A vertex-based finite volume method applied to non-linear material problems in computational solid mechanics
A vertex‐based finite volume (FV) method is presented for the computational solution of quasi‐static solid mechanics problems involving material non‐linearity and infinitesimal strains. The problems are analysed numerically with fully unstructured meshes that consist of a variety of two‐ and three‐d...
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| Vydané v: | International journal for numerical methods in engineering Ročník 56; číslo 4; s. 507 - 529 |
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| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Chichester, UK
John Wiley & Sons, Ltd
28.01.2003
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| ISSN: | 0029-5981, 1097-0207 |
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| Abstract | A vertex‐based finite volume (FV) method is presented for the computational solution of quasi‐static solid mechanics problems involving material non‐linearity and infinitesimal strains. The problems are analysed numerically with fully unstructured meshes that consist of a variety of two‐ and three‐dimensional element types. A detailed comparison between the vertex‐based FV and the standard Galerkin FE methods is provided with regard to discretization, solution accuracy and computational efficiency. For some problem classes a direct equivalence of the two methods is demonstrated, both theoretically and numerically. However, for other problems some interesting advantages and disadvantages of the FV formulation over the Galerkin FE method are highlighted. Copyright © 2002 John Wiley & Sons, Ltd. |
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| AbstractList | A vertex-based finite volume (FV) method is presented for the computational solution of quasi-static solid mechanics problems involving material non-linearity and infinitesimal strains. The problems are analysed numerically with fully unstructured meshes that consist of a variety of two- and three-dimensional element types. A detailed comparison between the vertex-based FV and the standard Galerkin FE methods is provided with regard to discretization, solution accuracy and computational efficiency. For some problem classes a direct equivalence of the two methods is demonstrated, both theoretically and numerically. However, for other problems some interesting advantages and disadvantages of the FV formulation over the Galerkin FE method are highlighted. A vertex-based finite volume (FV) method is presented for the computational solution of quasi-static solid mechanics problems involving material non-linearity and infinitesimal strains. The problems are analyzed numerically with fully unstructured meshes that consist of a variety of 2D and 3D element types. A detailed comparison between the vertex-based FV and the standard Galerkin FE methods is provided with regard to discretization, solution accuracy and computational efficiency. For some problem classes a direct equivalence of the two methods is demonstrated, both theoretically and numerically. However, for other problems some interesting advantages and disadvantages of the FV formulation over the Galerkin FE method are highlighted. A vertex‐based finite volume (FV) method is presented for the computational solution of quasi‐static solid mechanics problems involving material non‐linearity and infinitesimal strains. The problems are analysed numerically with fully unstructured meshes that consist of a variety of two‐ and three‐dimensional element types. A detailed comparison between the vertex‐based FV and the standard Galerkin FE methods is provided with regard to discretization, solution accuracy and computational efficiency. For some problem classes a direct equivalence of the two methods is demonstrated, both theoretically and numerically. However, for other problems some interesting advantages and disadvantages of the FV formulation over the Galerkin FE method are highlighted. Copyright © 2002 John Wiley & Sons, Ltd. |
| Author | Taylor, G. A. Cross, M. Bailey, C. |
| Author_xml | – sequence: 1 givenname: G. A. surname: Taylor fullname: Taylor, G. A. email: gareth.taylor@brunel.ac.uk organization: Centre for Numerical Modelling and Process Analysis, University of Greenwich, London SE10 9JH, U.K – sequence: 2 givenname: C. surname: Bailey fullname: Bailey, C. organization: Centre for Numerical Modelling and Process Analysis, University of Greenwich, London SE10 9JH, U.K – sequence: 3 givenname: M. surname: Cross fullname: Cross, M. organization: Centre for Numerical Modelling and Process Analysis, University of Greenwich, London SE10 9JH, U.K |
