On accurate and time efficient solution of primal-mixed finite element equations in multiscale solid mechanics
In order to identify the best technique to solve a class of geometrically multiscale model problems in thermoelasticity, we examine a combination of a primal‐mixed finite element approach and direct sparse solvers and matrix scaling routines. The criteria for optimality are robustness, accuracy and...
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| Published in | International journal for numerical methods in biomedical engineering Vol. 27; no. 1; pp. 95 - 112 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Chichester, UK
John Wiley & Sons, Ltd
01.01.2011
Wiley |
| Subjects | |
| Online Access | Get full text |
| ISSN | 2040-7939 2040-7947 2040-7947 |
| DOI | 10.1002/cnm.1296 |
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| Abstract | In order to identify the best technique to solve a class of geometrically multiscale model problems in thermoelasticity, we examine a combination of a primal‐mixed finite element approach and direct sparse solvers and matrix scaling routines. The criteria for optimality are robustness, accuracy and execution time. It will be shown that the present finite element approach, where displacement and stress variables are simultaneously solved from large‐scale indefinite poorly scaled systems of equations using the sparse HSL solver MA57 with the aid of the matrix scaling routines MC64 or MC30 during the factorization process, enables a reliable solution even if hexahedral finite elements in a mesh differ in size up to six orders of magnitude. A number of tests in multiscale elasticity and thermoelasticity are examined to test the accuracy and execution time efficiency of the proposed solution approach on a standard PC computing platform. Copyright © 2009 John Wiley & Sons, Ltd. |
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| AbstractList | In order to identify the best technique to solve a class of geometrically multiscale model problems in thermoelasticity, we examine a combination of a primal-mixed finite element approach and direct sparse solvers and matrix scaling routines. The criteria for optimality are robustness, accuracy and execution time. It will be shown that the present finite element approach, where displacement and stress variables are simultaneously solved from large-scale indefinite poorly scaled systems of equations using the sparse HSL solver MA57 with the aid of the matrix scaling routines MC64 or MC30 during the factorization process, enables a reliable solution even if hexahedral finite elements in a mesh differ in size up to six orders of magnitude. A number of tests in multiscale elasticity and thermoelasticity are examined to test the accuracy and execution time efficiency of the proposed solution approach on a standard PC computing platform. In order to identify the best technique to solve a class of geometrically multiscale model problems in thermoelasticity, we examine a combination of a primal‐mixed finite element approach and direct sparse solvers and matrix scaling routines. The criteria for optimality are robustness, accuracy and execution time. It will be shown that the present finite element approach, where displacement and stress variables are simultaneously solved from large‐scale indefinite poorly scaled systems of equations using the sparse HSL solver MA57 with the aid of the matrix scaling routines MC64 or MC30 during the factorization process, enables a reliable solution even if hexahedral finite elements in a mesh differ in size up to six orders of magnitude. A number of tests in multiscale elasticity and thermoelasticity are examined to test the accuracy and execution time efficiency of the proposed solution approach on a standard PC computing platform. Copyright © 2009 John Wiley & Sons, Ltd. In order to identify the best technique to solve a class of geometrically multiscale model problems in thermoelasticity, we examine a combination of a primal‐mixed finite element approach and direct sparse solvers and matrix scaling routines. The criteria for optimality are robustness, accuracy and execution time. It will be shown that the present finite element approach, where displacement and stress variables are simultaneously solved from large‐scale indefinite poorly scaled systems of equations using the sparse HSL solver MA57 with the aid of the matrix scaling routines MC64 or MC30 during the factorization process, enables a reliable solution even if hexahedral finite elements in a mesh differ in size up to six orders of magnitude. A number of tests in multiscale elasticity and thermoelasticity are examined to test the accuracy and execution time efficiency of the proposed solution approach on a standard PC computing platform. Copyright © 2009 John Wiley & Sons, Ltd. |
