Multiscale Nonconforming Finite Element Computation to Small Periodic Composite Materials of Elastic Structures on Anisotropic Meshes

The small periodic elastic structures of composite materials with the multiscale asymptotic expansion and homogenized method are discussed. A nonconforming Crouzeix-Raviart finite element is applied to calculate every term of the asymptotic expansion on anisotropic meshes. The approximation scheme to the higher derivatives of the homogenized solution is also derived. Finally, the optimal error estimate in ‖ ⋅ ‖h for displacement vector is obtained.


Introduction
Because the composite materials are rapidly oscillating and the period of oscillation is very small, the calculation of the small periodic elastic structures of composite materials is rather complex.It is hard to obtain an analytical solution of most problems.The heterogeneous multiscale method (HMM) [1][2][3][4] is a general method for efficient numerical computation of problems with rapidly oscillating coefficients.
The key concepts such as resonance, fast-slow scale interactions, averaging, and techniques for transformations to nonstiff forms have been discussed in [1].There are a number of numerical examples of the HMM applied to long time integration of wave propagation problems in both periodic and nonperiodic medium found in [2].The finite difference heterogeneous multiscale method (FD-HMM) [3] is used to solve multiscale parabolic problems.The heterogeneous multiscale finite element method (HM-FEM) is used to solve the elliptic problems in perforated domains for the first time in [4].
According to the inhomogeneous anisotropic properties of the composite materials and the complexities in geometric forms, the elements of A  (/) are rapidly oscillating when 0 <  < 1.The computational resources required to solve numerically for the smallest scales are prohibitive.To solve problem (1), a two-scale asymptotic analysis method is presented in [5], where the solution can be written as the form where  = (1, 2, . . ., ) is multiple index and ⟨⟩ = 1 + 2 + ⋅ ⋅ ⋅ + .N  () = N 1⋅⋅⋅ () are 1-periodic matrix functions that can be solved in the periodic cell .These are the solutions of the following auxiliary periodic problems: where Generally, when ⟨⟩ =  ≥ 3, it can be obtained that Let u 0 () be the solution of homogenized problem in the whole domain Ω: where is the solution of (3).
As can be seen from ( 2), solving the elastic problem using the multiscale technique involves three major steps: (1) the periodic solution N 1⋅⋅⋅ () is solved in the periodic cell ; (2) the solution u 0 () about homogenized problem is available in the whole domain Ω; (3) the high order derivatives of u 0 () are calculated.
The asymptotic expansion of the solution and the associated truncation errors have been deeply investigated in [6][7][8].However, there is little reference focusing on how to calculate every term of the asymptotic expansion.
Based on [5,8], an achievable finite element computational scheme about the small periodic composite materials of elastic structures on anisotropic meshes by combining multiscale technique and finite element method is presented in this paper.A nonconforming Crouzeix-Raviart finite element is applied to estimate every term of the asymptotic expansion on anisotropic meshes [9].At the same time, the approximation scheme to the higher derivative of the homogenized solution is derived.Finally, the optimal error estimate in ‖ ⋅ ‖ ℎ for displacement vector is obtained.

Construction of the Element
Let  ℎ be an anisotropic triangle subdivision of the unit cube  with  = ⋃ ∈ ℎ .Given  ∈  ℎ , we denote the length of edges parallel to the -axis and the -axis by 2ℎ  and 2ℎ  , respectively.Given  ∈  ℎ , we note that  is a triangle element of - plan,  1 (0, 0),  2 (ℎ  , 0), and  3 (0, ℎ  ) are the three vertices, and   =    +1 ( 4 =  1 ),  = 1, 2, 3, are the three edges of .And  ℎ do not meet the regular condition; that is, ℎ  ≪ ℎ  .Let K be a reference element of - plan, the three vertices are M1 (0, 0), M2 (1, 0), and M3 (0, 1), and the three edges are l = M M+1 .There exists an inverse mapping   : K → : The finite element ( K, P, Σ) on K can be defined as For any V ∈  1 ( K), it can be easily checked that the interpolation function ÎV can be expressed as Then, the following lemma can be obtained.
Lemma 1 (see [10]).The interpolation operator shows the anisotropic characteristic.That is to say, when the multiple index || = 1, there exists a constant  which satisfies, ∀V ∈  2 ( K), Throughout this paper,  denotes a general positive constant whose value may be different at different places but remains independent of ℎ and ℎ  /  .
The associated finite element space is defined as where [V  ] denotes the jump of V  across the boundary   and The interpolation operator is defined as We note that  ℎ 0 is the nonconforming finite element space Lemma 2 (see [10]).∀V ∈  ℎ 0 , one has ‖V‖ 0 ≤ ‖V‖ ℎ .