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| References | Hirsch C. Numerical Computation of Internal and External Flows: Fundamentals of Numerical Discretisation, vol. 1. Wiley: New York, 1988. Owen DRJ, Hinton E. Finite Elements in Plasticity: Theory and Practice. Pineridge Press Ltd.: Swansea, UK, 1980. Hua Lu, Bailey C, Cross M. Reliability analysis of flip chip designs via computer simulation. Journal of Electronic Packaging 2000; 122:214-219. Taylor GA, Bailey C, Cross M. Solution of the elastic/visco-plastic constitutive equations: a finite volume approach. Applied Mathematical Modelling 1995; 19:746-760. Perzyna P. Fundamental problems in visco-plasticity. Advances in Applied Mechanics 1966; 9:243-377. Wheel MA. A mixed finite volume formulation for determining the small strain deformation of incompressible materials. International Journal for Numerical Methods in Engineering 1999; 44:1843-1861. Zienkiewicz OC. Origins, milestones and directions of the finite element method-a personal view. Archives of Computational Methods in Engineering 1995; 2:1-48. Idelsohn SR, Og̃ate E. Finite volumes and finite elements: two 'good friends'. International Journal for Numerical Methods in Engineering 1994; 37:3323-3341. Selmin V. The node-centred finite volume approach: bridge between finite differences and finite elements. Computer Methods in Applied Mechanics and Engineering 1992; 102:107-138. Demirdzic I, Martinovic D. Finite volume method for thermo-elasto-plastic stress analysis. Computer Methods in Applied Mechanics and Engineering 1992; 109:331-349. Baliga BR, Patanker SV. A new finite-element formulation for convection-diffusion problems. Numerical Heat Transfer 1980; 3:393-409. Bourne DE, Kendal PC. Vector Analysis and Cartesian Tensors. The Camelot Press Ltd.: Southampton, UK, 1980. Bailey C, Cross M. A finite volume procedure to solve elastic solid mechanics problems in three dimensions on an unstructured mesh. International Journal for Numerical Methods in Engineering 1995; 38:1757-1776. Zienkiewicz OC, Cormeau IC. Visco-plasticity-plasticity and creep in elastic solids-a unified numerical solution approach. International Journal for Numerical Methods in Engineering 1974; 8:821-845. Patanker SV. Numerical Heat Transfer and Fluid Flow. Hemisphere: Washington DC, 1980. Zienkiewicz OC, Taylor RL. The Finite Element Method: Volume 1: Basic Formulation and Linear Problems. McGraw-Hill: Maidenhead, UK, 1989. Theocaris PS, Marketos E. Elastic-plastic analysis of perforated thin strips of a strain-hardening material. Journal of the Mechanics and Physics of Solids 1964; 12:377-390. Hill R. The Mathematical Theory of Plasticity. Clarendon Press: Oxford, UK, 1950. Og̃ate E, Cervera M, Zienkiewicz OC. A finite volume format for structural mechanics. International Journal for Numerical Methods in Engineering 1994; 37:181-201. Fryer YD, Bailey C, Cross M, Lai C-H. A control volume procedure for solving the elastic stress-strain equations on an unstructured mesh. Applied Mathematical Modelling 1991; 15:639-645. Wheel MA. A geometrically versatile finite volume formulation for plane elastostatic stress analysis. Journal of Strain Analysis 1996; 31(2):111-116. Zienkiewicz OC, Taylor RL. The Finite Element Method: Volume 2: Solid and Fluid Mechanics, Dynamics and Non-linearity. McGraw-Hill: Maidenhead, UK, 1991. Jasak H, Weller HG. Application of the finite volume method and unstructured meshes to linear elasticity. International Journal for Numerical Methods in Engineering 2000; 48:267-287. Demirdzic I, Muzaferija S. Finite volume method for stress analysis in complex domains. International Journal for Numerical Methods in Engineering 1994; 37:3751-3766. Hattel JH, Hansen PN. A control volume-based finite difference method for solving the equilibrium equations in terms of displacements. Applied Mathematical Modelling 1995; 19:210-243. Bailey C, Wheeler D, Cross M. An integrated modeling approach to solder joint formation. IEEE Transactions on Components and Packaging Technologies 1999; 22(4):497-502. 1991; 15 2000; 48 1995; 38 1966; 9 1992; 102 1998 1996 1999; 44 1999; 22 1995 1992; 109 1950 1993 1995; 19 1991 1995; 2 1974; 8 1996; 31 1988; 1 1980; 3 1994; 37 1964; 12 2000; 122 1980 1989 Zienkiewicz OC (e_1_2_1_25_2) 1991 Hirsch C (e_1_2_1_5_2) 1988 Zienkiewicz OC (e_1_2_1_3_2) 1989 e_1_2_1_22_2 e_1_2_1_20_2 e_1_2_1_21_2 e_1_2_1_26_2 e_1_2_1_27_2 e_1_2_1_24_2 Bourne DE (e_1_2_1_23_2) 1980 e_1_2_1_29_2 Patanker SV (e_1_2_1_4_2) 1980 Owen DRJ (e_1_2_1_2_2) 1980 Hill R (e_1_2_1_28_2) 1950 e_1_2_1_6_2 e_1_2_1_30_2 e_1_2_1_7_2 e_1_2_1_11_2 e_1_2_1_34_2 e_1_2_1_12_2 e_1_2_1_33_2 e_1_2_1_32_2 e_1_2_1_10_2 e_1_2_1_31_2 e_1_2_1_15_2 e_1_2_1_16_2 e_1_2_1_13_2 e_1_2_1_14_2 e_1_2_1_19_2 e_1_2_1_8_2 e_1_2_1_17_2 e_1_2_1_9_2 e_1_2_1_18_2 |
