Skip to main content
×
×
Home

Variationally consistent discretization schemes and numerical algorithms for contact problems*

  • Barbara Wohlmuth (a1)
Abstract

We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities, such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance. The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain. Having a uniform inf-sup bound, one can then establish optimal low-order a priori convergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations. Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal–dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual- and equilibrated-based a posteriori error estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time-integration schemes. However, the differential algebraic character of the system can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.

Copyright
References
Hide All
Achdou, Y. and Pironneau, O. (2005), Computational Methods for Option Pricing, SIAM.
Acosta, M., Merten, C., Eigenberger, G., Class, H., Helmig, R., Thoben, B. and Müller-Steinhagen, H. (2006), ‘Modeling non-isothermal two-phase multicom-ponent flow in the cathode of PEM fuel cells’, J. Power Sour. 159, 11231141.
Adams, R. (1975), Sobolev Spaces, Academic Press.
Ahrens, J., Geveci, B. and Law, C. (2005), ParaView: An end-user tool for large data visualization. In The Visualization Handbook (Hansen, C. D. and Johnson, C. R., eds), Elsevier, pp. 717732. Available at: www.paraview.org.
Ainsworth, M. and Oden, J. (1993), ‘A posteriori error estimators for 2nd order elliptic systems II: An optimal order process for calculating self-equilibrated fluxes’, Comput. Math. Appl. 26, 7587.
Ainsworth, M. and Oden, J. (2000), A Posteriori Error Estimation in Finite Element Analysis, Wiley.
Ainsworth, M., Oden, J. and Lee, C. (1993), ‘Local a posteriori error estimators for variational inequalities’, Numer. Methods Partial Diff. Equations 9, 2333.
Alart, P. and Curnier, A. (1991), ‘A mixed formulation for frictional contact problems prone to Newton like solution methods’, Comput. Methods Appl. Mech. Engrg 92, 353375.
Alberty, J., Carstensen, C. and Zarrabi, D. (1999), ‘Adaptive numerical analysis in primal elastoplasticity with hardening’, Comput. Methods Appl. Mech. Engrg 171, 175204.
Andersen, K., Christiansen, E., Conn, A. and Overton, M. (2000), ‘An efficient primal–dual interior point method for minimizing a sum of Euclidean norms’, SIAM J. Sci. Comput. 22, 243262.
Armero, F. and Petöcz, E. (1998), ‘A new class of conserving algorithms for dynamic contact problems.’, Comput. Methods Appl. Mech. Engrg 158, 269300.
Armero, F. and Petöcz, E. (1999), ‘A new dissipative time-stepping algorithm for frictional contact problems: Formulation and analysis’, Comput. Methods Appl. Mech. Engrg 179, 151178.
Arnold, D. and Awanou, G. (2005), ‘Rectangular mixed finite elements for elasticity’, Math. Models Meth. Appl. Sci. 15, 14171429.
Arnold, D. and Winther, R. (2002), ‘Mixed finite element methods for elasticity’, Numer. Math. 92, 401419.
Arnold, D. and Winther, R. (2003), Mixed finite elements for elasticity in the stress-displacement formulation. In Current Trends in Scientific Computing (Chen, Z., Glowinski, R. and Li, K., eds), Vol. 329 of Contemporary Mathematics, AMS, pp. 3342.
Arnold, D., Awanou, G. and Winther, R. (2008), ‘Finite elements for symmetric tensors in three dimensions’, Math. Comput. 77, 12291251.
Arnold, D., Falk, R. and Winther, R. (2006), Differential complexes and stability of finite element methods II: The elasticity complex. In Compatible Spatial Discretizations (Arnold, D. N.et al., eds), Vol. 142 of The IMA Volumes in Mathematics and its Applications, Springer, pp. 4767.
Babuška, I. and Strouboulis, T. (2001), The Finite Element Method and its Reliability., Clarendon.
Bajer, A. and Demkowicz, L. (2002), ‘Dynamic contact/impact problems, energy conservation, and planetary gear trains’, Comput. Methods Appl. Mech. En-gng 191, 41594191.
Baker, G. and Dougalis, V. (1976), ‘The effect of quadrature errors on finite element approximations for second order hyperbolic equations’, SIAM J. Numer. Anal. 13, 577598.
Ballard, P. (1999), ‘A counter-example to uniqueness in quasi-static elastic contact problems with small friction’, Internat. J. Engrg Sci. 37, 163178.
Ballard, P. and Basseville, S. (2005), ‘Existence and uniqueness for dynamical unilateral contact with Coulomb friction: A model problem’, M2AN: Math. Model. Numer. Anal. 39, 5977.
Ballard, P., Léger, A. and Pratt, E. (2006), Stability of discrete systems involving shocks and friction. In Analysis and Simulation of Contact Problems (Wrig-gers, P. and Nackenhorst, U., eds), Vol. 27 of Lecture Notes in Applied and Computational Mechanics, Springer, pp. 343350.
Bastian, P., Birken, K., Johannsen, K., Lang, S., Neuβ, N., Rentz-Reichert, H. and Wieners, C. (1997), ‘UG: A flexible software toolbox for solving partial differential equations’, Comput. Vis. Sci. 1, 2740.
Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M. and Sander, O. (2008), ‘A generic grid interface for parallel and adaptive scientific computing II: Implementation and tests in DUNE’, Computing 82, 121138.
Bayada, G., Sabil, J. and Sassi, T. (2002), ‘Neumann–Dirichlet algorithm for unilateral contact problems: Convergence results’, CR Math. Acad. Sci. Paris 335, 381386.
Bayada, G., Sabil, J. and Sassi, T. (2008), ‘Convergence of a Neumann–Dirichlet algorithm for tow-body contact problems with nonlocal Coulomb's friction law’, ESAIM: Math. Model. Numer. Anal. 42, 243262.
Belhachmi, Z. (2003), ‘A posteriori error estimates for the 3D stabilized mortar finite element method applied to the Laplace equation’, Math. Model. Numer. Anal. 37, 9911011.
Belhachmi, Z. (2004), ‘Residual a posteriori error estimates for a 3D mortar finite-element method: The Stokes system’, IMA J. Numer. Anal. 24, 521546.
Belhachmi, Z. and Ben Belgacem, F. (2000), ‘Finite elements of order two for Signorini's variational inequality’, CR Acad. Sci. Paris, Sér. I: Math. 331, 727732.
Ben Belgacem, F. (2000), ‘Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods’, SIAM J. Numer. Anal. 37, 11981216.
Ben Belgacem, F. and Maday, Y. (1997), ‘The mortar element method for three dimensional finite elements’, M2AN: Math. Model. Numer. Anal. 31, 289– 302.
Ben Belgacem, F. and Renard, Y. (2003), ‘Hybrid finite element methods for the Signorini problem’, Math. Comp. 72, 11171145.
Ben Belgacem, F., Hild, P. and Laborde, P. (1997), ‘Approximation of the unilateral contact problem by the mortar finite element method’, CR Acad. Sci. Paris, S ér. I 324, 123127.
Ben Belgacem, F., Hild, P. and Laborde, P. (1998), ‘The mortar finite element method for contact problems’, Math. Comput. Modelling 28, 263271.
Ben Belgacem, F., Hild, P. and Laborde, P. (1999), ‘Extension of the mortar finite element method to a variational inequality modeling unilateral contact’, Math. Models Methods Appl. Sci. 9, 287303.
Bergam, A., Bernardi, C., Hecht, F. and Mghazli, Z. (2003), ‘Error indicators for the mortar finite element discretization of a parabolic problem’, Numer. Algorithms 34, 187201.
Bernardi, C. and Hecht, F. (2002), ‘Error indicators for the mortar finite element discretization of the Laplace equation’, Math. Comput. 71, 13711403.
Bernardi, C., Maday, Y. and Patera, A. (1993), Domain decomposition by the mortar element method. In Asymptotic and Numerical Methods for Partial Differential Equations With Critical Parameters (Kaper, H.et al., eds), Reidel, pp. 269286.
