Stochastic optimization methods in optimal engineering design under stochastic uncertainty
Problems from optimal plastic design are based on the convex, linear or linearized yield/strength condition and the linear equilibrium equation for the generic stress (state) vector. Having to take into account, in practice, stochastic variations of the model parameters (e.g. yield stresses, plastic...
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| Veröffentlicht in: | Zeitschrift für angewandte Mathematik und Mechanik Jg. 83; H. 12; S. 795 - 811 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Berlin
WILEY-VCH Verlag
01.12.2003
WILEY‐VCH Verlag |
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| ISSN: | 0044-2267, 1521-4001 |
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| Abstract | Problems from optimal plastic design are based on the convex, linear or linearized yield/strength condition and the linear equilibrium equation for the generic stress (state) vector. Having to take into account, in practice, stochastic variations of the model parameters (e.g. yield stresses, plastic capacities) and external loadings, the basic stochastic optimal plastic design problem must be replaced by an appropriate deterministic substitute problem. After considering stochastic optimization methods based on failure/survival probabilities and presenting differentiation techniques and differentiation formulas for probability of failure/survival functions, a direct approach is presented using the construction and failure costs (e.g. costs for damage, repair, compensation for weakness within the structure, etc.). Based on the basic mechanical survival conditions, the failure costs may be represented by the minimum value of a convex and often linear program. Several mathematical properties of this program are shown. Minimizing then the total expected costs subject to the remaining (simple) deterministic constraints, a stochastic optimization problem is obtained which may be represented by a “Stochastic Convex Program (SCP) with recourse”. Working with linearized yield/strength conditions, a “Stochastic Linear Program (SLP) with complete fixed recourse” results. In case of a discretely distributed probability distribution or after the discretization of a more general probability distribution of the random structural parameters and loadings as well as certain random cost factors one has a linear program (LP) with a so‐called “dual decomposition data” structure. For stochastic programs of this type many theoretical results and efficient numerical solution procedures (LP‐solver) are available. The mathematical properties of theses substitute problems are considered, and numerical solution procedures are described. |
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| AbstractList | Problems from optimal plastic design are based on the convex, linear or linearized yield/strength condition and the linear equilibrium equation for the generic stress (state) vector. Having to take into account, in practice, stochastic variations of the model parameters (e.g. yield stresses, plastic capacities) and external loadings, the basic stochastic optimal plastic design problem must be replaced by an appropriate deterministic substitute problem. After considering stochastic optimization methods based on failure/survival probabilities and presenting differentiation techniques and differentiation formulas for probability of failure/survival functions, a direct approach is presented using the construction and failure costs (e.g. costs for damage, repair, compensation for weakness within the structure, etc.). Based on the basic mechanical survival conditions, the failure costs may be represented by the minimum value of a convex and often linear program. Several mathematical properties of this program are shown. Minimizing then the total expected costs subject to the remaining (simple) deterministic constraints, a stochastic optimization problem is obtained which may be represented by a “Stochastic Convex Program (SCP) with recourse”. Working with linearized yield/strength conditions, a “Stochastic