Sa

Sa

b

c

d

a

ARRAA

1

fpa2rbs(tfctctt

iKAoO

R

0h

Mechanics Research Communications 52 (2013) 11– 18

Contents lists available at SciVerse ScienceDirect

Mechanics Research Communications

j o ur na l ho me pag e: www.elsev ier .com/ locate /mechrescom

ystematic construction of higher order bases for the finite elementnalysis of multiscale elliptic problems

oheil Soghrati a,d, Ilinca Stanciulescub,c,∗

Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USADepartment of Civil and Environmental Engineering, Rice University, 6100 Main Street, Houston, TX 77005, USADepartment of Mechanical Engineering and Materials Science, Rice University, 6100 Main Street, Houston, TX 77005, USADepartment of Mechanical and Aerospace Engineering, The Ohio State University, 201 W 19th Ave., Columbus, OH 43210, USA

r t i c l e i n f o

rticle history:eceived 11 May 2011

a b s t r a c t

We introduce a new approach to deriving higher order basis functions implemented in the MultiscaleFinite Element Method (MsFEM) for elliptic problems. MsFEM relies on capturing small scale features of

eceived in revised form 11 May 2013ccepted 7 June 2013vailable online 19 June 2013

the system through bases utilized in the coarse scale solution. The proposed technique for the derivation ofsuch bases is completely systematic and the increase in the associated computational cost is insignificant.We also show that the implementation of higher order bases in MsFEM leads to similar advantagesas using higher order Lagrangian shape functions in the conventional finite element method. Variousnumerical examples for heat transfer problems with periodic or heterogeneous thermal properties aregiven to demonstrate the efficiency and improved characteristics of the proposed higher order bases.

. Introduction

Second order elliptic differential equations with multiple-scaleeatures appear in a variety of materials science and engineeringroblems, including heat conduction in heterogeneous materialsnd flow in porous media (Gonthier and Jogi, 2005; Corsini et al.,009). In such phenomena, variations and fluctuations of mate-ial properties at the fine scale have a significant impact on theehavior of the system at the coarse level. Obtaining an accurateolution for such problems via the standard finite element methodFEM) requires meshes that resolve the smallest scale of the sys-em, which often leads to extremely large numbers of degrees ofreedom. Hence, the large size of the resulting discrete problemonsiderably increases the computational cost, which, at best, leadso long running times, but in many cases exceeds available memoryapacities. While the parallel implementation alleviates the limita-ions in the computational resources, it cannot reduce the size ofhe discrete problem.

To address the problems discussed above, several techniquesncluding those based on the homogenization theory (Dykaar anditanidis, 1992) and upscaling methods (Luce and Perez, 2002;

rbogast, 2004) have emerged to determine the effective propertiesf the system and incorporate them in the coarse-scale simulation.ther approaches presented in the literature aim at developing

∗ Corresponding author at: Department of Civil and Environmental Engineering,ice University, 6100 Main Street, Houston, TX 77005, USA. Tel.: +1 7133484704.

E-mail address: ilinca@rice.edu (I. Stanciulescu).

093-6413/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.mechrescom.2013.06.002

© 2013 Elsevier Ltd. All rights reserved.

multiscale analysis methods by incorporating the fine scale infor-mation in the coarse scale solution. Hughes (1995) and Hugheset al. (1998) proposed a variational multiscale method (VMM) inwhich the solution is evaluated at two steps: the analytical solu-tion of the fine scale and its implementation for solving the coarsescale problem. Babuska et al. (1994) developed a generalized finiteelement method with special non-polynomial basis functions forapproximating second-order elliptical problems. In the heteroge-neous multiscale method (HMM), introduced by Engquist and Tsai(2005) and Engquist and Huang (2003), different scales of thesystem are separated through solving the macro-problem and eval-uating the missing information by performing local analyses at finerscales.

The Multiscale Finite Element Method (MsFEM) is among themost successful approaches for solving elliptic problems with oscil-latory coefficients of governing equations (Hou and Wu, 1997; Houet al., 1999; Efendiev and Wu, 2002). The main idea in this methodis to transfer the effect of the fine scale properties to the coarse levelby replacing the Lagrangian shape functions with appropriate basesthat contain this information. A great advantage of the method isthat it decouples the calculation of the basis functions in differ-ent elements, which suites parallel analysis implementation. Oneof the major issues in MsFEM, however, is the slow rate of conver-gence due to the resonance between the grid size and the physicalscale of the problem. This leads to a mismatch between the fine and

coarse scale solutions over element boundaries. To reduce the res-onance error, an oversampling technique is proposed in Hou andWu (1997), where the sampling is performed over a larger domainand interior data are used to evaluate the basis functions. Ming and

1 s Rese

Ysp

Mbntsiaabthci

wotpbbtoee

2

�acnts

T∫

wHEorthbwNf

∇srnf

2 S. Soghrati, I. Stanciulescu / Mechanic

ue (2006) presents a comprehensive overview on recent multi-cale methods including MsFEM and VMM for second-order ellipticroblems.

