Figure 1.
A rectangular mesh with Gauss integration points: midpoints are used for evaluation of Q1 elements (left) and four points for evaluation of Bogner-Fox-Schmit elements (right)
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January
2021, 14(1): 299-319.
doi: 10.3934/dcdss.2020322
Numerical approximation of von Kármán viscoelastic plates
| 1. | Institute for Computational and Applied Mathematics, University of Münster, Einsteinstr. 62, D-48149 Münster, Germany |
| 2. | Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodárenskou vĕží 4, CZ-182 00 Praha 8, Czechia, Faculty of Civil Engineering, Czech Technical University, Thákurova 7, CZ-166 29 Praha 6, Czechia |
| 3. | Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodárenskou vĕží 4, CZ-182 00 Praha 8, Czechia, Institute of Mathematics, University of South Bohemia, Branišovská 1760, CZ-370 05 České Budĕjovice, Czechia |
We consider metric gradient flows and their discretizations in time and space. We prove an abstract convergence result for time-space discretizations and identify their limits as curves of maximal slope. As an application, we consider a finite element approximation of a quasistatic evolution for viscoelastic von Kármán plates [
Keywords:
Viscoelasticity,
metric gradient flows,
$ \Gamma $-convergence,
dissipative distance,
minimizing movements,
numerical approximation.
Mathematics Subject Classification:
Primary: 74D05, 74D10, 35A15, 35Q74, 49J45; Secondary: 49S05.
Citation:
Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete and Continuous Dynamical Systems - S,
2021, 14
(1)
: 299-319.
doi: 10.3934/dcdss.2020322
![]()
References:
| [1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces (2nd ed), Elsevier, Amsterdam, 2003. |
| [2] |
L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 19 (1995), 191–246. |
| [3] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2005. |
| [4] |
I. Anjam and J. Valdman,
Fast MATLAB assembly of FEM matrices in 2D and 3D: edge elements, Appl. Math. Comput., 267 (2015), 252-263.
doi: 10.1016/j.amc.2015.03.105. |
| [5] |
S. S. Antman,
Physically unacceptable viscous stresses, Z. Angew. Math. Phys., 49 (1998), 980-988.
doi: 10.1007/s000330050134. |
| [6] |
S. S. Antman, Nonlinear Problems of Elasticity, Springer, New York, 2005. |
| [7] |
J. M. Ball,
Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1977), 337-403.
doi: 10.1007/BF00279992. |
| [8] |
J. M. Ball, J. C. Currie and P. L. Olver,
Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal., 41 (1981), 135-174.
doi: 10.1016/0022-1236(81)90085-9. |
| [9] |
R. C. Batra,
Thermodynamics of non-simple elastic materials, J. Elasticity, 6 (1976), 451-456.
doi: 10.1007/BF00040904. |
| [10] |
I. Bock,
On Von Kármán equations for viscoelastic plates, J. Comp. Appl. Math., 63 (1995), 277-282.
doi: 10.1016/0377-0427(95)00082-8. |
| [11] |
I. Bock and J. Jarušek,
Solvability of dynamic contact problems for elastic von Kármán plates, SIAM J. Math. Anal., 41 (2009), 37-45.
doi: 10.1137/080712179. |
| [12] |
I. Bock, J. Jarušek and M. Šilhavý,
On the solutions of a dynamic contact problem for a thermoelastic von Kármán plate, Nonlin. Anal.: Real World Appl., 32 (2016), 111-135.
doi: 10.1016/j.nonrwa.2016.04.004. |
| [13] |
F. K. Bogner, R. L. Fox and L. A. Schmit, The generation of inter-element compatible stiffness and mass matrices by the use of interpolation formulas, Proceedings of the Conference on Matrix Methods in Structural Mechanics, (1965), 397–444. |
| [14] |
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Springer, New York, 2008.
doi: 10.1007/978-0-387-75934-0. |
| [15] |
G. Capriz,
Continua with latent microstructure, Arch. Ration. Mech. Anal., 90 (1985), 43-56.
doi: 10.1007/BF00281586. |
| [16] |
V. Casarino and D. Percivale,
A variational model for nonlinear elastic plates, J. Convex Anal., 3 (1996), 221-243.
|
| [17] |
P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-dimensional Elasticity, North-Holland, Amsterdam, 1988. |
| [18] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, 2002.
doi: 10.1137/1.9780898719208. |
| [19] |
G. Dal Maso, An Introduction to $\Gamma$-convergence, Birkhäuser, Boston $\cdot$ Basel $\cdot$ Berlin, 1993.
doi: 10.1007/978-1-4612-0327-8. |
| [20] |
E. De Giorgi, A. Marino and M. Tosques,
Problems of evolution in metric spaces and maximal decreasing curve, Att. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 68 (1980), 180-187.
