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Nodally integrated thermomechanical RKPM: Part I—Thermoelasticity

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Abstract

In this two-part paper, a stable and efficient nodally-integrated reproducing kernel particle method (RKPM) is introduced for solving the governing equations of generalized thermomechanical theories. Part I investigates quadrature in the weak form using coupled and uncoupled classical thermoelasticity as model problems. It is first shown that nodal integration of these equations results in spurious oscillations in the solution many orders of magnitude greater than pure elasticity. A naturally stabilized nodal integration is then proposed for the coupled equations. The variational consistency conditions for nth order exactness and convergence in the two-field problem are then derived, and a uniform correction on the test function approximations is proposed to achieve these conditions. Several benchmark problems are solved to demonstrate the effectiveness of the proposed method. In the sequel, these methods are developed for generalized thermoelasticity and generalized finite-strain thermoplasticity theories of the hyperbolic type that are amenable to efficient explicit time integration.

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References

  1. Baek J, Chen J-S, Zhou G, Arnett K, Hillman M, Hegemier G, Hardesty S (2021) A semi-Lagrangian reproducing kernel particle method with particle-based shock algorithm for explosive welding simulation. Comput Mech 67:1601–1627

    Article  MathSciNet  MATH  Google Scholar 

  2. Beissel S, Belytschko T (1996) Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng 139:49–74

    Article  MathSciNet  MATH  Google Scholar 

  3. Belytschko T, Guo Y, Liu WK, Xiao SP (2000) A unifieded stability analysis of meshless particle methods. Int J Numer Methods Eng 48(9):1359–1400

    Article  MATH  Google Scholar 

  4. Bobaru F, Mukherjee S (2002) Meshless approach to shape optimization of linear thermoelastic solids. Int J Numer Methods Eng 53(4):765–796

    Article  Google Scholar 

  5. Boley BA, Tolins IS (1962) Transient coupled thermoelastic boundary value problems in the half-space. J Appl Mech 29(4):637–646

    Article  MathSciNet  MATH  Google Scholar 

  6. Cannarozzi AA, Ubertini F (2001) A mixed variational method for linear coupled thermoelastic analysis. Int J Solids Struct 38(4):717–739

    Article  MATH  Google Scholar 

  7. Carter JP, Booker JR (1989) Finite element analysis of coupled thermoelasticity. Comput Struct 31(1):73–80

    Article  Google Scholar 

  8. Chen J, Dargush GF (1995) Boundary element method for dynamic poroelastic and thermoelastic analyses. Int J Solids Struct 32(15):2257–2278

    Article  MATH  Google Scholar 

  9. Chen J, Hu W, Puso M, Wu Y, Zhang X (2007) Strain smoothing for stabilization and regularization of galerkin meshfree methods. In Meshfree methods for partial differential equations III, pages 57–75. Springer

  10. Chen J-S, Hillman M, Chi S-W (2017) Meshfree methods: progress made after 20 years. J Eng Mech 143(4):04017001

    Article  Google Scholar 

  11. Chen J-S, Hillman M, Rüter M (2013) An arbitrary order variationally consistent integration for Galerkin meshfree methods. Int J Numer Methods Eng 95(5):387–418

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen J-S, Pan C, Wu C-T, Liu WK (1996) Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 139(1–4):195–227

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen J-S, Wang H-P (2000) New boundary condition treatments in meshfree computation of contact problems. Comput Methods Appl Mech Eng 187(3–4):441–468

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen J-S, Wu C-T, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50(2):435–466

    Article  MATH  Google Scholar 

  15. Chen J-S, Yoon S, Wu C-T (2002) Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 53(12):2587–2615

    Article  MATH  Google Scholar 

  16. Chen J-S, Zhang X, Belytschko T (2004) An implicit gradient model by a reproducing kernel strain regularization in strain localization problems. Comput Methods Appl Mech Eng 193(27–29):2827–2844

