A mixed FEM approach to stress‐constrained topology optimization

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. Copyright © 2007 John Wiley & Sons, Ltd.     

Heaviside projection–based aggregation in stress‐constrained topology optimization

The paper introduces an approach to stress‐constrained topology optimization through Heaviside projection–based constraint aggregation. The aggregation is calculated by integrating Heaviside projected local stresses over the design domain, and then, it is normalized over the total material volume. Effectively, the normalized integral measures the volume fraction of the material that has violated the stress constraint. Hence, with the Heaviside aggregated constraint, we can remove the stress failed material from the final design by constraining the integral to a threshold value near zero. An adaptive strategy is developed to select the threshold value for ensuring that the optimized design is conservative. By adding a stress penalty factor to the integrand, the Heaviside aggregated constraint can further penalize high stresses and becomes more stable and less sensitive to the selection of the threshold value. Our two‐dimensional and three‐dimensional numerical experiments demonstrate that the single Heaviside aggregated stress constraint can efficiently control the local stress level. Compared with the traditional approaches based on the Kreisselmeier‐Steinhauser and p‐norm aggregations, the Heaviside aggregation–based single constraint can substantially reduce computational cost on sensitivity analysis. These advantages make it possible to apply the proposed approach to large‐scale stress‐constrained problems.     

A mixed integer programming for robust truss topology optimization with stress constraints

This paper presents a mixed integer programming (MIP) formulation for robust topology optimization of trusses subjected to the stress constraints under the uncertain load. A design‐dependent uncertainty model of the external load is proposed for dealing with the variation of truss topology in the course of optimization. For a truss with the discrete member cross‐sectional areas, it is shown that the robust topology optimization problem can be reduced to an MIP problem, which is solved globally. Numerical examples illustrate that the robust optimal topology of a truss depends on the magnitude of uncertainty. Copyright © 2010 John Wiley & Sons, Ltd.     

Global optimization of minimum weight truss topology problems with stress,displacement, and local buckling constraints using branch‐and‐bound

We present a convergent continuous branch‐and‐bound algorithm for global optimization of minimum weight truss topology problems with displacement, stress, and local buckling constraints. Valid inequalities which strengthen the problem formulation are derived. The inequalities are generated by solving well‐defined convex optimization problems. Computational results are reported on a large collection of problems taken from the literature. Most of these problems are, for the first time, solved with a proof of global optimality. Copyright © 2004 John Wiley & Sons, Ltd.     

A dual algorithm for the topology optimization of non‐linear elastic structures

Dual algorithms are ideally suited for the purpose of topology optimization since they work in the space of Lagrange multipliers associated with the constraints. To date, dual algorithms have been applied only for linear structures. Here we extend this methodology to the case of non‐linear structures. The perimeter constraint is used to make the topology problem well‐posed. We show that the proposed algorithm yields a value of perimeter that is close to that specified by the user. We also address the issue of manufacturability of these designs, by proposing a variant of the standard dual algorithm, which generates designs that are two‐dimensional although the loading and the geometry are three‐dimensional. Copyright © 2008 John Wiley & Sons, Ltd.     

Modelling topology optimization problems as linear mixed 0–1 programs

This paper deals with topology optimization of discretized continuum structures. It is shown that a large class of non‐linear 0–1 topology optimization problems, including stress‐ and displacement‐constrained minimum weight problems, can equivalently be modelled as linear mixed 0–1 programs. The modelling approach is applied to some test problems which are solved to global optimality. Copyright © 2003 John Wiley & Sons, Ltd.     

