Numerical construction of the transform of the kernel of the integral representation of the Poincaré-Steklov operator for an elastic strip. (English. Russian original) Zbl 1515.74009
Differ. Equ. 59, No. 1, 119-134 (2023); translation from Differ. Uravn. 59, No. 1, 115-129 (2023).
Summary: For an isotropic stratified elastic strip we consider the Poincaré-Steklov operator that maps normal stresses into normal displacements on part of the boundary. A new approach is proposed for constructing the transform of the kernel of the integral representation of this operator. A variational formulation of the boundary value problem for the transforms of displacements is obtained. A definition is given and the existence and uniqueness are proved for a generalized solution of the problem. An iteration method for solving variational equations is constructed, and conditions for its convergence are obtained based on the contraction mapping principle. The variational equations are approximated by the finite element method. As a result, at each step of the iteration method, it is required to solve two independent systems of linear algebraic equations, which are solved using the tridiagonal matrix algorithm. A heuristic algorithm is proposed for choosing the sequence of parameters of the iteration method that ensures its convergence. Verification of the developed computational algorithm is carried out, and recommendations on the use of adaptive finite element grids are given.
MSC:
| 74B05 | Classical linear elasticity |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
| 74G22 | Existence of solutions of equilibrium problems in solid mechanics |
| 74G30 | Uniqueness of solutions of equilibrium problems in solid mechanics |
| 74S05 | Finite element methods applied to problems in solid mechanics |
Keywords:
Fourier transform; energy functional minimization; iteration method; contraction mapping principle; existence; uniqueness; finite element methodReferences:
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