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

Abstract

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.

Keywords

• NI-RPIM, RBF, Nodal integration, Taylor expansion

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

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.

Anahtar Kelimeler

• NI-RPIM, RBF, Noktasal integrasyon, Taylor açılımı

Kaynakça

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