Abstract
A weighted strong form collocation framework with mixed radial basis approximations for the pressure and displacement fields is proposed for incompressible and nearly incompressible linear elasticity. It is shown that with the proper choice of independent source points and collocation points for the radial basis approximations in the pressure and displacement fields, together with the analytically derived weights associated with the incompressibility constraint and boundary condition collocation equations, optimal convergence can be achieved. The optimal weights associated with the collocation equations are derived based on achieving balanced errors resulting from domain, boundaries, and constraint equations. Since in the proposed method the overdetermined system of the collocation equations is solved by a least squares method, independent pressure and displacement approximations can be selected without suffering from instability due to violation of the LBB stability condition. The numerical solutions verify that the solution of the proposed method does not exhibit volumetric locking and pressure oscillation, and that the solution converges exponentially in both L\(_{2}\) norm and H\(_{1}\) semi-norm, consistent with the error analysis results presented in this paper.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig1_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig2_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig3_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig4_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig5_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig6_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig7_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig8_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig9_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig10_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig11_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig12_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig13_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00466-013-0909-9/MediaObjects/466_2013_909_Fig14_HTML.gif)
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Method Eng 37(2):229–256
Liu WK, Jun S, Zhang YF (1995) Reproducing Kernel particle methods. Int J Numer Method Fluid 20(8–9):1081–1106
Babuska I, Melenk JM (1997) The partition of unity method. Int J Numer Method Eng 40(4):727–758
Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics—theory and application to non-spherical stars. Mon Notices R Astron Soc 181(2):375–389
Atluri SN, Zhu TL (2000) The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics. Comput Mech 25(2–3):169–179
Chen JS, Pan CH, Wu CT, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Method Appl Mech Eng 139(1–4):195–227
Kansa EJ (1990) Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics.1. Comput Math Appl 19(8–9):127–145
Kansa EJ (1990) Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics.2. Solutions to parabolic, hyperbolic and elliptic partial-differential equations. Comput Math Appl 19(8–9):147–161
Aluru NR (2000) A point collocation method based on reproducing kernel approximations. Int J Numer Method Eng 47(6):1083–1121
Hu HY, Chen JS, Hu W (2011) Error analysis of collocation method based on reproducing Kernel approximation. Numer Method Partial Differ Equ 27(3):554–580
Chen JS, Hu HY, Hu W (2007) Weighted radial basis collocation method for boundary value problems. Int J Numer Method Eng 69(13):2736–2757
Chen JS, Wang LH, Hu HY, Chi SW (2009) Subdomain radial basis collocation method for heterogeneous media. Int J Numer Method Eng 80(2):163–190
Franke C, Schaback R (1998) Solving partial differential equations by collocation using radial basis functions. Appl Math Comput 93(1):73–82
Hu HY, Li ZC, Cheng AHD (2005) Radial basis collocation methods for elliptic boundary value problems. Comput Math Appl 50(1–2):289–320
Wendland H (1999) Meshless Galerkin methods using radial basis functions. Math Comput 68(228):1521–1531
Cecil T, Qian JL, Osher S (2004) Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions. J Comput Phys 196(1):327–347
Pollandt R (1997) Solving nonlinear differential equations of mechanics with the boundary element method and radial basis functions. Int J Numer Method Eng 40(1):61–73
Sonar T (1996) Optimal recovery using thin plate splines in finite volume methods for the numerical solution of hyperbolic conservation laws. Ima J Numer Anal 16(4):549–581
Chi SW, Chen JS, Luo H, Hu HY, Wang L (2012) Dispersion and stability properties of radial basis collocation method for elastodynamics. In: Numerical methods for partial differential equations. Springer, New York
Wang LH, Chen JS, Hu HY (2010) Subdomain radial basis collocation method for fracture mechanics. Int J Numer Method Eng 83(7):851–876
Wendland H (1998) Error estimates for interpolation by compactly supported radial basis functions of minimal degree. J Approx Theory 93(2):258–272
Chen JS, Hu W, Hu HY (2008) Reproducing kernel enhanced local radial basis collocation method. Int J Numer Method Eng 75(5):600–627
Herrmann LR (1965) Elasticity equations for incompressible and nearly incompressible materials by a variational theorem. AIAA J 3(10):1896–1900