A stress-influence-function with adaptive strength feature approach for stress constrained continuum topology optimization via small-loop sequential strategy

Recently, the stress-influence-function (SIF) approach is presented for stress constrained continuum topology optimization. The SIF approach provides an alternative for continuum topology optimization with stress constraints. However, the SIF approach is not good at controlling the maximum stress of the elements compared to the conventional approach. In the study, the stress-influence-function with adaptive strength feature (SIF-ASF) approach via small-loop sequential strategy is proposed to achieve better control on the maximum elemental stress. First, the stress constrained continuum topology optimization formulation is given and the SIF approach is briefly introduced. Then the SIF-ASF approach is proposed for stress constrained continuum topology optimization, in which the strength feature in the stress influence function is adjusted in each iterative step of the optimization process. The adjoint-vector based sensitivity analysis to the design variables is further discussed. Three numerical examples are given to illustrate the applicability and validity of the proposed SIF-ASF approach. It is shown that the proposed SIF-ASF approach can achieve better control on the maximum elemental stress than the SIF approach. Moreover, the proposed SIF-ASF approach may obtain a lighter structure than the conventional approach.     

Level set topology optimization for design‐dependent pressure load problems

This work presents a level set framework to solve the compliance topology optimization problem considering design‐dependent pressure loads. One of the major technical difficulties related to this class of problem is the adequate association between the moving boundary and the pressure acting on it. This difficulty is easily overcome by the level set method that allows for a clear tracking of the boundary along the optimization process. In the present approach, a reaction‐diffusion equation substitutes the classical Hamilton‐Jacobi equation to control the level set evolution. This choice has the advantages of allowing the nucleation of holes inside the domain and the elimination of the undesirable reinitialization steps. Moreover, the proposed algorithm allows merging pressurized (wet) boundaries with traction‐free boundaries during level set movements. This last property, allied to the simplicity of the level set representation and successful combination with the reaction‐diffusion based evolution applied to a design‐dependent pressure load problem, represents the main contribution of this paper. Numerical examples provide successful results, many of which comparable with others found in the literature and solved with different techniques.     

Higher‐order multi‐resolution topology optimization using the finite cell method

This article presents a detailed study on the potential and limitations of performing higher‐order multi‐resolution topology optimization with the finite cell method. To circumvent stiffness overestimation in high‐contrast topologies, a length‐scale is applied on the solution using filter methods. The relations between stiffness overestimation, the analysis system, and the applied length‐scale are examined, while a high‐resolution topology is maintained. The computational cost associated with nested topology optimization is reduced significantly compared with the use of first‐order finite elements. This reduction is caused by exploiting the decoupling of density and analysis mesh, and by condensing the higher‐order modes out of the stiffness matrix. Copyright © 2016 John Wiley & Sons, Ltd.     

Multi-material topology optimization for the transient heat conduction problem using a sequential quadratic programming algorithm

Transient heat conduction analysis involves extensive computational cost. It becomes more serious for multi-material topology optimization, in which many design variables are involved and hundreds of iterations are usually required for convergence. This article aims to provide an efficient quadratic approximation for multi-material topology optimization of transient heat conduction problems. Reciprocal-type variables, instead of relative densities, are introduced as design variables. The sequential quadratic programming approach with explicit Hessians can be utilized as the optimizer for the computationally demanding optimization problem, by setting up a sequence of quadratic programs, in which the thermal compliance and weight can be explicitly approximated by the first and second order Taylor series expansion in terms of design variables. Numerical examples show clearly that the present approach can achieve better performance in terms of computational efficiency and iteration number than the solid isotropic material with penalization method solved by the commonly used method of moving asymptotes. In addition, a more lightweight design can be achieved by using multi-phase materials for the transient heat conductive problem, which demonstrates the necessity for multi-material topology optimization.     

Maximizing opto‐mechanical interaction using topology optimization

This paper studies topology optimization of a coupled opto‐mechanical problem with the goal of finding the material layout which maximizes the optical modulation, i.e. the difference between the optical response for the mechanically deformed and undeformed configuration. The optimization is performed on a periodic cell and the periodic modeling of the optical and mechanical fields have been carried out using transverse electric Bloch waves and homogenization theory in a plane stress setting, respectively. Two coupling effects are included being the photoelastic effect and the geometric effect caused by the mechanical deformation. For the studied objective and material choice it is concluded that the photoelastic effect and the geometric effect counteract each other, which yields designs which are fundamentally different if the optimization takes only one effect into account. When both effects are active a compromise is found; however, a strong regularization is needed in order to achieve reasonable 0–1 designs with a clear physical interpretation. Copyright © 2011 John Wiley & Sons, Ltd.     