Bernardi, C., Maday, Y. and Patera, A. (1994), A new nonconforming approach to domain decomposition: The mortar element method. In Nonlinear Partial Differential Equations and their Applications (Brezis, H. and Lions, J.-L., eds), Vol. XI of Collège de France Seminar, Pitman, pp. 1351.
Betsch, P. and Hesch, C. (2007), Energy-momentum conserving schemes for friction-less contact problem I: NTS method. In Computational Methods in Contact Mechanics, Vol. 3 of IUTAM, Springer, pp. 7796.
Betsch, P. and Steinmann, P. (2002 a), ‘Conservation properties of a time FE method III: Mechanical systems with holonomic constraints’, Internat. J. Numer. Methods Engrg 53, 22712304.
Betsch, P. and Steinmann, P. (2002 b), ‘A DAE approach to flexible multibody dynamics’, Multibody Syst. Dyn. 8, 367391.
Bildhauer, M., Fuchs, M. and Repin, S. (2008), ‘Duality based a posteriori error estimates for higher order variational inequalities with power growth functionals’, Ann. Acad. Sci. Fenn., Math. 33, 475490.
Binev, P., Dahmen, W. and DeVore, R. (2004), ‘Adaptive finite element methods with convergence rates’, Numer. Math. 97, 219268.
Binning, P. and Celia, M. (1999), ‘Practical implementation of the fractional flow approach to multi-phase flow simulation’, Adv. Water Resour. 22, 461478.
Black, F. and Scholes, M. (1973), ‘The pricing of options and corporate liabilities’, J. Pol. Econ. 81, 637659.
Blum, H. and Suttmeier, F. (2000), ‘An adaptive finite element discretisation for a simplified Signorini problem’, Calcolo 37, 6577.
Boieri, P., Gastaldi, F. and Kinderlehrer, D. (1987), ‘Existence, uniqueness, and regularity results for the two-body contact problem’, Appl. Math. Optim. 15, 251277.
Borri, M., Bottasso, C. and Trainelli, L. (2001), ‘Integration of elastic multibody systems by invariant conserving/dissipating algorithms II: Numerical schemes and applications’, Comput. Methods Appl. Mech. Engrg 190, 37013733.
Bostan, V. and Han, W. (2006), ‘A posteriori error analysis for finite element solutions of a frictional contact problem’, Comput. Methods Appl. Mech. Engrg 195, 12521274.
Bostan, V., Han, W. and Reddy, B. (2005), ‘A posteriori error estimation and adaptive solution of elliptic variational inequalities of the second kind’, Appl. Numer. Math. 52, 1338.
Braess, D. (2005), ‘A posteriori error estimators for obstacle problems: Another look’, Numer. Math. 101, 523549.
Braess, D. and Dahmen, W. (1998), ‘Stability estimates of the mortar finite element method for 3-dimensional problems’, East–West J. Numer. Math. 6, 249263.
Braess, D. and Dahmen, W. (2002), The mortar element method revisited: What are the right norms? In Domain Decomposition Methods in Science and Engineering: Thirteenth International Conference on Domain Decomposition Methods (Debit, N.et al., eds), CIMNE, pp. 2740.
Braess, D., Carstensen, C. and Hoppe, R. (2007), ‘Convergence analysis of a conforming adaptive finite element method for an obstacle problem’, Numer. Math. 107, 455471.
Braess, D., Carstensen, C. and Hoppe, R. (2009 a), ‘Error reduction in adaptive finite element approximations of elliptic obstacle problems’, J. Comput. Math. 27, 148169.
Braess, D., Carstensen, C. and Reddy, B. (2004), ‘Uniform convergence and a posteriori error estimators for the enhanced strain finite element method’, Numer. Math. 96, 461479.
Braess, D., Hoppe, R. and Schöberl, J. (2008), ‘A posteriori estimators for obstacle problems by the hypercircle method’, Comput. Visual. Sci. 11, 351362.
Braess, D., Pillwein, V. and Schöberl, J. (2009 b), ‘Equilibrated residual error estimates are p-robust’, Comput. Methods Appl. Mech. Engrg 198, 11891197.
Brandt, A. and Cryer, C. (1983), ‘Multigrid algorithms for the solution of linear complementarity problems arising from free boundary problems’, SIAM J. Sci. Statist. Comput. 4, 655684.
Brenan, K., Campbell, S. and Petzold, L. (1989), Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North-Holland.
Brezis, H. (1971), ‘Problèmes unilatéraux’, J. Math. Pures Appl. 9, 1168.
Brezzi, F. and Fortin, M. (1991), Mixed and Hybrid Finite Element Methods, Springer.
Brezzi, F. and Marini, D. (2001), ‘Error estimates for the three-field formulation with bubble stabilization’, Math. Comput. 70, 911934.
Brezzi, F., Hager, W. and Raviart, P. (1977), ‘Error estimates for the finite element solution of variational inequalities’, Numer. Math. 28, 431443.
Brink, U. and Stein, E. (1998), ‘A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems’, Comput. Methods Appl. Mech. Engrg 161, 77101.
Brooks, R. and Corey, A. (1964), ‘Hydraulic properties of porous media’, Colorado State University, Fort Collins, Hydrology Paper 3, 2227.
Brunβen, S. and Wohlmuth, B. (2009), ‘An overlapping domain decomposition method for the simulation of elastoplastic incremental forming processes’, Internat. J. Numer. Methods Engrg 77, 12241246.
Brunβen, S., Hager, C., Wohlmuth, B. and Schmid, F. (2008), Simulation of elasto-plastic forming processes using overlapping domain decomposition and inexact Newton methods. In IUTAM Symposium on Theoretical, Computational an d Model l ing A spects o f In e lastic Media (Reddy, B. D., ed.), Springer Science and Business media, pp. 155164.
Buscaglia, G., Duran, R., Fancello, E., Feijoo, R. and Padra, C. (2001), ‘An adaptive finite element approach for frictionless contact problems’, Internat. J. Numer. Methods Engrg 50, 394418.
Carstensen, C., Scherf, O. and Wriggers, P. (1999), ‘Adaptive finite elements for elastic bodies in contact’, SIAM J. Sci. Comput. 20, 16051626.
Cascon, M., Kreuzer, C., Nochetto, R. and Siebert, K. (2008), ‘Quasi-optimal convergence rate for an adaptive finite element method’, SIAM J. Numer. Anal. 46, 25242550.
Chan, T., Golub, G. and Mulet, P. (1999), ‘A nonlinear primal–dual method for total variation-based image restoration’, SIAM J. Sci. Comput. 20, 19641977.
Chapelle, D. and Bathe, K. (1993), ‘The inf-sup test’, Comput. Struct. 47, 537545.
Chavent, G. and Jaffré, J. (1986), Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland.
Chawla, V. and Laursen, T. (1998), ‘Energy consistent algorithms for frictional contact problems’, Internat. J. Numer. Methods Engrg 42, 799827.
Cheddadi, I., Fučík, R., Prieto, M. and Vohralík, M. (2008), ‘Computable a posteriori error estimates in the finite element method based on its local conservativity: Improvements using local minimization’, ESAIM: Proc. 24, 7796.
Cheddadi, I., Fučík, R., Prieto, M. and Vohralík, M. (2009), ‘Guaranteed and robust a posteriori error estimates for singularly perturbed reaction–diffusion problems’, ESAIM: Math. Model. Numer. Anal. 43, 867888.
Chen, B., Chen, X. and Kanzow, C. (2000), ‘A penalized Fischer–Burmeister NCP-function’, Math. Program., Ser. A 88, 211216.
Chen, J. (2007), ‘On some NCP-functions based on the generalized Fischer–Burmeister function’, Asia–Pac. J. Oper. Res. 24, 401420.
Chen, Z. and Nochetto, R. (2000), ‘Residual type a posteriori error estimates for elliptic obstacle problems’, Numer. Math. 84, 527548.
