VIETNAM NATIONAL UNIVERSITY, HO CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF SCIENCE HOANG TUONG THE ISOGEOMETRIC MULTISCALE FINITE ELEMENT METHOD FOR HOMOGENIZATION PROBLEMS MSc THESIS IN MATHEMATICS Ho Chi Minh city - 2012 Acknowledgement First and foremost, I would like to express my sincere gratitude to my advisor Dr Nguyen Xuan Hung for supporting my research, for his patience, encouragement, and enthusiasm Without his guidance, this thesis would not have been completed I would like to thank Prof Dang Duc Trong - Dean of the Faculty of Mathematics and Computer Science, and Dr Nguyen Thanh Long for introducing me to Partial Differential Equations (PDEs) I will never forget the wonderful lectures I have learned during the course as a Master student I would like to thank all of my classmates, who were side by side with me in the Master program The time being with you is one of the most memorable time in my life To Mr Do Huy Hoang, the oldest member, his passion of learning will always inspires me I would like to thank Mr Thai Hoang Chien, Mr Tran Vinh Loc and all the members in Division of Computational Mechanics, Ton Duc Thang University for the helpful discussion of Isogeometric Analysis (IGA) when I first begin my journey doing the research Last but not the least, I would like to thank my parents for giving birth to me and continuously supporting me spiritually throughout my life Ho Chi Minh city, September, 2012 Hoang Tuong iv Contents Abstract ii Publications iii List of Figures xi List of Tables xiii Notations xiv Introduction 1.1 Heterogeneous material 1.2 Multiscale modeling 1.3 Homogenization theory 1.3.1 Setting of the problem 1.4 Finite Element Analysis (FEA) 1.5 Isogeometric Analysis (IGA) 1.6 The Finite Element Heterogeneous Multiscale Method (FE-HMM) v 1.7 The Isogeometric Analysis Heterogeneous Multiscale Method (IGAHMM) Preliminary results on homogenization theory 13 2.1 Main convergence results 15 2.2 Proof of the main convergence results 16 2.3 Convergence of the energy 20 The Finite Element Heterogeneous Multiscale Method (FE-HMM) 22 3.0.1 Model problems 22 The finite element heterogeneous multiscale method (FE-HMM) 23 3.1.1 Macro finite element space 24 3.1.2 Micro finite element space 25 3.1.3 The FE-HMM method 26 3.2 The motivation behind the FE-HMM 26 3.3 Convergence of the FE-HMM method 28 3.3.1 Priori estimates 28 3.3.2 Optimal micro refinement strategies 29 Numerical experiments 29 3.4.1 2D-elliptic problem with non-uniformly periodic tensor 29 3.4.2 2D-elliptic problem with uniform periodic tensor 31 3.1 3.4 11 The Isogeometric Analysis Heterogeneous Multiscale Method (IGAHMM) 41 vi 4.1 4.2 NURBS-based isogeometric analysis fundamentals 42 4.1.1 Knot vectors and basis functions 42 4.1.2 NURBS curves and surfaces 43 4.1.3 Refinement 45 An isogeometric analysis heterogeneous multiscale method (IGAHMM) 46 4.2.1 Model problems 46 4.2.2 Drawbacks of the FE-HMM method 47 4.2.3 The isogeometric analysis heterogeneous multiscale method (IGA-HMM) 47 Priori Error Estimates 51 Numerical validation 53 4.3.1 Problem 53 4.3.2 Problem 2: IGA-HMM applied for curved boundary domains 60 4.3.3 Problem 3: An efficiency of IGA-HMM with a flexible de- 4.2.4 4.3 gree elevation 4.3.4 63 An higher order of IGA-HMM in both macro and micro patch space 67 Conclusions and future work 73 Appendix 75 Control data for NURBS objects Bibliography 75 80 vii List of Figures 1.1 Heterogeneous material (www.advancedproductslab.com) 1.2 Multiscale systems (www.scorec.rpi.edu) 1.3 Multiscale modeling (http://www.efrc.udel.edu) 1.4 Periodic domain modelling 1.5 An illustration of CAD objects using NURBS (www.tsplines.com) 1.6 An illustration