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An automatic differentiation-enhanced meshfree finite block method for nonlinear problems

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  • Huang, W.
  • Yang, J.J.
  • Wen, P.H.

Abstract

This paper presents an Automatic Differentiation–Enhanced Meshfree Finite Block Method (AD-FBM) for solving strongly nonlinear partial differential equations (PDEs). The physical domain is divided into blocks, each mapped to a normalized standard domain, where shape functions are constructed via Lagrange polynomials. The automatic differentiation method computes exact derivatives of nonlinear material constitutive laws and PDE operators, significantly reducing the human effort and errors often associated with manual coding of Jacobians. The AD-FBM is validated through several benchmark problems, including a steady-state nonlinear heat conduction example, a bi-material scenario with thermal contact resistance, a large-deflection cantilever beam under follower loads, and a rectangular plate with a circular hole made of hypo-elastic materials. Each of which demonstrates excellent agreement with analytical or finite element solutions. The results show that the AD-FBM converges efficiently via Newton’s iteration, underscoring the advantages of integrating automatic differentiation with meshfree finite block method. The AD-FBM significantly reduces the coding complexity and the risk of errors associated with manual derivative computations for robust and flexible simulations of complex nonlinear PDEs.

Suggested Citation

  • Huang, W. & Yang, J.J. & Wen, P.H., 2025. "An automatic differentiation-enhanced meshfree finite block method for nonlinear problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 238(C), pages 388-402.
    • Handle: RePEc:eee:matcom:v:238:y:2025:i:c:p:388-402
    DOI: 10.1016/j.matcom.2025.06.032
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    References listed on IDEAS

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    1. Wei, H. & Pan, Q.X. & Adetoro, O.B. & Avital, E. & Yuan, Y. & Wen, P.H., 2020. "Dynamic large deformation analysis of a cantilever beam," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 174(C), pages 183-204.
    2. Lin, Ji & Reutskiy, S.Y. & Lu, Jun, 2018. "A novel meshless method for fully nonlinear advection–diffusion-reaction problems to model transfer in anisotropic media," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 459-476.
    3. Ahsan, Muhammad & Ahmad, Imtiaz & Ahmad, Masood & Hussian, Iltaf, 2019. "A numerical Haar wavelet-finite difference hybrid method for linear and non-linear Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 13-25.
    4. Tomar, Saurabh & Singh, Mehakpreet & Vajravelu, Kuppalapalle & Ramos, Higinio, 2023. "Simplifying the variational iteration method: A new approach to obtain the Lagrange multiplier," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 640-644.
    5. Stepanova, Irina V., 2019. "Group analysis of variable coefficients heat and mass transfer equations with power nonlinearity of thermal diffusivity," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 57-66.
    6. Aristizabal, Mauricio & Hernández-Estrada, José L. & Garcia, Manuel & Millwater, Harry, 2023. "Solution and sensitivity analysis of nonlinear equations using a hypercomplex-variable Newton-Raphson method," Applied Mathematics and Computation, Elsevier, vol. 451(C).
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