Chernov, A., Geyn, S., Maischak, M. and Stephan, E. (2006), Finite element/boundary element coupling for two-body elastoplastic contact problems with friction. In Analysis and Simulation of Contact Problems (Wriggers, P. and Nackenhorst, U., eds), Vol. 27 of Lecture Notes in Applied and Computational Mechanics, Springer, pp. 171178.
Chernov, A., Maischak, M. and Stephan, E. (2008), ‘hp-mortar boundary element method for two-body contact problems with friction’, Math. Meth. Appl. Sci. 31, 20292054.
Christensen, P. (2002 a), ‘A nonsmooth Newton method for elastoplastic problems’, Comput. Methods Appl. Mech. Engrg 191, 11891219.
Christensen, P. (2002 b), ‘A semi-smooth Newton method for elasto-plastic contact problems’, Internat. J. Solids Structures 39, 23232341.
Christensen, P. and Pang, J. (1999), Frictional contact algorithms based on semis-mooth Newton methods. In Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods (Fukushima, M. and Qi, L., eds), Kluwer, pp. 81116.
Christensen, P., Klarbring, A., Pang, J. and Strömberg, N. (1998), ‘Formulation and comparison of algorithms for frictional contact problems’, Internat. J. Numer. Methods Engrg 42, 145173.
Chung, J. and Hulbert, G. (1993), ‘A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized α-method’, J. Appl. Mech. 60, 371375.
Ciarlet, P. (1991), Basic error estimates for elliptic problems. In Finite Element Methods, Part 1 (Ciarlet, P. and Lions, J., eds), Vol. 2 of Handbook of Numerical Analysis, North-Holland, pp. 19351.
Ciarlet, P. (1998), Mathematical Elasticity, Vol. I, North-Holland.
Class, H. (2001), Theorie und numerische Modellierung nichtisothermer Mehrphasenprozesse in NAPL-kontaminierten porösen Medien. PhD thesis, Institut für Wasserbau, Universität Stuttgart.
Class, H. and Helmig, R. (2002), ‘Numerical simulation of non-isothermal multiphase multicomponent processes in porous media 2: Applications for the injection of steam and air’, Adv. Water Resour. 25, 551564.
Class, H., Helmig, R. and Bastian, P. (2002), ‘Numerical simulation of non-isothermal multiphase multicomponent processes in porous media 1: An efficient solution technique’, Adv. Water Resour. 25, 533550.
Coorevits, P., Hild, P. and Pelle, J. (2000), ‘A posteriori error estimation for unilateral contact with matching and non-matching meshes’, Comput. Methods Appl. Mech. Engrg 186, 6583.
Coorevits, P., Hild, P., Lhalouani, K. and Sassi, T. (2001), ‘Mixed finite element methods for unilateral problems: Convergence analysis and numerical studies’, Math. Comp. 71, 125.
Dautray, R. and Lions, J. (1992), Mathematical Analysis and Numerical Methods for Science and Technology: Evolution Problems, Vol. 5, Springer.
de Neef, M. (2000), Modelling capillary effects in heterogeneous porous media. PhD thesis, University of Delft, Netherlands.
De Saxcé, G. and Feng, Z. (1991), ‘New inequality and functional for contact with friction: The implicit standard material approach’, Mech. Based Des. Struct. Mach. 19, 301325.
Dembo, R., Eisenstat, S. and Steinhaug, T. (1982), ‘Inexact Newton methods’, SIAM J. Numer. Anal. 19, 400408.
Demkowicz, L. (1982), ‘On some results concerning the reciprocal formulation for the Signorini's problem’, Comput. Math. Appl. 8, 5774.
Demkowicz, L. and Bajer, A. (2001), ‘Conservative discretization of contact/impact problems for nearly rigid bodies’, Comput. Methods Appl. Mech. Engng 190, 19031924.
Demkowicz, L. and Oden, T. (1982), ‘On some existence and uniqueness results in contact problems with nonlocal friction’, Nonlinear Anal.: Theory Methods Appl. 6, 10751093.
Deuflhard, P. (2004), Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer.
Deuflhard, P., Krause, R. and Ertel, S. (2008), ‘A contact-stabilized Newmark method for dynamical contact problems’, Internat. J. Numer. Methods Engrg 73, 12741290.
Dickopf, T. and Krause, R. (2009 a), ‘Efficient simulation of multi-body contact problems on complex geometries: A flexible decomposition approach using constrained minimization’, Internat. J. Numer. Methods Engrg 77, 18341862.
Dickopf, T. and Krause, R. (2009 b), ‘Weak information transfer between non-matching warped interfaces.’, Bercovier, Michel (ed.) et al., Domain decomposition methods in science and engineering XVIII. Selected papers based on the presentations at the 18th international conference of domain decomposition methods, Jerusalem, Israel, January 12–17, 2008. Berlin: Springer. Lecture Notes in Computational Science and Engineering 70, 283–290 (2009).
Dohrmann, C., Key, S. and Heinstein, M. (2000), ‘A method for connecting dissimilar finite element meshes in two dimensions’, Internat. J. Numer. Methods Engrg 48, 655678.
Dörfler, W. (1996), ‘A convergent adaptive algorithm for Poisson's equation’, SIAM J. Numer. Anal. 33, 11061124.
Dörsek, P. and Melenk, J. (2010), ‘Adaptive hp-FEM for the contact problem with Tresca friction in linear elasticity: The primal–dual formulation and a posteriori error estimation’, Appl. Numer. Math. 60, 689704.
Dostál, Z. (2009), Optimal Quadratic Programming Algorithms, with Applications to Variational Inequalities, Vol. 23 of Springer Optimization and its Applications, Springer.
Dostál, Z. and Horák, D. (2003), ‘Scalability and FETI based algorithm for large discretized variational inequalities’, Math. Comput. Simul. 61, 347357.
Dostál, Z., Friedlander, A. and Santos, S. (1998), ‘Solution of coercive and semico-ercive contact problems by FETI domain decomposition’, Contemp. Math. 218, 8293.
Dostál, Z., Gomes Neto, F. and Santos, S. (2000), ‘Solution of contact problems by FETI domain decomposition with natural coarse space projections’, Comput. Methods Appl. Mech. Engrg 190, 16111627.
Dostál, Z., Horák, D. and Stefanica, D. (2007), ‘A scalable FETI-DP algorithm for a semi-coercive variational inequality’, Comput. Methods Appl. Mech. Engrg 196, 13691379.
Dostál, Z., Horák, D. and Stefanica, D. (2009), ‘A scalable FETI-DP algorithm with non-penetration mortar conditions on contact interface’, J. Comput. Appl. Math. 231, 577591.
Dostál, Z., Horák, D., Kučera, R., Vondrák, V., Haslinger, J., Dobiaš, J. and Pták, S. (2005), ‘FETI based algorithms for contact problems: Scalability, large displacements and 3D Coulomb friction’, Comput. Methods Appl. Mech. Engrg 194, 395409.
Doyen, D. and Ern, A. (2009), ‘Convergence of a space semi-discrete modified mass method for the dynamic Signorini problem’, Commun. Math. Sci. 7, 1063– 1072.
Duvaut, G. and Lions, J. (1976), Inequalities in Mechanics and Physics, Springer. Translation by John, C. W..
Eck, C. (2002), ‘Existence of solutions to a thermo-viscoelastic contact problem with Coulomb friction’, Math. Models Methods Appl. Sci. 12, 14911511.
Eck, C. and Jarušek, J. (1998), ‘Existence results for the static contact problem with Coulomb friction’, Math. Models Methods Appl. Sci. 8, 445468.
Eck, C. and Jarušek, J. (2001), ‘On the thermal aspect of dynamic contact problems’, Math. Bohem. 126, 337352.
Eck, C. and Jarušek, J. (2003), ‘Existence of solutions for the dynamic frictional contact problem of isotropic viscoelastic bodies’, Nonlinear Anal., Theory Methods Appl. 53, 157181.
