamental error in two papers of C.-C. Wang "on representations for isotropic functions, parts I and II, Archive for Rational Mechanics and Analysis, vol.36, issue.3, pp.161-165, 1970.

A. Solovev, E. Soloveva, V. Tishkin, A. Favorskii, and M. Y. Shashkov, Approximation of finite-difference operators on a mesh of Dirichlet cells, Differential Equations, vol.22, issue.7, pp.863-872, 1986.

A. Sommariva and M. Vianello, Meshless cubature by Green's formula, Applied Mathematics and Computation, vol.183, issue.2, pp.1098-1107, 2006.
DOI : 10.1016/j.amc.2006.05.211
URL : http://www.math.unipd.it/~marcov/pdf/green.pdf

A. Sommariva and M. Vianello, Numerical cubature on scattered data by radial basis functions, Computing, p.295, 2006.
DOI : 10.1007/s00607-005-0142-2

N. H. Son, An enforced essential boundary condition by penalty method in the element-free Galerkin (EFG) methods, Vietnam Journal of Mechanics, vol.31, issue.2, pp.122-132, 2009.

S. Sternberg, Supersymmetry and Equivariant de Rham Theory, 1999.

G. W. Stewart, On the perturbation of pseudo-inverses, projections and linear least squares problems, SIAM review, vol.19, issue.4, pp.634-662, 1977.

E. Stiemke, Über positive Lösungen homogener linearer Gleichungen. Mathematische Annalen, vol.76, pp.340-342, 1915.

G. Strang and G. Fix, An analysis of the finite element method, vol.212, 1973.

F. Stummel, The limitations of the patch test, International Journal for Numerical Methods in Engineering, vol.15, issue.2, pp.177-188, 1980.

X. Suo, Analyse mathématique et numérique de la propagation des fissures par le modèle de multi-couronnes, 1990.

J. Swegle, D. Hicks, and S. Attaway, Smoothed particle hydrodynamics stability analysis, Journal of Computational Physics, vol.116, issue.1, pp.123-134, 1995.

J. W. Swegle, Conservation of momentum and tensile instability in particle methods, 2000.

A. Sylvie and L. Jacques-olivier, Equivalence between order and cell complex representations, Proceedings of the Computer Vision Winter Workshop, pp.222-233, 2002.

A. Syrakos, S. Varchanis, Y. Dimakopoulos, A. Goulas, T. et al., On the order of accuracy of the divergence theorem (Green-Gauss) method for calculating the gradient in finite volume methods, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01532882

R. Taylor, J. Simo, O. Zienkiewicz, C. , and A. , The patch test-a condition for assessing FEM convergence, International Journal for Numerical Methods in Engineering, vol.22, issue.1, pp.39-62, 1986.

. Trevorgoodchild, User Trevorgoodchild on Wikipedia, 2018.

H. K. Versteeg and W. Malalasekera, An introduction to computational fluid dynamics: the finite volume method, 2007.

J. P. Vila, SPH renormalized hybrid methods for conservation laws: applications to free surface flows, Meshfree Methods for Partial Differential Equations II, pp.207-229, 2005.

C. C. Wang, On representations for isotropic functions. part I. isotropic functions of symmetric tensors and vectors. Archive for Rational Mechanics and Analysis, vol.33, pp.249-267, 1969.

D. S. Watkins, A generalization of the Bramble-Hilbert lemma and applications to multivariate interpolation, Journal of Approximation Theory, vol.26, issue.3, pp.219-231, 1979.

H. Wendland, Konstruktion und Untersuchung radialer Basisfunktionen mit kompaktem Träger. Göttingen, Georg-August, 1996.

H. Wendland, Meshless Galerkin methods using radial basis functions. Mathematics of Computation of the, vol.68, pp.1521-1531, 1999.

H. Wendland, Local polynomial reproduction and moving least squares approximation, IMA Journal of Numerical Analysis, vol.21, issue.1, pp.285-300, 2001.

H. Wendland, Scattered data approximation, vol.17, 2004.

Z. I. Wo?nicki, Solving Linear Systems: An Analysis of Matrix Prefactorization Iterative Methods. Matrix Editions, 2009.

W. Zeng and G. Liu, Smoothed finite element methods (S-FEM): an overview and recent developments, Archives of Computational Methods in Engineering, vol.25, issue.2, pp.397-435, 2018.

Q. Zhang, Quadrature for meshless Nitsche's method. Numerical Methods for Partial Differential Equations, vol.30, pp.265-288, 2014.

X. Zhang, X. Liu, M. Lu, C. , and Y. , Imposition of essential boundary conditions by displacement constraint equations in meshless methods, Communications in Numerical Methods in Engineering, vol.17, issue.3, pp.165-178, 2001.

X. Zhang, K. Z. Song, M. W. Lu, and X. Liu, Meshless methods based on collocation with radial basis functions, Computational Mechanics, vol.26, issue.4, pp.333-343, 2000.

T. Zhu and S. Atluri, A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method, Computational Mechanics, vol.21, issue.3, pp.211-222, 1998.