Q2. How many basic cells have to be considered for the computation of the homogenized properties?
In order to take into account the randomness of the spatial distribution of the inclusions, many basic cells have to be considered for the computation of the homogenized properties.
Q3. What is the representative volume element in a periodic medium?
In the case of a periodic medium, the representative volume element is simply3a basic cell Ω, which forms the composite by spatial repetition.
Q4. What is the advantage of the eXtended Finite Element Method?
The advantage of this approach is to retain all the advantages of the finite element approach (applicability to nonlinear and anisotropic constitutive laws, wide range of codes already written, robustness, ...) while considerably easing the meshing step.
Q5. What is the way to improve the enrichment functions for material interfaces?
In this paper, the authors improve the enrichment functions for material interfaces and obtain a convergence rate very close to the one obtained with regular finite elements (i.e. conforming meshes).
Q6. What is the X-FEM approximation of a finite element?
If the finite element mesh conforms to the material interface, the (linear) approximation above yields an O(h) convergence rate in the energy norm (provided the solution is smooth) [8].
Q7. What is the main drawback of the eXtended Finite Element Method?
The main drawback of such an approach is the explicit microstructure modeling, which leads to problems for generating the mesh for complex geometries, and requires the use of sophisticated tools, see e.g. [26].
Q8. What is the rate of convergence of a mesh?
For instance, a two-phase 1D problem with a mesh non-conforming to the interfaces gives an asymptotic rate of convergence as poor as O( √ h) [16].
Q9. How can the authors solve the basic cell problem for composites with periodic microstructure?
The formulation of the basic cell problem for composites with periodic microstructure can be derived in a systematic way using the two-scale asymptotic expansion method [3], [22], or following a process which is also valid for random media [25], [12].
Q10. What is the microstructure description for periodic materials?
The microstructure description is achieved through the definition of a representative volume element, containing all the geometrical and material data.
Q11. What is the X-FEM approach to solve the basic cell problem?
At the macroscopic scale, stress and strain fields are denoted by Σ and E, which are the average of the corresponding microscopic field, σ and e, on the basic cell (the macroscopic and microscopic scales are denoted by x and y, respectively):Σ(x) =< σ(x, y) >= 1 |Ω| ∫ Ω σ(x, y)dΩE(x) =< e(x, y) >
Q12. What is the method for analyzing structures with complex geometry?
The eXtended Finite Element Method simplifies greatly the analysis of structures with complex geometry since the mesh is not required to match this geometry.
Q13. What are the advantages of the eXtended Finite Element Method in micromechanical analysis?
The advantages of the FEM in micromechanical analysis are indeed the same as in standard engineering problems: its flexibility, and its applicability to nonlinear problems, anisotropic materials, and arbitrary geometries.
Q14. What is the way to solve the basic cell problem?
As the authors shall see in the numerical experiments, this strategy yields a reasonable rate of convergence for the homogenized parameters of the basic cell but suffers a slow rate of convergence for the quality of the overall stress distribution over the basic cell (energy norm in the stresses).
Q15. What is the difference between the basic cell and the boundary value problem?
Since the basic cell problem is a boundary value problem, classical numerical methods can be used for computing their solution, see [4] for a state of the art in this matter.