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Effect of Higher Order Taylor Series Expansion Terms of the NI-RPIM on the Solution Accuracy of 2D Elastic Problems

Yıl 2015, Cilt: 21 Sayı: 1, 1 - 10, 24.02.2015

Öz

In this study, effects of higher order Taylor series expansion terms in the nodal integration scheme of radial point interpolation method (NI-RPIM) are investigated on the solution accuracy of 2D elastic problems. The nodal integration scheme is proposed by Liu et al. [1] and based on the Taylor series expansion. It is used with increasing the order of terms up to 4th order in this study. 3 different case studies are applied and the results are compared with analytical, FEM and RPIM with Gaussian integration solutions. Also the effect of number of nodes is investigated. It can be accepted that the usage of Taylor series expansion and Gaussian method in integration of RPIM give similar solution times. However NI-RPIM with higher order Taylor series expansion terms has better solution speed than using Gaussian integration, especially in the solutions of model which has higher number of nodes. It is detected that 2nd order terms of nodal integration give sufficient results. If stress values are investigated, 4th order terms of nodal integration can be used for accuracy of the solution.

Kaynakça

  • Liu GR, Zhang G, Wang YY, Zhong ZH, Li GY, Han X. "A Nodal Integration Technique for Meshfree Radial Point Interpolation Method (NI-RPIM)". International Journal of Solids and Structures, 44(11-12), 3840-3860, 2007.
  • Rao BN, Rahman, S. "A Coupled Meshless-Finite Element Method for Fracture Analysis of Cracks". International Journal of Pressure Vessels and Piping, 78(9), 647-657, 2001.
  • Gu YT, Zhang, LC. "Coupling of the Meshfree and Finite Element Methods for Determination of the Crack Tip Fields". Engineering Fracture Mechanics, 75(5), 986-1004, 2008.
  • Wang YF, Yang ZG. "A Coupled Finite Element and Meshfree Analysis of Erosive Wear". Tribology International, 42(2), 373-377, 2009.
  • Wang HP, Wu CT, Guo Y, Botkin ME. "A Coupled Meshfree/Finite Element Method for Automotive Crashworthiness Simulations". International Journal of Impact Engineering, 36(10-11), 1210–1222, 2009.
  • Rao BN. "Coupled Meshfree and Fractal Finite Element Method for Unbounded Problems". Computers and Geotechnics, 38(5), 697–708, 2011.
  • Lucy LB. "A Numerical Approach to the Testing of the Fission 82(12), 1013-1024, 1977. The Astronomical Journal,
  • Nayroles B, Touzot G, Villon P. "Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements". Computational Mechanics, 10(5), 307-318, 1992.
  • Belytschko T, Lu YY, Gu L. "Element-Free Galerkin Methods". International Journal for Numerical Methods in Engineering, 37(2), 229-256, 1994.
  • Liu WK, Chen Y, Chang CT and Belytschko T. "Advances in Kernel Multiple Computational Mechanics, 18, 73-111, 1996. Particle Methods".
  • Atluri SN, Zhu T. "A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics". Computational Mechanics, 22, 117-127, 1998.
  • Liu GR, Gu YT. "A Point Interpolation Method for Two-Dimensional Solids". International Journal for Numerical Methods in Engineering, 50(4), 937-951, 2001.
  • Liu GR, Gu YT. "A Local Radial Point Interpolation Method (LR-PIM) for Free Vibration Analyses of 2-D Solids". Journal of Sound and Vibration, 246(1), 29-46, 2001.
  • Liu GR, Zhang GY, Dai KY. "A Linearly Conforming Point Interpolation Method (LC-PIM) for 2D Solid Mechanics Problems". International Journal of Computational Methods, 2(4), 645-665, 2005.
  • Wu SC, Liu GR, Zhang HO, Xu X, Li ZR. "A Node-Based Smoothed Point Interpolation Method (NS-PIM) for Three-Dimensional International 48(7), 1367-1376, 2009. Transfer Problems". Journal of Thermal Sciences,
  • Cui XY, Liu GR, Li GY. "A Cell-Based Smoothed Radial Point Interpolation Method (CS-RPIM) for Static and Free Vibration of Solids". Engineering Analysis with Boundary Elements, 34(2), 144-157, 2010.
  • Liu GR, Gu YT. An Introduction to Meshfree Methods and Their Programming. Berlin, Germany, Springer Science and Business Media, 2005.
  • Wang JG, Liu GR. "A Point Interpolation Meshless Method Based on Radial Basis Functions". International Journal for Numerical Methods in Engineering, 54(11), 1623-1648, 2002.
  • Wang JG, Liu GR. "On the Optimal Shape Parameters of Radial Basis Functions Used for 2-D Meshless Methods". Computer Methods Applied Mechanics Engineering, 191(23-24), 2611–2630, 2002.
  • Kanber B, Bozkurt OY, Erklig A. "Investigation of RPIM Shape Parameter Effects on the Solution Accuracy of 2D Elastoplastic Problems". International Journal for Computational Methods in Engineering Science and Mechanics, 14(4), 354-366, 2013.
  • Bozkurt OY, Kanber B, Asik MZ. "Assessment of RPIM Shape Parameters for Solution Accuracy of 2D Geometrically Nonlinear Problems". International Journal of Computational Methods, 10(3), 1-26, 2013.
  • Dinis LMJS, Jorgea, RMN, Belinha J. "Analysis of 3D Solids Using the Natural Neighbour Radial Point Interpolation Method". Computer Methods in Applied Mechanics and Engineering, 196(13-16), 2009–2028, 2007.
  • Dinis LMJS, Jorgea RMN, Belinha J. "Analysis of Plates and Laminates Using the Natural Neighbour Radial Point Interpolation Method". Engineering Analysis with Boundary Elements, 32(3), 267–279, 2008.
  • Xia P, Long SY, Cui HX, Li GY. "The Static and Free Vibration Analysis of a Non-Homogeneous Moderately Thick Plate Using the Meshless Local Radial Point Interpolation Method". Engineering Analysis with Boundary Elements, 33(6), 770-777, 2009.
  • Wagner D, Millwater H. "2D Weight Function Development Using a Complex Taylor Series Expansion Method". Engineering Fracture Mechanics, 86, 23-37, 2012.
  • Liu GR, Zhang J, Li H, Lam KY, Kee BBT. "Radial Point Interpolation Based Finite Difference Method for Mechanics Problems". International Journal for Numerical Methods in Engineering, 68(7), 728-754, 2006. [27] Anderson JD. Computational The Basics With Applications. 1st ed. New York, USA, McGraw-Hill Press, 1995. Fluid Dynamics,