The algorithm presented in Hou and Wu (1997) for evaluatingsFEM basis functions is limited to modifying standard linear or

ilinear shape functions. The extension to higher order bases is theext step, where among previous related efforts we can mentionhe work by Allaire and Brizzi (2005). However, the method pre-ented in Allaire and Brizzi (2005) for evaluating higher order basess mathematically complex and the desired results are achieved at

higher computational cost. In the current paper, we introduce simpler and yet general algorithm for constructing higher orderases that is applicable to elements with any desired configura-ion and number of nodes. We also show that while the proposedigher order bases considerably improve the accuracy and rate ofonvergence, such improvements are obtained without significantncrease in the computational cost.

The remainder of the paper is organized as follows. In Section 2,e present the elliptic multiscale problem and briefly explain the

riginal MsFEM formulation in Hou and Wu (1997). In Section 3,he proposed derivation of higher order MsFEM basis functions isresented. We first introduce a systematic approach for developingasis functions for 1D elements and then show how the resulting1Dases can be employed for constructing 2D basis functions. In Sec-ion 4, we analyze the computational cost, accuracy and the ratef convergence of these elements by studying several heat transferxample problems with periodic or heterogeneous thermal prop-rties.

. Second order elliptic problems and MsFEM

Consider an open domain ̋ ⊂ Rn and its boundary

= ̋ \ ̋ = �u ∪ �q with outward unit normal n. Let g : �u → R

nd h : �q → R define the Dirichlet and Neumann boundaryonditions for this system, respectively. Given the positive defi-ite tensor a(x) : �q → R

n × Rn (which, for instance, consists of

hermal conductivity coefficients for heat transfer problems), theecond-order elliptic equation studied in this work is expressed as

−∇ · (a(x)∇u) = f in ˝,

u = g on �u,

a(x)∇u · n = h on �q.

(2.1)

he weak form of Eq. (2.1) is written as: Find u ∈ U such that

˝

a(x)∇u · ∇v d ̋ =∫

˝

fv d ̋ +∫

�q

gv d� ∀v ∈ V, (2.2)

here U = {u(x), u|�u = g} ⊂ H1(˝) and V = {v(x), v|�u = 0} ⊂10(˝) are the solution and weight function spaces, respectively.q. (2.2) can be used to derive the Galerkin approximation andbtain the associated FEM formulation. However, an FEM solutionequires highly refined meshes to resolve the smallest scale ofhe problem, which can be extremely expensive if a(x) is veryeterogeneous or highly oscillatory. MsFEM addresses this issuey evaluating basis functions that capture the small scale effectsithin a coarse finite element mesh. To evaluate basis function(k)i

corresponding to element ˝k, we solve the homogeneousorm of Eq. (2.1),

· (a(x)∇N(k)i

) = 0 in ˝k ⊂ ˝, (2.3)

ubject to appropriate boundary conditions. This requires that theesulting bases satisfy the Kronecker delta property at the elementodes N(k)

i(xj) = ıij . Determining appropriate boundary conditions

or Eq. (2.3) is not trivial; choices made here significantly affect the

arch Communications 52 (2013) 11– 18

accuracy of the MsFEM solution. Two choices for the boundary con-ditions (for four node 2D elements) are proposed in Hou and Wu(1997). The first choice is to consider 1D Lagrangian shape func-tions, which are similar to boundaries of standard bilinear shapefunctions. This choice does not lead to satisfactory results becauseit neglects the oscillations in the fine scale over the boundaries ofthe coarse mesh. A second option is proposed, where the effectof oscillations in the fine scale properties is included by obtainingthe solution N�

iof the reduced form of Eq. (2.3) over each bound-

ary of the course element: d/ds(a(s)dN�i /ds) = 0 on � k

i= ∂˝k,

subject to N�i

(xj) = ıij and with s being the coordinate along � ki

.In the special case where the conductivity tensor is a separablefunction, a(x) = a1(x)a2(y), the 2D basis functions can be computeddirectly from the tensor product of the boundary condition func-tions. The interested reader can find a detailed description of theMsFEM, including the estimation of the resonance error in Hou andWu (1997) and Hou et al. (1999).