Eck, C. and Wendland, W. (2003), ‘A residual-based error estimator for BEM discretizations of contact problems’, Numer. Math. 95, 253282.
Eck, C. and Wohlmuth, B. (2003), ‘Convergence of a contact-Neumann iteration for the solution of two-body contact problems’, Math. Models Methods Appl. Sci. 13, 11031118.
Eck, C., Jarušek, J. and Krbec, M. (2005), Unilateral Contact Problems: Variational Methods and Existence Theorems, CRC Press.
Eisenstat, S. and Walker, H. (1996), ‘Choosing the forcing terms in an inexact Newton method’, SIAM J. Sci. Comput. 17, 1632.
Erdmann, B., Frei, M., Hoppe, R., Kornhuber, R. and Wiest, U. (1993), ‘Adaptive finite element methods for variational inequalities’, East–West J. Numer. Math. 1, 165197.
Ern, A. and Vohralík, M. (2009), ‘Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids’, CR Math. Acad. Sci. Paris 347, 441444.
Evans, L. (1998), Partial Differential Equations, AMS.
Facchinei, F. and Pang, J. (2003 a), Finite-Dimensional Variational Inequalities and Complementary Problems, Vol. I, Springer Series in Operations Research.
Facchinei, F. and Pang, J. (2003 b), Finite-Dimensional Variational Inequalities and Complementary Problems, Vol. II, Springer Series in Operations Research.
Falk, R. (1974), ‘Error estimates for the approximation of a class of variational inequalities’, Math. Comp. 28, 963971.
Felippa, C. (2000), ‘On the original publication of the general canonical functional of linear elasticity’, J. Appl. Mech. 67, 217219.
Fichera, G. (1964), ‘Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno’, Mem. Accad. Naz. Lincei 8, 91140.
Fischer, A. (1992), ‘A special Newton-type optimization method’, Optimization 24, 269284.
Fischer-Cripps, A. (2000), Introduction to Contact Mechanics, Springer Mechanical Engineering Series.
Fischer, K. and Wriggers, P. (2006), ‘Mortar based frictional contact formulation for higher order interpolations using the moving friction cone’, Comput. Methods Appl. Mech. Engrg 195, 50205036.
Flemisch, B., Fritz, J., Helmig, R., Niessner, J. and Wohlmuth, B. (2007), DUMUX: A multi-scale multi-physics toolbox for flow and transport processes in porous media. In ECCOMAS Thematic Conference on Multi-Scale Computational Methods for Solids and Fluids (Ibrahimbegovic, A. and Dias, F., eds), Cachan, France, pp. 8287.
Flemisch, B., Melenk, J. and Wohlmuth, B. (2005 a), ‘Mortar methods with curved interfaces’, Appl. Numer. Math. 54, 339361.
Flemisch, B., Puso, M. and Wohlmuth, B. (2005 b), ‘A new dual mortar method for curved interfaces: 2D elasticity’, Internat. J. Numer. Methods Engrg 63, 813832.
Fluegge, S., ed. (1972), Handbuch der Physik, Vol. VIa, chapter on Linear Ther-moelasticity, Springer, pp. 297346.
French, D., Larsson, S. and Nochetto, R. (2001), ‘Pointwise a posteriori error analysis for an adaptive penalty finite element method for the obstacle problem’, Comput. Methods Appl. Math. 1, 1838.
Fuchs, M. and Repin, S. (2010), ‘Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids’, Math. Methods Appl. Sci. 33, 11361147.
Geiger, C. and Kanzow, C. (2002), Theorie und Numerik Restringierter Optimierungsaufgaben, Springer.
Gitterle, M., Popp, A., Gee, M. and Wall, W. (2010), ‘Finite deformation frictional mortar contact using a semi-smooth Newton method with consistent linearization’, Internat. J. Numer. Methods Engrg 84, 543571.
Glowinski, R. (1984), Numerical Methods for Nonlinear Variational Problems, Springer.
Glowinski, R. and Le Tallec, P. (1989), Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics, Vol. 9 of SIAM Studies in Applied Mathematics.
Glowinski, R., Lions, J. and Trémolières, R. (1981), Numerical Analysis of Variational Inequalities, North-Holland.
Gonzales, M., Schmidt, B. and Ortiz, M. (2010), ‘Energy-stepping integrators in Lagrangian mechanics’, Internat. J. Numer. Methods Engrg 82, 205241.
Gonzalez, O. (2000), ‘Exact energy and momentum conserving algorithms for general models in nonlinear elasticity’, Comput. Methods Appl. Mech. Engrg 190, 17631783.
Gordon, W. and Hall, C. (1973 a), ‘Construction of curvilinear co-ordinate systems and applications to mesh generation’, Internat. J. Numer. Methods Engng 7, 461477.
Gordon, W. and Hall, C. (1973 b), ‘Transfinite element methods: Blending-function interpolation over arbitrary curved element domains’, Numer. Math. 21, 109– 129.
Gwinner, J. (2009), ‘On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction’, Appl. Numer. Math. 59, 27742784.
Hackbusch, W. (1985), Multi-Grid Methods and Applications, Springer.
Hackbusch, W. and Mittelmann, H. (1983), ‘On multigrid methods for variational inequalities’, Numer. Math. 42, 6576.
Hager, C. (2010), Robust numerical algorithms for dynamic frictional contact problems with different time and space scales. PhD thesis, IANS, Universität Stuttgart.
Hager, C. and Wohlmuth, B. (2009 a), ‘Analysis of a space-time discretization for dynamic elasticity problems based on mass-free surface elements’, SIAM J. Numer. Anal. 47, 18631885.
Hager, C. and Wohlmuth, B. (2009 b), ‘Nonlinear complementarity functions for plasticity problems with frictional contact’, Comput. Methods Appl. Mech. Engrg 198, 34113427.
Hager, C. and Wohlmuth, B. (2010), ‘Semismooth Newton methods for variational problems with inequality constraints’, GAMM–Mitt. 33, 824.
Hager, C., Hauret, P., Le Tallec, P. and Wohlmuth, B. (2010 a), Overlapping domain decomposition for multiscale dynamic contact problems. Technical report IANS Preprint 2010/007, Universität Stuttgart.
Hager, C., Huëber, S. and Wohlmuth, B. (2008), ‘A stable energy conserving approach for frictional contact problems based on quadrature formulas’, Internat. J. Numer. Methods Engrg 73, 205225.
Hager, C., Huëber, S. and Wohlmuth, B. (2010 b), ‘Numerical techniques for the valuation of basket options and its Greeks’, J. Comput. Fin. 13, 131.
Hairer, E. and Wanner, G. (1991), Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer.
Han, W. (2005), A Posteriori Error Analysis via Duality Theory: With Applications in Modeling and Numerical Approximations, Springer.
Han, W. and Reddy, B. (1995), ‘Computational plasticity: The variational basis and numerical analysis’, Comput. Mech. Advances 2, 283400.
Han, W. and Reddy, B. (1999), Plasticity: Mathematical Theory and Numerical Analysis, Springer.
Han, W. and Sofonea, M. (2000), ‘Numerical analysis of a frictionless contact problem for elastic-viscoplastic materials’, Comput. Methods Appl. Mech. Engrg 190, 179191.
Han, W. and Sofonea, M. (2002), Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, AMS, International Press.
Harker, P. and Pang, J. (1990), ‘Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications’, Math. Progr. 48, 161220.
Hartmann, S., Brunβen, S., Ramm, E. and Wohlmuth, B. (2007), ‘Unilateral nonlinear dynamic contact of thin-walled structures using a primal–dual active set strategy’, Internat. J. Numer. Meth. Engrg 70, 883912.
Haslinger, J. and Hlaváček, I. (1981), ‘Contact between two elastic bodies II: Finite element analysis’, Aplikace Mathematiky 26, 263290.
Haslinger, J., Hlaváček, I. and Nečas, J. (1996), Numerical methods for unilateral problems in solid mechanics. In Handbook of Numerical Analysis (Ciarlet, P. and Lions, J.-L., eds), Vol. IV, North-Holland, pp. 313485.
