×

A survey on numerical methods for the simulation of initial value problems with sDAEs. (English) Zbl 1402.65085

Ilchmann, Achim (ed.) et al., Surveys in differential-algebraic equations IV. Cham: Springer (ISBN 978-3-319-46617-0/pbk; 978-3-319-46618-7/ebook). Differential-Algebraic Equations Forum, 221-300 (2017).
Summary: This paper provides an overview on numerical aspects in the simulation of differential-algebraic equations (DAEs). Amongst others we discuss the basic construction principles of frequently used discretization schemes, such as BDF methods, Runge-Kutta methods, and ROW methods, as well as their adaption to DAEs. Moreover, topics like consistent initialization, stabilization, parametric sensitivity analysis, co-simulation techniques, aspects of real-time simulation, and contact problems are covered. Finally, some illustrative numerical examples are presented.
For the entire collection see [Zbl 1369.65004].

MSC:

65L80 Numerical methods for differential-algebraic equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amodio, P., Mazzia, F.: Numerical solution of differential algebraic equations and computation of consistent initial/boundary conditions. J. Comput. Appl. Math. 87, 135-146 (1997) · Zbl 0894.65031 · doi:10.1016/S0377-0427(97)00178-7
[2] Anitescu, M.: Optimization-based simulation of nonsmooth rigid multibody dynamics. Math. Program. 105 (1(A)), 113-143 (2006) · Zbl 1085.70008
[3] Anitescu, M., Potra, F.A.: Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dyn. 14 (3), 231-247 (1997) · Zbl 0899.70005 · doi:10.1023/A:1008292328909
[4] Anitescu, M., Potra, F.A.: A time-stepping method for stiff multibody dynamics with contact and friction. Int. J. Numer. Methods Eng. 55 (7), 753-784 (2002) · Zbl 1027.70001 · doi:10.1002/nme.512
[5] Anitescu, M., Tasora, A.: An iterative approach for cone complementarity problems for nonsmooth dynamics. Comput. Optim. Appl. 47 (2), 207-235 (2010) · Zbl 1200.90160 · doi:10.1007/s10589-008-9223-4
[6] Anitescu, M., Potra, F.A., Stewart, D.E.: Time-stepping for three-dimensional rigid body dynamics. Comput. Methods Appl. Mech. Eng. 177 (3-4), 183-197 (1999) · Zbl 0967.70003 · doi:10.1016/S0045-7825(98)00380-6
[7] Arnold, M.: Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2. BIT 38 (3), 415-438 (1998) · Zbl 0924.65068 · doi:10.1007/BF02510252
[8] Arnold, M.: Multi-rate time integration for large scale multibody system models. In: IUTAM Symposium on Multiscale Problems in Multibody System Contacts: Proceedings of the IUTAM Symposium held in Stuttgart, Germany, February 20-23, 2006, pp. 1-10. Springer, Dordrecht (2007) · Zbl 1208.70002
[9] Arnold, M.: Stability of sequential modular time integration methods for coupled multibody system models. J. Comput. Nonlinear Dyn. 5, 031003 (2010) · doi:10.1115/1.4001389
[10] Arnold, M.: Modular time integration of block-structured coupled systems without algebraic loops. In: Schöps, S., Bartel, A., Günther, M., ter Maten, E.J.W., Müller, P.C. (eds.) Progress in Differential-Algebraic Equations. Differential-Algebraic Equations Forum, pp. 97-106. Springer, Berlin/Heidelberg (2014)
[11] Arnold, M., Günther, M.: Preconditioned dynamic iteration for coupled differential-algebraic systems. BIT Numer. Math. 41 (1), 001-025 (2001) · Zbl 0986.65076 · doi:10.1023/A:1021909032551
[12] Arnold, M., Murua, A.: Non-stiff integrators for differential-algebraic systems of index 2. Numer. Algorithm. 19 (1-4), 25-41 (1998) · Zbl 0917.65066 · doi:10.1023/A:1019123010801
[13] Arnold, M., Strehmel, K., Weiner, R.: Half-explicit Runge-Kutta methods for semi-explicit differential-algebraic equations of index 1. Numer. Math. 64 (1), 409-431 (1993) · Zbl 0791.65055 · doi:10.1007/BF01388697
[14] Arnold, M., Burgermeister, B., Eichberger, A.: Linearly implicit time integration methods in real-time applications: DAEs and stiff ODEs. Multibody Syst. Dyn. 17 (2-3), 99-117 (2007) · Zbl 1123.70002 · doi:10.1007/s11044-007-9036-8