|
| [21] |
J. E. Dunn and J. Serrin,
On the thermomechanics of interstitial working, Arch. Ration. Mech. Anal., 88 (1985), 95-133.
doi: 10.1007/BF00250907. |
| [22] |
M. Friedrich and M. Kružík,
On the passage from nonlinear to linearized viscoelasticity, SIAM J. Math. Anal., 50 (2018), 4426-4456.
doi: 10.1137/17M1131428. |
| [23] |
M. Friedrich and M. Kružík, Derivation of von Kármán plate theory in the framework of three-dimensional viscoelasticity, preprint, arXiv: 1902.10037. |
| [24] |
G. Friesecke, R. D. James and S. Müller,
A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002), 1461-1506.
doi: 10.1002/cpa.10048. |
| [25] |
G. Friesecke, R. D. James and S. Müller,
A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence, Arch. Ration. Mech. Anal., 180 (2006), 183-236.
doi: 10.1007/s00205-005-0400-7. |
| [26] |
P. Harasim and J. Valdman,
Verification of functional a posteriori error estimates for an obstacle problem in 2D, Kybernetika, 50 (2014), 978-1002.
doi: 10.14736/kyb-2014-6-0978. |
| [27] |
S. Krömer and J. Valdman,
Global injectivity in second-gradient nonlinear elasticity and its approximation with penalty terms, Mathematics and Mechanics of Solids, 24 (2019), 3644-3673.
doi: 10.1177/1081286519851554. |
| [28] |
M. Lecumberry and S. Müller,
Stability of slender bodies under compression and validity of the von Kármán theory, Arch. Ration. Mech. Anal., 193 (2009), 255-310.
doi: 10.1007/s00205-009-0232-y. |
| [29] |
H. Le Dret and A. Raoult,
The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., 74 (1995), 549-578.
|
| [30] |
H. Le Dret and A. Raoult, The membrane shell model in nonlinear elasticity: A variational asymptotic derivation,, J. Nonl. Sci., 6 (1996), 59–84.
doi: 10.1007/BF02433810. |
| [31] |
A. Mielke, C. Ortner and Y. Şengül,
An approach to nonlinear viscoelasticity via metric gradient flows, SIAM J. Math. Anal., 46 (2014), 1317-1347.
doi: 10.1137/130927632. |
| [32] |
A. Mielke and T. Roubíček, Rate-Independent Systems - Theory and Application, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2706-7. |
| [33] |
A. Mielke and T. Roubíček,
Rate-independent elastoplasticity at finite strains and its numerical approximation, Math. Models & Methods in Appl. Sci., 26 (2016), 2203-2236.
doi: 10.1142/S0218202516500512. |
| [34] |
C. Ortner, Two Variational Techniques for the Approximation of Curves of Maximal Slope, , Technical report NA05/10, Oxford University Computing Laboratory, Oxford, UK, 2005. |
| [35] |
O. Pantz,
On the justification of the nonlinear inextensional plate model, Arch. Ration. Mech. Anal., 167 (2003), 179-209.
doi: 10.1007/s00205-002-0238-1. |
| [36] |
J. Y. Park and J. R. Kang,
Uniform decay of solutions for von Karman equations of dynamic viscoelasticity with memory, Acta Appl. Math., 110 (2010), 1461-1474.
doi: 10.1007/s10440-009-9520-7. |
| [37] |
P. Podio-Guidugli,
Contact interactions, stress, and material symmetry for nonsimple elastic materials, Theor. Appl. Mech., 28/29 (2002), 261-276.
|
| [38] |
T. Rahman and J. Valdman,
Fast MATLAB assembly of FEM matrices in 2D and 3D: Nodal elements, Appl. Math. Comput., 219 (2013), 7151-7158.
doi: 10.1016/j.amc.2011.08.043. |
| [39] |
E. Sandier and S. Serfaty,
Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.
doi: 10.1002/cpa.20046. |
| [40] |
S. Serfaty,
Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. Ser. A, 31 (2011), 1427-1451.
doi: 10.3934/dcds.2011.31.1427. |
| [41] |
R. A. Toupin,
Elastic materials with couple stresses, Arch. Ration. Mech. Anal., 11 (1962), 385-414.
doi: 10.1007/BF00253945. |
| [42] |
R. A. Toupin,
Theory of elasticity with couple stress, Arch. Ration. Mech. Anal., 17 (1964), 85-112.