of geometry description in IGA 1.7 Communication with CAD: a comparision between FEA and IGA (Hughes-Cottrell-Bazilevs, CMAME, 2005) 3.1 10 [4] In every element of the FE-HMM, the contribution to the stiffness matrix of the macroelements (a) is given by the solutions of the microproblems (b), which are computed using numerical quadrature on every microelement (c) 24 3.2 Domain of the problem (3.13) 30 3.3 Conductivity tensor and its oscillation of problem (3.13) with ε = 0.1 30 3.4 H (energy) error between Dirichlet coupling conditions for δ = 1.1ε, δ = 35 ε and Periodic coupling condition for δ = ε viii 33 3.5 L2 error between Dirichlet coupling conditions for δ = 1.1ε, δ = 53 ε and Periodic coupling condition for δ = ε 3.6 H (energy) norm between Dirichlet coupling conditions for δ = 1.1ε, δ = 35 ε and Periodic coupling condition for δ = ε 3.7 34 H error when using periodic coupling conditions for δ = 1.1ε, δ = 3ε 3.9 34 L2 norm between Dirichlet coupling conditions for δ = 1.1ε, δ = 53 ε and Periodic coupling condition for δ = ε 3.8 33 and δ = ε 35 L2 error when using periodic coupling conditions for δ = 1.1ε, δ = 3ε and δ = ε 36 3.10 H (energy) norm when using periodic coupling conditions for δ = 1.1ε, δ = 35 ε and δ = ε 36 3.11 L2 norm when using periodic coupling conditions for δ = 1.1ε, δ = 3ε and δ = ε 37 3.12 The convergence in H (energy) norm of the solution of problem (3.14) 38 3.13 The convergence in L2 norm of the solution of problem (3.14) 38 3.14 The convergence rates of H error match the result 39 3.15 The convergence rates of L2 errors match the predicted result 39 3.16 The convergence rates of the error in H and L2 norms match the predicted result when using H micro refinement strategy 40 4.1 This figure illustrate cubic shape basis of B-spline 43 4.2 2D quadratic, cubic and quartic 2D B-spline basis functions 44 4.3 Physical mesh and control mesh with quadratic NURBS surface 45 ix 4.4 Solution of the problem 4.3.1 4.5 L2 error of the thermal square problem, using L2 micro refinement strategy 4.6 57 L2 error of the thermal square problem, using H micro refinement strategy 4.9 57 H error of the thermal square problem, using H micro refinement strategy 4.8 56 H error of the thermal square problem, using L2 micro refinement strategy 4.7 55 59 Coarse mesh of the domain described in problem 4.3.2 and its control net 61 4.10 Meshes for the quarter circular annulus test cases 62 4.11 Solution of the problem 4.3.2 62 4.12 H error of the thermal quarter annulus problem, using H micro refinement strategy 63 4.13 L2 error of the thermal quarter annulus problem, using H micro refinement strategy 64 4.14 Coarse mesh of the geometry domain Ω and its control net with degree p = 66 4.15 Solution distribution of the problem 4.3.3 66 4.17 Convergence of the energy norm and max norm, purely using degree elevation 66 4.16 Left: IGA-HMM solutions of the problem 4.3.3 on semi circle boundary for various degrees Right: zoom into the solutions x 68 4.18 H error of the thermal quarter annulus problem, using NURBS of degree in micro space 70 4.19 L2 error of the thermal quarter annulus problem, using NURBS of degree in micro space 71 4.20 CPU time solving thermal quarter annulus problem, using NURBS of degree in micro space xi 72 List of Tables 1.1 Significant of u and f in application 3.1 The independence on ε of the HMM solution (δ = ε, periodic micro constraint, Nmac fixed.) 