Approximated approximations for SAO

SUMMARYStress-related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. Continuum structural topological optimization with stress constraints is of wide engineering application prospect, in which there still are many problems to solve, such as the stress concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization method of continuum structures with the stress constraints and the objective function being the structural volume has been presented in this paper. To solve the stress concentration issue, an approximate stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient constraint is constructed for the optimized structure under the r th load case. To obtain stable convergent series solutions and enhance the control on the stress level, two p-norm global stress constraint functions with different indexes are adopted, and some weighting p-norm global stress constraint functions are introduced for any load case. And an equivalent topological optimization model with reduced stress constraints is constructed,being incorporated with the rational approximation for material properties, an active constraint technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed, based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level optimization problem with artificial variables and many possible non-active design variables is proposed by adopting an inequality constrained nonlinear programming method with simple trust regions, based on the primal-dual theory, in which the non-smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a twolevel optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed to deal with the aggregation constraints that always are of loose constraint (non active constraint) features in the conventional structural optimization method. A novel structural topological optimization method with stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed method is feasible and very effective.

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“…are the upper limit and lower limit of moving asymptotes, and other notations are referred to the references [31][32][33].…”

Section: Explicit Approximations Of Aggregation Functionsmentioning

confidence: 99%

Continuum structural topological optimizations with stress constraints based on an active constraint technique

Rong

Xiao

et al. 2016

Numerical Meth Engineering

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“…We now show how an SQP type method can be developed from the SCP class of methods by means of the approximated approximations concept presented in Groenwold et al (2010).…”

Section: Diagonal Qp Subproblems For Scpmentioning

confidence: 99%

“…For other diagonal quadratic approximations to a number of different well known nonlinear approximations, including the exponential, CONLIN, MMA and TANA approximations, see Groenwold et al (2010).…”

Section: Diagonal Quadratic Function Approximationsmentioning

confidence: 99%

“…In Groenwold and Etman (2010b), we have demonstrated that the diagonal quadratic approximation to the reciprocal and exponential approximate objective functions can successfully be used in topology optimization. We have generalized this observation even further in Groenwold et al (2010), where diagonal quadratic approximations to arbitrary nonlinear approximate objective and constraint functions were constructed; the resulting subproblems are convexified (when necessary), and cast into the dual statement. This gives an SCP type of algorithm which uses diagonal quadratic instead of reciprocal type of approximations.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the form of the dual subproblem does not depend on which nonlinear approximations or intervening variables are selected for quadratic treatment: all the approximated approximations can be used simultaneously in a single dual statement. In Groenwold et al (2010), diagonal quadratic replacements for the reciprocal, exponential, CONLIN, MMA, and TANA (Xu and Grandhi 1998) approximations were presented. We have described this approach with the term "approximated-approximations".…”

Section: Introductionmentioning

confidence: 99%

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First-order sequential convex programming using approximate diagonal QP subproblems

Etman

Groenwold

Rooda

2011

Struct Multidisc Optim

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Optimization algorithms based on convex separable approximations for optimal structural design often use reciprocal-like approximations in a dual setting; CONLIN and the method of moving asymptotes (MMA) are wellknown examples of such sequential convex programming (SCP) algorithms. We have previously demonstrated that replacement of these nonlinear (reciprocal) approximations by their own second order Taylor series expansion provides a powerful new algorithmic option within the SCP class of algorithms. This note shows that the quadratic treatment of the original nonlinear approximations also enables the restatement of the SCP as a series of Lagrange-Newton QP subproblems. This results in a diagonal trust-region SQP type of algorithm, in which the second order diagonal terms are estimated from the nonlinear (reciprocal) intervening variables, rather than from historic information using an exact or a quasi-Newton Hessian approach. The QP formulation seems particularly attractive for problems with far more constraints than variables (when pure dual methods are at a disadvantage), or when both the number of design variables and the number of (active) constraints is very large.

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A quadratic approximation for structural topology optimization

Groenwold

Etman

2009

Numerical Meth Engineering

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SUMMARYIn topology optimization, it is customary to use reciprocal-like approximations, which result in monotonically decreasing approximate objective functions. In this paper, we demonstrate that efficient quadratic approximations for topology optimization can also be derived, if the approximate Hessian terms are chosen with care. To demonstrate this, we construct a dual SAO algorithm for topology optimization based on a strictly convex, diagonal quadratic approximation to the objective function. Although the approximation is purely quadratic, it does contain essential elements of reciprocal-like approximations: for self-adjoint problems, our approximation is identical to the quadratic or second-order Taylor series approximation to the exponential approximation. We present both a single-point and a two-point variant of the new quadratic approximation.

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Approximated approximations for SAO

Cited by 28 publications

References 24 publications

Continuum structural topological optimizations with stress constraints based on an active constraint technique

Continuum structural topological optimizations with stress constraints based on an active constraint technique

First-order sequential convex programming using approximate diagonal QP subproblems

A quadratic approximation for structural topology optimization

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