Główne pojęcia
Mixed finite element methods provide stable and convergent solutions for linear Cosserat equations.
Streszczenie
The content discusses the equilibrium equations for linearized Cosserat materials, introducing differential complexes and mixed finite element methods. It covers theoretical results, numerical verifications, and convergence rates. The analysis includes well-posedness proofs, relationship to linear elasticity, and discrete approximations. Key concepts involve Hodge-Laplace operators, bounded cochain projections, and strongly coupled MFE spaces.
Introduction:
Extends linearized elasticity equations.
Focuses on complex material modeling.
Discusses numerical approximation methods.
Notation and Problem Definition:
Defines differential operators.
Introduces weighted inner product.
Establishes boundary conditions.
Analysis and MFE Discretization:
Constructs a Cosserat complex.
Identifies isomorphism with de Rham complex.
Presents well-posedness theorems.
Strongly Coupled Mixed Finite Element Approximation:
Defines strongly coupled spaces.
Introduces bounded cochain projections.
States optimal approximation properties.
Theorem - Optimal Approximation:
Ensures unique solvability.
Guarantees standard approximation rates.
Theorem - Strongly Coupled MFE for Cosserat:
Defines finite element spaces.
Establishes bounded cochain projections.
Demonstrates convergence rates.