Haslinger, J., Hlaváček, I., Nečas, J. and Lovíšsek, J. (1988), Solution of Variational Inequalities in Mechanics, Springer.
Hassanizadeh, S. and Gray, W. (1993), ‘Thermodynamic basis of capillary pressure in porous media’, Water Resour. Research 29, 33893405.
Hassanizadeh, S., Celia, M. and Dahle, H. (2002), ‘Experimental measurements of saturation overshoot on infiltration’, Vadose Zone J. 1, 3857.
Hauret, P. and Le Tallec, P. (2006), ‘Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact’, Comput. Methods Appl. Mech. Engrg 195, 48904916.
Hauret, P. and Le Tallec, P. (2007), ‘A discontinuous stabilized mortar method for general 3D elastic problems’, Comput. Methods Appl. Mech. Engng 196, 48814900.
Hauret, P., Salomon, J., Weiss, A. and Wohlmuth, B. (2008), ‘Energy consistent co-rotational schemes for frictional contact problems’, SIAM J. Sci. Comput. 30, 24882511.
Helmig, R. (1997), Multiphase Flow and Transport Processes in the Subsurface, Springer.
Helmig, R., Weiss, A. and Wohlmuth, B. (2009), ‘Variational inequalities for modeling flow in heterogeneous porous media with entry pressure’, Comput. Geosci. 13, 373390.
Hertz, H. (1882), ‘Über die Berührung fester elastischer Körper’, J. Reine Angew. Math. 92, 156171.
Hesch, C. and Betsch, P. (2006), ‘A comparison of computational methods for large deformation contact problems of flexible bodies’, ZAMM: Z. Angew. Math. Mech. 86, 818827.
Hesch, C. and Betsch, P. (2009), ‘A mortar method for energy-momentum conserving schemes in frictionless dynamic contact problems’, Internat. J. Numer. Methods Engrg 77, 14681500.
Hesch, C. and Betsch, P. (2010), ‘Transient three-dimensional domain decomposition problems: Frame-indifferent mortar constraints and conserving integration’, Internat. J. Numer. Methods Engrg 82, 329358.
Hilber, H., Hughes, T. and Taylor, R. (1977), ‘Improved numerical dissipation for time integration algorithms in structural dynamics’, Earthquake Engrg Struct. Dyn. 5, 283292.
Hild, P. (2000), ‘Numerical implementation of two nonconforming finite element methods for unilateral contact’, Comput. Methods Appl. Mech. Engrg 184, 99123.
Hild, P. (2003), ‘An example of nonuniqueness for the continuous static unilateral contact model with Coulomb friction’, CR Math. Acad. Sci. Paris 337, 685– 688.
Hild, P. (2004), ‘Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity’, Q. J. Mech. Appl. Math. 57, 225235.
Hild, P. and Laborde, P. (2002), ‘Quadratic finite element methods for unilateral contact problems’, Appl. Numer. Math. 41, 410421.
Hild, P. and Lleras, V. (2009), ‘Residual error estimators for Coulomb friction’, SIAM J. Numer. Anal. 47, 35503583.
Hild, P. and Nicaise, S. (2005), ‘A posteriori error estimations of residual type for Signorini's problem’, Numer. Math. 101, 523549.
Hild, P. and Nicaise, S. (2007), ‘Residual a posteriori error estimators for contact problems in elasticity’, Math. Model. Numer. Anal. 41, 897923.
Hild, P. and Renard, Y. (2006), Local uniqueness results for the discrete friction problem. In Analysis and Simulation of Contact Problems (Wriggers, P. and Nackenhorst, U., eds), Vol. 27 of Lecture Notes in Applied and Computational Mechanics, Springer, pp. 129136.
Hild, P. and Renard, Y. (2007), ‘An error estimate for the Signorini problem with Coulomb friction approximated by finite elements’, SIAM J. Numer. Anal. 45, 20122031.
Hild, P. and Renard, Y. (2010), ‘A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics’, Numer. Math. 115, 101129.
Hintermüller, M. and Stadler, G. (2006), ‘An infeasible primal–dual algorithm for total variation-based inf-convolution-type image restoration’, SIAM J. Sci. Comput. 28, 123.
Hintermüller, M., Ito, K. and Kunisch, K. (2002), ‘The primal–dual active set strategy as a semi–smooth Newton method’, SIAM J. Optim. 13, 865888.
Hintermüller, M., Kovtunenko, V. and Kunisch, K. (2004), ‘Semismooth Newton methods for a class of unilaterally constrained variational problems’, Adv. Math. Sci. Appl. 14, 513535.
Hoppe, R. (1987), ‘Multigrid algorithms for variational inequalities’, SIAM J. Numer. Anal. 24, 10461065.
Hoppe, R. and Kornhuber, R. (1994), ‘Adaptive multilevel methods for obstacle problems’, SIAM J. Numer. Anal. 31, 301323.
Hu, H. (1955), ‘On some variational principles in the theory of elasticity and the theory of plasticity’, Scientia Sinica 4, 3354.
Hu, S., Huang, Z. and Chen, J. (2009), ‘Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems’, J. Comput. Appl. Math. 230, 6982.
Huber, R. and Helmig, R. (2000), ‘Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media’, Comput. Geosci. 4, 141164.
Huëber, S. (2008), Discretization techniques and efficient algorithms for contact problems. PhD thesis, IANS, Universität Stuttgart.
Huëber, S. and Wohlmuth, B. (2005 a), ‘An optimal a priori error estimate for nonlinear multibody contact problems’, SIAM J. Numer. Anal. 43, 157173.
Huëber, S. and Wohlmuth, B. (2005 b), ‘A primal–dual active set strategy for nonlinear multibody contact problems’, Comput. Methods Appl. Mech. Engrg 194, 31473166.
Huëber, S. and Wohlmuth, B. (2009), ‘Thermo-mechanical contact problem on non-matching meshes’, Comput. Methods Appl. Mech. Engrg 198, 13381350.
Huëber, S. and Wohlmuth, B. (2010), Equilibration techniques for solving contact problems with Coulomb friction. Comput. Methods Appl. Mech. Engrg doi:10.1016/j.cma.2010.12.021.
Huëber, S., Mair, M. and Wohlmuth, B. (2005 a), ‘A priori error estimates and an inexact primal–dual active set strategy for linear and quadratic finite elements applied to multibody contact problems’, Appl. Numer. Math. 54, 555576.
Huëber, S., Matei, A. and Wohlmuth, B. (2005 b), ‘A mixed variational formulation and an optimal a priori error estimate for a frictional contact problem in elasto-piezoelectricity’, Bull. Math. Soc. Sci. Math. Roumanie 48, 209232.
Huëber, S., Matei, A. and Wohlmuth, B. (2007), ‘Efficient algorithms for problems with friction’, SIAM J. Sci. Comput. 29, 7092.
Huëber, S., Stadler, G. and Wohlmuth, B. (2008), ‘A primal–dual active set algorithm for three-dimensional contact problems with Coulomb friction’, SIAM J. Sci. Comput. 30, 572596.
Hughes, T. (1987), The Finite Element Method: Linear, Static and Dynamic Finite Element Analysis, Prentice-Hall.
Hulbert, G. (1992), ‘Time finite element methods for structural dynamics’, Internat. J. Numer. Methods Engrg 33, 307331.
Hull, J. (2006), Options, Futures, and Other Derivatives, sixth edition, Prentice-Hall.
Ito, K. and Kunisch, K. (2003), ‘Semi-smooth Newton methods for variational inequalities of the first kind’, M2AN: Math. Model. Numer. Anal. 37, 4162.
Ito, K. and Kunisch, K. (2004), ‘The primal–dual active set method for nonlinear optimal control problems with bilateral constraints’, SIAM J. Control. Optim. 43, 357376.
Ito, K. and Kunisch, K. (2008 a), Lagrange Multiplier Approach to Variational Problems and Applications, SIAM.