[15] Arnold, M., Burgermeister, B., Führer, C., Hippmann, G., Rill, G.: Numerical methods in vehicle system dynamics: state of the art and current developments. Veh. Syst. Dyn. 49 (7), 1159-1207 (2011) · doi:10.1080/00423114.2011.582953
[16] Arnold, M., Clauß, C., Schierz, T.: Error analysis and error estimates for co-simulation in FMI for model exchange and co-simulation v2.0. Arch. Mech. Eng. LX, 75-94 (2013) · Zbl 1319.65074
[17] Arnold, M., Hante, S., Köbis, M.A.: Error analysis for co-simulation with force-displacement coupling. Proc. Appl. Math. Mech. 14 (1), 43-44 (2014) · doi:10.1002/pamm.201410014
[18] Ascher, U.M., Petzold, L.R.: Projected implicit Runge-Kutta methods for differential-algebraic equations. SIAM J. Numer. Anal. 28 (4), 1097-1120 (1991) · Zbl 0732.65067 · doi:10.1137/0728059
[19] Balzer, M., Burger, M., Däuwel, T., Ekevid, T., Steidel, S., Weber, D.: Coupling DEM particles to MBS wheel loader via co-simulation. In: Proceedings of the 4th Commercial Vehicle Technology Symposium (CVT 2016), pp. 479-488 (2016)
[20] Bartel, A., Brunk, M., Günther, M., Schöps, S.: Dynamic iteration for coupled problems of electric circuits and distributed devices. SIAM J. Sci. Comput. 35 (2), B315-B335 (2013) · Zbl 1266.65121 · doi:10.1137/120867111
[21] Bartel, A., Brunk, M., Schöps, S.: On the convergence rate of dynamic iteration for coupled problems with multiple subsystems. J. Comput. Appl. Math. 262, 14-24 (2014). Selected Papers from NUMDIFF-13 · Zbl 1301.65086
[22] Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1, 1-16 (1972) · Zbl 0262.70017 · doi:10.1016/0045-7825(72)90018-7
[23] Becker, U.: Efficient time integration and nonlinear model reduction for incompressible hyperelastic materials. Ph.D. thesis, TU Kaiserslautern (2012)
[24] Becker, U., Simeon, B., Burger, M.: On rosenbrock methods for the time integration of nearly incompressible materials and their usage for nonlinear model reduction. J. Comput. Appl. Math. 262, 333-345 (2014). Selected Papers from NUMDIFF-13 · Zbl 1302.74156
[25] Bock, H.G.: Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen, vol. 183. Bonner Mathematische Schriften, Bonn (1987) · Zbl 0622.65064
[26] Brasey, V., Hairer, E.: Half-explicit RungeKutta methods for differential-algebraic systems of index 2. SIAM J. Numer. Anal. 30 (2), 538-552 (1993) · Zbl 0785.65082 · doi:10.1137/0730025
[27] Brenan, K.E., Engquist, B.E.: Backward differentiation approximations of nonlinear differential/algebraic systems. Math. Comput. 51 (184), 659-676 (1988) · Zbl 0699.65059 · doi:10.1090/S0025-5718-1988-0930221-3
[28] Brenan, K.E., Petzold, L.R.: The numerical solution of higher index differential/algebraic equations by implicit methods. SIAM J. Numer. Anal. 26 (4), 976-996 (1989) · Zbl 0681.65050 · doi:10.1137/0726054
[29] Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Classics in Applied Mathematics, vol. 14. SIAM, Philadelphia (1996) · Zbl 0844.65058
[30] Brown, P.N., Hindmarsh, A.C., Petzold, L.R.: Consistent initial condition calculation for differential-algebraic systems. SIAM J. Sci. Comput. 19 (5), 1495-1512 (1998) · Zbl 0915.65079 · doi:10.1137/S1064827595289996
[31] Burgermeister, B., Arnold, M., Esterl, B.: DAE time integration for real-time applications in multi-body dynamics. Z. Angew. Math. Mech. 86 (10), 759-771 (2006) · Zbl 1105.70003 · doi:10.1002/zamm.200610284
[32] Burgermeister, B., Arnold, M., Eichberger, A.: Smooth velocity approximation for constrained systems in real-time simulation. Multibody Syst. Dyn. 26 (1), 1-14 (2011) · Zbl 1287.70002 · doi:10.1007/s11044-011-9243-1
[33] Büskens, C., Gerdts, M.: Differentiability of consistency functions for DAE systems. J. Optim. Theory Appl. 125 (1), 37-61 (2005) · Zbl 1114.34003 · doi:10.1007/s10957-004-1710-x