| Author | Duff, Iain Mijuca, Dubravka |
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| Keywords | Solid mechanics time efficient geometrically multiscale Thermomechanical properties Large displacement Factorization Modeling Mixed method indefinite Finite element method reliable Multiscale method sparse Sparse matrix Thermoelasticity primal-mixed finite element |
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| References | Haider J, Rahman M, Corcoran B, Hashmi MSJ. Simulation of thermal stress in magnetron sputtered thin coating by finite element analysis. Journal of Materials Processing Technology 2005; 168(1):36-41. DOI: 10.1016/j.jmatprotec.2004.09.093. Roark RJ, Young WC. Formulas for Stress and Strain (5th edn). McGraw-Hill: New York, 1975. Lu S, Dong M. An advanced BEM for thermal and stress analyses of components with thermal barrier coating. Electronic Journal of Boundary Elements 2003; 1(2):302-315. Zivkovic I, Maksimovic S, Aleksic R. Numerical and experimental analysis of initial failure of composite laminates with embedded fibreoptic sensors. Structural Integrity and Life 2004; 4(3):137-148. Kim DN, Bathe KJ. A 4-node 3D-shell element to model shell surface tractions and incompressible behavior. Computers and Structures 2008; 86:2027-2041. Brezzi F, Fortin M, Marini D. Mixed finite element methods with continuous stresses. Mathematical Models and Methods in Applied Sciences 1993; 3(2):275-287. Cannarozzi AA, Ubertini F. A mixed variational method for linear coupled thermoelastic analysis. International Journal of Solids and Structures 2001; 38(4):717-739. DOI: 10.1016/S0020-7683(00)00061-5. Green AE, Lindsay KE. Thermoelasticity. Journal of Elasticity 1972; 2(1):1-7. Benzi M, Golub GH, Liesen J. Numerical solution of saddle point problems. Acta Numerica 2005; 14:1-137. DOI: 10.1017/S0962492904000212. Mijuca D. On a reliable finite element approach in multiscale multimaterial solid thermo-mechanics. Journal of the Serbian Society for Computational Mechanics 2008; 2(1):44-62. Duff IS, Koster J. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM Journal on Matrix Analysis and Applications 2001; 22(4):973-996. DOI: 10.1137/S0895479899358443. Belytschko T, Xiao SP. Coupling methods for continuum model with molecular model. International Journal for Multiscale Computational Engineering 2003; 1(1):115-126. DOI: 10.1615/IntJMultCompEng.v1.i1.100. Mijuca D, Berkovic M. On the main properties of the primal-mixed finite element formulation. Facta Universitatis Series Mechanics, Automatic Control and Robotics 1999; 2(9):903-920. Duff IS. MA57-a code for the solution of sparse symmetric indefinite systems. ACM Transactions on Mathematical Software 2004; 30(2):118-144. DOI: 10.1145/992200.992202. Duff IS, Pralet S. Strategies for scaling and pivoting for sparse symmetric indefinite problems. SIAM Journal on Matrix Analysis and Applications 2005; 27(2):313-340. DOI: 10.1137/04061043X. Mijuca D. On hexahedral finite element HC8/27 in elasticity. Computational Mechanics 2004; 33(6):466-480. DOI: 10.1007/s00466-003-0546-9. Bathe KJ. The inf-sup condition and its evaluation for mixed finite element methods. Computers and Structures 2001; 79:243-252. Shepherd MS. Finite element modelling within an integrated geometric modelling environment: Part II attribute specification, domain differences and indirect element types. Engineering with Computers 1985; 1:61-71. DOI: 10.1007/BF01200065. Askes H, Morata I, Aifantis EC. Finite element analysis with staggered gradient elasticity. Computers and Structures 2008; 86:1266-1279. Ghoniem NM, Esteban PB, Busso EP, Kioussis N, Huang H. Multiscale modelling of nanomechanics and micromechanics: an overview. Philosophical Magazine 2003; 83(31):3475-3528. DOI: 10.1080/14786430310001607388. Amestoy PR, Davis TA, Duff IS. An approximate minimum degree ordering algorithm. SIAM Journal on Matrix Analysis and Applications 1996; 17(4):886-905. DOI: 10.1137/S0895479894278952. Onate E. Multiscale computational analysis in mechanics using finite calculus: an introduction. Computer Methods in Applied Mechanics and Engineering 2003; 192(28-30):3043-3059. DOI: 10.1016/S0045-7825(03)00340-2. Mesarovic S, Padbidri J. Minimal kinematic boundary conditions for simulations