Ito, K. and Kunisch, K. (2008 b), ‘On a semi-smooth Newton method for the Signorini problem’, Appl. Math. 53, 455468.
Jarušek, J. (1983), ‘Contact problems with bounded friction: Coercive case’, Czech. Math. J. 33, 237261.
Johnson, C. (1992), ‘Adaptive finite element methods for the obstacle problem’, Math. Models Methods Appl. Sci. 2, 483487.
Johnson, K. (1985), Contact Mechanics, Cambridge University Press.
Kane, C., Marsden, J., Ortiz, M. and West, M. (2000), ‘Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems’, Internat. J. Numer. Methods Engrg 49, 12951325.
Kanzow, C., Yamashita, N. and Fukushima, M. (1997), ‘New NCP-functions and their properties’, J. Optimization Theory Appl. 94, 115135.
Karypis, G. and Kumar, V. (1998), ‘A fast and high quality multilevel scheme for partitioning irregular graphs’, SIAM J. Sci. Comput. 20, 359392.
Kasper, E. and Taylor, R. (2000 a), ‘A mixed-enhanced strain method I: Geometrically linear problems’, Computers and Structures 75, 237250.
Kasper, E. and Taylor, R. (2000 b), ‘A mixed-enhanced strain method II: Geometrically nonlinear problems’, Computers and Structures 75, 251260.
Kelly, D. (1984), ‘The self-equilibration of residuals and complementary a posteriori error estimates in the finite element method’, Internat. J. Numer. Methods Engrg 20, 14911506.
Kelly, D. and Isles, J. (1989), ‘Procedures for residual equilibration and local error estimation in the finite element method’, Commun. Appl. Numer. Methods 5, 497505.
Khenous, H., Laborde, P. and Renard, Y. (2006 a), ‘Comparison of two approaches for the discretization of elastodynamic contact problems’, CR Math. Acad. Sci. Paris 342, 791796.
Khenous, H., Laborde, P. and Renard, Y. (2006 b), On the discretization of contact problems in elastodynamics. In Analysis and Simulation of Contact Problems (Wriggers, P. and Nackenhorst, U., eds), Vol. 27 of Lecture Notes in Applied and Computational Mechanics, Springer, pp. 3138.
Khenous, H., Laborde, P. and Renard, Y. (2008), ‘Mass redistribution method for finite element contact problems in elastodynamics’, Eur. J. Mech., A, Solids 27, 918932.
Kikuchi, N. and Oden, J. (1988), Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Vol. 8 of SIAM Studies in Applied Mathematics.
Kinderlehrer, D. and Stampacchia, G. (2000), An Introduction to Variational Inequalities and their Applications, SIAM.
Klapproth, C., Deuflhard, P. and Schiela, A. (2009), ‘A perturbation result for dynamical contact problems’, Numer. Math., Theory Methods Appl. 2, 237257.
Klapproth, C., Schiela, A. and Deuflhard, P. (2010), ‘Consistency results on Newmark methods for dynamical contact problems’, Numer. Math. 116, 6594.
Kornhuber, R. (1994), ‘Monotone multigrid methods for elliptic variational inequalities I’, Numer. Math. 69, 167184.
Kornhuber, R. (1996), ‘Monotone multigrid methods for elliptic variational inequalities II’, Numer. Math. 72, 481499.
Kornhuber, R. (1997), Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems, Teubner.
Kornhuber, R. and Krause, R. (2001), ‘Adaptive multigrid methods for Signorini's problem in linear elasticity’, Comput. Vis. Sci. 4, 920.
Kornhuber, R. and Zou, Q. (2011), ‘Efficient and reliable hierarchical error estimates for the discretization error of elliptic obstacle problems’, Math. Comp. 80, 69– 88.
Kornhuber, R., Krause, R., Sander, O., Deuflhard, P. and Ertel, S. (2007), ‘A monotone multigrid solver for two body contact problems in biomechanics’, Comput. Vis. Sci. 11, 315.
Koziara, T. and Bicanic, N. (2008), ‘Semismooth Newton method for frictional contact between pseudo-rigid bodies’, Comput. Methods Appl. Mech. Engrg 197, 27632777.
Krause, R. (2008), On the multiscale solution of constrained minimization problems. In Domain Decomposition Methods in Science and Engineering XVII (Langer, U.et al. eds), Vol. 60 of Lecture Notes in Computational Science and Engineering, Springer, pp. 93104.
Krause, R. (2009), ‘A nonsmooth multiscale method for solving frictional two-body contact problems in 2D and 3D with multigrid efficiency’, SIAM J. Sci. Comput. 31, 13991423.
Krause, R. and Mohr, C. (2011), ‘Level set based multi-scale methods for large deformation contact problems’, Appl. Numer. Math. 61, 428442.
Krause, R. and Walloth, M. (2009), ‘A time discretization scheme based on Rothe's method for dynamical contact problems with friction’, Comput. Methods Appl. Mech. Engrg 199, 119.
Krause, R. and Wohlmuth, B. (2002), ‘A Dirichlet–Neumann type algorithm for contact problems with friction’, Comput. Vis. Sci. 5, 139148.
Kuczma, M. and Demkowicz, L. (1992), ‘An adaptive algorithm for unilateral visco-elastic contact problems for beams and plates’, Comput. Methods Appl. Mech. Engng 101, 183196.
Kuhl, D. and Ramm, E. (1999), ‘Generalized energy-momentum method for nonlinear adaptive shell dynamics’, Comput. Methods Appl. Mech. Engrg pp. 343366.
Lacour, C. and Ben Belgacem, F. (2011), The Mortar Finite Element Method: Basics, Theory and Implementation, Chapman & Hall/CRC Press. To appear.
Ladevèze, P. and Leguillon, D. (1983), ‘Error estimate procedure in the finite element method and applications’, SIAM J. Numer. Anal. 20, 485509.
Ladevèze, P. and Maunder, E. (1996), ‘A general method for recovering equilibrating element tractions’, Comput. Methods Appl. Mech. Engrg 137, 111151.
Ladevèze, P. and Rougeot, P. (1997), ‘New advances on a posteriori error on constitutive relation in f.e. analysis’, Comput. Methods Appl. Mech. Engrg 150, 239249.
Lamichhane, B. and Wohlmuth, B. (2007), ‘Biorthogonal bases with local support and approximation properties’, Math. Comp. 76, 233249.
Lamichhane, B., Reddy, B. and Wohlmuth, B. (2006), ‘Convergence in the incompressible limit of finite element approximations based on the Hu–Washizu formulation’, Numer. Math. 104, 151175.
Laursen, T. (2002), Computational Contact and Impact Mechanics, Springer.
Laursen, T. and Chawla, V. (1997), ‘Design of energy conserving algorithms for frictionless dynamic contact problems’, Internat. J. Numer. Methods Engrg 40, 836886.
Laursen, T. and Love, G. (2002), ‘Improved implicit integrators for transient impact problems: Geometric admissibility within the conserving framework’, Internat. J. Numer. Methods Engrg 53, 245274.
Laursen, T. and Meng, X. (2001), ‘A new solution procedure for application of energy-conserving algorithms to general constitutive models in nonlinear elas-todynamics’, Comput. Methods Appl. Mech. Engrg 190, 63096322.
Laursen, T. and Simo, J. (1993 a), ‘A continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact problems’, Internat. J. Numer. Methods Engrg 36, 34513485.
Laursen, T. and Simo, J. (1993 b), ‘Algorithmic symmetrization of Coulomb frictional problems using augmented Lagrangians’, Comput. Methods Appl. Mech. Engng 108, 133146.
Lauser, A., Hager, C., Helmig, R. and Wohlmuth, B. (2010), A new approach for phase transitions in miscible multi-phase flow in porous media. SimTech-Preprint 2010–34, Universität Stuttgart. To appear In Adv. Water Resour.
Lee, C. and Oden, J. (1994), ‘A posteriori error estimation of h-p finite element approximations of frictional contact problems.’, Comput. Methods Appl. Mech. Engrg 113, 1145.