3.2 56 H error of the thermal square problem, using H micro refinement strategy 4.4 55 H error of the thermal square problem using L2 micro refinement strategy 4.3 32 L2 error of the thermal square problem, using L2 micro refinement strategy 4.2 31 Periodic micro boundary condition performs better than the Dirichlet one ( δ = 35 ε, ε = 0.005, Nmac fixed) 4.1 58 L2 error of the thermal square problem, using H micro refinement strategy 58 4.5 Performance comparision between FE-HMM and IGA-HMM 60 4.6 H error of the thermal quarter annulus problem, using H micro refinement strategy xii 63 Figure 4.14: Coarse mesh of the geometry domain Ω and its control net with degree p = 0.08 0.06 0.04 0.02 0.5 0.5 0 −0.5 −1 −0.5 Figure 4.15: Solution distribution of the problem 4.3.3 0.22 0.075 0.07 0.2 max norm energy norm 0.21 0.19 0.065 0.06 0.18 0.055 IGA−HMM Reference 0.17 10 12 degree 14 16 18 20 IGA−HMM Reference 0.05 10 12 degree 14 16 18 20 Figure 4.17: Convergence of the energy norm and max norm, purely using degree 66 elevation Table 4.8: Max norm, energy norm of the solution of problem 4.3.3 and its CPU-time Degree Max norm Energy norm Time(s) 4.3.4 p=2 0.05220 0.17780 0.05 p=3 0.06198 0.19294 0.06 p=4 0.06303 0.19828 0.08 p=5 0.06621 0.20229 0.28 p=6 0.06741 0.20512 0.35 p=7 0.06870 0.20685 0.68 p=8 0.06935 0.20785 0.99 p=9 0.07008 0.20912 1.92 p=10 0.07062 0.21001 3.53 p=11 0.07106 0.21069 5.93 p=12 0.07139 0.21121 9.65 p=13 0.07172 0.21171 18.11 p=14 0.07194 0.21208 31.70 p=15 0.07215 0.21238 52.48 p=16 0.07230 0.21262 83.63 p=17 0.07245 0.21285 150.99 p=18 0.07257 0.21302 234.96 p=19 0.07265 0.21314 374.85 An higher order of IGA-HMM in both macro and micro patch space In the previous examples, we have already see that the IGA-HMM performs much better than FE-HMM when using NURBS with higher order in macro patch space 67 0.08 0.073 0.07 0.06 0.0725 0.05 0.072 0.04 0.0715 0.03 0.071 Degree=10 Degree=12 Degree=15 Degree=17 Degree=19 FE−HMM reference 0.02 0.01 −0.5 0.0705 0.5 0.07 −0.02 −0.01 0.01 0.02 The FE-HMM reference solution is obtained by using 1137 macro dofs with Nmic=32 Figure 4.16: Left: IGA-HMM solutions of the problem 4.3.3 on semi circle boundary for various degrees Right: zoom into the solutions and NURBS with degree q = in micro space In this section, we continue to test the IGA-HMM when ultizing high order NURBS in both micro and macro space Total computational cost of the IGA-HMM and the choice of macro-micro degree Let Nmac denote the number of macro elements per dimension, p, q the degree of macro and micro basis functions, respectively Using L2 micro refinement strategy, we have Total computational cost= number of macro elements × number of micro problem per macro element × number of dofs per micro problem Thus, the total computational cost is p+1 2+ p+1 2q Nmac × (p + 1)2 × (qNmac + 1)2 ∼ O Nmac q (4.34) From Eq (4.34) we see that if the degree of the micro space is fixed at q = 1, 68 the total cost of the IGA-HMM will increase with the degree elevation of p: p+3 The total cost ∼ O(Nmac ) (q = 1), (4.35) which is very inefficient when using high order of basis function at macro space Therefore, instead of fixing q = at micro level, we choose q ≥ p + Then, ) The total cost ∼ O(Nmac (4.36) ), for In this case, the complexity of the IGA-HMM will be hold at about O(Nmac every p Remark 4.3.1 The formula 4.34 also holds in FE-HMM, but in this standard approach, it is very hard to increase the basis function degree when doing implementation due to the different connection between various kinds of nodes such as center, internal, edges, Let consider again the problem given in section 4.3.3 For the micro space we use NURBS of fixed degree 5, and for the macro space we use NURBS of degree ranging from to Here, L2 micro refinement strategy is used The results in Table 4.9, Table 4.10 , Fig 4.18 and Fig 4.19 show that this approach far outperforms the standard FE-HMM in terms of convergence rate , the