On projection methods,convergence and robust formulations in topology optimization

Mesh convergence and manufacturability of topology optimized designs have previously mainly been assured using density or sensitivity based filtering techniques. The drawback of these techniques has been gray transition regions between solid and void parts, but this problem has recently been alleviated using various projection methods. In this paper we show that simple projection methods do not ensure local mesh-convergence and propose a modified robust topology optimization formulation based on erosion, intermediate and dilation projections that ensures both global and local mesh-convergence.     

On the validity of ESO type methods in topology optimization

It is shown on a simple test example that ESO’s rejection criteria may result in a highly nonoptimal design. Reasons for this failure are also discussed. Received September 12, 2000     

Heaviside projection based topology optimization by a PDE-filtered scalar function

This paper deals with topology optimization based on the Heaviside projection method using a scalar function as design variables. The scalar function is then regularized by a PDE based filter. Several image-processing based filtering techniques have so far been proposed for regularization or restricting the minimum length scale. They are conventionally applied to the design sensitivities rather than the design variables themselves. However, it causes discrepancies between the filtered sensitivities and the actual sensitivities that may confuse the optimization process and disturb the convergence. In this paper, we propose a Heaviside projection based topology optimization method with a scalar function that is filtered by a Helmholtz type partial differential equation. Therefore, the optimality can be strictly discussed in terms of the KKT condition. In order to demonstrate the effectiveness of the proposed method, a minimum compliance problem is solved.     

Aeroelastic topology optimization of membrane structures for micro air vehicles

This work considers the aeroelastic optimization of a membrane micro air vehicle wing through topology optimization. The low aspect ratio wing is discretized into panels: a two material formulation on the wetted surface is used, where each panel can be membrane (wing skin) or carbon fiber (laminate reinforcement). An analytical sensitivity analysis of the aeroelastic system is used for the gradient-based optimization of aerodynamic objective functions. An explicit penalty is added, as needed, to force the structure to a 0–1 distribution. The dependence of the solution upon initial design, angle of attack, mesh density, and objective function are presented. Deformation and pressure distributions along the wing are studied for various load-augmenting and load-alleviating designs (both baseline and optimized), in order to establish a link between stiffness distribution and aerodynamic performance of membrane micro air vehicle wings. The work concludes with an experimental validation of the superiority of select optimal designs.     

Interior point multigrid methods for topology optimization

In this paper, a new multigrid interior point approach to topology optimization problems in the context of the homogenization method is presented. The key observation is that nonlinear interior point methods lead to linear-quadratic subproblems with structures that can be favourably exploited within multigrid methods. Primal as well as primal-dual formulations are discussed. The multigrid approach is based on the transformed smoother paradigm. Numerical results for an example problem are presented. Received February 15, 1999     

Parallel methods for optimality criteria-based topology optimization

Topology optimization problems require the repeated solution of finite element problems that are often extremely ill-conditioned due to highly heterogeneous material distributions. This makes the use of iterative linear solvers inefficient unless appropriate preconditioning is used. Even then, the solution time for topology optimization problems is typically very high. These problems are addressed by considering the use of non-overlapping domain decomposition-based parallel methods for the solution of topology optimization problems. The parallel algorithms presented here are based on the solid isotropic material with penalization (SIMP) formulation of the topology optimization problem and use the optimality criteria method for iterative optimization. We consider three parallel linear solvers to solve the equilibrium problem at each step of the iterative optimization procedure. These include two preconditioned conjugate gradient (PCG) methods: one using a diagonal preconditioner and one using an incomplete LU factorization preconditioner with a drop tolerance. A third substructuring solver that employs a hybrid of direct and iterative (PCG) techniques is also studied. This solver is found to be the most effective of the three solvers studied, both in terms of parallel efficiency and in terms of its ability to mitigate the effects of ill-conditioning. In addition to examining parallel linear solvers, we consider the parallelization of the iterative optimality criteria method. To tackle checkerboarding and mesh dependence, we propose a multi-pass filtering technique that limits the number of “ghost” elements that need to be exchanged across interprocessor boundaries.     