Yüksek Dereceli NI-RPIM Taylor Serisi Açılımı Terimlerinin 2 Boyutlu Elastik Problemlerin Çözüm Hassasiyetine Etkisi

Yıl 2015, Cilt: 21 Sayı: 1, 1 - 10, 24.02.2015

Öz

Bu çalışmada, yüksek mertebeden Taylor serisi açılım terimlerinin radyal nokta interpolasyon yöntemi düğüm entegrasyon şeması (NI-RPIM) açısından 2 boyutlu elastik problemlerin çözüm doğruluğuna etkileri incelenmiştir. Düğüm integrasyon şeması Liu ve diğ. [1] tarafından tasarlanmıştır ve Taylor serisi açılımı üzerinedir. Bu çalışmada 4. mertebeye kadar terimleri arttırılarak kullanılmıştır. 3 farklı alan çalışması yapılmış ve sonuçları analitik, sonlu elemanlar yöntemi ve Gauss integrasyonlu RPIM sonuçları ile kısaylanmıştır. Ayrıca nokta sayısının etkisi araştırılmıştır. RPIM integrasyonu içerisinde kullanılan Taylor serisi açılımı ve Gauss metodu benzer çözüm zamanları verdiği kabul edilebilir. Bununla birlikte özellikle yüksek sayıda nokta içeren modellerin çözümünde, NI-RPIM ile yüksek mertebeden Taylor serisi açılımı terimleri Gauss integrasyonundan daha iyi çözüm hızına sahiptir. 2. mertebeden düğüm integrasyon terimlerinin yeterli sonuç verdiği belirlenmiştir. Eğer gerilme değerleri incelenmekteyse, çözüm hassasiyeti için 4. mertebe düğüm integrasyon terimleri kullanılabilinir.