Lenhard, R., Parker, J. and Mishra, S. (1989), ‘On the correspondence between Brooks–Corey and Van Genuchten models’, J. Irrig. and Drain. Engrg 115, 744751.
Leverett, M. (1941), ‘Capillary behavior in porous solids’, AIME Petroleum Transactions 142, 152169.
Lhalouani, K. and Sassi, T. (1999), ‘Nonconforming mixed variational inequalities and domain decomposition for unilateral problems’, East–West J. Numer. Math. 7, 2330.
Li, J., Melenk, J., Wohlmuth, B. and Zou, J. (2010), ‘Optimal a priori estimates for higher order finite elements for elliptic interface problems’, Appl. Numer. Math. 60, 1937.
Lions, J. and Stampacchia, G. (1967), ‘Variational inequalities’, Comm. Pure Appl. Math. XX, 493519.
Liu, W. and Yan, N. (2000), ‘A posteriori error estimators for a class of variational inequalities’, J. Sci. Comput. 15, 361393.
Luce, R. and Wohlmuth, B. (2004), ‘A local a posteriori error estimator based on equilibrated fluxes’, SIAM J. Numer. Anal. 42, 13941414.
Lunk, C. and Simeon, B. (2006), ‘Solving constrained mechanical systems by the family of Newmark and α-methods’, Z. Angew. Math. Mech. 86, 772784.
Maischak, M. and Stephan, E. (2005), ‘Adaptive hp-versions of BEM for Signorini problems’, Appl. Numer. Math. 54, 425449.
Maischak, M. and Stephan, E. (2007), ‘Adaptive hp-versions of boundary element methods for elastic contact problems’, Comput. Mech. 39, 597607.
Martins, J., Barbarin, S., Raous, M. and Pinto da Costa, A. (1999), ‘Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction’, Comput. Methods Appl. Mech. Engrg 177, 289328.
Melenk, M. and Wohlmuth, B. (2011), On the convergence of surface based Lagrange multipliers in finite element methods. In preparation.
Moon, K., Nochetto, R., von Petersdorff, T. and Zhang, C. (2007), ‘A posteriori error analysis for parabolic variational inequalities’, ESAIM: Math. Model. Numer. Anal. 41, 485511.
Moreau, J. (1977), ‘Evolution problem associated with a moving convex set in a Hilbert space’, J. Differential Equations 26, 347374.
Morin, P., Nochetto, R. and Siebert, K. (2002), ‘Convergence of adaptive finite element methods’, SIAM Rev. 44, 631658.
Nečas, J., Jarušek, J. and Haslinger, J. (1980), ‘On the solution of the variational inequality to the Signorini problem with small friction’, Boll. Unione Mat. Ital., V. Ser., B 17, 796811.
Nicaise, S., Witowski, K. and Wohlmuth, B. (2008), ‘An a posteriori error estimator for the Lame equation based on H(div)-conforming stress approximations’, IMA J. Numer. Anal. 28, 331353.
Nicolaides, R. (1982), ‘Existence, uniqueness and approximation for generalized saddle point problems’, SIAM J. Numer. Anal. 19, 349357.
Niessner, J. and Helmig, R. (2007), ‘Multi-scale modeling of three-phase-three-component processes in heterogeneous porous media’, Adv. Water Resour. 30, 23092325.
Nochetto, R. and Wahlbin, L. (2002), ‘Positivity preserving finite element approximation’, Math. Comput. 71, 14051419.
Nochetto, R., von Petersdorff, T. and Zhang, C. (2010), ‘A posteriori error analysis for a class of integral equations and variational inequalities’, Numer. Math. 116, 519552.
Nochetto, R., Siebert, K. and Veeser, A. (2003), ‘Pointwise a posteriori error control for elliptic obstacle problems’, Numer. Math. 95, 163195.
Nochetto, R., Siebert, K. and Veeser, A. (2005), ‘Fully localized a posteriori error estimators and barrier sets for contact problems’, SIAM J. Numer. Anal. 42, 21182135.
Nochetto, R., Siebert, K. and Veeser, A. (2009), Theory of adaptive finite element methods: An introduction. In Multiscale, Nonlinear and Adaptive Approximation: Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday (DeVore, R.et al., eds), Springer, pp. 409542.
Oden, T., Becker, E., Lin, T. and Demkowicz, L. (1985), Formulation and finite element analysis of a general class of rolling contact problems with finite elastic deformations. In The Mathematics of Finite Elements and Applications V: MAFELAP 1984, pp. 505532.
Pandolfi, A., Kane, C., Marsden, J. and Ortiz, M. (2002), ‘Time-discretized variational formulation of non-smooth frictional contact’, Internat. J. Numer. Methods Engrg 53, 18011829.
Pang, J. (1990), ‘Newton's method for B-differentiable equations’, Math. Oper. Res. 15, 311341.
Pang, J. and Gabriel, S. (1993), ‘NE/SQP: A robust algorithm for the nonlinear complementarity problem’, Math. Progr. 60, 295337.
Pang, J. and Qi, L. (1993), ‘Nonsmooth equations: Motivation and algorithms’, SIAM J. Optim. 3, 443465.
Pironneau, O. and Achdou, Y. (2009), Partial differential equations for option pricing. In Handbook of Numerical Analysis, Vol XV: Mathematical Modeling and Numerical Methods in Finance (Bensoussan, A.et al, eds), Elsevier/North-Holland, pp. 369495.
Popp, A., Gee, M. and Wall, W. (2009), ‘A finite deformation mortar contact formulation using a primal–dual active set strategy’, Internat. J. Numer. Methods Engrg 79, 13541391.
Popp, A., Gitterle, M., Gee, M. and Wall, W. (2010), ‘A dual mortar approach for 3D finite deformation contact with consistent linearization’, Internat. J. Numer. Methods Engrg 83, 14281465.
Pousin, J. and Sassi, T. (2005), ‘A posteriori error estimates and domain decomposition with nonmatching grids’, Adv. Comput. Math. 23, 241263.
Prager, W. and Synge, J. (1947), ‘Approximations in elasticity based on concepts of function spaces’, Quart. Appl. Math. 5, 241269.
Puso, M. (2004), ‘A 3D mortar method for solid mechanics’, Internat. J. Numer. Methods Engrg 59, 315336.
Puso, M. and Laursen, T. (2004 a), ‘A mortar segment-to-segment contact method for large deformation solid mechanics’, Comput. Methods Appl. Mech. Engrg 193, 601629.
Puso, M. and Laursen, T. (2004 b), ‘A mortar segment-to-segment frictional contact method for large deformations’, Comput. Methods Appl. Mech. Engrg 193, 48914913.
Puso, M., Laursen, T. and Solberg, J. (2008), ‘A segment-to-segment mortar contact method for quadratic elements and large deformations’, Comput. Methods Appl. Mech. Engrg 197, 555566.
Raous, M., Barbarin, S. and Vola, D. (2002), Numerical characterization and computation of dynamic instabilities for frictional contact problems. In Friction and Instabilities (Martinis, J. A. C.et al., ed.), Vol. 457 of CISM Courses Lect., Springer, pp. 233291.
Raviart, P. and Thomas, J. (1983), Introduction à l'Analyse Numérique des Equations aux Dérivées Partielles, Collection Mathématiques Appliquées pour la Maîtrise, Masson.
Renard, Y. (2006), ‘A uniqueness criterion for the Signorini problem with Coulomb friction’, SIAM J. Math. Anal. 38, 452467.
Renard, Y. (2010), ‘The singular dynamic method for constrained second order hyperbolic equations: Application to dynamic contact problems’, J. Comput. Appl. Math. 234, 906923.
Repin, S. (2008), A Posteriori Estimates for Partial Differential Equations, Radon Series on Computational and Applied Mathematics, de Gruyter.
Repin, S., Sauter, S. and Smolianski, A. (2003), ‘A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions’, Computing 70, 205233.