doi: 10.1007/BF00253050. |
| [43] |
J. Valdman, MATLAB Implementation of C1 finite elements: Bogner-Fox-Schmit rectangle, In: Wyrzykowski R., Dongarra J., Deelman E., Karczewski K. (eds) Parallel Processing and Applied Mathematics. PPAM 2019, LNCS, Springer, 12044 (2020), 256–266.
doi: 10.1007/978-3-030-43222-5_22. |
| [44] |
T. von Kármán,
Festigkeitsprobleme im maschinenbau in encyclopädie der mathematischen wissenschaften, Leipzig, 4 (1910), 311-385.
|
show all references
References:
| [1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces (2nd ed), Elsevier, Amsterdam, 2003. |
| [2] |
L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 19 (1995), 191–246. |
| [3] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2005. |
| [4] |
I. Anjam and J. Valdman,
Fast MATLAB assembly of FEM matrices in 2D and 3D: edge elements, Appl. Math. Comput., 267 (2015), 252-263.
doi: 10.1016/j.amc.2015.03.105. |
| [5] |
S. S. Antman,
Physically unacceptable viscous stresses, Z. Angew. Math. Phys., 49 (1998), 980-988.
doi: 10.1007/s000330050134. |
| [6] |
S. S. Antman, Nonlinear Problems of Elasticity, Springer, New York, 2005. |
| [7] |
J. M. Ball,
Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1977), 337-403.
doi: 10.1007/BF00279992. |
| [8] |
J. M. Ball, J. C. Currie and P. L. Olver,
Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal., 41 (1981), 135-174.
doi: 10.1016/0022-1236(81)90085-9. |
| [9] |
R. C. Batra,
Thermodynamics of non-simple elastic materials, J. Elasticity, 6 (1976), 451-456.
doi: 10.1007/BF00040904. |
| [10] |
I. Bock,
On Von Kármán equations for viscoelastic plates, J. Comp. Appl. Math., 63 (1995), 277-282.
doi: 10.1016/0377-0427(95)00082-8. |
| [11] |
I. Bock and J. Jarušek,
Solvability of dynamic contact problems for elastic von Kármán plates, SIAM J. Math. Anal., 41 (2009), 37-45.
doi: 10.1137/080712179. |
| [12] |
I. Bock, J. Jarušek and M. Šilhavý,
On the solutions of a dynamic contact problem for a thermoelastic von Kármán plate, Nonlin. Anal.: Real World Appl., 32 (2016), 111-135.
doi: 10.1016/j.nonrwa.2016.04.004. |
| [13] |
F. K. Bogner, R. L. Fox and L. A. Schmit, The generation of inter-element compatible stiffness and mass matrices by the use of interpolation formulas, Proceedings of the Conference on Matrix Methods in Structural Mechanics, (1965), 397–444. |
| [14] |
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Springer, New York, 2008.
doi: 10.1007/978-0-387-75934-0. |
| [15] |
G. Capriz,
Continua with latent microstructure, Arch. Ration. Mech. Anal., 90 (1985), 43-56.
doi: 10.1007/BF00281586. |
| [16] |
V. Casarino and D. Percivale,
A variational model for nonlinear elastic plates, J. Convex Anal., 3 (1996), 221-243.
|
| [17] |
P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-dimensional Elasticity, North-Holland, Amsterdam, 1988. |
| [18] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, 2002.
doi: 10.1137/1.9780898719208. |
| [19] |
G. Dal Maso, An Introduction to $\Gamma$-convergence, Birkhäuser, Boston $\cdot$ Basel $\cdot$ Berlin, 1993.
doi: 10.1007/978-1-4612-0327-8. |
| [20] |
E. De Giorgi, A. Marino and M. Tosques,
Problems of evolution in metric spaces and maximal decreasing curve, Att. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 68 (1980), 180-187.
|
| [21] |
J. E. Dunn and J. Serrin,
On the thermomechanics of interstitial working, Arch. Ration. Mech. Anal., 88 (1985), 95-133.
doi: 10.1007/BF00250907. |
| [22] |
M. Friedrich and M. Kružík,
On the passage from nonlinear to linearized viscoelasticity, SIAM J. Math. Anal., 50 (2018), 4426-4456.
doi: 10.1137/17M1131428. |
| [23] |
M. Friedrich and M. Kružík, Derivation of von Kármán plate theory in the framework of three-dimensional viscoelasticity, preprint, arXiv: 1902.10037. |
| [24] |
G. Friesecke, R. D. James and S. Müller,
A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002), 1461-1506.
doi: 10.1002/cpa.10048. |
| [25] |
G. Friesecke, R. D. James and S. Müller,
A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence, Arch. Ration. Mech. Anal., 180 (2006), 183-236.