| References_xml | – reference: Bourne DE, Kendal PC. Vector Analysis and Cartesian Tensors. The Camelot Press Ltd.: Southampton, UK, 1980. – reference: Bailey C, Cross M. A finite volume procedure to solve elastic solid mechanics problems in three dimensions on an unstructured mesh. International Journal for Numerical Methods in Engineering 1995; 38:1757-1776. – reference: Bailey C, Wheeler D, Cross M. An integrated modeling approach to solder joint formation. IEEE Transactions on Components and Packaging Technologies 1999; 22(4):497-502. – reference: Hirsch C. Numerical Computation of Internal and External Flows: Fundamentals of Numerical Discretisation, vol. 1. Wiley: New York, 1988. – reference: Og̃ate E, Cervera M, Zienkiewicz OC. A finite volume format for structural mechanics. International Journal for Numerical Methods in Engineering 1994; 37:181-201. – reference: Fryer YD, Bailey C, Cross M, Lai C-H. A control volume procedure for solving the elastic stress-strain equations on an unstructured mesh. Applied Mathematical Modelling 1991; 15:639-645. – reference: Zienkiewicz OC, Taylor RL. The Finite Element Method: Volume 1: Basic Formulation and Linear Problems. McGraw-Hill: Maidenhead, UK, 1989. – reference: Demirdzic I, Muzaferija S. Finite volume method for stress analysis in complex domains. International Journal for Numerical Methods in Engineering 1994; 37:3751-3766. – reference: Taylor GA, Bailey C, Cross M. Solution of the elastic/visco-plastic constitutive equations: a finite volume approach. Applied Mathematical Modelling 1995; 19:746-760. – reference: Hattel JH, Hansen PN. A control volume-based finite difference method for solving the equilibrium equations in terms of displacements. Applied Mathematical Modelling 1995; 19:210-243. – reference: Idelsohn SR, Og̃ate E. Finite volumes and finite elements: two 'good friends'. International Journal for Numerical Methods in Engineering 1994; 37:3323-3341. – reference: Perzyna P. Fundamental problems in visco-plasticity. Advances in Applied Mechanics 1966; 9:243-377. – reference: Hua Lu, Bailey C, Cross M. Reliability analysis of flip chip designs via computer simulation. Journal of Electronic Packaging 2000; 122:214-219. – reference: Baliga BR, Patanker SV. A new finite-element formulation for convection-diffusion problems. Numerical Heat Transfer 1980; 3:393-409. – reference: Zienkiewicz OC, Taylor RL. The Finite Element Method: Volume 2: Solid and Fluid Mechanics, Dynamics and Non-linearity. McGraw-Hill: Maidenhead, UK, 1991. – reference: Zienkiewicz OC. Origins, milestones and directions of the finite element method-a personal view. Archives of Computational Methods in Engineering 1995; 2:1-48. – reference: Wheel MA. A mixed finite volume formulation for determining the small strain deformation of incompressible materials. International Journal for Numerical Methods in Engineering 1999; 44:1843-1861. – reference: Wheel MA. A geometrically versatile finite volume formulation for plane elastostatic stress analysis. Journal of Strain Analysis 1996; 31(2):111-116. – reference: Demirdzic I, Martinovic D. Finite volume method for thermo-elasto-plastic stress analysis. Computer Methods in Applied Mechanics and Engineering 1992; 109:331-349. – reference: Zienkiewicz OC, Cormeau IC. Visco-plasticity-plasticity and creep in elastic solids-a unified numerical solution approach. International Journal for Numerical Methods in Engineering 1974; 8:821-845. – reference: Owen DRJ, Hinton E. Finite Elements in Plasticity: Theory and Practice. Pineridge Press Ltd.: Swansea, UK, 1980. – reference: Selmin V. The node-centred finite volume approach: bridge between finite differences and finite elements. Computer Methods in Applied Mechanics and Engineering 1992; 102:107-138. – reference: Jasak H, Weller HG. Application of the finite volume method and unstructured meshes to linear elasticity. International Journal for Numerical Methods in Engineering 2000; 48:267-287. – reference: Theocaris PS, Marketos E. Elastic-plastic analysis of perforated thin strips of a strain-hardening material. Journal of the Mechanics and Physics of Solids 1964; 12:377-390. – reference: Hill R. The