3. Extension to higher order elements

In this section, we introduce a general approach for evaluatingthe MsFEM basis functions for higher order elements with differentnumber of nodes. We aim to preserve some of the properties of theLagrangian shape functions in the higher order MsFEM bases. Theapproach we propose not only avoids heavy mathematical deriva-tion and the use of homogenization theory in computing the MsFEMbases, but also provides more insight over the features of thesebases. We follow a direct extension of the procedure explained inSection 2 by solving an FEM problem over the coarse elements sub-ject to appropriate boundary conditions. The boundary conditionsassociated with Eq. (2.3) determine how well the characteristics ofthe fine scale are captured in the basis and thus transferred to thecoarse-scale solution. In order to achieve an acceptable accuracy,we first need to incorporate the fine scale information in the bound-ary conditions of higher order bases. To achieve this goal, we firstdevise an approach for evaluating higher order bases for 1D prob-lems, and then employ these 1D-bases as boundary conditions forthe 2D basis functions.

3.1. Higher order bases for 1D problems

Consider the 1D elliptic problem

− d

dx

(a(x)

du

dx

)= f ∀x ∈ [0, L]. (3.1)

Constructing 2-node bases for the MsFEM solution of this problemusing the method outlined in Section 2 is quite straightforward:local 2-node basis functions are obtained form the solutions ofd/d�(a(�)du/d�) = 0 for all � ∈ [−1, 1] in the local coordinate sys-tem and subject to Nk

i(�j) = ıij boundary conditions. In order to

extend this algorithm to higher order bases, we require that thebasis functions are C0-continuous and satisfy the partition of unity,i.e.,

∑ni=1Nk

i= 1.

Before introducing an algorithm for creating higher order 1Dbasis functions, we study some of the important characteristics ofstandard Lagrangian shape functions to mimic similar properties inthe higher order bases. Unlike linear shape functions, higher orderLagrangian shape functions do not satisfy the homogeneous formof a second order elliptic equation. Similarly, we see no reason torequire higher order multiscale bases to satisfy this equation. The2nd order Lagrangian shape functions are constructed through the

linear combination of 1st order Lagrangian shape functions and aparabolic function. This parabolic function can be the solution of aparticular instance of Eq. (3.1), a0

d2ud�2 = 1, subject to zero essen-

tial boundary conditions. Similarly, multiscale bases for 3-node

S. Soghrati, I. Stanciulescu / Mechanics Rese

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

0.5

1

ξ

N1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

0.5

1

ξ

N2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

0.5

1

N3

e

ofne3

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩Idctsoct

3ba

fC

ξ

Fig. 1. Multiscale (solid lines) and Lagrangian (dotted lines) shape functions.

lements, N(3)i

, can be constructed through a linear combination

f 2-node basis functions, N(2)e , and the solution Np of Eq. (3.1) with

= 1, subject to zero essential boundary conditions. Using e and iotations to refer to degrees of freedom of 2-node and higher orderlements, respectively, we can express the higher order bases for-node elements as N(3)

i= ıeiN

(2)e + ˛iNp:

N(3)1 = N(2)

1 + ˛1Np → ˛1 = −(N(2)1 /Np)|�=0

N(3)2 = ˛2Np → ˛2 = (1/Np)|�=0

N(3)3 = N(2)

2 + ˛3Np → ˛3 = −(N(2)2 /Np)|�=0,

(3.2)

n the expression above, ˛i are determined such that Kroneckerelta condition is satisfied at the mid-point, i.e., N(3)

i|�j

= ıij . Fig. 1ompares second order standard Lagrangian shape functions withhe multiscale basis functions evaluated for the conductivity ten-or a(x) = 2000(1.3 + sin 400�x) (1.3 + cos(400�x + 2�/5)), with fivescillations per coarse element. Note that the resulted functionsonverge to standard Lagrangian shape functions as oscillations inhe fine scale properties are damped out.

The proposed approach for evaluating the shape functions of-node elements can be easily extended to even higher orderases. In general, 1D bases for an n-node element are expressed

s N(n)

i= ıeiN

(2)e +

∑n−2j=1 ˛(j)

iN(j)

p , where functions N(j)p are evaluated

rom solving Eq. (3.1) with the right-hand-side function f(j)(�) = �j−1.oefficients ˛(j)

iare then computed such that the resulting basis

Fig. 2. Boundary conditions and resulting MsFE

= - 0.5

Fig. 3. Construction of MsFEM basis functions

arch Communications 52 (2013) 11– 18 13

functions satisfy the Kronecker delta property at the interior nodes,i.e., N(n)

i(�k) = ıik.

3.2. Higher order bases for 2D problems

The extension of the algorithm described in Section 2 to higherorder 2D bases is not difficult provided that good choices are madefor boundary conditions of coarse elements. The approach adoptedhere is to solve Eq. (2.1) over a coarse element with 1D basis func-tions corresponding to the number of nodes along edges of theelement edges as the boundary conditions. The resulting solutioncan be then used as basis functions for elements with nodes dis-tributed only over the boundaries, i.e., serendipity-type elements.The basis functions calculated with this method for an 8-nodequadrilateral element are illustrated in Fig. 2, where bases of 3-node1D elements define the boundary conditions along element’s edges.