[34] Campbell, S.L., Gear, C.W.: The index of general nonlinear DAEs. Numer. Math. 72, 173-196 (1995) · Zbl 0844.34007 · doi:10.1007/s002110050165
[35] Campbell, S.L., Kelley, C.T., Yeomans, K.D.: Consistent initial conditions for unstructured higher index DAEs: a computational study. In: Computational Engineering in Systems Applications, France, pp. 416-421 (1996)
[36] Cao, Y., Li, S., Petzold, L.R., Serban, R.: Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution. SIAM J. Sci. Comput. 24 (3), 1076-1089 (2003) · Zbl 1034.65066 · doi:10.1137/S1064827501380630
[37] Caracotsios, M., Stewart, W.E.: Sensitivity analysis of initial-boundary-value problems with mixed PDEs and algebraic equations. Comput. Chem. Eng. 19 (9), 1019-1030 (1985) · doi:10.1016/0098-1354(94)00090-B
[38] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) · Zbl 0582.49001
[39] Cuadrado, J., Cardenal, J., Bayo, E.: Modeling and solution methods for efficient real-time simulation of multibody dynamics. Multibody Syst. Dyn. 1 (3), 259-280 (1997) · Zbl 0908.70003 · doi:10.1023/A:1009754006096
[40] Curtiss, C.F., Hirschfelder, J.O.: Integration of stiff equations. Proc. Nat. Acad. Sci. U.S.A. 38, 235-243 (1952) · Zbl 0046.13602 · doi:10.1073/pnas.38.3.235
[41] Deuflhard, P., Hairer, E., Zugck, J.: One-step and extrapolation methods for differential-algebraic systems. Numer. Math. 51 (5), 501-516 (1987) · Zbl 0635.65083 · doi:10.1007/BF01400352
[42] Diehl, M., Bock, H.G., Schlöder, J.P., Findeisen, R., Nagy, Z., Allgöwer, F.: Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations. J. Process Control 12 (4), 577-585 (2002) · doi:10.1016/S0959-1524(01)00023-3
[43] Diehl, M., Bock, H.G., Schlöder, J.P.: A real-time iteration scheme for nonlinear optimization in optimal feedback control. SIAM J. Control Optim. 43 (5), 1714-1736 (2005) · Zbl 1078.65060 · doi:10.1137/S0363012902400713
[44] Dopico, D., Lugris, U., Gonzalez, M., Cuadrado, J.: Two implementations of IRK integrators for real-time multibody dynamics. Int. J. Numer. Methods Eng. 65 (12), 2091-2111 (2006) · Zbl 1174.70301 · doi:10.1002/nme.1544
[45] Duff, I.S., Gear, C.W.: Computing the structural index. SIAM J. Algebr. Discrete Methods 7 (4), 594-603 (1986) · Zbl 0619.65064 · doi:10.1137/0607066
[46] Ebrahimi, S., Eberhard, P.: A linear complementarity formulation on position level for frictionless impact of planar deformable bodies. Z. Angew. Math. Mech. 86 (10), 807-817 (2006) · Zbl 1201.74233 · doi:10.1002/zamm.200510288
[47] Eich, E.: Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM J. Numer. Anal. 30 (5), 1467-1482 (1993) · Zbl 0785.65079 · doi:10.1137/0730076
[48] Eichberger, A., Rulka, W.: Process save reduction by macro joint approach: the key to real time and efficient vehicle simulation. Veh. Syst. Dyn. 41 (5), 401-413 (2004) · doi:10.1080/00423110412331300354
[49] Engelhardt, L., Burger, M., Bitsch, G.: Real-time simulation of multibody systems for on-board applications. In: Proceedings of the First Joint International Conference on Multibody System Dynamics (IMSD2010) (2010)
[50] Esterl, B., Butz, T., Simeon, B., Burgermeister, B.: Real-time capable vehicletrailer coupling by algorithms for differential-algebraic equations. Veh. Syst. Dyn. 45 (9), 819-834 (2007) · doi:10.1080/00423110601132588
[51] Estévez Schwarz, D.: Consistent initialization for index-2 differential-algebraic equations and its application to circuit simulation. Ph.D. thesis, Mathematisch-Naturwissenschaftlichen Fakultät II, Humboldt-Universität Berlin (2000)
[52] Feehery, W.F., Tolsma, J.E., Barton, P.I.: Efficient sensitivity analysis of large-scale differential-algebraic systems. Appl. Numer. Math. 25, 41-54 (1997) · Zbl 0884.65086 · doi:10.1016/S0168-9274(97)00050-0
[53] Feng, A., Holland, C.D., Gallun, S.E.: Development and comparison of a generalized semi-implicit Runge-Kutta method with Gear’s method for systems of coupled differential and algebraic equations. Comput. Chem. Eng. 8 (1), 51-59 (1984) · doi:10.1016/0098-1354(84)80015-0