accuracy as well as the time efficiency In Table 4.11 and Fig 4.20, the results also confirm the nearly optimal computational cost given by 4.34 We note that in FE-HMM, to obtain the accuracy about 10−6 , 10−7 , 10−8 in H error or 10−7 , 10−8 , 10−9 in L2 error, it is (nearly) impossible because as we has seen in Table 4.5, to reach the accuracy of 10− in H , the total number of macro degree of freedom needed is 655362 which approximately more than × 109 , and even if we suppose that there is enough memory for the computer, the CPU time will exceed 1010 seconds (which is approximately more than 300 years!) Here, in IGA-HMM with high order of NURBS basis functions in both macro and micro space we can obtain these accuracy (very) easily: about 1000 seconds with 69 mesh × for the accuracy of 10−6 in H error and × 10−7 in L2 error; for accuracy of 2.7 × 10−8 in H error and 2.7 × 10−9 , it takes about 9000 seconds with mesh 16 × 16 So far, we haven’t seen any numerical method in the literature solving these kind of equation can obtain such accuracy with such efficiency Table 4.9: H error of the thermal quarter annulus problem, using NURBS of degree in micro space ❍❍ ❍❍ ❍ degree mesh ❍❍ ❍❍ ❍ 2x2 4x4 8x8 16x16 0.00040 p=2 0.04501 0.00724 0.00164 p=3 0.00450 0.00116 1.13E-04 1.33E-05 p=4 0.00241 2.72E-04 1.04E-05 5.79E-07 p=5 0.00012 3.80E-05 1.11E-06 2.71E-08 −2 10 slope=2.03 −3 10 −4 u0 −uH H1 u0 H 10 slope=3.09 −5 10 slope=4.17 −6 10 −7 10 −8 slope=5.36 deg=2 deg=3 deg=4 deg=5 10 10 1/Hmax Figure 4.18: H error of the thermal quarter annulus problem, using NURBS of degree in micro space 70 Table 4.10: L2 error of the thermal quarter annulus problem, using NURBS of degree in micro space ❍ ❍❍ ❍ mesh 2x2 4x4 p=2 0.02784 0.002084 0.00022 p=3 0.00211 0.00041 p=4 0.00108 1.10E-04 1.84E-06 5.78E-08 p=5 4.43E-05 1.19E-05 2.08E-07 2.71E-09 ❍❍ ❍ degree 8x8 16x16 ❍❍ ❍ 2.58E-05 1.89E-05 1.08E-06 −2 10 −3 10 slope=3.07 −4 u0 −uH L2 u L2 10 −5 10 slope=4.12 −6 10 slope=4.98 −7 10 −8 10 −9 deg=2 deg=3 deg=4 deg=5 slope=6.26 10 10 1/hmax Figure 4.19: L2 error of the thermal quarter annulus problem, using NURBS of degree in micro space 71 Table 4.11: CPU time solving the thermal quarter annulus problem, using NURBS of degree in micro space ❍ ❍❍ ❍ mesh 2x2 4x4 8x8 16x16 p=2 4.27 17.01 68.02 512.62 p=3 7.96 32.33 235.54 1638.44 p=4 12.48 49.84 374.17 2592.46 p=5 18.84 141.07 956.76 9055.52 ❍❍ ❍ degree ❍❍ ❍ 10 slope=3.24 CPU-time (seconds) slope=2.80 slope=2.80 slope=2.91 10 10 deg=2 deg=3 deg=4 deg=5 10 10 number of macro elements per dimension Figure 4.20: CPU time solving thermal quarter annulus problem, using NURBS of degree in micro space 72 Conclusions and future work Summary In this thesis, we have presented an efficient isogeometric analysis heterogeneous multiscale method (IGA-HMM) for elliptic homogenization problems The method is capable of capturing the exact geometric representation and is very flexible with refinement and degree elevation of using NURBS basis functions As a result, the high-order IGA-HMM macroscopic and microscopic solver are designed simply and effectively Numerical results showed the IGA-HMM achieves high order accuracy with the optimally convergence rate for both L2 and H micro refinement strategies It is thus very promising to provide an effective tool for analysis of homogenization problems in practice Future developments There are some directions to improve and extend the results in this work • Macro: T-Splines [7, 14] can be used instead of FEM or NURBS (to use adaptivity) 73 • Micro: using spectral method (to obtain superconvergence for simple do- main) Moreover, to apply for problems with a more higher complexity, a combination