To construct basis functions for elements with interior nodes,similar to the approach discussed for 1D elements, we first solve Eq.(2.1) in each coarse element subject to zero boundary conditionsand appropriate right-hand-side polynomial functions. In general,2D bases of a standard n-node quadrilateral element with m interior

nodes are given by N(n)i

= ıeiN(n−m)e +

∑√m

j=1

∑√m

k=1˛(j,k)i

N(j,k)p , where

N(n−m)e are the basis functions of the corresponding serendipity-

type element with n − m nodes and subscript e refers to the degreesof freedom in this element. N(j,k)

p are evaluated from solving Eq.(2.1) with the right-hand-side function f(j,k)(�, �) = �j−1�k−1. Coeffi-cient ˛(j,k)

icorresponding to the ith basis function is determined

by satisfying the Kronecker delta property at element’s nodes,i.e., N(n)

i(�, �)p = ıip. Following this algorithm, basis functions for

a standard 9-node quadrilateral element, for example, is writ-ten as N(9)

e = ıeiN(8)e + ˛iNp, where Np is obtained from solving

− ∇ · (a(�) ∇ Np) = 1 in ˝k and subject to Np = 0 on ∂˝k as boundaryconditions.

If the conductivity tensor is composed of separable functions inthe Euclidian space, i.e., a(x) = a1(x)a2(y), we can use a computa-tionally less expensive method: 2D shape functions are obtainedfrom the tensor product of 1D bases evaluated over the bound-aries of the coarse element (Fig. 3.) Note that the method discussedearlier for evaluating the bases from the numerical solution of Eq.(2.1) and the current approach yield different basis functions. How-ever, the results presented in Section 4 will demonstrate that bothschemes yield similar level of accuracy in problems with separableconductivity tensors.

4. Numerical experiments

In this section, we examine the performance of higher orderMsFEM bases by studying variations of the l2 and L∞ norms of the

M basis functions for an 8-node element.

- 0.5

through the tensor product of 1D bases.

14 S. Soghrati, I. Stanciulescu / Mechanics Research Communications 52 (2013) 11– 18

Table 1MsFEM results (Example 1).

1st order 2nd order

Nf ||E||l2 Nf ||E||l2100 9.163 × 10−4 10 8.723 × 10−3

200 2.287 × 10−4 20 6.218 × 10−4

300 1.012 × 10−4 30 9.914 × 10−5

400 5.660 × 10−5 40 3.179 × 10−5

500 3.589 × 10−5 50 1.266 × 10−5

−5 −6

elfi

suNFbMd∀sdirt

csusTotitmoFseFon

lfolFisestm

filst

Table 2Two- and three-scale MsFEM solutions (Example 1).

Two-scale solution Three-scale solution Nodes

Nf ||E||l2 Nf1 Nf2 ||E||l2100 9.163 × 10−4 10 5 5.219 × 10−4 10,000200 2.287 × 10−4 20 5 8.220 × 10−5 20,000300 1.012 × 10−4 30 5 5.012 × 10−5 30,000400 5.660 × 10−5 40 5 2.815 × 10−5 40,000500 3.589 × 10−5 25 10 9.411 × 10−6 50,000600 2.467 × 10−5 30 10 7.215 × 10−6 60,000

−5 −6

the error in this case is approximately 10 times larger than theerror obtained with 9-node basis functions (Table 3).

600 2.467 × 10 60 5.692 × 10700 1.789 × 10−5 70 2.689 × 10−6

800 1.350 × 10−5 80 1.226 × 10−6

rror in several numerical examples. Note that the values of the2 norm of the error reported below are computed at nodes of thene-scale mesh to incorporate the sub-grid effect on the error.

Example 1 (1D problem): For 1D problems, MsFEM yieldsimilar results as the standard FEM, provided that the prod-ct of the number of elements in the fine and course meshes,f × Nc, is equal to the number of elements used in the standardEM solution. The current example sheds more light on theehavior of these bases and offers additional insight into thesFEM algorithm. For this reason, we study the problem

/dx((1.3 + sin(2�x/ε))(1.3 + cos(2�x/ε + 2�/5)) · du/dx) = −1000, x ∈ [0, L], where u(0) = u(L) = 0, and ε = 0.025 characterizes themall scale properties (40 oscillations per unit length of theomain). The length of the domain is L = 5 and the coarse mesh

s composed of Nc = 100 nodes. A finite element solution using aefined mesh with 2 × 105 second-order elements is adopted ashe reference solution for this problem.

