Variational formulation of linear problems with nonhomogeneous boundary conditions and internal discontinuities
A variational formulation applicable to linear operators with nonhomogeneous boundary conditions and jump discontinuities is presented. For the formulation to be applicable, the boundary condition and discontinuities have to be consistent with the operator governing the field problem. The problem is...
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| Published in | Computer methods in applied mechanics and engineering Vol. 7; no. 1; pp. 75 - 91 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
1976
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| Online Access | Get full text |
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| Abstract | A variational formulation applicable to linear operators with nonhomogeneous boundary conditions and jump discontinuities is presented. For the formulation to be applicable, the boundary condition and discontinuities have to be consistent with the operator governing the field problem. The problem is set up in a space of suitable continuous bilinear mapping. Thus, operators on inner product spaces, convolution spaces and energy spaces are included as specializations. The basic construction can be used to generate dual-complementary variational principles. Implementation is illustrated by examples. The role of boundary terms in finite element discretizations based on interpolants of limited smoothness is discussed. |
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| AbstractList | A variational formulation applicable to linear operators with nonhomogeneous boundary conditions and jump discontinuities is presented. For the formulation to be applicable, the boundary condition and discontinuities have to be consistent with the operator governing the field problem. The problem is set up in a space of suitable continuous bilinear mapping. Thus, operators on inner product spaces, convolution spaces and energy spaces are included as specializations. The basic construction can be used to generate dual-complementary variational principles. Implementation is illustrated by examples. The role of boundary terms in finite element discretizations based on interpolants of limited smoothness is discussed. A variational formulation applicable to linear operators with nonhomogeneous boundary conditions and jump discontinuities is presented. For the formulation to be applicable, the boundary condition and discontinuities have to be consistent with the operator governing the field problem. The problem is set up in a space of suitable continuous bilinear mapping. Thus, operators on inner product spaces, convolution spaces and energy spaces are included as specializations. The basic construction can be used to generate dual-complementary variational principles. Implementation is illustrated by examples. The role of boundary terms in finite element discretization based on interpolants of limited smoothness is discussed. |
| Author | Sandhu, Ranbir S. Salaam, U. |
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| Cites_doi | 10.1016/0020-7225(74)90073-1 10.1007/BF01262691 10.1090/qam/99951 10.1007/BF00248489 10.1007/BF02153074 10.1016/0020-7225(70)90081-9 10.1002/nme.1620010103 10.1016/0020-7683(68)90081-4 10.1016/0020-7683(71)90085-0 |
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| References | Magri (BIB20) 1972 Oden (BIB7) 1972 Mikhlin (BIB4) 1970 Pian (BIB11) 1972 Tonti (BIB10) 1967; 2 Gurtin (BIB14) 1964; 16 Gurtin (BIB13) 1963; 13 Oden (BIB18) 1971 Arthurs (BIB9) 1970 Tikhonov, Samarskii (BIB27) 1963 Prager (BIB24) 1967 Oden, Reddy (BIB12) 1974; 12 Tonti (BIB8) 1972 Salaam (BIB28) 1974 Pian, Tong (BIB29) 1969; 1 Tonti (BIB19) 1972 Washizu (BIB22) 1968 Gurtin (BIB15) 1964; 12 Sandhu, Pister (BIB17) 1971; 7 Vainberg (BIB5) 1964 Finlayson (BIB6) 1972 Nemat-Nasser (BIB26) 1974 Mikhlin (BIB1) 1965 Reissner (BIB21) 1958; 8 Sandhu, Pister (BIB16) 1970; 8 Mikhlin (BIB2) 