[54] Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Mathematics in Science and Engineering, vol. 165. Academic Press, New York (1983) · Zbl 0543.90075
[55] Fischer, A.: A special Newton-type optimization method. Optimization 24, 269-284 (1992) · Zbl 0814.65063 · doi:10.1080/02331939208843795
[56] Führer, C.: Differential-algebraische Gleichungssysteme in mechanischen Mehrkörpersystemen: Theorie, numerische Ansätze und Anwendungen. Ph.D. thesis, Fakultät für Mathematik und Informatik, Technische Universität München (1988)
[57] Führer, C., Leimkuhler, B.J.: Numerical solution of differential-algebraic equations for constraint mechanical motion. Numer. Math. 59, 55-69 (1991) · Zbl 0701.70003 · doi:10.1007/BF01385770
[58] Gallrein, A., Baecker, M., Burger, M., Gizatullin, A.: An advanced flexible realtime tire model and its integration into Fraunhofer’s driving simulator. SAE Technical Paper 2014-01-0861 (2014)
[59] Garavello, M., Piccoli, B.: Hybrid necessary principle. SIAM J. Control Optim. 43 (5), 1867-1887 (2005) · Zbl 1084.49021 · doi:10.1137/S0363012903416219
[60] Gavrea, B.I., Anitescu, M., Potra, F.A.: Convergence of a class of semi-implicit time-stepping schemes for nonsmooth rigid multibody dynamics. SIAM J. Optim. 19 (2), 969-1001 (2008) · Zbl 1162.70003 · doi:10.1137/060675745
[61] Gear, C.W.: Simultaneous numerical solution of differential-algebraic equations. IEEE Trans. Circuit Theory 18 (1), 89-95 (1971) · doi:10.1109/TCT.1971.1083221
[62] Gear, C.W.: Differential-algebraic equation index transformations. SIAM J. Sci. Stat. Comput. 9, 39-47 (1988) · Zbl 0637.65072 · doi:10.1137/0909004
[63] Gear, C.W., Petzold, L.R.: ODE methods for the solution of differential/algebraic systems. SIAM J. Numer. Anal. 21 (4), 716-728 (1984) · Zbl 0557.65053 · doi:10.1137/0721048
[64] Gear, C.W., Leimkuhler, B., Gupta, G.K.: Automatic integration of Euler-Lagrange equations with constraints. J. Comput. Appl. Math. 12 (13), 77-90 (1985) · Zbl 0576.65072 · doi:10.1016/0377-0427(85)90008-1
[65] Geier, T., Foerg, M., Zander, R., Ulbrich, H., Pfeiffer, F., Brandsma, A., van der Velde, A.: Simulation of a push belt CVT considering uni- and bilateral constraints. Z. Angew. Math. Mech. 86 (10), 795-806 (2006) · Zbl 1105.74027 · doi:10.1002/zamm.200610287
[66] Gerdts, M.: Optimal control and real-time optimization of mechanical multi-body systems. Z. Angew. Math. Mech. 83 (10), 705-719 (2003) · Zbl 1052.70016 · doi:10.1002/zamm.200310067
[67] Gerdts, M.: Parameter optimization in mechanical multibody systems and linearized runge-kutta methods. In: Buikis, A., Ciegis, R., Flitt, A.D. (eds.) Progress in Industrial Mathematics at ECMI 2002. Mathematics in Industry, vol. 5, pp. 121-126. Springer, Heidelberg (2004) · Zbl 1125.70302 · doi:10.1007/978-3-662-09510-2_12
[68] Gerdts, M.: Optimal Control of ODEs and DAEs. Walter de Gruyter, Berlin/Boston (2012) · Zbl 1275.49001 · doi:10.1515/9783110249996
[69] Gerdts, M., Büskens, C.: Consistent initialization of sensitivity matrices for a class of parametric DAE systems. BIT Numer. Math. 42 (4), 796-813 (2002) · Zbl 1026.65064 · doi:10.1023/A:1021952420623
[70] Gerdts, M., Kunkel, M.: A nonsmooth Newton’s method for discretized optimal control problems with state and control constraints. J. Ind. Manag. Optim. 4 (2), 247-270 (2008) · Zbl 1157.49036 · doi:10.3934/jimo.2008.4.247
[71] Gopal, V., Biegler, L.T.: A successive linear programming approach for initialization and reinitialization after discontinuities of differential-algebraic equations. SIAM J. Sci. Comput. 20 (2), 447-467 (1998) · Zbl 0927.65092 · doi:10.1137/S1064827596307725
[72] Griewank, A., Walther, A.: Evaluating Derivatives. Principles and Techniques of Algorithmic Differentiation, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2008) · Zbl 1159.65026