Finite Element Approximations for the Auxiliary Periodic Problems
The equivalent variational formulation for auxiliary periodic problem (3) is to find N 1 such that where The discrete problem of variational formulation (20) consists of finding N ℎ 0 1 such that where the bilinear form  ℎ 0 and the linear function 1 are defined on elements.
It is known that (21) is the linearly elastic system [11].Using the Lax-Milgram Lemma, discrete problem (21) presents unique solution in  ℎ 0 .Theorem 5.If N 1 ∈  2 () 2×2 is the weak solution for (20) and N ℎ 0 1 is the corresponding finite element solution, one has Proof.It can be proved by the Strong Lemma and Lemma 3.
The equivalent variational formulation of auxiliary periodic problem (4) is to find N 12 such that where the linear function is We solve the following modified variational problem in practical calculation: where the corresponding linear function is Theorem 6.If N 12 and Ñ12 are the weak solutions for ( 23) and ( 25), respectively, one has Proof.One has The discrete problem of variational formulation (25) consists of finding N

Finite Element Approximate Schemes for the Homogenized Problem
Take the two-dimensional problem; for instance, we solve the following modified homogenized problem computationally: where Let  ℎ = {} be an anisotropic triangle subdivision of the region Ω, and the finite element space is still nonconforming Crouzeix-Raviart.Let ℎ = max{ℎ  } be the scale of subdivision.
The equivalent variational formulation of (33) is to find ũ0 such that where the bilinear form is The linear function is It is known that (35) is the linearly elastic system [11].Using the Lax-Milgram Lemma, variational formulation (35) presents unique solution in  ℎ .Theorem 10.If u 0 and ũ0 are the weak solutions of ( 6) and ( 35), respectively, one has Proof.It is given in Theorem 4.1 of [11].

The discrete problem of variational formulation (35) consists of finding ũ0
ℎ such that where the bilinear form  ℎ and the linear function  ℎ are defined on elements.
Proof.One has Then the theorem is obtained by the reduction of a fraction and the prior error estimation.
Theorem 12.If u 0 is the weak solution for ( 6) and ũ0 ℎ is the finite element solution for (35), one has Proof.It can be proved using Theorems 5 and 6 and the triangle inequality.
Notice that ℎ 0 and ℎ are the scales of subdivisions of  and Ω, respectively.

Higher-Order Derivatives and Error Estimation
In this section we provide the approximation formulae for all the partial derivatives of the vector function u 0 and give their error estimations.Let where (  ) shows the integral of triangle unit with side of   , (  ) shows the quantity of units in (  ), then ũ0 ℎ is solved from problem (38), and [ũ 0 ℎ /  ]  shows the partial derivative value of vector function ũ0 ℎ on the unit .Define a new interpolation function ∏ : ( 0 (Ω)) 2 →  ℎ , which satisfies In the same way, let By that analogy, let Finally, the approximate solution of the original problem is obtained as follows: The main conclusion of this paper is as follows.
Proof.One has By [5], we have The proof is completed by combining Theorems 9, 12, and 13 into (50).Table 1 shows that the optimal error estimates in  2norms and ℎ-norm for displacement vector are obtained when ℎ → 0.   (52)

Numerical Examples
The domain Ω and the periodic cell  are shown in Figures 1 and 2.