1964 Prager (BIB25) 1968; 4 Nickelland, Sackman (BIB23) 1968; 26 Mikhlin (BIB3) 1971 Gurtin (10.1016/0045-7825(76)90006-2_BIB15) 1964; 12 Salaam (10.1016/0045-7825(76)90006-2_BIB28) 1974 Mikhlin (10.1016/0045-7825(76)90006-2_BIB2) 1964 Mikhlin (10.1016/0045-7825(76)90006-2_BIB3) 1971 Nickelland (10.1016/0045-7825(76)90006-2_BIB23) 1968; 26 Prager (10.1016/0045-7825(76)90006-2_BIB25) 1968; 4 Tonti (10.1016/0045-7825(76)90006-2_BIB19) 1972 Washizu (10.1016/0045-7825(76)90006-2_BIB22) 1968 Vainberg (10.1016/0045-7825(76)90006-2_BIB5) 1964 Tonti (10.1016/0045-7825(76)90006-2_BIB10) 1967; 2 Tikhonov (10.1016/0045-7825(76)90006-2_BIB27) 1963 Mikhlin (10.1016/0045-7825(76)90006-2_BIB1) 1965 Pian (10.1016/0045-7825(76)90006-2_BIB11) 1972 Oden (10.1016/0045-7825(76)90006-2_BIB18) 1971 Oden (10.1016/0045-7825(76)90006-2_BIB12) 1974; 12 Nemat-Nasser (10.1016/0045-7825(76)90006-2_BIB26) 1974 Sandhu (10.1016/0045-7825(76)90006-2_BIB17) 1971; 7 Tonti (10.1016/0045-7825(76)90006-2_BIB8) 1972 Reissner (10.1016/0045-7825(76)90006-2_BIB21) 1958; 8 Arthurs (10.1016/0045-7825(76)90006-2_BIB9) 1970 Mikhlin (10.1016/0045-7825(76)90006-2_BIB4) 1970 Pian (10.1016/0045-7825(76)90006-2_BIB29) 1969; 1 Magri (10.1016/0045-7825(76)90006-2_BIB20) 1972 Prager (10.1016/0045-7825(76)90006-2_BIB24) 1967 Gurtin (10.1016/0045-7825(76)90006-2_BIB14) 1964; 16 Sandhu (10.1016/0045-7825(76)90006-2_BIB16) 1970; 8 Gurtin (10.1016/0045-7825(76)90006-2_BIB13) 1963; 13 Finlayson (10.1016/0045-7825(76)90006-2_BIB6) 1972 Oden (10.1016/0045-7825(76)90006-2_BIB7) 1972 |
| References_xml | – start-page: 463 year: 1967 end-page: 474 ident: BIB24 article-title: Variational principles of linear elastostatics for discontinuous displacements, strains and stresses publication-title: Recent progress in applied mechanics contributor: fullname: Prager – volume: 1 start-page: 3 year: 1969 end-page: 28 ident: BIB29 article-title: Basis of finite element methods for solid continua publication-title: Int. J. Num. Math. Eng. contributor: fullname: Tong – year: 1968 ident: BIB22 article-title: Variational methods in elasticity and plasticity contributor: fullname: Washizu – volume: 2 start-page: 201 year: 1967 end-page: 209 ident: BIB10 article-title: Variational principles in elastostatics publication-title: Meccanica contributor: fullname: Tonti – year: 1972 ident: BIB19 article-title: On the variational formulation for linear initial value problems publication-title: Report Instituto di Matematica del Politechnico di Milano 32-20133 contributor: fullname: Tonti – year: 1972 ident: BIB20 article-title: Variational formulation for any linear eqiation publication-title: Variational methods in engineering contributor: fullname: Magri – year: 1970 ident: BIB4 article-title: An advanced course in mathematical physics contributor: fullname: Mikhlin – volume: 8 start-page: 989 year: 1970 end-page: 999 ident: BIB16 article-title: A variational principle for linear coupled field problems publication-title: Int. J. Eng. Sci. contributor: fullname: Pister – year: 1965 ident: BIB1 article-title: The problem of the minimum of a quadratic functional contributor: fullname: Mikhlin – year: 1971 ident: BIB18 article-title: Finite element models of nonlinear operator equations publication-title: Proceedings, Third Conference on matrix methods in structural mechanics contributor: fullname: Oden – year: 1972 ident: BIB7 article-title: Variational principles in nonlinear continuum mechanics publication-title: Variational methods in engineering contributor: fullname: Oden – volume: 26 start-page: 11 year: 1968 end-page: 26 ident: BIB23 article-title: Variational principles for linear coupled thermoelasticity publication-title: App. Math. contributor: fullname: Sackman – year: 1970 ident: BIB9 article-title: Complementary variational principles contributor: fullname: Arthurs – volume: 7 start-page: 639 year: 1971 end-page: 654 ident: BIB17 article-title: Variational principles for boundary value and initial