Kernkonzepte
This paper presents a comparative study of mixed finite element and two-point stress approximation finite volume methods for the numerical simulation of linearized elasticity and Cosserat materials. The methods are assessed for their accuracy, robustness, and computational efficiency.
Zusammenfassung
The paper presents and compares three numerical methods for solving the equations of linearized elasticity and Cosserat materials:
Mixed Finite Element Method (MFEM):
Formulated as a four-field method for Cosserat materials, with variables for displacement, rotation, Cauchy stress, and couple stress.
Shown to be stable and convergent with optimal rates on shape-regular, simplicial grids.
Leads to a relatively large saddle-point system that can be computationally demanding to solve.
Two-Point Stress Approximation Finite Volume Method (TPSA):
Formulated using a minimal stencil, with primary variables for displacement, rotation, and solid pressure.
Shown to be robust in the incompressible and Cauchy limits.
Simpler to implement and applicable to a wide range of grid types, but convergence rates depend on grid quality.
Multi-Point Stress Approximation Finite Volume Method (MPSA):
Employs a larger stencil compared to TPSA, with only displacement as the primary variable.
Shown to be robust for nearly incompressible materials, but can suffer in the extreme incompressible limit.
Requires a more complex assembly process involving local static condensation of stress degrees of freedom.
The performance of the three methods is assessed through numerical experiments on three test cases:
Homogeneous linear elastic medium, testing robustness in the incompressible limit.
Heterogeneous linear elastic medium, testing robustness to material discontinuities.
Composite Cosserat/elastic material, testing the methods on Cosserat equations.
The results demonstrate that all three methods exhibit good accuracy and robustness, with the MFEM and TPSA methods being the most stable across the different parameter regimes and material heterogeneities considered.
Statistiken
The paper does not provide any specific numerical data or statistics to support the key arguments. The results are presented in the form of convergence plots comparing the error norms of the different methods.