A mixed parameter formulation with applications to linear viscoelastic slender structures. (English) Zbl 1541.65099
Summary: We present the analysis of an abstract parameter-dependent mixed variational formulation based on Volterra integrals of second kind. Adapting the classic mixed theory in the Volterra equations setting, we prove the well posedness of the resulting system. Stability and error estimates are derived, where all the estimates are uniform with respect to the perturbation parameter. We provide applications of the developed analysis for a viscoelastic Timoshenko beam and report numerical tests for this problem. We also comment, numerically, the performance of a viscoelastic Reissner-Mindlin plate.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 45D05 | Volterra integral equations |
| 65R20 | Numerical methods for integral equations |
| 35A15 | Variational methods applied to PDEs |
| 76A10 | Viscoelastic fluids |
| 74K20 | Plates |
| 74B10 | Linear elasticity with initial stresses |
| 74D05 | Linear constitutive equations for materials with memory |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
Software:
FEniCSReferences:
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