boundary value problems publication-title: Int. J. Solids Struc. contributor: fullname: Pister – volume: 4 start-page: 837 year: 1968 end-page: 844 ident: BIB25 article-title: Variational principles for elastic plates with relaxed continuity requirements publication-title: Int. J. Solids Struc. contributor: fullname: Prager – volume: 12 start-page: 252 year: 1964 end-page: 256 ident: BIB15 article-title: Variational principles for linear initial value problems publication-title: Quar. App. Math. contributor: fullname: Gurtin – year: 1964 ident: BIB5 article-title: Variational methods for the study of nonlinear operators contributor: fullname: Vainberg – year: 1963 ident: BIB27 article-title: Equations of mathematical physics contributor: fullname: Samarskii – year: 1972 ident: BIB6 article-title: The method of Weighted residuals and variational principles contributor: fullname: Finlayson – year: 1974 ident: BIB26 article-title: General variational principles in nonlinear and linear elasticity with applications publication-title: Mechanics Today contributor: fullname: Nemat-Nasser – year: 1964 ident: BIB2 article-title: Variational methods in mathematical physics contributor: fullname: Mikhlin – year: 1972 ident: BIB8 article-title: A systematic approach to the search for variational principles publication-title: Variational methods in engineering contributor: fullname: Tonti – year: 1972 ident: BIB11 article-title: Finite element formulation by variational principles with relaxed continuity requirements publication-title: The mathematical foundations of the finite element method with applications to partial differential equations contributor: fullname: Pian – volume: 8 start-page: 1 year: 1958 end-page: 6 ident: BIB21 article-title: On variational principles in elasticity publication-title: Proc. Sym. App. Math. contributor: fullname: Reissner – volume: 12 start-page: 1 year: 1974 end-page: 29 ident: BIB12 article-title: On dual-complementary variational principles in mathematical physics publication-title: Int. J. Eng. Sci. contributor: fullname: Reddy – volume: 16 start-page: 234 year: 1964 end-page: 254 ident: BIB14 article-title: Variational principles in linear elastodynamics publication-title: Arch. Rat. Mech. Anal. contributor: fullname: Gurtin – volume: 13 start-page: 179 year: 1963 end-page: 191 ident: BIB13 article-title: Variational principles in the linear theory of viscoelasticity publication-title: Arch. Rat. Mech. Anal. contributor: fullname: Gurtin – year: 1974 ident: BIB28 article-title: Finite elements with relaxed continuity, an investigation of the variational basis and numerical performance publication-title: Ph.D. Thesis contributor: fullname: Salaam – year: 1971 ident: BIB3 article-title: The numerical performance of variational methods contributor: fullname: Mikhlin – volume: 12 start-page: 1 year: 1974 ident: 10.1016/0045-7825(76)90006-2_BIB12 article-title: On dual-complementary variational principles in mathematical physics publication-title: Int. J. Eng. Sci. doi: 10.1016/0020-7225(74)90073-1 contributor: fullname: Oden – year: 1972 ident: 10.1016/0045-7825(76)90006-2_BIB20 article-title: Variational formulation for any linear eqiation contributor: fullname: Magri – year: 1972 ident: 10.1016/0045-7825(76)90006-2_BIB6 contributor: fullname: Finlayson – year: 1974 ident: 10.1016/0045-7825(76)90006-2_BIB26 article-title: General variational principles in nonlinear and linear elasticity with applications contributor: fullname: Nemat-Nasser – year: 1974 ident: 10.1016/0045-7825(76)90006-2_BIB28 article-title: Finite elements with relaxed continuity, an investigation of the variational basis and numerical performance contributor: fullname: Salaam – year: 1972 ident: 10.1016/0045-7825(76)90006-2_BIB11 article-title: Finite element formulation by variational principles with relaxed continuity requirements contributor: fullname: Pian – year: 1971 ident: 10.1016/0045-7825(76)90006-2_BIB18 article-title: Finite