of IGA-HMM with techniques introduced in [26] can be considered In this way, not only the number of macro elements will be reduced but also that of micro problems As a result, the computational cost will dramatically be minimized This will be our forthcoming development 74 Bibliography [1] A Abdulle On a priori error analysis of fully discrete heterogeneous multiscale FEM Multiscale Modeling and Simulation, 4(2):447–459, 2005 [2] A Abdulle The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs GAKUTO Int Ser Math Sci Appl, 31:135–184, 2009 [3] A Abdulle and B Engquist Finite element heterogeneous multiscale methods with near optimal computational complexity Multiscale modeling & simulation, 6(4):1059, 2007 [4] A Abdulle and A Nonnenmacher A short and versatile finite element multiscale code for homogenization problems Computer Methods in Applied 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partial differential equations MIT, 2007 [27] L A Piegl and W Tiller The NURBS book Springer Verlag, 1997 [28] P.-B Ming R Li and F.-Y Tang An efficient high order heterogeneous multiscale method for elliptic problems Multiscale Model Simul., 10(1):259– 283, 2012 79 [29] O.A.Oleinik V V Zhikov, S M Kozlov Homogenization of differential operators and integral functionals Springer-Verlag, 1994 [30] E Weinan Principles of Multiscale Modeling Cambridge University Press, 2011 [31] E Weinan and B Engquist Multiscale modeling and computation Notices of the AMS, 50(9):1062–1070, 2003 [32] E Weinan, B Engquist, and Z Huang Heterogeneous multiscale method: a general methodology for multiscale modeling Physical Review B, 67(9):092101, 2003 [33] E Weinan, P Ming, and P Zhang Analysis of the heterogeneous multiscale method for elliptic homogenization problems J Amer Math Soc, 18(1):121– 156, 2005 80 Appendix Control data for NURBS objects Table A.1: Control points and weights for problem 4.3.2 i Bi wi (0,1) (1,1) 1/ (1,0) (0,3/2) (3/2,3/2) 1/ (3/2,0) (0,2) (2,2) 1/ (2,0) √ √ √ 75 Table A.2: Control points and weights for problem 4.3.3 i Bi wi (−0.500000000000000, −1.000000000000000) (−0.250000000000000, −1.000000000000000) (0.000000000000000, −1.000000000000000) (0.250000000000000, −1.000000000000000) (0.500000000000000, −1.000000000000000) (−0.500000000000000, −0.500000000000000) (0.000000000000000, −0.250000000000000) √ (0.353553390593274, −0.378679656440357) (1 + 1/ 2)/2 10 (0.500000000000000, −0.500000000000000) 11 (−0.500000000000000, 0.000000000000000) 12 (−0.500000000000000, 0.500000000000000) 1/ 13 (0.000000000000000, 0.500000000000000) 14 (0.500000000000000, 0.500000000000000) 1/ 15 (0.500000000000000, 0.000000000000000) √ (−0.353553390593274, −0.378679656440357) (1 + 1/ 2)/2 76 √ √ [...]... 21 f u0 dx (2.38) Chapter 3 The Finite Element Heterogeneous Multiscale Method (FE-HMM) In this chapter, we present a numerical approach to the homogenization problem: the Finite Element Heterogeneous Multiscale Method (FE-HMM) This method was first proposed by Weinan and Engquist in [31], and latter developed in [3], [23, 1] For a review, we refer to [6, 2] 3.0.1 Model problems Let be a domain in... these questions is the aim of the mathematical theory of homogenization These questions are very important in the applications since, if one can give positive answers, then the limit coefficients, as it is well known from engineers and physicists, are good approximations of the global characteristics of the composite material, when regarded as an homogeneous one Moreover, replacing the problem by the. .. IGA The geometrical map F and the weight w are fixed at the coarsest level of discretization! R V azquez (IMATI-CNR Italy) 1.6 Introduction to Isogeometric