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Biomimetic approach to compliance optimization and multiple load cases. (English) Zbl 1432.74184
Summary: The variational approach to shape optimization in linearized elasticity is used in order to improve convergence of a known heuristic algorithm. The speed method of shape optimization is applied to obtain necessary optimality conditions for representative test examples. The algorithm originates from the biomimetic approach to compliance optimization. The trabecular bone adapts its form to mechanical loads and is able to form structures that are lightweight and very stiff at the same time. In this sense, it is a problem pertaining to both the nature or living entities which is similar to structural optimization, especially topology optimization. The paper presents the biomimetic approach, based on the trabecular bone remodeling phenomenon, with the aim of minimizing the compliance in multiple load cases. The method employed aims at minimizing the energy and combines structural evolution inspired by trabecular bone remodeling and the shape gradient framework, with strict analysis based on functionals in the 3-dimensional elasticity model. The method is enhanced to handle the problem of structural optimization under multiple loads. The new biomimetic approach does not require volume constraints. Instead of imposing volume constraints, shapes are parameterized by the assumed strain energy density on the structural surface. The stiffest design is obtained by adding or removing material on the structural surface in virtual space. Structural evolution is based on shape gradient approximation by the speed method, and it is separated from the finite element method of the model solution. Numerical examples confirm that the heuristic algorithm for structural optimization is efficient.
74P10 Optimization of other properties in solid mechanics
35C20 Asymptotic expansions of solutions to PDEs
35J15 Second-order elliptic equations
35S05 Pseudodifferential operators as generalizations of partial differential operators
49J40 Variational inequalities
49Q12 Sensitivity analysis for optimization problems on manifolds
Matlab; top.m
Full Text: DOI
[1] Novotny, Aa; Sokołowski, J.; Żochowski, A., Topological derivatives of shape functionals. Part I: theory in singularly perturbed geometrical domains, J. Optim. Theory Appl., 180, 341-373 (2019) · Zbl 1409.35018
[2] Novotny, Aa; Sokołowski, J.; Żochowski, A., Topological derivatives of shape functionals. Part II: first-order method and applications, J. Optim. Theory Appl., 180, 683-710 (2019) · Zbl 1414.49045
[3] Novotny, Antonio André; Sokołowski, Jan; Żochowski, Antoni, Topological Derivatives of Shape Functionals. Part III: Second-Order Method and Applications, Journal of Optimization Theory and Applications, 181, 1, 1-22 (2018) · Zbl 1419.35016
[4] Arora, Rk, Optimization—Algorithms and Applications (2015), Boca Raton: CRC Press, Boca Raton
[5] Haftka, R.; Gürdal, Z., Elements of Structural Optimization (1992), Dordrecht: Kluwer, Dordrecht · Zbl 0782.73004
[6] Wolff, J., Das Gesetz der Transformation der Knochen (1892), New York: Hirschwald, New York
[7] Maquet, P.; Furlong, R., The Law of Bone Remodeling (English Translation of Wolffs 1892 Article) (1986), Berlin: Springer, Berlin
[8] Cowin, Sc; Hegedus, Dh, Bone remodeling I: a theory of adaptive elasticity, J. Elast., 6, 313-326 (1976) · Zbl 0335.73028
[9] Huiskes, R., Effects of mechanical forces on maintenance and adaptation of form in trabecular bone, Nature, 404, 704-706 (2000)
[10] Wasiutyński, Z., On the congruency of the forming according to the minimum potential energy with that according to equal strength, Bull. Acad. Polon. Sci. Sér. Sci. Techbol., 7, 259-268 (1960) · Zbl 0097.17403
[11] Pedersen P.: Optimal designs—structures and materials—problems and tools (2003). ISBN 87-90416-06-6
[12] Nowak, M.; Sokołowski, J.; Żochowski, A., Justification of a certain algorithm for shape optimization in 3D elasticity, Struct. Multidiscip. Optim., 57, 721-734 (2018)
[13] Sokołowski, J.; Zolesio, J., Introduction to Shape Optimization. Shape Sensitivity Analysis (1992), Berlin: Springer, Berlin · Zbl 0761.73003
[14] Sigmund, O., A 99 line topology optimization code written in Matlab, Struct. Multidiscip. Optim., 21, 120-127 (2001)
[15] Bendsoe, M.; Olhoff, N.; Taylor, J., A variational formulation for multicriteria structural optimization, J. Struct. Mech., 11, 523-544 (1984)
[16] Bendsoe, M.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng., 71, 197-224 (1988) · Zbl 0671.73065
[17] Krog, L., Tucker, A., Kemp, M., Boyd, R.: Topology optimization of aircraft wing box ribs. AIAA Paper 2004-4481 (2004)
[18] Nishiwaki, S., Optimal structural design considering flexibility, Comput. Methods Appl. Mech. Eng., 190, 4457-4504 (2001)
[19] James, K.; Hansen, J.; Martins, J., Structural topology optimization for multiple load cases using a dynamic aggregation technique, Eng. Optim., 41, 1103-1118 (2009)
[20] Bruggi, M., On an alternative approach to stress constraints relaxation in topology optimization, Struct. Multidiscip. Optim., 36, 125-141 (2008) · Zbl 1273.74397
[21] Le, C.; Norato, J.; Bruns, T.; Ha, C.; Tortorelli, D., Stress-based topology optimization for continua, Struct. Multidiscip. Optim., 41, 605-620 (2010)
[22] Carter, Dr, Mechanical loading histories and cortical bone remodeling, Calcif. Tissue Int., 36, Suppl. 1, S19-S24 (1984)
[23] Frost, Hm, The Laws of Bone Structure (1964), Springfield: C.C. Thomas, Springfield
[24] Huiskes, R., Adaptive bone-remodeling theory applied to prosthetic-design analysis, J. Biomech., 20, 1135-1150 (1987)
[25] Nowak, M., Structural optimization system based on trabecular bone surface adaptation, J. Struct. Multidiscip. Optim., 32, 241-251 (2006)
[26] Nowak, M., On some properties of bone functional adaptation phenomenon useful in mechanical design, Acta Bioeng. Biomech., 12, 49-54 (2010)
[27] Rozvany, G., Exact analytical solutions for some popular benchmark problems in topology optimization, Struct. Optim., 15, 42-48 (1998) · Zbl 1245.74075
[28] Diaz, Ar; Bendsoe, Mp, Shape optimization of structures for multiple load conditions using a homogenization method, Struct. Optim., 4, 17-22 (1992)
[29] Beckers, M., Topology optimization using a dual method with discrete variables, Struct. Optim., 17, 14-24 (1999)
[30] Picelli, R.; Townsend, S.; Brampton, C.; Norato, J.; Kim, Ha, Stress-based shape and topology optimization with the level set method, Comput. Methods Appl. Mech. Eng., 329, 1-23 (2018)
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