    Article  MATH  Google Scholar 

  17. Ching HK, Yen SC (2006) Transient thermoelastic deformations of 2-D functionally graded beams under nonuniformly convective heat supply. Compos Struct 73(4):381–393

    Article  Google Scholar 

  18. Danilouskaya V (1950) Thermal stresses in elastic half space due to sudden heating of its boundary. Pelageya Yakovlevna Kochina 14:316–321

    Google Scholar 

  19. Danilouskaya V (1952) On a dynamic problem of thermoelasticity. Prikladnaya Matematika i Mekhanika 16:341–344

    Google Scholar 

  20. Dolbow J, Belytschko T (1999) Numerical integration of the Galerkin weak form in meshfree methods. Comput Mech 23(3):219–230

    Article  MathSciNet  MATH  Google Scholar 

  21. Duan Q, Li X, Zhang H, Belytschko T (2012) Second-order accurate derivatives and integration schemes for meshfree methods. Int J Numer Methods Eng 92(4):399–424

    Article  MathSciNet  MATH  Google Scholar 

  22. Fries T-P, Belytschko T (2008) Convergence and stabilization of stress-point integration in mesh-free and particle methods. Int J Numer Methods Eng 74(7):1067–1087

    Article  MathSciNet  MATH  Google Scholar 

  23. Hasanpour K, Mirzaei D (2018) A fast meshfree technique for the coupled thermoelasticity problem. Acta Mech 229(6):2657–2673

    Article  MathSciNet  MATH  Google Scholar 

  24. Hillman M, Chen J-S (2016) An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics. Int J Numer Methods Eng 107:603–630

    Article  MathSciNet  MATH  Google Scholar 

  25. Hillman M, Chen J-S (2016) Nodally integrated implicit gradient reproducing kernel particle method for convection dominated problems. Comput Methods Appl Mech Eng 299:381–400

    Article  MathSciNet  MATH  Google Scholar 

  26. Hillman M, Chen J-S (2018) Performance comparison of nodally integrated galerkin meshfree methods and nodally collocated strong form meshfree methods. In Advances in Computational Plasticity, Vol. 46, pages 145–164. Springer

  27. Hillman M, Chen J-S, Chi S-W (2014) Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems. Comput Part Mech 1(3):245–256

    Article  Google Scholar 

  28. Hillman M, Lin K-C (2021) Consistent weak forms for meshfree methods: Full realization of \(h\)-refinement, \(p\)-refinement, and \(a\)-refinement in strong-type essential boundary condition enforcement. Comput Methods Appl Mech Eng 373:113448

    Article  MathSciNet  MATH  Google Scholar 

  29. Hosseini-Tehrani P, Eslami MR (2000) BEM analysis of thermal and mechanical shock in a two-dimensional finite domain considering coupled thermoelasticity. Eng Anal Bound Elem 24(3):249–257

    Article  MATH  Google Scholar 

  30. Hu H-Y, Lai C-K, Chen J-S (2009) A study on convergence and complexity of reproducing kernel collocation method. Interact Multiscale Mech 2(3):295–319

    Article  Google Scholar 

  31. Hughes TJ (2012) The finite element method: linear static and dynamic finite element analysis. Dover Publications, Mineola

    Google Scholar 

  32. Tamma KK, Railkar SB (1988) On heat displacement based hybrid transfinite element formulations for uncoupled/coupled thermally induced stress wave propagation. Comput Struct 30:1025–1036

    Article  MATH  Google Scholar 

  33. Keramidas GA, Ting EC (1976) A finite element formulation for thermal stress analysis. Part I: variational formulation. Nucl Eng Des 39(2–3):267–275

    Article  Google Scholar 

  34. Keramidas GA, Ting EC (1976) A finite element formulation for thermal stress analysis. Part II: finite element formulation. Nucl Eng Des 39(2–3):277–287

