Abstract
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 for displacement vector is obtained.
1. 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) [1–4] 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].
The elastic problems of periodic composite materials can be formulated mathematically as the following boundary value problems of elliptic equations by using Einstein summation notation:where is a family of matrices. Let ; its elements can be presented as , which is 1-periodic functions of . denotes the displacement column vector and is the body force column vector.
According to the inhomogeneous anisotropic properties of the composite materials and the complexities in geometric forms, the elements of are rapidly oscillating when . 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 formwhere is multiple index and .
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 , it can be obtained that
Let be the solution of homogenized problem in the whole domain :where and 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 is solved in the periodic cell ;(2)the solution about homogenized problem is available in the whole domain ;(3)the high order derivatives of are calculated.
The asymptotic expansion of the solution and the associated truncation errors have been deeply investigated in [6–8]. 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 for displacement vector is obtained.
2. Construction of the Element
Let be an anisotropic triangle subdivision of the unit cube with . Given , we denote the length of edges parallel to the -axis and the -axis by and , respectively. Given , we note that is a triangle element of plan, , , and are the three vertices, and , , are the three edges of . And do not meet the regular condition; that is, . Let be a reference element of plan, the three vertices are , , and , and the three edges are . There exists an inverse mapping :
The finite element on can be defined as
For any , it can be easily checked that the interpolation function 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 , there exists a constant which satisfies, ,Throughout this paper, denotes a general positive constant whose value may be different at different places but remains independent of and .
The associated finite element space is defined aswhere denotes the jump of across the boundary and if . Let .
The interpolation operator is defined as :We note that is the nonconforming finite element space since . Define , , and .
Lemma 2 (see [10]). , one has .
Lemma 3. If , for any on anisotropic meshes, one has
Proof. Let ; then one has Now we show the affine equivalence property of :Therefore
Lemma 4. Let ; one has
Proof. We only prove (18).
It can be seen that and is a constant about the above nonconforming Crouzeix-Raviart triangle element. Using the interpolation condition, we have
3. Finite Element Approximations for the Auxiliary Periodic Problems
The equivalent variational formulation for auxiliary periodic problem (3) is to find such thatwhere is the bilinear form and is the linear function.
The discrete problem of variational formulation (20) consists of finding such thatwhere the bilinear form and the linear function 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 .
Theorem 5. If is the weak solution for (20) and 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 such thatwhere the linear function is We solve the following modified variational problem in practical calculation:where the corresponding linear function is
Theorem 6. If and are the weak solutions for (23) and (25), respectively, one has
Proof. One hasThe discrete problem of variational formulation (25) consists of finding such that
Theorem 7. If is the weak solution for (25) and is the finite element solution for (25), one has
Proof. It can be proved by the Strong Lemma.
Theorem 8. If is the weak solution for (23) and is the finite element solution for (25), one has
Proof. It can be proved using Theorems 6 and 7 and the triangle inequality.
Using the same procedure we also obtain the following.
Theorem 9. If is the weak solution for (5) and is the finite element solution, one has
4. Finite Element Approximate Schemes for the Homogenized Problem
Take the two-dimensional problem; for instance, we solve the following modified homogenized problem computationally:where and is the finite element solution of .
Let be an anisotropic triangle subdivision of the region , and the finite element space is still nonconforming Crouzeix-Raviart. Let be the scale of subdivision.
The equivalent variational formulation of (33) is to find such that where the bilinear form isThe 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 .
Theorem 10. If and 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 such thatwhere the bilinear form and the linear function are defined on elements.
Theorem 11. If is the weak solution for (35) and is the corresponding finite element solution, one has
Proof. One hasThen the theorem is obtained by the reduction of a fraction and the prior error estimation.
Theorem 12. If is the weak solution for (6) and 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 and are the scales of subdivisions of and , respectively.
5. Higher-Order Derivatives and Error Estimation
In this section we provide the approximation formulae for all the partial derivatives of the vector function and give their error estimations.
Letwhere shows the integral of triangle unit with side of , shows the quantity of units in , then is solved from problem (38), and shows the partial derivative value of vector function on the unit .
Define a new interpolation function , which satisfiesIn the same way, letBy that analogy, let
Theorem 13. One has
Proof. One has
Finally, the approximate solution of the original problem is obtained as follows:
The main conclusion of this paper is as follows.
Theorem 14. For , one has the error estimation
Proof. One hasBy [5], we haveThe proof is completed by combining Theorems 9, 12, and 13 into (50).
6. Numerical Examples
Example 1. Consider (1), in which , and is the identity matrix with the exact solution . We divide the domain into uniform rectangles. The errors of displacement under -norms and -norm with , , , , , and are obtained, respectively.
Table 1 shows that the optimal error estimates in -norms and -norm for displacement vector are obtained when .
Example 2. Consider the following plane stress problem:
The domain and the periodic cell are shown in Figures 1 and 2.
Let . The elements in the family of matrices () are defined asin which () are Lame constants that can be expressed by Elasticity modulus and Poisson ratio Let , , , and . The coefficient matrix family of the homogenized problem is
In two-scale asymptotic formulation (2), the homogenized solution is the main part that reflects the macroscopic response of composite materials, whereas the periodic solution and the higher derivatives of the homogenized solution are the correction part that reflect the local micromechanical behavior of composite materials. In general, only the first-order and second-order multiscale asymptotic solution produce satisfactory results in engineering.
Since it is difficult to obtain analytical solutions of (52), we take the finite element solution on the fine-mesh as the exact solution . The scale of fine-mesh for calculating the original problem is , and the scale for the homogenized problem and periodic problem is .
Table 2 shows that the computing resources can be saved significantly and the computational efficiency can be improved greatly by multiscale method in this paper compared with the fine-mesh finite element method. Particularly when is very small, the computational resources required for the finite element method scale rapidly, whereas the multiscale finite element algorithm is relatively stable.
The comparison between the true solution and the second-order multiscale asymptotic solution is shown in Figure 3.
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Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work is supported by National Natural Science Foundation of China (11402090 and 51209094) and National Basic Research Program of China (Grant 2012CB025904).