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Abstract <p>This paper discusses the features and properties of matrix-free implicit algorithms using the example of numerical solution of initial-boundary value hyperbolic and parabolic problems of dynamic and quasi-static elasticity theory. A parabolic problem is understood as a pre-parabolized formulation of a quasi-static elastic problem, which is solved by the relaxation method. The algorithm is based on a matrix-free implementation of implicit schemes based on the conjugate gradient method. Its advantage is that the implementation of the iterative process of the conjugate gradient method does not require the formation, memorization and any operations with the matrix of the implicit approximation of the system of equations. The described method has a fairly large generality and can be applied for the numerical economic solution of systems of equations of continuum mechanics with various, including non-linear, rheology using a wide class of implicit finite-difference and finite-volume, as well as the finite element schemes. It can also be used for through calculation of coupled systems of equations consisting of subsystems of different types (for example, non-stationary thermomechanics).</p>

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Using Implicit Matrix-Free Schemes for Solving Dynamic and Static Problems of Linear Elasticity

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • V. I. Golubev,
  • I. S. Nikitin,
  • A. V. Shevchenko,
  • V. S. Vaziulia,
  • I. B. Petrov

摘要

Abstract

This paper discusses the features and properties of matrix-free implicit algorithms using the example of numerical solution of initial-boundary value hyperbolic and parabolic problems of dynamic and quasi-static elasticity theory. A parabolic problem is understood as a pre-parabolized formulation of a quasi-static elastic problem, which is solved by the relaxation method. The algorithm is based on a matrix-free implementation of implicit schemes based on the conjugate gradient method. Its advantage is that the implementation of the iterative process of the conjugate gradient method does not require the formation, memorization and any operations with the matrix of the implicit approximation of the system of equations. The described method has a fairly large generality and can be applied for the numerical economic solution of systems of equations of continuum mechanics with various, including non-linear, rheology using a wide class of implicit finite-difference and finite-volume, as well as the finite element schemes. It can also be used for through calculation of coupled systems of equations consisting of subsystems of different types (for example, non-stationary thermomechanics).