Numerical method for solving of ill-conditioned boundary value problems in nonlinear elasticity

The article discusses the boundary value problem (BVP) of nonlinear elasticity for mapping (deformation) in two weak variational forms: as the static equilibrium equation and as the minimization problem for energy. Some questions of a mathematical correctness of the nonlinear elasticity BVP are disc...

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Bibliographic Details
Published in:Acta mechanica Vol. 234; no. 3; pp. 1293 - 1303
Main Author: Brigadnov, Igor A.
Format: Journal Article
Language:English
Published: Vienna Springer Vienna 01.03.2023
Springer
Springer Nature B.V
Subjects:
Boundary value problems
Classical and Continuum Physics
Control
Decomposition
Deformation
Discontinuity
Dynamical Systems
Elasticity
Engineering
Engineering Fluid Dynamics
Engineering Thermodynamics
Equilibrium equations
Heat and Mass Transfer
Heterogeneity
Iterative methods
Nonlinear systems
Numerical methods
Optimization
Solid Mechanics
Static equilibrium
Theoretical and Applied Mechanics
Vibration
74S05
74B20
65L03
ISSN:0001-5970, 1619-6937
Online Access:Get full text
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Summary:The article discusses the boundary value problem (BVP) of nonlinear elasticity for mapping (deformation) in two weak variational forms: as the static equilibrium equation and as the minimization problem for energy. Some questions of a mathematical correctness of the nonlinear elasticity BVP are discussed. Two simple examples prove that for some nonlinear elastic models the appropriate BVP either can have not a solution, or can have a discontinuous solution with discontinuities of sliding type. Both effects correspond to ill-conditioned BVP. Using finite element approximation the initial variational problems are transformed into a nonlinear system of algebraic equations and a finite-dimensional minimization problem, respectively. These problems can be ill-conditioned. The phenomenon is observed in problems where deformation has a large gradient heterogeneity. For numerically solving the original decomposition method of adaptive block relaxation is proposed. The main idea of this method consists of the iterative improvement in zones with “proportional” deformation by special decomposition of domain (variables) and separate calculation in these zones (on these variables). The numerical results show that for finding the deformed configuration this technique has obvious benefits over the standard methods.
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ISSN:0001-5970
1619-6937
DOI:10.1007/s00707-022-03441-x