The gradient projection method for structural topology optimization including density-dependent force

This paper proposes a modified gradient projection method (GPM) that can solve the structural topology optimization problem including density-dependent force efficiently. The particular difficulty of the considered problem is the non-monotonicity of the objective function and consequently the optimization problem is not definitely constrained. Transformation of variables technique is used to eliminate the constraints of the design variables, and thus the volume is the only possible constraint. The negative gradient of the objective function is adopted as the most promising search direction when the point is inside the feasible domain, while the projected negative gradient is used instead on condition that the point is on the hypersurface of the constraint. A rational step size is given via a self-adjustment mechanism that ensures the step size is a good compromising between efficiency and reliability. Furthermore, some image processing techniques are employed to improve the layouts. Numerical examples with different prescribed volume fractions and different load ratios are tested respectively to illustrate the characteristics of the topology optimization with density-dependent load.     

A short numerical study on the optimization methods influence on topology optimization

Structural and Multidisciplinary Optimization - Structural topology optimization problems are commonly defined using continuous design variables combined with material interpolation schemes. One of...     

Automatic mesh generation and transformation for topology optimization methods

This paper presents automatic tools aimed at the generation and adaptation of unstructured tetrahedral meshes in the context of composite or heterogeneous geometry. These tools are primarily intended for applications in the domain of topology optimization methods but the approach introduced presents great potential in a wider context. Indeed, various fields of application can be foreseen for which meshing heterogeneous geometry is required, such as finite element simulations (in the case of heterogeneous materials and assemblies, for example), animation and visualization (medical imaging, for example). Using B-Rep concepts as well as specific adaptations of advancing front mesh generation algorithms, the mesh generation approach presented guarantees, in a simple and natural way, mesh continuity and conformity across interior boundaries when trying to mesh a composite domain. When applied in the context of topology optimization methods, this approach guarantees that design and non-design sub-domains are meshed so that finite elements are tagged as design and non-design elements and so that continuity and conformity are guaranteed at the interface between design and non-design sub-domains. The paper also presents how mesh transformation and mesh smoothing tools can be successfully used when trying to derive a functional shape from raw topology optimization results.     

On multigrid-CG for efficient topology optimization

This article presents a computational approach that facilitates the efficient solution of 3-D structural topology optimization problems on a standard PC. Computing time associated with solving the nested analysis problem is reduced significantly in comparison to other existing approaches. The cost reduction is obtained by exploiting specific characteristics of a multigrid preconditioned conjugate gradients (MGCG) solver. In particular, the number of MGCG iterations is reduced by relating it to the geometric parameters of the problem. At the same time, accurate outcome of the optimization process is ensured by linking the required accuracy of the design sensitivities to the progress of optimization. The applicability of the proposed procedure is demonstrated on several 2-D and 3-D examples involving up to hundreds of thousands of degrees of freedom. Implemented in MATLAB, the MGCG-based program solves 3-D topology optimization problems in a matter of minutes. This paves the way for efficient implementations in computational environments that do not enjoy the benefits of high performance computing, such as applications on mobile devices and plug-ins for modeling software.     

On nonunique solutions in topology optimization

This paper deals with the problem of non-unique solutions in topology optimization. Depending on the optimization path, the solutions, in other words the topologies of a structure, are different. The nonuniqueness problem in topology optimization is presented in connection with the testing of different lower material mass value bounding functions and the use of different material properties updating functions and different threshold functions. The structure strain energy minimum criterion is applied to find the optimum topology. A comparison of the topologies obtained from the energy criterion point of view is made.     

On symmetry and non-uniqueness in exact topology optimization

The aim of this article is to initiate an exchange of ideas on symmetry and non-uniqueness in topology optimization. These concepts are discussed in the context of 2D trusses and grillages, but could be extended to other structures and design constraints, including 3D problems and numerical solutions. The treatment of the subject is pitched at the background of engineering researchers, and principles of mechanics are given preference to those of pure mathematics. The author hopes to provide some new insights into fundamental properties of exact optimal topologies. Combining elements of the optimal layout theory (of Prager and the author) with those of linear programming, it is concluded that for the considered problems the optimal topology is in general unique and symmetric if the loads, domain boundaries and supports are symmetric. However, in some special cases the number of optimal solutions may be infinite, and some of these may be non-symmetric. The deeper reasons for the above findings are explained in the light of the above layout theory.     