Kaynakça

  • Liu GR, Zhang G, Wang YY, Zhong ZH, Li GY, Han X. "A Nodal Integration Technique for Meshfree Radial Point Interpolation Method (NI-RPIM)". International Journal of Solids and Structures, 44(11-12), 3840-3860, 2007.
  • Rao BN, Rahman, S. "A Coupled Meshless-Finite Element Method for Fracture Analysis of Cracks". International Journal of Pressure Vessels and Piping, 78(9), 647-657, 2001.
  • Gu YT, Zhang, LC. "Coupling of the Meshfree and Finite Element Methods for Determination of the Crack Tip Fields". Engineering Fracture Mechanics, 75(5), 986-1004, 2008.
  • Wang YF, Yang ZG. "A Coupled Finite Element and Meshfree Analysis of Erosive Wear". Tribology International, 42(2), 373-377, 2009.
  • Wang HP, Wu CT, Guo Y, Botkin ME. "A Coupled Meshfree/Finite Element Method for Automotive Crashworthiness Simulations". International Journal of Impact Engineering, 36(10-11), 1210–1222, 2009.
  • Rao BN. "Coupled Meshfree and Fractal Finite Element Method for Unbounded Problems". Computers and Geotechnics, 38(5), 697–708, 2011.
  • Lucy LB. "A Numerical Approach to the Testing of the Fission 82(12), 1013-1024, 1977. The Astronomical Journal,
  • Nayroles B, Touzot G, Villon P. "Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements". Computational Mechanics, 10(5), 307-318, 1992.
  • Belytschko T, Lu YY, Gu L. "Element-Free Galerkin Methods". International Journal for Numerical Methods in Engineering, 37(2), 229-256, 1994.
  • Liu WK, Chen Y, Chang CT and Belytschko T. "Advances in Kernel Multiple Computational Mechanics, 18, 73-111, 1996. Particle Methods".
  • Atluri SN, Zhu T. "A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics". Computational Mechanics, 22, 117-127, 1998.
  • Liu GR, Gu YT. "A Point Interpolation Method for Two-Dimensional Solids". International Journal for Numerical Methods in Engineering, 50(4), 937-951, 2001.
  • Liu GR, Gu YT. "A Local Radial Point Interpolation Method (LR-PIM) for Free Vibration Analyses of 2-D Solids". Journal of Sound and Vibration, 246(1), 29-46, 2001.
  • Liu GR, Zhang GY, Dai KY. "A Linearly Conforming Point Interpolation Method (LC-PIM) for 2D Solid Mechanics Problems". International Journal of Computational Methods, 2(4), 645-665, 2005.
  • Wu SC, Liu GR, Zhang HO, Xu X, Li ZR. "A Node-Based Smoothed Point Interpolation Method (NS-PIM) for Three-Dimensional International 48(7), 1367-1376, 2009. Transfer Problems". Journal of Thermal Sciences,
  • Cui XY, Liu GR, Li GY. "A Cell-Based Smoothed Radial Point Interpolation Method (CS-RPIM) for Static and Free Vibration of Solids". Engineering Analysis with Boundary Elements, 34(2), 144-157, 2010.
  • Liu GR, Gu YT. An Introduction to Meshfree Methods and Their Programming. Berlin, Germany, Springer Science and Business Media, 2005.
  • Wang JG, Liu GR. "A Point Interpolation Meshless Method Based on Radial Basis Functions". International Journal for Numerical Methods in Engineering, 54(11), 1623-1648, 2002.
  • Wang JG, Liu GR. "On the Optimal Shape Parameters of Radial Basis Functions Used for 2-D Meshless Methods". Computer Methods Applied Mechanics Engineering, 191(23-24), 2611–2630, 2002.
  • Kanber B, Bozkurt OY, Erklig A. "Investigation of RPIM Shape Parameter Effects on the Solution Accuracy of 2D Elastoplastic Problems". International Journal for Computational Methods in Engineering Science and Mechanics, 14(4), 354-366, 2013.
  • Bozkurt OY, Kanber B, Asik MZ. "Assessment of RPIM Shape Parameters for Solution Accuracy of 2D Geometrically Nonlinear Problems". International Journal of Computational Methods, 10(3), 1-26, 2013.
  • Dinis LMJS, Jorgea, RMN, Belinha J. "Analysis of 3D Solids Using the Natural Neighbour Radial Point Interpolation Method". Computer Methods in Applied Mechanics and Engineering, 196(13-16), 2009–2028, 2007.
  • Dinis LMJS, Jorgea RMN, Belinha J. "Analysis of Plates and Laminates Using the Natural Neighbour Radial Point Interpolation Method". Engineering Analysis with Boundary Elements, 32(3), 267–279, 2008.
  • Xia P, Long SY, Cui HX, Li GY. "The Static and Free Vibration Analysis of a Non-Homogeneous Moderately Thick Plate Using the Meshless Local Radial Point Interpolation Method". Engineering Analysis with Boundary Elements, 33(6), 770-777, 2009.
  • Wagner D, Millwater H. "2D Weight Function Development Using a Complex Taylor Series Expansion Method". Engineering Fracture Mechanics, 86, 23-37, 2012.
  • Liu GR, Zhang J, Li H, Lam KY, Kee BBT. "Radial Point Interpolation Based Finite Difference Method for Mechanics Problems". International Journal for Numerical Methods in Engineering, 68(7), 728-754, 2006. [27] Anderson JD. Computational The Basics With Applications. 1st ed. New York, USA, McGraw-Hill Press, 1995. Fluid Dynamics,
Toplam 26 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Makale
Yazarlar