[73] Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, vol. 14, 2nd edn. Springer, Berlin/Heidelberg/New York (1996) · Zbl 0859.65067
[74] Hairer, E., Lubich, C., Roche, M.: Error of Rosenbrock methods for stiff problems studied via differential algebraic equations. BIT 29 (1), 77-90 (1989) · Zbl 0674.65039 · doi:10.1007/BF01932707
[75] Hairer, E., Lubich, C., Roche, M.: The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Lecture Notes in Mathematics, vol. 1409. Springer, Berlin/Heidelberg/New York (1989) · Zbl 0683.65050
[76] Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics, vol. 8, 2nd edn. Springer, Berlin/Heidelberg/New York (1993) · Zbl 0789.65048
[77] Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Reprint of the Second 2006 edition. Springer, Berlin (2010) · Zbl 1094.65125
[78] Hansen, B.: Computing consistent initial values for nonlinear index-2 differential-algebraic equations. Seminarberichte Humboldt-Universität Berlin, 92-1, 142-157 (1992)
[79] Heim, A.: Parameteridentifizierung in differential-algebraischen Gleichungssystemen. Master’s thesis, Mathematisches Institut, Technische Universität München (1992)
[80] Hindmarsh, A.C., Brown, P.N., Grant, K.E., Lee, S.L., Serban, R., Shumaker, D.E., Woodward, C.S.: Sundials: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. 31 (3), 363-396 (2005) · Zbl 1136.65329 · doi:10.1145/1089014.1089020
[81] INTEC GmbH. SIMPACK - Analysis and Design of General Mechanical Systems. Weßling
[82] Jackiewicz, Z., Kwapisz, M.L Convergence of waveform relaxation methods for differential-algebraic systems. SIAM J. Numer. Anal. 33 (6), 2303-2317 (1996) · Zbl 0889.34064
[83] Jackson, K.R.: A survey of parallel numerical methods for initial value problems for ordinary differential equations. IEEE Trans. Magn. 27 (5), 3792-3797 (1991) · doi:10.1109/20.104928
[84] Jay, L.: Collocation methods for differential-algebraic equations of index 3. Numer. Math. 65, 407-421 (1993) · Zbl 0791.65056 · doi:10.1007/BF01385759
[85] Jay, L.: Convergence of Runge-Kutta methods for differential-algebraic systems of index 3. Appl. Numer. Math. 17, 97-118 (1995) · Zbl 0832.65078 · doi:10.1016/0168-9274(95)00013-K
[86] Jiang, H.: Global convergence analysis of the generalized Newton and Gauss-Newton methods of the Fischer-Burmeister equation for the complementarity problem. Math. Oper. Res. 24 (3), 529-543 (1999) · Zbl 0944.90094 · doi:10.1287/moor.24.3.529
[87] Kiehl, M.: Sensitivity analysis of ODEs and DAEs - theory and implementation guide. Optim. Methods Softw. 10 (6), 803-821 (1999) · Zbl 0939.34003 · doi:10.1080/10556789908805742
[88] Kleinert, J., Simeon, B., Dreßler, K.: Nonsmooth contact dynamics for the large-scale simulation of granular material. Technical report, Fraunhofer ITWM, Kaiserslautern, Germany. J. Comput. Appl. Math. (2015, in press). http://dx.doi.org/10.1016/j.cam.2016.09.037 · Zbl 1376.49021 · doi:10.1016/j.cam.2016.09.037
[89] Kübler, R., Schiehlen, W.: Two methods of simulator coupling. Math. Comput. Model. Dyn. Syst. 6 (2), 93-113 (2000) · Zbl 0962.65107 · doi:10.1076/1387-3954(200006)6:2;1-M;FT093
[90] Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution, vol. viii, 377 p. European Mathematical Society Publishing House, Zürich (2006) · Zbl 1095.34004
[91] Küsters, F., Ruppert, M.G.-M., Trenn, S.: Controllability of switched differential-algebraic equations. Syst. Control Lett. 78, 32-39 (2015) · Zbl 1320.93020 · doi:10.1016/j.sysconle.2015.01.011
[92] Lamour, R., März, R., Tischendorf, C.: Differential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum. Springer, Berlin (2013) · Zbl 1276.65045 · doi:10.1007/978-3-642-27555-5
[93] Leimkuhler, B., Petzold, L.R., Gear, C.W.: Approximation methods for the consistent initialization of differential-algebraic equations. SIAM J. Numer. Anal. 28 (1), 205-226 (1991) · Zbl 0725.65076 · doi:10.1137/0728011
