Checkerboard‐free topology optimization using non‐conforming finite elements

Jang, Gang-Won; Jeong, Je Hyun; Kim, Yoon Young; Sheen, Dongwoo; Park, Chunjae; Kim, Myoungnyoun

doi:10.1002/nme.738

Cited by 66 publications

(45 citation statements)

References 21 publications

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“…About the feasibility of the reached optimum with regard to the effectiveness of constraints, reference is made to the MMA algorithm that provides at each iteration the r.h.s. values of the inequalities used to impose stresses and volume constraints in formulation (16). For all solutions presented in this work, the MMA has converged with all r.h.s.…”

Section: Numerical Observationsmentioning

confidence: 96%

A mixed FEM approach to stress‐constrained topology optimization

Bruggi

Venini

2007

Numerical Meth Engineering

106

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We present an alternative topology optimization formulation capable of handling the presence of stress constraints in a straightforward fashion. The main idea is to adopt a mixed finite-element discretization scheme wherein not only displacements (as usual) but also stresses are the variables entering the formulation. By doing so, any stress constraint may be handled within the optimization procedure without resorting to post-processing operation typical of displacement-based techniques that may also cause a loss in accuracy in stress computation if no smoothing of the stress is performed. Two dual variational principles of Hellinger–Reissner type are presented in continuous and discrete form that, which included in a rather general topology optimization problem in the presence of stress constraints that is solved by the method of moving asymptotes (Int. J. Numer. Meth. Engng. 1984; 24(3):359–373). Extensive numerical simulations are performed and ongoing extensions outlined, including the optimization of elastoplastic and incompressible media

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Section: Numerical Observationsmentioning

confidence: 96%

A mixed FEM approach to stress‐constrained topology optimization

Bruggi

Venini

2007

Numerical Meth Engineering

106

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“…Checkerboards can be removed through smoothing or inhibited by using higher order finite elements [2,3], non-conforming finite elements [4], or additional constraints [5]. A popular approach to eliminating mesh dependence is to restrict the design space so that a solution exists for the original continuum problem.…”

Section: Introductionmentioning

confidence: 99%

Achieving minimum length scale in topology optimization using nodal design variables and projection functions

Guest

Prévost

Belytschko

2004

Numerical Meth Engineering

1,037

576

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SUMMARYA methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described. Nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology. The projection is made via mesh independent functions that are based upon the minimum length scale. A simple linear projection scheme and a non-linear scheme using a regularized Heaviside step function to achieve nearly 0 -1 solutions are examined. The new approach is demonstrated on the minimum compliance problem and the popular SIMP method is used to penalize the stiffness of intermediate volume fraction elements. Solutions are shown to meet user-defined length scale criterion without additional constraints, penalty functions or sensitivity filters. No instances of mesh dependence or checkerboard patterns have been observed.

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“…If the design parameters are approximated by continuous functions [8], it is known that a numerical instability, such as the socalled island phenomena, is observed [9]. In addition, although many numerical schemes have been proposed to overcome such numerical instabilities [10,11], regularity in the sense of functional analysis has not been shown.…”

Section: Problem 1 (Topology Optimization Problem)mentioning

confidence: 99%

Regular solution to topology optimization problems of continua

Azegami

Kaizu

Takeuchi³

2011

JSIAM Letters

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The present paper describes a numerical solution to topology optimization problems of domains in which boundary value problems of partial differential equations are defined. Density raised to a power is used instead of the characteristic function of the domain. A design variable is set by a function on a fixed domain which is converted to the density by a sigmoidal function. Evaluation of derivatives of cost functions with respect to the design variable appear as stationary conditions of the Lagrangians. A numerical solution is constructed by a gradient method in a design space for the design variable.

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Checkerboard‐free topology optimization using non‐conforming finite elements

Cited by 66 publications

References 21 publications

A mixed FEM approach to stress‐constrained topology optimization

A mixed FEM approach to stress‐constrained topology optimization

Achieving minimum length scale in topology optimization using nodal design variables and projection functions

Regular solution to topology optimization problems of continua

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