Mustafa Yavuz Bu kişi benim

Bahattin Kanber Bu kişi benim

Yayımlanma Tarihi 24 Şubat 2015
Yayımlandığı Sayı Yıl 2015 Cilt: 21 Sayı: 1

Kaynak Göster

APA Yavuz, M., & Kanber, B. (2015). Effect of Higher Order Taylor Series Expansion Terms of the NI-RPIM on the Solution Accuracy of 2D Elastic Problems. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi, 21(1), 1-10. https://doi.org/10.5505/pajes.2014.29591
AMA Yavuz M, Kanber B. Effect of Higher Order Taylor Series Expansion Terms of the NI-RPIM on the Solution Accuracy of 2D Elastic Problems. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi. Şubat 2015;21(1):1-10. doi:10.5505/pajes.2014.29591
Chicago Yavuz, Mustafa, ve Bahattin Kanber. “Effect of Higher Order Taylor Series Expansion Terms of the NI-RPIM on the Solution Accuracy of 2D Elastic Problems”. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi 21, sy. 1 (Şubat 2015): 1-10. https://doi.org/10.5505/pajes.2014.29591.
EndNote Yavuz M, Kanber B (01 Şubat 2015) Effect of Higher Order Taylor Series Expansion Terms of the NI-RPIM on the Solution Accuracy of 2D Elastic Problems. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi 21 1 1–10.
IEEE M. Yavuz ve B. Kanber, “Effect of Higher Order Taylor Series Expansion Terms of the NI-RPIM on the Solution Accuracy of 2D Elastic Problems”, Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi, c. 21, sy. 1, ss. 1–10, 2015, doi: 10.5505/pajes.2014.29591.
ISNAD Yavuz, Mustafa - Kanber, Bahattin. “Effect of Higher Order Taylor Series Expansion Terms of the NI-RPIM on the Solution Accuracy of 2D Elastic Problems”. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi 21/1 (Şubat 2015), 1-10. https://doi.org/10.5505/pajes.2014.29591.
JAMA Yavuz M, Kanber B. Effect of Higher Order Taylor Series Expansion Terms of the NI-RPIM on the Solution Accuracy of 2D Elastic Problems. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi. 2015;21:1–10.
MLA Yavuz, Mustafa ve Bahattin Kanber. “Effect of Higher Order Taylor Series Expansion Terms of the NI-RPIM on the Solution Accuracy of 2D Elastic Problems”. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi, c. 21, sy. 1, 2015, ss. 1-10, doi:10.5505/pajes.2014.29591.
Vancouver Yavuz M, Kanber B. Effect of Higher Order Taylor Series Expansion Terms of the NI-RPIM on the Solution Accuracy of 2D Elastic Problems. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi. 2015;21(1):1-10.





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