[94] Lelarasmee, E., Ruehli, A.E., Sangiovanni-Vincentelli, A.L.: The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 1 (3), 131-145 (1982) · doi:10.1109/TCAD.1982.1270004
[95] Lemke, C.E.: The dual method of solving the linear programming problem. Naval Res. Log. Q. 1, 36-47 (1954) · doi:10.1002/nav.3800010107
[96] Leyendecker, S., Ober-Blöbaum, S.: A variational approach to multirate integration for constrained systems. In: Multibody Dynamics. Computational Methods and Applications. Selected Papers Based on the Presentations at the ECCOMAS Thematic Conference, Brussels, Belgium, 4-7 July 2011, pp. 97-121. Springer, Dordrecht (2013) · Zbl 1311.70026
[97] Liberzon, D., Trenn, S.: Switched nonlinear differential algebraic equations: solution theory, Lyapunov functions, and stability. Automatica 48 (5), 954-963 (2012) · Zbl 1246.93064 · doi:10.1016/j.automatica.2012.02.041
[98] Linn, J., Stephan, T., Carlson, J.S., Bohlin, R.: Fast simulation of quasistatic rod deformations for VR applications. In: Bonilla, L.L., Moscoso, M., Platero, G., Vega, J.M. (eds.) Progress in Industrial Mathematics at ECMI 2006. Springer, New York (2007) · Zbl 1308.74084
[99] Lötstedt, P., Petzold, L.R.: Numerical solution of nonlinear differential equations with algebraic constraints I: convergence results for backward differentiation formulas. Math. Comput. 46, 491-516 (1986) · Zbl 0601.65060 · doi:10.2307/2007989
[100] Lubich, C., Engstler, C., Nowak, U., Pöhle, U.: Numerical integration of constrained mechanical systems using MEXX*. Mech. Struct. Mach. 23 (4), 473-495 (1995) · doi:10.1080/08905459508905248
[101] Maly, T., Petzold, L.R.: Numerical methods and software for sensitivity analysis of differential-algebraic systems. Appl. Numer. Math. 20 (1), 57-79 (1996) · Zbl 0854.65056 · doi:10.1016/0168-9274(95)00117-4
[102] Michael, J., Gerdts, M.: A method to model impulsive multi-body-dynamics using Riemann-Stieltjes- Integrals. In: 8th Vienna International Conference on Mathematical Modelling, International Federation of Automatic Control, pp. 629-634 (2015)
[103] Michael, J., Chudej, K., Gerdts, M., Pannek, J.: Optimal rendezvous path planning to an uncontrolled tumbling target. In: IFAC Proceedings Volumes (IFAC-PapersOnline), 19th IFAC Symposium on Automatic Control in Aerospace, ACA 2013, Wurzburg, Germany, 2-6 September 2013, vol. 19, pp. 347-352 (2013)
[104] Miekkala, U., Nevanlinna, O.: Convergence of dynamic iteration methods for initial value problems. SIAM J. Sci. Stat. Comput. 8 (4), 459-482 (1987) · Zbl 0625.65063 · doi:10.1137/0908046
[105] Murua, A.: Partitioned half-explicit Runge-Kutta methods for differential-algebraic systems of index 2. Computing 59 (1), 43-61 (1997) · Zbl 0881.65071 · doi:10.1007/BF02684403
[106] Negrut, D., Sandu, A., Haug, E.J., Potra, F.A., Sandu, C.: A Rosenbrock-Nystrom state space implicit approach for the dynamic analysis of mechanical systems: II -the method and numerical examples. J. Multi-body Dyn. 217 (4), 273-281 (2003)
[107] Ostermann, A.: A class of half-explicit Runge-Kutta methods for differential-algebraic systems of index 3. Appl. Numer. Math. 13 (1), 165-179 (1993) · Zbl 0788.65084 · doi:10.1016/0168-9274(93)90140-M
[108] Pantelides, C.C.: The consistent initialization of differential-algebraic systems. SIAM J. Sci. Stat. Comput. 9 (2), 213-231 (1988) · Zbl 0643.65039 · doi:10.1137/0909014
[109] Petzold, L.R.: A description of DASSL: a differential/algebraic system solver. Rep. Sand 82-8637, Sandia National Laboratory, Livermore (1982)
[110] Petzold, L.R.: Differential/algebraic equations are not ODE’s. SIAM J. Sci. Stat. Comput. 3 (3), 367-384 (1982) · Zbl 0482.65041 · doi:10.1137/0903023
[111] Petzold, L.R.: Recent developments in the numerical solution of differential/algebraic systems. Comput. Methods Appl. Mech. Eng. 75, 77-89 (1989) · Zbl 0695.65048 · doi:10.1016/0045-7825(89)90016-9
