Topology Optimization Under Linear Thermo-Elastic Buckling

2016 ◽  
Author(s):  
Shiguang Deng ◽  
Krishnan Suresh
Keyword(s):  
Topology Optimization ◽  
Augmented Lagrangian ◽  
Thermal Environment ◽  
Augmented Lagrangian Method ◽  
Compressive Load ◽  
Elastic Buckling ◽  
Buckling Modes ◽  
Thermally Induced ◽  
Stiffness Matrices ◽  
Level Set Approach

This paper focuses on topology optimization of structures subject to a compressive load in a thermal environment. Such problems are important, for example, in aerospace, where structures are prone to thermally induced buckling. Popular strategies for thermo-elastic topology optimization include Solid Isotropic Material with Penalization (SIMP) and Rational Approximation of Material Properties (RAMP). However, since both methods fundamentally rely on material parameterization, they are often challenged by: (1) pseudo buckling modes in low-density regions, and (2) ill-conditioned stiffness matrices. To overcome these, we consider here an alternate level-set approach that relies discrete topological sensitivity. Buckling sensitivity analysis is carried out via direct and adjoint formulations. Augmented Lagrangian method is then used to solve a buckling constrained compliance minimization problem. Finally, 3D numerical experiments illustrate the efficiency of the proposed method.

Stress-Constrained Thermo-Elastic Topology Optimization: A Topological Sensitivity Approach

2014 ◽  
Author(s):  
Shiguang Deng ◽  
Krishnan Suresh ◽  
James Joo
Keyword(s):  
Topology Optimization ◽  
Material Properties ◽  
Ad Hoc ◽  
Thermal Gradients ◽  
Aircraft Industry ◽  
Topological Sensitivity ◽  
Stiffness Matrices ◽  
Optimization Parameters ◽  
Sensitivity Approach ◽  
Level Set Approach

The focus of this paper is on thermo-elastic topology optimization where the structure is subject to both mechanical and thermal loads. Such problems are of significant importance, for example, in the aircraft industry where structures subject to aerodynamic forces and thermal-gradients must be optimized. A popular strategy for solving such problems is Solid Isotropic Material with Penalization (SIMP) where pseudo-densities serve as optimization parameters. Yet another strategy is the Rational Approximation of Material Properties (RAMP) that overcomes some of the deficiencies of SIMP. Both methods fundamentally rely on parameterization of the material properties as a function of the pseudo-densities. Here we consider an alternate level-set approach that relies on the concept of topological sensitivity. The advantages of the proposed method over SIMP and RAMP are: (1) ad hoc material parameterization is not required (2) the stresses are well-defined at all points within the evolving topology and (3) the underlying stiffness matrices are always well-conditioned. The proposed method is illustrated through numerical experiments.

scholarly journals A Level Set-Based Topology Optimization Method Using the Augmented Lagrangian Method

2011 ◽  
Vol 6 (2) ◽  
pp. 387-399
Author(s):  
Shintaro YAMASAKI ◽  
Tsuyoshi NOMURA ◽  
Atsushi KAWAMOTO ◽  
Kazuo SATO ◽  
Kazuhiro IZUI ◽  
...  
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Optimization Method ◽  
Lagrangian Method ◽  
Topology Optimization Method

scholarly journals Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals

2021 ◽  
Author(s):  
Jingmin Xia ◽  
Patrick E. Farrell ◽  
Florian Wechsung
Keyword(s):  
Liquid Crystals ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Mass Matrix ◽  
Unit Length ◽  
Mesh Size ◽  
Schur Complement ◽  
Cholesteric Liquid Crystals ◽  
Multigrid Algorithm ◽  
The Cost

AbstractWe propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen–Frank model arising in nematic and cholesteric liquid crystals. By applying the augmented Lagrangian method, the Schur complement of the director block can be better approximated by the weighted mass matrix of the Lagrange multiplier, at the cost of making the augmented director block harder to solve. In order to solve the augmented director block, we develop a robust multigrid algorithm which includes an additive Schwarz relaxation that captures a pointwise version of the kernel of the semi-definite term. Furthermore, we prove that the augmented Lagrangian term improves the discrete enforcement of the unit-length constraint. Numerical experiments verify the efficiency of the algorithm and its robustness with respect to problem-related parameters (Frank constants and cholesteric pitch) and the mesh size.

scholarly journals An augmented Lagrangian method for solving total variation (TV)-based image registration model

2020 ◽  
Vol 14 ◽  
pp. 174830262097353
Author(s):  
Noppadol Chumchob ◽  
Ke Chen
Keyword(s):  
Image Registration ◽  
Augmented Lagrangian ◽  
Numerical Solutions ◽  
Augmented Lagrangian Method ◽  
Closed Form Solution ◽  
Numerical Algorithms ◽  
Lagrangian Method ◽  
Form Solution ◽  
The Other ◽  
Other Hand

Variational methods for image registration basically involve a regularizer to ensure that the resulting well-posed problem admits a solution. Different choices of regularizers lead to different deformations. On one hand, the conventional regularizers, such as the elastic, diffusion and curvature regularizers, are able to generate globally smooth deformations and generally useful for many applications. On the other hand, these regularizers become poor in some applications where discontinuities or steep gradients in the deformations are required. As is well-known, the total (TV) variation regularizer is more appropriate to preserve discontinuities of the deformations. However, it is difficult in developing an efficient numerical method to ensure that numerical solutions satisfy this requirement because of the non-differentiability and non-linearity of the TV regularizer. In this work we focus on computational challenges arising in approximately solving TV-based image registration model. Motivated by many efficient numerical algorithms in image restoration, we propose to use augmented Lagrangian method (ALM). At each iteration, the computation of our ALM requires to solve two subproblems. On one hand for the first subproblem, it is impossible to obtain exact solution. On the other hand for the second subproblem, it has a closed-form solution. To this end, we propose an efficient nonlinear multigrid (NMG) method to obtain an approximate solution to the first subproblem. Numerical results on real medical images not only confirm that our proposed ALM is more computationally efficient than some existing methods, but also that the proposed ALM delivers the accurate registration results with the desired property of the constructed deformations in a reasonable number of iterations.

scholarly journals An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

2021 ◽  
Author(s):  
Christian Kanzow ◽  
Andreas B. Raharja ◽  
Alexandra Schwartz
Keyword(s):  
Constrained Optimization ◽  
Numerical Experiments ◽  
Penalty Method ◽  
Augmented Lagrangian ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Constrained Optimization Problems ◽  
Strongly Stationary ◽  
Type Constraint

AbstractA reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.

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