Table 1 shows the performance of the MsFEM solution. The twoases correspond to utilizing linear and second-order Lagrangianhape functions for evaluating the fine scale solution. The val-es of the l2-norm of the error in this problem are exactly theame for all coarse scale elements (2, 3 or 4-node) presented inable 1. At the first glance, this might indicate a low performancef higher order bases as despite increasing the computational cost,he accuracy is not improved. However, this behavior can be eas-ly explained. MsFEM is the generalization of the standard FEMhrough modifying shape functions to include the small scale infor-

ation. Accordingly, the behavior of MsFEM bases is similar to thatf Lagrangian shape functions of the same order in the standardEM for the same problem but with a constant conductivity ten-or. Employing any Lagrangian shape functions for solving a 1Dlliptic problem with constant conductivity tensor with standardEM yields super-convergent results at the nodes. Similarly, higherrder bases of MsFEM yield the same nodal solution at the commonodes of coarse meshes used to discretize the domain.

The main advantage of higher order 1D bases appears in prob-ems with more than two scales. The order of Lagrangian shapeunctions used in the solution at the small scale has a crucial effectn the accuracy. The reason can be traced back to the effect of oscil-ations in the small scale properties, which prevents the standardEM from yielding super-convergent results. In this case, as shownn Table 1, using 2nd order Lagrangian shape functions for theolution at the small scale yields a better accuracy than the lin-ar functions for comparable computational costs. For instance, 50econd-order elements in the small scale provide better accuracyhan 800 linear elements, while the size of the resulted stiffness

atrix is reduced by a factor of 8.The same characteristic of the standard FEM can be used to

urther reduce the size of the discrete model in the MsFEM. For

nstance, one can solve the current problem in three scales, usinginear shape functions in the finest scale, 3-node bases in the smallcale and 2-node bases in the coarse mesh. In this case, we expecto achieve more accurate results compared to the solution obtained

700 1.789 × 10 35 10 5.076 × 10 70,000800 1.350 × 10−5 40 10 3.128 × 10−6 80,000

if only two scales with linear and modified linear shape functionsare used. Table 2 confirms this expectation: while the total numberof degrees of freedom is kept constant in each row, the three-scalesolution described above yields significantly more accurate results.Note that using three scales for solving this problem is only a choicefor the numerical solution and does not correspond to the physicalscales of the problem.

Example 2 (periodic separable conductivity tensor): In thisexample problem, we solve Eq. (2.1) for a(x) = (1.3 + sin 2�x/ε)(1.3 + sin 2�y/ε) (1.3 + cos(2�x/ε + 2�/5)) (1.3 + cos(2�y/ε + 2�/5))and f = −1000 over a 5 × 5 square domain with zero essential bound-ary condition and ε = 1/12, which corresponds to 60 oscillations ineach direction over the domain (Fig. 4). A finite element mesh with8.1 × 105 second-order elements is employed to evaluate the refer-ence solution in the current and following example problems. TheMsFEM solutions using 4 and 9-node elements (using the tensorproduct approach) are reported in Table 3 and the convergence inl2-norm of the error is illustrated in Figs. 5–6. Note that for clar-ity, the range of error in the two graphs differs by a factor of 10.The accuracy of the MsFEM solution is limited by the size of thecoarse mesh, which is independent of the resolution of the dis-cretization in the fine scale. Using 10 × 10 or 20 × 20 coarse mesheswith modified bilinear shape functions, i.e., 4-node elements, doesnot provide results with a satisfactory level of accuracy, even forextremely refined meshes in the fine scale. The l2-norm norm of

Fig. 4. Oscillations in the conductivity tensor (Example 2).

S. Soghrati, I. Stanciulescu / Mechanics Research Communications 52 (2013) 11– 18 15

Table 3Example 2 (periodic separable conductivity tensor), performance of higher order elements.

