Regularization of optimal design problems for bars and plates, part 2

Lurie, Konstantin A.; Cherkaev, Andrej; Fedorov, A. V.

doi:10.1007/bf00934954

Cited by 62 publications

(29 citation statements)

References 2 publications

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“…In the case of 2D isotropic non-linear materials a condition equivalent to (1.6) was implicitly obtained byŠilhavý [70] (see also [69]). Various special forms of (1.6) appeared earlier in physical and optimization literature where the functional was minimized with respect to the orientation of the layered microstructure, see [65,53,51,67,45]. A need for an additional equality on the equilibrium phase boundary was also realized in the series of papers on ellipsoidal inclusions of a new phase appearing in an elastic matrix ( see e.g.…”

Section: Introductionmentioning

confidence: 99%

Roughening Instability of Broken Extremals

Grabovsky

Truskinovsky

2010

Arch Rational Mech Anal

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We derive a new general jump condition on a broken Weierstrass-Erdmann extremal of a vectorial variational problem. Such extremals, containing surfaces of gradient discontinuity, are ubiquitous in shape optimization and in the theory of elastic phase transformations. The new condition, which does not have a one dimensional analog, reflects the stationarity of the singular surface with respect to two-scale variations that are nontrivial generalizations of Weierstrass needles. The over-determinacy of the ensuing free boundary problem suggests that typical stable configurations must involve microstructures or chattering controls.

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Section: Introductionmentioning

confidence: 99%

Roughening Instability of Broken Extremals

Grabovsky

Truskinovsky

2010

Arch Rational Mech Anal

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“…It has been used extensively to analyze composite materials (Suquet, 1985;Guedes, 1990), and also in predicting optimal topology of microstructured materials (Bendsoe and Kikuchi, 1988;Lurie et al, 1982). In biomechanics, Crolet et al (1988Crolet et al ( , 1990 applied the homogenization theory to model cortical bone mechanics.…”

Section: Introductionmentioning

confidence: 99%

Application of homogenization theory to the study of trabecular bone mechanics

Hollister

Fyhrie

Jepsen

et al. 1991

Journal of Biomechanics

137

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Ah&act-It is generally accepted that the strength and stiffness of trabecular bone is strongly affected by trabecular microstructure. It has also been hvoothesized that stress induced adaptation of trabecular bone is affected by trabecular tissue level stress and;& strain. At this time, however, there is no generally accepted (or easily accomplished) technique for predicting the effect of microstructure on trabecular bone apparent stiffness and strength or estimating tissue level stress or strain. In this paper, a recently developed mechanics theory specifically designed to analyze microstructured materials, called the homogenization theory, is presented and applied to analyze trabecular bone mechanics. Using the homogenization theory it is possible to perform microstructural and continuum analyses separately and then combine them in a systematic manner. Stiffness predictions from two different microstructural models of trabecular bone show reasonable agreement with experimental results, depending on metaphyseal region, (R">0.5 for proximai humerus specimens, R* ~0.5 for distal femur and proximal tibia specimens). Estimates of both microstructural strain energy density (SED) and apparent SED show that there are large differences (up to 30 times) between apparent SED (as calculated by standard continuum finite element analyses) and the maximum microstructural or tissue SED. Furthermore, a strut and spherical void microstructure gave very different estimates of maximum tissue SED for the same bone volume fraction (BV/TV). The estimates from the spherical void microstructure are between 2 and 20 times greater than the strut microstructure at lo-20% W/TV INTRODUCTION Stress and strain fields in individual trabeculae are determined by both overall metaphyseal geometry and the microstructural organization of a local region of trabeculae. However, directly calculating stress and strain fields for each trabecula of a whole metaphysis is impossible with current computational technology. Current techniques for determining trabecular stress have focused on either analyzing very small regions of bone with a limited number of trabeculae, or analyzing a much larger region of bone assuming it to be a solid with apparent material properties. Neither technique provides direct information about trabecular tissue stresses and strains in an entire metaphyseal region. The homogenization theory is a recently developed method which provides a systematic way to combine stress analyses at the metaphyseal and individual trabecula level. This paper presents the initial application of homogenization theory to the study of trabecular bone mechanics.Microstructural analyses are generally used to predict apparent moduli and stress distributions within representative pieces of a given microstructure. These analyses assume that the whole microstructure within a material is constructed by repeating the representative piece analyzed. This type of analysis is quite Received infinalform 14 February 1991.

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“…This approach was inspired by theoretical studies on generalized shape design in conduction and torsion problems and by numerical and theoretical work related to plate design (see, e.g., [11,17,23]). Initially, composites consisting of square or rectangular holes in periodically repeated square cells were used for planar problems.…”

Section: Homogenization Models With Anisotropymentioning

confidence: 99%

Topology Design of Structures, Materials and Mechanisms — Status and Perspectives

Bendsøe

2000

System Modelling and Optimization

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INTRODUCTIONThe field of variable-topology shape design in structural optimization has its origins in theoretical studies of existence of solutions in variational problems, in particular shape optimization problems, and in studies in theoretical material science of variational bounds on material properties. Progress in computational methods and the ever increasing computer power has oriented the area towards applications, with significant developments being achieved over the last decade, leading to a fairly widespread use of the methodology in industry.In this short paper we outline some of the basic ideas and methods of existing methods, but it is not our purpose to cover all work and approaches in this field. Instead we refer to existing literat ure containing rather comprehensive surveys, see e.g., [7,8,31]. Moreover, note that reference is mostly made to recent papers that include bibliographies useful for on overview of the area. Thus the presentation does not try to present a complete historie al perspective.The area of computational variable-topology shape design of continuum structures is presently dominated by methods which employ a material distribution approach for a fixed reference domain, in the spirit of the so-called 'homogenization method' for topology design ([3, 9]). That is, the geometrie representation of a structure is similar to a grey-scale rendering of an image, in discrete form corresponding to araster representation of the geometry on a fixed reference domain. The physics of the problem is also represented by boundary conditions and forcing The original version of this chapter was revised: The copyright line was incorrect. This has been corrected. The Erratum to this chapter is available at DOI: 10.1007/978-0-387-35514-6_15 2 Martin P. BendsrJe terms defined on this fixed reference domain, much in analogy to fictitious domain methods for FEM analysis.One can normally distinguish between three versions of raster based geometry models for continuum topology optimization. The basic problem is an unrestricted '0-1' integer design problem (generalized shape optimization), that is, a design specifies unambiguously whether there is solid material or void at every point in a candidate design region. Otherwise, there are no restrictions on the shape. Unfortunately, in general, this dass of problems is ill-posed in the continuum setting (cf., [11,20]). Well-posed problems can be obtained by either extending the space of admissible solutions to obtain their relaxed versions, usually by incorporating microstructure (see, e.g., [3]), or by restricting the space of admissible solutions. The latter can be accomplished by enforcing an upper bound on the perimeter of the structure (see [28], and references therein), by imposing constraints on the slopes of the parameters defining geometry (see [29], and references therein), by the introduction of a filtering function limiting the minimum scale (see [10,36] for an overview), or one can introduce a ground structure with a fixed number of design degrees of...

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Regularization of optimal design problems for bars and plates, part 2

Cited by 62 publications

References 2 publications

Roughening Instability of Broken Extremals

Roughening Instability of Broken Extremals

Application of homogenization theory to the study of trabecular bone mechanics

Topology Design of Structures, Materials and Mechanisms — Status and Perspectives

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