Level‐set topology optimization with many linear buckling constraints using an efficient and robust eigensolver

Linear buckling constraints are important in structural topology optimization for obtaining designs that can support the required loads without failure. During the optimization process, the critical buckling eigenmode can change; this poses a challenge to gradient‐based optimization and can require the computation of a large number of linear buckling eigenmodes. This is potentially both computationally difficult to achieve and prohibitively expensive. In this paper, we motivate the need for a large number of linear buckling modes and show how several features of the block Jacobi conjugate gradient (BJCG) eigenvalue method, including optimal shift estimates, the reuse of eigenvectors, adaptive eigenvector tolerances and multiple shifts, can be used to efficiently and robustly compute a large number of buckling eigenmodes. This paper also introduces linear buckling constraints for level‐set topology optimization. In our approach, the velocity function is defined as a weighted sum of the shape sensitivities for the objective and constraint functions. The weights are found by solving an optimization sub‐problem to reduce the mass while maintaining feasibility of the buckling constraints. The effectiveness of this approach in combination with the BJCG method is demonstrated using a 3D optimization problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Howard's algorithm in a phase‐field topology optimization approach

The topology optimization problem is formulated in a phase‐field approach. The solution procedure is based on the Allan–Cahn diffusion model where the conservation of volume is enforced by a global constraint. The functional defining the minimization problem is selected such that no penalization is imposed for full and void materials. Upper and lower bounds of the density function are enforced by infinite penalty at the bounds. A gradient term that introduces cost for boundaries and thereby regularizing the problem is also included in the objective functional. Conditions for stationarity of the functional are derived, and it is shown that the problem can be stated as a variational inequality or a max–min problem, both defining a double obstacle problem. The numerical examples used to demonstrate the method are solved using the FEM, whereas the double obstacle problem is solved using Howard's algorithm. Copyright © 2012 John Wiley & Sons, Ltd.     

A fast method for binary programming using first‐order derivatives,with application to topology optimization with buckling constraints

We present a method for finding solutions of large‐scale binary programming problems where the calculation of derivatives is very expensive. We then apply this method to a topology optimization problem of weight minimization subject to compliance and buckling constraints. We derive an analytic expression for the derivative of the stress stiffness matrix with respect to the density of an element in the finite‐element setting. Results are presented for a number of two‐dimensional test problems.Copyright © 2012 John Wiley & Sons, Ltd.     

A simple and efficient post‐processing method for the computation of interfaces by local Reissner method

Sandwich materials are currently much valued in industry, especially in transport (automotive, aeronautics, shipbuilding and railroads) and civil engineering. Because of this rise of interest, it becomes more and more important to develop analysis tools able to take their specificities into account. Herein, we introduce a very simple and efficient post‐processing method especially developed for heterogeneous materials such as sandwich structures. The method, based on Reissner principle, permits to fulfill the force equilibrium at interfaces between different layers, the skins and the core whose mechanical properties are very different, and yields accurate results for continuous stress components even with a coarse meshing through the thickness of the structure. Only small matrices are involved in the computation, leading to an easy programming and implementation even in existing finite element packages. Moreover, the problem caused by nodal extrapolation of stress components from Gauss points is removed. The present formulation, detailed in 2D for a simple illustration, is also valid in 3D. Finally, the method is assessed in the classical case of a simply supported beam under uniform pressure as well as in the non‐standard case like a U‐beam. Copyright © 2007 John Wiley & Sons, Ltd.     