Mesh 4-Node elements 9-Node elements

Nc Nf El2 E∞ El2 E∞

10 10 1.986 × 10−1 3.550 × 10−1 1.372 × 10−2 2.452 × 10−2

10 20 8.521 × 10−2 1.525 × 10−1 6.958 × 10−4 9.512 × 10−4

10 30 3.489 × 10−2 6.110 × 10−2 4.561 × 10−4 7.308 × 10−4

10 40 2.220 × 10−2 4.010 × 10−2 3.961 × 10−4 6.149 × 10−4

10 50 1.286 × 10−2 2.343 × 10−2 3.186 × 10−4 5.851 × 10−4

20 10 8.825 × 10−2 1.576 × 10−1 4.599 × 10−4 7.484 × 10−4

20 20 2.564 × 10−2 4.588 × 10−2 3.030 × 10−4 5.538 × 10−4

20 30 1.073 × 10−2 1.924 × 10−2 2.146 × 10−4 3.958 × 10−4

20 40 5.744 × 10−3 1.029 × 10−2 2.343 × 10−4 4.098 × 10−4

20 50 3.109 × 10−3 5.720 × 10−3 2.342 × 10−4 4.088 × 10−4

30 10 3.786 × 10−2 6.772 × 10−2 8.219 × 10−4 1.420 × 10−3

30 20 1.134 × 10−2 2.028 × 10−2 3.155 × 10−4 4.968 × 10−4

30 30 4.820 × 10−3 8.720 × 10−3 2.380 × 10−4 4.153 × 10−4

30 40 2.463 × 10−3 4.381 × 10−3 2.350 × 10−4 4.149 × 10−4

30 50 1.206 × 10−3 2.341 × 10−3 2.350 × 10−4 4.149 × 10−4

60 10 1.173 × 10−2 2.094 × 10−2 3.475 × 10−4 6.247 × 10−4

60 20 2.781 × 10−3 5.038 × 10−3 2.266 × 10−4 4.087 × 10−4

60 30 1.031 × 10−3 1 −3 −4 −4

60 40 3.909 × 10−4 760 50 2.549 × 10−4 4

10 15 20 25 30 35 40 45 500

0.02

0.04

0.06

0.08

0.1

Nf

l2 -nor

m e

rror

Nc=5

Nc = 10

Nc = 15

Nc = 20

Nc = 30

Nc = 60

Fig. 5. l2-Norm of error for 4-node basis functions (Example 2).

10 15 20 25 30 35 40 45 500

0.002

0.004

0.006

0.008

0.01

Nf

l2 -nor

m e

rror

Nc=5

Nc = 10

Nc = 15

Nc = 20

Nc = 30

Nc = 60

Fig. 6. l2-Norm of error for 9-node basis functions (Example 2).

.793 × 10 2.266 × 10 4.087 × 10

.074 × 10−4 2.266 × 10−4 4.087 × 10−4

.277 × 10−4 2.266 × 10−4 4.087 × 10−4

Table 3 and Fig. 7 show that 10 × 10 meshes with 9-node ele-ments provide almost the same level of accuracy as 60 × 60 mesheswith 4-node elements. Moreover, comparing the results of thesetwo coarse meshes shows a faster convergence with increase in Nffor the 9-node coarse mesh (Fig. 7). Fig. 8 provides a typical depend-ence of the l2-norm of the error on the refinement of the coarsescale. Here too, for a similar refinement on the small scale, the 9-node elements provide better accuracy than the 4-node elements.Fig. 9 summarizes the results discussed here in a 3D representationof the error with respect to number of elements in both the fineand coarse meshes.

This example clearly shows that for similar number of degreesof freedom, increasing the order of basis functions improves theaccuracy. Consequently, using higher order bases decreases theminimum required number of elements in the coarse mesh forachieving a certain level of the accuracy. This feature is extremelyvaluable not only to decrease the mesh size, but also to provide abetter chance to select an appropriate mesh that matches the period

of oscillations (it uses fewer elements in the coarse mesh for prob-lems with periodic oscillations). Thus, we evaluate the bases onlyonce and eliminate the need for parallel computation in this step.

10 14 18 22 26 300

0.04

0.08

0.12

0.16

0.2

Nf

l2-n

orm

err

or

Nc = 10, 4-node

Nc = 20, 4-node

Nc = 10, 9-node

Nc = 20, 9-node

Fig. 7. Dependence of l2-norm of error on fine scale refinement (Example 2).

16 S. Soghrati, I. Stanciulescu / Mechanics Research Communications 52 (2013) 11– 18

10 20 30 40 50 600

0.04

0.08

0.12

0.16

0.2

Nc

l2 -nor

m e

rror

Nf = 10, 4-node

Nf = 10, 9-node

Fig. 8. Dependence of l2-norm of error on coarse scale refinement (Example 2).

ioawmcmes

1011

101210

-4

10-3

10-2

10-1

Analytical measure of computational cost

l2 -nor

m e

rror

4-node9-node

Fig. 10. l2-Norm of error dependence on the computational cost (Example 2).

0 0.5 1 1.5 2 2.5 3-3

0

0.01

0.02

0.03

0.04

l2 -nor

m e

rror

Nc = 10, 4-node

Nc = 20, 4-node

Nc = 30, 4-node

Nc = 10, 9-node

Nc = 20, 9-node

Nc = 30, 9-node

TE

Fig. 9. variation of l2-norm of error for 4 and 9-node elements (Example 2).