element models of nonlinear operator equations contributor: fullname: Oden – volume: 13 start-page: 179 year: 1963 ident: 10.1016/0045-7825(76)90006-2_BIB13 article-title: Variational principles in the linear theory of viscoelasticity publication-title: Arch. Rat. Mech. Anal. doi: 10.1007/BF01262691 contributor: fullname: Gurtin – year: 1971 ident: 10.1016/0045-7825(76)90006-2_BIB3 contributor: fullname: Mikhlin – volume: 12 start-page: 252 year: 1964 ident: 10.1016/0045-7825(76)90006-2_BIB15 article-title: Variational principles for linear initial value problems publication-title: Quar. App. Math. doi: 10.1090/qam/99951 contributor: fullname: Gurtin – year: 1964 ident: 10.1016/0045-7825(76)90006-2_BIB5 contributor: fullname: Vainberg – volume: 26 start-page: 11 year: 1968 ident: 10.1016/0045-7825(76)90006-2_BIB23 article-title: Variational principles for linear coupled thermoelasticity publication-title: App. Math. contributor: fullname: Nickelland – year: 1963 ident: 10.1016/0045-7825(76)90006-2_BIB27 contributor: fullname: Tikhonov – year: 1970 ident: 10.1016/0045-7825(76)90006-2_BIB4 contributor: fullname: Mikhlin – year: 1972 ident: 10.1016/0045-7825(76)90006-2_BIB19 article-title: On the variational formulation for linear initial value problems publication-title: Report Instituto di Matematica del Politechnico di Milano 32-20133 contributor: fullname: Tonti – year: 1972 ident: 10.1016/0045-7825(76)90006-2_BIB8 article-title: A systematic approach to the search for variational principles contributor: fullname: Tonti – year: 1970 ident: 10.1016/0045-7825(76)90006-2_BIB9 contributor: fullname: Arthurs – volume: 16 start-page: 234 year: 1964 ident: 10.1016/0045-7825(76)90006-2_BIB14 article-title: Variational principles in linear elastodynamics publication-title: Arch. Rat. Mech. Anal. doi: 10.1007/BF00248489 contributor: fullname: Gurtin – year: 1965 ident: 10.1016/0045-7825(76)90006-2_BIB1 contributor: fullname: Mikhlin – year: 1972 ident: 10.1016/0045-7825(76)90006-2_BIB7 article-title: Variational principles in nonlinear continuum mechanics contributor: fullname: Oden – volume: 2 start-page: 201 year: 1967 ident: 10.1016/0045-7825(76)90006-2_BIB10 article-title: Variational principles in elastostatics publication-title: Meccanica doi: 10.1007/BF02153074 contributor: fullname: Tonti – volume: 8 start-page: 989 year: 1970 ident: 10.1016/0045-7825(76)90006-2_BIB16 article-title: A variational principle for linear coupled field problems publication-title: Int. J. Eng. Sci. doi: 10.1016/0020-7225(70)90081-9 contributor: fullname: Sandhu – year: 1968 ident: 10.1016/0045-7825(76)90006-2_BIB22 contributor: fullname: Washizu – year: 1964 ident: 10.1016/0045-7825(76)90006-2_BIB2 contributor: fullname: Mikhlin – volume: 8 start-page: 1 year: 1958 ident: 10.1016/0045-7825(76)90006-2_BIB21 article-title: On variational principles in elasticity contributor: fullname: Reissner – volume: 1 start-page: 3 year: 1969 ident: 10.1016/0045-7825(76)90006-2_BIB29 article-title: Basis of finite element methods for solid continua publication-title: Int. J. Num. Math. Eng. doi: 10.1002/nme.1620010103 contributor: fullname: Pian – volume: 4 start-page: 837 year: 1968 ident: 10.1016/0045-7825(76)90006-2_BIB25 article-title: Variational principles for elastic plates with relaxed continuity requirements publication-title: Int. J. Solids Struc. doi: 10.1016/0020-7683(68)90081-4 contributor: fullname: Prager – start-page: 463 year: 1967 ident: 10.1016/0045-7825(76)90006-2_BIB24 article-title: Variational principles of linear elastostatics for discontinuous displacements, strains and stresses contributor: fullname: Prager – volume: 7 start-page: 639 year: 1971 ident: 10.1016/0045-7825(76)90006-2_BIB17 article-title: Variational principles for boundary value and initial boundary value problems publication-title: Int. J. Solids Struc. doi: 10.1016/0020-7683(71)90085-0 contributor: fullname: Sandhu |
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