Linear Program (SLP) with complete fixed recourse” results. In case of a discretely distributed probability distribution or after the discretization of a more general probability distribution of the random structural parameters and loadings as well as certain random cost factors one has a linear program (LP) with a so‐called “dual decomposition data” structure. For stochastic programs of this type many theoretical results and efficient numerical solution procedures (LP‐solver) are available. The mathematical properties of theses substitute problems are considered, and numerical solution procedures are described. |
| Author | Marti, K. |
| Author_xml | – sequence: 1 givenname: K. surname: Marti fullname: Marti, K. email: kurt.marti@unibw-muenchen.de organization: Federal Armed Forces University Munich, Aerospace Engineering and Technology, 85577 Neubiberg/München, Germany |
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| References | K. Marti, Computation of descent directions in stochastic optimization problems with invariant distributions, Z. Angew. Math. Mech. 65, 355-378 (1985). K. Marti, Approximationen der Entscheidungprobleme mit linearer Ergebnisfunktion und positiv homogener, subadditiver Verlustfunktion, Z. Wahrscheinlichkeitstheor. Verwandte Geb. 31, 203-233 (1975). K. Marti, Diskretisierung stochastischer Programme unter Berücksichtigung der Problemstruktur, Z. Angew. Math. Mech. 59, T105-T108 (1979). G. Cocchetti and G. Maier, Static shakedown theorems in piecewise linearized poroplasticity, Arch. Appl. Mech. 68, 651-661 (1988). L. Corradi and A. Zavelani, A linear programming approach to shakedown analysis of structures, Comput. Methods Appl. Mech. Eng. (Netherlands) 3, 37-53 (1974). D.M. Frangopol, Sensitivity of reliability-based optimum design, J. Struct. Div. 111, 1703-1721 (1985). W. Prager and R.T. Shield, A general theory of optimal plastic design, J. Appl. Mech. 34(1), 184-186 (1967). D.L. Smith and J. Munro, Plastic analysis and synthesis of frames subject to multiple loadings, Eng. Optim. 2, 145-157 (1976). F. Tin-Loi, Plastic limit analysis of plane frames and grids using GAMS, Comput. Struct. 54, 15-25 (1995). K. Marti, Stochastic optimization methods in structural mechanics, Z. Angew. Math. Mech. 70, T742-T745 (1990). Y. Murotsu and S. Shao, Optimum shape design of truss structures based on reliability, Struct. Optim. 2(2), 65-76 (1990). O. Ditlevsen, Narrow reliability bounds for structural systems, J. Struct. Mech. 7, 453-472 (1979). K. Marti, Approximation and derivatives of probabilities of survival in structural analysis and design, Struct. Optim. 13, 230-243 (1997). D.M. Frangopol et al. (eds.), Reliability oriented structural design, Special Issue of Reliab. Eng. Syst. Saf. 73(3), 105-301 (2001). K. Marti, Differentiation formulas for probability functions: the transformation method, Math. Program. B 75, 201-220 (1996). E.G. Kounias, Bounds for the probability of a union, with applications, Ann. Math. Statist. 39, 2154-2158 (1968). G. Maier and J. Munro, Mathematical programming applications to engineering plastic analysis, Appl. Mech. Rev. 35(12), 1631-1643 (1982). J. Galambos, Bonferroni inequalities, The Annals of Probability 5, 577-581 (1977). K. Marti, Computation of efficient solutions of discretely distributed stochastic optimization problems, Math. Methods Oper. Res. 36, 259-294 (1992). K. Marti, Differentiation of probability functions: the transformation method, Comput. Math. Appl. 30, 361-382 (1995). F. Tin-Loi, On the optimal plastic synthesis of frames, Eng. Optim. 16, 91-108 (1990). C.A. Cornell, Bounds on the reliability of structural systems, J. Struct. Div., ASCE 93, 171-200 (1967). J. Heyman, Plastic design of beams and plane frames for minimum material consumption, Q. Appl. Math. 8, 373-381 (1950/51). K. Marti, Entscheidungsprobleme mit linearem Aktionen- und Ergebnisraum, Z. Wahrscheinlichkeitstheor. Verwandte Geb. 23, 133-147 (1972). 