Numerical solution of the scalar double-well problem allowing microstructure

DOI10.1090/S0025-5718-97-00849-1zbMath0870.65055OpenAlexW2054535420MaRDI QIDQ4340853

Petr Plecháč, Carsten Carstensen

Publication date: 15 June 1997

Published in: Mathematics of Computation (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1090/s0025-5718-97-00849-1

zbMATH Keywords

microstructureYoung measuresnumerical experimentsfinite elementrapid oscillationsdouble-well problemnon-convex minimisationautomatic mesh refinements

Mathematics Subject Classification ID

Numerical optimization and variational techniques (65K10) Existence theories for optimal control problems involving ordinary differential equations (49J15) Existence theories for optimal control problems involving partial differential equations (49J20)

Related Items (42)

NUMERICAL SOLUTION OF A CLASS OF NON-CONVEX VARIATIONAL PROBLEMS BY SQP ⋮ Convergence for stabilisation of degenerately convex minimisation problems ⋮ A BV functional and its relaxation for joint motion estimation and image sequence recovery ⋮ On optimality conditions of relaxed non-convex variational problems ⋮ Modeling of Microstructures in a Cosserat Continuum Using Relaxed Energies: Analytical and Numerical Aspects ⋮ Adaptive Least-Squares, Discontinuous Petrov-Galerkin, and Hybrid High-Order Methods ⋮ Numerical Algorithms for the Simulation of Finite Plasticity with Microstructures ⋮ Approximations of Energy Minimization in Cell-Induced Phase Transitions of Fibrous Biomaterials: $\Gamma$-Convergence Analysis ⋮ Young-Measure approximations for elastodynamics with non-monotone stress-strain relations ⋮ Linear convergence in the approximation of rank-one convex envelopes ⋮ Relaxation of a model energy for the cubic to tetragonal phase transformation in two dimensions ⋮ Laminated microstructure in a variational problem with a non-rank-one connected double well potential ⋮ Adaptive algorithms for scalar non-convex variational problems ⋮ Convergent adaptive hybrid higher-order schemes for convex minimization ⋮ Stabilized FEM for some optimal design problem ⋮ Analytical and numerical relaxation results for models in soil mechanics ⋮ A two-scale finite element relaxation analysis of shear bands in non-convex inelastic solids: Small-strain theory for standard dissipative materials. ⋮ Analysis of microstructures and phase transition phenomena in one-dimensional, non-linear elasticity by convex optimization ⋮ A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials. ⋮ Material instabilities in nematic elastomers ⋮ Error estimation and adaptive meshing in strongly nonlinear dynamic problems ⋮ An averaging scheme for macroscopic numerical simulation of nonconvex minimization problems ⋮ Global stress regularity of convex and some nonconvex variational problems ⋮ Deformation patterning in finite-strain crystal plasticity by spectral homogenization with application to magnesium ⋮ A convergent adaptive finite element method for an optimal design problem ⋮ Optimal laminates in single-slip elastoplasticity ⋮ A comparison of classical and new finite element methods for the computation of laminate microstructure ⋮ Nonconforming finite element approximation of crystalline microstructure ⋮ Linear-programming approach to nonconvex variational problems ⋮ Coarse-Convex-Compactification Approach to Numerical Solution of Nonconvex Variational Problems ⋮ On the Numerical Modeling of Deformations of Pressurized Martensitic Thin Films ⋮ Numerical Analysis of a Relaxed Variational Model of Hysteresis in Two-Phase Solids ⋮ Ten remarks on nonconvex minimisation for phase transition simulations ⋮ Computational Microstructures in Phase Transition Solids and Finite-Strain Elastoplasticity ⋮ Unstabilized Hybrid High-order Method for a Class of Degenerate Convex Minimization Problems ⋮ Stability of microstructure for tetragonal to monoclinic martensitic transformations ⋮ Effective relaxation for microstructure simulations: algorithms and applications ⋮ Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity ⋮ Nonconforming FEMs for an Optimal Design Problem ⋮ Convergence of adaptive finite element methods for a nonconvex double-well minimization problem ⋮ Application of a partially relaxed shape memory free energy function to estimate the phase diagram and predict global microstructure evolution ⋮ Stability of microstructures for some martensitic transformations

Uses Software

• tn

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Numerical analysis of oscillations in nonconvex problems
• Numerical study of a relaxed variational problem from optimal design
• Fine phase mixtures as minimizers of energy
• On the anti-plane shear problem in finite elasticity
• Sets which are well-distributed and invariant relative to all isometry invariant total extensions of Lebesgue measure
• Characterizations of Young measures generated by gradients
• A nonconvex variational problem related to change of phase
• Jensen's inequality in the calculus of variations
• Newton-Type Minimization via the Lanczos Method
• On the Convergence of Finite-Element Approximations of a Relaxed Variational Problem
• Optimal-Order Error Estimates for the Finite Element Approximation of the Solution of a Nonconvex Variational Problem
• Weak Convergence of Integrands and the Young Measure Representation
• Proposed experimental tests of a theory of fine microstructure and the two-well problem
• Numerical Approximations in Variational Problems with Potential Wells
• A Posteriori Error Estimates for Nonlinear Problems. Finite Element Discretizations of Elliptic Equations
• Approximated Convex Envelope of a Function
• A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems
• Strong Convergence of Numerical Solutions to Degenerate Variational Problems
• Direct methods in the calculus of variations

This page was built for publication: Numerical solution of the scalar double-well problem allowing microstructure