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

A 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. Copyright © 2004 John Wiley & Sons, Ltd.     

A novel approach for length scale control in structural topology optimization

This article proposes a novel method for topology optimization with length scale control. In this method, the structural skeleton based on the level set framework is employed. On this basis, the concept of the skeleton feature is proposed. The skeleton feature is defined as a circle, of which the centre is the skeleton point and the radius is the length scale. The signed distance function based on the skeleton feature is applied to achieve minimum and maximum length scale control. The length scale constraint is determined according to the location relationship between the structure and the skeleton feature. An increasing Lagrange multiplier is applied for length scale constraints. Several simple examples are presented to demonstrate the effectiveness of the skeleton features in length scale control.     

Minimum length scale constraints in multi-scale topology optimisation for additive manufacturing

ABSTRACT

This paper performs a combined numerical and experimental study to explore the role of minimum length scale constraints in multi-scale topology optimisation. Multi-scale topology optimisation is generally performed without considering the actual unit cell size, while an arbitrary value considerably smaller than the part is selected afterwards. However, this procedure would be problematic if including geometric constraints, e.g. minimum length scale constraints, since geometric constraints cannot be applied without knowing the unit cell dimensions. To address this issue, unit cell size should be defined beforehand, and guidelines will be provided in this work through a thorough numerical exploration, i.e. compliance minimisation multi-scale topology optimisation with different unit cell sizes and a consistent minimum length scale limit will be performed. The numerical results indicate that selecting the unit cell size considerably smaller than the part and larger than the length scale limit would be recommended. Then, experiments are conducted to explore the effect of minimum length scale limit on the stiffness and strength of the multi-scale design. It is observed that increasing the minimum length scale limit would reduce the structural mechanical performance in both aspects.     

Reducing dimensionality in topology optimization using adaptive design variable fields

Topology optimization methodologies typically use the same discretization for the design variable and analysis meshes. Analysis accuracy and expense are thus directly tied to design dimensionality and optimization expense. This paper proposes leveraging properties of the Heaviside projection method (HPM) to separate the design variable field from the analysis mesh in continuum topology optimization. HPM projects independent design variables onto element space over a prescribed length scale. A single design variable therefore influences several elements, creating a redundancy within the design that can be exploited to reduce the number of independent design variables without significantly restricting the design space. The algorithm begins with sparse design variable fields and adapts these fields as the optimization progresses. The technique is demonstrated on minimum compliance (maximum stiffness) problems solved using continuous optimization and genetic algorithms. For the former, the proposed algorithm typically identifies solutions having objective functions within 1% of those found using full design variable fields. Computational savings are minor to moderate for the minimum compliance formulation with a single constraint, and are substantial for formulations having many local constraints. When using genetic algorithms, solutions are consistently obtained on mesh resolutions that were previously considered intractable. Copyright © 2009 John Wiley & Sons, Ltd.     

Large‐scale topology optimization using preconditioned Krylov subspace methods with recycling

The computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis. We propose fast iterative solvers for large three‐dimensional topology optimization problems to address this problem. Since the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems. In addition, we introduce a MINRES (minimum residual method) version with recycling (and a short‐term recurrence) to make recycling more efficient for symmetric problems. Furthermore, we discuss preconditioning to ensure fast convergence. We show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density. We demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast PC. Copyright © 2006 John Wiley & Sons, Ltd.     

A density field parametrization for topology optimization using Bernstein elements

A new way of describing the density field in density‐based topology optimization is introduced. The new method uses finite elements constructed from Bernstein polynomials rather than the more common Lagrange polynomials. Use of the Bernstein finite elements allows higher‐order elements to be used in the density‐field interpolation without producing unrealistic density values, ie, values lower than zero or higher than one. Results on several test problems indicate that using the higher‐order Bernstein elements produces optimal designs with sharper estimates of the optimal boundary on coarse design meshes. However, higher‐order elements are also required in the structural analysis to prevent the appearance of unrealistic material distributions. The Bernstein element density interpolation can be combined with adaptive mesh refinement to further improve design accuracy even on design domains with complex geometry.     

A material based model for topology optimization of thermoelastic structures

This paper presents the development of a computational model for the topology optimization problem, using a material distribution approach, of a 2-D linear-elastic solid subjected to thermal loads, with a compliance objective function and an isoperimetric constraint on volume. Defining formally the augmented Lagrangian associated with the optimization problem, the optimality conditions are derived analytically. The results of analysis are implemented in a computer code to produce numerical solutions for the optimal topology, considering the temperature distribution independent of design. The design optimization problem is solved via a sequence of linearized subproblems. The computational model developed is tested in example problems. The influence of both the temperature and the finite element model on the optimal solution obtained is analysed.     