Level-set methods for structural topology optimization: a review

This review paper provides an overview of different level-set methods for structural topology optimization. Level-set methods can be categorized with respect to the level-set-function parameterization, the geometry mapping, the physical/mechanical model, the information and the procedure to update the design and the applied regularization. Different approaches for each of these interlinked components are outlined and compared. Based on this categorization, the convergence behavior of the optimization process is discussed, as well as control over the slope and smoothness of the level-set function, hole nucleation and the relation of level-set methods to other topology optimization methods. The importance of numerical consistency for understanding and studying the behavior of proposed methods is highlighted. This review concludes with recommendations for future research.     

A simple and effective inverse projection scheme for void distribution control in topology optimization

The ability to control both the minimum size of holes and the minimum size of structural members are essential requirements in the topology optimization design process for manufacturing. This paper addresses both requirements by means of a unified approach involving mesh-independent projection techniques. An inverse projection is developed to control the minimum hole size while a standard direct projection scheme is used to control the minimum length of structural members. In addition, a heuristic scheme combining both contrasting requirements simultaneously is discussed. Two topology optimization implementations are contributed: one in which the projection (either inverse or direct) is used at each iteration; and the other in which a two-phase scheme is explored. In the first phase, the compliance minimization is carried out without any projection until convergence. In the second phase, the chosen projection scheme is applied iteratively until a solution is obtained while satisfying either the minimum member size or minimum hole size. Examples demonstrate the various features of the projection-based techniques presented.     

On simultaneous optimization of truss geometry and topology

The paper addresses the classical problem of optimal truss design where cross-sectional areas and the positions of joints are simultaneously optimized. Se-veral approaches are discussed from a general point of view. In particular, we focus on the difference between simultaneous and alternating optimization of geometry and topology. We recall a rigorously mathematical approach based on the implicit programming technique which considers the classical single load minimum compliance problem subject to a volume constraint. This approach is refined leading to three new problem formulations which can be treated by methods of Mathematical Programming. In particular, these formulations cover the effect of melting end nodes, i.e., vanishing potential bars due to changes in the geometry. In one of these new problem formulations, the objective function is a polynomial of degree three and the constraints are bilinear or just sign constraints. Because heuristics is avoided, certain optimality properties can be proven for resulting structures. The paper closes with two numerical test examples.     

Sequential integer programming methods for stress constrained topology optimization

This paper deals with topology optimization of load carrying structures defined on a discretized design domain where binary design variables are used to indicate material or void in the various finite elements. The main contribution is the development of two iterative methods which are guaranteed to find a local optimum with respect to a 1-neighbourhood. Each new iteration point is obtained as the optimal solution to an integer linear programming problem which is an approximation of the original problem at the previous iteration point. The proposed methods are quite general and can be applied to a variety of topology optimization problems defined by 0-1 design variables. Most of the presented numerical examples are devoted to problems involving stresses which can be handled in a natural way since the design variables are kept binary in the subproblems.     

Lubrication analysis of the viscous micro/nano pump with slip

A viscous pump is a device such that a cylindrical rotor is eccentrically placed in a channel, so that the viscous resistance between the small and large gaps between the cylinder and the channel walls generate a net flow along the channel. Assuming that the gaps between the cylinder and the channel walls are small compared to the radius of the rotor, the hydrodynamic theory of lubrication may be utilized to study the viscous pump. Here lubrication theory is used to obtain an analytical solution which relates the flowrate, rotation rate, pressure drop and applied torque as functions of two geometric parameters for a viscous pump. This analysis differs from a previous similar study in two ways. Firstly, certain integrals are evaluated explicitly, and secondly the standard no-slip boundary condition of fluid mechanics has been replaced with the Navier boundary condition which allows a degree of tangential velocity slip on all solid boundaries. Comparison with the prior known solution shows that the solution obtained in this study predicts a slightly improved pump performance for the case of no-slip. For the case of slip, our results demonstrate that the performance of the pump is significantly improved.