Analysis Santiago de Compostela, 2010 7 / 33 The Finite Element Heterogeneous Multiscale Method (FE-HMM) To solve the homogenization problems, analytic approaches such as in [9], [29] homogenized equations are derived However, the coefficients of these... standard finite element methods are used Details on FE-HMM can be found in Chapter 3 1.7 The Isogeometric Analysis Heterogeneous Multiscale Method (IGA-HMM) Although very popular, the standard FEM still has some shortcomings which affect the efficiency of the FE-HMM method Firstly, the discretized geometry through mesh generation is required This process often leads to the geometrical error even using the. .. value problems with heterogeneous media If the period of the 3 structure is small in comparison with the size of the region in which the system is studying, it can be characterized by a small parameter which is the ratio of the period of the structure to a typical length in the region Starting from the microscopic description of the problem, we want to find the macroscopic or effective behavior of the. .. equations for a0 (x) are available But even in this case, at each point x , d cell- problems must be solved and a0 (x) is obtained by taking appro- priate integration Therefore, analytic a0 (x) cannot be obtained due to infinite number of problems we have to solve The FE-HMM method gives a numerical solution to the homogenization problem from a different viewpoint 3.1 The finite element heterogeneous multiscale. .. problems The simplest one is the finite element heterogeneous multiscale method (FE-HMM), which uses standard finite elements such as simplicial or quadrilateral ones in both macroscopic and microscopic level Solving the so-called micro problems (with a suitable set up) in sampling domains around traditional Gauss integration points allows one to approximate missing effective 10 information for the macro... In the following, we present the main convergence result, which plays an important role in the homogenization theory Then, we will give a proof for this result 2.1 Main convergence results Theorem 2.1.1 Let f L2 () and u is the solution of (2.4) with a defined by (2.5)(2.7) Then, i) ii) u u0 weakly in H01 () a u a0 u0 weakly in (2.8) (L2 ()N where u0 is the unique solution in H01 () of the homogenization. .. Some references for FEA are [10, 20] 1.5 Isogeometric Analysis (IGA) The isogeometric analysis was first proposed by Hughes and co-workers [21] and now has attracted the attention of academic as well as industrial engineering community all over the world The IGA allows to closely link the gap between Computer Aided Design (CAD) and Finite Element Analysis (FEA) It means that the IGA uses the same basis... periodic in y, i.e w(x, y) L2per (Y ; C ()) Then, the sequence w(x, x/) converges weakly in L2 () to Y w(x, y)dy In the next section, we consider the convergence of the energy 2.3 Convergence of the energy One interesting consequence of Theorem 2.1.1 is the convergence of the energy associated to the problem (2.4), namely of the quantity E (u ) = a u u dx (2.35) The following result was originally proved ... 20 The Finite Element Heterogeneous Multiscale Method (FE-HMM) 22 3.0.1 Model problems 22 The finite element heterogeneous multiscale method (FE-HMM) 23 3.1.1 Macro finite. .. FE-HMM method gives a numerical solution to the homogenization problem from a different viewpoint 3.1 The finite element heterogeneous multiscale method (FE-HMM) The so-called finite element. .. which are small parts of the whole computational domain Next, we describe the Finite Element Heterogeneous Multiscale Method 25 3.1.3 The FE-HMM method The solution uH of the homogenization problem