A new hole insertion method for level set based structural topology optimization

Structural shape and topology optimization using level set functions is becoming increasingly popular. However, traditional methods do not naturally allow for new hole creation and solutions can be dependent on the initial design. Various methods have been proposed that enable new hole insertion; however, the link between hole insertion and boundary optimization can be unclear. The new method presented in this paper utilizes a secondary level set function that represents a pseudo third dimension in two‐dimensional problems to facilitate new hole insertion. The update of the secondary function is connected to the primary level set function forming a meaningful link between boundary optimization and hole creation. The performance of the method is investigated to identify suitable parameters that produce good solutions for a range of problems. Copyright © 2012 John Wiley & Sons, Ltd.     

A novel minimum weight formulation of topology optimization implemented with reanalysis approach

In this paper, we develop an efficient diagonal quadratic optimization formulation for minimum weight design problem subject to multiple constraints. A high-efficiency computational approach of topology optimization is implemented within the framework of approximate reanalysis. The key point of the formulation is the introduction of the reciprocal-type variables. The topology optimization seeking for minimum weight can be transformed as a sequence of quadratic program with separable and strictly positive definite Hessian matrix, thus can be solved by a sequential quadratic programming approach. A modified sensitivity filtering scheme is suggested to remove undesirable checkerboard patterns and mesh dependence. Several typical examples are provided to validate the presented approach. It is observed that the optimized structure can achieve lighter weight than those from the established method by the demonstrative numerical test. Considerable computational savings can be achieved without loss of accuracy of the final design for 3D structure. Moreover, the effects of multiple constraints and upper bound of the allowable compliance upon the optimized designs are investigated by numerical examples.     

A new level set method for topology optimization of distributed compliant mechanisms

A new level set method for topology optimization of distributed compliant mechanism is presented in this study. By taking two types of mean compliance into consideration, several new objective functions are developed and built in the conventional level set method to avoid generating the de facto hinges in the created mechanisms. Aimed at eliminating the costly reinitialization procedure during the evolution of the level set function, an accelerated level set evolution algorithm is developed by adding an extra energy function, which can force the level set function to close to a signed distance function during the evolution. Two widely studied numerical examples in topology optimization of compliant mechanisms are studied to demonstrate the effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

On a cellular division method for topology optimization

This paper concerns a comparative analysis of a novel biologically inspired method for topology optimization. The proposed methodology develops each individual topology according to a set of rules that regulate a ‘cellular division’ process. These rules are then evolved using a genetic algorithm to minimize objective functions while satisfying a set of constraints. The results reported in this work show that the methodology suits engineering design and represents an improvement over existing topology optimization methods based on evolutionary algorithms. Copyright © 2011 John Wiley & Sons, Ltd.     

A level set approach for topology optimization with local stress constraints

The purpose of this work is to present a level set‐based approach for the structural topology optimization problem of mass minimization submitted to local stress constraints. The main contributions are threefold. First, the inclusion of local stress constraints by means of an augmented Lagrangian approach within the level set context. Second, the proposition of a constraint procedure that accounts for a continuous activation/deactivation of a finite number of local stress constraints during the optimization sequence. Finally, the proposition of a logarithmic scaling of the level set normal velocity as an additional regularization technique in order to improve the minimization sequence. A set of benchmark tests in two dimensions achieving successful numerical results assesses the good behavior of the proposed method. In these examples, it is verified that the algorithm is able to identify stress concentrations and drive the design to a feasible local minimum. Copyright © 2014 John Wiley & Sons, Ltd.     

Topology optimization for nano‐scale heat transfer

We consider the problem of optimal design of nano‐scale heat conducting systems using topology optimization techniques. At such small scales the empirical Fourier's law of heat conduction no longer captures the underlying physical phenomena because the mean‐free path of the heat carriers, phonons in our case, becomes comparable with, or even larger than, the feature sizes of considered material distributions. A more accurate model at nano‐scales is given by kinetic theory, which provides a compromise between the inaccurate Fourier's law and precise, but too computationally expensive, atomistic simulations. We analyze the resulting optimal control problem in a continuous setting, briefly describing the computational approach to the problem based on discontinuous Galerkin methods, algebraic multigrid preconditioned generalized minimal residual method, and a gradient‐based mathematical programming algorithm. Numerical experiments with our implementation of the proposed numerical scheme are reported. Copyright © 2008 John Wiley & Sons, Ltd.     