    Article  Google Scholar 

  35. Li S, Liu WK (1998) Synchronized reproducing kernel interpolant via multiple wavelet expansion. Comput Mech 21:28–47

    Article  MathSciNet  MATH  Google Scholar 

  36. Li S, Liu WK (1999) Reproducing kernel hierarchical partition of unity, part I—formulation and theory. Int J Numer Methods Eng 288(July 1998):251–288

    Article  MATH  Google Scholar 

  37. Li S, Liu WK (1999) Reproducing kernel hierarchical partition of unity, Part II—applications. Int J Numer Methods Eng 45(3):289–317

    Article  Google Scholar 

  38. Liu G-R, Zhang GY, Wang YY, Zhong ZH, Li GY, Han X (2007) A nodal integration technique for meshfree radial point interpolation method (NI-RPIM). Int J Solids Struct 44(11–12):3840–3860

    Article  MATH  Google Scholar 

  39. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20(8–9):1081–1106

    Article  MathSciNet  MATH  Google Scholar 

  40. Liu W-K, Li S, Belytschko T (1997) Moving least-square reproducing kernel methods (i) methodology and convergence. Comput Methods Appl Mech Eng 143(1–2):113–154

    Article  MathSciNet  MATH  Google Scholar 

  41. Liu WK, Ong JS-J, Uras RA (1985) Finite element stabilization matrices—a unification approach. Comput Methods Appl Mech Eng 53(1):13–46

    Article  MathSciNet  MATH  Google Scholar 

  42. Mahdavi A, Chi S-W, Atif MM (2020) A two-field semi-Lagrangian reproducing kernel model for impact and penetration simulation into geo-materials. Comput Part Mech 7(2):351–364

    Article  Google Scholar 

  43. Mahdavi A, Chi S-W, Zhu H (2019) A gradient reproducing kernel collocation method for high order differential equations. Comput Mech 64(5):1421–1454

    Article  MathSciNet  MATH  Google Scholar 

  44. Moutsanidis G, Li W, Bazilevs Y (2021) Reduced quadrature for FEM, IGA and meshfree methods. Comput Methods Appl Mech Eng 373:113521

    Article  MathSciNet  MATH  Google Scholar 

  45. Nagashima T (1999) Node-by-node meshless approach and its applications to structural analyses. Int J Numer Methods Eng 46(3):341–385

    Article  MATH  Google Scholar 

  46. Nickell JL, Robert E, Sackman R (1968) Approximate solutions in linear, coupled thermoelasticity. J Appl Mech 35(2):255–266

    Article  MATH  Google Scholar 

  47. Nowacki W (1975) Dynamic problems of thermoelasticity. Springer, New York

    MATH  Google Scholar 

  48. Prevost J-H, Tao D (1983) Finite element analysis of dynamic coupled thermoelasticity problems with relaxation times. J Appl Mech 50(4a):817–822

    Article  MATH  Google Scholar 

  49. Puso MA, Chen J-S, Zywicz E, Elmer W (2008) Meshfree and finite element nodal integration methods. Int J Numer Methods Eng 74(3):416–446

    Article  MathSciNet  MATH  Google Scholar 

  50. Qian LF, Batra RC (2004) Transient thermoelastic deformations of a thick functionally graded plate. J Therm Stresses 27(8):705–740

    Article  Google Scholar 

  51. Randles PW, Libersky LD (2000) Normalized SPH with stress points. Int J Numer Methods Eng 48(May 1999):1445–1462

    Article  MATH  Google Scholar 

  52. Rüter M, Hillman M, Chen J-S (2013) Corrected stabilized non-conforming nodal integration in meshfree methods. In Meshfree methods for partial differential equations VI, pages 75–92. Springer

  53. Siriaksorn T, Chi S-W, Foster C, Mahdavi A (2018) \(u\)-\(p\) semi-Lagrangian reproducing kernel formulation for landslide modeling. Int J Numer Anal Methods Geomech 42(2):231–255