Three‐dimensional grayscale‐free topology optimization using a level‐set based r‐refinement method

In this paper, we propose a three‐dimensional (3D) grayscale‐free topology optimization method using a conforming mesh to the structural boundary, which is represented by the level‐set method. The conforming mesh is generated in an r‐refinement manner; that is, it is generated by moving the nodes of the Eulerian mesh that maintains the level‐set function. Although the r‐refinement approach for the conforming mesh generation has many benefits from an implementation aspect, it has been considered as a difficult task to stably generate 3D conforming meshes in the r‐refinement manner. To resolve this task, we propose a new level‐set based r‐refinement method. Its main novelty is a procedure for minimizing the number of the collapsed elements whose nodes are moved to the structural boundary in the conforming mesh; in addition, we propose a new procedure for improving the quality of the conforming mesh, which is inspired by Laplacian smoothing. Because of these novelties, the proposed r‐refinement method can generate 3D conforming meshes at a satisfactory level, and 3D grayscale‐free topology optimization is realized. The usefulness of the proposed 3D grayscale‐free topology optimization method is confirmed through several numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.     

A level set‐based topology optimization method targeting metallic waveguide design problems

In this paper, we propose a level set‐based topology optimization method targeting metallic waveguide design problems, where the skin effect must be taken into account since the metallic waveguides are generally used in the high‐frequency range where this effect critically affects performance. One of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal. To implement this approach, we develop a boundary‐tracking scheme for the arbitrary Lagrangian Eulerian (ALE) mesh pertaining to the zero iso‐contour of the level set function that is given in an Eulerian mesh, and impose Dirichlet boundary conditions at the nodes on the zero iso‐contour in the ALE mesh to compute the electric field. Since the ALE mesh accurately tracks the zero iso‐contour at every optimization iteration, the electric field is always appropriately computed during optimization. For the sensitivity analysis, we compute the nodal coordinate sensitivities in the ALE mesh and smooth them by solving a Helmholtz‐type partial differential equation. The obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the Eulerian mesh, and the level set function is updated based on the computed normal velocity. Finally, the utility of the proposed method is discussed through several numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.     

P1‐nonconforming quadrilateral finite element for topology optimization

This investigation focuses on an alternative approach to topology optimization problems involving incompressible materials using the P1‐nonconforming finite element. Instead of using the mixed displacement‐pressure formulation, a pure displacement‐based approach can be employed for finite element formulation owing to the Poisson locking‐free property of the P1‐nonconforming element. Moreover, because the P1‐nonconforming element has linear shape functions that are defined at element vertices, it has considerably fewer degrees of freedom than other quadrilateral nonconforming elements and its implementation is as simple as that of the conforming bilinear element. Various problems dealing with incompressible materials and pressure‐loaded structures found in published works are solved to verify the applicability of the proposed method. The application of the method is extended to the optimal design of fluid channels in the Stokes flow. This is done by expressing pressure in terms of volumetric strain rates and developing a velocity‐field‐only finite element formulation. The optimization results obtained from all the problems considered in this study are in close agreement with those found in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

Triangular checkerboard control using a wavelet‐based method in topology optimization

Topology optimization has been carried out mainly with structured quadrilateral finite elements. However, triangular elements facilitate mesh generation especially for problems having complex geometries that often appear in practical industrial problems. The use of triangular elements, especially low‐order triangular elements, causes a serious numerical trouble that is equivalent to the rectangular checkerboard pattern formation. The objective of this investigation is to develop a triangular checkerboard‐freeing method that directly restricts the design space. To this end, we use the multiscale design space that is mapped from the standard single‐scale density space. To facilitate the mapping, we employ the triangular mesh subdivision and propose a bi‐orthogonal wavelet transform suitable for a triangulated domain. For checkerboard‐free designs, a shrinkage method based on the wavelet frame appropriate for triangular mesh is proposed. Typical benchmark problems and a simplified roof‐reinforcing problem in an automobile body are considered to check the effectiveness of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.