Theoretical aspects of the internal element connectivity parameterization approach for topology optimization

The internal element connectivity parameterization (I‐ECP) method is an alternative approach to overcome numerical instabilities associated with low‐stiffness element states in non‐linear problems. In I‐ECP, elements are connected by zero‐length links while their link stiffness values are varied. Therefore, it is important to interpolate link stiffness properly to obtain stably converging results. The main objective of this work is two‐fold (1) the investigation of the relationship between the link stiffness and the stiffness of a domain‐discretizing patch by using a discrete model and a homogenized model and (2) the suggestion of link stiffness interpolation functions. The effects of link stiffness penalization on solution convergence are then tested with several numerical examples. The developed homogenized I‐ECP model can also be used to physically interpret an intermediate design variable state. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical methods for the topology optimization of structures that exhibit snap‐through

The inclusion of non‐linear elastic analyses into the topology optimization problem is necessary to capture the finite deformation response, e.g. the geometric non‐linear response of compliant mechanisms. In previous work, the non‐linear response is computed by standard non‐linear elastic finite element analysis. Here, we incorporate a load–displacement constraint method to traverse non‐linear equilibrium paths with limit points to design structures that exhibit snap‐through behaviour. To accomplish this, we modify the basic arc length algorithm and embed this analysis into the topology optimization problem. Copyright © 2002 John Wiley & Sons, Ltd.     

A new method to generate the ground structure in truss topology optimization

The aim of this article is to provide an effective method to generate the ground structure in truss topology optimization. The core of this method is to place nodal points for the ground structure at the intersection of the first and third principal stress trajectories, which are obtained by solving the equivalent static problem in the design domain with a homogeneous isotropic material property. It is applicable to generate the ground structure for arbitrary regular and irregular geometric design domains. The proposed method is tested on some benchmark examples in truss topology optimization. The optimization model is a standard linear programming problem based on plastic design and solved by the interior point algorithm. Compared with other methods, the proposed method may use a well-defined ground structure with fewer nodes and bars, resulting in faster solution convergence, which shows it to be efficient.     

Approximate reanalysis in topology optimization

In the nested approach to structural optimization, most of the computational effort is invested in the solution of finite element analysis equations. In this study, the integration of an approximate reanalysis procedure into the framework of topology optimization of continuum structures is investigated. The nested optimization problem is reformulated to accommodate the use of an approximate displacement vector and the design sensitivities are derived accordingly. It is shown that relatively rough approximations are acceptable since the errors are taken into account in the sensitivity analysis. The implementation is tested on several small and medium scale problems, including 2‐D and 3‐D minimum compliance problems and 2‐D compliant force inverter problems. Accurate results are obtained and the savings in computation time are promising. Copyright © 2008 John Wiley & Sons, Ltd.     

Two‐scale topology optimization in computational material design: An integrated approach

In this work, a new strategy for solving multiscale topology optimization problems is presented. An alternate direction algorithm and a precomputed offline microstructure database (Computational Vademecum) are used to efficiently solve the problem. In addition, the influence of considering manufacturable constraints is examined. Then, the strategy is extended to solve the coupled problem of designing both the macroscopic and microscopic topologies. Full details of the algorithms and numerical examples to validate the methodology are provided.     

A hybrid topology optimization algorithm for static and vibrating shell structures

Structural designers are reconsidering traditional design procedures using structural optimization techniques. Although shape and sizing optimization techniques have facilitated a great improvement in the emergence of new optimum designs, they are still limited by the fact that a suitable topology must be assumed initially. In this paper a hybrid algorithm entitled constrained adaptive topology optimization, or CATO is introduced. The algorithm, based on an artificial material model and an adaptive updating scheme, combines ideas from the mathematically rigorous homogenization (h) methods and the intuitive evolutionary (e) methods. The algorithm is applied to shell structures under static or free vibration situations. For the static situation, the objective is to produce the stiffest structure subject to given loading conditions, boundary conditions and material properties. For the free vibration situation, the objective is to maximize or minimize a chosen frequency. In both cases, a constraint on the structural volume/mass is applied and the optimization process is achieved by redistributing the material through the shell structure. The efficiency of the proposed algorithm is illustrated through several numerical examples of shells under either static or free vibration situations. Copyright © 2002 John Wiley & Sons, Ltd.