- مبلغ: ۸۶,۰۰۰ تومان
- مبلغ: ۹۱,۰۰۰ تومان
In this paper we study numerical solutions to linear elasticity equations with highly varying coefficients. Such equations typically occur when modeling the deformation of a heterogeneous material, for instance a composite material. Problems with this type of coefficients are commonly referred to as multiscale problems. The convergence of classical finite element methods based on continuous piecewise polynomials depends on (at least) the spatial H 2 -norm of the solution u. However, for problems with multiscale features this norm may be very large. Indeed, if the coefficient varies at a scale of size ϵ, then ∥u∥H2 ∼ ϵ −1 . Thus, to achieve convergence the mesh size must be small (h < ϵ). In many applications this condition leads to issues with computational cost and available memory. To overcome this difficulty several methods have been proposed, see, for instance, [1–8], and for multiscale methods particularly addressing elasticity problems, see [9–12]. However, the analysis of many of these methods relies on structural assumptions on the diffusion coefficient, such as periodicity or scale separation. In  general L∞-coefficients are considered, but the construction leads to larger localization domains.
5. Numerical experiments
In this section we perform two numerical experiments to test the convergence rate obtained in Theorem 4.1. The first experiment shows that linear convergence is obtained, in the H 1 -norm, for a problem with multiscale data. The second experiment shows that the locking effect is reduced for a problem with high value of λ. We refer to  for a discussion on how to implement this type of generalized finite elements efficiently. We consider an isotropic medium, see Remark 2.1, on the unit square in R 2 . Recall that the stress tensor in the isotropic case takes the form σ (u) = 2µε(u) + λ(∇ · u)I, where µ and λ are the Lame coefficients. For simplicity we consider only homogeneous Dirichlet boundary conditions, ´ that is, ΓD = ∂Ω and g = 0. The body forces are set to f = [1 1] | . In the first experiment, we test the convergence on two different setups for the Lame coefficients, one with ´ multiscale features, and one with constant coefficients µ = λ = 1. For the problem with multiscale features we choose µ and λ to be discontinuous on a Cartesian grid of size 2−5 . The values at the cells are chosen randomly between 0.1 and 10. The resulting coefficients are shown in Fig. 1. For the numerical approximations we discretize the domain with a uniform triangulation. The reference solution uh in (3.1) is computed using a mesh of size h = √ 2 · 2 −6 , which is small enough to resolve the multiscale coefficients in Fig. 1. The generalized finite element (GFEM) solution in (4.4) is computed on several meshes of decreasing size, H = √ 2 · 2 −1 , . . . , √ 2 · 2 −5 with k = 1, 1, 2, 2, 3, which corresponds to k = ⌈0.8 log H −1 ⌉. These solutions are compared to the reference solution. For comparison we also compute the classical piecewise linear finite element (P1- FEM) solution on the meshes of size H = √ 2 · 2 −1 , . . . , √ 2 · 2 −5 . The error is computed using the H 1 semi-norm ∥∇ · ∥ and plotted in Fig. 2.