doi: 10.1007/s00205-005-0400-7. |
| [26] |
P. Harasim and J. Valdman,
Verification of functional a posteriori error estimates for an obstacle problem in 2D, Kybernetika, 50 (2014), 978-1002.
doi: 10.14736/kyb-2014-6-0978. |
| [27] |
S. Krömer and J. Valdman,
Global injectivity in second-gradient nonlinear elasticity and its approximation with penalty terms, Mathematics and Mechanics of Solids, 24 (2019), 3644-3673.
doi: 10.1177/1081286519851554. |
| [28] |
M. Lecumberry and S. Müller,
Stability of slender bodies under compression and validity of the von Kármán theory, Arch. Ration. Mech. Anal., 193 (2009), 255-310.
doi: 10.1007/s00205-009-0232-y. |
| [29] |
H. Le Dret and A. Raoult,
The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., 74 (1995), 549-578.
|
| [30] |
H. Le Dret and A. Raoult, The membrane shell model in nonlinear elasticity: A variational asymptotic derivation,, J. Nonl. Sci., 6 (1996), 59–84.
doi: 10.1007/BF02433810. |
| [31] |
A. Mielke, C. Ortner and Y. Şengül,
An approach to nonlinear viscoelasticity via metric gradient flows, SIAM J. Math. Anal., 46 (2014), 1317-1347.
doi: 10.1137/130927632. |
| [32] |
A. Mielke and T. Roubíček, Rate-Independent Systems - Theory and Application, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2706-7. |
| [33] |
A. Mielke and T. Roubíček,
Rate-independent elastoplasticity at finite strains and its numerical approximation, Math. Models & Methods in Appl. Sci., 26 (2016), 2203-2236.
doi: 10.1142/S0218202516500512. |
| [34] |
C. Ortner, Two Variational Techniques for the Approximation of Curves of Maximal Slope, , Technical report NA05/10, Oxford University Computing Laboratory, Oxford, UK, 2005. |
| [35] |
O. Pantz,
On the justification of the nonlinear inextensional plate model, Arch. Ration. Mech. Anal., 167 (2003), 179-209.
doi: 10.1007/s00205-002-0238-1. |
| [36] |
J. Y. Park and J. R. Kang,
Uniform decay of solutions for von Karman equations of dynamic viscoelasticity with memory, Acta Appl. Math., 110 (2010), 1461-1474.
doi: 10.1007/s10440-009-9520-7. |
| [37] |
P. Podio-Guidugli,
Contact interactions, stress, and material symmetry for nonsimple elastic materials, Theor. Appl. Mech., 28/29 (2002), 261-276.
|
| [38] |
T. Rahman and J. Valdman,
Fast MATLAB assembly of FEM matrices in 2D and 3D: Nodal elements, Appl. Math. Comput., 219 (2013), 7151-7158.
doi: 10.1016/j.amc.2011.08.043. |
| [39] |
E. Sandier and S. Serfaty,
Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.
doi: 10.1002/cpa.20046. |
| [40] |
S. Serfaty,
Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. Ser. A, 31 (2011), 1427-1451.
doi: 10.3934/dcds.2011.31.1427. |
| [41] |
R. A. Toupin,
Elastic materials with couple stresses, Arch. Ration. Mech. Anal., 11 (1962), 385-414.
doi: 10.1007/BF00253945. |
| [42] |
R. A. Toupin,
Theory of elasticity with couple stress, Arch. Ration. Mech. Anal., 17 (1964), 85-112.
doi: 10.1007/BF00253050. |
| [43] |
J. Valdman, MATLAB Implementation of C1 finite elements: Bogner-Fox-Schmit rectangle, In: Wyrzykowski R., Dongarra J., Deelman E., Karczewski K. (eds) Parallel Processing and Applied Mathematics. PPAM 2019, LNCS, Springer, 12044 (2020), 256–266.
doi: 10.1007/978-3-030-43222-5_22. |
| [44] |
T. von Kármán,
Festigkeitsprobleme im maschinenbau in encyclopädie der mathematischen wissenschaften, Leipzig, 4 (1910), 311-385.
|
Figure 2.
Time sequence of energy minimizers $ (u, v) $ in Benchmark Ⅰ. To emphasize the mesh deformation, the nodes displacement is magnified by a factor of 4
Figure 3.
Time sequence of energy minimizers $ (u, v) $ in Benchmark Ⅱ. To emphasize the mesh deformation, the nodes displacement is magnified by a factor of 7
Figure 4.
Example of $ C^1 $ approximation by the Bogner-Fox-Schmit rectangular elements: the function $ v_0 = (1-x_1^2)^2(1-x_2^2)^2 $ from Benchmark 1 is represented by its value (A), gradient components (C), (D) and the second mixed derivative (B) in all mesh nodes.
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