of disordered microstructures. Philosophical Magazine 2005; 85(1):65-78. DOI: 10.1080/14786430412331313321. Duff IS, Erisman AM, Reid JK. Direct Methods for Sparse Matrices. Oxford University Press: London, 1986. Schmauder S, Weber U, Soppa E. Computational mechanics of heterogeneous materials-influence of residual stresses. Computational Materials Science 2003; 26:142-153. DOI: 10.1016/S0927-0256(02)00414-7. Mijuca D, Ziberna A, Medjo B. A new multifield finite element method in steady state heat analysis. Thermal Science 2005; 9(1):111-130. Mijuca D, Ziberna A, Medjo B. A novel primal-mixed finite element approach for heat transfer in solids. Computational Mechanics 2007; 39(4):367-381. DOI: 10.1007/s00466-006-0034-0. Reid JK, Curtis AR. On the automatic scaling of matrices for Gaussian elimination. Journal of the Institute of Mathematics and its Applications 1972; 10(1):118-124. DOI: 10.1093/imamat/10.1.118. Arnold DN. Mixed finite element methods for elliptic problems. Computer Methods in Applied Mechanics and Engineering 1990; 82:281-300. Rannacher R, Suttmeier FT. A feed-back approach to error control in finite element methods: application to linear elasticity. Computational Mechanics 1997; 19(5):434-446. Arioli M, Demmel JW, Duff IS. Solving sparse linear systems with sparse backward error. SIAM Journal on Matrix Analysis and Applications 1989; 10(2):165-190. DOI: 10.1137/0610013. Oden JT, Brauchli HJ. On the calculation of consistent stress distributions in finite element approximations. International Journal for Numerical Methods in Engineering 1971; 3:317-325. Babuska I, Vogelius M. On a dimensional reduction method. III. A-posteriori error estimation and an adaptive approach. Mathematics of Computation 1981; 37(56):361-383. DOI: 10.2307/2007498. Brezzi F, Bathe KJ. A discourse on the stability conditions for mixed finite element formulations. Journal of Computer Methods in Applied Mechanics and Engineering 1990; 82:27-57. Bathe KJ. Finite Element Procedures. Prentice-Hall: Englewood Cliffs, NJ, 1996. 2007; 39 1985; 1 1996; 17 1998 2004; 4 1975 2005; 85 1996 2007 1995 2003; 192 1999; 2 2003 2001; 22 2008; 2 2005; 27 1993; 3 1990; 82 1972; 2 1999 2004; 33 2004; 30 1989; 10 2005; 168 2005; 9 1997; 19 1986 2003; 26 1981; 37 1972; 10 2001; 38 1982 2003; 1 2008; 86 2003; 83 2001; 79 1971; 3 2005; 14 e_1_2_9_30_2 e_1_2_9_10_2 e_1_2_9_33_2 e_1_2_9_34_2 e_1_2_9_12_2 e_1_2_9_31_2 Lu S (e_1_2_9_40_2) 2003; 1 e_1_2_9_11_2 e_1_2_9_32_2 Mijuca D (e_1_2_9_19_2) 2008; 2 Roark RJ (e_1_2_9_39_2) 1975 e_1_2_9_14_2 e_1_2_9_37_2 e_1_2_9_13_2 e_1_2_9_38_2 Zivkovic I (e_1_2_9_45_2) 2004; 4 e_1_2_9_16_2 e_1_2_9_35_2 e_1_2_9_15_2 e_1_2_9_36_2 e_1_2_9_17_2 e_1_2_9_41_2 e_1_2_9_21_2 e_1_2_9_44_2 e_1_2_9_20_2 e_1_2_9_23_2 e_1_2_9_42_2 e_1_2_9_22_2 Bathe KJ (e_1_2_9_24_2) 1996 e_1_2_9_43_2 e_1_2_9_7_2 e_1_2_9_6_2 e_1_2_9_5_2 Duff IS (e_1_2_9_29_2) 1986 e_1_2_9_4_2 e_1_2_9_3_2 e_1_2_9_2_2 Mijuca D (e_1_2_9_18_2) 1999; 2 e_1_2_9_9_2 e_1_2_9_8_2 e_1_2_9_25_2 e_1_2_9_27_2 e_1_2_9_26_2 e_1_2_9_28_2 |
| References_xml | – reference: Reid JK, Curtis AR. On the automatic scaling of matrices for Gaussian elimination. Journal of the Institute of Mathematics and its Applications 1972; 10(1):118-124. DOI: 10.1093/imamat/10.1.118. – reference: Brezzi F, Fortin M, Marini D. Mixed finite element methods with continuous stresses. Mathematical Models and Methods in Applied Sciences 1993; 3(2):275-287. – reference: Duff IS, Pralet S. Strategies for scaling and pivoting for sparse symmetric indefinite problems. SIAM Journal on Matrix Analysis and Applications 2005; 27(2):313-340. DOI: 10.1137/04061043X. – reference: Mijuca D. On a reliable finite element approach in multiscale multimaterial solid thermo-mechanics. Journal of the Serbian Society for Computational Mechanics 2008; 2(1):44-62. – reference: Kim DN, Bathe KJ. A 4-node 3D-shell element to model shell surface tractions and incompressible behavior. Computers and Structures 2008; 86:2027-2041. – reference: Mijuca D, Berkovic M. On the main properties of the primal-mixed finite element formulation. Facta Universitatis Series Mechanics, Automatic Control and Robotics 1999; 2(9):903-920. – reference: Cannarozzi AA, Ubertini F. A mixed variational method for linear coupled thermoelastic analysis. International Journal of Solids and Structures 2001; 38(4):717-739. DOI: 10.1016/S0020-7683(00)00061-5. – reference: Schmauder S, Weber U, Soppa E. Computational mechanics of heterogeneous materials-influence of residual stresses. Computational Materials Science 2003; 