Rivière, B. (2008), Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, SIAM.
Salomon, J., Weiss, A. and Wohlmuth, B. (2008), ‘Energy conserving algorithms for a corotational formulation’, SIAM J. Numer. Anal. 46, 18421866.
Scheidegger, A. (1960), The Physics of Flow through Porous Media, University of Toronto Press.
Schenk, O. and Gärtner, K. (2004), ‘Solving unsymmetric sparse systems of linear equations with PARDISO’, J. Future Generation Computer Systems 20, 475487.
Schenk, O. and Gärtner, K. (2006), ‘On fast factorization pivoting methods for symmetric indefinite systems’, Elec. Trans. Numer. Anal. 23, 158179.
Schöberl, J. (1997), ‘An advancing front 2D/3D-mesh generator based on abstract rules’, Comput. Visual. Sci. 1, 4152.
Schöberl, J. (1998), ‘Solving the Signorini problem on the basis of domain decomposition techniques’, Computing 60, 323344.
Scott, L. and Zhang, S. (1990), ‘Finite element interpolation of nonsmooth functions satisfying boundary conditions’, Math. Comp. 54, 483493.
Simeon, B. (2006), ‘On Lagrange multipliers in flexible multibody dynamics’, Comput. Methods Appl. Mech. Engrg 195, 69937005.
Simo, J. (1998), Local behavior in finite element methods. In Numerical Methods for Solids, Part 3 and Numerical Methods for Fluids, Part 1 (Ciarlet, P. and Lions, J., eds), Vol. VI of Handbook of Numerical Analysis, North-Holland, pp. 183499.
Simo, J. and Armero, F. (1992), ‘Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes’, Internat. J. Numer. Methods Engrg 33, 14131449.
Simo, J. and Hughes, T. (1998), Computational Inelasticity, Springer.
Simo, J. and Laursen, T. (1992), ‘Augmented Lagrangian treatment of contact problems involving friction’, Comput. Struct. 42, 97116.
Simo, J. and Rifai, M. (1990), ‘A class of assumed strain methods and the method of incompatible modes’, Internat. J. Numer. Methods Engrg 29, 15951638.
Simo, J. and Tarnow, N. (1992), ‘The discrete energy-momentum method: Conserving algorithms for nonlinear elastodynamics’, Z. Angew. Math. Phys. 43, 757792.
Simo, J., Armero, F. and Taylor, R. (1993), ‘Improved versions of assumed enhanced trilinear elements for 3D finite deformation problems’, Comput. Methods Appl. Mech. Engrg 110, 359386.
Stein, E. and Ohnimus, S. (1997), ‘Equilibrium method for postprocessing and error estimation in the finite element method’, Comput. Assist. Mech. Engrg Sci. 4, 645666.
Stein, E. and Ohnimus, S. (1999), ‘Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems’, Comput. Methods Appl. Mech. Engrg 176, 363385.
Stevenson, R. (2005), ‘An optimal adaptive finite element method’, SIAM J. Numer. Anal. 42, 21882217.
Stevenson, R. (2007), ‘Optimality of a standard adaptive finite element method’, Found. Comput. Math. 7, 245269.
Sun, D. and Qi, L. (1999), ‘On NCP-functions’, Comput. Optim. Appl. 13, 201220.
Suttmeier, F. (2005), ‘On a direct approach to adaptive FE-discretisations for elliptic variational inequalities’, J. Numer. Math. 13, 7380.
Thomée, V. (1997), Galerkin Finite Element Methods for Parabolic Problems, Springer.
Toselli, A. and Widlund, O. (2005), Domain Decomposition Methods: Algorithms and Theory, Springer.
Van Genuchten, M. (1980), ‘A closed-form equation for predicting the hydraulic conductivity of unsaturated soils’, Soil Sci. Soc. Am. J. 44, 892898.
Veeser, A. (2001), On a posteriori error estimation for constant obstacle problems. In Numerical Methods for Viscosity Solutions and Applications (Falcone, M. and Makridakis, C., eds), Vol. 59 of Advances in Mathematics for Applied Sciences, World Scientific, pp. 221234.
Verfürth, R. (1994), ‘A posteriori error estimation and adaptive mesh-refinement techniques’, J. Comput. Appl. Math. 50, 6783.
Verfürth, R. (1996), A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Advances in Numerical Mathematics, Wiley–Teubner.
Vohralík, M. (2008), ‘A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization’, CR Math. Acad. Sci. Paris 346, 687690.
Washizu, K. (1955), On the variational principles of elasticity and plasticity. Report 25–18, Massachusetts Institute of Technology.
Weiss, A. and Wohlmuth, B. (2009), ‘A posteriori error estimator and error control for contact problems’, Math. Comp. 78, 12371267.
Weiss, A. and Wohlmuth, B. (2010), ‘A posteriori error estimator for obstacle problems’, SIAM J. Sci. Comput. 32, 26272658.
Wheeler, M. and Yotov, I. (2005), ‘A posteriori error estimates for the mortar mixed finite element method’, SIAM J. Numer. Anal. 43, 10211042.
Wieners, C. (2007), ‘Nonlinear solution methods for infinitesimal perfect plasticity’, Z. Angew. Math. Mech. 87, 643660.
Wieners, C. and Wohlmuth, B. (2011), ‘A primal–dual finite element approximation for a nonlocal model in plasticity’, SIAM J. Sci. Comput. 49, 692710.
Willner, K. (2003), Kontinuums- und Kontaktmechanik, Springer.
Wilmott, P., Dewynne, J. and Howison, S. (1997), Option Pricing: Mathematical Models and Computation, Oxford Financial Press.
Wohlmuth, B. (1999 a), ‘Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers’, SIAM J. Numer. Anal. 36, 16361658.
Wohlmuth, B. (1999 b), ‘A residual based error-estimator for mortar finite element discretizations’, Numer. Math. 84, 143171.
Wohlmuth, B. (2000), ‘A mortar finite element method using dual spaces for the Lagrange multiplier’, SIAM J. Numer. Anal. 38, 9891012.
Wohlmuth, B. (2001), Discretization Methods and Iterative Solvers Based on Domain Decomposition, Springer.
Wohlmuth, B. (2005), ‘A V-cycle multigrid approach for mortar finite elements’, SIAM J. Numer. Anal. 42, 24762495.
Wohlmuth, B. (2007), ‘An a posteriori error estimator for two-body contact problems on non-matching meshes’, J. Sci. Comput. 33, 2545.
Wohlmuth, B. and Krause, R. (2001), ‘Multigrid methods based on the unconstrained product space for mortar finite element discretizations’, SIAM J. Numer. Anal. 39, 192213.
Wohlmuth, B. and Krause, R. (2003), ‘Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems’, SIAM J. Sci. Comput. 25, 324347.
Wooding, R. and Morel-Seytoux, H. (1976), ‘Multiphase fluid flow through porous media’, Annu. Rev. Fluid Mech. 8, 233274.
Wriggers, P. (2006), Computational Contact Mechanics, second edition, Springer.
Wriggers, P. and Nackenhorst, U., eds (2007), Computational Methods in Contact Mechanics, Vol. 3 of IUTAM Bookseries, Springer.
Wriggers, P. and Scherf, O. (1998), ‘Different a posteriori error estimators and indicators for contact problems’, Math. Comput. Modelling 28, 437447.
Wright, S. (1997), Primal–Dual Interior Point Methods, SIAM.
Yang, B. and Laursen, T. (2008 a), ‘A contact searching algorithm including bounding volume trees applied to finite sliding mortar formulations’, Comput. Mech. 41, 189205.
Yang, B. and Laursen, T. (2008 b), ‘A large deformation mortar formulation of self contact with finite sliding’, Comput. Methods Appl. Mech. Engrg 197, 756– 772.
Yang, B., Laursen, T. and Meng, X. (2005), ‘Two dimensional mortar contact methods for large deformation frictional sliding’, Internat. J. Numer. Methods Engrg 62, 11831225.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed