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Vectorized MATLAB Implementation of the Incremental Minimization Principle for Rate-Independent Dissipative Solids Using FEM: A Constitutive Model of Shape Memory Alloys

Author

Listed:
  • Miroslav Frost

    (Institute of Thermomechanics, Czech Academy of Sciences, Dolejškova 5, CZ-18200 Prague, Czech Republic)

  • Jan Valdman

    (Institute of Information Theory and Automation, Czech Academy of Sciences, Pod Vodárenskou věží 4, CZ-18200 Prague, Czech Republic
    Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, CZ-16000 Prague, Czech Republic)

Abstract

The incremental energy minimization principle provides a compact variational formulation for evolutionary boundary problems based on constitutive models of rate-independent dissipative solids. In this work, we develop and implement a versatile computational tool for the resolution of these problems via the finite element method (FEM). The implementation is coded in the MATLAB programming language and benefits from vector operations, allowing all local energy contributions to be evaluated over all degrees of freedom at once. The monolithic solution scheme combined with gradient-based optimization methods is applied to the inherently nonlinear, non-smooth convex minimization problem. An advanced constitutive model for shape memory alloys, which features a strongly coupled rate-independent dissipation function and several constraints on internal variables, is implemented as a benchmark example. Numerical simulations demonstrate the capabilities of the computational tool, which is suited for the rapid development and testing of advanced constitutive laws of rate-independent dissipative solids.

Suggested Citation

  • Miroslav Frost & Jan Valdman, 2022. "Vectorized MATLAB Implementation of the Incremental Minimization Principle for Rate-Independent Dissipative Solids Using FEM: A Constitutive Model of Shape Memory Alloys," Mathematics, MDPI, vol. 10(23), pages 1-17, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:23:p:4412-:d:981296
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    References listed on IDEAS

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    1. Moskovka, Alexej & Valdman, Jan, 2022. "Fast MATLAB evaluation of nonlinear energies using FEM in 2D and 3D: Nodal elements," Applied Mathematics and Computation, Elsevier, vol. 424(C).
    2. Anjam, I. & Valdman, J., 2015. "Fast MATLAB assembly of FEM matrices in 2D and 3D: Edge elements," Applied Mathematics and Computation, Elsevier, vol. 267(C), pages 252-263.
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