26:142-153. DOI: 10.1016/S0927-0256(02)00414-7. – reference: Mijuca D. On hexahedral finite element HC8/27 in elasticity. Computational Mechanics 2004; 33(6):466-480. DOI: 10.1007/s00466-003-0546-9. – reference: Askes H, Morata I, Aifantis EC. Finite element analysis with staggered gradient elasticity. Computers and Structures 2008; 86:1266-1279. – reference: Duff IS, Erisman AM, Reid JK. Direct Methods for Sparse Matrices. Oxford University Press: London, 1986. – reference: Onate E. Multiscale computational analysis in mechanics using finite calculus: an introduction. Computer Methods in Applied Mechanics and Engineering 2003; 192(28-30):3043-3059. DOI: 10.1016/S0045-7825(03)00340-2. – reference: Arnold DN. Mixed finite element methods for elliptic problems. Computer Methods in Applied Mechanics and Engineering 1990; 82:281-300. – reference: Duff IS, Koster J. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM Journal on Matrix Analysis and Applications 2001; 22(4):973-996. DOI: 10.1137/S0895479899358443. – reference: Brezzi F, Bathe KJ. A discourse on the stability conditions for mixed finite element formulations. Journal of Computer Methods in Applied Mechanics and Engineering 1990; 82:27-57. – reference: Mijuca D, Ziberna A, Medjo B. A novel primal-mixed finite element approach for heat transfer in solids. Computational Mechanics 2007; 39(4):367-381. DOI: 10.1007/s00466-006-0034-0. – reference: Shepherd MS. Finite element modelling within an integrated geometric modelling environment: Part II attribute specification, domain differences and indirect element types. Engineering with Computers 1985; 1:61-71. DOI: 10.1007/BF01200065. – reference: Benzi M, Golub GH, Liesen J. Numerical solution of saddle point problems. Acta Numerica 2005; 14:1-137. DOI: 10.1017/S0962492904000212. – reference: Belytschko T, Xiao SP. Coupling methods for continuum model with molecular model. International Journal for Multiscale Computational Engineering 2003; 1(1):115-126. DOI: 10.1615/IntJMultCompEng.v1.i1.100. – reference: Lu S, Dong M. An advanced BEM for thermal and stress analyses of components with thermal barrier coating. Electronic Journal of Boundary Elements 2003; 1(2):302-315. – reference: Oden JT, Brauchli HJ. On the calculation of consistent stress distributions in finite element approximations. International Journal for Numerical Methods in Engineering 1971; 3:317-325. – reference: Duff IS. MA57-a code for the solution of sparse symmetric indefinite systems. ACM Transactions on Mathematical Software 2004; 30(2):118-144. DOI: 10.1145/992200.992202. – reference: Bathe KJ. Finite Element Procedures. Prentice-Hall: Englewood Cliffs, NJ, 1996. – reference: Arioli M, Demmel JW, Duff IS. Solving sparse linear systems with sparse backward error. SIAM Journal on Matrix Analysis and Applications 1989; 10(2):165-190. DOI: 10.1137/0610013. – reference: Roark RJ, Young WC. Formulas for Stress and Strain (5th edn). McGraw-Hill: New York, 1975. – reference: Rannacher R, Suttmeier FT. A feed-back approach to error control in finite element methods: application to linear elasticity. Computational Mechanics 1997; 19(5):434-446. – reference: Zivkovic I, Maksimovic S, Aleksic R. Numerical and experimental analysis of initial failure of composite laminates with embedded fibreoptic sensors. 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Multiscale modelling of nanomechanics and micromechanics: an overview. Philosophical Magazine 2003; 83(31):3475-3528. DOI: 10.1080/14786430310001607388. – reference: Mesarovic S, Padbidri J. Minimal kinematic boundary conditions for simulations of disordered microstructures. Philosophical Magazine 2005; 85(1):65-78. DOI: 10.1080/14786430412331313321. – reference: Babuska I, Vogelius M. On a dimensional reduction method. III. A-posteriori error estimation and an adaptive approach. Mathematics of Computation 1981; 37(56):361-383. 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| SubjectTerms | Accuracy Computing time Exact sciences and technology Finite element method Fundamental areas of phenomenology (including applications) geometrically multiscale indefinite Mathematical analysis Mathematical models Physics primal-mixed finite element reliable Routines Solid mechanics Solvers sparse Static elasticity (thermoelasticity...) Structural and continuum mechanics Thermoelasticity time efficient |
| Title | On accurate and time efficient solution of primal-mixed finite element equations in multiscale solid mechanics |
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