    Article  Google Scholar 

  54. Sladek J, Sladek V, Solek P, Tan CL, Zhang C (2009) Two- and three-dimensional transient thermoelastic analysis by the MLPG method. Computer Modeling in Engineering and Sciences 47

  55. Sládek V, Sládek J (1985) Boundary element method in micropolar thermoelasticity. Part II: boundary integro-differential equations. Eng Anal 2(2):81–91

    Article  MATH  Google Scholar 

  56. Sternberg ELI, Chakravorty JG (1958) On inertia effects in a transient thermoelastic problem. Brown University, Providence

    Google Scholar 

  57. Tanaka M, Matsumoto T, Moradi M (1995) Application of boundary element method to 3-D problems of coupled thermoelasticity. Eng Anal Bound Elem 16(4):297–303

    Article  Google Scholar 

  58. Thornton EA (1996) Thermal structures for aerospace applications. American Institute of Aeronautics and Astronautics, New York

    Book  Google Scholar 

  59. Tosaka N, Suh IG (1991) Boundary element analysis of dynamic coupled thermoelasticity problems. Comput Mech 8(5):331–342

  60. Wang D, Wu J (2019) An inherently consistent reproducing kernel gradient smoothing framework toward efficient Galerkin meshfree formulation with explicit quadrature. Comput Methods Appl Mech Eng 349:628–672

    Article  MathSciNet  MATH  Google Scholar 

  61. Wei H, Chen J-S, Beckwith F, Baek J (2020) A naturally stabilized semi-Lagrangian meshfree formulation for multiphase porous media with application to landslide modeling. J Eng Mech 146(4):04020012

    Article  Google Scholar 

  62. Wei H, Chen J-S, Hillman M (2016) A stabilized nodally integrated meshfree formulation for fully coupled hydro-mechanical analysis of fluid-saturated porous media. Comput Fluids 141:105–115

    Article  MathSciNet  MATH  Google Scholar 

  63. Wu C-T, Chi S-W, Koishi M, Wu Y (2016) Strain gradient stabilization with dual stress points for the meshfree nodal integration method in inelastic analyses. Int J Numer Methods Eng 107(1):3–30

  64. Wu C-T, Koishi M, Hu W (2015) A displacement smoothing induced strain gradient stabilization for the meshfree Galerkin nodal integration method. Computational Mechanics

  65. Wu CT, Wu Y, Lyu D, Pan X, Hu W (2020) The momentum-consistent smoothed particle Galerkin (MC-SPG) method for simulating the extreme thread forming in the flow drill screw-driving process. Comput Part Mech 7(2):177–191

    Article  Google Scholar 

  66. Wu J, Wang D (2021) An accuracy analysis of Galerkin meshfree methods accounting for numerical integration. Comput Methods Appl Mech Eng 375:113631

    Article  MathSciNet  MATH  Google Scholar 

  67. Zheng BJ, Gao XW, Yang K, Zhang CZ (2015) A novel meshless local Petrov-Galerkin method for dynamic coupled thermoelasticity analysis under thermal and mechanical shock loading. Eng Anal Bound Elem 60:154–161

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors greatly acknowledge the support of this work by Penn State, and the endowment of the L. Robert and Mary L. Kimball Early Career Professorship. Proofing of this manuscript by Jennifer Dougal is also acknowledged and appreciated.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA, 16802, USA

    Michael Hillman

  2. The Pennsylvania State University, 224A Sackett Building, University Park, PA, 16802, USA

    Michael Hillman

  3. Department of Civil Engineering, National Cheng Kung University, Tainan, 70101, Taiwan, ROC

    Kuan-Chung Lin

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Hillman, M., Lin, KC. Nodally integrated thermomechanical RKPM: Part I—Thermoelasticity. Comput Mech 68, 795–820 (2021). https://doi.org/10.1007/s00466-021-02047-9

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