IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i7p1572-d1105636.html
   My bibliography  Save this article

Efficient Method for Derivatives of Nonlinear Stiffness Matrix

Author

Listed:
  • Tuan Anh Bui

    (Department of Aeronautic, Mechanical and Electrical Convergence Engineering, Kumoh National Institute of Technology, 61 Daehak-ro, Gumi 39177, Gyeongbuk, Republic of Korea
    Department of Engineering Mechanics, Faculty of Mechanical Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi 10000, Vietnam)

  • Jun-Sik Kim

    (Department of Mechanical System Engineering, Kumoh National Institute of Technology, 61 Daehak-ro, Gumi 39177, Gyeongbuk, Republic of Korea)

  • Junyoung Park

    (Department of Aeronautic, Mechanical and Electrical Convergence Engineering, Kumoh National Institute of Technology, 61 Daehak-ro, Gumi 39177, Gyeongbuk, Republic of Korea
    Department of Mechanical Design Engineering, Kumoh National Institute of Technology, 61 Daehak-ro, Gumi 39177, Gyeongbuk, Republic of Korea)

Abstract

Structural design often includes geometrically nonlinear analysis to reduce structural weight and increase energy efficiency. The full-order finite element model can perform the geometrically nonlinear analysis, but its computational cost is expensive. Therefore, nonlinear reduced-order models (NLROMs) have been developed to reduce costs. The non-intrusive NLROM has a lower cost than the other due to the approximation of the nonlinear internal force by a polynomial of reduced coordinates based on the Taylor expansion. The constants in the polynomial, named reduced stiffnesses, are derived from the derivative of the structure’s tangential stiffness matrix with respect to the reduced coordinates. The precision of the derivative of the tangential stiffness affects the reduced stiffness, which in turn significantly influences the accuracy of the NLROM. Therefore, this study evaluates the accuracy of the derivative of the tangential stiffness calculated by the methods: finite difference, complex step, and hyper-dual step. Analytical derivatives of the nonlinear stiffness are developed to provide references for evaluating the accuracy of the numerical methods. We propose using the central difference method to calculate the stiffness coefficients of NLROM due to its advantages, such as accuracy, low computational cost, and compatibility with commercial finite element software.

Suggested Citation

  • Tuan Anh Bui & Jun-Sik Kim & Junyoung Park, 2023. "Efficient Method for Derivatives of Nonlinear Stiffness Matrix," Mathematics, MDPI, vol. 11(7), pages 1-20, March.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1572-:d:1105636
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/7/1572/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/11/7/1572/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Sakal M. & Slavich A. & Cheretyko G., 2013. "The Y-generation through the prism of HRM," Вестник Омского университета. Серия «Экономика», CyberLeninka;Федеральное государственное бюджетное образовательное учреждение высшего образования «Омский государственный университет им. Ф.М. Достоевского», issue 2, pages 6-12.
    2. Jun Yang & Fanqiang Kong & Jianchao Xi & Quansheng Ge & Xueming Li & Peng Xie, 2013. "Land Use Patch Generalization Based on Semantic Priority," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, April.
    3. Universidad Nacional de Mar del Plata, Facultad de Ciencias Económicas y Sociales, Grupo Estudios del Trabajo, 2013. "Informe Sociolaboral del Partido de General Pueyrredon," Nülan. Deposited Documents 1858, Universidad Nacional de Mar del Plata, Facultad de Ciencias Económicas y Sociales, Centro de Documentación.
    4. Savin Treanţă, 2022. "Recent Advances of Constrained Variational Problems Involving Second-Order Partial Derivatives: A Review," Mathematics, MDPI, vol. 10(15), pages 1-13, July.
    5. Kristian Behrens & Gianmarco I. P. Ottaviano, 2013. "General Equilibrium Trade Theory and Firm Behaviour," Palgrave Macmillan Books, in: Daniel Bernhofen & Rod Falvey & David Greenaway & Udo Kreickemeier (ed.), Palgrave Handbook of International Trade, chapter 5, pages 119-159, Palgrave Macmillan.
    6. Gauss M. Cordeiro & Rodrigo R. Pescim & Edwin M. M. Ortega & Clarice G. B. Demétrio, 2013. "The Beta Generalized Half-Normal Distribution: New Properties," Journal of Probability and Statistics, Hindawi, vol. 2013, pages 1-18, December.
    7. Min Cai & Changpin Li, 2020. "Numerical Approaches to Fractional Integrals and Derivatives: A Review," Mathematics, MDPI, vol. 8(1), pages 1-53, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Tao, Sui & Rohde, David & Corcoran, Jonathan, 2014. "Examining the spatial–temporal dynamics of bus passenger travel behaviour using smart card data and the flow-comap," Journal of Transport Geography, Elsevier, vol. 41(C), pages 21-36.
    2. Alois S. Mlambo, 2017. "From an Industrial Powerhouse to a Nation of Vendors: Over Two Decades of Economic Decline and Deindustrialization in Zimbabwe 1990–2015," Journal of Developing Societies, , vol. 33(1), pages 99-125, March.
    3. Savin Treanţă, 2022. "Variational Problems and Applications," Mathematics, MDPI, vol. 11(1), pages 1-4, December.
    4. Ashish Rayal & Bhagawati Prasad Joshi & Mukesh Pandey & Delfim F. M. Torres, 2023. "Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets," Mathematics, MDPI, vol. 11(11), pages 1-22, May.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1572-:d:1105636. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.