The benefits discussed here are obtained without significantncrease in the computational cost. Fig. 10 depicts the dependencef the l2-norm of the error on the total number of arithmetic oper-tions as a measure of the computational cost for coarse meshesith 4-node and 9-node elements. Note how the use of 9-node ele-ents leads to a better accuracy while reducing the computational

ost. Fig. 11 shows the dependence of the error on the small scaleesh size. Note that for coarse meshes, implementing higher order

lements allows for a significant reduction in the error for similarubcell sizes.

able 4xample 3 (periodic nonseparable conductivity tensor). Performance of higher order elem

Mesh 4-Node elements 8/9-No

Nc Nf El2 E∞ El2

10 10 2.974 × 10−2 5.342 × 10−2 6.219 ×10 20 6.734 × 10−3 1.230 × 10−2 2.684 ×10 40 1.769 × 10−3 3.921 × 10−3 1.279 ×10 50 4.738 × 10−3 4.305 × 10−3 1.262 ×20 10 1.020 × 10−2 1.834 × 10−2 2.477 ×20 20 2.217 × 10−3 4.049 × 10−3 1.089 ×20 40 4.369 × 10−4 1.206 × 10−3 1.067 ×20 50 6.254 × 10−4 1.418 × 10−3 1.067 ×30 10 4.162 × 10−3 7.320 × 10−3 5.060 ×30 20 1.269 × 10−3 2.194 × 10−3 1.039 ×30 40 4.791 × 10−4 9.567 × 10−4 1.037 ×30 50 4.611 × 10−4 9.228 × 10−4 1.037 ×

x1subcell size

Fig. 11. l2-Norm of error dependence on fine scale element size (Example 2).

Example 3 (periodic non-separable conductivity tensor): Usingsimilar dimensions and boundary conditions as the previousexample, the conductivity tensor for this problem is given bya(x) = a0(1.5 + sin 2�x/ε)−1 + a0(1.5 + cos 2�y/ε)−1, with ε = 1/12 (60oscillations in each direction over one element). Due to the non-

separable function used for the conductivity tensor, we use thestandard FEM in the fine scale to evaluate the corresponding bases.Table 4 illustrates the performance of the MsFEM solution of thisproblem for different discretization levels in the coarse and fine

ents.

de elements 12/16-Node elements

E∞ El2 E∞

10−3 1.138 × 10−2 4.721 × 10−3 8.820 × 10−3

10−4 4.252 × 10−4 1.413 × 10−4 3.110 × 10−4

10−4 3.434 × 10−4 6.954 × 10−5 1.317 × 10−4

10−4 3.427 × 10−4 6.887 × 10−5 1.308 × 10−4

10−4 4.368 × 10−4 1.772 × 10−4 3.170 × 10−4

10−4 1.819 × 10−4 7.969 × 10−5 1.761 × 10−4

10−4 1.777 × 10−4 7.346 × 10−5 1.752 × 10−4

10−4 1.777 × 10−4 7.346 × 10−5 1.752 × 10−4

10−4 8.038 × 10−4 5.161 × 10−4 8.203 × 10−4

10−4 1.365 × 10−4 7.112 × 10−5 1.632 × 10−4

10−4 1.358 × 10−4 7.091 × 10−5 1.627 × 10−4

10−4 1.358 × 10−4 7.091 × 10−5 1.627 × 10−4

S. Soghrati, I. Stanciulescu / Mechanics Research Communications 52 (2013) 11– 18 17

10 15 20 25 30 35 40 45 500

0.004

0.008

0.012l2 -n

orm

err

or

Nc = 20, 4 -node

Nc = 20, 8 -node

Nc = 20, 12-node

sciamwacstsN

tdvptifom

0 0.00 2 0.00 4 0.00 6 0.00 8 0.01

2

4

6

8

10

12

14 x10-3

subcell size

l2 -nor

m e

rror

4-node8-node12-node

Fig. 14. Variation of l2-norm of error for a 10 × 10 coarse mesh (Example 4).

1

2

3

4

5

6 x10-3

l2 -nor

m e

rror

4-node8-node12-node

Nf

Fig. 12. Performance of different order bases (Example 3).

cales. These results confirm that using higher order MsFEM basesan significantly improve the accuracy of the solution. Table 4ndicates that regardless of the refinement in the fine scale, theccuracy reached with 12-node elements can never be achieved byodified bilinear bases. A better accuracy for the MsFEM solutionith 4-node elements can only be obtained with techniques such

s oversampling to reduce the resonance error, which impose aonsiderable extra cost to the solution. Moreover, compared to theimple proposed approach for constructing higher order elements,he upscaling approach requires a more complicated computationalcheme. Fig. 12 depicts the variation of the l2-norm of the error forc = 20.