Mathematical Theory of Plasticity. Clarendon Press: Oxford, UK, 1950. – reference: Patanker SV. Numerical Heat Transfer and Fluid Flow. Hemisphere: Washington DC, 1980. – volume: 37 start-page: 3751 year: 1994 end-page: 3766 article-title: Finite volume method for stress analysis in complex domains publication-title: International Journal for Numerical Methods in Engineering – volume: 8 start-page: 821 year: 1974 end-page: 845 article-title: Visco‐plasticity—plasticity and creep in elastic solids—a unified numerical solution approach publication-title: International Journal for Numerical Methods in Engineering – start-page: 1459 year: 1995 end-page: 1470 – volume: 44 start-page: 1843 year: 1999 end-page: 1861 article-title: A mixed finite volume formulation for determining the small strain deformation of incompressible materials publication-title: International Journal for Numerical Methods in Engineering – volume: 31 start-page: 111 issue: 2 year: 1996 end-page: 116 article-title: A geometrically versatile finite volume formulation for plane elastostatic stress analysis publication-title: Journal of Strain Analysis – volume: 22 start-page: 497 issue: 4 year: 1999 end-page: 502 article-title: An integrated modeling approach to solder joint formation publication-title: IEEE Transactions on Components and Packaging Technologies – volume: 122 start-page: 214 year: 2000 end-page: 219 article-title: Reliability analysis of flip chip designs via computer simulation publication-title: Journal of Electronic Packaging – year: 1989 – year: 1996 – volume: 37 start-page: 181 year: 1994 end-page: 201 article-title: A finite volume format for structural mechanics publication-title: International Journal for Numerical Methods in Engineering – volume: 37 start-page: 3323 year: 1994 end-page: 3341 article-title: Finite volumes and finite elements: two ‘good friends’ publication-title: International Journal for Numerical Methods in Engineering – year: 1950 – volume: 38 start-page: 1757 year: 1995 end-page: 1776 article-title: A finite volume procedure to solve elastic solid mechanics problems in three dimensions on an unstructured mesh publication-title: International Journal for Numerical Methods in Engineering – volume: 48 start-page: 267 year: 2000 end-page: 287 article-title: Application of the finite volume method and unstructured meshes to linear elasticity publication-title: International Journal for Numerical Methods in Engineering – volume: 19 start-page: 746 year: 1995 end-page: 760 article-title: Solution of the elastic/visco‐plastic constitutive equations: a finite volume approach publication-title: Applied Mathematical Modelling – volume: 15 start-page: 639 year: 1991 end-page: 645 article-title: A control volume procedure for solving the elastic stress–strain equations on an unstructured mesh publication-title: Applied Mathematical Modelling – volume: 1 year: 1988 – year: 1980 – volume: 3 start-page: 393 year: 1980 end-page: 409 article-title: A new finite‐element formulation for convection‐diffusion problems publication-title: Numerical Heat Transfer – volume: 19 start-page: 210 year: 1995 end-page: 243 article-title: A control volume‐based finite difference method for solving the equilibrium equations in terms of displacements publication-title: Applied Mathematical Modelling – volume: 9 start-page: 243 year: 1966 end-page: 377 article-title: Fundamental problems in visco‐plasticity publication-title: Advances in Applied Mechanics – volume: 12 start-page: 377 year: 1964 end-page: 390 article-title: Elastic–plastic analysis of perforated thin strips of a strain‐hardening material publication-title: Journal of the Mechanics and Physics of Solids – year: 1991 – volume: 2 start-page: 1 year: 1995 end-page: 48 article-title: Origins, milestones and directions of the finite element method—a personal view publication-title: Archives of Computational Methods in Engineering – year: 1993 – volume: 109 start-page: 331 year: 1992 end-page: 349 article-title: Finite volume method for thermo‐elasto‐plastic stress analysis publication-title: Computer Methods in Applied 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| SubjectTerms | Computation Computational efficiency computational solid mechanics Finite element method finite volume Galerkin methods Mathematical models Nonlinearity Solid mechanics Three dimensional vertex-based |
| Title | A vertex-based finite volume method applied to non-linear material problems in computational solid mechanics |
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