Murakawa H, Atluri SN (1979) inite elasticity solutions using hybrid finite-elements based on a complementary energy principle.2. Incompressible materials. J Appl Mech Trans ASME 46(1):71–77
Pian THH, Sumihara K (1984) Rational approach for assumed stress finite-elements. Int J Numer Method Eng 20(9):1685–1695
Simo JC, Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Method Eng 29(8):1595–1638
Liu WK, Belytschko T, Chen JS (1988) Nonlinear versions of flexurally superconvergent elements. Comput Method Appl Mech Eng 71(3):241–258
Chen JS, Satyamurthy K, Hirschfelt LR (1994) Consistent finite-element procedures for nonlinear rubber elasticity with a higher-order strain-energy function. Comput Struct 50(6):715–727
Babuska I (1973) Finite-element method with Lagrangian multipliers. Numer Math 20(3):179–192
Brezzi F (1974) Existence, uniqueness and approximation of Saddle-point problems arising from Lagrangian multipliers. Revue Francaise D Automatique Informatique Recherche Operationnelle 8(Nr2): 129–151
Sani RL, Gresho PM, Lee RL, Griffiths DF (1981) The cause and cure (questionable) of the spurious pressures generated by certain fem solutions of the incompressible Navier-stokes equations.1. Int J Numer Method Fluid 1(1):17–43
Sussman T, Bathe KJ (1987) A finite-element formulation for nonlinear incompressible elastic and inelastic analysis. Comput Struct 26(1–2):357–409
Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover, New York
Chen JS, Han W, Wu CT, Duan W (1997) On the perturbed Lagrangian formulation for nearly incompressible and incompressible hyperelasticity. Comput Method Appl Mech Eng 142(3–4):335–351
Chen JS, Pan CH (1996) A pressure projection method for nearly incompressible rubber hyperelasticity.1. J Appl Mech Trans ASME 63(4):862–868
Chen JS, Wu CT, Pan CH (1996) A pressure projection method for nearly incompressible rubber hyperelasticity.2. Applications. J Appl Mech Trans ASME 63(4):869–876
Fried I (1974) Finite element analysis of incompressible material by residual energy balancing. Int J Solid Struct 10(9):993–1002
Zienkiewicz OC, Taylor RL, Too JM (1971) Reduced integration technique in general analysis of plates and shells. Int J Numer Method Eng 3(2):275–290
Malkus DS, Hughes TJR (1978) Mixed finite-element methods—reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng 15(1):63–81
Hughes TJR (1980) Generalization of selective integration procedures to anisotropic and nonlinear media. Int J Numer Method Eng 15(9):1413–1418
Belytschko T, Ong JSJ, Liu WK, Kennedy JM (1984) Hourglass control in linear and nonlinear problems. Comput Method Appl Mech Eng 43(3):251–276
Liu WK, Ong JSJ, Uras RA (1985) Finite-element stabilization matrices—a unification approach. Comput Method Appl Mech Eng 53(1):13–46
Vidal Y, Villon P, Huerta A (2003) Locking in the incompressible limit: pseudo-divergence-free element free Galerkin. Commun Numer Method Eng 19(9):725–735
Lovadina C, Auricchio F, da Veiga LB, Buffa A, Reali A, Sangalli G (2007) A fully “locking-free” isogeometric approach for plane linear elasticity problems: a stream function formulation. Comput Method Appl Mech Eng 197(1–4):160–172
Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Method Appl Mech Eng 194(39–41):4135–4195
Wu CT, Hu W, Chen JS (2012) A meshfree-enriched finite element method for compressible and near-incompressible elasticity. Int J Numer Method Eng 90(7):882–914
Franca LP, Stenberg R (1991) Error analysis of some Galerkin least-squares methods for the elasticity equations. SIAM J Numer Anal 28(6):1680–1697
Cai ZQ, Starke G (2004) Least-squares methods for linear elasticity. SIAM J Numer Anal 42(2):826–842
Micchelli CA (1986) Interpolation of scattered data—distance matrices and conditionally positive definite functions. Constr Approx 2(1):11–22
Madych WR, Nelson SA (1990) Multivariate interpolation and conditionally positive definite functions.2. Math Comput 54(189):211–230
Madych WR (1992) Miscellaneous error-bounds for multiquadric and related interpolators. Comput Math Appl 24(12):121–138
Ciarlet PG (1978) Mathematical elasticity vol I: three dimensional elasticity. North-Holland, Amsterdam
Timoshenko SP, Goodier JN (1934) Theory of elasticity. McGraw-Hill, New York
Chen JS, Pan C, Chang TYP (1995) On the control of pressure oscillation in bilinear-displacement constant-pressure element. Comput Method Appl Mech Eng 128(1–2):137–152
Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element method: its basis and fundamentals, 6th edn. Elsevier, New York
Acknowledgments
The support of this work by the University of Illinois at Chicago to the first author, the support by US Army ERDC under contract W912HZ-07-C-0019 to the second author, and the support by National Science Council (Taiwan) 100-2115-M-029-002 to the third author are greatly acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chi, SW., Chen, JS. & Hu, HY. A weighted collocation on the strong form with mixed radial basis approximations for incompressible linear elasticity. Comput Mech 53, 309–324 (2014). https://doi.org/10.1007/s00466-013-0909-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-013-0909-9