[112] Pfeiffer, A.: Numerische Sensitivitätsanalyse unstetiger multidisziplinärer Modelle mit Anwendungen in der gradientenbasierten Optimierung. Fortschritt-Berichte VDI Reihe 20, Nr. 417. VDI-Verlag, Düsseldorf (2008)
[113] Potra, F.A., Anitescu, M., Gavrea, B., Trinkle, J.: A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact, joints, and friction. Int. J. Numer. Methods Eng. 66 (7), 1079-1124 (2006) · Zbl 1110.70303 · doi:10.1002/nme.1582
[114] Pytlak, R., Suski, D.: On solving hybrid optimal control problems with higher index DAEs. Institute of Automatic Control and Robotics, Warsaw University of Technology, Warsaw, Poland (2015, Preprint) · Zbl 1379.49026
[115] Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18 (1), 227-244 (1993) · Zbl 0776.65037 · doi:10.1287/moor.18.1.227
[116] Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58 (3), 353-367 (1993) · Zbl 0780.90090 · doi:10.1007/BF01581275
[117] Rentrop, P., Roche, M., Steinebach, G.: The application of Rosenbrock-Wanner type methods with stepsize control in differential-algebraic equations. Numer. Math. 55 (5), 545-563 (1989) · Zbl 0672.65046 · doi:10.1007/BF01398915
[118] Rill, G.: A modified implicit Euler algorithm for solving vehicle dynamic equations. Multibody Syst. Dyn. 15 (1), 1-24 (2006) · Zbl 1146.70004 · doi:10.1007/s11044-006-2359-z
[119] Rill, G., Chucholowski, C.: Real time simulation of large vehicle systems. In: Proceedings of Multibody Dynamics 2007 (ECCOMAS Thematic Conference) (2007)
[120] Roche, M.: Rosenbrock methods for differential algebraic equations. Numer. Math. 52 (1), 45-63 (1988) · Zbl 0613.65076 · doi:10.1007/BF01401021
[121] Rosenbrock, H.H.: Some general implicit processes for the numerical solution of differential equations. Comput. J. 5 (4), 329-330 (1963) · Zbl 0112.07805 · doi:10.1093/comjnl/5.4.329
[122] Rulka, W., Pankiewicz, E.: MBS approach to generate equations of motions for hil-simulations in vehicle dynamics. Multibody Syst. Dyn. 14 (3), 367-386 (2005) · Zbl 1146.70324 · doi:10.1007/s11044-005-1144-8
[123] Sandu, A., Negrut, D., Haug, E.J., Potra, F.A., Sandu, C.: A Rosenbrock-Nystrom state space implicit approach for the dynamic analysis of mechanical systems: I—theoretical formulation. J. Multi-body Dyn. 217 (4), 263-271 (2003)
[124] Schaub, M., Simeon, B.: Blended Lobatto methods in multibody dynamics. Z. Angew. Math. Mech. 83 (10), 720-728 (2003) · Zbl 1037.70002 · doi:10.1002/zamm.200310069
[125] Schierz, T., Arnold, M.: Stabilized overlapping modular time integration of coupled differential-algebraic equations. Appl. Numer. Math. 62 (10), 1491-1502 (2012). Selected Papers from NUMDIFF-12 · Zbl 1251.65116
[126] Schneider, F., Burger, M., Arnold, M., Simeon, B.: A new approach for force-displacement co-simulation using kinematic coupling constraints. Submitted to Z. Angew. Math. Mech. (2016)
[127] Schulz, V.H., Bock, H.G., Steinbach, M.C.: Exploiting invariants in the numerical solution of multipoint boundary value problems for DAE. SIAM J. Sci. Comput. 19 (2), 440-467 (1998) · Zbl 0947.65096 · doi:10.1137/S1064827594261917
[128] Schwartz, W., Frik, S., Leister, G.: Simulation of the IAVSD Road Vehicle Benchmark Bombardier Iltis with FASIM, MEDYNA, NEWEUL and SIMPACK. Technical Report IB 515/92-20, Robotik und Systemdynamik, Deutsche Forschungsanstalt für Luft- und Raumfahrt (1992)
[129] Schweizer, B., Lu, D.: Stabilized index-2 co-simulation approach for solver coupling with algebraic constraints. Multibody Syst. Dyn. 34 (2), 129-161 (2014) · Zbl 1368.70037 · doi:10.1007/s11044-014-9422-y
[130] Schweizer, B., Li, P., Lu, D.: Implicit co-simulation methods: stability and convergence analysis for solver coupling approaches with algebraic constraints. Z. Angew. Math. Mech. 96 (8), 986-1012 (2016) · doi:10.1002/zamm.201400087
[131] Stetter, H.J.: Analysis of Discretization Methods for Ordinary Differential Equations. Springer Tracts in Natural Philosophy, vol. 23. Springer, Berlin/Heidelberg/New York (1973) · Zbl 0276.65001