Example 4 (non-periodic non-separable conductivity tensor): Inhis last example, we use the same domain and boundary con-itions as before but a different pattern (shown in Fig. 13) forariations of the conductivity tensor. The results of solving thisroblem for two coarse meshes with different orders of basis func-ions are illustrated in Figs. 14–15. The faster convergence and

mprovement of the accuracy of the solution is evident as we switchrom first order to higher order bases. This behavior can be easilybserved by comparing the results obtained using 8-node ele-ents with a 10 × 10 coarse mesh and those obtained with 4-node

Fig. 13. Oscillations of the conductivity tensor (Example 4).

0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 4. 5 5x1

-3subcell size

Fig. 15. Variation of l2-norm of error for a 20 × 20 coarse mesh (Example 4).

elements on a 20 × 20 mesh. While the total number of degrees offreedom is similar for both cases, the solution with 8-node elementsis considerably more accurate. Also, note from Figs. 14 and 15 thatrefining the fine scale mesh for 4-node elements does not improvethe accuracy considerably.

5. Conclusions

A robust and systematic algorithm for the construction of higherorder MsFEM basis functions is introduced. The proposed schemecreates bases for elements with any arbitrary configuration or num-ber of nodes. We discuss the importance of the boundary conditionsof the coarse scale elements and demonstrate that using higherorder 1D MsFEM bases as the boundary conditions for 2D elementsimproves the performance of the resulting 2D bases. Numericalexamples are presented to illustrate the improved performance ofthe higher order MsFEM basis functions. We show that using theproposed higher order bases not only leads to a significant improve-

ment in accuracy, but also considerably reduces the computationalcost.

1 s Rese

R

A

A

B

C

D

E

E

E

Luce, R., Perez, S., 2002. A numerical upscaling method for an elliptic equation withheterogeneous tensorial coefficients. International Journal for Numerical Meth-

8 S. Soghrati, I. Stanciulescu / Mechanic

eferences

llaire, G., Brizzi, R., 2005. A multiscale finite element method for numerical homog-enization. Multiscale Modeling & Simulation 4 (3), 790–812.

rbogast, T., 2004. Analysis of two-scale, locally conservative subgrid upscaling forelliptic problems. SIAM Journal on Numerical Analysis 42, 576–598.

abuska, I., Caloz, G., Osborn, J.E., 1994. Special finite element methods for a class ofsecond order elliptic problems with rough coefficients. SIAM Journal on Numer-ical Analysis 31 (4), 945–981.

orsini, A., Menichini, F., Rispoli, F., Santoriello, A., Tezduyar, T.E., 2009. A multiscalefinite element formulation with discontinuity capturing for turbulence modelswith dominant reactionlike terms. Journal of Applied Mechanics 76 (2).

ykaar, B.B., Kitanidis, P.K., 1992. Determination of the effective hydraulic conduc-tivity for heterogeneous porous media using a numerical specteral approach 1.Method. Water Resources Reseach 28 (4), 1155–1166.

fendiev, Y.R., Wu, H.J., 2002. Multiscale finite element method for problems withhighly oscilatory. Numerische Mathematik 90, 459–486.

ngquist, W.E.B., Huang, Z., 2003. Heterogeneous multiscale method: a generalmethodology for multiscale modeling. Condensed Matters and Material Physics67 (9), 92101-1-4.

ngquist, B., Tsai, Y.H., 2005. Heterogeneous multiscale methods for stiff ordinarydifferential equations. Mathematics of Computations 74 (252), 1707–1742.

arch Communications 52 (2013) 11– 18

Gonthier, K.A., Jogi, V., 2005. Multiscale shock heating analysis of a granular explo-sive. Journal of Applied Mechanics 72 (4), 538–552.

Hou, T.Y., Wu, X.H., 1997. A multiscale finite element method for elliptic problems incomposite materials and porous media. Journal of Computational Physics 134,169–189.

Hou, T.Y., Wu, X.H., Cai, Z., 1999. Convergence of a multiscale finite element methodfor elliptic problems with rapidly oscillating coefficients. Mathematics of Com-putation 68 (227), 913–943.

Hughes, T.J.R., Feijoo, G.R., Mazzei, L., Quincy, J.B., 1998. The variational multi-scale method—a paradigm for computational mechanics. Computer Methodsin Applied Mechanics and Engineering 166, 3–24.

Hughes, T.J.R., 1995. Multiscale phenomena: Green’s functions, the Drichlet toNeumann formulation, subgrid scale models, bubbles and the origin of the sta-blized methods. Computer Methods in Applied Mechanics and Engineering 127,387–401.

ods in Engineering 54, 537–556.Ming, P., Yue, X., 2006. Numerical methods for multiscale elliptic problems. Journal

of Computational Physics 214, 421–445.