1995; 30 1982; 35(12) 1979; 59 1968; 39 1990; 16 1967; 93 1997; 13 1995; 54 1990; 2(2) 1976; 2 1988; 68 1967; 34(1) 1975; 31 1992; 36 2001; 73(3) 8 1972; 23 1985; 65 1974; 3 1990; 70 1996; 75 1977; 5 1979; 7 1985; 111 e_1_2_1_41_2 e_1_2_1_45_2 e_1_2_1_20_2 e_1_2_1_43_2 Marti K. (e_1_2_1_26_2) 1979; 59 e_1_2_1_49_2 e_1_2_1_24_2 e_1_2_1_47_2 Frangopol D.M. (e_1_2_1_9_2) 2001; 73 e_1_2_1_28_2 e_1_2_1_6_2 Marti K. (e_1_2_1_29_2) 1990; 70 e_1_2_1_2_2 e_1_2_1_12_2 e_1_2_1_33_2 e_1_2_1_50_2 e_1_2_1_10_2 e_1_2_1_31_2 e_1_2_1_16_2 e_1_2_1_37_2 e_1_2_1_14_2 e_1_2_1_35_2 Heyman J. (e_1_2_1_13_2); 8 e_1_2_1_8_2 e_1_2_1_18_2 e_1_2_1_39_2 Cornell C.A. (e_1_2_1_4_2) 1967; 93 e_1_2_1_40_2 e_1_2_1_23_2 e_1_2_1_44_2 e_1_2_1_21_2 e_1_2_1_42_2 e_1_2_1_27_2 e_1_2_1_48_2 e_1_2_1_25_2 e_1_2_1_46_2 Maier G. (e_1_2_1_22_2) 1982; 35 e_1_2_1_30_2 e_1_2_1_7_2 e_1_2_1_5_2 e_1_2_1_11_2 e_1_2_1_34_2 e_1_2_1_3_2 e_1_2_1_32_2 e_1_2_1_51_2 e_1_2_1_15_2 e_1_2_1_38_2 e_1_2_1_36_2 e_1_2_1_19_2 e_1_2_1_17_2 |
| References_xml | – reference: D.L. Smith and J. Munro, Plastic analysis and synthesis of frames subject to multiple loadings, Eng. Optim. 2, 145-157 (1976). – reference: C.A. Cornell, Bounds on the reliability of structural systems, J. Struct. Div., ASCE 93, 171-200 (1967). – reference: W. Prager and R.T. Shield, A general theory of optimal plastic design, J. Appl. Mech. 34(1), 184-186 (1967). – reference: K. Marti, Differentiation formulas for probability functions: the transformation method, Math. Program. B 75, 201-220 (1996). – reference: F. Tin-Loi, On the optimal plastic synthesis of frames, Eng. Optim. 16, 91-108 (1990). – reference: K. Marti, Computation of efficient solutions of discretely distributed stochastic optimization problems, Math. Methods Oper. Res. 36, 259-294 (1992). – reference: G. Cocchetti and G. Maier, Static shakedown theorems in piecewise linearized poroplasticity, Arch. Appl. Mech. 68, 651-661 (1988). – reference: J. Heyman, Plastic design of beams and plane frames for minimum material consumption, Q. Appl. Math. 8, 373-381 (1950/51). – reference: K. Marti, Differentiation of probability functions: the transformation method, Comput. Math. Appl. 30, 361-382 (1995). – reference: K. Marti, Diskretisierung stochastischer Programme unter Berücksichtigung der Problemstruktur, Z. Angew. Math. Mech. 59, T105-T108 (1979). – reference: Y. Murotsu and S. Shao, Optimum shape design of truss structures based on reliability, Struct. Optim. 2(2), 65-76 (1990). – reference: K. Marti, Approximation and derivatives of probabilities of survival in structural analysis and design, Struct. Optim. 13, 230-243 (1997). – reference: F. Tin-Loi, Plastic limit analysis of plane frames and grids using GAMS, Comput. Struct. 54, 15-25 (1995). – reference: J. Galambos, Bonferroni inequalities, The Annals of Probability 5, 577-581 (1977). – reference: K. Marti, Entscheidungsprobleme mit linearem Aktionen- und Ergebnisraum, Z. Wahrscheinlichkeitstheor. Verwandte Geb. 23, 133-147 (1972). – reference: D.M. Frangopol et al. (eds.), Reliability oriented structural design, Special Issue of Reliab. Eng. Syst. Saf. 73(3), 105-301 (2001). – reference: O. Ditlevsen, Narrow reliability bounds for structural systems, J. Struct. Mech. 7, 453-472 (1979). – reference: D.M. Frangopol, Sensitivity of reliability-based optimum design, J. Struct. Div. 111, 1703-1721 (1985). – reference: E.G. Kounias, Bounds for the probability of a union, with applications, Ann. Math. Statist. 39, 2154-2158 (1968). – reference: K. Marti, Stochastic optimization methods in structural mechanics, Z. Angew. Math. Mech. 70, T742-T745 (1990). – reference: K. Marti, Computation of descent directions in stochastic optimization problems with invariant distributions, Z. Angew. Math. Mech. 65, 355-378 (1985). – reference: G. Maier and J. Munro, Mathematical programming applications to engineering plastic analysis, Appl. Mech. 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| SubjectTerms | expected costs of failure limit load analysis optimal structural design probability of failure stochastic linear programming stochastic optimization stochastic uncertainty |
| Title | Stochastic optimization methods in optimal engineering design under stochastic uncertainty |
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