[132] Stewart, D.E.: Rigid-body dynamics with friction and impact. SIAM Rev. 42 (1), 3-39 (2000) · Zbl 0962.70010 · doi:10.1137/S0036144599360110
[133] Stewart, D.E., Anitescu, M.: Optimal control of systems with discontinuous differential equations. Numer. Math. 114 (4), 653-695 (2010) · Zbl 1183.49032 · doi:10.1007/s00211-009-0262-2
[134] Strehmel, K., Weiner, R.: Numerik gewöhnlicher Differentialgleichungen. Teubner, Stuttgart (1995) · Zbl 0841.65055
[135] Strehmel, K., Weiner, R., Dannehl, I.: On error behaviour of partitioned linearly implicit Runge-Kutta methods for stiff and differential algebraic systems. BIT 30 (2), 358-375 (1990) · Zbl 0702.65073 · doi:10.1007/BF02017354
[136] Sussmann, H.J.: A nonsmooth hybrid maximum principle. In: Stability and Stabilization of Nonlinear Systems. Proceedings of the 1st Workshop on Nonlinear Control Network, Held in Gent, Belgium, 15-16 March 1999, pp. 325-354. Springer, London (1999)
[137] Tasora, A., Anitescu, M.: A fast NCP solver for large rigid-body problems with contacts, friction, and joints. In: Multibody Dynamics. Computational Methods and Applications. Revised, extended and selected papers of the ECCOMAS Thematic Conference on Multibody Dynamics 2007, Milano, Italy, 25-28 June 2007, pp. 45-55. Springer, Dordrecht (2009)
[138] Tasora, A., Anitescu, M.: A matrix-free cone complementarity approach for solving large-scale, nonsmooth, rigid body dynamics. Comput. Methods Appl. Mech. Eng. 200 (5-8), 439-453 (2011) · Zbl 1225.70004 · doi:10.1016/j.cma.2010.06.030
[139] Tasora, A., Anitescu, M.: A complementarity-based rolling friction model for rigid contacts. Meccanica 48 (7), 1643-1659 (2013) · Zbl 1293.70052 · doi:10.1007/s11012-013-9694-y
[140] Tasora, A., Negrut, D., Anitescu, M.: GPU-based parallel computing for the simulation of complex multibody systems with unilateral and bilateral constraints: an overview. In: Multibody Dynamics. Computational Methods and Applications. Selected papers based on the presentations at the ECCOMAS Conference on Multibody Dynamics, Warsaw, Poland, June 29-July 2, 2009, pp. 283-307. Springer, New York, NY (2011) · Zbl 1385.70001
[141] Trenn, S.: Solution concepts for linear DAEs: a survey. In: Ilchmann, A., Reis, T. (eds.) Surveys in Differential-Algebraic Equations I. Differential-Algebraic Equations Forum, pp. 137-172. Springer, Berlin (2013) · Zbl 1277.34010 · doi:10.1007/978-3-642-34928-7_4
[142] van der Schaft, A., Schumacher, H.: An Introduction to Hybrid Dynamical Systems. Springer, London (1989) · Zbl 0940.93004
[143] Veitl, A., Gordon, T., van de Sand, A., Howell, M., Valasek, M., Vaculin, O., Steinbauer, P.: Methodologies for coupling simulation models and codes in mechatronic system analysis and design. In: Proceedings of the 16th IAVSD Symposium on Dynamics of Vehicles on Roads and Tracks. Pretoria. Supplement to Vehicle System Dynamics, vol. 33, pp. 231-243. Swets & Zeitlinger (1999)
[144] von Schwerin, R.: Multibody System Simulation: Numerical Methods, Algorithms, and Software. Lecture Notes in Computational Science and Engineering, vol. 7. Springer, Berlin/Heidelberg/New York (1999) · Zbl 0945.65092
[145] Wensch, J.: An eight stage fourth order partitioned Rosenbrock method for multibody systems in index-3 formulation. Appl. Numer. Math. 27 (2), 171-183 (1998) · Zbl 0932.65083 · doi:10.1016/S0168-9274(98)00007-5
[146] Wensch, J., Strehmel, K., Weiner, R.: A class of linearly-implicit Runge-Kutta methods for multibody systems. Appl. Numer. Math. 22 (13), 381-398 (1996). Special Issue Celebrating the Centenary of Runge-Kutta Methods · Zbl 0868.65046
[147] Wolfbrandt, A., Steihaug, T.: An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations. Math. Comput. 33 (146), 521-534 (1979) · Zbl 0451.65055 · doi:10.1090/S0025-5718-1979-0521273-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.